general solution for double integral of square root of quadratic polynomials I wonder if there is a closed-form solution for double integral of square root of quadratic polynomials. Such as 
$$
\int _0 ^1 \int _0 ^1 \sqrt{a \cdot x^2 + b \cdot y^2 + c \cdot x \cdot y + d \cdot x+e\cdot y +1 } \ dx dy 
$$
I have tried to solve 
$$
 \int _0 ^1 \sqrt{a \cdot x^2 + b \cdot y^2 + c \cdot x \cdot y + d \cdot x+e\cdot y +1 } \ dx 
$$
and this has a closed form of solution, but is very complicated that makes it nearlly impossible to get the integral further on y. 
So I wonder if a closed form solution really exists, does anyone have some ideas?
 A: MAPLE got:
$\int _0^1\int_0^1\sqrt{a\cdot x^2+b\cdot y^2+c\cdot x\cdot y+d\cdot x+e\cdot y +1 }\ dx\ dy$
$$=\frac{1}{8}\left(-4\,ab\ln\left({\frac{c+d+2\,\sqrt{b+e+1}\sqrt{a}}{\sqrt{a}}}\right)+4\,ab\ln\left({\frac{2\,a+c+d+2\,\sqrt{a+b+c+d+e+1}\sqrt{a}}{\sqrt{a}}}\right)-4\,ae\ln\left({\frac{c+d+2\,\sqrt{b+e+1}\sqrt{a}}{\sqrt{a}}}\right)+4\,ae\ln\left({\frac{2\,a+c+d+2\,\sqrt{a+b+c+d+e+1}\sqrt{a}}{\sqrt{a}}}\right)+{c}^{2}\ln\left({\frac{c+d+2\,\sqrt{b+e+1}\sqrt{a}}{\sqrt{a}}}\right)-{c}^{2}\ln\left({\frac{2\,a+c+d+2\,\sqrt{a+b+c+d+e+1}\sqrt{a}}{\sqrt{a}}}\right)+2\,cd\ln\left({\frac{c+d+2\,\sqrt{b+e+1}\sqrt{a}}{\sqrt{a}}}\right)-2\,cd\ln\left({\frac{2\,a+c+d+2\,\sqrt{a+b+c+d+e+1}\sqrt{a}}{\sqrt{a}}}\right)-{d}^{2}\ln\left({\frac{d+2\,\sqrt{a}}{\sqrt{a}}}\right)+{d}^{2}\ln\left({\frac{2\,a+d+2\,\sqrt{a+d+1}\sqrt{a}}{\sqrt{a}}}\right)+{d}^{2}\ln\left({\frac{c+d+2\,\sqrt{b+e+1}\sqrt{a}}{\sqrt{a}}}\right)-{d}^{2}\ln\left({\frac{2\,a+c+d+2\,\sqrt{a+b+c+d+e+1}\sqrt{a}}{\sqrt{a}}}\right)+4\,a\ln\left({\frac{d+2\,\sqrt{a}}{\sqrt{a}}}\right)-4\,a\ln\left({\frac{2\,a+d+2\,\sqrt{a+d+1}\sqrt{a}}{\sqrt{a}}}\right)-4\,a\ln\left({\frac{c+d+2\,\sqrt{b+e+1}\sqrt{a}}{\sqrt{a}}}\right)+4\,a\ln\left({\frac{2\,a+c+d+2\,\sqrt{a+b+c+d+e+1}\sqrt{a}}{\sqrt{a}}}\right)+4\,\sqrt{a+b+c+d+e+1}{a}^{3/2}-4\,\sqrt{a+d+1}{a}^{3/2}+2\,\sqrt{a+b+c+d+e+1}\sqrt{a}c-2\,\sqrt{b+e+1}\sqrt{a}c+2\,\sqrt{a}d+2\,\sqrt{a+b+c+d+e+1}\sqrt{a}d-2\,\sqrt{b+e+1}\sqrt{a}d-2\,\sqrt{a+d+1}\sqrt{a}d\right){a}^{-3/2}$$ 
