How to solve this definite integral $\int_0^1{\frac{at+b}{\sqrt{ct^2+dt+f}}\arctan(\frac{gt+h}{\sqrt{ct^2+dt+f}})dt}$ I'm trying to analytically integrate the following to get a closed form expression if possible. I have tried integrating by parts and it was getting pretty hairy. MATLAB symbolic toolkit was no help with this either. If anyone has good suggestions please contribute.
$$\int_0^1{\frac{at+b}{\sqrt{ct^2+dt+f}}\arctan(\frac{gt+h}{\sqrt{ct^2+dt+f}})dt}$$
$a$ through $h$ are real-valued constants.
 A: First, we get rid of two unnecessary parameters, and rename the rest ($a,b,c,f,h$):
$$I=\int_0^1{\frac{t+a}{\sqrt{t^2+bt+c}}\arctan \left( f\frac{t+h}{\sqrt{t^2+bt+c}} \right) dt}$$
This integral is the same as in the OP up to a constant factor.
Now we use the celebrated Euler substitution:
$$u=\sqrt{t^2+bt+c}-t$$
$$t=\frac{u^2-c}{b-2u}$$

After some elementary algebra we obtain:
$$dt=-2 \frac{u^2-bu+c}{(b-2u)^2}du$$
$$\frac{t+a}{\sqrt{t^2+bt+c}}=- \frac{u^2-2au+ab-c}{u^2-bu+c}$$
And finally:

$$I=-2 \int_{\sqrt{c}}^{\sqrt{b+c+1}-1} \frac{u^2-2au+ab-c}{(b-2u)^2} \arctan \left(f \frac{u^2-2hu+hb-c}{u^2-bu+c} \right)du$$
Now we have an integral of $\arctan$ of a rational argument multiplied by a rational function, which, using integration by parts, reduces to an integral of just a rational function, which always evaluates in elementary functions.

Thus, I consider the problem solved, unless the values of the parameters are specified.
Set:
$$P(u)=\frac{u^2-2au+ab-c}{(b-2u)^2}$$
$$Q(u)=f \frac{u^2-2hu+hb-c}{u^2-bu+c}$$
$$\int P(u) \arctan Q(u) du=\arctan Q(u) \int P(u) du-\int  \frac{Q'(u)}{1+Q^2(u)} \left(\int_\alpha^u P(v)dv \right) du$$
Here $\left(\int_\alpha^u P(v)dv \right)$ denotes a primitive of $P(u)$ (with $\alpha$ being some constant), defined according to the usual rules of integration by parts.
