Integrand size = 40, antiderivative size = 589 \[ \int (a g+b g x) (c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx=\frac {B^2 (b c-a d)^3 g i^2 x}{12 b^2 d}+\frac {B^2 (b c-a d)^2 g i^2 (c+d x)^2}{12 b d^2}-\frac {B^2 (b c-a d)^4 g i^2 \log \left (\frac {a+b x}{c+d x}\right )}{12 b^3 d^2}-\frac {B (b c-a d)^3 g i^2 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6 b^3 d}-\frac {B (b c-a d)^2 g i^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6 b^3}+\frac {B (b c-a d)^2 g i^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 b d^2}-\frac {B (b c-a d) g i^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6 d^2}+\frac {(b c-a d)^2 g i^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{12 b^3}+\frac {(b c-a d) g i^2 (a+b x)^2 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{6 b^2}+\frac {g i^2 (a+b x)^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{4 b}-\frac {B (b c-a d)^4 g i^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6 b^3 d^2}-\frac {B^2 (b c-a d)^4 g i^2 \log (c+d x)}{4 b^3 d^2}-\frac {B^2 (b c-a d)^4 g i^2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{6 b^3 d^2} \] Output:
1/12*B^2*(-a*d+b*c)^3*g*i^2*x/b^2/d+1/12*B^2*(-a*d+b*c)^2*g*i^2*(d*x+c)^2/ b/d^2-1/12*B^2*(-a*d+b*c)^4*g*i^2*ln((b*x+a)/(d*x+c))/b^3/d^2-1/6*B*(-a*d+ b*c)^3*g*i^2*(b*x+a)*(A+B*ln(e*(b*x+a)/(d*x+c)))/b^3/d-1/6*B*(-a*d+b*c)^2* g*i^2*(b*x+a)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))/b^3+1/4*B*(-a*d+b*c)^2*g*i^2*( d*x+c)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))/b/d^2-1/6*B*(-a*d+b*c)*g*i^2*(d*x+c)^ 3*(A+B*ln(e*(b*x+a)/(d*x+c)))/d^2+1/12*(-a*d+b*c)^2*g*i^2*(b*x+a)^2*(A+B*l n(e*(b*x+a)/(d*x+c)))^2/b^3+1/6*(-a*d+b*c)*g*i^2*(b*x+a)^2*(d*x+c)*(A+B*ln (e*(b*x+a)/(d*x+c)))^2/b^2+1/4*g*i^2*(b*x+a)^2*(d*x+c)^2*(A+B*ln(e*(b*x+a) /(d*x+c)))^2/b-1/6*B*(-a*d+b*c)^4*g*i^2*ln((-a*d+b*c)/b/(d*x+c))*(A+B+B*ln (e*(b*x+a)/(d*x+c)))/b^3/d^2-1/4*B^2*(-a*d+b*c)^4*g*i^2*ln(d*x+c)/b^3/d^2- 1/6*B^2*(-a*d+b*c)^4*g*i^2*polylog(2,d*(b*x+a)/b/(d*x+c))/b^3/d^2
Time = 0.60 (sec) , antiderivative size = 677, normalized size of antiderivative = 1.15 \[ \int (a g+b g x) (c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx=\frac {g i^2 \left (-4 (b c-a d) (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2+3 b (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2+\frac {4 B (b c-a d)^2 \left (2 A b d (b c-a d) x-B (b c-a d) (b d x+(b c-a d) \log (a+b x))+2 B d (b c-a d) (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )+b^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+2 (b c-a d)^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-2 B (b c-a d)^2 \log (c+d x)-B (b c-a d)^2 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )\right )}{b^3}-\frac {B (b c-a d) \left (6 A b d (b c-a d)^2 x-3 B (b c-a d)^2 (b d x+(b c-a d) \log (a+b x))-B (b c-a d) \left (2 b d (b c-a d) x+b^2 (c+d x)^2+2 (b c-a d)^2 \log (a+b x)\right )+6 B d (b c-a d)^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )+3 b^2 (b c-a d) (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+2 b^3 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+6 (b c-a d)^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-6 B (b c-a d)^3 \log (c+d x)-3 B (b c-a d)^3 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )\right )}{b^3}\right )}{12 d^2} \] Input:
Integrate[(a*g + b*g*x)*(c*i + d*i*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x) ])^2,x]
Output:
(g*i^2*(-4*(b*c - a*d)*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 + 3*b*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 + (4*B*(b*c - a*d )^2*(2*A*b*d*(b*c - a*d)*x - B*(b*c - a*d)*(b*d*x + (b*c - a*d)*Log[a + b* x]) + 2*B*d*(b*c - a*d)*(a + b*x)*Log[(e*(a + b*x))/(c + d*x)] + b^2*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 2*(b*c - a*d)^2*Log[a + b*x] *(A + B*Log[(e*(a + b*x))/(c + d*x)]) - 2*B*(b*c - a*d)^2*Log[c + d*x] - B *(b*c - a*d)^2*(Log[a + b*x]*(Log[a + b*x] - 2*Log[(b*(c + d*x))/(b*c - a* d)]) - 2*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)])))/b^3 - (B*(b*c - a*d)* (6*A*b*d*(b*c - a*d)^2*x - 3*B*(b*c - a*d)^2*(b*d*x + (b*c - a*d)*Log[a + b*x]) - B*(b*c - a*d)*(2*b*d*(b*c - a*d)*x + b^2*(c + d*x)^2 + 2*(b*c - a* d)^2*Log[a + b*x]) + 6*B*d*(b*c - a*d)^2*(a + b*x)*Log[(e*(a + b*x))/(c + d*x)] + 3*b^2*(b*c - a*d)*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 2*b^3*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 6*(b*c - a*d)^ 3*Log[a + b*x]*(A + B*Log[(e*(a + b*x))/(c + d*x)]) - 6*B*(b*c - a*d)^3*Lo g[c + d*x] - 3*B*(b*c - a*d)^3*(Log[a + b*x]*(Log[a + b*x] - 2*Log[(b*(c + d*x))/(b*c - a*d)]) - 2*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)])))/b^3)) /(12*d^2)
Time = 1.47 (sec) , antiderivative size = 670, normalized size of antiderivative = 1.14, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {2962, 2783, 2782, 27, 86, 2009, 2783, 2773, 49, 2009, 2781, 2784, 2754, 2838}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int (a g+b g x) (c i+d i x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2 \, dx\) |
| \(\Big \downarrow \) 2962 |
| \(\displaystyle g i^2 (b c-a d)^4 \int \frac {(a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )^5}d\frac {a+b x}{c+d x}\) |
| \(\Big \downarrow \) 2783 |
| \(\displaystyle g i^2 (b c-a d)^4 \left (-\frac {B \int \frac {(a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )^4}d\frac {a+b x}{c+d x}}{2 b}+\frac {\int \frac {(a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )^4}d\frac {a+b x}{c+d x}}{2 b}+\frac {(a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{4 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}\right )\) |
| \(\Big \downarrow \) 2782 |
| \(\displaystyle g i^2 (b c-a d)^4 \left (-\frac {B \left (-B \int -\frac {(c+d x) \left (b-\frac {3 d (a+b x)}{c+d x}\right )}{6 d^2 (a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^3}d\frac {a+b x}{c+d x}-\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{2 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {b \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}\right )}{2 b}+\frac {\int \frac {(a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )^4}d\frac {a+b x}{c+d x}}{2 b}+\frac {(a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{4 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle g i^2 (b c-a d)^4 \left (-\frac {B \left (\frac {B \int \frac {(c+d x) \left (b-\frac {3 d (a+b x)}{c+d x}\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^3}d\frac {a+b x}{c+d x}}{6 d^2}-\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{2 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {b \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}\right )}{2 b}+\frac {\int \frac {(a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )^4}d\frac {a+b x}{c+d x}}{2 b}+\frac {(a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{4 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}\right )\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle g i^2 (b c-a d)^4 \left (-\frac {B \left (\frac {B \int \left (\frac {d}{b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {d}{b \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {2 d}{\left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {c+d x}{b^2 (a+b x)}\right )d\frac {a+b x}{c+d x}}{6 d^2}-\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{2 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {b \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}\right )}{2 b}+\frac {\int \frac {(a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )^4}d\frac {a+b x}{c+d x}}{2 b}+\frac {(a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{4 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle g i^2 (b c-a d)^4 \left (\frac {\int \frac {(a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )^4}d\frac {a+b x}{c+d x}}{2 b}-\frac {B \left (-\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{2 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {b \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {B \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^2}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^2}+\frac {1}{b \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right )^2}\right )}{6 d^2}\right )}{2 b}+\frac {(a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{4 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}\right )\) |
| \(\Big \downarrow \) 2783 |
| \(\displaystyle g i^2 (b c-a d)^4 \left (\frac {-\frac {2 B \int \frac {(a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )^3}d\frac {a+b x}{c+d x}}{3 b}+\frac {\int \frac {(a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )^3}d\frac {a+b x}{c+d x}}{3 b}+\frac {(a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{3 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}}{2 b}-\frac {B \left (-\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{2 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {b \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {B \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^2}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^2}+\frac {1}{b \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right )^2}\right )}{6 d^2}\right )}{2 b}+\frac {(a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{4 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}\right )\) |
| \(\Big \downarrow \) 2773 |
| \(\displaystyle g i^2 (b c-a d)^4 \left (\frac {-\frac {2 B \left (\frac {(a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B \int \frac {a+b x}{(c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )^2}d\frac {a+b x}{c+d x}}{2 b}\right )}{3 b}+\frac {\int \frac {(a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )^3}d\frac {a+b x}{c+d x}}{3 b}+\frac {(a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{3 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}}{2 b}-\frac {B \left (-\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{2 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {b \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {B \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^2}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^2}+\frac {1}{b \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right )^2}\right )}{6 d^2}\right )}{2 b}+\frac {(a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{4 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}\right )\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle g i^2 (b c-a d)^4 \left (\frac {-\frac {2 B \left (\frac {(a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B \int \left (\frac {b}{d \left (\frac {d (a+b x)}{c+d x}-b\right )^2}+\frac {1}{d \left (\frac {d (a+b x)}{c+d x}-b\right )}\right )d\frac {a+b x}{c+d x}}{2 b}\right )}{3 b}+\frac {\int \frac {(a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )^3}d\frac {a+b x}{c+d x}}{3 b}+\frac {(a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{3 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}}{2 b}-\frac {B \left (-\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{2 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {b \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {B \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^2}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^2}+\frac {1}{b \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right )^2}\right )}{6 d^2}\right )}{2 b}+\frac {(a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{4 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle g i^2 (b c-a d)^4 \left (\frac {\frac {\int \frac {(a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )^3}d\frac {a+b x}{c+d x}}{3 b}-\frac {2 B \left (\frac {(a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B \left (\frac {b}{d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{d^2}\right )}{2 b}\right )}{3 b}+\frac {(a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{3 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}}{2 b}-\frac {B \left (-\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{2 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {b \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {B \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^2}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^2}+\frac {1}{b \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right )^2}\right )}{6 d^2}\right )}{2 b}+\frac {(a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{4 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}\right )\) |
| \(\Big \downarrow \) 2781 |
| \(\displaystyle g i^2 (b c-a d)^4 \left (\frac {\frac {\frac {(a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{2 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B \int \frac {(a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )^2}d\frac {a+b x}{c+d x}}{b}}{3 b}-\frac {2 B \left (\frac {(a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B \left (\frac {b}{d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{d^2}\right )}{2 b}\right )}{3 b}+\frac {(a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{3 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}}{2 b}-\frac {B \left (-\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{2 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {b \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {B \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^2}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^2}+\frac {1}{b \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right )^2}\right )}{6 d^2}\right )}{2 b}+\frac {(a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{4 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}\right )\) |
| \(\Big \downarrow \) 2784 |
| \(\displaystyle g i^2 (b c-a d)^4 \left (\frac {\frac {\frac {(a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{2 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B \left (\frac {(a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{d (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {\int \frac {A+B+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{b-\frac {d (a+b x)}{c+d x}}d\frac {a+b x}{c+d x}}{d}\right )}{b}}{3 b}-\frac {2 B \left (\frac {(a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B \left (\frac {b}{d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{d^2}\right )}{2 b}\right )}{3 b}+\frac {(a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{3 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}}{2 b}-\frac {B \left (-\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{2 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {b \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {B \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^2}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^2}+\frac {1}{b \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right )^2}\right )}{6 d^2}\right )}{2 b}+\frac {(a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{4 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}\right )\) |
| \(\Big \downarrow \) 2754 |
| \(\displaystyle g i^2 (b c-a d)^4 \left (\frac {\frac {\frac {(a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{2 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B \left (\frac {(a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{d (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {\frac {B \int \frac {(c+d x) \log \left (1-\frac {d (a+b x)}{b (c+d x)}\right )}{a+b x}d\frac {a+b x}{c+d x}}{d}-\frac {\log \left (1-\frac {d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A+B\right )}{d}}{d}\right )}{b}}{3 b}-\frac {2 B \left (\frac {(a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B \left (\frac {b}{d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{d^2}\right )}{2 b}\right )}{3 b}+\frac {(a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{3 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}}{2 b}-\frac {B \left (-\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{2 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {b \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {B \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^2}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^2}+\frac {1}{b \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right )^2}\right )}{6 d^2}\right )}{2 b}+\frac {(a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{4 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}\right )\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle g i^2 (b c-a d)^4 \left (-\frac {B \left (-\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{2 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {b \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {B \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^2}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^2}+\frac {1}{b \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right )^2}\right )}{6 d^2}\right )}{2 b}+\frac {-\frac {2 B \left (\frac {(a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B \left (\frac {b}{d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{d^2}\right )}{2 b}\right )}{3 b}+\frac {\frac {(a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{2 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B \left (\frac {(a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{d (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {-\frac {\log \left (1-\frac {d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A+B\right )}{d}-\frac {B \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d}}{d}\right )}{b}}{3 b}+\frac {(a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{3 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}}{2 b}+\frac {(a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{4 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}\right )\) |
Input:
Int[(a*g + b*g*x)*(c*i + d*i*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2,x ]
Output:
(b*c - a*d)^4*g*i^2*(((a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/ (4*b*(c + d*x)^2*(b - (d*(a + b*x))/(c + d*x))^4) - (B*((b*(A + B*Log[(e*( a + b*x))/(c + d*x)]))/(3*d^2*(b - (d*(a + b*x))/(c + d*x))^3) - (A + B*Lo g[(e*(a + b*x))/(c + d*x)])/(2*d^2*(b - (d*(a + b*x))/(c + d*x))^2) + (B*( -(b - (d*(a + b*x))/(c + d*x))^(-2) + 1/(b*(b - (d*(a + b*x))/(c + d*x))) + Log[(a + b*x)/(c + d*x)]/b^2 - Log[b - (d*(a + b*x))/(c + d*x)]/b^2))/(6 *d^2)))/(2*b) + (((a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(3*b *(c + d*x)^2*(b - (d*(a + b*x))/(c + d*x))^3) - (2*B*(((a + b*x)^2*(A + B* Log[(e*(a + b*x))/(c + d*x)]))/(2*b*(c + d*x)^2*(b - (d*(a + b*x))/(c + d* x))^2) - (B*(b/(d^2*(b - (d*(a + b*x))/(c + d*x))) + Log[b - (d*(a + b*x)) /(c + d*x)]/d^2))/(2*b)))/(3*b) + (((a + b*x)^2*(A + B*Log[(e*(a + b*x))/( c + d*x)])^2)/(2*b*(c + d*x)^2*(b - (d*(a + b*x))/(c + d*x))^2) - (B*(((a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(d*(c + d*x)*(b - (d*(a + b*x ))/(c + d*x))) - (-(((A + B + B*Log[(e*(a + b*x))/(c + d*x)])*Log[1 - (d*( a + b*x))/(b*(c + d*x))])/d) - (B*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))]) /d)/d))/b)/(3*b))/(2*b))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symb ol] :> Simp[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^p/e), x] - Simp[b*n*(p/e) Int[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)* (x_)^(r_.))^(q_), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^r)^(q + 1)*((a + b*Log[c*x^n])/(d*f*(m + 1))), x] - Simp[b*(n/(d*(m + 1))) Int[(f*x)^m*(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && Eq Q[m + r*(q + 1) + 1, 0] && NeQ[m, -1]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_))^(q_), x_Symbol] :> Simp[(-(f*x)^(m + 1))*(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/(d*f*(q + 1))), x] + Simp[b*n*(p/(d*(q + 1))) Int[(f*x) ^m*(d + e*x)^(q + 1)*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, q}, x] && EqQ[m + q + 2, 0] && IGtQ[p, 0] && LtQ[q, -1]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_))^(q _), x_Symbol] :> With[{u = IntHide[x^m*(d + e*x)^q, x]}, Simp[(a + b*Log[c* x^n]) u, x] - Simp[b*n Int[SimplifyIntegrand[u/x, x], x], x]] /; FreeQ[ {a, b, c, d, e, n}, x] && ILtQ[m + q + 2, 0] && IGtQ[m, 0]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_))^(q_), x_Symbol] :> Simp[(-(f*x)^(m + 1))*(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/(d*f*(q + 1))), x] + (Simp[(m + q + 2)/(d*(q + 1)) Int[ (f*x)^m*(d + e*x)^(q + 1)*(a + b*Log[c*x^n])^p, x], x] + Simp[b*n*(p/(d*(q + 1))) Int[(f*x)^m*(d + e*x)^(q + 1)*(a + b*Log[c*x^n])^(p - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, n}, x] && ILtQ[m + q + 2, 0] && IGtQ[p, 0] && L tQ[q, -1] && GtQ[m, 0]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)* (x_))^(q_.), x_Symbol] :> Simp[(f*x)^m*(d + e*x)^(q + 1)*((a + b*Log[c*x^n] )/(e*(q + 1))), x] - Simp[f/(e*(q + 1)) Int[(f*x)^(m - 1)*(d + e*x)^(q + 1)*(a*m + b*n + b*m*Log[c*x^n]), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && ILtQ[q, -1] && GtQ[m, 0]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_ )]*(B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.)*((h_.) + (i_.)*(x_))^(q_.), x_Sy mbol] :> Simp[(b*c - a*d)^(m + q + 1)*(g/b)^m*(i/d)^q Subst[Int[x^m*((A + B*Log[e*x^n])^p/(b - d*x)^(m + q + 2)), x], x, (a + b*x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, A, B, n, p}, x] && EqQ[n + mn, 0] && IGt Q[n, 0] && NeQ[b*c - a*d, 0] && EqQ[b*f - a*g, 0] && EqQ[d*h - c*i, 0] && I ntegersQ[m, q]
\[\int \left (b g x +a g \right ) \left (d i x +c i \right )^{2} \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )^{2}d x\]
Input:
int((b*g*x+a*g)*(d*i*x+c*i)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))^2,x)
Output:
int((b*g*x+a*g)*(d*i*x+c*i)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))^2,x)
\[ \int (a g+b g x) (c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx=\int { {\left (b g x + a g\right )} {\left (d i x + c i\right )}^{2} {\left (B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}^{2} \,d x } \] Input:
integrate((b*g*x+a*g)*(d*i*x+c*i)^2*(A+B*log(e*(b*x+a)/(d*x+c)))^2,x, algo rithm="fricas")
Output:
integral(A^2*b*d^2*g*i^2*x^3 + A^2*a*c^2*g*i^2 + (2*A^2*b*c*d + A^2*a*d^2) *g*i^2*x^2 + (A^2*b*c^2 + 2*A^2*a*c*d)*g*i^2*x + (B^2*b*d^2*g*i^2*x^3 + B^ 2*a*c^2*g*i^2 + (2*B^2*b*c*d + B^2*a*d^2)*g*i^2*x^2 + (B^2*b*c^2 + 2*B^2*a *c*d)*g*i^2*x)*log((b*e*x + a*e)/(d*x + c))^2 + 2*(A*B*b*d^2*g*i^2*x^3 + A *B*a*c^2*g*i^2 + (2*A*B*b*c*d + A*B*a*d^2)*g*i^2*x^2 + (A*B*b*c^2 + 2*A*B* a*c*d)*g*i^2*x)*log((b*e*x + a*e)/(d*x + c)), x)
Timed out. \[ \int (a g+b g x) (c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx=\text {Timed out} \] Input:
integrate((b*g*x+a*g)*(d*i*x+c*i)**2*(A+B*ln(e*(b*x+a)/(d*x+c)))**2,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 2259 vs. \(2 (562) = 1124\).
Time = 0.17 (sec) , antiderivative size = 2259, normalized size of antiderivative = 3.84 \[ \int (a g+b g x) (c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx=\text {Too large to display} \] Input:
integrate((b*g*x+a*g)*(d*i*x+c*i)^2*(A+B*log(e*(b*x+a)/(d*x+c)))^2,x, algo rithm="maxima")
Output:
1/4*A^2*b*d^2*g*i^2*x^4 + 2/3*A^2*b*c*d*g*i^2*x^3 + 1/3*A^2*a*d^2*g*i^2*x^ 3 + 1/2*A^2*b*c^2*g*i^2*x^2 + A^2*a*c*d*g*i^2*x^2 + 2*(x*log(b*e*x/(d*x + c) + a*e/(d*x + c)) + a*log(b*x + a)/b - c*log(d*x + c)/d)*A*B*a*c^2*g*i^2 + (x^2*log(b*e*x/(d*x + c) + a*e/(d*x + c)) - a^2*log(b*x + a)/b^2 + c^2* log(d*x + c)/d^2 - (b*c - a*d)*x/(b*d))*A*B*b*c^2*g*i^2 + 2*(x^2*log(b*e*x /(d*x + c) + a*e/(d*x + c)) - a^2*log(b*x + a)/b^2 + c^2*log(d*x + c)/d^2 - (b*c - a*d)*x/(b*d))*A*B*a*c*d*g*i^2 + 2/3*(2*x^3*log(b*e*x/(d*x + c) + a*e/(d*x + c)) + 2*a^3*log(b*x + a)/b^3 - 2*c^3*log(d*x + c)/d^3 - ((b^2*c *d - a*b*d^2)*x^2 - 2*(b^2*c^2 - a^2*d^2)*x)/(b^2*d^2))*A*B*b*c*d*g*i^2 + 1/3*(2*x^3*log(b*e*x/(d*x + c) + a*e/(d*x + c)) + 2*a^3*log(b*x + a)/b^3 - 2*c^3*log(d*x + c)/d^3 - ((b^2*c*d - a*b*d^2)*x^2 - 2*(b^2*c^2 - a^2*d^2) *x)/(b^2*d^2))*A*B*a*d^2*g*i^2 + 1/12*(6*x^4*log(b*e*x/(d*x + c) + a*e/(d* x + c)) - 6*a^4*log(b*x + a)/b^4 + 6*c^4*log(d*x + c)/d^4 - (2*(b^3*c*d^2 - a*b^2*d^3)*x^3 - 3*(b^3*c^2*d - a^2*b*d^3)*x^2 + 6*(b^3*c^3 - a^3*d^3)*x )/(b^3*d^3))*A*B*b*d^2*g*i^2 + A^2*a*c^2*g*i^2*x - 1/12*(7*a^2*b*c^2*d^2*g *i^2 - 2*a^3*c*d^3*g*i^2 - (2*g*i^2*log(e) - g*i^2)*b^3*c^4 + 2*(4*g*i^2*l og(e) - 3*g*i^2)*a*b^2*c^3*d)*B^2*log(d*x + c)/(b^2*d^2) + 1/6*(b^4*c^4*g* i^2 - 4*a*b^3*c^3*d*g*i^2 + 6*a^2*b^2*c^2*d^2*g*i^2 - 4*a^3*b*c*d^3*g*i^2 + a^4*d^4*g*i^2)*(log(b*x + a)*log((b*d*x + a*d)/(b*c - a*d) + 1) + dilog( -(b*d*x + a*d)/(b*c - a*d)))*B^2/(b^3*d^2) + 1/12*(3*B^2*b^4*d^4*g*i^2*...
\[ \int (a g+b g x) (c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx=\int { {\left (b g x + a g\right )} {\left (d i x + c i\right )}^{2} {\left (B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}^{2} \,d x } \] Input:
integrate((b*g*x+a*g)*(d*i*x+c*i)^2*(A+B*log(e*(b*x+a)/(d*x+c)))^2,x, algo rithm="giac")
Output:
integrate((b*g*x + a*g)*(d*i*x + c*i)^2*(B*log((b*x + a)*e/(d*x + c)) + A) ^2, x)
Timed out. \[ \int (a g+b g x) (c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx=\int \left (a\,g+b\,g\,x\right )\,{\left (c\,i+d\,i\,x\right )}^2\,{\left (A+B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\right )}^2 \,d x \] Input:
int((a*g + b*g*x)*(c*i + d*i*x)^2*(A + B*log((e*(a + b*x))/(c + d*x)))^2,x )
Output:
int((a*g + b*g*x)*(c*i + d*i*x)^2*(A + B*log((e*(a + b*x))/(c + d*x)))^2, x)
\[ \int (a g+b g x) (c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx =\text {Too large to display} \] Input:
int((b*g*x+a*g)*(d*i*x+c*i)^2*(A+B*log(e*(b*x+a)/(d*x+c)))^2,x)
Output:
(g*( - 2*int((log((a*e + b*e*x)/(c + d*x))*x)/(a*c + a*d*x + b*c*x + b*d*x **2),x)*a**4*b**2*d**5 + 8*int((log((a*e + b*e*x)/(c + d*x))*x)/(a*c + a*d *x + b*c*x + b*d*x**2),x)*a**3*b**3*c*d**4 - 12*int((log((a*e + b*e*x)/(c + d*x))*x)/(a*c + a*d*x + b*c*x + b*d*x**2),x)*a**2*b**4*c**2*d**3 + 8*int ((log((a*e + b*e*x)/(c + d*x))*x)/(a*c + a*d*x + b*c*x + b*d*x**2),x)*a*b* *5*c**3*d**2 - 2*int((log((a*e + b*e*x)/(c + d*x))*x)/(a*c + a*d*x + b*c*x + b*d*x**2),x)*b**6*c**4*d - 2*log(c + d*x)*a**5*d**4 + 8*log(c + d*x)*a* *4*b*c*d**3 + log(c + d*x)*a**4*b*d**4 - 12*log(c + d*x)*a**3*b**2*c**2*d* *2 - 4*log(c + d*x)*a**3*b**2*c*d**3 + 8*log(c + d*x)*a**2*b**3*c**3*d + 6 *log(c + d*x)*a**2*b**3*c**2*d**2 - 2*log(c + d*x)*a*b**4*c**4 - 4*log(c + d*x)*a*b**4*c**3*d + log(c + d*x)*b**5*c**4 + log((a*e + b*e*x)/(c + d*x) )**2*a**3*b**2*c*d**3 - 3*log((a*e + b*e*x)/(c + d*x))**2*a**2*b**3*c**2*d **2 - log((a*e + b*e*x)/(c + d*x))**2*a*b**4*c**3*d - 12*log((a*e + b*e*x) /(c + d*x))**2*a*b**4*c**2*d**2*x - 12*log((a*e + b*e*x)/(c + d*x))**2*a*b **4*c*d**3*x**2 - 4*log((a*e + b*e*x)/(c + d*x))**2*a*b**4*d**4*x**3 - 6*l og((a*e + b*e*x)/(c + d*x))**2*b**5*c**2*d**2*x**2 - 8*log((a*e + b*e*x)/( c + d*x))**2*b**5*c*d**3*x**3 - 3*log((a*e + b*e*x)/(c + d*x))**2*b**5*d** 4*x**4 - 2*log((a*e + b*e*x)/(c + d*x))*a**5*d**4 + 8*log((a*e + b*e*x)/(c + d*x))*a**4*b*c*d**3 + log((a*e + b*e*x)/(c + d*x))*a**4*b*d**4 - 12*log ((a*e + b*e*x)/(c + d*x))*a**3*b**2*c**2*d**2 - 2*log((a*e + b*e*x)/(c ...