Integrand size = 43, antiderivative size = 635 \[ \int (a g+b g x) (c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\frac {B^2 (b c-a d)^3 g i^2 n^2 x}{12 b^2 d}+\frac {B^2 (b c-a d)^2 g i^2 n^2 (c+d x)^2}{12 b d^2}-\frac {B (b c-a d)^3 g i^2 n (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 b^3 d}-\frac {B (b c-a d)^2 g i^2 n (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 b^3}+\frac {B (b c-a d)^2 g i^2 n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 b d^2}-\frac {B (b c-a d) g i^2 n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 d^2}+\frac {(b c-a d)^2 g i^2 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{12 b^3}+\frac {(b c-a d) g i^2 (a+b x)^2 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{6 b^2}+\frac {g i^2 (a+b x)^2 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 b}-\frac {B (b c-a d)^4 g i^2 n \left (A+B n+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{6 b^3 d^2}-\frac {B^2 (b c-a d)^4 g i^2 n^2 \log \left (\frac {a+b x}{c+d x}\right )}{12 b^3 d^2}-\frac {B^2 (b c-a d)^4 g i^2 n^2 \log (c+d x)}{4 b^3 d^2}-\frac {B^2 (b c-a d)^4 g i^2 n^2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{6 b^3 d^2} \] Output:
1/12*B^2*(-a*d+b*c)^3*g*i^2*n^2*x/b^2/d+1/12*B^2*(-a*d+b*c)^2*g*i^2*n^2*(d *x+c)^2/b/d^2-1/6*B*(-a*d+b*c)^3*g*i^2*n*(b*x+a)*(A+B*ln(e*((b*x+a)/(d*x+c ))^n))/b^3/d-1/6*B*(-a*d+b*c)^2*g*i^2*n*(b*x+a)^2*(A+B*ln(e*((b*x+a)/(d*x+ c))^n))/b^3+1/4*B*(-a*d+b*c)^2*g*i^2*n*(d*x+c)^2*(A+B*ln(e*((b*x+a)/(d*x+c ))^n))/b/d^2-1/6*B*(-a*d+b*c)*g*i^2*n*(d*x+c)^3*(A+B*ln(e*((b*x+a)/(d*x+c) )^n))/d^2+1/12*(-a*d+b*c)^2*g*i^2*(b*x+a)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n) )^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)) ^n))^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))^n))^2 /b-1/6*B*(-a*d+b*c)^4*g*i^2*n*(A+B*n+B*ln(e*((b*x+a)/(d*x+c))^n))*ln((-a*d +b*c)/b/(d*x+c))/b^3/d^2-1/12*B^2*(-a*d+b*c)^4*g*i^2*n^2*ln((b*x+a)/(d*x+c ))/b^3/d^2-1/4*B^2*(-a*d+b*c)^4*g*i^2*n^2*ln(d*x+c)/b^3/d^2-1/6*B^2*(-a*d+ b*c)^4*g*i^2*n^2*polylog(2,d*(b*x+a)/b/(d*x+c))/b^3/d^2
Time = 0.67 (sec) , antiderivative size = 713, normalized size of antiderivative = 1.12 \[ \int (a g+b g x) (c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\frac {g i^2 \left (-4 (b c-a d) (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2+3 b (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2+\frac {4 B (b c-a d)^2 n \left (2 A b d (b c-a d) x-B (b c-a d) n (b d x+(b c-a d) \log (a+b x))+2 B d (b c-a d) (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+b^2 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+2 (b c-a d)^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-2 B (b c-a d)^2 n \log (c+d x)-B (b c-a d)^2 n \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )\right )}{b^3}-\frac {B (b c-a d) n \left (6 A b d (b c-a d)^2 x-3 B (b c-a d)^2 n (b d x+(b c-a d) \log (a+b x))-B (b c-a d) n \left (2 b d (b c-a d) x+b^2 (c+d x)^2+2 (b c-a d)^2 \log (a+b x)\right )+6 B d (b c-a d)^2 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+3 b^2 (b c-a d) (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+2 b^3 (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+6 (b c-a d)^3 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-6 B (b c-a d)^3 n \log (c+d x)-3 B (b c-a d)^3 n \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )\right )}{b^3}\right )}{12 d^2} \] Input:
Integrate[(a*g + b*g*x)*(c*i + d*i*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x)) ^n])^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))^n])^ 2 + 3*b*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2 + (4*B*(b*c - a*d)^2*n*(2*A*b*d*(b*c - a*d)*x - B*(b*c - a*d)*n*(b*d*x + (b*c - a*d)*Lo g[a + b*x]) + 2*B*d*(b*c - a*d)*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n] + b^2*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 2*(b*c - a*d)^2* Log[a + b*x]*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) - 2*B*(b*c - a*d)^2*n* Log[c + d*x] - B*(b*c - a*d)^2*n*(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)*n*(6*A*b*d*(b*c - a*d)^2*x - 3*B*(b*c - a*d)^2*n*(b*d*x + (b*c - a*d)*Log[a + b*x]) - B*(b*c - a*d)*n*(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))^n] + 3*b^2*(b*c - a*d)*(c + d*x)^2*(A + B*Log [e*((a + b*x)/(c + d*x))^n]) + 2*b^3*(c + d*x)^3*(A + B*Log[e*((a + b*x)/( c + d*x))^n]) + 6*(b*c - a*d)^3*Log[a + b*x]*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) - 6*B*(b*c - a*d)^3*n*Log[c + d*x] - 3*B*(b*c - a*d)^3*n*(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 = 1.55 (sec) , antiderivative size = 702, normalized size of antiderivative = 1.11, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.326, Rules used = {2961, 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.
| \(\displaystyle \int (a g+b g x) (c i+d i x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2 \, dx\) |
| \(\Big \downarrow \) 2961 |
| \(\displaystyle g i^2 (b c-a d)^4 \int \frac {(a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )^5}d\frac {a+b x}{c+d x}\) |
| \(\Big \downarrow \) 2783 |
| \(\displaystyle g i^2 (b c-a d)^4 \left (-\frac {B n \int \frac {(a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )^4}d\frac {a+b x}{c+d x}}{2 b}+\frac {\int \frac {(a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )^4}d\frac {a+b x}{c+d x}}{2 b}+\frac {(a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}\right )\) |
| \(\Big \downarrow \) 2782 |
| \(\displaystyle g i^2 (b c-a d)^4 \left (-\frac {B n \left (-B n \int -\frac {(c+d x) \left (b-\frac {3 d (a+b x)}{c+d x}\right )}{6 d^2 (a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^3}d\frac {a+b x}{c+d x}-\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{2 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}\right )}{2 b}+\frac {\int \frac {(a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )^4}d\frac {a+b x}{c+d x}}{2 b}+\frac {(a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle g i^2 (b c-a d)^4 \left (-\frac {B n \left (\frac {B n \int \frac {(c+d x) \left (b-\frac {3 d (a+b x)}{c+d x}\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^3}d\frac {a+b x}{c+d x}}{6 d^2}-\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{2 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}\right )}{2 b}+\frac {\int \frac {(a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )^4}d\frac {a+b x}{c+d x}}{2 b}+\frac {(a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}\right )\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle g i^2 (b c-a d)^4 \left (-\frac {B n \left (\frac {B n \int \left (\frac {d}{b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {d}{b \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {2 d}{\left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {c+d x}{b^2 (a+b x)}\right )d\frac {a+b x}{c+d x}}{6 d^2}-\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{2 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}\right )}{2 b}+\frac {\int \frac {(a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )^4}d\frac {a+b x}{c+d x}}{2 b}+\frac {(a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle g i^2 (b c-a d)^4 \left (\frac {\int \frac {(a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )^4}d\frac {a+b x}{c+d x}}{2 b}-\frac {B n \left (-\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{2 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^2}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^2}+\frac {1}{b \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right )^2}\right )}{6 d^2}\right )}{2 b}+\frac {(a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}\right )\) |
| \(\Big \downarrow \) 2783 |
| \(\displaystyle g i^2 (b c-a d)^4 \left (\frac {-\frac {2 B n \int \frac {(a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )^3}d\frac {a+b x}{c+d x}}{3 b}+\frac {\int \frac {(a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )^3}d\frac {a+b x}{c+d x}}{3 b}+\frac {(a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{3 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}}{2 b}-\frac {B n \left (-\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{2 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^2}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^2}+\frac {1}{b \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right )^2}\right )}{6 d^2}\right )}{2 b}+\frac {(a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}\right )\) |
| \(\Big \downarrow \) 2773 |
| \(\displaystyle g i^2 (b c-a d)^4 \left (\frac {-\frac {2 B n \left (\frac {(a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \int \frac {a+b x}{(c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )^2}d\frac {a+b x}{c+d x}}{2 b}\right )}{3 b}+\frac {\int \frac {(a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )^3}d\frac {a+b x}{c+d x}}{3 b}+\frac {(a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{3 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}}{2 b}-\frac {B n \left (-\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{2 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^2}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^2}+\frac {1}{b \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right )^2}\right )}{6 d^2}\right )}{2 b}+\frac {(a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}\right )\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle g i^2 (b c-a d)^4 \left (\frac {-\frac {2 B n \left (\frac {(a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \int \left (\frac {b}{d \left (\frac {d (a+b x)}{c+d x}-b\right )^2}+\frac {1}{d \left (\frac {d (a+b x)}{c+d x}-b\right )}\right )d\frac {a+b x}{c+d x}}{2 b}\right )}{3 b}+\frac {\int \frac {(a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )^3}d\frac {a+b x}{c+d x}}{3 b}+\frac {(a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{3 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}}{2 b}-\frac {B n \left (-\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{2 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^2}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^2}+\frac {1}{b \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right )^2}\right )}{6 d^2}\right )}{2 b}+\frac {(a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle g i^2 (b c-a d)^4 \left (\frac {\frac {\int \frac {(a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )^3}d\frac {a+b x}{c+d x}}{3 b}-\frac {2 B n \left (\frac {(a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {b}{d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{d^2}\right )}{2 b}\right )}{3 b}+\frac {(a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{3 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}}{2 b}-\frac {B n \left (-\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{2 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^2}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^2}+\frac {1}{b \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right )^2}\right )}{6 d^2}\right )}{2 b}+\frac {(a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}\right )\) |
| \(\Big \downarrow \) 2781 |
| \(\displaystyle g i^2 (b c-a d)^4 \left (\frac {\frac {\frac {(a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \int \frac {(a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )^2}d\frac {a+b x}{c+d x}}{b}}{3 b}-\frac {2 B n \left (\frac {(a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {b}{d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{d^2}\right )}{2 b}\right )}{3 b}+\frac {(a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{3 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}}{2 b}-\frac {B n \left (-\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{2 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^2}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^2}+\frac {1}{b \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right )^2}\right )}{6 d^2}\right )}{2 b}+\frac {(a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}\right )\) |
| \(\Big \downarrow \) 2784 |
| \(\displaystyle g i^2 (b c-a d)^4 \left (\frac {\frac {\frac {(a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {(a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{d (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {\int \frac {A+B n+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b-\frac {d (a+b x)}{c+d x}}d\frac {a+b x}{c+d x}}{d}\right )}{b}}{3 b}-\frac {2 B n \left (\frac {(a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {b}{d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{d^2}\right )}{2 b}\right )}{3 b}+\frac {(a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{3 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}}{2 b}-\frac {B n \left (-\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{2 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^2}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^2}+\frac {1}{b \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right )^2}\right )}{6 d^2}\right )}{2 b}+\frac {(a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}\right )\) |
| \(\Big \downarrow \) 2754 |
| \(\displaystyle g i^2 (b c-a d)^4 \left (\frac {\frac {\frac {(a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {(a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{d (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {\frac {B n \int \frac {(c+d x) \log \left (1-\frac {d (a+b x)}{b (c+d x)}\right )}{a+b x}d\frac {a+b x}{c+d x}}{d}-\frac {\log \left (1-\frac {d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A+B n\right )}{d}}{d}\right )}{b}}{3 b}-\frac {2 B n \left (\frac {(a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {b}{d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{d^2}\right )}{2 b}\right )}{3 b}+\frac {(a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{3 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}}{2 b}-\frac {B n \left (-\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{2 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^2}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^2}+\frac {1}{b \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right )^2}\right )}{6 d^2}\right )}{2 b}+\frac {(a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}\right )\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle g i^2 (b c-a d)^4 \left (-\frac {B n \left (-\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{2 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^2}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^2}+\frac {1}{b \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right )^2}\right )}{6 d^2}\right )}{2 b}+\frac {-\frac {2 B n \left (\frac {(a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {b}{d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{d^2}\right )}{2 b}\right )}{3 b}+\frac {\frac {(a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {(a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{d (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {-\frac {\log \left (1-\frac {d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A+B n\right )}{d}-\frac {B n \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d}}{d}\right )}{b}}{3 b}+\frac {(a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{3 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}}{2 b}+\frac {(a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}\right )\) |
Input:
Int[(a*g + b*g*x)*(c*i + d*i*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^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))^n])^2 )/(4*b*(c + d*x)^2*(b - (d*(a + b*x))/(c + d*x))^4) - (B*n*((b*(A + B*Log[ e*((a + b*x)/(c + d*x))^n]))/(3*d^2*(b - (d*(a + b*x))/(c + d*x))^3) - (A + B*Log[e*((a + b*x)/(c + d*x))^n])/(2*d^2*(b - (d*(a + b*x))/(c + d*x))^2 ) + (B*n*(-(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)) ^n])^2)/(3*b*(c + d*x)^2*(b - (d*(a + b*x))/(c + d*x))^3) - (2*B*n*(((a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*b*(c + d*x)^2*(b - (d*(a + b*x))/(c + d*x))^2) - (B*n*(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*L og[e*((a + b*x)/(c + d*x))^n])^2)/(2*b*(c + d*x)^2*(b - (d*(a + b*x))/(c + d*x))^2) - (B*n*(((a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(d*(c + d*x)*(b - (d*(a + b*x))/(c + d*x))) - (-(((A + B*n + B*Log[e*((a + b*x) /(c + d*x))^n])*Log[1 - (d*(a + b*x))/(b*(c + d*x))])/d) - (B*n*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_))/((c_.) + (d_.)*(x_)))^(n_.)]*( B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.)*((h_.) + (i_.)*(x_))^(q_.), x_Symbol ] :> Simp[(b*c - a*d)^(m + q + 1)*(g/b)^m*(i/d)^q Subst[Int[x^m*((A + B*L og[e*x^n])^p/(b - d*x)^(m + q + 2)), x], x, (a + b*x)/(c + d*x)], x] /; Fre eQ[{a, b, c, d, e, f, g, h, i, A, B, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[ b*f - a*g, 0] && EqQ[d*h - c*i, 0] && IntegersQ[m, q]
\[\int \left (b g x +a g \right ) \left (d i x +c i \right )^{2} {\left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )}^{2}d x\]
Input:
int((b*g*x+a*g)*(d*i*x+c*i)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^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))^n))^2,x)
\[ \int (a g+b g x) (c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\int { {\left (b g x + a g\right )} {\left (d i x + c i\right )}^{2} {\left (B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{2} \,d x } \] Input:
integrate((b*g*x+a*g)*(d*i*x+c*i)^2*(A+B*log(e*((b*x+a)/(d*x+c))^n))^2,x, algorithm="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(e*((b*x + a)/(d*x + c))^n)^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(e*((b*x + a)/(d*x + c))^n), x)
\[ \int (a g+b g x) (c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=g i^{2} \left (\int A^{2} a c^{2}\, dx + \int A^{2} a d^{2} x^{2}\, dx + \int A^{2} b c^{2} x\, dx + \int A^{2} b d^{2} x^{3}\, dx + \int B^{2} a c^{2} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}^{2}\, dx + \int 2 A B a c^{2} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}\, dx + \int 2 A^{2} a c d x\, dx + \int 2 A^{2} b c d x^{2}\, dx + \int B^{2} a d^{2} x^{2} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}^{2}\, dx + \int B^{2} b c^{2} x \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}^{2}\, dx + \int B^{2} b d^{2} x^{3} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}^{2}\, dx + \int 2 A B a d^{2} x^{2} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}\, dx + \int 2 A B b c^{2} x \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}\, dx + \int 2 A B b d^{2} x^{3} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}\, dx + \int 2 B^{2} a c d x \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}^{2}\, dx + \int 2 B^{2} b c d x^{2} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}^{2}\, dx + \int 4 A B a c d x \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}\, dx + \int 4 A B b c d x^{2} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}\, dx\right ) \] Input:
integrate((b*g*x+a*g)*(d*i*x+c*i)**2*(A+B*ln(e*((b*x+a)/(d*x+c))**n))**2,x )
Output:
g*i**2*(Integral(A**2*a*c**2, x) + Integral(A**2*a*d**2*x**2, x) + Integra l(A**2*b*c**2*x, x) + Integral(A**2*b*d**2*x**3, x) + Integral(B**2*a*c**2 *log(e*(a/(c + d*x) + b*x/(c + d*x))**n)**2, x) + Integral(2*A*B*a*c**2*lo g(e*(a/(c + d*x) + b*x/(c + d*x))**n), x) + Integral(2*A**2*a*c*d*x, x) + Integral(2*A**2*b*c*d*x**2, x) + Integral(B**2*a*d**2*x**2*log(e*(a/(c + d *x) + b*x/(c + d*x))**n)**2, x) + Integral(B**2*b*c**2*x*log(e*(a/(c + d*x ) + b*x/(c + d*x))**n)**2, x) + Integral(B**2*b*d**2*x**3*log(e*(a/(c + d* x) + b*x/(c + d*x))**n)**2, x) + Integral(2*A*B*a*d**2*x**2*log(e*(a/(c + d*x) + b*x/(c + d*x))**n), x) + Integral(2*A*B*b*c**2*x*log(e*(a/(c + d*x) + b*x/(c + d*x))**n), x) + Integral(2*A*B*b*d**2*x**3*log(e*(a/(c + d*x) + b*x/(c + d*x))**n), x) + Integral(2*B**2*a*c*d*x*log(e*(a/(c + d*x) + b* x/(c + d*x))**n)**2, x) + Integral(2*B**2*b*c*d*x**2*log(e*(a/(c + d*x) + b*x/(c + d*x))**n)**2, x) + Integral(4*A*B*a*c*d*x*log(e*(a/(c + d*x) + b* x/(c + d*x))**n), x) + Integral(4*A*B*b*c*d*x**2*log(e*(a/(c + d*x) + b*x/ (c + d*x))**n), x))
Leaf count of result is larger than twice the leaf count of optimal. 2662 vs. \(2 (608) = 1216\).
Time = 0.61 (sec) , antiderivative size = 2662, normalized size of antiderivative = 4.19 \[ \int (a g+b g x) (c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\text {Too large to display} \] Input:
integrate((b*g*x+a*g)*(d*i*x+c*i)^2*(A+B*log(e*((b*x+a)/(d*x+c))^n))^2,x, algorithm="maxima")
Output:
1/2*A*B*b*d^2*g*i^2*x^4*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/4*A^2*b *d^2*g*i^2*x^4 + 4/3*A*B*b*c*d*g*i^2*x^3*log(e*(b*x/(d*x + c) + a/(d*x + c ))^n) + 2/3*A*B*a*d^2*g*i^2*x^3*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 2 /3*A^2*b*c*d*g*i^2*x^3 + 1/3*A^2*a*d^2*g*i^2*x^3 + A*B*b*c^2*g*i^2*x^2*log (e*(b*x/(d*x + c) + a/(d*x + c))^n) + 2*A*B*a*c*d*g*i^2*x^2*log(e*(b*x/(d* x + c) + a/(d*x + c))^n) + 1/2*A^2*b*c^2*g*i^2*x^2 + A^2*a*c*d*g*i^2*x^2 - 1/12*A*B*b*d^2*g*i^2*n*(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)) + 2/3*A*B*b*c*d*g*i^2*n*(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)) + 1/3*A*B*a*d^2*g*i^2*n*(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^2*g*i^2*n*(a^2*log(b*x + a)/b^2 - c^2*log(d*x + c)/d^2 + (b*c - a*d)*x/(b*d)) - 2*A*B*a*c*d*g*i^2*n*(a^2*log(b*x + a)/b^2 - c^2*lo g(d*x + c)/d^2 + (b*c - a*d)*x/(b*d)) + 2*A*B*a*c^2*g*i^2*n*(a*log(b*x + a )/b - c*log(d*x + c)/d) + 2*A*B*a*c^2*g*i^2*x*log(e*(b*x/(d*x + c) + a/(d* x + c))^n) + A^2*a*c^2*g*i^2*x - 1/12*(7*a^2*b*c^2*d^2*g*i^2*n^2 - 2*a^3*c *d^3*g*i^2*n^2 + (g*i^2*n^2 - 2*g*i^2*n*log(e))*b^3*c^4 - 2*(3*g*i^2*n^2 - 4*g*i^2*n*log(e))*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*n^2 - 4*a*b^3*c^3*d*g*i^2*n^2 + 6*a^2*b^2*c^2*d^2*g*i^2*n^2 - 4*a...
Timed out. \[ \int (a g+b g x) (c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\text {Timed out} \] Input:
integrate((b*g*x+a*g)*(d*i*x+c*i)^2*(A+B*log(e*((b*x+a)/(d*x+c))^n))^2,x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int (a g+b g x) (c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\int \left (a\,g+b\,g\,x\right )\,{\left (c\,i+d\,i\,x\right )}^2\,{\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}^2 \,d x \] Input:
int((a*g + b*g*x)*(c*i + d*i*x)^2*(A + B*log(e*((a + b*x)/(c + d*x))^n))^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))^n))^2 , x)
\[ \int (a g+b g x) (c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\text {too large to display} \] Input:
int((b*g*x+a*g)*(d*i*x+c*i)^2*(A+B*log(e*((b*x+a)/(d*x+c))^n))^2,x)
Output:
(g*( - 2*int((log(((a + b*x)**n*e)/(c + d*x)**n)*x)/(a*c + a*d*x + b*c*x + b*d*x**2),x)*a**4*b**2*d**5*n + 8*int((log(((a + b*x)**n*e)/(c + d*x)**n) *x)/(a*c + a*d*x + b*c*x + b*d*x**2),x)*a**3*b**3*c*d**4*n - 12*int((log(( (a + b*x)**n*e)/(c + d*x)**n)*x)/(a*c + a*d*x + b*c*x + b*d*x**2),x)*a**2* b**4*c**2*d**3*n + 8*int((log(((a + b*x)**n*e)/(c + d*x)**n)*x)/(a*c + a*d *x + b*c*x + b*d*x**2),x)*a*b**5*c**3*d**2*n - 2*int((log(((a + b*x)**n*e) /(c + d*x)**n)*x)/(a*c + a*d*x + b*c*x + b*d*x**2),x)*b**6*c**4*d*n - 2*lo g(c + d*x)*a**5*d**4*n + 8*log(c + d*x)*a**4*b*c*d**3*n + log(c + d*x)*a** 4*b*d**4*n**2 - 12*log(c + d*x)*a**3*b**2*c**2*d**2*n - 4*log(c + d*x)*a** 3*b**2*c*d**3*n**2 + 8*log(c + d*x)*a**2*b**3*c**3*d*n + 6*log(c + d*x)*a* *2*b**3*c**2*d**2*n**2 - 2*log(c + d*x)*a*b**4*c**4*n - 4*log(c + d*x)*a*b **4*c**3*d*n**2 + log(c + d*x)*b**5*c**4*n**2 + log(((a + b*x)**n*e)/(c + d*x)**n)**2*a**3*b**2*c*d**3 - 3*log(((a + b*x)**n*e)/(c + d*x)**n)**2*a** 2*b**3*c**2*d**2 - log(((a + b*x)**n*e)/(c + d*x)**n)**2*a*b**4*c**3*d - 1 2*log(((a + b*x)**n*e)/(c + d*x)**n)**2*a*b**4*c**2*d**2*x - 12*log(((a + b*x)**n*e)/(c + d*x)**n)**2*a*b**4*c*d**3*x**2 - 4*log(((a + b*x)**n*e)/(c + d*x)**n)**2*a*b**4*d**4*x**3 - 6*log(((a + b*x)**n*e)/(c + d*x)**n)**2* b**5*c**2*d**2*x**2 - 8*log(((a + b*x)**n*e)/(c + d*x)**n)**2*b**5*c*d**3* x**3 - 3*log(((a + b*x)**n*e)/(c + d*x)**n)**2*b**5*d**4*x**4 - 2*log(((a + b*x)**n*e)/(c + d*x)**n)*a**5*d**4 + 8*log(((a + b*x)**n*e)/(c + d*x)...