Optimal. Leaf size=487 \[ \frac {B g^2 i n (b c-a d)^4 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+2 A+3 B n\right )}{12 b^2 d^3}+\frac {B g^2 i n (a+b x) (b c-a d)^3 \left (2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+2 A+B n\right )}{12 b^2 d^2}-\frac {B g^2 i n (a+b x)^2 (b c-a d)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{12 b^2 d}+\frac {g^2 i (a+b x)^3 (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{12 b^2}-\frac {B g^2 i n (a+b x)^3 (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{6 b^2}+\frac {g^2 i (a+b x)^3 (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 b}+\frac {B^2 g^2 i n^2 (b c-a d)^4 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{6 b^2 d^3}+\frac {B^2 g^2 i n^2 (b c-a d)^4 \log (c+d x)}{6 b^2 d^3}+\frac {B^2 g^2 i n^2 (c+d x)^2 (b c-a d)^2}{12 d^3}-\frac {B^2 g^2 i n^2 x (b c-a d)^3}{3 b d^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.59, antiderivative size = 578, normalized size of antiderivative = 1.19, number of steps used = 44, number of rules used = 13, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.302, Rules used = {2528, 2525, 12, 2486, 31, 43, 2524, 2418, 2394, 2393, 2391, 2390, 2301} \[ \frac {B^2 g^2 i n^2 (b c-a d)^4 \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{6 b^2 d^3}-\frac {B g^2 i n (b c-a d)^4 \log (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{6 b^2 d^3}-\frac {B g^2 i n (a+b x)^2 (b c-a d)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{12 b^2 d}+\frac {g^2 i (a+b x)^3 (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{3 b^2}-\frac {B g^2 i n (a+b x)^3 (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{6 b^2}+\frac {d g^2 i (a+b x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 b^2}+\frac {A B g^2 i n x (b c-a d)^3}{6 b d^2}+\frac {B^2 g^2 i n (a+b x) (b c-a d)^3 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{6 b^2 d^2}-\frac {B^2 g^2 i n^2 (b c-a d)^4 \log ^2(c+d x)}{12 b^2 d^3}-\frac {B^2 g^2 i n^2 (b c-a d)^4 \log (c+d x)}{12 b^2 d^3}+\frac {B^2 g^2 i n^2 (b c-a d)^4 \log (c+d x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{6 b^2 d^3}+\frac {B^2 g^2 i n^2 (a+b x)^2 (b c-a d)^2}{12 b^2 d}-\frac {B^2 g^2 i n^2 x (b c-a d)^3}{12 b d^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 31
Rule 43
Rule 2301
Rule 2390
Rule 2391
Rule 2393
Rule 2394
Rule 2418
Rule 2486
Rule 2524
Rule 2525
Rule 2528
Rubi steps
\begin {align*} \int (160 c+160 d x) (a g+b g x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx &=\int \left (\frac {160 (b c-a d) (a g+b g x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b}+\frac {160 d (a g+b g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b g}\right ) \, dx\\ &=\frac {(160 (b c-a d)) \int (a g+b g x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx}{b}+\frac {(160 d) \int (a g+b g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx}{b g}\\ &=\frac {160 (b c-a d) g^2 (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^2}+\frac {40 d g^2 (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2}-\frac {(80 B d n) \int \frac {(b c-a d) g^4 (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{c+d x} \, dx}{b^2 g^2}-\frac {(320 B (b c-a d) n) \int \frac {(b c-a d) g^3 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{c+d x} \, dx}{3 b^2 g}\\ &=\frac {160 (b c-a d) g^2 (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^2}+\frac {40 d g^2 (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2}-\frac {\left (80 B d (b c-a d) g^2 n\right ) \int \frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{c+d x} \, dx}{b^2}-\frac {\left (320 B (b c-a d)^2 g^2 n\right ) \int \frac {(a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{c+d x} \, dx}{3 b^2}\\ &=\frac {160 (b c-a d) g^2 (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^2}+\frac {40 d g^2 (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2}-\frac {\left (80 B d (b c-a d) g^2 n\right ) \int \left (\frac {b (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d^3}-\frac {b (b c-a d) (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d^2}+\frac {b (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d}+\frac {(-b c+a d)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d^3 (c+d x)}\right ) \, dx}{b^2}-\frac {\left (320 B (b c-a d)^2 g^2 n\right ) \int \left (-\frac {b (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d^2}+\frac {b (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d}+\frac {(-b c+a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d^2 (c+d x)}\right ) \, dx}{3 b^2}\\ &=\frac {160 (b c-a d) g^2 (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^2}+\frac {40 d g^2 (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2}-\frac {\left (80 B (b c-a d) g^2 n\right ) \int (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b}+\frac {\left (80 B (b c-a d)^2 g^2 n\right ) \int (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b d}-\frac {\left (320 B (b c-a d)^2 g^2 n\right ) \int (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{3 b d}-\frac {\left (80 B (b c-a d)^3 g^2 n\right ) \int \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b d^2}+\frac {\left (320 B (b c-a d)^3 g^2 n\right ) \int \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{3 b d^2}+\frac {\left (80 B (b c-a d)^4 g^2 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{b^2 d^2}-\frac {\left (320 B (b c-a d)^4 g^2 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{3 b^2 d^2}\\ &=\frac {80 A B (b c-a d)^3 g^2 n x}{3 b d^2}-\frac {40 B (b c-a d)^2 g^2 n (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^2 d}-\frac {80 B (b c-a d) g^2 n (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^2}+\frac {160 (b c-a d) g^2 (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^2}+\frac {40 d g^2 (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2}-\frac {80 B (b c-a d)^4 g^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3 b^2 d^3}-\frac {\left (80 B^2 (b c-a d)^3 g^2 n\right ) \int \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \, dx}{b d^2}+\frac {\left (320 B^2 (b c-a d)^3 g^2 n\right ) \int \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \, dx}{3 b d^2}+\frac {\left (80 B^2 (b c-a d) g^2 n^2\right ) \int \frac {(b c-a d) (a+b x)^2}{c+d x} \, dx}{3 b^2}-\frac {\left (40 B^2 (b c-a d)^2 g^2 n^2\right ) \int \frac {(b c-a d) (a+b x)}{c+d x} \, dx}{b^2 d}+\frac {\left (160 B^2 (b c-a d)^2 g^2 n^2\right ) \int \frac {(b c-a d) (a+b x)}{c+d x} \, dx}{3 b^2 d}-\frac {\left (80 B^2 (b c-a d)^4 g^2 n^2\right ) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{b^2 d^3}+\frac {\left (320 B^2 (b c-a d)^4 g^2 n^2\right ) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{3 b^2 d^3}\\ &=\frac {80 A B (b c-a d)^3 g^2 n x}{3 b d^2}+\frac {80 B^2 (b c-a d)^3 g^2 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{3 b^2 d^2}-\frac {40 B (b c-a d)^2 g^2 n (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^2 d}-\frac {80 B (b c-a d) g^2 n (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^2}+\frac {160 (b c-a d) g^2 (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^2}+\frac {40 d g^2 (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2}-\frac {80 B (b c-a d)^4 g^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3 b^2 d^3}+\frac {\left (80 B^2 (b c-a d)^2 g^2 n^2\right ) \int \frac {(a+b x)^2}{c+d x} \, dx}{3 b^2}-\frac {\left (40 B^2 (b c-a d)^3 g^2 n^2\right ) \int \frac {a+b x}{c+d x} \, dx}{b^2 d}+\frac {\left (160 B^2 (b c-a d)^3 g^2 n^2\right ) \int \frac {a+b x}{c+d x} \, dx}{3 b^2 d}-\frac {\left (80 B^2 (b c-a d)^4 g^2 n^2\right ) \int \left (\frac {b \log (c+d x)}{a+b x}-\frac {d \log (c+d x)}{c+d x}\right ) \, dx}{b^2 d^3}+\frac {\left (320 B^2 (b c-a d)^4 g^2 n^2\right ) \int \left (\frac {b \log (c+d x)}{a+b x}-\frac {d \log (c+d x)}{c+d x}\right ) \, dx}{3 b^2 d^3}+\frac {\left (80 B^2 (b c-a d)^4 g^2 n^2\right ) \int \frac {1}{c+d x} \, dx}{b^2 d^2}-\frac {\left (320 B^2 (b c-a d)^4 g^2 n^2\right ) \int \frac {1}{c+d x} \, dx}{3 b^2 d^2}\\ &=\frac {80 A B (b c-a d)^3 g^2 n x}{3 b d^2}+\frac {80 B^2 (b c-a d)^3 g^2 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{3 b^2 d^2}-\frac {40 B (b c-a d)^2 g^2 n (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^2 d}-\frac {80 B (b c-a d) g^2 n (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^2}+\frac {160 (b c-a d) g^2 (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^2}+\frac {40 d g^2 (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2}-\frac {80 B^2 (b c-a d)^4 g^2 n^2 \log (c+d x)}{3 b^2 d^3}-\frac {80 B (b c-a d)^4 g^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3 b^2 d^3}+\frac {\left (80 B^2 (b c-a d)^2 g^2 n^2\right ) \int \left (-\frac {b (b c-a d)}{d^2}+\frac {b (a+b x)}{d}+\frac {(-b c+a d)^2}{d^2 (c+d x)}\right ) \, dx}{3 b^2}-\frac {\left (40 B^2 (b c-a d)^3 g^2 n^2\right ) \int \left (\frac {b}{d}+\frac {-b c+a d}{d (c+d x)}\right ) \, dx}{b^2 d}+\frac {\left (160 B^2 (b c-a d)^3 g^2 n^2\right ) \int \left (\frac {b}{d}+\frac {-b c+a d}{d (c+d x)}\right ) \, dx}{3 b^2 d}-\frac {\left (80 B^2 (b c-a d)^4 g^2 n^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{b d^3}+\frac {\left (320 B^2 (b c-a d)^4 g^2 n^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{3 b d^3}+\frac {\left (80 B^2 (b c-a d)^4 g^2 n^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{b^2 d^2}-\frac {\left (320 B^2 (b c-a d)^4 g^2 n^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{3 b^2 d^2}\\ &=\frac {80 A B (b c-a d)^3 g^2 n x}{3 b d^2}-\frac {40 B^2 (b c-a d)^3 g^2 n^2 x}{3 b d^2}+\frac {40 B^2 (b c-a d)^2 g^2 n^2 (a+b x)^2}{3 b^2 d}+\frac {80 B^2 (b c-a d)^3 g^2 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{3 b^2 d^2}-\frac {40 B (b c-a d)^2 g^2 n (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^2 d}-\frac {80 B (b c-a d) g^2 n (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^2}+\frac {160 (b c-a d) g^2 (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^2}+\frac {40 d g^2 (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2}-\frac {40 B^2 (b c-a d)^4 g^2 n^2 \log (c+d x)}{3 b^2 d^3}+\frac {80 B^2 (b c-a d)^4 g^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b^2 d^3}-\frac {80 B (b c-a d)^4 g^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3 b^2 d^3}+\frac {\left (80 B^2 (b c-a d)^4 g^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{b^2 d^3}-\frac {\left (320 B^2 (b c-a d)^4 g^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{3 b^2 d^3}+\frac {\left (80 B^2 (b c-a d)^4 g^2 n^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b^2 d^2}-\frac {\left (320 B^2 (b c-a d)^4 g^2 n^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{3 b^2 d^2}\\ &=\frac {80 A B (b c-a d)^3 g^2 n x}{3 b d^2}-\frac {40 B^2 (b c-a d)^3 g^2 n^2 x}{3 b d^2}+\frac {40 B^2 (b c-a d)^2 g^2 n^2 (a+b x)^2}{3 b^2 d}+\frac {80 B^2 (b c-a d)^3 g^2 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{3 b^2 d^2}-\frac {40 B (b c-a d)^2 g^2 n (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^2 d}-\frac {80 B (b c-a d) g^2 n (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^2}+\frac {160 (b c-a d) g^2 (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^2}+\frac {40 d g^2 (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2}-\frac {40 B^2 (b c-a d)^4 g^2 n^2 \log (c+d x)}{3 b^2 d^3}+\frac {80 B^2 (b c-a d)^4 g^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b^2 d^3}-\frac {80 B (b c-a d)^4 g^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3 b^2 d^3}-\frac {40 B^2 (b c-a d)^4 g^2 n^2 \log ^2(c+d x)}{3 b^2 d^3}+\frac {\left (80 B^2 (b c-a d)^4 g^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{b^2 d^3}-\frac {\left (320 B^2 (b c-a d)^4 g^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{3 b^2 d^3}\\ &=\frac {80 A B (b c-a d)^3 g^2 n x}{3 b d^2}-\frac {40 B^2 (b c-a d)^3 g^2 n^2 x}{3 b d^2}+\frac {40 B^2 (b c-a d)^2 g^2 n^2 (a+b x)^2}{3 b^2 d}+\frac {80 B^2 (b c-a d)^3 g^2 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{3 b^2 d^2}-\frac {40 B (b c-a d)^2 g^2 n (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^2 d}-\frac {80 B (b c-a d) g^2 n (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^2}+\frac {160 (b c-a d) g^2 (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^2}+\frac {40 d g^2 (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2}-\frac {40 B^2 (b c-a d)^4 g^2 n^2 \log (c+d x)}{3 b^2 d^3}+\frac {80 B^2 (b c-a d)^4 g^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b^2 d^3}-\frac {80 B (b c-a d)^4 g^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3 b^2 d^3}-\frac {40 B^2 (b c-a d)^4 g^2 n^2 \log ^2(c+d x)}{3 b^2 d^3}+\frac {80 B^2 (b c-a d)^4 g^2 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{3 b^2 d^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.57, size = 716, normalized size = 1.47 \[ \frac {g^2 i \left (\frac {4 B n (b c-a d)^2 \left (-d^2 (a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )-2 (b c-a d)^2 \log (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )+2 A b d x (b c-a d)+2 B d (a+b x) (b c-a d) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+B n (b c-a d)^2 \left (2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac {d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )-2 B n (b c-a d)^2 \log (c+d x)+B n (b c-a d) ((a d-b c) \log (c+d x)+b d x)\right )}{d^3}-\frac {B n (b c-a d) \left (2 d^3 (a+b x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )+3 d^2 (a+b x)^2 (a d-b c) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )-6 (b c-a d)^3 \log (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )+6 A b d x (b c-a d)^2+B n (b c-a d) \left (2 b d x (b c-a d)-2 (b c-a d)^2 \log (c+d x)-d^2 (a+b x)^2\right )+6 B d (a+b x) (b c-a d)^2 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+3 B n (b c-a d)^3 \left (2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac {d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )-6 B n (b c-a d)^3 \log (c+d x)+3 B n (b c-a d)^2 ((a d-b c) \log (c+d x)+b d x)\right )}{d^3}+3 d (a+b x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2+4 (a+b x)^3 (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2\right )}{12 b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.82, size = 0, normalized size = 0.00 \[ {\rm integral}\left (A^{2} b^{2} d g^{2} i x^{3} + A^{2} a^{2} c g^{2} i + {\left (A^{2} b^{2} c + 2 \, A^{2} a b d\right )} g^{2} i x^{2} + {\left (2 \, A^{2} a b c + A^{2} a^{2} d\right )} g^{2} i x + {\left (B^{2} b^{2} d g^{2} i x^{3} + B^{2} a^{2} c g^{2} i + {\left (B^{2} b^{2} c + 2 \, B^{2} a b d\right )} g^{2} i x^{2} + {\left (2 \, B^{2} a b c + B^{2} a^{2} d\right )} g^{2} i x\right )} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )^{2} + 2 \, {\left (A B b^{2} d g^{2} i x^{3} + A B a^{2} c g^{2} i + {\left (A B b^{2} c + 2 \, A B a b d\right )} g^{2} i x^{2} + {\left (2 \, A B a b c + A B a^{2} d\right )} g^{2} i x\right )} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.30, size = 0, normalized size = 0.00 \[ \int \left (b g x +a g \right )^{2} \left (d i x +c i \right ) \left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 6.92, size = 2691, normalized size = 5.53 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (a\,g+b\,g\,x\right )}^2\,\left (c\,i+d\,i\,x\right )\,{\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________