Optimal. Leaf size=381 \[ -\frac {b^3 (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g^2 i^3 (a+b x) (b c-a d)^4}-\frac {3 b^2 d \log \left (\frac {a+b x}{c+d x}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g^2 i^3 (b c-a d)^4}-\frac {d^3 (a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 g^2 i^3 (c+d x)^2 (b c-a d)^4}+\frac {3 b d^2 (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g^2 i^3 (c+d x) (b c-a d)^4}-\frac {b^3 B n (c+d x)}{g^2 i^3 (a+b x) (b c-a d)^4}+\frac {3 b^2 B d n \log ^2\left (\frac {a+b x}{c+d x}\right )}{2 g^2 i^3 (b c-a d)^4}+\frac {B d^3 n (a+b x)^2}{4 g^2 i^3 (c+d x)^2 (b c-a d)^4}-\frac {3 b B d^2 n (a+b x)}{g^2 i^3 (c+d x) (b c-a d)^4} \]
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Rubi [C] time = 1.08, antiderivative size = 657, normalized size of antiderivative = 1.72, number of steps used = 30, number of rules used = 11, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.256, Rules used = {2528, 2525, 12, 44, 2524, 2418, 2390, 2301, 2394, 2393, 2391} \[ -\frac {3 b^2 B d n \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{g^2 i^3 (b c-a d)^4}-\frac {3 b^2 B d n \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{g^2 i^3 (b c-a d)^4}-\frac {3 b^2 d \log (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g^2 i^3 (b c-a d)^4}-\frac {b^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g^2 i^3 (a+b x) (b c-a d)^3}+\frac {3 b^2 d \log (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g^2 i^3 (b c-a d)^4}-\frac {2 b d \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g^2 i^3 (c+d x) (b c-a d)^3}-\frac {d \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 g^2 i^3 (c+d x)^2 (b c-a d)^2}-\frac {b^2 B n}{g^2 i^3 (a+b x) (b c-a d)^3}+\frac {3 b^2 B d n \log ^2(a+b x)}{2 g^2 i^3 (b c-a d)^4}+\frac {3 b^2 B d n \log ^2(c+d x)}{2 g^2 i^3 (b c-a d)^4}+\frac {3 b^2 B d n \log (a+b x)}{2 g^2 i^3 (b c-a d)^4}-\frac {3 b^2 B d n \log (c+d x)}{2 g^2 i^3 (b c-a d)^4}-\frac {3 b^2 B d n \log (c+d x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{g^2 i^3 (b c-a d)^4}-\frac {3 b^2 B d n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{g^2 i^3 (b c-a d)^4}+\frac {5 b B d n}{2 g^2 i^3 (c+d x) (b c-a d)^3}+\frac {B d n}{4 g^2 i^3 (c+d x)^2 (b c-a d)^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 44
Rule 2301
Rule 2390
Rule 2391
Rule 2393
Rule 2394
Rule 2418
Rule 2524
Rule 2525
Rule 2528
Rubi steps
\begin {align*} \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(156 c+156 d x)^3 (a g+b g x)^2} \, dx &=\int \left (\frac {b^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3796416 (b c-a d)^3 g^2 (a+b x)^2}-\frac {b^3 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{1265472 (b c-a d)^4 g^2 (a+b x)}+\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3796416 (b c-a d)^2 g^2 (c+d x)^3}+\frac {b d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{1898208 (b c-a d)^3 g^2 (c+d x)^2}+\frac {b^2 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{1265472 (b c-a d)^4 g^2 (c+d x)}\right ) \, dx\\ &=-\frac {\left (b^3 d\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{1265472 (b c-a d)^4 g^2}+\frac {\left (b^2 d^2\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{1265472 (b c-a d)^4 g^2}+\frac {b^3 \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^2} \, dx}{3796416 (b c-a d)^3 g^2}+\frac {\left (b d^2\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c+d x)^2} \, dx}{1898208 (b c-a d)^3 g^2}+\frac {d^2 \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c+d x)^3} \, dx}{3796416 (b c-a d)^2 g^2}\\ &=-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3796416 (b c-a d)^3 g^2 (a+b x)}-\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7592832 (b c-a d)^2 g^2 (c+d x)^2}-\frac {b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{1898208 (b c-a d)^3 g^2 (c+d x)}-\frac {b^2 d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{1265472 (b c-a d)^4 g^2}+\frac {b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{1265472 (b c-a d)^4 g^2}+\frac {\left (b^2 B d n\right ) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{1265472 (b c-a d)^4 g^2}-\frac {\left (b^2 B d n\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}{1265472 (b c-a d)^4 g^2}+\frac {\left (b^2 B n\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{3796416 (b c-a d)^3 g^2}+\frac {(b B d n) \int \frac {b c-a d}{(a+b x) (c+d x)^2} \, dx}{1898208 (b c-a d)^3 g^2}+\frac {(B d n) \int \frac {b c-a d}{(a+b x) (c+d x)^3} \, dx}{7592832 (b c-a d)^2 g^2}\\ &=-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3796416 (b c-a d)^3 g^2 (a+b x)}-\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7592832 (b c-a d)^2 g^2 (c+d x)^2}-\frac {b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{1898208 (b c-a d)^3 g^2 (c+d x)}-\frac {b^2 d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{1265472 (b c-a d)^4 g^2}+\frac {b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{1265472 (b c-a d)^4 g^2}+\frac {\left (b^2 B d n\right ) \int \left (\frac {b \log (a+b x)}{a+b x}-\frac {d \log (a+b x)}{c+d x}\right ) \, dx}{1265472 (b c-a d)^4 g^2}-\frac {\left (b^2 B d n\right ) \int \left (\frac {b \log (c+d x)}{a+b x}-\frac {d \log (c+d x)}{c+d x}\right ) \, dx}{1265472 (b c-a d)^4 g^2}+\frac {\left (b^2 B n\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{3796416 (b c-a d)^2 g^2}+\frac {(b B d n) \int \frac {1}{(a+b x) (c+d x)^2} \, dx}{1898208 (b c-a d)^2 g^2}+\frac {(B d n) \int \frac {1}{(a+b x) (c+d x)^3} \, dx}{7592832 (b c-a d) g^2}\\ &=-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3796416 (b c-a d)^3 g^2 (a+b x)}-\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7592832 (b c-a d)^2 g^2 (c+d x)^2}-\frac {b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{1898208 (b c-a d)^3 g^2 (c+d x)}-\frac {b^2 d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{1265472 (b c-a d)^4 g^2}+\frac {b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{1265472 (b c-a d)^4 g^2}+\frac {\left (b^3 B d n\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{1265472 (b c-a d)^4 g^2}-\frac {\left (b^3 B d n\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{1265472 (b c-a d)^4 g^2}-\frac {\left (b^2 B d^2 n\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{1265472 (b c-a d)^4 g^2}+\frac {\left (b^2 B d^2 n\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{1265472 (b c-a d)^4 g^2}+\frac {\left (b^2 B n\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^2}-\frac {b d}{(b c-a d)^2 (a+b x)}+\frac {d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{3796416 (b c-a d)^2 g^2}+\frac {(b B d n) \int \left (\frac {b^2}{(b c-a d)^2 (a+b x)}-\frac {d}{(b c-a d) (c+d x)^2}-\frac {b d}{(b c-a d)^2 (c+d x)}\right ) \, dx}{1898208 (b c-a d)^2 g^2}+\frac {(B d n) \int \left (\frac {b^3}{(b c-a d)^3 (a+b x)}-\frac {d}{(b c-a d) (c+d x)^3}-\frac {b d}{(b c-a d)^2 (c+d x)^2}-\frac {b^2 d}{(b c-a d)^3 (c+d x)}\right ) \, dx}{7592832 (b c-a d) g^2}\\ &=-\frac {b^2 B n}{3796416 (b c-a d)^3 g^2 (a+b x)}+\frac {B d n}{15185664 (b c-a d)^2 g^2 (c+d x)^2}+\frac {5 b B d n}{7592832 (b c-a d)^3 g^2 (c+d x)}+\frac {b^2 B d n \log (a+b x)}{2530944 (b c-a d)^4 g^2}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3796416 (b c-a d)^3 g^2 (a+b x)}-\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7592832 (b c-a d)^2 g^2 (c+d x)^2}-\frac {b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{1898208 (b c-a d)^3 g^2 (c+d x)}-\frac {b^2 d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{1265472 (b c-a d)^4 g^2}-\frac {b^2 B d n \log (c+d x)}{2530944 (b c-a d)^4 g^2}-\frac {b^2 B d n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{1265472 (b c-a d)^4 g^2}+\frac {b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{1265472 (b c-a d)^4 g^2}-\frac {b^2 B d n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{1265472 (b c-a d)^4 g^2}+\frac {\left (b^2 B d n\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{1265472 (b c-a d)^4 g^2}+\frac {\left (b^2 B d n\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{1265472 (b c-a d)^4 g^2}+\frac {\left (b^3 B d n\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{1265472 (b c-a d)^4 g^2}+\frac {\left (b^2 B d^2 n\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{1265472 (b c-a d)^4 g^2}\\ &=-\frac {b^2 B n}{3796416 (b c-a d)^3 g^2 (a+b x)}+\frac {B d n}{15185664 (b c-a d)^2 g^2 (c+d x)^2}+\frac {5 b B d n}{7592832 (b c-a d)^3 g^2 (c+d x)}+\frac {b^2 B d n \log (a+b x)}{2530944 (b c-a d)^4 g^2}+\frac {b^2 B d n \log ^2(a+b x)}{2530944 (b c-a d)^4 g^2}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3796416 (b c-a d)^3 g^2 (a+b x)}-\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7592832 (b c-a d)^2 g^2 (c+d x)^2}-\frac {b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{1898208 (b c-a d)^3 g^2 (c+d x)}-\frac {b^2 d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{1265472 (b c-a d)^4 g^2}-\frac {b^2 B d n \log (c+d x)}{2530944 (b c-a d)^4 g^2}-\frac {b^2 B d n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{1265472 (b c-a d)^4 g^2}+\frac {b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{1265472 (b c-a d)^4 g^2}+\frac {b^2 B d n \log ^2(c+d x)}{2530944 (b c-a d)^4 g^2}-\frac {b^2 B d n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{1265472 (b c-a d)^4 g^2}+\frac {\left (b^2 B d n\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{1265472 (b c-a d)^4 g^2}+\frac {\left (b^2 B d n\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{1265472 (b c-a d)^4 g^2}\\ &=-\frac {b^2 B n}{3796416 (b c-a d)^3 g^2 (a+b x)}+\frac {B d n}{15185664 (b c-a d)^2 g^2 (c+d x)^2}+\frac {5 b B d n}{7592832 (b c-a d)^3 g^2 (c+d x)}+\frac {b^2 B d n \log (a+b x)}{2530944 (b c-a d)^4 g^2}+\frac {b^2 B d n \log ^2(a+b x)}{2530944 (b c-a d)^4 g^2}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3796416 (b c-a d)^3 g^2 (a+b x)}-\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7592832 (b c-a d)^2 g^2 (c+d x)^2}-\frac {b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{1898208 (b c-a d)^3 g^2 (c+d x)}-\frac {b^2 d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{1265472 (b c-a d)^4 g^2}-\frac {b^2 B d n \log (c+d x)}{2530944 (b c-a d)^4 g^2}-\frac {b^2 B d n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{1265472 (b c-a d)^4 g^2}+\frac {b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{1265472 (b c-a d)^4 g^2}+\frac {b^2 B d n \log ^2(c+d x)}{2530944 (b c-a d)^4 g^2}-\frac {b^2 B d n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{1265472 (b c-a d)^4 g^2}-\frac {b^2 B d n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{1265472 (b c-a d)^4 g^2}-\frac {b^2 B d n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{1265472 (b c-a d)^4 g^2}\\ \end {align*}
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Mathematica [C] time = 0.79, size = 477, normalized size = 1.25 \[ \frac {-12 b^2 d \log (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )-\frac {4 b^2 (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{a+b x}+12 b^2 d \log (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )-\frac {8 b d (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{c+d x}-\frac {2 d (b c-a d)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{(c+d x)^2}-\frac {4 b^3 B c n}{a+b x}+6 b^2 B d n \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )\right )-6 b^2 B d n \left (2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac {d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )+\frac {4 a b^2 B d n}{a+b x}+6 b^2 B d n \log (a+b x)-\frac {8 a b B d^2 n}{c+d x}+\frac {2 b B d n (b c-a d)}{c+d x}+\frac {B d n (b c-a d)^2}{(c+d x)^2}+\frac {8 b^2 B c d n}{c+d x}-6 b^2 B d n \log (c+d x)}{4 g^2 i^3 (b c-a d)^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.98, size = 949, normalized size = 2.49 \[ -\frac {4 \, A b^{3} c^{3} + 6 \, A a b^{2} c^{2} d - 12 \, A a^{2} b c d^{2} + 2 \, A a^{3} d^{3} + 6 \, {\left (2 \, A b^{3} c d^{2} - 2 \, A a b^{2} d^{3} - {\left (B b^{3} c d^{2} - B a b^{2} d^{3}\right )} n\right )} x^{2} + 6 \, {\left (B b^{3} d^{3} n x^{3} + B a b^{2} c^{2} d n + {\left (2 \, B b^{3} c d^{2} + B a b^{2} d^{3}\right )} n x^{2} + {\left (B b^{3} c^{2} d + 2 \, B a b^{2} c d^{2}\right )} n x\right )} \log \left (\frac {b x + a}{d x + c}\right )^{2} + {\left (4 \, B b^{3} c^{3} - 15 \, B a b^{2} c^{2} d + 12 \, B a^{2} b c d^{2} - B a^{3} d^{3}\right )} n + 3 \, {\left (6 \, A b^{3} c^{2} d - 4 \, A a b^{2} c d^{2} - 2 \, A a^{2} b d^{3} - {\left (B b^{3} c^{2} d + 2 \, B a b^{2} c d^{2} - 3 \, B a^{2} b d^{3}\right )} n\right )} x + 2 \, {\left (2 \, B b^{3} c^{3} + 3 \, B a b^{2} c^{2} d - 6 \, B a^{2} b c d^{2} + B a^{3} d^{3} + 6 \, {\left (B b^{3} c d^{2} - B a b^{2} d^{3}\right )} x^{2} + 3 \, {\left (3 \, B b^{3} c^{2} d - 2 \, B a b^{2} c d^{2} - B a^{2} b d^{3}\right )} x + 6 \, {\left (B b^{3} d^{3} x^{3} + B a b^{2} c^{2} d + {\left (2 \, B b^{3} c d^{2} + B a b^{2} d^{3}\right )} x^{2} + {\left (B b^{3} c^{2} d + 2 \, B a b^{2} c d^{2}\right )} x\right )} \log \left (\frac {b x + a}{d x + c}\right )\right )} \log \relax (e) + 2 \, {\left (6 \, A a b^{2} c^{2} d - 3 \, {\left (B b^{3} d^{3} n - 2 \, A b^{3} d^{3}\right )} x^{3} - 3 \, {\left (3 \, B a b^{2} d^{3} n - 4 \, A b^{3} c d^{2} - 2 \, A a b^{2} d^{3}\right )} x^{2} + {\left (2 \, B b^{3} c^{3} - 6 \, B a^{2} b c d^{2} + B a^{3} d^{3}\right )} n + 3 \, {\left (2 \, A b^{3} c^{2} d + 4 \, A a b^{2} c d^{2} + {\left (2 \, B b^{3} c^{2} d - 4 \, B a b^{2} c d^{2} - B a^{2} b d^{3}\right )} n\right )} x\right )} \log \left (\frac {b x + a}{d x + c}\right )}{4 \, {\left ({\left (b^{5} c^{4} d^{2} - 4 \, a b^{4} c^{3} d^{3} + 6 \, a^{2} b^{3} c^{2} d^{4} - 4 \, a^{3} b^{2} c d^{5} + a^{4} b d^{6}\right )} g^{2} i^{3} x^{3} + {\left (2 \, b^{5} c^{5} d - 7 \, a b^{4} c^{4} d^{2} + 8 \, a^{2} b^{3} c^{3} d^{3} - 2 \, a^{3} b^{2} c^{2} d^{4} - 2 \, a^{4} b c d^{5} + a^{5} d^{6}\right )} g^{2} i^{3} x^{2} + {\left (b^{5} c^{6} - 2 \, a b^{4} c^{5} d - 2 \, a^{2} b^{3} c^{4} d^{2} + 8 \, a^{3} b^{2} c^{3} d^{3} - 7 \, a^{4} b c^{2} d^{4} + 2 \, a^{5} c d^{5}\right )} g^{2} i^{3} x + {\left (a b^{4} c^{6} - 4 \, a^{2} b^{3} c^{5} d + 6 \, a^{3} b^{2} c^{4} d^{2} - 4 \, a^{4} b c^{3} d^{3} + a^{5} c^{2} d^{4}\right )} g^{2} i^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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maple [F] time = 0.47, size = 0, normalized size = 0.00 \[ \int \frac {B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A}{\left (b g x +a g \right )^{2} \left (d i x +c i \right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 2.76, size = 1724, normalized size = 4.52 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 7.53, size = 1018, normalized size = 2.67 \[ \frac {\frac {4\,A\,b^2\,c^2-2\,A\,a^2\,d^2+B\,a^2\,d^2\,n+4\,B\,b^2\,c^2\,n+10\,A\,a\,b\,c\,d-11\,B\,a\,b\,c\,d\,n}{2\,\left (a\,d-b\,c\right )}+\frac {3\,x^2\,\left (2\,A\,b^2\,d^2-B\,b^2\,d^2\,n\right )}{a\,d-b\,c}+\frac {3\,x\,\left (2\,A\,a\,b\,d^2+6\,A\,b^2\,c\,d-3\,B\,a\,b\,d^2\,n-B\,b^2\,c\,d\,n\right )}{2\,\left (a\,d-b\,c\right )}}{x\,\left (4\,a^3\,c\,d^3\,g^2\,i^3-6\,a^2\,b\,c^2\,d^2\,g^2\,i^3+2\,b^3\,c^4\,g^2\,i^3\right )+x^2\,\left (2\,a^3\,d^4\,g^2\,i^3-6\,a\,b^2\,c^2\,d^2\,g^2\,i^3+4\,b^3\,c^3\,d\,g^2\,i^3\right )+x^3\,\left (2\,a^2\,b\,d^4\,g^2\,i^3-4\,a\,b^2\,c\,d^3\,g^2\,i^3+2\,b^3\,c^2\,d^2\,g^2\,i^3\right )+2\,a^3\,c^2\,d^2\,g^2\,i^3+2\,a\,b^2\,c^4\,g^2\,i^3-4\,a^2\,b\,c^3\,d\,g^2\,i^3}-\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,\left (\frac {\frac {B\,\left (a\,d+2\,b\,c\right )}{2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {3\,B\,b\,d\,x}{2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}}{x\,\left (b\,c^2\,g^2\,i^3+2\,a\,d\,c\,g^2\,i^3\right )+x^2\,\left (a\,d^2\,g^2\,i^3+2\,b\,c\,d\,g^2\,i^3\right )+a\,c^2\,g^2\,i^3+b\,d^2\,g^2\,i^3\,x^3}-\frac {3\,B\,b^2\,d\,\left (d\,g^2\,i^3\,n\,x^2\,\left (a\,d-b\,c\right )+\frac {a\,c\,g^2\,i^3\,n\,\left (a\,d-b\,c\right )}{b}+\frac {g^2\,i^3\,n\,x\,\left (a\,d+b\,c\right )\,\left (a\,d-b\,c\right )}{b}\right )}{g^2\,i^3\,n\,{\left (a\,d-b\,c\right )}^4\,\left (x\,\left (b\,c^2\,g^2\,i^3+2\,a\,d\,c\,g^2\,i^3\right )+x^2\,\left (a\,d^2\,g^2\,i^3+2\,b\,c\,d\,g^2\,i^3\right )+a\,c^2\,g^2\,i^3+b\,d^2\,g^2\,i^3\,x^3\right )}\right )-\frac {3\,B\,b^2\,d\,{\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}^2}{2\,g^2\,i^3\,n\,{\left (a\,d-b\,c\right )}^4}+\frac {b^2\,d\,\mathrm {atan}\left (\frac {b^2\,d\,\left (2\,A-B\,n\right )\,\left (\frac {a^4\,d^4\,g^2\,i^3-2\,a^3\,b\,c\,d^3\,g^2\,i^3+2\,a\,b^3\,c^3\,d\,g^2\,i^3-b^4\,c^4\,g^2\,i^3}{a^3\,d^3\,g^2\,i^3-3\,a^2\,b\,c\,d^2\,g^2\,i^3+3\,a\,b^2\,c^2\,d\,g^2\,i^3-b^3\,c^3\,g^2\,i^3}+2\,b\,d\,x\right )\,\left (a^3\,d^3\,g^2\,i^3-3\,a^2\,b\,c\,d^2\,g^2\,i^3+3\,a\,b^2\,c^2\,d\,g^2\,i^3-b^3\,c^3\,g^2\,i^3\right )\,3{}\mathrm {i}}{g^2\,i^3\,\left (6\,A\,b^2\,d-3\,B\,b^2\,d\,n\right )\,{\left (a\,d-b\,c\right )}^4}\right )\,\left (2\,A-B\,n\right )\,3{}\mathrm {i}}{g^2\,i^3\,{\left (a\,d-b\,c\right )}^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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