Optimal. Leaf size=389 \[ -\frac {b^3 (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 g^4 i (a+b x)^3 (b c-a d)^4}+\frac {3 b^2 d (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 g^4 i (a+b x)^2 (b c-a d)^4}-\frac {d^3 \log \left (\frac {a+b x}{c+d x}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g^4 i (b c-a d)^4}-\frac {3 b d^2 (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g^4 i (a+b x) (b c-a d)^4}-\frac {b^3 B n (c+d x)^3}{9 g^4 i (a+b x)^3 (b c-a d)^4}+\frac {3 b^2 B d n (c+d x)^2}{4 g^4 i (a+b x)^2 (b c-a d)^4}+\frac {B d^3 n \log ^2\left (\frac {a+b x}{c+d x}\right )}{2 g^4 i (b c-a d)^4}-\frac {3 b B d^2 n (c+d x)}{g^4 i (a+b x) (b c-a d)^4} \]
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Rubi [C] time = 1.08, antiderivative size = 646, normalized size of antiderivative = 1.66, 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} \[ -\frac {B d^3 n \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{g^4 i (b c-a d)^4}-\frac {B d^3 n \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{g^4 i (b c-a d)^4}-\frac {d^3 \log (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g^4 i (b c-a d)^4}+\frac {d^3 \log (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g^4 i (b c-a d)^4}-\frac {d^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g^4 i (a+b x) (b c-a d)^3}+\frac {d \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 g^4 i (a+b x)^2 (b c-a d)^2}-\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{3 g^4 i (a+b x)^3 (b c-a d)}-\frac {11 B d^2 n}{6 g^4 i (a+b x) (b c-a d)^3}+\frac {B d^3 n \log ^2(a+b x)}{2 g^4 i (b c-a d)^4}+\frac {B d^3 n \log ^2(c+d x)}{2 g^4 i (b c-a d)^4}-\frac {11 B d^3 n \log (a+b x)}{6 g^4 i (b c-a d)^4}+\frac {11 B d^3 n \log (c+d x)}{6 g^4 i (b c-a d)^4}-\frac {B d^3 n \log (c+d x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{g^4 i (b c-a d)^4}-\frac {B d^3 n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{g^4 i (b c-a d)^4}+\frac {5 B d n}{12 g^4 i (a+b x)^2 (b c-a d)^2}-\frac {B n}{9 g^4 i (a+b x)^3 (b c-a d)} \]
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 )}{(142 c+142 d x) (a g+b g x)^4} \, dx &=\int \left (\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d) g^4 (a+b x)^4}-\frac {b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d)^2 g^4 (a+b x)^3}+\frac {b d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d)^3 g^4 (a+b x)^2}-\frac {b d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d)^4 g^4 (a+b x)}+\frac {d^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d)^4 g^4 (c+d x)}\right ) \, dx\\ &=-\frac {\left (b d^3\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{142 (b c-a d)^4 g^4}+\frac {d^4 \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{142 (b c-a d)^4 g^4}+\frac {\left (b d^2\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^2} \, dx}{142 (b c-a d)^3 g^4}-\frac {(b d) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^3} \, dx}{142 (b c-a d)^2 g^4}+\frac {b \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^4} \, dx}{142 (b c-a d) g^4}\\ &=-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{426 (b c-a d) g^4 (a+b x)^3}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{284 (b c-a d)^2 g^4 (a+b x)^2}-\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d)^3 g^4 (a+b x)}-\frac {d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d)^4 g^4}+\frac {d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{142 (b c-a d)^4 g^4}+\frac {\left (B d^3 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}{142 (b c-a d)^4 g^4}-\frac {\left (B d^3 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}{142 (b c-a d)^4 g^4}+\frac {\left (B d^2 n\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{142 (b c-a d)^3 g^4}-\frac {(B d n) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{284 (b c-a d)^2 g^4}+\frac {(B n) \int \frac {b c-a d}{(a+b x)^4 (c+d x)} \, dx}{426 (b c-a d) g^4}\\ &=-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{426 (b c-a d) g^4 (a+b x)^3}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{284 (b c-a d)^2 g^4 (a+b x)^2}-\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d)^3 g^4 (a+b x)}-\frac {d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d)^4 g^4}+\frac {d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{142 (b c-a d)^4 g^4}+\frac {(B n) \int \frac {1}{(a+b x)^4 (c+d x)} \, dx}{426 g^4}+\frac {\left (B d^3 n\right ) \int \left (\frac {b \log (a+b x)}{a+b x}-\frac {d \log (a+b x)}{c+d x}\right ) \, dx}{142 (b c-a d)^4 g^4}-\frac {\left (B d^3 n\right ) \int \left (\frac {b \log (c+d x)}{a+b x}-\frac {d \log (c+d x)}{c+d x}\right ) \, dx}{142 (b c-a d)^4 g^4}+\frac {\left (B d^2 n\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{142 (b c-a d)^2 g^4}-\frac {(B d n) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{284 (b c-a d) g^4}\\ &=-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{426 (b c-a d) g^4 (a+b x)^3}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{284 (b c-a d)^2 g^4 (a+b x)^2}-\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d)^3 g^4 (a+b x)}-\frac {d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d)^4 g^4}+\frac {d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{142 (b c-a d)^4 g^4}+\frac {(B n) \int \left (\frac {b}{(b c-a d) (a+b x)^4}-\frac {b d}{(b c-a d)^2 (a+b x)^3}+\frac {b d^2}{(b c-a d)^3 (a+b x)^2}-\frac {b d^3}{(b c-a d)^4 (a+b x)}+\frac {d^4}{(b c-a d)^4 (c+d x)}\right ) \, dx}{426 g^4}+\frac {\left (b B d^3 n\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{142 (b c-a d)^4 g^4}-\frac {\left (b B d^3 n\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{142 (b c-a d)^4 g^4}-\frac {\left (B d^4 n\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{142 (b c-a d)^4 g^4}+\frac {\left (B d^4 n\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{142 (b c-a d)^4 g^4}+\frac {\left (B d^2 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}{142 (b c-a d)^2 g^4}-\frac {(B d n) \int \left (\frac {b}{(b c-a d) (a+b x)^3}-\frac {b d}{(b c-a d)^2 (a+b x)^2}+\frac {b d^2}{(b c-a d)^3 (a+b x)}-\frac {d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{284 (b c-a d) g^4}\\ &=-\frac {B n}{1278 (b c-a d) g^4 (a+b x)^3}+\frac {5 B d n}{1704 (b c-a d)^2 g^4 (a+b x)^2}-\frac {11 B d^2 n}{852 (b c-a d)^3 g^4 (a+b x)}-\frac {11 B d^3 n \log (a+b x)}{852 (b c-a d)^4 g^4}-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{426 (b c-a d) g^4 (a+b x)^3}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{284 (b c-a d)^2 g^4 (a+b x)^2}-\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d)^3 g^4 (a+b x)}-\frac {d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d)^4 g^4}+\frac {11 B d^3 n \log (c+d x)}{852 (b c-a d)^4 g^4}-\frac {B d^3 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{142 (b c-a d)^4 g^4}+\frac {d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{142 (b c-a d)^4 g^4}-\frac {B d^3 n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{142 (b c-a d)^4 g^4}+\frac {\left (B d^3 n\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{142 (b c-a d)^4 g^4}+\frac {\left (B d^3 n\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{142 (b c-a d)^4 g^4}+\frac {\left (b B d^3 n\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{142 (b c-a d)^4 g^4}+\frac {\left (B d^4 n\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{142 (b c-a d)^4 g^4}\\ &=-\frac {B n}{1278 (b c-a d) g^4 (a+b x)^3}+\frac {5 B d n}{1704 (b c-a d)^2 g^4 (a+b x)^2}-\frac {11 B d^2 n}{852 (b c-a d)^3 g^4 (a+b x)}-\frac {11 B d^3 n \log (a+b x)}{852 (b c-a d)^4 g^4}+\frac {B d^3 n \log ^2(a+b x)}{284 (b c-a d)^4 g^4}-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{426 (b c-a d) g^4 (a+b x)^3}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{284 (b c-a d)^2 g^4 (a+b x)^2}-\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d)^3 g^4 (a+b x)}-\frac {d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d)^4 g^4}+\frac {11 B d^3 n \log (c+d x)}{852 (b c-a d)^4 g^4}-\frac {B d^3 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{142 (b c-a d)^4 g^4}+\frac {d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{142 (b c-a d)^4 g^4}+\frac {B d^3 n \log ^2(c+d x)}{284 (b c-a d)^4 g^4}-\frac {B d^3 n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{142 (b c-a d)^4 g^4}+\frac {\left (B d^3 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 )}{142 (b c-a d)^4 g^4}+\frac {\left (B d^3 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 )}{142 (b c-a d)^4 g^4}\\ &=-\frac {B n}{1278 (b c-a d) g^4 (a+b x)^3}+\frac {5 B d n}{1704 (b c-a d)^2 g^4 (a+b x)^2}-\frac {11 B d^2 n}{852 (b c-a d)^3 g^4 (a+b x)}-\frac {11 B d^3 n \log (a+b x)}{852 (b c-a d)^4 g^4}+\frac {B d^3 n \log ^2(a+b x)}{284 (b c-a d)^4 g^4}-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{426 (b c-a d) g^4 (a+b x)^3}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{284 (b c-a d)^2 g^4 (a+b x)^2}-\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d)^3 g^4 (a+b x)}-\frac {d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d)^4 g^4}+\frac {11 B d^3 n \log (c+d x)}{852 (b c-a d)^4 g^4}-\frac {B d^3 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{142 (b c-a d)^4 g^4}+\frac {d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{142 (b c-a d)^4 g^4}+\frac {B d^3 n \log ^2(c+d x)}{284 (b c-a d)^4 g^4}-\frac {B d^3 n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{142 (b c-a d)^4 g^4}-\frac {B d^3 n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{142 (b c-a d)^4 g^4}-\frac {B d^3 n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{142 (b c-a d)^4 g^4}\\ \end {align*}
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Mathematica [C] time = 0.72, size = 518, normalized size = 1.33 \[ \frac {\frac {36 A d^2 (a d-b c)}{a+b x}+\frac {18 A d (b c-a d)^2}{(a+b x)^2}-\frac {12 A (b c-a d)^3}{(a+b x)^3}-36 A d^3 \log (a+b x)-36 B d^3 \log (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+36 B d^3 \log (c+d x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-36 B d^3 n \text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )-36 B d^3 n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )-36 B d^3 n \log (c+d x) \log \left (\frac {d (a+b x)}{a d-b c}\right )-36 B d^3 n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )+\frac {36 B d^2 (a d-b c) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x}+\frac {66 B d^2 n (a d-b c)}{a+b x}+\frac {18 B d (b c-a d)^2 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^2}-\frac {12 B (b c-a d)^3 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^3}+\frac {15 B d n (b c-a d)^2}{(a+b x)^2}-\frac {4 B n (b c-a d)^3}{(a+b x)^3}+18 B d^3 n \log ^2(a+b x)-66 B d^3 n \log (a+b x)+36 A d^3 \log (c+d x)+18 B d^3 n \log ^2(c+d x)+66 B d^3 n \log (c+d x)}{36 g^4 i (b c-a d)^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.99, size = 859, normalized size = 2.21 \[ -\frac {12 \, A b^{3} c^{3} - 54 \, A a b^{2} c^{2} d + 108 \, A a^{2} b c d^{2} - 66 \, A a^{3} d^{3} + 6 \, {\left (6 \, A b^{3} c d^{2} - 6 \, A a b^{2} d^{3} + 11 \, {\left (B b^{3} c d^{2} - B a b^{2} d^{3}\right )} n\right )} x^{2} + 18 \, {\left (B b^{3} d^{3} n x^{3} + 3 \, B a b^{2} d^{3} n x^{2} + 3 \, B a^{2} b d^{3} n x + B a^{3} d^{3} n\right )} \log \left (\frac {b x + a}{d x + c}\right )^{2} + {\left (4 \, B b^{3} c^{3} - 27 \, B a b^{2} c^{2} d + 108 \, B a^{2} b c d^{2} - 85 \, B a^{3} d^{3}\right )} n - 3 \, {\left (6 \, A b^{3} c^{2} d - 36 \, A a b^{2} c d^{2} + 30 \, A a^{2} b d^{3} + {\left (5 \, B b^{3} c^{2} d - 54 \, B a b^{2} c d^{2} + 49 \, B a^{2} b d^{3}\right )} n\right )} x + 6 \, {\left (2 \, B b^{3} c^{3} - 9 \, B a b^{2} c^{2} d + 18 \, B a^{2} b c d^{2} - 11 \, B a^{3} d^{3} + 6 \, {\left (B b^{3} c d^{2} - B a b^{2} d^{3}\right )} x^{2} - 3 \, {\left (B b^{3} c^{2} d - 6 \, B a b^{2} c d^{2} + 5 \, B a^{2} b d^{3}\right )} x + 6 \, {\left (B b^{3} d^{3} x^{3} + 3 \, B a b^{2} d^{3} x^{2} + 3 \, B a^{2} b d^{3} x + B a^{3} d^{3}\right )} \log \left (\frac {b x + a}{d x + c}\right )\right )} \log \relax (e) + 6 \, {\left (6 \, A a^{3} d^{3} + {\left (11 \, B b^{3} d^{3} n + 6 \, A b^{3} d^{3}\right )} x^{3} + 3 \, {\left (6 \, A a b^{2} d^{3} + {\left (2 \, B b^{3} c d^{2} + 9 \, B a b^{2} d^{3}\right )} n\right )} x^{2} + {\left (2 \, B b^{3} c^{3} - 9 \, B a b^{2} c^{2} d + 18 \, B a^{2} b c d^{2}\right )} n + 3 \, {\left (6 \, A a^{2} b d^{3} - {\left (B b^{3} c^{2} d - 6 \, B a b^{2} c d^{2} - 6 \, B a^{2} b d^{3}\right )} n\right )} x\right )} \log \left (\frac {b x + a}{d x + c}\right )}{36 \, {\left ({\left (b^{7} c^{4} - 4 \, a b^{6} c^{3} d + 6 \, a^{2} b^{5} c^{2} d^{2} - 4 \, a^{3} b^{4} c d^{3} + a^{4} b^{3} d^{4}\right )} g^{4} i x^{3} + 3 \, {\left (a b^{6} c^{4} - 4 \, a^{2} b^{5} c^{3} d + 6 \, a^{3} b^{4} c^{2} d^{2} - 4 \, a^{4} b^{3} c d^{3} + a^{5} b^{2} d^{4}\right )} g^{4} i x^{2} + 3 \, {\left (a^{2} b^{5} c^{4} - 4 \, a^{3} b^{4} c^{3} d + 6 \, a^{4} b^{3} c^{2} d^{2} - 4 \, a^{5} b^{2} c d^{3} + a^{6} b d^{4}\right )} g^{4} i x + {\left (a^{3} b^{4} c^{4} - 4 \, a^{4} b^{3} c^{3} d + 6 \, a^{5} b^{2} c^{2} d^{2} - 4 \, a^{6} b c d^{3} + a^{7} d^{4}\right )} g^{4} i\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 \[ \int \frac {B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A}{\left (b g x +a g \right )^{4} \left (d i x +c i \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 2.36, size = 1472, normalized size = 3.78 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 7.23, size = 986, normalized size = 2.53 \[ \frac {\frac {66\,A\,a^2\,d^2+12\,A\,b^2\,c^2+85\,B\,a^2\,d^2\,n+4\,B\,b^2\,c^2\,n-42\,A\,a\,b\,c\,d-23\,B\,a\,b\,c\,d\,n}{6\,\left (a\,d-b\,c\right )}+\frac {x\,\left (30\,A\,a\,b\,d^2-6\,A\,b^2\,c\,d+49\,B\,a\,b\,d^2\,n-5\,B\,b^2\,c\,d\,n\right )}{2\,\left (a\,d-b\,c\right )}+\frac {d\,x^2\,\left (6\,A\,b^2\,d+11\,B\,b^2\,d\,n\right )}{a\,d-b\,c}}{x\,\left (18\,i\,a^4\,b\,d^2\,g^4-36\,i\,a^3\,b^2\,c\,d\,g^4+18\,i\,a^2\,b^3\,c^2\,g^4\right )+x^2\,\left (18\,i\,a^3\,b^2\,d^2\,g^4-36\,i\,a^2\,b^3\,c\,d\,g^4+18\,i\,a\,b^4\,c^2\,g^4\right )+x^3\,\left (6\,i\,a^2\,b^3\,d^2\,g^4-12\,i\,a\,b^4\,c\,d\,g^4+6\,i\,b^5\,c^2\,g^4\right )+6\,a^5\,d^2\,g^4\,i+6\,a^3\,b^2\,c^2\,g^4\,i-12\,a^4\,b\,c\,d\,g^4\,i}-\frac {B\,d^3\,{\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}^2}{2\,g^4\,i\,n\,\left (a\,d-b\,c\right )\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {B\,d^3\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,\left (x\,\left (b\,\left (\frac {g^4\,i\,n\,\left (a\,d-b\,c\right )\,\left (3\,a\,d-b\,c\right )}{6\,d^2}+\frac {a\,g^4\,i\,n\,\left (a\,d-b\,c\right )}{3\,d}\right )+\frac {2\,a\,b\,g^4\,i\,n\,\left (a\,d-b\,c\right )}{3\,d}+\frac {b\,g^4\,i\,n\,\left (a\,d-b\,c\right )\,\left (3\,a\,d-b\,c\right )}{3\,d^2}\right )+a\,\left (\frac {g^4\,i\,n\,\left (a\,d-b\,c\right )\,\left (3\,a\,d-b\,c\right )}{6\,d^2}+\frac {a\,g^4\,i\,n\,\left (a\,d-b\,c\right )}{3\,d}\right )+\frac {g^4\,i\,n\,\left (a\,d-b\,c\right )\,\left (3\,a^2\,d^2-3\,a\,b\,c\,d+b^2\,c^2\right )}{3\,d^3}+\frac {b^2\,g^4\,i\,n\,x^2\,\left (a\,d-b\,c\right )}{d}\right )}{g^4\,i\,n\,\left (a\,d-b\,c\right )\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )\,\left (i\,a^3\,g^4+3\,i\,a^2\,b\,g^4\,x+3\,i\,a\,b^2\,g^4\,x^2+i\,b^3\,g^4\,x^3\right )}+\frac {d^3\,\mathrm {atan}\left (\frac {d^3\,\left (\frac {i\,a^4\,d^4\,g^4-2\,i\,a^3\,b\,c\,d^3\,g^4+2\,i\,a\,b^3\,c^3\,d\,g^4-i\,b^4\,c^4\,g^4}{i\,a^3\,d^3\,g^4-3\,i\,a^2\,b\,c\,d^2\,g^4+3\,i\,a\,b^2\,c^2\,d\,g^4-i\,b^3\,c^3\,g^4}+2\,b\,d\,x\right )\,\left (A+\frac {11\,B\,n}{6}\right )\,\left (i\,a^3\,d^3\,g^4-3\,i\,a^2\,b\,c\,d^2\,g^4+3\,i\,a\,b^2\,c^2\,d\,g^4-i\,b^3\,c^3\,g^4\right )\,6{}\mathrm {i}}{g^4\,i\,\left (6\,A\,d^3+11\,B\,d^3\,n\right )\,{\left (a\,d-b\,c\right )}^4}\right )\,\left (A+\frac {11\,B\,n}{6}\right )\,2{}\mathrm {i}}{g^4\,i\,{\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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