Optimal. Leaf size=326 \[ -\frac {d^3 i^3 \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b^4 g^4}-\frac {d^2 i^3 (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b^3 g^4 (a+b x)}-\frac {d i^3 (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 b^2 g^4 (a+b x)^2}-\frac {i^3 (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 b g^4 (a+b x)^3}+\frac {B d^3 i^3 n \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b^4 g^4}-\frac {B d^2 i^3 n (c+d x)}{b^3 g^4 (a+b x)}-\frac {B d i^3 n (c+d x)^2}{4 b^2 g^4 (a+b x)^2}-\frac {B i^3 n (c+d x)^3}{9 b g^4 (a+b x)^3} \]
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Rubi [A] time = 0.79, antiderivative size = 444, normalized size of antiderivative = 1.36, number of steps used = 22, 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} \[ \frac {B d^3 i^3 n \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{b^4 g^4}+\frac {d^3 i^3 \log (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b^4 g^4}-\frac {3 d^2 i^3 (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b^4 g^4 (a+b x)}-\frac {3 d i^3 (b c-a d)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 b^4 g^4 (a+b x)^2}-\frac {i^3 (b c-a d)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 b^4 g^4 (a+b x)^3}-\frac {11 B d^2 i^3 n (b c-a d)}{6 b^4 g^4 (a+b x)}+\frac {B d^3 i^3 n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^4 g^4}-\frac {7 B d i^3 n (b c-a d)^2}{12 b^4 g^4 (a+b x)^2}-\frac {B i^3 n (b c-a d)^3}{9 b^4 g^4 (a+b x)^3}-\frac {B d^3 i^3 n \log ^2(a+b x)}{2 b^4 g^4}-\frac {11 B d^3 i^3 n \log (a+b x)}{6 b^4 g^4}+\frac {11 B d^3 i^3 n \log (c+d x)}{6 b^4 g^4} \]
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 {(134 c+134 d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^4} \, dx &=\int \left (\frac {2406104 (b c-a d)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^4 (a+b x)^4}+\frac {7218312 d (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^4 (a+b x)^3}+\frac {7218312 d^2 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^4 (a+b x)^2}+\frac {2406104 d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^4 (a+b x)}\right ) \, dx\\ &=\frac {\left (2406104 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}{b^3 g^4}+\frac {\left (7218312 d^2 (b c-a d)\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^2} \, dx}{b^3 g^4}+\frac {\left (7218312 d (b c-a d)^2\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^3} \, dx}{b^3 g^4}+\frac {\left (2406104 (b c-a d)^3\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^4} \, dx}{b^3 g^4}\\ &=-\frac {2406104 (b c-a d)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^4 g^4 (a+b x)^3}-\frac {3609156 d (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^4 (a+b x)^2}-\frac {7218312 d^2 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^4 (a+b x)}+\frac {2406104 d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^4}-\frac {\left (2406104 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}{b^4 g^4}+\frac {\left (7218312 B d^2 (b c-a d) n\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^4 g^4}+\frac {\left (3609156 B d (b c-a d)^2 n\right ) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{b^4 g^4}+\frac {\left (2406104 B (b c-a d)^3 n\right ) \int \frac {b c-a d}{(a+b x)^4 (c+d x)} \, dx}{3 b^4 g^4}\\ &=-\frac {2406104 (b c-a d)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^4 g^4 (a+b x)^3}-\frac {3609156 d (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^4 (a+b x)^2}-\frac {7218312 d^2 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^4 (a+b x)}+\frac {2406104 d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^4}-\frac {\left (2406104 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}{b^4 g^4}+\frac {\left (7218312 B d^2 (b c-a d)^2 n\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{b^4 g^4}+\frac {\left (3609156 B d (b c-a d)^3 n\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{b^4 g^4}+\frac {\left (2406104 B (b c-a d)^4 n\right ) \int \frac {1}{(a+b x)^4 (c+d x)} \, dx}{3 b^4 g^4}\\ &=-\frac {2406104 (b c-a d)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^4 g^4 (a+b x)^3}-\frac {3609156 d (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^4 (a+b x)^2}-\frac {7218312 d^2 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^4 (a+b x)}+\frac {2406104 d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^4}-\frac {\left (2406104 B d^3 n\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{b^3 g^4}+\frac {\left (2406104 B d^4 n\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b^4 g^4}+\frac {\left (7218312 B d^2 (b c-a 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}{b^4 g^4}+\frac {\left (3609156 B d (b c-a d)^3 n\right ) \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}{b^4 g^4}+\frac {\left (2406104 B (b c-a d)^4 n\right ) \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}{3 b^4 g^4}\\ &=-\frac {2406104 B (b c-a d)^3 n}{9 b^4 g^4 (a+b x)^3}-\frac {4210682 B d (b c-a d)^2 n}{3 b^4 g^4 (a+b x)^2}-\frac {13233572 B d^2 (b c-a d) n}{3 b^4 g^4 (a+b x)}-\frac {13233572 B d^3 n \log (a+b x)}{3 b^4 g^4}-\frac {2406104 (b c-a d)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^4 g^4 (a+b x)^3}-\frac {3609156 d (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^4 (a+b x)^2}-\frac {7218312 d^2 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^4 (a+b x)}+\frac {2406104 d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^4}+\frac {13233572 B d^3 n \log (c+d x)}{3 b^4 g^4}+\frac {2406104 B d^3 n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^4 g^4}-\frac {\left (2406104 B d^3 n\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^4 g^4}-\frac {\left (2406104 B d^3 n\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^3 g^4}\\ &=-\frac {2406104 B (b c-a d)^3 n}{9 b^4 g^4 (a+b x)^3}-\frac {4210682 B d (b c-a d)^2 n}{3 b^4 g^4 (a+b x)^2}-\frac {13233572 B d^2 (b c-a d) n}{3 b^4 g^4 (a+b x)}-\frac {13233572 B d^3 n \log (a+b x)}{3 b^4 g^4}-\frac {1203052 B d^3 n \log ^2(a+b x)}{b^4 g^4}-\frac {2406104 (b c-a d)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^4 g^4 (a+b x)^3}-\frac {3609156 d (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^4 (a+b x)^2}-\frac {7218312 d^2 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^4 (a+b x)}+\frac {2406104 d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^4}+\frac {13233572 B d^3 n \log (c+d x)}{3 b^4 g^4}+\frac {2406104 B d^3 n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^4 g^4}-\frac {\left (2406104 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 )}{b^4 g^4}\\ &=-\frac {2406104 B (b c-a d)^3 n}{9 b^4 g^4 (a+b x)^3}-\frac {4210682 B d (b c-a d)^2 n}{3 b^4 g^4 (a+b x)^2}-\frac {13233572 B d^2 (b c-a d) n}{3 b^4 g^4 (a+b x)}-\frac {13233572 B d^3 n \log (a+b x)}{3 b^4 g^4}-\frac {1203052 B d^3 n \log ^2(a+b x)}{b^4 g^4}-\frac {2406104 (b c-a d)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^4 g^4 (a+b x)^3}-\frac {3609156 d (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^4 (a+b x)^2}-\frac {7218312 d^2 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^4 (a+b x)}+\frac {2406104 d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^4}+\frac {13233572 B d^3 n \log (c+d x)}{3 b^4 g^4}+\frac {2406104 B d^3 n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^4 g^4}+\frac {2406104 B d^3 n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^4 g^4}\\ \end {align*}
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Mathematica [A] time = 0.46, size = 326, normalized size = 1.00 \[ \frac {i^3 \left (36 d^3 \log (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )+\frac {108 d^2 (a d-b c) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{a+b x}-\frac {54 d (b c-a d)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{(a+b x)^2}-\frac {12 (b c-a d)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{(a+b x)^3}-18 B d^3 n \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )\right )+\frac {66 B d^2 n (a d-b c)}{a+b x}-\frac {21 B d n (b c-a d)^2}{(a+b x)^2}-\frac {4 B n (b c-a d)^3}{(a+b x)^3}-66 B d^3 n \log (a+b x)+66 B d^3 n \log (c+d x)\right )}{36 b^4 g^4} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.82, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {A d^{3} i^{3} x^{3} + 3 \, A c d^{2} i^{3} x^{2} + 3 \, A c^{2} d i^{3} x + A c^{3} i^{3} + {\left (B d^{3} i^{3} x^{3} + 3 \, B c d^{2} i^{3} x^{2} + 3 \, B c^{2} d i^{3} x + B c^{3} i^{3}\right )} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )}{b^{4} g^{4} x^{4} + 4 \, a b^{3} g^{4} x^{3} + 6 \, a^{2} b^{2} g^{4} x^{2} + 4 \, a^{3} b g^{4} x + a^{4} g^{4}}, x\right ) \]
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 = 0.45, size = 0, normalized size = 0.00 \[ \int \frac {\left (d i x +c i \right )^{3} \left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )}{\left (b g x +a g \right )^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (c\,i+d\,i\,x\right )}^3\,\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}{{\left (a\,g+b\,g\,x\right )}^4} \,d x \]
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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