Optimal. Leaf size=373 \[ \frac {d i^3 (a+b x) (b c-a d)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b^4 g}-\frac {i^3 (b c-a d)^3 \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b^4 g}+\frac {i^3 (c+d x)^2 (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 b^2 g}+\frac {i^3 (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 b g}+\frac {B i^3 n (b c-a d)^3 \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b^4 g}-\frac {5 B i^3 n (b c-a d)^3 \log \left (\frac {a+b x}{c+d x}\right )}{6 b^4 g}-\frac {11 B i^3 n (b c-a d)^3 \log (c+d x)}{6 b^4 g}-\frac {5 B d i^3 n x (b c-a d)^2}{6 b^3 g}-\frac {B i^3 n (c+d x)^2 (b c-a d)}{6 b^2 g} \]
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Rubi [A] time = 0.60, antiderivative size = 455, normalized size of antiderivative = 1.22, number of steps used = 22, number of rules used = 13, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.302, Rules used = {2528, 2486, 31, 2524, 2418, 2390, 12, 2301, 2394, 2393, 2391, 2525, 43} \[ \frac {B i^3 n (b c-a d)^3 \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{b^4 g}+\frac {i^3 (c+d x)^2 (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 b^2 g}+\frac {i^3 (b c-a d)^3 \log (a g+b g x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b^4 g}+\frac {A d i^3 x (b c-a d)^2}{b^3 g}+\frac {i^3 (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 b g}+\frac {B d i^3 (a+b x) (b c-a d)^2 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^4 g}-\frac {B i^3 n (c+d x)^2 (b c-a d)}{6 b^2 g}-\frac {5 B d i^3 n x (b c-a d)^2}{6 b^3 g}-\frac {B i^3 n (b c-a d)^3 \log ^2(g (a+b x))}{2 b^4 g}-\frac {5 B i^3 n (b c-a d)^3 \log (a+b x)}{6 b^4 g}-\frac {B i^3 n (b c-a d)^3 \log (c+d x)}{b^4 g}+\frac {B i^3 n (b c-a d)^3 \log (a g+b g x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^4 g} \]
Antiderivative was successfully verified.
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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 \frac {(131 c+131 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} \, dx &=\int \left (\frac {2248091 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}+\frac {17161 d (b c-a d) (131 c+131 d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g}+\frac {131 d (131 c+131 d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b g}+\frac {2248091 (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 (a g+b g x)}\right ) \, dx\\ &=\frac {\left (2248091 (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 g+b g x} \, dx}{b^3}+\frac {(131 d) \int (131 c+131 d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b g}+\frac {(17161 d (b c-a d)) \int (131 c+131 d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b^2 g}+\frac {\left (2248091 d (b c-a d)^2\right ) \int \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b^3 g}\\ &=\frac {2248091 A d (b c-a d)^2 x}{b^3 g}+\frac {2248091 (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^2 g}+\frac {2248091 (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b g}+\frac {2248091 (b c-a d)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (a g+b g x)}{b^4 g}+\frac {\left (2248091 B d (b c-a d)^2\right ) \int \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \, dx}{b^3 g}-\frac {(B n) \int \frac {2248091 (b c-a d) (c+d x)^2}{a+b x} \, dx}{3 b g}-\frac {(131 B (b c-a d) n) \int \frac {17161 (b c-a d) (c+d x)}{a+b x} \, dx}{2 b^2 g}-\frac {\left (2248091 B (b c-a 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 g+b g x)}{a+b x} \, dx}{b^4 g}\\ &=\frac {2248091 A d (b c-a d)^2 x}{b^3 g}+\frac {2248091 B d (b c-a d)^2 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^4 g}+\frac {2248091 (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^2 g}+\frac {2248091 (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b g}+\frac {2248091 (b c-a d)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (a g+b g x)}{b^4 g}-\frac {(2248091 B (b c-a d) n) \int \frac {(c+d x)^2}{a+b x} \, dx}{3 b g}-\frac {\left (2248091 B (b c-a d)^2 n\right ) \int \frac {c+d x}{a+b x} \, dx}{2 b^2 g}-\frac {\left (2248091 B (b c-a d)^3 n\right ) \int \left (\frac {b \log (a g+b g x)}{a+b x}-\frac {d \log (a g+b g x)}{c+d x}\right ) \, dx}{b^4 g}-\frac {\left (2248091 B d (b c-a d)^3 n\right ) \int \frac {1}{c+d x} \, dx}{b^4 g}\\ &=\frac {2248091 A d (b c-a d)^2 x}{b^3 g}+\frac {2248091 B d (b c-a d)^2 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^4 g}+\frac {2248091 (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^2 g}+\frac {2248091 (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b g}-\frac {2248091 B (b c-a d)^3 n \log (c+d x)}{b^4 g}+\frac {2248091 (b c-a d)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (a g+b g x)}{b^4 g}-\frac {(2248091 B (b c-a d) n) \int \left (\frac {d (b c-a d)}{b^2}+\frac {(b c-a d)^2}{b^2 (a+b x)}+\frac {d (c+d x)}{b}\right ) \, dx}{3 b g}-\frac {\left (2248091 B (b c-a d)^2 n\right ) \int \left (\frac {d}{b}+\frac {b c-a d}{b (a+b x)}\right ) \, dx}{2 b^2 g}-\frac {\left (2248091 B (b c-a d)^3 n\right ) \int \frac {\log (a g+b g x)}{a+b x} \, dx}{b^3 g}+\frac {\left (2248091 B d (b c-a d)^3 n\right ) \int \frac {\log (a g+b g x)}{c+d x} \, dx}{b^4 g}\\ &=\frac {2248091 A d (b c-a d)^2 x}{b^3 g}-\frac {11240455 B d (b c-a d)^2 n x}{6 b^3 g}-\frac {2248091 B (b c-a d) n (c+d x)^2}{6 b^2 g}-\frac {11240455 B (b c-a d)^3 n \log (a+b x)}{6 b^4 g}+\frac {2248091 B d (b c-a d)^2 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^4 g}+\frac {2248091 (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^2 g}+\frac {2248091 (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b g}-\frac {2248091 B (b c-a d)^3 n \log (c+d x)}{b^4 g}+\frac {2248091 (b c-a d)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (a g+b g x)}{b^4 g}+\frac {2248091 B (b c-a d)^3 n \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b^4 g}-\frac {\left (2248091 B (b c-a d)^3 n\right ) \int \frac {\log \left (\frac {b g (c+d x)}{b c g-a d g}\right )}{a g+b g x} \, dx}{b^3}-\frac {\left (2248091 B (b c-a d)^3 n\right ) \operatorname {Subst}\left (\int \frac {g \log (x)}{x} \, dx,x,a g+b g x\right )}{b^4 g^2}\\ &=\frac {2248091 A d (b c-a d)^2 x}{b^3 g}-\frac {11240455 B d (b c-a d)^2 n x}{6 b^3 g}-\frac {2248091 B (b c-a d) n (c+d x)^2}{6 b^2 g}-\frac {11240455 B (b c-a d)^3 n \log (a+b x)}{6 b^4 g}+\frac {2248091 B d (b c-a d)^2 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^4 g}+\frac {2248091 (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^2 g}+\frac {2248091 (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b g}-\frac {2248091 B (b c-a d)^3 n \log (c+d x)}{b^4 g}+\frac {2248091 (b c-a d)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (a g+b g x)}{b^4 g}+\frac {2248091 B (b c-a d)^3 n \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b^4 g}-\frac {\left (2248091 B (b c-a d)^3 n\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a g+b g x\right )}{b^4 g}-\frac {\left (2248091 B (b c-a d)^3 n\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c g-a d g}\right )}{x} \, dx,x,a g+b g x\right )}{b^4 g}\\ &=\frac {2248091 A d (b c-a d)^2 x}{b^3 g}-\frac {11240455 B d (b c-a d)^2 n x}{6 b^3 g}-\frac {2248091 B (b c-a d) n (c+d x)^2}{6 b^2 g}-\frac {11240455 B (b c-a d)^3 n \log (a+b x)}{6 b^4 g}-\frac {2248091 B (b c-a d)^3 n \log ^2(g (a+b x))}{2 b^4 g}+\frac {2248091 B d (b c-a d)^2 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^4 g}+\frac {2248091 (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^2 g}+\frac {2248091 (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b g}-\frac {2248091 B (b c-a d)^3 n \log (c+d x)}{b^4 g}+\frac {2248091 (b c-a d)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (a g+b g x)}{b^4 g}+\frac {2248091 B (b c-a d)^3 n \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b^4 g}+\frac {2248091 B (b c-a d)^3 n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^4 g}\\ \end {align*}
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Mathematica [A] time = 0.28, size = 368, normalized size = 0.99 \[ \frac {i^3 \left (2 b^3 (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )+3 b^2 (c+d x)^2 (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )+6 (b c-a d)^3 \log (g (a+b x)) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )+6 A b d x (b c-a d)^2-B n (b c-a d) \left (2 b d x (b c-a d)+2 (b c-a d)^2 \log (a+b x)+b^2 (c+d x)^2\right )+6 B d (a+b x) (b c-a d)^2 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-3 B n (b c-a d)^3 \left (\log (g (a+b x)) \left (\log (g (a+b x))-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )\right )-6 B n (b c-a d)^3 \log (c+d x)-3 B n (b c-a d)^2 ((b c-a d) \log (a+b x)+b d x)\right )}{6 b^4 g} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.92, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {A d^{3} i^{3} x^{3} + 3 \, A c d^{2} i^{3} x^{2} + 3 \, A c^{2} d i^{3} x + A c^{3} i^{3} + {\left (B d^{3} i^{3} x^{3} + 3 \, B c d^{2} i^{3} x^{2} + 3 \, B c^{2} d i^{3} x + B c^{3} i^{3}\right )} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )}{b g x + a g}, x\right ) \]
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 \[ \int \frac {\left (d i x +c i \right )^{3} \left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )}{b g x +a g}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 4.61, size = 935, normalized size = 2.51 \[ 3 \, A c^{2} d i^{3} {\left (\frac {x}{b g} - \frac {a \log \left (b x + a\right )}{b^{2} g}\right )} - \frac {1}{6} \, A d^{3} i^{3} {\left (\frac {6 \, a^{3} \log \left (b x + a\right )}{b^{4} g} - \frac {2 \, b^{2} x^{3} - 3 \, a b x^{2} + 6 \, a^{2} x}{b^{3} g}\right )} + \frac {3}{2} \, A c d^{2} i^{3} {\left (\frac {2 \, a^{2} \log \left (b x + a\right )}{b^{3} g} + \frac {b x^{2} - 2 \, a x}{b^{2} g}\right )} + \frac {A c^{3} i^{3} \log \left (b g x + a g\right )}{b g} - \frac {{\left (11 \, b^{2} c^{3} i^{3} n - 15 \, a b c^{2} d i^{3} n + 6 \, a^{2} c d^{2} i^{3} n\right )} B \log \left (d x + c\right )}{6 \, b^{3} g} + \frac {{\left (b^{3} c^{3} i^{3} n - 3 \, a b^{2} c^{2} d i^{3} n + 3 \, a^{2} b c d^{2} i^{3} n - a^{3} d^{3} i^{3} n\right )} {\left (\log \left (b x + a\right ) \log \left (\frac {b d x + a d}{b c - a d} + 1\right ) + {\rm Li}_2\left (-\frac {b d x + a d}{b c - a d}\right )\right )} B}{b^{4} g} + \frac {2 \, B b^{3} d^{3} i^{3} x^{3} \log \relax (e) - {\left ({\left (i^{3} n - 9 \, i^{3} \log \relax (e)\right )} b^{3} c d^{2} - {\left (i^{3} n - 3 \, i^{3} \log \relax (e)\right )} a b^{2} d^{3}\right )} B x^{2} - 3 \, {\left (b^{3} c^{3} i^{3} n - 3 \, a b^{2} c^{2} d i^{3} n + 3 \, a^{2} b c d^{2} i^{3} n - a^{3} d^{3} i^{3} n\right )} B \log \left (b x + a\right )^{2} - {\left ({\left (7 \, i^{3} n - 18 \, i^{3} \log \relax (e)\right )} b^{3} c^{2} d - 6 \, {\left (2 \, i^{3} n - 3 \, i^{3} \log \relax (e)\right )} a b^{2} c d^{2} + {\left (5 \, i^{3} n - 6 \, i^{3} \log \relax (e)\right )} a^{2} b d^{3}\right )} B x + {\left (6 \, b^{3} c^{3} i^{3} \log \relax (e) + 18 \, {\left (i^{3} n - i^{3} \log \relax (e)\right )} a b^{2} c^{2} d - 9 \, {\left (3 \, i^{3} n - 2 \, i^{3} \log \relax (e)\right )} a^{2} b c d^{2} + {\left (11 \, i^{3} n - 6 \, i^{3} \log \relax (e)\right )} a^{3} d^{3}\right )} B \log \left (b x + a\right ) + {\left (2 \, B b^{3} d^{3} i^{3} x^{3} + 3 \, {\left (3 \, b^{3} c d^{2} i^{3} - a b^{2} d^{3} i^{3}\right )} B x^{2} + 6 \, {\left (3 \, b^{3} c^{2} d i^{3} - 3 \, a b^{2} c d^{2} i^{3} + a^{2} b d^{3} i^{3}\right )} B x + 6 \, {\left (b^{3} c^{3} i^{3} - 3 \, a b^{2} c^{2} d i^{3} + 3 \, a^{2} b c d^{2} i^{3} - a^{3} d^{3} i^{3}\right )} B \log \left (b x + a\right )\right )} \log \left ({\left (b x + a\right )}^{n}\right ) - {\left (2 \, B b^{3} d^{3} i^{3} x^{3} + 3 \, {\left (3 \, b^{3} c d^{2} i^{3} - a b^{2} d^{3} i^{3}\right )} B x^{2} + 6 \, {\left (3 \, b^{3} c^{2} d i^{3} - 3 \, a b^{2} c d^{2} i^{3} + a^{2} b d^{3} i^{3}\right )} B x + 6 \, {\left (b^{3} c^{3} i^{3} - 3 \, a b^{2} c^{2} d i^{3} + 3 \, a^{2} b c d^{2} i^{3} - a^{3} d^{3} i^{3}\right )} B \log \left (b x + a\right )\right )} \log \left ({\left (d x + c\right )}^{n}\right )}{6 \, b^{4} g} \]
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 \[ \int \frac {{\left (c\,i+d\,i\,x\right )}^3\,\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}{a\,g+b\,g\,x} \,d x \]
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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