Optimal. Leaf size=289 \[ \frac {d i^2 (a+b x) (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b^3 g}-\frac {i^2 (b c-a d)^2 \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b^3 g}+\frac {i^2 (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 b g}+\frac {B i^2 n (b c-a d)^2 \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b^3 g}-\frac {B i^2 n (b c-a d)^2 \log \left (\frac {a+b x}{c+d x}\right )}{2 b^3 g}-\frac {3 B i^2 n (b c-a d)^2 \log (c+d x)}{2 b^3 g}-\frac {B d i^2 n x (b c-a d)}{2 b^2 g} \]
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Rubi [A] time = 0.49, antiderivative size = 369, normalized size of antiderivative = 1.28, number of steps used = 18, 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} \[ \frac {B i^2 n (b c-a d)^2 \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{b^3 g}+\frac {i^2 (b c-a d)^2 \log (a g+b g x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b^3 g}+\frac {A d i^2 x (b c-a d)}{b^2 g}+\frac {i^2 (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 b g}+\frac {B d i^2 (a+b x) (b c-a d) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^3 g}-\frac {B d i^2 n x (b c-a d)}{2 b^2 g}-\frac {B i^2 n (b c-a d)^2 \log ^2(g (a+b x))}{2 b^3 g}-\frac {B i^2 n (b c-a d)^2 \log (a+b x)}{2 b^3 g}-\frac {B i^2 n (b c-a d)^2 \log (c+d x)}{b^3 g}+\frac {B i^2 n (b c-a d)^2 \log (a g+b g x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g} \]
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 {(121 c+121 d x)^2 \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 {14641 d (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g}+\frac {121 d (121 c+121 d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b g}+\frac {14641 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 (a g+b g x)}\right ) \, dx\\ &=\frac {\left (14641 (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 g+b g x} \, dx}{b^2}+\frac {(121 d) \int (121 c+121 d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b g}+\frac {(14641 d (b c-a d)) \int \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b^2 g}\\ &=\frac {14641 A d (b c-a d) x}{b^2 g}+\frac {14641 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b g}+\frac {14641 (b c-a d)^2 \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^3 g}+\frac {(14641 B d (b c-a d)) \int \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \, dx}{b^2 g}-\frac {(B n) \int \frac {14641 (b c-a d) (c+d x)}{a+b x} \, dx}{2 b g}-\frac {\left (14641 B (b c-a d)^2 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^3 g}\\ &=\frac {14641 A d (b c-a d) x}{b^2 g}+\frac {14641 B d (b c-a d) (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^3 g}+\frac {14641 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b g}+\frac {14641 (b c-a d)^2 \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^3 g}-\frac {(14641 B (b c-a d) n) \int \frac {c+d x}{a+b x} \, dx}{2 b g}-\frac {\left (14641 B (b c-a d)^2 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^3 g}-\frac {\left (14641 B d (b c-a d)^2 n\right ) \int \frac {1}{c+d x} \, dx}{b^3 g}\\ &=\frac {14641 A d (b c-a d) x}{b^2 g}+\frac {14641 B d (b c-a d) (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^3 g}+\frac {14641 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b g}-\frac {14641 B (b c-a d)^2 n \log (c+d x)}{b^3 g}+\frac {14641 (b c-a d)^2 \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^3 g}-\frac {(14641 B (b c-a d) n) \int \left (\frac {d}{b}+\frac {b c-a d}{b (a+b x)}\right ) \, dx}{2 b g}-\frac {\left (14641 B (b c-a d)^2 n\right ) \int \frac {\log (a g+b g x)}{a+b x} \, dx}{b^2 g}+\frac {\left (14641 B d (b c-a d)^2 n\right ) \int \frac {\log (a g+b g x)}{c+d x} \, dx}{b^3 g}\\ &=\frac {14641 A d (b c-a d) x}{b^2 g}-\frac {14641 B d (b c-a d) n x}{2 b^2 g}-\frac {14641 B (b c-a d)^2 n \log (a+b x)}{2 b^3 g}+\frac {14641 B d (b c-a d) (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^3 g}+\frac {14641 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b g}-\frac {14641 B (b c-a d)^2 n \log (c+d x)}{b^3 g}+\frac {14641 (b c-a d)^2 \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^3 g}+\frac {14641 B (b c-a d)^2 n \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b^3 g}-\frac {\left (14641 B (b c-a d)^2 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^2}-\frac {\left (14641 B (b c-a d)^2 n\right ) \operatorname {Subst}\left (\int \frac {g \log (x)}{x} \, dx,x,a g+b g x\right )}{b^3 g^2}\\ &=\frac {14641 A d (b c-a d) x}{b^2 g}-\frac {14641 B d (b c-a d) n x}{2 b^2 g}-\frac {14641 B (b c-a d)^2 n \log (a+b x)}{2 b^3 g}+\frac {14641 B d (b c-a d) (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^3 g}+\frac {14641 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b g}-\frac {14641 B (b c-a d)^2 n \log (c+d x)}{b^3 g}+\frac {14641 (b c-a d)^2 \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^3 g}+\frac {14641 B (b c-a d)^2 n \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b^3 g}-\frac {\left (14641 B (b c-a d)^2 n\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a g+b g x\right )}{b^3 g}-\frac {\left (14641 B (b c-a d)^2 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^3 g}\\ &=\frac {14641 A d (b c-a d) x}{b^2 g}-\frac {14641 B d (b c-a d) n x}{2 b^2 g}-\frac {14641 B (b c-a d)^2 n \log (a+b x)}{2 b^3 g}-\frac {14641 B (b c-a d)^2 n \log ^2(g (a+b x))}{2 b^3 g}+\frac {14641 B d (b c-a d) (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^3 g}+\frac {14641 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b g}-\frac {14641 B (b c-a d)^2 n \log (c+d x)}{b^3 g}+\frac {14641 (b c-a d)^2 \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^3 g}+\frac {14641 B (b c-a d)^2 n \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b^3 g}+\frac {14641 B (b c-a d)^2 n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^3 g}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 264, normalized size = 0.91 \[ \frac {i^2 \left (b^2 (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )+2 (b c-a d)^2 \log (g (a+b x)) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )+2 A b d x (b c-a d)+2 B d (a+b x) (b c-a d) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+B n (b c-a d)^2 \left (2 \text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )-\log (g (a+b x)) \left (\log (g (a+b x))-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )\right )-2 B n (b c-a d)^2 \log (c+d x)-B n (b c-a d) ((b c-a d) \log (a+b x)+b d x)\right )}{2 b^3 g} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.93, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {A d^{2} i^{2} x^{2} + 2 \, A c d i^{2} x + A c^{2} i^{2} + {\left (B d^{2} i^{2} x^{2} + 2 \, B c d i^{2} x + B c^{2} i^{2}\right )} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )}{b g x + a g}, x\right ) \]
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 = 0.47, size = 0, normalized size = 0.00 \[ \int \frac {\left (d i x +c i \right )^{2} \left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )}{b g x +a g}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 5.31, size = 580, normalized size = 2.01 \[ 2 \, A c d i^{2} {\left (\frac {x}{b g} - \frac {a \log \left (b x + a\right )}{b^{2} g}\right )} + \frac {1}{2} \, A d^{2} i^{2} {\left (\frac {2 \, a^{2} \log \left (b x + a\right )}{b^{3} g} + \frac {b x^{2} - 2 \, a x}{b^{2} g}\right )} + \frac {A c^{2} i^{2} \log \left (b g x + a g\right )}{b g} - \frac {{\left (3 \, b c^{2} i^{2} n - 2 \, a c d i^{2} n\right )} B \log \left (d x + c\right )}{2 \, b^{2} g} + \frac {{\left (b^{2} c^{2} i^{2} n - 2 \, a b c d i^{2} n + a^{2} d^{2} i^{2} n\right )} {\left (\log \left (b x + a\right ) \log \left (\frac {b d x + a d}{b c - a d} + 1\right ) + {\rm Li}_2\left (-\frac {b d x + a d}{b c - a d}\right )\right )} B}{b^{3} g} + \frac {B b^{2} d^{2} i^{2} x^{2} \log \relax (e) - {\left (b^{2} c^{2} i^{2} n - 2 \, a b c d i^{2} n + a^{2} d^{2} i^{2} n\right )} B \log \left (b x + a\right )^{2} - {\left ({\left (i^{2} n - 4 \, i^{2} \log \relax (e)\right )} b^{2} c d - {\left (i^{2} n - 2 \, i^{2} \log \relax (e)\right )} a b d^{2}\right )} B x + {\left (2 \, b^{2} c^{2} i^{2} \log \relax (e) + 4 \, {\left (i^{2} n - i^{2} \log \relax (e)\right )} a b c d - {\left (3 \, i^{2} n - 2 \, i^{2} \log \relax (e)\right )} a^{2} d^{2}\right )} B \log \left (b x + a\right ) + {\left (B b^{2} d^{2} i^{2} x^{2} + 2 \, {\left (2 \, b^{2} c d i^{2} - a b d^{2} i^{2}\right )} B x + 2 \, {\left (b^{2} c^{2} i^{2} - 2 \, a b c d i^{2} + a^{2} d^{2} i^{2}\right )} B \log \left (b x + a\right )\right )} \log \left ({\left (b x + a\right )}^{n}\right ) - {\left (B b^{2} d^{2} i^{2} x^{2} + 2 \, {\left (2 \, b^{2} c d i^{2} - a b d^{2} i^{2}\right )} B x + 2 \, {\left (b^{2} c^{2} i^{2} - 2 \, a b c d i^{2} + a^{2} d^{2} i^{2}\right )} B \log \left (b x + a\right )\right )} \log \left ({\left (d x + c\right )}^{n}\right )}{2 \, b^{3} g} \]
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 \[ \int \frac {{\left (c\,i+d\,i\,x\right )}^2\,\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}{a\,g+b\,g\,x} \,d x \]
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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