Optimal. Leaf size=266 \[ -\frac{b^2 (c+d x)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{2 g^3 i (a+b x)^2 (b c-a d)^3}+\frac{d^2 \log \left (\frac{a+b x}{c+d x}\right ) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{g^3 i (b c-a d)^3}+\frac{2 b d (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{g^3 i (a+b x) (b c-a d)^3}-\frac{B d^2 n \log ^2\left (\frac{a+b x}{c+d x}\right )}{2 g^3 i (b c-a d)^3}-\frac{B n (c+d x)^2 \left (b-\frac{4 d (a+b x)}{c+d x}\right )^2}{4 g^3 i (a+b x)^2 (b c-a d)^3} \]
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Rubi [C] time = 0.834219, antiderivative size = 557, normalized size of antiderivative = 2.09, number of steps used = 26, 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} \[ \frac{B d^2 n \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{g^3 i (b c-a d)^3}+\frac{B d^2 n \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{g^3 i (b c-a d)^3}+\frac{d^2 \log (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{g^3 i (b c-a d)^3}-\frac{d^2 \log (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{g^3 i (b c-a d)^3}+\frac{d \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{g^3 i (a+b x) (b c-a d)^2}-\frac{B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A}{2 g^3 i (a+b x)^2 (b c-a d)}-\frac{B d^2 n \log ^2(a+b x)}{2 g^3 i (b c-a d)^3}-\frac{B d^2 n \log ^2(c+d x)}{2 g^3 i (b c-a d)^3}+\frac{3 B d^2 n \log (a+b x)}{2 g^3 i (b c-a d)^3}-\frac{3 B d^2 n \log (c+d x)}{2 g^3 i (b c-a d)^3}+\frac{B d^2 n \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{g^3 i (b c-a d)^3}+\frac{B d^2 n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{g^3 i (b c-a d)^3}+\frac{3 B d n}{2 g^3 i (a+b x) (b c-a d)^2}-\frac{B n}{4 g^3 i (a+b x)^2 (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 2528
Rule 2525
Rule 12
Rule 44
Rule 2524
Rule 2418
Rule 2390
Rule 2301
Rule 2394
Rule 2393
Rule 2391
Rubi steps
\begin{align*} \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(141 c+141 d x) (a g+b g x)^3} \, dx &=\int \left (\frac{b \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{141 (b c-a d) g^3 (a+b x)^3}-\frac{b d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{141 (b c-a d)^2 g^3 (a+b x)^2}+\frac{b d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{141 (b c-a d)^3 g^3 (a+b x)}-\frac{d^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{141 (b c-a d)^3 g^3 (c+d x)}\right ) \, dx\\ &=\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} \, dx}{141 (b c-a d)^3 g^3}-\frac{d^3 \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{141 (b c-a d)^3 g^3}-\frac{(b d) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^2} \, dx}{141 (b c-a d)^2 g^3}+\frac{b \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^3} \, dx}{141 (b c-a d) g^3}\\ &=-\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{282 (b c-a d) g^3 (a+b x)^2}+\frac{d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{141 (b c-a d)^2 g^3 (a+b x)}+\frac{d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{141 (b c-a d)^3 g^3}-\frac{d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{141 (b c-a d)^3 g^3}-\frac{\left (B 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+b x)}{a+b x} \, dx}{141 (b c-a d)^3 g^3}+\frac{\left (B 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 (c+d x)}{a+b x} \, dx}{141 (b c-a d)^3 g^3}-\frac{(B d n) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{141 (b c-a d)^2 g^3}+\frac{(B n) \int \frac{b c-a d}{(a+b x)^3 (c+d x)} \, dx}{282 (b c-a d) g^3}\\ &=-\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{282 (b c-a d) g^3 (a+b x)^2}+\frac{d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{141 (b c-a d)^2 g^3 (a+b x)}+\frac{d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{141 (b c-a d)^3 g^3}-\frac{d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{141 (b c-a d)^3 g^3}+\frac{(B n) \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{282 g^3}-\frac{\left (B d^2 n\right ) \int \left (\frac{b \log (a+b x)}{a+b x}-\frac{d \log (a+b x)}{c+d x}\right ) \, dx}{141 (b c-a d)^3 g^3}+\frac{\left (B d^2 n\right ) \int \left (\frac{b \log (c+d x)}{a+b x}-\frac{d \log (c+d x)}{c+d x}\right ) \, dx}{141 (b c-a d)^3 g^3}-\frac{(B d n) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{141 (b c-a d) g^3}\\ &=-\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{282 (b c-a d) g^3 (a+b x)^2}+\frac{d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{141 (b c-a d)^2 g^3 (a+b x)}+\frac{d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{141 (b c-a d)^3 g^3}-\frac{d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{141 (b c-a d)^3 g^3}+\frac{(B 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}{282 g^3}-\frac{\left (b B d^2 n\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{141 (b c-a d)^3 g^3}+\frac{\left (b B d^2 n\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{141 (b c-a d)^3 g^3}+\frac{\left (B d^3 n\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{141 (b c-a d)^3 g^3}-\frac{\left (B d^3 n\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{141 (b c-a d)^3 g^3}-\frac{(B d n) \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}{141 (b c-a d) g^3}\\ &=-\frac{B n}{564 (b c-a d) g^3 (a+b x)^2}+\frac{B d n}{94 (b c-a d)^2 g^3 (a+b x)}+\frac{B d^2 n \log (a+b x)}{94 (b c-a d)^3 g^3}-\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{282 (b c-a d) g^3 (a+b x)^2}+\frac{d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{141 (b c-a d)^2 g^3 (a+b x)}+\frac{d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{141 (b c-a d)^3 g^3}-\frac{B d^2 n \log (c+d x)}{94 (b c-a d)^3 g^3}+\frac{B d^2 n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{141 (b c-a d)^3 g^3}-\frac{d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{141 (b c-a d)^3 g^3}+\frac{B d^2 n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{141 (b c-a d)^3 g^3}-\frac{\left (B d^2 n\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{141 (b c-a d)^3 g^3}-\frac{\left (B d^2 n\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{141 (b c-a d)^3 g^3}-\frac{\left (b B d^2 n\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{141 (b c-a d)^3 g^3}-\frac{\left (B d^3 n\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{141 (b c-a d)^3 g^3}\\ &=-\frac{B n}{564 (b c-a d) g^3 (a+b x)^2}+\frac{B d n}{94 (b c-a d)^2 g^3 (a+b x)}+\frac{B d^2 n \log (a+b x)}{94 (b c-a d)^3 g^3}-\frac{B d^2 n \log ^2(a+b x)}{282 (b c-a d)^3 g^3}-\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{282 (b c-a d) g^3 (a+b x)^2}+\frac{d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{141 (b c-a d)^2 g^3 (a+b x)}+\frac{d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{141 (b c-a d)^3 g^3}-\frac{B d^2 n \log (c+d x)}{94 (b c-a d)^3 g^3}+\frac{B d^2 n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{141 (b c-a d)^3 g^3}-\frac{d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{141 (b c-a d)^3 g^3}-\frac{B d^2 n \log ^2(c+d x)}{282 (b c-a d)^3 g^3}+\frac{B d^2 n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{141 (b c-a d)^3 g^3}-\frac{\left (B d^2 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 )}{141 (b c-a d)^3 g^3}-\frac{\left (B d^2 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 )}{141 (b c-a d)^3 g^3}\\ &=-\frac{B n}{564 (b c-a d) g^3 (a+b x)^2}+\frac{B d n}{94 (b c-a d)^2 g^3 (a+b x)}+\frac{B d^2 n \log (a+b x)}{94 (b c-a d)^3 g^3}-\frac{B d^2 n \log ^2(a+b x)}{282 (b c-a d)^3 g^3}-\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{282 (b c-a d) g^3 (a+b x)^2}+\frac{d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{141 (b c-a d)^2 g^3 (a+b x)}+\frac{d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{141 (b c-a d)^3 g^3}-\frac{B d^2 n \log (c+d x)}{94 (b c-a d)^3 g^3}+\frac{B d^2 n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{141 (b c-a d)^3 g^3}-\frac{d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{141 (b c-a d)^3 g^3}-\frac{B d^2 n \log ^2(c+d x)}{282 (b c-a d)^3 g^3}+\frac{B d^2 n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{141 (b c-a d)^3 g^3}+\frac{B d^2 n \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{141 (b c-a d)^3 g^3}+\frac{B d^2 n \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{141 (b c-a d)^3 g^3}\\ \end{align*}
Mathematica [C] time = 0.380885, size = 434, normalized size = 1.63 \[ \frac{-2 B d^2 n (a+b x)^2 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )-2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )+2 B d^2 n (a+b x)^2 \left (2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )+4 d^2 (a+b x)^2 \log (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )-4 d^2 (a+b x)^2 \log (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )-2 (b c-a d)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+4 d (a+b x) (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )-B n \left (2 d^2 (a+b x)^2 \log (c+d x)+2 d (a+b x) (a d-b c)+(b c-a d)^2-2 d^2 (a+b x)^2 \log (a+b x)\right )+4 B d n (a+b x) (-d (a+b x) \log (c+d x)+d (a+b x) \log (a+b x)-a d+b c)}{4 g^3 i (a+b x)^2 (b c-a d)^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.764, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( bgx+ag \right ) ^{3} \left ( dix+ci \right ) } \left ( A+B\ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) \right ) }\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.42947, size = 1199, normalized size = 4.51 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.556775, size = 1023, normalized size = 3.85 \begin{align*} -\frac{2 \, A b^{2} c^{2} - 8 \, A a b c d + 6 \, A a^{2} d^{2} - 2 \,{\left (B b^{2} d^{2} n x^{2} + 2 \, B a b d^{2} n x + B a^{2} d^{2} n\right )} \log \left (\frac{b x + a}{d x + c}\right )^{2} +{\left (B b^{2} c^{2} - 8 \, B a b c d + 7 \, B a^{2} d^{2}\right )} n - 2 \,{\left (2 \, A b^{2} c d - 2 \, A a b d^{2} + 3 \,{\left (B b^{2} c d - B a b d^{2}\right )} n\right )} x + 2 \,{\left (B b^{2} c^{2} - 4 \, B a b c d + 3 \, B a^{2} d^{2} - 2 \,{\left (B b^{2} c d - B a b d^{2}\right )} x - 2 \,{\left (B b^{2} d^{2} x^{2} + 2 \, B a b d^{2} x + B a^{2} d^{2}\right )} \log \left (\frac{b x + a}{d x + c}\right )\right )} \log \left (e\right ) - 2 \,{\left (2 \, A a^{2} d^{2} +{\left (3 \, B b^{2} d^{2} n + 2 \, A b^{2} d^{2}\right )} x^{2} -{\left (B b^{2} c^{2} - 4 \, B a b c d\right )} n + 2 \,{\left (2 \, A a b d^{2} +{\left (B b^{2} c d + 2 \, B a b d^{2}\right )} n\right )} x\right )} \log \left (\frac{b x + a}{d x + c}\right )}{4 \,{\left ({\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} g^{3} i x^{2} + 2 \,{\left (a b^{4} c^{3} - 3 \, a^{2} b^{3} c^{2} d + 3 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}\right )} g^{3} i x +{\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} g^{3} i\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{B \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) + A}{{\left (b g x + a g\right )}^{3}{\left (d i x + c i\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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