Optimal. Leaf size=255 \[ -\frac{b^2 (c+d x)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{2 g^3 i (a+b x)^2 (b c-a d)^3}+\frac{d^2 \log \left (\frac{a+b x}{c+d x}\right ) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{g^3 i (b c-a d)^3}+\frac{2 b d (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{g^3 i (a+b x) (b c-a d)^3}-\frac{B d^2 \log ^2\left (\frac{a+b x}{c+d x}\right )}{2 g^3 i (b c-a d)^3}-\frac{B (c+d x)^2 \left (b-\frac{4 d (a+b x)}{c+d x}\right )^2}{4 g^3 i (a+b x)^2 (b c-a d)^3} \]
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Rubi [C] time = 0.877028, antiderivative size = 535, normalized size of antiderivative = 2.1, number of steps used = 28, number of rules used = 11, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.275, Rules used = {2528, 2525, 12, 44, 2524, 2418, 2390, 2301, 2394, 2393, 2391} \[ \frac{B d^2 \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{g^3 i (b c-a d)^3}+\frac{B d^2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{g^3 i (b c-a d)^3}+\frac{d^2 \log (a+b x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{g^3 i (b c-a d)^3}-\frac{d^2 \log (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{g^3 i (b c-a d)^3}+\frac{d \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{g^3 i (a+b x) (b c-a d)^2}-\frac{B \log \left (\frac{e (a+b x)}{c+d x}\right )+A}{2 g^3 i (a+b x)^2 (b c-a d)}-\frac{B d^2 \log ^2(a+b x)}{2 g^3 i (b c-a d)^3}-\frac{B d^2 \log ^2(c+d x)}{2 g^3 i (b c-a d)^3}+\frac{3 B d^2 \log (a+b x)}{2 g^3 i (b c-a d)^3}+\frac{B d^2 \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{g^3 i (b c-a d)^3}-\frac{3 B d^2 \log (c+d x)}{2 g^3 i (b c-a d)^3}+\frac{B d^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{g^3 i (b c-a d)^3}+\frac{3 B d}{2 g^3 i (a+b x) (b c-a d)^2}-\frac{B}{4 g^3 i (a+b x)^2 (b c-a d)} \]
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 (\frac{e (a+b x)}{c+d x}\right )}{(37 c+37 d x) (a g+b g x)^3} \, dx &=\int \left (\frac{b \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{37 (b c-a d) g^3 (a+b x)^3}-\frac{b d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{37 (b c-a d)^2 g^3 (a+b x)^2}+\frac{b d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{37 (b c-a d)^3 g^3 (a+b x)}-\frac{d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{37 (b c-a d)^3 g^3 (c+d x)}\right ) \, dx\\ &=\frac{\left (b d^2\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{37 (b c-a d)^3 g^3}-\frac{d^3 \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{37 (b c-a d)^3 g^3}-\frac{(b d) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{37 (b c-a d)^2 g^3}+\frac{b \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{37 (b c-a d) g^3}\\ &=-\frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{74 (b c-a d) g^3 (a+b x)^2}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{37 (b c-a d)^2 g^3 (a+b x)}+\frac{d^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{37 (b c-a d)^3 g^3}-\frac{d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{37 (b c-a d)^3 g^3}-\frac{\left (B d^2\right ) \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \log (a+b x)}{e (a+b x)} \, dx}{37 (b c-a d)^3 g^3}+\frac{\left (B d^2\right ) \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \log (c+d x)}{e (a+b x)} \, dx}{37 (b c-a d)^3 g^3}-\frac{(B d) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{37 (b c-a d)^2 g^3}+\frac{B \int \frac{b c-a d}{(a+b x)^3 (c+d x)} \, dx}{74 (b c-a d) g^3}\\ &=-\frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{74 (b c-a d) g^3 (a+b x)^2}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{37 (b c-a d)^2 g^3 (a+b x)}+\frac{d^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{37 (b c-a d)^3 g^3}-\frac{d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{37 (b c-a d)^3 g^3}+\frac{B \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{74 g^3}-\frac{(B d) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{37 (b c-a d) g^3}-\frac{\left (B d^2\right ) \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{37 (b c-a d)^3 e g^3}+\frac{\left (B d^2\right ) \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{37 (b c-a d)^3 e g^3}\\ &=-\frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{74 (b c-a d) g^3 (a+b x)^2}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{37 (b c-a d)^2 g^3 (a+b x)}+\frac{d^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{37 (b c-a d)^3 g^3}-\frac{d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{37 (b c-a d)^3 g^3}+\frac{B \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}{74 g^3}-\frac{(B d) \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}{37 (b c-a d) g^3}-\frac{\left (B d^2\right ) \int \left (\frac{b e \log (a+b x)}{a+b x}-\frac{d e \log (a+b x)}{c+d x}\right ) \, dx}{37 (b c-a d)^3 e g^3}+\frac{\left (B d^2\right ) \int \left (\frac{b e \log (c+d x)}{a+b x}-\frac{d e \log (c+d x)}{c+d x}\right ) \, dx}{37 (b c-a d)^3 e g^3}\\ &=-\frac{B}{148 (b c-a d) g^3 (a+b x)^2}+\frac{3 B d}{74 (b c-a d)^2 g^3 (a+b x)}+\frac{3 B d^2 \log (a+b x)}{74 (b c-a d)^3 g^3}-\frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{74 (b c-a d) g^3 (a+b x)^2}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{37 (b c-a d)^2 g^3 (a+b x)}+\frac{d^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{37 (b c-a d)^3 g^3}-\frac{3 B d^2 \log (c+d x)}{74 (b c-a d)^3 g^3}-\frac{d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{37 (b c-a d)^3 g^3}-\frac{\left (b B d^2\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{37 (b c-a d)^3 g^3}+\frac{\left (b B d^2\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{37 (b c-a d)^3 g^3}+\frac{\left (B d^3\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{37 (b c-a d)^3 g^3}-\frac{\left (B d^3\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{37 (b c-a d)^3 g^3}\\ &=-\frac{B}{148 (b c-a d) g^3 (a+b x)^2}+\frac{3 B d}{74 (b c-a d)^2 g^3 (a+b x)}+\frac{3 B d^2 \log (a+b x)}{74 (b c-a d)^3 g^3}-\frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{74 (b c-a d) g^3 (a+b x)^2}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{37 (b c-a d)^2 g^3 (a+b x)}+\frac{d^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{37 (b c-a d)^3 g^3}-\frac{3 B d^2 \log (c+d x)}{74 (b c-a d)^3 g^3}+\frac{B d^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{37 (b c-a d)^3 g^3}-\frac{d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{37 (b c-a d)^3 g^3}+\frac{B d^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{37 (b c-a d)^3 g^3}-\frac{\left (B d^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{37 (b c-a d)^3 g^3}-\frac{\left (B d^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{37 (b c-a d)^3 g^3}-\frac{\left (b B d^2\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{37 (b c-a d)^3 g^3}-\frac{\left (B d^3\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{37 (b c-a d)^3 g^3}\\ &=-\frac{B}{148 (b c-a d) g^3 (a+b x)^2}+\frac{3 B d}{74 (b c-a d)^2 g^3 (a+b x)}+\frac{3 B d^2 \log (a+b x)}{74 (b c-a d)^3 g^3}-\frac{B d^2 \log ^2(a+b x)}{74 (b c-a d)^3 g^3}-\frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{74 (b c-a d) g^3 (a+b x)^2}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{37 (b c-a d)^2 g^3 (a+b x)}+\frac{d^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{37 (b c-a d)^3 g^3}-\frac{3 B d^2 \log (c+d x)}{74 (b c-a d)^3 g^3}+\frac{B d^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{37 (b c-a d)^3 g^3}-\frac{d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{37 (b c-a d)^3 g^3}-\frac{B d^2 \log ^2(c+d x)}{74 (b c-a d)^3 g^3}+\frac{B d^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{37 (b c-a d)^3 g^3}-\frac{\left (B d^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{37 (b c-a d)^3 g^3}-\frac{\left (B d^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{37 (b c-a d)^3 g^3}\\ &=-\frac{B}{148 (b c-a d) g^3 (a+b x)^2}+\frac{3 B d}{74 (b c-a d)^2 g^3 (a+b x)}+\frac{3 B d^2 \log (a+b x)}{74 (b c-a d)^3 g^3}-\frac{B d^2 \log ^2(a+b x)}{74 (b c-a d)^3 g^3}-\frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{74 (b c-a d) g^3 (a+b x)^2}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{37 (b c-a d)^2 g^3 (a+b x)}+\frac{d^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{37 (b c-a d)^3 g^3}-\frac{3 B d^2 \log (c+d x)}{74 (b c-a d)^3 g^3}+\frac{B d^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{37 (b c-a d)^3 g^3}-\frac{d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{37 (b c-a d)^3 g^3}-\frac{B d^2 \log ^2(c+d x)}{74 (b c-a d)^3 g^3}+\frac{B d^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{37 (b c-a d)^3 g^3}+\frac{B d^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{37 (b c-a d)^3 g^3}+\frac{B d^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{37 (b c-a d)^3 g^3}\\ \end{align*}
Mathematica [C] time = 0.378282, size = 418, normalized size = 1.64 \[ \frac{-2 B d^2 (a+b x)^2 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )-2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )+2 B d^2 (a+b x)^2 \left (2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )+4 d^2 (a+b x)^2 \log (a+b x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )-4 d^2 (a+b x)^2 \log (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )-2 (b c-a d)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )+4 d (a+b x) (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )-B \left (2 d^2 (a+b x)^2 \log (c+d x)+2 d (a+b x) (a d-b c)+(b c-a d)^2-2 d^2 (a+b x)^2 \log (a+b x)\right )+4 B d (a+b x) (-d (a+b x) \log (c+d x)+d (a+b x) \log (a+b x)-a d+b c)}{4 g^3 i (a+b x)^2 (b c-a d)^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.055, size = 1040, normalized size = 4.1 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.5394, size = 1195, normalized size = 4.69 \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 = 0.48864, size = 732, normalized size = 2.87 \begin{align*} -\frac{{\left (2 \, A + B\right )} b^{2} c^{2} - 8 \,{\left (A + B\right )} a b c d +{\left (6 \, A + 7 \, B\right )} a^{2} d^{2} - 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 e x + a e}{d x + c}\right )^{2} - 2 \,{\left ({\left (2 \, A + 3 \, B\right )} b^{2} c d -{\left (2 \, A + 3 \, B\right )} a b d^{2}\right )} x - 2 \,{\left ({\left (2 \, A + 3 \, B\right )} b^{2} d^{2} x^{2} - B b^{2} c^{2} + 4 \, B a b c d + 2 \, A a^{2} d^{2} + 2 \,{\left (B b^{2} c d + 2 \,{\left (A + B\right )} a b d^{2}\right )} x\right )} \log \left (\frac{b e x + a e}{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 [B] time = 7.53382, size = 889, normalized size = 3.49 \begin{align*} \text{result too large to display} \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 (\frac{{\left (b x + a\right )} e}{d x + c}\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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