Optimal. Leaf size=287 \[ -\frac{e^2 p \text{PolyLog}\left (2,\frac{b}{a x}+1\right )}{d^3}+\frac{e^2 p \text{PolyLog}\left (2,\frac{a (d+e x)}{a d-b e}\right )}{d^3}-\frac{e^2 p \text{PolyLog}\left (2,\frac{e x}{d}+1\right )}{d^3}+\frac{a^2 p \log \left (a+\frac{b}{x}\right )}{2 b^2 d}-\frac{e^2 \log \left (-\frac{b}{a x}\right ) \log \left (c \left (a+\frac{b}{x}\right )^p\right )}{d^3}-\frac{e^2 \log (d+e x) \log \left (c \left (a+\frac{b}{x}\right )^p\right )}{d^3}+\frac{e \left (a+\frac{b}{x}\right ) \log \left (c \left (a+\frac{b}{x}\right )^p\right )}{b d^2}-\frac{\log \left (c \left (a+\frac{b}{x}\right )^p\right )}{2 d x^2}+\frac{e^2 p \log (d+e x) \log \left (-\frac{e (a x+b)}{a d-b e}\right )}{d^3}-\frac{a p}{2 b d x}-\frac{e^2 p \log \left (-\frac{e x}{d}\right ) \log (d+e x)}{d^3}-\frac{e p}{d^2 x}+\frac{p}{4 d x^2} \]
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Rubi [A] time = 0.32949, antiderivative size = 287, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 13, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.565, Rules used = {2466, 2454, 2395, 43, 2389, 2295, 2394, 2315, 2462, 260, 2416, 2393, 2391} \[ -\frac{e^2 p \text{PolyLog}\left (2,\frac{b}{a x}+1\right )}{d^3}+\frac{e^2 p \text{PolyLog}\left (2,\frac{a (d+e x)}{a d-b e}\right )}{d^3}-\frac{e^2 p \text{PolyLog}\left (2,\frac{e x}{d}+1\right )}{d^3}+\frac{a^2 p \log \left (a+\frac{b}{x}\right )}{2 b^2 d}-\frac{e^2 \log \left (-\frac{b}{a x}\right ) \log \left (c \left (a+\frac{b}{x}\right )^p\right )}{d^3}-\frac{e^2 \log (d+e x) \log \left (c \left (a+\frac{b}{x}\right )^p\right )}{d^3}+\frac{e \left (a+\frac{b}{x}\right ) \log \left (c \left (a+\frac{b}{x}\right )^p\right )}{b d^2}-\frac{\log \left (c \left (a+\frac{b}{x}\right )^p\right )}{2 d x^2}+\frac{e^2 p \log (d+e x) \log \left (-\frac{e (a x+b)}{a d-b e}\right )}{d^3}-\frac{a p}{2 b d x}-\frac{e^2 p \log \left (-\frac{e x}{d}\right ) \log (d+e x)}{d^3}-\frac{e p}{d^2 x}+\frac{p}{4 d x^2} \]
Antiderivative was successfully verified.
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Rule 2466
Rule 2454
Rule 2395
Rule 43
Rule 2389
Rule 2295
Rule 2394
Rule 2315
Rule 2462
Rule 260
Rule 2416
Rule 2393
Rule 2391
Rubi steps
\begin{align*} \int \frac{\log \left (c \left (a+\frac{b}{x}\right )^p\right )}{x^3 (d+e x)} \, dx &=\int \left (\frac{\log \left (c \left (a+\frac{b}{x}\right )^p\right )}{d x^3}-\frac{e \log \left (c \left (a+\frac{b}{x}\right )^p\right )}{d^2 x^2}+\frac{e^2 \log \left (c \left (a+\frac{b}{x}\right )^p\right )}{d^3 x}-\frac{e^3 \log \left (c \left (a+\frac{b}{x}\right )^p\right )}{d^3 (d+e x)}\right ) \, dx\\ &=\frac{\int \frac{\log \left (c \left (a+\frac{b}{x}\right )^p\right )}{x^3} \, dx}{d}-\frac{e \int \frac{\log \left (c \left (a+\frac{b}{x}\right )^p\right )}{x^2} \, dx}{d^2}+\frac{e^2 \int \frac{\log \left (c \left (a+\frac{b}{x}\right )^p\right )}{x} \, dx}{d^3}-\frac{e^3 \int \frac{\log \left (c \left (a+\frac{b}{x}\right )^p\right )}{d+e x} \, dx}{d^3}\\ &=-\frac{e^2 \log \left (c \left (a+\frac{b}{x}\right )^p\right ) \log (d+e x)}{d^3}-\frac{\operatorname{Subst}\left (\int x \log \left (c (a+b x)^p\right ) \, dx,x,\frac{1}{x}\right )}{d}+\frac{e \operatorname{Subst}\left (\int \log \left (c (a+b x)^p\right ) \, dx,x,\frac{1}{x}\right )}{d^2}-\frac{e^2 \operatorname{Subst}\left (\int \frac{\log \left (c (a+b x)^p\right )}{x} \, dx,x,\frac{1}{x}\right )}{d^3}-\frac{\left (b e^2 p\right ) \int \frac{\log (d+e x)}{\left (a+\frac{b}{x}\right ) x^2} \, dx}{d^3}\\ &=-\frac{\log \left (c \left (a+\frac{b}{x}\right )^p\right )}{2 d x^2}-\frac{e^2 \log \left (c \left (a+\frac{b}{x}\right )^p\right ) \log \left (-\frac{b}{a x}\right )}{d^3}-\frac{e^2 \log \left (c \left (a+\frac{b}{x}\right )^p\right ) \log (d+e x)}{d^3}+\frac{e \operatorname{Subst}\left (\int \log \left (c x^p\right ) \, dx,x,a+\frac{b}{x}\right )}{b d^2}+\frac{(b p) \operatorname{Subst}\left (\int \frac{x^2}{a+b x} \, dx,x,\frac{1}{x}\right )}{2 d}-\frac{\left (b e^2 p\right ) \int \left (\frac{\log (d+e x)}{b x}-\frac{a \log (d+e x)}{b (b+a x)}\right ) \, dx}{d^3}+\frac{\left (b e^2 p\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{b x}{a}\right )}{a+b x} \, dx,x,\frac{1}{x}\right )}{d^3}\\ &=-\frac{e p}{d^2 x}+\frac{e \left (a+\frac{b}{x}\right ) \log \left (c \left (a+\frac{b}{x}\right )^p\right )}{b d^2}-\frac{\log \left (c \left (a+\frac{b}{x}\right )^p\right )}{2 d x^2}-\frac{e^2 \log \left (c \left (a+\frac{b}{x}\right )^p\right ) \log \left (-\frac{b}{a x}\right )}{d^3}-\frac{e^2 \log \left (c \left (a+\frac{b}{x}\right )^p\right ) \log (d+e x)}{d^3}-\frac{e^2 p \text{Li}_2\left (1+\frac{b}{a x}\right )}{d^3}+\frac{(b p) \operatorname{Subst}\left (\int \left (-\frac{a}{b^2}+\frac{x}{b}+\frac{a^2}{b^2 (a+b x)}\right ) \, dx,x,\frac{1}{x}\right )}{2 d}-\frac{\left (e^2 p\right ) \int \frac{\log (d+e x)}{x} \, dx}{d^3}+\frac{\left (a e^2 p\right ) \int \frac{\log (d+e x)}{b+a x} \, dx}{d^3}\\ &=\frac{p}{4 d x^2}-\frac{a p}{2 b d x}-\frac{e p}{d^2 x}+\frac{a^2 p \log \left (a+\frac{b}{x}\right )}{2 b^2 d}+\frac{e \left (a+\frac{b}{x}\right ) \log \left (c \left (a+\frac{b}{x}\right )^p\right )}{b d^2}-\frac{\log \left (c \left (a+\frac{b}{x}\right )^p\right )}{2 d x^2}-\frac{e^2 \log \left (c \left (a+\frac{b}{x}\right )^p\right ) \log \left (-\frac{b}{a x}\right )}{d^3}-\frac{e^2 \log \left (c \left (a+\frac{b}{x}\right )^p\right ) \log (d+e x)}{d^3}-\frac{e^2 p \log \left (-\frac{e x}{d}\right ) \log (d+e x)}{d^3}+\frac{e^2 p \log \left (-\frac{e (b+a x)}{a d-b e}\right ) \log (d+e x)}{d^3}-\frac{e^2 p \text{Li}_2\left (1+\frac{b}{a x}\right )}{d^3}+\frac{\left (e^3 p\right ) \int \frac{\log \left (-\frac{e x}{d}\right )}{d+e x} \, dx}{d^3}-\frac{\left (e^3 p\right ) \int \frac{\log \left (\frac{e (b+a x)}{-a d+b e}\right )}{d+e x} \, dx}{d^3}\\ &=\frac{p}{4 d x^2}-\frac{a p}{2 b d x}-\frac{e p}{d^2 x}+\frac{a^2 p \log \left (a+\frac{b}{x}\right )}{2 b^2 d}+\frac{e \left (a+\frac{b}{x}\right ) \log \left (c \left (a+\frac{b}{x}\right )^p\right )}{b d^2}-\frac{\log \left (c \left (a+\frac{b}{x}\right )^p\right )}{2 d x^2}-\frac{e^2 \log \left (c \left (a+\frac{b}{x}\right )^p\right ) \log \left (-\frac{b}{a x}\right )}{d^3}-\frac{e^2 \log \left (c \left (a+\frac{b}{x}\right )^p\right ) \log (d+e x)}{d^3}-\frac{e^2 p \log \left (-\frac{e x}{d}\right ) \log (d+e x)}{d^3}+\frac{e^2 p \log \left (-\frac{e (b+a x)}{a d-b e}\right ) \log (d+e x)}{d^3}-\frac{e^2 p \text{Li}_2\left (1+\frac{b}{a x}\right )}{d^3}-\frac{e^2 p \text{Li}_2\left (1+\frac{e x}{d}\right )}{d^3}-\frac{\left (e^2 p\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{a x}{-a d+b e}\right )}{x} \, dx,x,d+e x\right )}{d^3}\\ &=\frac{p}{4 d x^2}-\frac{a p}{2 b d x}-\frac{e p}{d^2 x}+\frac{a^2 p \log \left (a+\frac{b}{x}\right )}{2 b^2 d}+\frac{e \left (a+\frac{b}{x}\right ) \log \left (c \left (a+\frac{b}{x}\right )^p\right )}{b d^2}-\frac{\log \left (c \left (a+\frac{b}{x}\right )^p\right )}{2 d x^2}-\frac{e^2 \log \left (c \left (a+\frac{b}{x}\right )^p\right ) \log \left (-\frac{b}{a x}\right )}{d^3}-\frac{e^2 \log \left (c \left (a+\frac{b}{x}\right )^p\right ) \log (d+e x)}{d^3}-\frac{e^2 p \log \left (-\frac{e x}{d}\right ) \log (d+e x)}{d^3}+\frac{e^2 p \log \left (-\frac{e (b+a x)}{a d-b e}\right ) \log (d+e x)}{d^3}-\frac{e^2 p \text{Li}_2\left (1+\frac{b}{a x}\right )}{d^3}+\frac{e^2 p \text{Li}_2\left (\frac{a (d+e x)}{a d-b e}\right )}{d^3}-\frac{e^2 p \text{Li}_2\left (1+\frac{e x}{d}\right )}{d^3}\\ \end{align*}
Mathematica [A] time = 0.21616, size = 241, normalized size = 0.84 \[ -\frac{4 e^2 p \left (-\text{PolyLog}\left (2,\frac{a (d+e x)}{a d-b e}\right )+\text{PolyLog}\left (2,\frac{e x}{d}+1\right )+\log (d+e x) \left (\log \left (-\frac{e x}{d}\right )-\log \left (\frac{e (a x+b)}{b e-a d}\right )\right )\right )+4 e^2 p \text{PolyLog}\left (2,\frac{b}{a x}+1\right )-\frac{d^2 p \left (2 a^2 x^2 \log \left (a+\frac{b}{x}\right )+b (b-2 a x)\right )}{b^2 x^2}+\frac{2 d^2 \log \left (c \left (a+\frac{b}{x}\right )^p\right )}{x^2}+4 e^2 \log (d+e x) \log \left (c \left (a+\frac{b}{x}\right )^p\right )-\frac{4 d e \left (a+\frac{b}{x}\right ) \log \left (c \left (a+\frac{b}{x}\right )^p\right )}{b}+4 e^2 \log \left (-\frac{b}{a x}\right ) \log \left (c \left (a+\frac{b}{x}\right )^p\right )+\frac{4 d e p}{x}}{4 d^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.733, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3} \left ( ex+d \right ) }\ln \left ( c \left ( a+{\frac{b}{x}} \right ) ^{p} \right ) }\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.28857, size = 414, normalized size = 1.44 \begin{align*} \frac{1}{4} \,{\left (4 \, e{\left (\frac{a \log \left (a x + b\right )}{b^{2} d^{2}} - \frac{a \log \left (x\right )}{b^{2} d^{2}} - \frac{1}{b d^{2} x}\right )} - \frac{4 \,{\left (\log \left (\frac{a x}{b} + 1\right ) \log \left (x\right ) +{\rm Li}_2\left (-\frac{a x}{b}\right )\right )} e^{2}}{b d^{3}} + \frac{4 \,{\left (\log \left (\frac{e x}{d} + 1\right ) \log \left (x\right ) +{\rm Li}_2\left (-\frac{e x}{d}\right )\right )} e^{2}}{b d^{3}} + \frac{4 \,{\left (\log \left (e x + d\right ) \log \left (-\frac{a e x + a d}{a d - b e} + 1\right ) +{\rm Li}_2\left (\frac{a e x + a d}{a d - b e}\right )\right )} e^{2}}{b d^{3}} + \frac{2 \, a^{2} \log \left (a x + b\right )}{b^{3} d} - \frac{2 \, a^{2} \log \left (x\right )}{b^{3} d} - \frac{2 \,{\left (2 \, e^{2} \log \left (e x + d\right ) \log \left (x\right ) - e^{2} \log \left (x\right )^{2}\right )}}{b d^{3}} - \frac{2 \, a x - b}{b^{2} d x^{2}}\right )} b p - \frac{1}{2} \,{\left (\frac{2 \, e^{2} \log \left (e x + d\right )}{d^{3}} - \frac{2 \, e^{2} \log \left (x\right )}{d^{3}} - \frac{2 \, e x - d}{d^{2} x^{2}}\right )} \log \left ({\left (a + \frac{b}{x}\right )}^{p} c\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\log \left (c \left (\frac{a x + b}{x}\right )^{p}\right )}{e x^{4} + d x^{3}}, x\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{\log \left ({\left (a + \frac{b}{x}\right )}^{p} c\right )}{{\left (e x + d\right )} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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