Optimal. Leaf size=198 \[ \frac {p}{d x}-\frac {\left (a+\frac {b}{x}\right ) \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{b d}+\frac {e \log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log \left (-\frac {b}{a x}\right )}{d^2}+\frac {e \log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log (d+e x)}{d^2}+\frac {e p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d^2}-\frac {e p \log \left (-\frac {e (b+a x)}{a d-b e}\right ) \log (d+e x)}{d^2}+\frac {e p \text {Li}_2\left (1+\frac {b}{a x}\right )}{d^2}-\frac {e p \text {Li}_2\left (\frac {a (d+e x)}{a d-b e}\right )}{d^2}+\frac {e p \text {Li}_2\left (1+\frac {e x}{d}\right )}{d^2} \]
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Rubi [A]
time = 0.20, antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps
used = 16, number of rules used = 11, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {2516, 2504,
2436, 2332, 2441, 2352, 2512, 266, 2463, 2440, 2438} \begin {gather*} \frac {e p \text {PolyLog}\left (2,\frac {b}{a x}+1\right )}{d^2}-\frac {e p \text {PolyLog}\left (2,\frac {a (d+e x)}{a d-b e}\right )}{d^2}+\frac {e p \text {PolyLog}\left (2,\frac {e x}{d}+1\right )}{d^2}+\frac {e \log \left (-\frac {b}{a x}\right ) \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{d^2}+\frac {e \log (d+e x) \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{d^2}-\frac {\left (a+\frac {b}{x}\right ) \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{b d}-\frac {e p \log (d+e x) \log \left (-\frac {e (a x+b)}{a d-b e}\right )}{d^2}+\frac {e p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d^2}+\frac {p}{d x} \end {gather*}
Antiderivative was successfully verified.
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Rule 266
Rule 2332
Rule 2352
Rule 2436
Rule 2438
Rule 2440
Rule 2441
Rule 2463
Rule 2504
Rule 2512
Rule 2516
Rubi steps
\begin {align*} \int \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{x^2 (d+e x)} \, dx &=\int \left (\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{d x^2}-\frac {e \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{d^2 x}+\frac {e^2 \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{d^2 (d+e x)}\right ) \, dx\\ &=\frac {\int \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{x^2} \, dx}{d}-\frac {e \int \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{x} \, dx}{d^2}+\frac {e^2 \int \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{d+e x} \, dx}{d^2}\\ &=\frac {e \log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log (d+e x)}{d^2}-\frac {\text {Subst}\left (\int \log \left (c (a+b x)^p\right ) \, dx,x,\frac {1}{x}\right )}{d}+\frac {e \text {Subst}\left (\int \frac {\log \left (c (a+b x)^p\right )}{x} \, dx,x,\frac {1}{x}\right )}{d^2}+\frac {(b e p) \int \frac {\log (d+e x)}{\left (a+\frac {b}{x}\right ) x^2} \, dx}{d^2}\\ &=\frac {e \log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log \left (-\frac {b}{a x}\right )}{d^2}+\frac {e \log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log (d+e x)}{d^2}-\frac {\text {Subst}\left (\int \log \left (c x^p\right ) \, dx,x,a+\frac {b}{x}\right )}{b d}+\frac {(b e p) \int \left (\frac {\log (d+e x)}{b x}-\frac {a \log (d+e x)}{b (b+a x)}\right ) \, dx}{d^2}-\frac {(b e p) \text {Subst}\left (\int \frac {\log \left (-\frac {b x}{a}\right )}{a+b x} \, dx,x,\frac {1}{x}\right )}{d^2}\\ &=\frac {p}{d x}-\frac {\left (a+\frac {b}{x}\right ) \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{b d}+\frac {e \log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log \left (-\frac {b}{a x}\right )}{d^2}+\frac {e \log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log (d+e x)}{d^2}+\frac {e p \text {Li}_2\left (1+\frac {b}{a x}\right )}{d^2}+\frac {(e p) \int \frac {\log (d+e x)}{x} \, dx}{d^2}-\frac {(a e p) \int \frac {\log (d+e x)}{b+a x} \, dx}{d^2}\\ &=\frac {p}{d x}-\frac {\left (a+\frac {b}{x}\right ) \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{b d}+\frac {e \log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log \left (-\frac {b}{a x}\right )}{d^2}+\frac {e \log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log (d+e x)}{d^2}+\frac {e p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d^2}-\frac {e p \log \left (-\frac {e (b+a x)}{a d-b e}\right ) \log (d+e x)}{d^2}+\frac {e p \text {Li}_2\left (1+\frac {b}{a x}\right )}{d^2}-\frac {\left (e^2 p\right ) \int \frac {\log \left (-\frac {e x}{d}\right )}{d+e x} \, dx}{d^2}+\frac {\left (e^2 p\right ) \int \frac {\log \left (\frac {e (b+a x)}{-a d+b e}\right )}{d+e x} \, dx}{d^2}\\ &=\frac {p}{d x}-\frac {\left (a+\frac {b}{x}\right ) \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{b d}+\frac {e \log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log \left (-\frac {b}{a x}\right )}{d^2}+\frac {e \log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log (d+e x)}{d^2}+\frac {e p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d^2}-\frac {e p \log \left (-\frac {e (b+a x)}{a d-b e}\right ) \log (d+e x)}{d^2}+\frac {e p \text {Li}_2\left (1+\frac {b}{a x}\right )}{d^2}+\frac {e p \text {Li}_2\left (1+\frac {e x}{d}\right )}{d^2}+\frac {(e p) \text {Subst}\left (\int \frac {\log \left (1+\frac {a x}{-a d+b e}\right )}{x} \, dx,x,d+e x\right )}{d^2}\\ &=\frac {p}{d x}-\frac {\left (a+\frac {b}{x}\right ) \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{b d}+\frac {e \log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log \left (-\frac {b}{a x}\right )}{d^2}+\frac {e \log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log (d+e x)}{d^2}+\frac {e p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d^2}-\frac {e p \log \left (-\frac {e (b+a x)}{a d-b e}\right ) \log (d+e x)}{d^2}+\frac {e p \text {Li}_2\left (1+\frac {b}{a x}\right )}{d^2}-\frac {e p \text {Li}_2\left (\frac {a (d+e x)}{a d-b e}\right )}{d^2}+\frac {e p \text {Li}_2\left (1+\frac {e x}{d}\right )}{d^2}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 166, normalized size = 0.84 \begin {gather*} \frac {\frac {d p}{x}-\frac {d \left (a+\frac {b}{x}\right ) \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{b}+e \log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log \left (-\frac {b}{a x}\right )+e \log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log (d+e x)+e p \text {Li}_2\left (1+\frac {b}{a x}\right )+e p \left (\left (\log \left (-\frac {e x}{d}\right )-\log \left (\frac {e (b+a x)}{-a d+b e}\right )\right ) \log (d+e x)-\text {Li}_2\left (\frac {a (d+e x)}{a d-b e}\right )+\text {Li}_2\left (1+\frac {e x}{d}\right )\right )}{d^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.23, size = 0, normalized size = 0.00 \[\int \frac {\ln \left (c \left (a +\frac {b}{x}\right )^{p}\right )}{x^{2} \left (e x +d \right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.34, size = 246, normalized size = 1.24 \begin {gather*} \frac {1}{2} \, b p {\left (\frac {2 \, {\left (\log \left (\frac {a x}{b} + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-\frac {a x}{b}\right )\right )} e}{b d^{2}} - \frac {2 \, {\left (\log \left (x\right ) \log \left (\frac {x e}{d} + 1\right ) + {\rm Li}_2\left (-\frac {x e}{d}\right )\right )} e}{b d^{2}} - \frac {2 \, {\left (\log \left (x e + d\right ) \log \left (-\frac {a x e + a d}{a d - b e} + 1\right ) + {\rm Li}_2\left (\frac {a x e + a d}{a d - b e}\right )\right )} e}{b d^{2}} - \frac {2 \, a \log \left (a x + b\right )}{b^{2} d} + \frac {2 \, a \log \left (x\right )}{b^{2} d} + \frac {2 \, e \log \left (x e + d\right ) \log \left (x\right ) - e \log \left (x\right )^{2}}{b d^{2}} + \frac {2}{b d x}\right )} + {\left (\frac {e \log \left (x e + d\right )}{d^{2}} - \frac {e \log \left (x\right )}{d^{2}} - \frac {1}{d x}\right )} \log \left ({\left (a + \frac {b}{x}\right )}^{p} c\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\ln \left (c\,{\left (a+\frac {b}{x}\right )}^p\right )}{x^2\,\left (d+e\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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