Optimal. Leaf size=159 \[ -\frac {\log (d+e x) \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{d}-\frac {\log \left (-\frac {b}{a x}\right ) \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{d}+\frac {p \text {Li}_2\left (\frac {a (d+e x)}{a d-b e}\right )}{d}+\frac {p \log (d+e x) \log \left (-\frac {e (a x+b)}{a d-b e}\right )}{d}-\frac {p \text {Li}_2\left (\frac {b}{a x}+1\right )}{d}-\frac {p \text {Li}_2\left (\frac {e x}{d}+1\right )}{d}-\frac {p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d} \]
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Rubi [A] time = 0.25, antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {2466, 2454, 2394, 2315, 2462, 260, 2416, 2393, 2391} \[ \frac {p \text {PolyLog}\left (2,\frac {a (d+e x)}{a d-b e}\right )}{d}-\frac {p \text {PolyLog}\left (2,\frac {b}{a x}+1\right )}{d}-\frac {p \text {PolyLog}\left (2,\frac {e x}{d}+1\right )}{d}-\frac {\log (d+e x) \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{d}-\frac {\log \left (-\frac {b}{a x}\right ) \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{d}+\frac {p \log (d+e x) \log \left (-\frac {e (a x+b)}{a d-b e}\right )}{d}-\frac {p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d} \]
Antiderivative was successfully verified.
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Rule 260
Rule 2315
Rule 2391
Rule 2393
Rule 2394
Rule 2416
Rule 2454
Rule 2462
Rule 2466
Rubi steps
\begin {align*} \int \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{x (d+e x)} \, dx &=\int \left (\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{d x}-\frac {e \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{d (d+e x)}\right ) \, dx\\ &=\frac {\int \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{x} \, dx}{d}-\frac {e \int \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{d+e x} \, dx}{d}\\ &=-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log (d+e x)}{d}-\frac {\operatorname {Subst}\left (\int \frac {\log \left (c (a+b x)^p\right )}{x} \, dx,x,\frac {1}{x}\right )}{d}-\frac {(b p) \int \frac {\log (d+e x)}{\left (a+\frac {b}{x}\right ) x^2} \, dx}{d}\\ &=-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log \left (-\frac {b}{a x}\right )}{d}-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log (d+e x)}{d}-\frac {(b p) \int \left (\frac {\log (d+e x)}{b x}-\frac {a \log (d+e x)}{b (b+a x)}\right ) \, dx}{d}+\frac {(b p) \operatorname {Subst}\left (\int \frac {\log \left (-\frac {b x}{a}\right )}{a+b x} \, dx,x,\frac {1}{x}\right )}{d}\\ &=-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log \left (-\frac {b}{a x}\right )}{d}-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log (d+e x)}{d}-\frac {p \text {Li}_2\left (1+\frac {b}{a x}\right )}{d}-\frac {p \int \frac {\log (d+e x)}{x} \, dx}{d}+\frac {(a p) \int \frac {\log (d+e x)}{b+a x} \, dx}{d}\\ &=-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log \left (-\frac {b}{a x}\right )}{d}-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log (d+e x)}{d}-\frac {p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d}+\frac {p \log \left (-\frac {e (b+a x)}{a d-b e}\right ) \log (d+e x)}{d}-\frac {p \text {Li}_2\left (1+\frac {b}{a x}\right )}{d}+\frac {(e p) \int \frac {\log \left (-\frac {e x}{d}\right )}{d+e x} \, dx}{d}-\frac {(e p) \int \frac {\log \left (\frac {e (b+a x)}{-a d+b e}\right )}{d+e x} \, dx}{d}\\ &=-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log \left (-\frac {b}{a x}\right )}{d}-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log (d+e x)}{d}-\frac {p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d}+\frac {p \log \left (-\frac {e (b+a x)}{a d-b e}\right ) \log (d+e x)}{d}-\frac {p \text {Li}_2\left (1+\frac {b}{a x}\right )}{d}-\frac {p \text {Li}_2\left (1+\frac {e x}{d}\right )}{d}-\frac {p \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {a x}{-a d+b e}\right )}{x} \, dx,x,d+e x\right )}{d}\\ &=-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log \left (-\frac {b}{a x}\right )}{d}-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log (d+e x)}{d}-\frac {p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d}+\frac {p \log \left (-\frac {e (b+a x)}{a d-b e}\right ) \log (d+e x)}{d}-\frac {p \text {Li}_2\left (1+\frac {b}{a x}\right )}{d}+\frac {p \text {Li}_2\left (\frac {a (d+e x)}{a d-b e}\right )}{d}-\frac {p \text {Li}_2\left (1+\frac {e x}{d}\right )}{d}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 139, normalized size = 0.87 \[ -\frac {\log (d+e x) \log \left (c \left (a+\frac {b}{x}\right )^p\right )+\log \left (-\frac {b}{a x}\right ) \log \left (c \left (a+\frac {b}{x}\right )^p\right )-p \text {Li}_2\left (\frac {a (d+e x)}{a d-b e}\right )-p \log (d+e x) \log \left (\frac {e (a x+b)}{b e-a d}\right )+p \text {Li}_2\left (\frac {b}{a x}+1\right )+p \text {Li}_2\left (\frac {e x}{d}+1\right )+p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left (c \left (\frac {a x + b}{x}\right )^{p}\right )}{e x^{2} + d x}, x\right ) \]
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 \[ \int \frac {\log \left ({\left (a + \frac {b}{x}\right )}^{p} c\right )}{{\left (e x + d\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.42, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (c \left (a +\frac {b}{x}\right )^{p}\right )}{\left (e x +d \right ) x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.05, size = 179, normalized size = 1.13 \[ -\frac {1}{2} \, b p {\left (\frac {2 \, \log \left (e x + d\right ) \log \relax (x) - \log \relax (x)^{2}}{b d} + \frac {2 \, {\left (\log \left (\frac {a x}{b} + 1\right ) \log \relax (x) + {\rm Li}_2\left (-\frac {a x}{b}\right )\right )}}{b d} - \frac {2 \, {\left (\log \left (\frac {e x}{d} + 1\right ) \log \relax (x) + {\rm Li}_2\left (-\frac {e x}{d}\right )\right )}}{b d} - \frac {2 \, {\left (\log \left (e x + d\right ) \log \left (-\frac {a e x + a d}{a d - b e} + 1\right ) + {\rm Li}_2\left (\frac {a e x + a d}{a d - b e}\right )\right )}}{b d}\right )} - {\left (\frac {\log \left (e x + d\right )}{d} - \frac {\log \relax (x)}{d}\right )} \log \left ({\left (a + \frac {b}{x}\right )}^{p} c\right ) \]
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 \[ \int \frac {\ln \left (c\,{\left (a+\frac {b}{x}\right )}^p\right )}{x\,\left (d+e\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log {\left (c \left (a + \frac {b}{x}\right )^{p} \right )}}{x \left (d + e x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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