Optimal. Leaf size=151 \[ -\frac {d \log (d+e x) \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e^2}+\frac {x \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e}+\frac {d p \text {Li}_2\left (\frac {a (d+e x)}{a d-b e}\right )}{e^2}+\frac {d p \log (d+e x) \log \left (-\frac {e (a x+b)}{a d-b e}\right )}{e^2}+\frac {b p \log (a x+b)}{a e}-\frac {d p \text {Li}_2\left (\frac {e x}{d}+1\right )}{e^2}-\frac {d p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e^2} \]
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Rubi [A] time = 0.21, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 11, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {2466, 2448, 263, 31, 2462, 260, 2416, 2394, 2315, 2393, 2391} \[ \frac {d p \text {PolyLog}\left (2,\frac {a (d+e x)}{a d-b e}\right )}{e^2}-\frac {d p \text {PolyLog}\left (2,\frac {e x}{d}+1\right )}{e^2}-\frac {d \log (d+e x) \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e^2}+\frac {x \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e}+\frac {d p \log (d+e x) \log \left (-\frac {e (a x+b)}{a d-b e}\right )}{e^2}+\frac {b p \log (a x+b)}{a e}-\frac {d p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e^2} \]
Antiderivative was successfully verified.
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Rule 31
Rule 260
Rule 263
Rule 2315
Rule 2391
Rule 2393
Rule 2394
Rule 2416
Rule 2448
Rule 2462
Rule 2466
Rubi steps
\begin {align*} \int \frac {x \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{d+e x} \, dx &=\int \left (\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e}-\frac {d \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e (d+e x)}\right ) \, dx\\ &=\frac {\int \log \left (c \left (a+\frac {b}{x}\right )^p\right ) \, dx}{e}-\frac {d \int \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{d+e x} \, dx}{e}\\ &=\frac {x \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e}-\frac {d \log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log (d+e x)}{e^2}-\frac {(b d p) \int \frac {\log (d+e x)}{\left (a+\frac {b}{x}\right ) x^2} \, dx}{e^2}+\frac {(b p) \int \frac {1}{\left (a+\frac {b}{x}\right ) x} \, dx}{e}\\ &=\frac {x \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e}-\frac {d \log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log (d+e x)}{e^2}-\frac {(b d p) \int \left (\frac {\log (d+e x)}{b x}-\frac {a \log (d+e x)}{b (b+a x)}\right ) \, dx}{e^2}+\frac {(b p) \int \frac {1}{b+a x} \, dx}{e}\\ &=\frac {x \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e}+\frac {b p \log (b+a x)}{a e}-\frac {d \log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log (d+e x)}{e^2}-\frac {(d p) \int \frac {\log (d+e x)}{x} \, dx}{e^2}+\frac {(a d p) \int \frac {\log (d+e x)}{b+a x} \, dx}{e^2}\\ &=\frac {x \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e}+\frac {b p \log (b+a x)}{a e}-\frac {d \log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log (d+e x)}{e^2}-\frac {d p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e^2}+\frac {d p \log \left (-\frac {e (b+a x)}{a d-b e}\right ) \log (d+e x)}{e^2}+\frac {(d p) \int \frac {\log \left (-\frac {e x}{d}\right )}{d+e x} \, dx}{e}-\frac {(d p) \int \frac {\log \left (\frac {e (b+a x)}{-a d+b e}\right )}{d+e x} \, dx}{e}\\ &=\frac {x \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e}+\frac {b p \log (b+a x)}{a e}-\frac {d \log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log (d+e x)}{e^2}-\frac {d p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e^2}+\frac {d p \log \left (-\frac {e (b+a x)}{a d-b e}\right ) \log (d+e x)}{e^2}-\frac {d p \text {Li}_2\left (1+\frac {e x}{d}\right )}{e^2}-\frac {(d p) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {a x}{-a d+b e}\right )}{x} \, dx,x,d+e x\right )}{e^2}\\ &=\frac {x \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e}+\frac {b p \log (b+a x)}{a e}-\frac {d \log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log (d+e x)}{e^2}-\frac {d p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e^2}+\frac {d p \log \left (-\frac {e (b+a x)}{a d-b e}\right ) \log (d+e x)}{e^2}+\frac {d p \text {Li}_2\left (\frac {a (d+e x)}{a d-b e}\right )}{e^2}-\frac {d p \text {Li}_2\left (1+\frac {e x}{d}\right )}{e^2}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 149, normalized size = 0.99 \[ -\frac {d \log (d+e x) \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e^2}+\frac {x \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e}-\frac {d p \left (-\text {Li}_2\left (\frac {a (d+e x)}{a d-b e}\right )-\log (d+e x) \log \left (-\frac {e (a x+b)}{a d-b e}\right )+\text {Li}_2\left (\frac {d+e x}{d}\right )+\log \left (-\frac {e x}{d}\right ) \log (d+e x)\right )}{e^2}+\frac {b p \left (\frac {\log \left (a+\frac {b}{x}\right )}{a}+\frac {\log (x)}{a}\right )}{e} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x \log \left (c \left (\frac {a x + b}{x}\right )^{p}\right )}{e x + d}, x\right ) \]
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 \[ \int \frac {x \log \left ({\left (a + \frac {b}{x}\right )}^{p} c\right )}{e x + d}\,{d x} \]
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 \[ \int \frac {x \ln \left (c \left (a +\frac {b}{x}\right )^{p}\right )}{e x +d}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \log \left ({\left (a + \frac {b}{x}\right )}^{p} c\right )}{e x + d}\,{d x} \]
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 \[ \int \frac {x\,\ln \left (c\,{\left (a+\frac {b}{x}\right )}^p\right )}{d+e\,x} \,d x \]
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 \[ \int \frac {x \log {\left (c \left (a + \frac {b}{x}\right )^{p} \right )}}{d + e x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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