Optimal. Leaf size=146 \[ -\frac {e \log \left (-\frac {b x}{a}\right ) \log \left (c (a+b x)^p\right )}{d^2}+\frac {e \log \left (c (a+b x)^p\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{d^2}-\frac {\log \left (c (a+b x)^p\right )}{d x}+\frac {e p \text {Li}_2\left (-\frac {e (a+b x)}{b d-a e}\right )}{d^2}-\frac {e p \text {Li}_2\left (\frac {b x}{a}+1\right )}{d^2}+\frac {b p \log (x)}{a d}-\frac {b p \log (a+b x)}{a d} \]
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Rubi [A] time = 0.17, antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {44, 2416, 2395, 36, 29, 31, 2394, 2315, 2393, 2391} \[ \frac {e p \text {PolyLog}\left (2,-\frac {e (a+b x)}{b d-a e}\right )}{d^2}-\frac {e p \text {PolyLog}\left (2,\frac {b x}{a}+1\right )}{d^2}-\frac {e \log \left (-\frac {b x}{a}\right ) \log \left (c (a+b x)^p\right )}{d^2}+\frac {e \log \left (c (a+b x)^p\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{d^2}-\frac {\log \left (c (a+b x)^p\right )}{d x}+\frac {b p \log (x)}{a d}-\frac {b p \log (a+b x)}{a d} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 44
Rule 2315
Rule 2391
Rule 2393
Rule 2394
Rule 2395
Rule 2416
Rubi steps
\begin {align*} \int \frac {\log \left (c (a+b x)^p\right )}{x^2 (d+e x)} \, dx &=\int \left (\frac {\log \left (c (a+b x)^p\right )}{d x^2}-\frac {e \log \left (c (a+b x)^p\right )}{d^2 x}+\frac {e^2 \log \left (c (a+b x)^p\right )}{d^2 (d+e x)}\right ) \, dx\\ &=\frac {\int \frac {\log \left (c (a+b x)^p\right )}{x^2} \, dx}{d}-\frac {e \int \frac {\log \left (c (a+b x)^p\right )}{x} \, dx}{d^2}+\frac {e^2 \int \frac {\log \left (c (a+b x)^p\right )}{d+e x} \, dx}{d^2}\\ &=-\frac {\log \left (c (a+b x)^p\right )}{d x}-\frac {e \log \left (-\frac {b x}{a}\right ) \log \left (c (a+b x)^p\right )}{d^2}+\frac {e \log \left (c (a+b x)^p\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{d^2}+\frac {(b p) \int \frac {1}{x (a+b x)} \, dx}{d}+\frac {(b e p) \int \frac {\log \left (-\frac {b x}{a}\right )}{a+b x} \, dx}{d^2}-\frac {(b e p) \int \frac {\log \left (\frac {b (d+e x)}{b d-a e}\right )}{a+b x} \, dx}{d^2}\\ &=-\frac {\log \left (c (a+b x)^p\right )}{d x}-\frac {e \log \left (-\frac {b x}{a}\right ) \log \left (c (a+b x)^p\right )}{d^2}+\frac {e \log \left (c (a+b x)^p\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{d^2}-\frac {e p \text {Li}_2\left (1+\frac {b x}{a}\right )}{d^2}+\frac {(b p) \int \frac {1}{x} \, dx}{a d}-\frac {\left (b^2 p\right ) \int \frac {1}{a+b x} \, dx}{a d}-\frac {(e p) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {e x}{b d-a e}\right )}{x} \, dx,x,a+b x\right )}{d^2}\\ &=\frac {b p \log (x)}{a d}-\frac {b p \log (a+b x)}{a d}-\frac {\log \left (c (a+b x)^p\right )}{d x}-\frac {e \log \left (-\frac {b x}{a}\right ) \log \left (c (a+b x)^p\right )}{d^2}+\frac {e \log \left (c (a+b x)^p\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{d^2}+\frac {e p \text {Li}_2\left (-\frac {e (a+b x)}{b d-a e}\right )}{d^2}-\frac {e p \text {Li}_2\left (1+\frac {b x}{a}\right )}{d^2}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 139, normalized size = 0.95 \[ \frac {a e x \log \left (c (a+b x)^p\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )-a d \log \left (c (a+b x)^p\right )-a e x \log \left (-\frac {b x}{a}\right ) \log \left (c (a+b x)^p\right )+a e p x \text {Li}_2\left (\frac {e (a+b x)}{a e-b d}\right )-b d p x \log (a+b x)-a e p x \text {Li}_2\left (\frac {b x}{a}+1\right )+b d p x \log (x)}{a d^2 x} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left ({\left (b x + a\right )}^{p} c\right )}{e x^{3} + d x^{2}}, x\right ) \]
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 \[ \int \frac {\log \left ({\left (b x + a\right )}^{p} c\right )}{{\left (e x + d\right )} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.25, size = 615, normalized size = 4.21 \[ \frac {i \pi e \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right ) \ln \relax (x )}{2 d^{2}}-\frac {i \pi e \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right ) \ln \left (e x +d \right )}{2 d^{2}}-\frac {i \pi e \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2} \ln \relax (x )}{2 d^{2}}+\frac {i \pi e \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2} \ln \left (e x +d \right )}{2 d^{2}}-\frac {i \pi e \,\mathrm {csgn}\left (i \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2} \ln \relax (x )}{2 d^{2}}+\frac {i \pi e \,\mathrm {csgn}\left (i \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2} \ln \left (e x +d \right )}{2 d^{2}}+\frac {i \pi e \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{3} \ln \relax (x )}{2 d^{2}}-\frac {i \pi e \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{3} \ln \left (e x +d \right )}{2 d^{2}}+\frac {e p \ln \relax (x ) \ln \left (\frac {b x +a}{a}\right )}{d^{2}}-\frac {e p \ln \left (\frac {a e -b d +\left (e x +d \right ) b}{a e -b d}\right ) \ln \left (e x +d \right )}{d^{2}}+\frac {b p \ln \relax (x )}{a d}-\frac {b p \ln \left (b x +a \right )}{a d}+\frac {i \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )}{2 d x}-\frac {i \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2}}{2 d x}-\frac {i \pi \,\mathrm {csgn}\left (i \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2}}{2 d x}+\frac {i \pi \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{3}}{2 d x}+\frac {e p \dilog \left (\frac {b x +a}{a}\right )}{d^{2}}-\frac {e p \dilog \left (\frac {a e -b d +\left (e x +d \right ) b}{a e -b d}\right )}{d^{2}}-\frac {e \ln \relax (c ) \ln \relax (x )}{d^{2}}+\frac {e \ln \relax (c ) \ln \left (e x +d \right )}{d^{2}}-\frac {e \ln \relax (x ) \ln \left (\left (b x +a \right )^{p}\right )}{d^{2}}+\frac {e \ln \left (\left (b x +a \right )^{p}\right ) \ln \left (e x +d \right )}{d^{2}}-\frac {\ln \relax (c )}{d x}-\frac {\ln \left (\left (b x +a \right )^{p}\right )}{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.65, size = 156, normalized size = 1.07 \[ b p {\left (\frac {{\left (\log \left (\frac {b x}{a} + 1\right ) \log \relax (x) + {\rm Li}_2\left (-\frac {b x}{a}\right )\right )} e}{b d^{2}} - \frac {{\left (\log \left (e x + d\right ) \log \left (-\frac {b e x + b d}{b d - a e} + 1\right ) + {\rm Li}_2\left (\frac {b e x + b d}{b d - a e}\right )\right )} e}{b d^{2}} - \frac {\log \left (b x + a\right )}{a d} + \frac {\log \relax (x)}{a d}\right )} + {\left (\frac {e \log \left (e x + d\right )}{d^{2}} - \frac {e \log \relax (x)}{d^{2}} - \frac {1}{d x}\right )} \log \left ({\left (b x + a\right )}^{p} c\right ) \]
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 \[ \int \frac {\ln \left (c\,{\left (a+b\,x\right )}^p\right )}{x^2\,\left (d+e\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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