Optimal. Leaf size=91 \[ -\frac{d p \text{PolyLog}\left (2,-\frac{e (a+b x)}{b d-a e}\right )}{e^2}-\frac{d \log \left (c (a+b x)^p\right ) \log \left (\frac{b (d+e x)}{b d-a e}\right )}{e^2}+\frac{(a+b x) \log \left (c (a+b x)^p\right )}{b e}-\frac{p x}{e} \]
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Rubi [A] time = 0.107653, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368, Rules used = {43, 2416, 2389, 2295, 2394, 2393, 2391} \[ -\frac{d p \text{PolyLog}\left (2,-\frac{e (a+b x)}{b d-a e}\right )}{e^2}-\frac{d \log \left (c (a+b x)^p\right ) \log \left (\frac{b (d+e x)}{b d-a e}\right )}{e^2}+\frac{(a+b x) \log \left (c (a+b x)^p\right )}{b e}-\frac{p x}{e} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2416
Rule 2389
Rule 2295
Rule 2394
Rule 2393
Rule 2391
Rubi steps
\begin{align*} \int \frac{x \log \left (c (a+b x)^p\right )}{d+e x} \, dx &=\int \left (\frac{\log \left (c (a+b x)^p\right )}{e}-\frac{d \log \left (c (a+b x)^p\right )}{e (d+e x)}\right ) \, dx\\ &=\frac{\int \log \left (c (a+b x)^p\right ) \, dx}{e}-\frac{d \int \frac{\log \left (c (a+b x)^p\right )}{d+e x} \, dx}{e}\\ &=-\frac{d \log \left (c (a+b x)^p\right ) \log \left (\frac{b (d+e x)}{b d-a e}\right )}{e^2}+\frac{\operatorname{Subst}\left (\int \log \left (c x^p\right ) \, dx,x,a+b x\right )}{b e}+\frac{(b d p) \int \frac{\log \left (\frac{b (d+e x)}{b d-a e}\right )}{a+b x} \, dx}{e^2}\\ &=-\frac{p x}{e}+\frac{(a+b x) \log \left (c (a+b x)^p\right )}{b e}-\frac{d \log \left (c (a+b x)^p\right ) \log \left (\frac{b (d+e x)}{b d-a e}\right )}{e^2}+\frac{(d p) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{e x}{b d-a e}\right )}{x} \, dx,x,a+b x\right )}{e^2}\\ &=-\frac{p x}{e}+\frac{(a+b x) \log \left (c (a+b x)^p\right )}{b e}-\frac{d \log \left (c (a+b x)^p\right ) \log \left (\frac{b (d+e x)}{b d-a e}\right )}{e^2}-\frac{d p \text{Li}_2\left (-\frac{e (a+b x)}{b d-a e}\right )}{e^2}\\ \end{align*}
Mathematica [A] time = 0.0327032, size = 79, normalized size = 0.87 \[ \frac{-b d p \text{PolyLog}\left (2,\frac{e (a+b x)}{a e-b d}\right )+\log \left (c (a+b x)^p\right ) \left (-b d \log \left (\frac{b (d+e x)}{b d-a e}\right )+a e+b e x\right )-b e p x}{b e^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.622, size = 427, normalized size = 4.7 \begin{align*}{\frac{\ln \left ( \left ( bx+a \right ) ^{p} \right ) x}{e}}-{\frac{\ln \left ( \left ( bx+a \right ) ^{p} \right ) d\ln \left ( ex+d \right ) }{{e}^{2}}}-{\frac{px}{e}}-{\frac{dp}{{e}^{2}}}+{\frac{ap\ln \left ( b \left ( ex+d \right ) +ae-bd \right ) }{be}}+{\frac{dp}{{e}^{2}}{\it dilog} \left ({\frac{b \left ( ex+d \right ) +ae-bd}{ae-bd}} \right ) }+{\frac{dp\ln \left ( ex+d \right ) }{{e}^{2}}\ln \left ({\frac{b \left ( ex+d \right ) +ae-bd}{ae-bd}} \right ) }+{\frac{{\frac{i}{2}}\pi \,{\it csgn} \left ( ic \right ){\it csgn} \left ( i \left ( bx+a \right ) ^{p} \right ){\it csgn} \left ( ic \left ( bx+a \right ) ^{p} \right ) d\ln \left ( ex+d \right ) }{{e}^{2}}}-{\frac{{\frac{i}{2}}\pi \,{\it csgn} \left ( ic \right ) \left ({\it csgn} \left ( ic \left ( bx+a \right ) ^{p} \right ) \right ) ^{2}d\ln \left ( ex+d \right ) }{{e}^{2}}}-{\frac{{\frac{i}{2}}\pi \, \left ({\it csgn} \left ( ic \left ( bx+a \right ) ^{p} \right ) \right ) ^{3}x}{e}}-{\frac{{\frac{i}{2}}\pi \,{\it csgn} \left ( ic \right ){\it csgn} \left ( i \left ( bx+a \right ) ^{p} \right ){\it csgn} \left ( ic \left ( bx+a \right ) ^{p} \right ) x}{e}}+{\frac{{\frac{i}{2}}\pi \,{\it csgn} \left ( ic \right ) \left ({\it csgn} \left ( ic \left ( bx+a \right ) ^{p} \right ) \right ) ^{2}x}{e}}+{\frac{{\frac{i}{2}}\pi \, \left ({\it csgn} \left ( ic \left ( bx+a \right ) ^{p} \right ) \right ) ^{3}d\ln \left ( ex+d \right ) }{{e}^{2}}}-{\frac{{\frac{i}{2}}\pi \,{\it csgn} \left ( i \left ( bx+a \right ) ^{p} \right ) \left ({\it csgn} \left ( ic \left ( bx+a \right ) ^{p} \right ) \right ) ^{2}d\ln \left ( ex+d \right ) }{{e}^{2}}}+{\frac{{\frac{i}{2}}\pi \,{\it csgn} \left ( i \left ( bx+a \right ) ^{p} \right ) \left ({\it csgn} \left ( ic \left ( bx+a \right ) ^{p} \right ) \right ) ^{2}x}{e}}+{\frac{\ln \left ( c \right ) x}{e}}-{\frac{\ln \left ( c \right ) d\ln \left ( ex+d \right ) }{{e}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \log \left ({\left (b x + a\right )}^{p} c\right )}{e x + d}\,{d x} \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{x \log \left ({\left (b x + a\right )}^{p} c\right )}{e x + d}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \log{\left (c \left (a + b x\right )^{p} \right )}}{d + e x}\, dx \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{x \log \left ({\left (b x + a\right )}^{p} c\right )}{e x + d}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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