Optimal. Leaf size=68 \[ -\frac{\log \left (c (a+b x)^p\right )}{e (d+e x)}+\frac{b p \log (a+b x)}{e (b d-a e)}-\frac{b p \log (d+e x)}{e (b d-a e)} \]
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Rubi [A] time = 0.0274055, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2395, 36, 31} \[ -\frac{\log \left (c (a+b x)^p\right )}{e (d+e x)}+\frac{b p \log (a+b x)}{e (b d-a e)}-\frac{b p \log (d+e x)}{e (b d-a e)} \]
Antiderivative was successfully verified.
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Rule 2395
Rule 36
Rule 31
Rubi steps
\begin{align*} \int \frac{\log \left (c (a+b x)^p\right )}{(d+e x)^2} \, dx &=-\frac{\log \left (c (a+b x)^p\right )}{e (d+e x)}+\frac{(b p) \int \frac{1}{(a+b x) (d+e x)} \, dx}{e}\\ &=-\frac{\log \left (c (a+b x)^p\right )}{e (d+e x)}-\frac{(b p) \int \frac{1}{d+e x} \, dx}{b d-a e}+\frac{\left (b^2 p\right ) \int \frac{1}{a+b x} \, dx}{e (b d-a e)}\\ &=\frac{b p \log (a+b x)}{e (b d-a e)}-\frac{\log \left (c (a+b x)^p\right )}{e (d+e x)}-\frac{b p \log (d+e x)}{e (b d-a e)}\\ \end{align*}
Mathematica [A] time = 0.0482561, size = 52, normalized size = 0.76 \[ \frac{\frac{b p (\log (a+b x)-\log (d+e x))}{b d-a e}-\frac{\log \left (c (a+b x)^p\right )}{d+e x}}{e} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.358, size = 329, normalized size = 4.8 \begin{align*} -{\frac{\ln \left ( \left ( bx+a \right ) ^{p} \right ) }{ \left ( ex+d \right ) e}}-{\frac{-i\pi \,ae{\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 ) +i\pi \,ae{\it csgn} \left ( ic \right ) \left ({\it csgn} \left ( ic \left ( bx+a \right ) ^{p} \right ) \right ) ^{2}+i\pi \,ae{\it csgn} \left ( i \left ( bx+a \right ) ^{p} \right ) \left ({\it csgn} \left ( ic \left ( bx+a \right ) ^{p} \right ) \right ) ^{2}-i\pi \,ae \left ({\it csgn} \left ( ic \left ( bx+a \right ) ^{p} \right ) \right ) ^{3}+i\pi \,bd{\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 ) -i\pi \,bd{\it csgn} \left ( ic \right ) \left ({\it csgn} \left ( ic \left ( bx+a \right ) ^{p} \right ) \right ) ^{2}-i\pi \,bd{\it csgn} \left ( i \left ( bx+a \right ) ^{p} \right ) \left ({\it csgn} \left ( ic \left ( bx+a \right ) ^{p} \right ) \right ) ^{2}+i\pi \,bd \left ({\it csgn} \left ( ic \left ( bx+a \right ) ^{p} \right ) \right ) ^{3}-2\,\ln \left ( -ex-d \right ) bepx+2\,\ln \left ( bx+a \right ) bepx-2\,\ln \left ( -ex-d \right ) bdp+2\,\ln \left ( bx+a \right ) bdp+2\,\ln \left ( c \right ) ae-2\,\ln \left ( c \right ) bd}{ \left ( 2\,ex+2\,d \right ) e \left ( ae-bd \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01787, size = 88, normalized size = 1.29 \begin{align*} \frac{b p{\left (\frac{\log \left (b x + a\right )}{b d - a e} - \frac{\log \left (e x + d\right )}{b d - a e}\right )}}{e} - \frac{\log \left ({\left (b x + a\right )}^{p} c\right )}{{\left (e x + d\right )} e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.96902, size = 176, normalized size = 2.59 \begin{align*} \frac{{\left (b e p x + a e p\right )} \log \left (b x + a\right ) -{\left (b e p x + b d p\right )} \log \left (e x + d\right ) -{\left (b d - a e\right )} \log \left (c\right )}{b d^{2} e - a d e^{2} +{\left (b d e^{2} - a e^{3}\right )} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19767, size = 123, normalized size = 1.81 \begin{align*} \frac{b p x e \log \left (b x + a\right ) - b p x e \log \left (x e + d\right ) + a p e \log \left (b x + a\right ) - b d p \log \left (x e + d\right ) - b d \log \left (c\right ) + a e \log \left (c\right )}{b d x e^{2} + b d^{2} e - a x e^{3} - a d e^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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