Optimal. Leaf size=68 \[ \frac {b p \log (a+b x)}{e (b d-a e)}-\frac {\log \left (c (a+b x)^p\right )}{e (d+e x)}-\frac {b p \log (d+e x)}{e (b d-a e)} \]
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Rubi [A]
time = 0.02, antiderivative size = 68, normalized size of antiderivative = 1.00, 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 = {2442, 36, 31}
\begin {gather*} -\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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 36
Rule 2442
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*}
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Mathematica [A]
time = 0.04, size = 52, normalized size = 0.76 \begin {gather*} \frac {-\frac {\log \left (c (a+b x)^p\right )}{d+e x}+\frac {b p (\log (a+b x)-\log (d+e x))}{b d-a e}}{e} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.54, size = 329, normalized size = 4.84
method | result | size |
risch | \(-\frac {\ln \left (\left (b x +a \right )^{p}\right )}{e \left (e x +d \right )}-\frac {i \pi a e \,\mathrm {csgn}\left (i \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2}-i \pi a e \,\mathrm {csgn}\left (i \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \right )-i \pi a e \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{3}+i \pi a e \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right )-i \pi b d \,\mathrm {csgn}\left (i \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2}+i \pi b d \,\mathrm {csgn}\left (i \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \right )+i \pi b d \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{3}-i \pi b d \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right )+2 \ln \left (b x +a \right ) b e p x -2 \ln \left (-e x -d \right ) b e p x +2 \ln \left (b x +a \right ) b d p -2 \ln \left (-e x -d \right ) b d p +2 \ln \left (c \right ) a e -2 d b \ln \left (c \right )}{2 \left (e x +d \right ) e \left (a e -b d \right )}\) | \(329\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.36, size = 67, normalized size = 0.99 \begin {gather*} b p {\left (\frac {\log \left (b x + a\right )}{b d - a e} - \frac {\log \left (x e + d\right )}{b d - a e}\right )} e^{\left (-1\right )} - \frac {e^{\left (-1\right )} \log \left ({\left (b x + a\right )}^{p} c\right )}{x e + d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.43, size = 80, normalized size = 1.18 \begin {gather*} \frac {{\left (b p x + a p\right )} e \log \left (b x + a\right ) - {\left (b p x e + b d p\right )} \log \left (x e + d\right ) - {\left (b d - a e\right )} \log \left (c\right )}{b d^{2} e - a x e^{3} + {\left (b d x - a d\right )} e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.24, size = 91, normalized size = 1.34 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.07, size = 70, normalized size = 1.03 \begin {gather*} -\frac {\ln \left (c\,{\left (a+b\,x\right )}^p\right )}{e\,\left (d+e\,x\right )}+\frac {b\,p\,\mathrm {atan}\left (\frac {a\,e\,1{}\mathrm {i}+b\,d\,1{}\mathrm {i}+b\,e\,x\,2{}\mathrm {i}}{a\,e-b\,d}\right )\,2{}\mathrm {i}}{a\,e^2-b\,d\,e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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