Optimal. Leaf size=74 \[ \frac {b e n \log (d+e x)}{g (e f-d g)}-\frac {a+b \log \left (c (d+e x)^n\right )}{g (f+g x)}-\frac {b e n \log (f+g x)}{g (e f-d g)} \]
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Rubi [A]
time = 0.02, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {2442, 36, 31}
\begin {gather*} -\frac {a+b \log \left (c (d+e x)^n\right )}{g (f+g x)}+\frac {b e n \log (d+e x)}{g (e f-d g)}-\frac {b e n \log (f+g x)}{g (e f-d g)} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 36
Rule 2442
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c (d+e x)^n\right )}{(f+g x)^2} \, dx &=-\frac {a+b \log \left (c (d+e x)^n\right )}{g (f+g x)}+\frac {(b e n) \int \frac {1}{(d+e x) (f+g x)} \, dx}{g}\\ &=-\frac {a+b \log \left (c (d+e x)^n\right )}{g (f+g x)}-\frac {(b e n) \int \frac {1}{f+g x} \, dx}{e f-d g}+\frac {\left (b e^2 n\right ) \int \frac {1}{d+e x} \, dx}{g (e f-d g)}\\ &=\frac {b e n \log (d+e x)}{g (e f-d g)}-\frac {a+b \log \left (c (d+e x)^n\right )}{g (f+g x)}-\frac {b e n \log (f+g x)}{g (e f-d g)}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 57, normalized size = 0.77 \begin {gather*} \frac {-\frac {a+b \log \left (c (d+e x)^n\right )}{f+g x}+\frac {b e n (\log (d+e x)-\log (f+g x))}{e f-d g}}{g} \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 = 0.38, size = 354, normalized size = 4.78
method | result | size |
risch | \(-\frac {b \ln \left (\left (e x +d \right )^{n}\right )}{g \left (g x +f \right )}-\frac {i \pi b e f \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}-i \pi b e f \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}-i \pi b d g \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}+i \pi b d g \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}-i \pi b d g \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )-i \pi b e f \,\mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}+i \pi b d g \,\mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}+i \pi b e f \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )-2 \ln \left (-g x -f \right ) b e g n x +2 \ln \left (e x +d \right ) b e g n x -2 \ln \left (-g x -f \right ) b e f n +2 \ln \left (e x +d \right ) b e f n +2 \ln \left (c \right ) b d g -2 \ln \left (c \right ) b e f +2 a d g -2 a e f}{2 \left (g x +f \right ) g \left (d g -e f \right )}\) | \(354\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 90, normalized size = 1.22 \begin {gather*} b n {\left (\frac {\log \left (g x + f\right )}{d g^{2} - f g e} - \frac {\log \left (x e + d\right )}{d g^{2} - f g e}\right )} e - \frac {b \log \left ({\left (x e + d\right )}^{n} c\right )}{g^{2} x + f g} - \frac {a}{g^{2} x + f g} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 99, normalized size = 1.34 \begin {gather*} -\frac {a d g - a f e - {\left (b g n x + b f n\right )} e \log \left (g x + f\right ) + {\left (b g n x e + b d g n\right )} \log \left (x e + d\right ) + {\left (b d g - b f e\right )} \log \left (c\right )}{d g^{3} x + d f g^{2} - {\left (f g^{2} x + f^{2} g\right )} e} \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 = 5.81, size = 111, normalized size = 1.50 \begin {gather*} \frac {b g n x e \log \left (g x + f\right ) - b g n x e \log \left (x e + d\right ) + b f n e \log \left (g x + f\right ) - b d g n \log \left (x e + d\right ) - b d g \log \left (c\right ) + b f e \log \left (c\right ) - a d g + a f e}{d g^{3} x - f g^{2} x e + d f g^{2} - f^{2} g e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.14, size = 84, normalized size = 1.14 \begin {gather*} -\frac {a}{x\,g^2+f\,g}-\frac {b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )}{g\,\left (f+g\,x\right )}+\frac {b\,e\,n\,\mathrm {atan}\left (\frac {e\,f\,2{}\mathrm {i}+e\,g\,x\,2{}\mathrm {i}}{d\,g-e\,f}+1{}\mathrm {i}\right )\,2{}\mathrm {i}}{g\,\left (d\,g-e\,f\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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