Optimal. Leaf size=74 \[ -\frac {a+b \log \left (c (d+e x)^n\right )}{g (f+g x)}+\frac {b e n \log (d+e x)}{g (e f-d g)}-\frac {b e n \log (f+g x)}{g (e f-d g)} \]
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Rubi [A] time = 0.03, 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 = {2395, 36, 31} \[ -\frac {a+b \log \left (c (d+e x)^n\right )}{g (f+g x)}+\frac {b e n \log (d+e x)}{g (e f-d g)}-\frac {b e n \log (f+g x)}{g (e f-d g)} \]
Antiderivative was successfully verified.
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Rule 31
Rule 36
Rule 2395
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 = 0.06, size = 57, normalized size = 0.77 \[ \frac {\frac {b e n (\log (d+e x)-\log (f+g x))}{e f-d g}-\frac {a+b \log \left (c (d+e x)^n\right )}{f+g x}}{g} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 95, normalized size = 1.28 \[ -\frac {a e f - a d g - {\left (b e g n x + b d g n\right )} \log \left (e x + d\right ) + {\left (b e g n x + b e f n\right )} \log \left (g x + f\right ) + {\left (b e f - b d g\right )} \log \relax (c)}{e f^{2} g - d f g^{2} + {\left (e f g^{2} - d g^{3}\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 111, normalized size = 1.50 \[ \frac {b g n x e \log \left (g x + f\right ) - b g n x e \log \left (x e + d\right ) + b f n e \log \left (g x + f\right ) - b d g n \log \left (x e + d\right ) - b d g \log \relax (c) + b f e \log \relax (c) - a d g + a f e}{d g^{3} x - f g^{2} x e + d f g^{2} - f^{2} g e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.34, size = 354, normalized size = 4.78 \[ -\frac {b \ln \left (\left (e x +d \right )^{n}\right )}{\left (g x +f \right ) g}-\frac {-i \pi b d g \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )+i \pi b d g \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}+i \pi b d g \,\mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}-i \pi b d g \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}+i \pi b e f \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )-i \pi b e f \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}-i \pi b e f \,\mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}+i \pi b e f \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}+2 b e g n x \ln \left (e x +d \right )-2 b e g n x \ln \left (-g x -f \right )+2 b e f n \ln \left (e x +d \right )-2 b e f n \ln \left (-g x -f \right )+2 b d g \ln \relax (c )-2 b e f \ln \relax (c )+2 a d g -2 a e f}{2 \left (g x +f \right ) \left (d g -e f \right ) g} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.01, size = 85, normalized size = 1.15 \[ b e n {\left (\frac {\log \left (e x + d\right )}{e f g - d g^{2}} - \frac {\log \left (g x + f\right )}{e f g - d g^{2}}\right )} - \frac {b \log \left ({\left (e x + d\right )}^{n} c\right )}{g^{2} x + f g} - \frac {a}{g^{2} x + f g} \]
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 \[ -\frac {a}{x\,g^2+f\,g}-\frac {b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )}{g\,\left (f+g\,x\right )}+\frac {b\,e\,n\,\mathrm {atan}\left (\frac {e\,f\,2{}\mathrm {i}+e\,g\,x\,2{}\mathrm {i}}{d\,g-e\,f}+1{}\mathrm {i}\right )\,2{}\mathrm {i}}{g\,\left (d\,g-e\,f\right )} \]
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 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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