Optimal. Leaf size=138 \[ \frac {f \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^2 (f+g x)}+\frac {\log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^2}+\frac {b n \text {Li}_2\left (-\frac {g (d+e x)}{e f-d g}\right )}{g^2}-\frac {b e f n \log (d+e x)}{g^2 (e f-d g)}+\frac {b e f n \log (f+g x)}{g^2 (e f-d g)} \]
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Rubi [A] time = 0.15, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {43, 2416, 2395, 36, 31, 2394, 2393, 2391} \[ \frac {b n \text {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{g^2}+\frac {f \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^2 (f+g x)}+\frac {\log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^2}-\frac {b e f n \log (d+e x)}{g^2 (e f-d g)}+\frac {b e f n \log (f+g x)}{g^2 (e f-d g)} \]
Antiderivative was successfully verified.
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Rule 31
Rule 36
Rule 43
Rule 2391
Rule 2393
Rule 2394
Rule 2395
Rule 2416
Rubi steps
\begin {align*} \int \frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )}{(f+g x)^2} \, dx &=\int \left (-\frac {f \left (a+b \log \left (c (d+e x)^n\right )\right )}{g (f+g x)^2}+\frac {a+b \log \left (c (d+e x)^n\right )}{g (f+g x)}\right ) \, dx\\ &=\frac {\int \frac {a+b \log \left (c (d+e x)^n\right )}{f+g x} \, dx}{g}-\frac {f \int \frac {a+b \log \left (c (d+e x)^n\right )}{(f+g x)^2} \, dx}{g}\\ &=\frac {f \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^2 (f+g x)}+\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g^2}-\frac {(b e n) \int \frac {\log \left (\frac {e (f+g x)}{e f-d g}\right )}{d+e x} \, dx}{g^2}-\frac {(b e f n) \int \frac {1}{(d+e x) (f+g x)} \, dx}{g^2}\\ &=\frac {f \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^2 (f+g x)}+\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g^2}-\frac {(b n) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {g x}{e f-d g}\right )}{x} \, dx,x,d+e x\right )}{g^2}-\frac {\left (b e^2 f n\right ) \int \frac {1}{d+e x} \, dx}{g^2 (e f-d g)}+\frac {(b e f n) \int \frac {1}{f+g x} \, dx}{g (e f-d g)}\\ &=-\frac {b e f n \log (d+e x)}{g^2 (e f-d g)}+\frac {f \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^2 (f+g x)}+\frac {b e f n \log (f+g x)}{g^2 (e f-d g)}+\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g^2}+\frac {b n \text {Li}_2\left (-\frac {g (d+e x)}{e f-d g}\right )}{g^2}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 114, normalized size = 0.83 \[ \frac {\log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac {f \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x}+b n \text {Li}_2\left (\frac {g (d+e x)}{d g-e f}\right )-\frac {b e f n (\log (d+e x)-\log (f+g x))}{e f-d g}}{g^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b x \log \left ({\left (e x + d\right )}^{n} c\right ) + a x}{g^{2} x^{2} + 2 \, f g x + f^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} x}{{\left (g x + f\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.23, size = 519, normalized size = 3.76 \[ \frac {b e f n \ln \left (d g -e f +\left (g x +f \right ) e \right )}{\left (d g -e f \right ) g^{2}}-\frac {b e f n \ln \left (g x +f \right )}{\left (d g -e f \right ) g^{2}}-\frac {i \pi b 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 \left (g x +f \right ) g^{2}}+\frac {i \pi b f \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}}{2 \left (g x +f \right ) g^{2}}+\frac {i \pi b 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}}{2 \left (g x +f \right ) g^{2}}-\frac {i \pi b f \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}}{2 \left (g x +f \right ) g^{2}}-\frac {i \pi b \,\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 ) \ln \left (g x +f \right )}{2 g^{2}}+\frac {i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} \ln \left (g x +f \right )}{2 g^{2}}+\frac {i \pi b \,\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 )}{2 g^{2}}-\frac {i \pi b \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3} \ln \left (g x +f \right )}{2 g^{2}}-\frac {b n \ln \left (\frac {d g -e f +\left (g x +f \right ) e}{d g -e f}\right ) \ln \left (g x +f \right )}{g^{2}}+\frac {b f \ln \relax (c )}{\left (g x +f \right ) g^{2}}+\frac {b f \ln \left (\left (e x +d \right )^{n}\right )}{\left (g x +f \right ) g^{2}}-\frac {b n \dilog \left (\frac {d g -e f +\left (g x +f \right ) e}{d g -e f}\right )}{g^{2}}+\frac {b \ln \relax (c ) \ln \left (g x +f \right )}{g^{2}}+\frac {b \ln \left (\left (e x +d \right )^{n}\right ) \ln \left (g x +f \right )}{g^{2}}+\frac {a f}{\left (g x +f \right ) g^{2}}+\frac {a \ln \left (g x +f \right )}{g^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ a {\left (\frac {f}{g^{3} x + f g^{2}} + \frac {\log \left (g x + f\right )}{g^{2}}\right )} + b \int \frac {x \log \left ({\left (e x + d\right )}^{n}\right ) + x \log \relax (c)}{g^{2} x^{2} + 2 \, f g x + f^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x\,\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}{{\left (f+g\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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