Optimal. Leaf size=179 \[ -\frac {b e n \log (d+e x)}{f (e f-d g)}+\frac {a+b \log \left (c (d+e x)^n\right )}{f (f+g x)}+\frac {\log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^2}+\frac {b e n \log (f+g x)}{f (e f-d g)}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{f^2}-\frac {b n \text {Li}_2\left (-\frac {g (d+e x)}{e f-d g}\right )}{f^2}+\frac {b n \text {Li}_2\left (1+\frac {e x}{d}\right )}{f^2} \]
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Rubi [A]
time = 0.15, antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 9, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {46, 2463,
2441, 2352, 2442, 36, 31, 2440, 2438} \begin {gather*} -\frac {b n \text {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{f^2}+\frac {b n \text {PolyLog}\left (2,\frac {e x}{d}+1\right )}{f^2}-\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 )}{f^2}+\frac {\log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^2}+\frac {a+b \log \left (c (d+e x)^n\right )}{f (f+g x)}-\frac {b e n \log (d+e x)}{f (e f-d g)}+\frac {b e n \log (f+g x)}{f (e f-d g)} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 36
Rule 46
Rule 2352
Rule 2438
Rule 2440
Rule 2441
Rule 2442
Rule 2463
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c (d+e x)^n\right )}{x (f+g x)^2} \, dx &=\int \left (\frac {a+b \log \left (c (d+e x)^n\right )}{f^2 x}-\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )}{f (f+g x)^2}-\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^2 (f+g x)}\right ) \, dx\\ &=\frac {\int \frac {a+b \log \left (c (d+e x)^n\right )}{x} \, dx}{f^2}-\frac {g \int \frac {a+b \log \left (c (d+e x)^n\right )}{f+g x} \, dx}{f^2}-\frac {g \int \frac {a+b \log \left (c (d+e x)^n\right )}{(f+g x)^2} \, dx}{f}\\ &=\frac {a+b \log \left (c (d+e x)^n\right )}{f (f+g x)}+\frac {\log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^2}-\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 )}{f^2}-\frac {(b e n) \int \frac {\log \left (-\frac {e x}{d}\right )}{d+e x} \, dx}{f^2}+\frac {(b e n) \int \frac {\log \left (\frac {e (f+g x)}{e f-d g}\right )}{d+e x} \, dx}{f^2}-\frac {(b e n) \int \frac {1}{(d+e x) (f+g x)} \, dx}{f}\\ &=\frac {a+b \log \left (c (d+e x)^n\right )}{f (f+g x)}+\frac {\log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^2}-\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 )}{f^2}+\frac {b n \text {Li}_2\left (1+\frac {e x}{d}\right )}{f^2}+\frac {(b n) \text {Subst}\left (\int \frac {\log \left (1+\frac {g x}{e f-d g}\right )}{x} \, dx,x,d+e x\right )}{f^2}-\frac {\left (b e^2 n\right ) \int \frac {1}{d+e x} \, dx}{f (e f-d g)}+\frac {(b e g n) \int \frac {1}{f+g x} \, dx}{f (e f-d g)}\\ &=-\frac {b e n \log (d+e x)}{f (e f-d g)}+\frac {a+b \log \left (c (d+e x)^n\right )}{f (f+g x)}+\frac {\log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^2}+\frac {b e n \log (f+g x)}{f (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 )}{f^2}-\frac {b n \text {Li}_2\left (-\frac {g (d+e x)}{e f-d g}\right )}{f^2}+\frac {b n \text {Li}_2\left (1+\frac {e x}{d}\right )}{f^2}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 152, normalized size = 0.85 \begin {gather*} \frac {\frac {f \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x}+\log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac {b e f n (\log (d+e x)-\log (f+g x))}{e f-d g}-\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )-b n \text {Li}_2\left (\frac {g (d+e x)}{-e f+d g}\right )+b n \text {Li}_2\left (1+\frac {e x}{d}\right )}{f^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.33, size = 694, normalized size = 3.88
method | result | size |
risch | \(-\frac {i b \pi \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3} \ln \left (x \right )}{2 f^{2}}+\frac {i b \pi \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3} \ln \left (g x +f \right )}{2 f^{2}}-\frac {i b \pi \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}}{2 f \left (g x +f \right )}-\frac {b e n \ln \left (g x +f \right )}{f \left (d g -e f \right )}-\frac {b \ln \left (\left (e x +d \right )^{n}\right ) \ln \left (g x +f \right )}{f^{2}}+\frac {b \ln \left (\left (e x +d \right )^{n}\right )}{f \left (g x +f \right )}-\frac {b n \dilog \left (\frac {e x +d}{d}\right )}{f^{2}}+\frac {b n \dilog \left (\frac {\left (g x +f \right ) e +d g -e f}{d g -e f}\right )}{f^{2}}+\frac {a \ln \left (x \right )}{f^{2}}+\frac {a}{f \left (g x +f \right )}-\frac {a \ln \left (g x +f \right )}{f^{2}}+\frac {b \ln \left (\left (e x +d \right )^{n}\right ) \ln \left (x \right )}{f^{2}}-\frac {b \ln \left (c \right ) \ln \left (g x +f \right )}{f^{2}}+\frac {b \ln \left (c \right )}{f \left (g x +f \right )}+\frac {b \ln \left (c \right ) \ln \left (x \right )}{f^{2}}-\frac {i b \pi \,\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 f \left (g x +f \right )}-\frac {i b \pi \,\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 (x \right )}{2 f^{2}}+\frac {i b \pi \,\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 f^{2}}-\frac {b n \ln \left (x \right ) \ln \left (\frac {e x +d}{d}\right )}{f^{2}}+\frac {b n \ln \left (g x +f \right ) \ln \left (\frac {\left (g x +f \right ) e +d g -e f}{d g -e f}\right )}{f^{2}}+\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}}{2 f \left (g x +f \right )}+\frac {b e n \ln \left (e x +d \right )}{f \left (d g -e f \right )}-\frac {i b \pi \,\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 f^{2}}+\frac {i b \pi \,\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 (x \right )}{2 f^{2}}-\frac {i b \pi \,\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 f^{2}}+\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} \ln \left (x \right )}{2 f^{2}}+\frac {i b \pi \,\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 f \left (g x +f \right )}\) | \(694\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )}{x\,{\left (f+g\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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