Optimal. Leaf size=240 \[ \frac {b e n \log (x)}{d f^2}-\frac {b e n \log (d+e x)}{d f^2}+\frac {b e g n \log (d+e x)}{f^2 (e f-d g)}-\frac {a+b \log \left (c (d+e x)^n\right )}{f^2 x}-\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^2 (f+g x)}-\frac {2 g \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^3}-\frac {b e g n \log (f+g x)}{f^2 (e f-d g)}+\frac {2 g \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{f^3}+\frac {2 b g n \text {Li}_2\left (-\frac {g (d+e x)}{e f-d g}\right )}{f^3}-\frac {2 b g n \text {Li}_2\left (1+\frac {e x}{d}\right )}{f^3} \]
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Rubi [A]
time = 0.18, antiderivative size = 240, normalized size of antiderivative = 1.00, number of steps
used = 15, number of rules used = 10, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {46, 2463,
2442, 36, 29, 31, 2441, 2352, 2440, 2438} \begin {gather*} \frac {2 b g n \text {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{f^3}-\frac {2 b g n \text {PolyLog}\left (2,\frac {e x}{d}+1\right )}{f^3}-\frac {2 g \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^3}+\frac {2 g \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^3}-\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^2 (f+g x)}-\frac {a+b \log \left (c (d+e x)^n\right )}{f^2 x}+\frac {b e g n \log (d+e x)}{f^2 (e f-d g)}-\frac {b e g n \log (f+g x)}{f^2 (e f-d g)}+\frac {b e n \log (x)}{d f^2}-\frac {b e n \log (d+e x)}{d f^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 29
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^2 (f+g x)^2} \, dx &=\int \left (\frac {a+b \log \left (c (d+e x)^n\right )}{f^2 x^2}-\frac {2 g \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^3 x}+\frac {g^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^2 (f+g x)^2}+\frac {2 g^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^3 (f+g x)}\right ) \, dx\\ &=\frac {\int \frac {a+b \log \left (c (d+e x)^n\right )}{x^2} \, dx}{f^2}-\frac {(2 g) \int \frac {a+b \log \left (c (d+e x)^n\right )}{x} \, dx}{f^3}+\frac {\left (2 g^2\right ) \int \frac {a+b \log \left (c (d+e x)^n\right )}{f+g x} \, dx}{f^3}+\frac {g^2 \int \frac {a+b \log \left (c (d+e x)^n\right )}{(f+g x)^2} \, dx}{f^2}\\ &=-\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^2 (f+g x)}-\frac {2 g \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^3}+\frac {2 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 )}{f^3}+\frac {(b e n) \int \frac {1}{x (d+e x)} \, dx}{f^2}+\frac {(2 b e g n) \int \frac {\log \left (-\frac {e x}{d}\right )}{d+e x} \, dx}{f^3}-\frac {(2 b e g n) \int \frac {\log \left (\frac {e (f+g x)}{e f-d g}\right )}{d+e x} \, dx}{f^3}+\frac {(b e g n) \int \frac {1}{(d+e x) (f+g x)} \, dx}{f^2}\\ &=-\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^2 (f+g x)}-\frac {2 g \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^3}+\frac {2 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 )}{f^3}-\frac {2 b g n \text {Li}_2\left (1+\frac {e x}{d}\right )}{f^3}+\frac {(b e n) \int \frac {1}{x} \, dx}{d f^2}-\frac {\left (b e^2 n\right ) \int \frac {1}{d+e x} \, dx}{d f^2}-\frac {(2 b g 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^3}+\frac {\left (b e^2 g n\right ) \int \frac {1}{d+e x} \, dx}{f^2 (e f-d g)}-\frac {\left (b e g^2 n\right ) \int \frac {1}{f+g x} \, dx}{f^2 (e f-d g)}\\ &=\frac {b e n \log (x)}{d f^2}-\frac {b e n \log (d+e x)}{d f^2}+\frac {b e g n \log (d+e x)}{f^2 (e f-d g)}-\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^2 (f+g x)}-\frac {2 g \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^3}-\frac {b e g n \log (f+g x)}{f^2 (e f-d g)}+\frac {2 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 )}{f^3}+\frac {2 b g n \text {Li}_2\left (-\frac {g (d+e x)}{e f-d g}\right )}{f^3}-\frac {2 b g n \text {Li}_2\left (1+\frac {e x}{d}\right )}{f^3}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 199, normalized size = 0.83 \begin {gather*} \frac {\frac {b e f n (\log (x)-\log (d+e x))}{d}-\frac {f \left (a+b \log \left (c (d+e x)^n\right )\right )}{x}-\frac {f g \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x}-2 g \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac {b e f g n (\log (d+e x)-\log (f+g x))}{e f-d g}+2 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 )+2 b g n \text {Li}_2\left (\frac {g (d+e x)}{-e f+d g}\right )-2 b g n \text {Li}_2\left (1+\frac {e x}{d}\right )}{f^3} \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 = 0.38, size = 936, normalized size = 3.90
method | result | size |
risch | \(\text {Expression too large to display}\) | \(936\) |
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a + b \log {\left (c \left (d + e x\right )^{n} \right )}}{x^{2} \left (f + g x\right )^{2}}\, dx \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 = -0.00 \begin {gather*} \int \frac {a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )}{x^2\,{\left (f+g\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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