Optimal. Leaf size=335 \[ \frac {3 g^2 \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^4}-\frac {3 g^2 \log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^4}+\frac {g^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^3 (f+g x)}+\frac {2 g \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^3 x}-\frac {a+b \log \left (c (d+e x)^n\right )}{2 f^2 x^2}-\frac {b e^2 n \log (x)}{2 d^2 f^2}+\frac {b e^2 n \log (d+e x)}{2 d^2 f^2}-\frac {3 b g^2 n \text {Li}_2\left (-\frac {g (d+e x)}{e f-d g}\right )}{f^4}+\frac {3 b g^2 n \text {Li}_2\left (\frac {e x}{d}+1\right )}{f^4}-\frac {b e g^2 n \log (d+e x)}{f^3 (e f-d g)}+\frac {b e g^2 n \log (f+g x)}{f^3 (e f-d g)}-\frac {2 b e g n \log (x)}{d f^3}+\frac {2 b e g n \log (d+e x)}{d f^3}-\frac {b e n}{2 d f^2 x} \]
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Rubi [A] time = 0.31, antiderivative size = 335, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 10, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {44, 2416, 2395, 36, 29, 31, 2394, 2315, 2393, 2391} \[ -\frac {3 b g^2 n \text {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{f^4}+\frac {3 b g^2 n \text {PolyLog}\left (2,\frac {e x}{d}+1\right )}{f^4}+\frac {3 g^2 \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^4}+\frac {g^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^3 (f+g x)}-\frac {3 g^2 \log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^4}+\frac {2 g \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^3 x}-\frac {a+b \log \left (c (d+e x)^n\right )}{2 f^2 x^2}-\frac {b e^2 n \log (x)}{2 d^2 f^2}+\frac {b e^2 n \log (d+e x)}{2 d^2 f^2}-\frac {b e g^2 n \log (d+e x)}{f^3 (e f-d g)}+\frac {b e g^2 n \log (f+g x)}{f^3 (e f-d g)}-\frac {2 b e g n \log (x)}{d f^3}+\frac {2 b e g n \log (d+e x)}{d f^3}-\frac {b e n}{2 d f^2 x} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 44
Rule 2315
Rule 2391
Rule 2393
Rule 2394
Rule 2395
Rule 2416
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c (d+e x)^n\right )}{x^3 (f+g x)^2} \, dx &=\int \left (\frac {a+b \log \left (c (d+e x)^n\right )}{f^2 x^3}-\frac {2 g \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^3 x^2}+\frac {3 g^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^4 x}-\frac {g^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^3 (f+g x)^2}-\frac {3 g^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^4 (f+g x)}\right ) \, dx\\ &=\frac {\int \frac {a+b \log \left (c (d+e x)^n\right )}{x^3} \, dx}{f^2}-\frac {(2 g) \int \frac {a+b \log \left (c (d+e x)^n\right )}{x^2} \, dx}{f^3}+\frac {\left (3 g^2\right ) \int \frac {a+b \log \left (c (d+e x)^n\right )}{x} \, dx}{f^4}-\frac {\left (3 g^3\right ) \int \frac {a+b \log \left (c (d+e x)^n\right )}{f+g x} \, dx}{f^4}-\frac {g^3 \int \frac {a+b \log \left (c (d+e x)^n\right )}{(f+g x)^2} \, dx}{f^3}\\ &=-\frac {a+b \log \left (c (d+e x)^n\right )}{2 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^3 (f+g x)}+\frac {3 g^2 \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^4}-\frac {3 g^2 \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^4}+\frac {(b e n) \int \frac {1}{x^2 (d+e x)} \, dx}{2 f^2}-\frac {(2 b e g n) \int \frac {1}{x (d+e x)} \, dx}{f^3}-\frac {\left (3 b e g^2 n\right ) \int \frac {\log \left (-\frac {e x}{d}\right )}{d+e x} \, dx}{f^4}+\frac {\left (3 b e g^2 n\right ) \int \frac {\log \left (\frac {e (f+g x)}{e f-d g}\right )}{d+e x} \, dx}{f^4}-\frac {\left (b e g^2 n\right ) \int \frac {1}{(d+e x) (f+g x)} \, dx}{f^3}\\ &=-\frac {a+b \log \left (c (d+e x)^n\right )}{2 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^3 (f+g x)}+\frac {3 g^2 \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^4}-\frac {3 g^2 \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^4}+\frac {3 b g^2 n \text {Li}_2\left (1+\frac {e x}{d}\right )}{f^4}+\frac {(b e n) \int \left (\frac {1}{d x^2}-\frac {e}{d^2 x}+\frac {e^2}{d^2 (d+e x)}\right ) \, dx}{2 f^2}-\frac {(2 b e g n) \int \frac {1}{x} \, dx}{d f^3}+\frac {\left (2 b e^2 g n\right ) \int \frac {1}{d+e x} \, dx}{d f^3}+\frac {\left (3 b g^2 n\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {g x}{e f-d g}\right )}{x} \, dx,x,d+e x\right )}{f^4}-\frac {\left (b e^2 g^2 n\right ) \int \frac {1}{d+e x} \, dx}{f^3 (e f-d g)}+\frac {\left (b e g^3 n\right ) \int \frac {1}{f+g x} \, dx}{f^3 (e f-d g)}\\ &=-\frac {b e n}{2 d f^2 x}-\frac {b e^2 n \log (x)}{2 d^2 f^2}-\frac {2 b e g n \log (x)}{d f^3}+\frac {b e^2 n \log (d+e x)}{2 d^2 f^2}+\frac {2 b e g n \log (d+e x)}{d f^3}-\frac {b e g^2 n \log (d+e x)}{f^3 (e f-d g)}-\frac {a+b \log \left (c (d+e x)^n\right )}{2 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^3 (f+g x)}+\frac {3 g^2 \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^4}+\frac {b e g^2 n \log (f+g x)}{f^3 (e f-d g)}-\frac {3 g^2 \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^4}-\frac {3 b g^2 n \text {Li}_2\left (-\frac {g (d+e x)}{e f-d g}\right )}{f^4}+\frac {3 b g^2 n \text {Li}_2\left (1+\frac {e x}{d}\right )}{f^4}\\ \end {align*}
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Mathematica [A] time = 0.50, size = 269, normalized size = 0.80 \[ -\frac {\frac {f^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{x^2}-\frac {2 f g^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x}+6 g^2 \log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac {4 f g \left (a+b \log \left (c (d+e x)^n\right )\right )}{x}-6 g^2 \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac {b e f^2 n (-e x \log (d+e x)+d+e x \log (x))}{d^2 x}+6 b g^2 n \text {Li}_2\left (\frac {g (d+e x)}{d g-e f}\right )+\frac {2 b e f g^2 n (\log (d+e x)-\log (f+g x))}{e f-d g}+\frac {4 b e f g n (\log (x)-\log (d+e x))}{d}-6 b g^2 n \text {Li}_2\left (\frac {e x}{d}+1\right )}{2 f^4} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}{g^{2} x^{5} + 2 \, f g x^{4} + f^{2} x^{3}}, x\right ) \]
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 \[ \int \frac {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}{{\left (g x + f\right )}^{2} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.26, size = 1224, normalized size = 3.65 \[ \text {result too large to display} \]
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 \[ \frac {1}{2} \, a {\left (\frac {6 \, g^{2} x^{2} + 3 \, f g x - f^{2}}{f^{3} g x^{3} + f^{4} x^{2}} - \frac {6 \, g^{2} \log \left (g x + f\right )}{f^{4}} + \frac {6 \, g^{2} \log \relax (x)}{f^{4}}\right )} + b \int \frac {\log \left ({\left (e x + d\right )}^{n}\right ) + \log \relax (c)}{g^{2} x^{5} + 2 \, f g x^{4} + f^{2} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )}{x^3\,{\left (f+g\,x\right )}^2} \,d x \]
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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