Optimal. Leaf size=342 \[ \frac {3 b^2 e^2 n^2 \text {Li}_2\left (-\frac {e f-d g}{g (d+e x)}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g (e f-d g)^2}+\frac {3 b^2 e^2 n^2 \log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g (e f-d g)^2}-\frac {3 b e^2 n \log \left (\frac {e f-d g}{g (d+e x)}+1\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g (e f-d g)^2}-\frac {3 b e n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 (f+g x) (e f-d g)^2}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{2 g (f+g x)^2}+\frac {3 b^3 e^2 n^3 \text {Li}_2\left (-\frac {g (d+e x)}{e f-d g}\right )}{g (e f-d g)^2}+\frac {3 b^3 e^2 n^3 \text {Li}_3\left (-\frac {e f-d g}{g (d+e x)}\right )}{g (e f-d g)^2} \]
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Rubi [A] time = 0.62, antiderivative size = 370, normalized size of antiderivative = 1.08, number of steps used = 12, number of rules used = 11, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {2398, 2411, 2347, 2344, 2302, 30, 2317, 2374, 6589, 2318, 2391} \[ -\frac {3 b^2 e^2 n^2 \text {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g (e f-d g)^2}+\frac {3 b^3 e^2 n^3 \text {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{g (e f-d g)^2}+\frac {3 b^3 e^2 n^3 \text {PolyLog}\left (3,-\frac {g (d+e x)}{e f-d g}\right )}{g (e f-d g)^2}+\frac {3 b^2 e^2 n^2 \log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g (e f-d g)^2}-\frac {3 b e^2 n \log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g (e f-d g)^2}+\frac {e^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{2 g (e f-d g)^2}-\frac {3 b e n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 (f+g x) (e f-d g)^2}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{2 g (f+g x)^2} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2302
Rule 2317
Rule 2318
Rule 2344
Rule 2347
Rule 2374
Rule 2391
Rule 2398
Rule 2411
Rule 6589
Rubi steps
\begin {align*} \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(f+g x)^3} \, dx &=-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{2 g (f+g x)^2}+\frac {(3 b e n) \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(d+e x) (f+g x)^2} \, dx}{2 g}\\ &=-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{2 g (f+g x)^2}+\frac {(3 b n) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x \left (\frac {e f-d g}{e}+\frac {g x}{e}\right )^2} \, dx,x,d+e x\right )}{2 g}\\ &=-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{2 g (f+g x)^2}-\frac {(3 b n) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{\left (\frac {e f-d g}{e}+\frac {g x}{e}\right )^2} \, dx,x,d+e x\right )}{2 (e f-d g)}+\frac {(3 b e n) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x \left (\frac {e f-d g}{e}+\frac {g x}{e}\right )} \, dx,x,d+e x\right )}{2 g (e f-d g)}\\ &=-\frac {3 b e n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 (e f-d g)^2 (f+g x)}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{2 g (f+g x)^2}-\frac {(3 b e n) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{\frac {e f-d g}{e}+\frac {g x}{e}} \, dx,x,d+e x\right )}{2 (e f-d g)^2}+\frac {\left (3 b e^2 n\right ) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx,x,d+e x\right )}{2 g (e f-d g)^2}+\frac {\left (3 b^2 e n^2\right ) \operatorname {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{\frac {e f-d g}{e}+\frac {g x}{e}} \, dx,x,d+e x\right )}{(e f-d g)^2}\\ &=-\frac {3 b e n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 (e f-d g)^2 (f+g x)}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{2 g (f+g x)^2}+\frac {3 b^2 e^2 n^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 )}{g (e f-d g)^2}-\frac {3 b e^2 n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e (f+g x)}{e f-d g}\right )}{2 g (e f-d g)^2}+\frac {\left (3 e^2\right ) \operatorname {Subst}\left (\int x^2 \, dx,x,a+b \log \left (c (d+e x)^n\right )\right )}{2 g (e f-d g)^2}+\frac {\left (3 b^2 e^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {g x}{e f-d g}\right )}{x} \, dx,x,d+e x\right )}{g (e f-d g)^2}-\frac {\left (3 b^3 e^2 n^3\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {g x}{e f-d g}\right )}{x} \, dx,x,d+e x\right )}{g (e f-d g)^2}\\ &=-\frac {3 b e n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 (e f-d g)^2 (f+g x)}+\frac {e^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{2 g (e f-d g)^2}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{2 g (f+g x)^2}+\frac {3 b^2 e^2 n^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 )}{g (e f-d g)^2}-\frac {3 b e^2 n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e (f+g x)}{e f-d g}\right )}{2 g (e f-d g)^2}+\frac {3 b^3 e^2 n^3 \text {Li}_2\left (-\frac {g (d+e x)}{e f-d g}\right )}{g (e f-d g)^2}-\frac {3 b^2 e^2 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (-\frac {g (d+e x)}{e f-d g}\right )}{g (e f-d g)^2}+\frac {\left (3 b^3 e^2 n^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {g x}{e f-d g}\right )}{x} \, dx,x,d+e x\right )}{g (e f-d g)^2}\\ &=-\frac {3 b e n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 (e f-d g)^2 (f+g x)}+\frac {e^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{2 g (e f-d g)^2}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{2 g (f+g x)^2}+\frac {3 b^2 e^2 n^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 )}{g (e f-d g)^2}-\frac {3 b e^2 n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e (f+g x)}{e f-d g}\right )}{2 g (e f-d g)^2}+\frac {3 b^3 e^2 n^3 \text {Li}_2\left (-\frac {g (d+e x)}{e f-d g}\right )}{g (e f-d g)^2}-\frac {3 b^2 e^2 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (-\frac {g (d+e x)}{e f-d g}\right )}{g (e f-d g)^2}+\frac {3 b^3 e^2 n^3 \text {Li}_3\left (-\frac {g (d+e x)}{e f-d g}\right )}{g (e f-d g)^2}\\ \end {align*}
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Mathematica [A] time = 0.75, size = 620, normalized size = 1.81 \[ -\frac {3 b^2 n^2 \left (2 e^2 (f+g x)^2 \text {Li}_2\left (\frac {g (d+e x)}{d g-e f}\right )-2 e^2 (f+g x)^2 \log \left (\frac {e (f+g x)}{e f-d g}\right )+g (d+e x) \log ^2(d+e x) (d g-e (2 f+g x))+2 e (f+g x) \log (d+e x) \left (e (f+g x) \log \left (\frac {e (f+g x)}{e f-d g}\right )+g (d+e x)\right )\right ) \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )-3 b e^2 n (f+g x)^2 \log (d+e x) \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )^2+3 b e^2 n (f+g x)^2 \log (f+g x) \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )^2-3 b e n (f+g x) (e f-d g) \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )^2+3 b n (e f-d g)^2 \log (d+e x) \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )^2+(e f-d g)^2 \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )^3+b^3 n^3 \left (-6 e^2 (f+g x)^2 \text {Li}_2\left (\frac {g (d+e x)}{d g-e f}\right )-6 e^2 (f+g x)^2 \text {Li}_3\left (\frac {g (d+e x)}{d g-e f}\right )-6 e^2 (f+g x)^2 \log (d+e x) \left (\log \left (\frac {e (f+g x)}{e f-d g}\right )-\text {Li}_2\left (\frac {g (d+e x)}{d g-e f}\right )\right )+g (d+e x) \log ^3(d+e x) (d g-e (2 f+g x))+3 e (f+g x) \log ^2(d+e x) \left (e (f+g x) \log \left (\frac {e (f+g x)}{e f-d g}\right )+g (d+e x)\right )\right )}{2 g (f+g x)^2 (e f-d g)^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{3} \log \left ({\left (e x + d\right )}^{n} c\right )^{3} + 3 \, a b^{2} \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + 3 \, a^{2} b \log \left ({\left (e x + d\right )}^{n} c\right ) + a^{3}}{g^{3} x^{3} + 3 \, f g^{2} x^{2} + 3 \, f^{2} g x + f^{3}}, x\right ) \]
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 \[ \int \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{3}}{{\left (g x + f\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 2.13, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \left (e x +d \right )^{n}\right )+a \right )^{3}}{\left (g x +f \right )^{3}}\, dx \]
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 \[ \frac {3}{2} \, a^{2} b e n {\left (\frac {e \log \left (e x + d\right )}{e^{2} f^{2} g - 2 \, d e f g^{2} + d^{2} g^{3}} - \frac {e \log \left (g x + f\right )}{e^{2} f^{2} g - 2 \, d e f g^{2} + d^{2} g^{3}} + \frac {1}{e f^{2} g - d f g^{2} + {\left (e f g^{2} - d g^{3}\right )} x}\right )} - \frac {b^{3} \log \left ({\left (e x + d\right )}^{n}\right )^{3}}{2 \, {\left (g^{3} x^{2} + 2 \, f g^{2} x + f^{2} g\right )}} - \frac {3 \, a^{2} b \log \left ({\left (e x + d\right )}^{n} c\right )}{2 \, {\left (g^{3} x^{2} + 2 \, f g^{2} x + f^{2} g\right )}} - \frac {a^{3}}{2 \, {\left (g^{3} x^{2} + 2 \, f g^{2} x + f^{2} g\right )}} + \int \frac {2 \, b^{3} d g \log \relax (c)^{3} + 6 \, a b^{2} d g \log \relax (c)^{2} + 3 \, {\left (2 \, a b^{2} d g + {\left (e f n + 2 \, d g \log \relax (c)\right )} b^{3} + {\left (2 \, a b^{2} e g + {\left (e g n + 2 \, e g \log \relax (c)\right )} b^{3}\right )} x\right )} \log \left ({\left (e x + d\right )}^{n}\right )^{2} + 2 \, {\left (b^{3} e g \log \relax (c)^{3} + 3 \, a b^{2} e g \log \relax (c)^{2}\right )} x + 6 \, {\left (b^{3} d g \log \relax (c)^{2} + 2 \, a b^{2} d g \log \relax (c) + {\left (b^{3} e g \log \relax (c)^{2} + 2 \, a b^{2} e g \log \relax (c)\right )} x\right )} \log \left ({\left (e x + d\right )}^{n}\right )}{2 \, {\left (e g^{4} x^{4} + d f^{3} g + {\left (3 \, e f g^{3} + d g^{4}\right )} x^{3} + 3 \, {\left (e f^{2} g^{2} + d f g^{3}\right )} x^{2} + {\left (e f^{3} g + 3 \, d f^{2} g^{2}\right )} x\right )}}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^3}{{\left (f+g\,x\right )}^3} \,d x \]
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 \[ \int \frac {\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{3}}{\left (f + g x\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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