Optimal. Leaf size=295 \[ -\frac {3 b^2 n^2 (d g+e f) \text {Li}_2\left (-\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 e^2}+\frac {3 b^2 n^2 (e f-d g) \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 e^2}-\frac {3 b n (d g+e f) \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 e^2}+\frac {f^2 \left (a+b \log \left (c x^n\right )\right )^3}{2 d^2 (e f-d g)}-\frac {3 b n x (e f-d g) \left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 e (d+e x)}-\frac {(f+g x)^2 \left (a+b \log \left (c x^n\right )\right )^3}{2 (d+e x)^2 (e f-d g)}+\frac {3 b^3 n^3 (e f-d g) \text {Li}_2\left (-\frac {e x}{d}\right )}{d^2 e^2}+\frac {3 b^3 n^3 (d g+e f) \text {Li}_3\left (-\frac {e x}{d}\right )}{d^2 e^2} \]
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Rubi [A] time = 0.62, antiderivative size = 408, normalized size of antiderivative = 1.38, number of steps used = 17, number of rules used = 11, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {2357, 2319, 2347, 2344, 2302, 30, 2317, 2374, 6589, 2318, 2391} \[ -\frac {3 b^2 n^2 (e f-d g) \text {PolyLog}\left (2,-\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 e^2}-\frac {6 b^2 g n^2 \text {PolyLog}\left (2,-\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{d e^2}+\frac {3 b^3 n^3 (e f-d g) \text {PolyLog}\left (2,-\frac {e x}{d}\right )}{d^2 e^2}+\frac {3 b^3 n^3 (e f-d g) \text {PolyLog}\left (3,-\frac {e x}{d}\right )}{d^2 e^2}+\frac {6 b^3 g n^3 \text {PolyLog}\left (3,-\frac {e x}{d}\right )}{d e^2}+\frac {3 b^2 n^2 (e f-d g) \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 e^2}-\frac {3 b n (e f-d g) \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 e^2}+\frac {(e f-d g) \left (a+b \log \left (c x^n\right )\right )^3}{2 d^2 e^2}-\frac {3 b n x (e f-d g) \left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 e (d+e x)}-\frac {(e f-d g) \left (a+b \log \left (c x^n\right )\right )^3}{2 e^2 (d+e x)^2}-\frac {3 b g n \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d e^2}+\frac {g x \left (a+b \log \left (c x^n\right )\right )^3}{d e (d+e x)} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2302
Rule 2317
Rule 2318
Rule 2319
Rule 2344
Rule 2347
Rule 2357
Rule 2374
Rule 2391
Rule 6589
Rubi steps
\begin {align*} \int \frac {(f+g x) \left (a+b \log \left (c x^n\right )\right )^3}{(d+e x)^3} \, dx &=\int \left (\frac {(e f-d g) \left (a+b \log \left (c x^n\right )\right )^3}{e (d+e x)^3}+\frac {g \left (a+b \log \left (c x^n\right )\right )^3}{e (d+e x)^2}\right ) \, dx\\ &=\frac {g \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{(d+e x)^2} \, dx}{e}+\frac {(e f-d g) \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{(d+e x)^3} \, dx}{e}\\ &=-\frac {(e f-d g) \left (a+b \log \left (c x^n\right )\right )^3}{2 e^2 (d+e x)^2}+\frac {g x \left (a+b \log \left (c x^n\right )\right )^3}{d e (d+e x)}-\frac {(3 b g n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx}{d e}+\frac {(3 b (e f-d g) n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)^2} \, dx}{2 e^2}\\ &=-\frac {(e f-d g) \left (a+b \log \left (c x^n\right )\right )^3}{2 e^2 (d+e x)^2}+\frac {g x \left (a+b \log \left (c x^n\right )\right )^3}{d e (d+e x)}-\frac {3 b g n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{d e^2}+\frac {(3 b (e f-d g) n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)} \, dx}{2 d e^2}-\frac {(3 b (e f-d g) n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2} \, dx}{2 d e}+\frac {\left (6 b^2 g n^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{x} \, dx}{d e^2}\\ &=-\frac {3 b (e f-d g) n x \left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 e (d+e x)}-\frac {(e f-d g) \left (a+b \log \left (c x^n\right )\right )^3}{2 e^2 (d+e x)^2}+\frac {g x \left (a+b \log \left (c x^n\right )\right )^3}{d e (d+e x)}-\frac {3 b g n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{d e^2}-\frac {6 b^2 g n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )}{d e^2}+\frac {(3 b (e f-d g) n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx}{2 d^2 e^2}-\frac {(3 b (e f-d g) n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx}{2 d^2 e}+\frac {\left (3 b^2 (e f-d g) n^2\right ) \int \frac {a+b \log \left (c x^n\right )}{d+e x} \, dx}{d^2 e}+\frac {\left (6 b^3 g n^3\right ) \int \frac {\text {Li}_2\left (-\frac {e x}{d}\right )}{x} \, dx}{d e^2}\\ &=-\frac {3 b (e f-d g) n x \left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 e (d+e x)}-\frac {(e f-d g) \left (a+b \log \left (c x^n\right )\right )^3}{2 e^2 (d+e x)^2}+\frac {g x \left (a+b \log \left (c x^n\right )\right )^3}{d e (d+e x)}+\frac {3 b^2 (e f-d g) n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d^2 e^2}-\frac {3 b g n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{d e^2}-\frac {3 b (e f-d g) n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{2 d^2 e^2}-\frac {6 b^2 g n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )}{d e^2}+\frac {6 b^3 g n^3 \text {Li}_3\left (-\frac {e x}{d}\right )}{d e^2}+\frac {(3 (e f-d g)) \operatorname {Subst}\left (\int x^2 \, dx,x,a+b \log \left (c x^n\right )\right )}{2 d^2 e^2}+\frac {\left (3 b^2 (e f-d g) n^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{x} \, dx}{d^2 e^2}-\frac {\left (3 b^3 (e f-d g) n^3\right ) \int \frac {\log \left (1+\frac {e x}{d}\right )}{x} \, dx}{d^2 e^2}\\ &=-\frac {3 b (e f-d g) n x \left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 e (d+e x)}+\frac {(e f-d g) \left (a+b \log \left (c x^n\right )\right )^3}{2 d^2 e^2}-\frac {(e f-d g) \left (a+b \log \left (c x^n\right )\right )^3}{2 e^2 (d+e x)^2}+\frac {g x \left (a+b \log \left (c x^n\right )\right )^3}{d e (d+e x)}+\frac {3 b^2 (e f-d g) n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d^2 e^2}-\frac {3 b g n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{d e^2}-\frac {3 b (e f-d g) n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{2 d^2 e^2}+\frac {3 b^3 (e f-d g) n^3 \text {Li}_2\left (-\frac {e x}{d}\right )}{d^2 e^2}-\frac {6 b^2 g n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )}{d e^2}-\frac {3 b^2 (e f-d g) n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )}{d^2 e^2}+\frac {6 b^3 g n^3 \text {Li}_3\left (-\frac {e x}{d}\right )}{d e^2}+\frac {\left (3 b^3 (e f-d g) n^3\right ) \int \frac {\text {Li}_2\left (-\frac {e x}{d}\right )}{x} \, dx}{d^2 e^2}\\ &=-\frac {3 b (e f-d g) n x \left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 e (d+e x)}+\frac {(e f-d g) \left (a+b \log \left (c x^n\right )\right )^3}{2 d^2 e^2}-\frac {(e f-d g) \left (a+b \log \left (c x^n\right )\right )^3}{2 e^2 (d+e x)^2}+\frac {g x \left (a+b \log \left (c x^n\right )\right )^3}{d e (d+e x)}+\frac {3 b^2 (e f-d g) n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d^2 e^2}-\frac {3 b g n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{d e^2}-\frac {3 b (e f-d g) n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{2 d^2 e^2}+\frac {3 b^3 (e f-d g) n^3 \text {Li}_2\left (-\frac {e x}{d}\right )}{d^2 e^2}-\frac {6 b^2 g n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )}{d e^2}-\frac {3 b^2 (e f-d g) n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )}{d^2 e^2}+\frac {6 b^3 g n^3 \text {Li}_3\left (-\frac {e x}{d}\right )}{d e^2}+\frac {3 b^3 (e f-d g) n^3 \text {Li}_3\left (-\frac {e x}{d}\right )}{d^2 e^2}\\ \end {align*}
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Mathematica [A] time = 0.39, size = 339, normalized size = 1.15 \[ \frac {\frac {(e f-d g) \left (-3 b n (d+e x) \left (\left (a+b \log \left (c x^n\right )\right ) \left (a+b \log \left (c x^n\right )-2 b n \log \left (\frac {e x}{d}+1\right )\right )-2 b^2 n^2 \text {Li}_2\left (-\frac {e x}{d}\right )\right )-6 b^2 n^2 (d+e x) \left (\text {Li}_2\left (-\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )-b n \text {Li}_3\left (-\frac {e x}{d}\right )\right )+(d+e x) \left (a+b \log \left (c x^n\right )\right )^3-3 b n (d+e x) \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2+3 b d n \left (a+b \log \left (c x^n\right )\right )^2\right )}{d^2 (d+e x)}+\frac {2 g \left (-6 b^2 n^2 \text {Li}_2\left (-\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )+\left (a+b \log \left (c x^n\right )\right )^2 \left (a+b \log \left (c x^n\right )-3 b n \log \left (\frac {e x}{d}+1\right )\right )+6 b^3 n^3 \text {Li}_3\left (-\frac {e x}{d}\right )\right )}{d}-\frac {(e f-d g) \left (a+b \log \left (c x^n\right )\right )^3}{(d+e x)^2}-\frac {2 g \left (a+b \log \left (c x^n\right )\right )^3}{d+e x}}{2 e^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.73, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a^{3} g x + a^{3} f + {\left (b^{3} g x + b^{3} f\right )} \log \left (c x^{n}\right )^{3} + 3 \, {\left (a b^{2} g x + a b^{2} f\right )} \log \left (c x^{n}\right )^{2} + 3 \, {\left (a^{2} b g x + a^{2} b f\right )} \log \left (c x^{n}\right )}{e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}}, x\right ) \]
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 \[ \int \frac {{\left (g x + f\right )} {\left (b \log \left (c x^{n}\right ) + a\right )}^{3}}{{\left (e x + d\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.75, size = 0, normalized size = 0.00 \[ \int \frac {\left (g x +f \right ) \left (b \ln \left (c \,x^{n}\right )+a \right )^{3}}{\left (e x +d \right )^{3}}\, dx \]
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 \[ \frac {3}{2} \, a^{2} b f n {\left (\frac {1}{d e^{2} x + d^{2} e} - \frac {\log \left (e x + d\right )}{d^{2} e} + \frac {\log \relax (x)}{d^{2} e}\right )} - \frac {3}{2} \, a^{2} b g n {\left (\frac {1}{e^{3} x + d e^{2}} + \frac {\log \left (e x + d\right )}{d e^{2}} - \frac {\log \relax (x)}{d e^{2}}\right )} - \frac {3 \, {\left (2 \, e x + d\right )} a^{2} b g \log \left (c x^{n}\right )}{2 \, {\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}} - \frac {{\left (2 \, e x + d\right )} a^{3} g}{2 \, {\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}} - \frac {3 \, a^{2} b f \log \left (c x^{n}\right )}{2 \, {\left (e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e\right )}} - \frac {a^{3} f}{2 \, {\left (e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e\right )}} - \frac {{\left (2 \, b^{3} e g x + {\left (e f + d g\right )} b^{3}\right )} \log \left (x^{n}\right )^{3}}{2 \, {\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}} + \int \frac {2 \, {\left (b^{3} e^{2} g \log \relax (c)^{3} + 3 \, a b^{2} e^{2} g \log \relax (c)^{2}\right )} x^{2} + 3 \, {\left ({\left (d e f n + d^{2} g n\right )} b^{3} + 2 \, {\left (a b^{2} e^{2} g + {\left (e^{2} g n + e^{2} g \log \relax (c)\right )} b^{3}\right )} x^{2} + {\left (2 \, a b^{2} e^{2} f + {\left (e^{2} f n + 3 \, d e g n + 2 \, e^{2} f \log \relax (c)\right )} b^{3}\right )} x\right )} \log \left (x^{n}\right )^{2} + 2 \, {\left (b^{3} e^{2} f \log \relax (c)^{3} + 3 \, a b^{2} e^{2} f \log \relax (c)^{2}\right )} x + 6 \, {\left ({\left (b^{3} e^{2} g \log \relax (c)^{2} + 2 \, a b^{2} e^{2} g \log \relax (c)\right )} x^{2} + {\left (b^{3} e^{2} f \log \relax (c)^{2} + 2 \, a b^{2} e^{2} f \log \relax (c)\right )} x\right )} \log \left (x^{n}\right )}{2 \, {\left (e^{5} x^{4} + 3 \, d e^{4} x^{3} + 3 \, d^{2} e^{3} x^{2} + d^{3} e^{2} x\right )}}\,{d x} \]
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 \[ \int \frac {\left (f+g\,x\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3}{{\left (d+e\,x\right )}^3} \,d x \]
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 \[ \int \frac {\left (a + b \log {\left (c x^{n} \right )}\right )^{3} \left (f + g x\right )}{\left (d + e x\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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