Optimal. Leaf size=372 \[ -\frac {6 b^2 n^2 \text {Li}_3\left (-\frac {g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g h-f i}+\frac {6 b^2 n^2 \text {Li}_3\left (-\frac {i (d+e x)}{e h-d i}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g h-f i}+\frac {3 b n \text {Li}_2\left (-\frac {g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g h-f i}-\frac {3 b n \text {Li}_2\left (-\frac {i (d+e x)}{e h-d i}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g h-f i}+\frac {\log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{g h-f i}-\frac {\log \left (\frac {e (h+i x)}{e h-d i}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{g h-f i}+\frac {6 b^3 n^3 \text {Li}_4\left (-\frac {g (d+e x)}{e f-d g}\right )}{g h-f i}-\frac {6 b^3 n^3 \text {Li}_4\left (-\frac {i (d+e x)}{e h-d i}\right )}{g h-f i} \]
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Rubi [A] time = 0.52, antiderivative size = 372, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 6, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {2418, 2396, 2433, 2374, 2383, 6589} \[ -\frac {6 b^2 n^2 \text {PolyLog}\left (3,-\frac {g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g h-f i}+\frac {6 b^2 n^2 \text {PolyLog}\left (3,-\frac {i (d+e x)}{e h-d i}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g h-f i}+\frac {3 b n \text {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g h-f i}-\frac {3 b n \text {PolyLog}\left (2,-\frac {i (d+e x)}{e h-d i}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g h-f i}+\frac {6 b^3 n^3 \text {PolyLog}\left (4,-\frac {g (d+e x)}{e f-d g}\right )}{g h-f i}-\frac {6 b^3 n^3 \text {PolyLog}\left (4,-\frac {i (d+e x)}{e h-d i}\right )}{g h-f i}+\frac {\log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{g h-f i}-\frac {\log \left (\frac {e (h+i x)}{e h-d i}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{g h-f i} \]
Antiderivative was successfully verified.
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Rule 2374
Rule 2383
Rule 2396
Rule 2418
Rule 2433
Rule 6589
Rubi steps
\begin {align*} \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(h+232 x) (f+g x)} \, dx &=\int \left (\frac {232 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(232 f-g h) (h+232 x)}-\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(232 f-g h) (f+g x)}\right ) \, dx\\ &=\frac {232 \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{h+232 x} \, dx}{232 f-g h}-\frac {g \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{f+g x} \, dx}{232 f-g h}\\ &=\frac {\log \left (-\frac {e (h+232 x)}{232 d-e h}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{232 f-g h}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac {e (f+g x)}{e f-d g}\right )}{232 f-g h}-\frac {(3 b e n) \int \frac {\log \left (\frac {e (h+232 x)}{-232 d+e h}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d+e x} \, dx}{232 f-g h}+\frac {(3 b e n) \int \frac {\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 )}{d+e x} \, dx}{232 f-g h}\\ &=\frac {\log \left (-\frac {e (h+232 x)}{232 d-e h}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{232 f-g h}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac {e (f+g x)}{e f-d g}\right )}{232 f-g h}-\frac {(3 b n) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (\frac {e \left (\frac {-232 d+e h}{e}+\frac {232 x}{e}\right )}{-232 d+e h}\right )}{x} \, dx,x,d+e x\right )}{232 f-g h}+\frac {(3 b n) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (\frac {e \left (\frac {e f-d g}{e}+\frac {g x}{e}\right )}{e f-d g}\right )}{x} \, dx,x,d+e x\right )}{232 f-g h}\\ &=\frac {\log \left (-\frac {e (h+232 x)}{232 d-e h}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{232 f-g h}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac {e (f+g x)}{e f-d g}\right )}{232 f-g h}-\frac {3 b n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text {Li}_2\left (-\frac {g (d+e x)}{e f-d g}\right )}{232 f-g h}+\frac {3 b n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text {Li}_2\left (\frac {232 (d+e x)}{232 d-e h}\right )}{232 f-g h}+\frac {\left (6 b^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {g x}{e f-d g}\right )}{x} \, dx,x,d+e x\right )}{232 f-g h}-\frac {\left (6 b^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {232 x}{-232 d+e h}\right )}{x} \, dx,x,d+e x\right )}{232 f-g h}\\ &=\frac {\log \left (-\frac {e (h+232 x)}{232 d-e h}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{232 f-g h}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac {e (f+g x)}{e f-d g}\right )}{232 f-g h}-\frac {3 b n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text {Li}_2\left (-\frac {g (d+e x)}{e f-d g}\right )}{232 f-g h}+\frac {3 b n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text {Li}_2\left (\frac {232 (d+e x)}{232 d-e h}\right )}{232 f-g h}+\frac {6 b^2 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_3\left (-\frac {g (d+e x)}{e f-d g}\right )}{232 f-g h}-\frac {6 b^2 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_3\left (\frac {232 (d+e x)}{232 d-e h}\right )}{232 f-g h}-\frac {\left (6 b^3 n^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3\left (-\frac {g x}{e f-d g}\right )}{x} \, dx,x,d+e x\right )}{232 f-g h}+\frac {\left (6 b^3 n^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3\left (-\frac {232 x}{-232 d+e h}\right )}{x} \, dx,x,d+e x\right )}{232 f-g h}\\ &=\frac {\log \left (-\frac {e (h+232 x)}{232 d-e h}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{232 f-g h}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac {e (f+g x)}{e f-d g}\right )}{232 f-g h}-\frac {3 b n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text {Li}_2\left (-\frac {g (d+e x)}{e f-d g}\right )}{232 f-g h}+\frac {3 b n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text {Li}_2\left (\frac {232 (d+e x)}{232 d-e h}\right )}{232 f-g h}+\frac {6 b^2 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_3\left (-\frac {g (d+e x)}{e f-d g}\right )}{232 f-g h}-\frac {6 b^2 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_3\left (\frac {232 (d+e x)}{232 d-e h}\right )}{232 f-g h}-\frac {6 b^3 n^3 \text {Li}_4\left (-\frac {g (d+e x)}{e f-d g}\right )}{232 f-g h}+\frac {6 b^3 n^3 \text {Li}_4\left (\frac {232 (d+e x)}{232 d-e h}\right )}{232 f-g h}\\ \end {align*}
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Mathematica [A] time = 0.55, size = 599, normalized size = 1.61 \[ \frac {6 b^2 n^2 \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right ) \left (-\text {Li}_3\left (\frac {g (d+e x)}{d g-e f}\right )+\log (d+e x) \text {Li}_2\left (\frac {g (d+e x)}{d g-e f}\right )+\frac {1}{2} \log ^2(d+e x) \log \left (\frac {e (f+g x)}{e f-d g}\right )+\text {Li}_3\left (\frac {i (d+e x)}{d i-e h}\right )-\log (d+e x) \text {Li}_2\left (\frac {i (d+e x)}{d i-e h}\right )-\frac {1}{2} \log ^2(d+e x) \log \left (\frac {e (h+i x)}{e h-d i}\right )\right )+3 b n \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )^2 \left (\log (d+e x) \left (\log \left (\frac {e (f+g x)}{e f-d g}\right )-\log \left (\frac {e (h+i x)}{e h-d i}\right )\right )+\text {Li}_2\left (\frac {g (d+e x)}{d g-e f}\right )-\text {Li}_2\left (\frac {i (d+e x)}{d i-e h}\right )\right )+\log (f+g x) \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )^3-\log (h+i x) \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )^3+b^3 n^3 \left (6 \text {Li}_4\left (\frac {g (d+e x)}{d g-e f}\right )+3 \log ^2(d+e x) \text {Li}_2\left (\frac {g (d+e x)}{d g-e f}\right )-6 \log (d+e x) \text {Li}_3\left (\frac {g (d+e x)}{d g-e f}\right )+\log ^3(d+e x) \log \left (\frac {e (f+g x)}{e f-d g}\right )-6 \text {Li}_4\left (\frac {i (d+e x)}{d i-e h}\right )-3 \log ^2(d+e x) \text {Li}_2\left (\frac {i (d+e x)}{d i-e h}\right )+6 \log (d+e x) \text {Li}_3\left (\frac {i (d+e x)}{d i-e h}\right )-\log ^3(d+e x) \log \left (\frac {e (h+i x)}{e h-d i}\right )\right )}{g h-f i} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{3} \log \left ({\left (e x + d\right )}^{n} c\right )^{3} + 3 \, a b^{2} \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + 3 \, a^{2} b \log \left ({\left (e x + d\right )}^{n} c\right ) + a^{3}}{g i x^{2} + f h + {\left (g h + f i\right )} x}, x\right ) \]
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 \[ \int \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{3}}{{\left (g x + f\right )} {\left (i x + h\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.12, size = 21696, normalized size = 58.32 \[ \text {output too large to display} \]
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 \[ a^{3} {\left (\frac {\log \left (g x + f\right )}{g h - f i} - \frac {\log \left (i x + h\right )}{g h - f i}\right )} + \int \frac {b^{3} \log \left ({\left (e x + d\right )}^{n}\right )^{3} + b^{3} \log \relax (c)^{3} + 3 \, a b^{2} \log \relax (c)^{2} + 3 \, a^{2} b \log \relax (c) + 3 \, {\left (b^{3} \log \relax (c) + a b^{2}\right )} \log \left ({\left (e x + d\right )}^{n}\right )^{2} + 3 \, {\left (b^{3} \log \relax (c)^{2} + 2 \, a b^{2} \log \relax (c) + a^{2} b\right )} \log \left ({\left (e x + d\right )}^{n}\right )}{g i x^{2} + f h + {\left (g h + f i\right )} x}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^3}{\left (f+g\,x\right )\,\left (h+i\,x\right )} \,d x \]
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 \[ \int \frac {\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{3}}{\left (f + g x\right ) \left (h + i x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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