Optimal. Leaf size=602 \[ \frac {6 b^2 e n^2 \text {Li}_2\left (-\frac {i (d+e x)}{e h-d i}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(e h-d i) (g h-f i)}-\frac {6 b^2 g n^2 \text {Li}_3\left (-\frac {g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(g h-f i)^2}+\frac {6 b^2 g n^2 \text {Li}_3\left (-\frac {i (d+e x)}{e h-d i}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(g h-f i)^2}+\frac {3 b g n \text {Li}_2\left (-\frac {g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(g h-f i)^2}-\frac {3 b g n \text {Li}_2\left (-\frac {i (d+e x)}{e h-d i}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(g h-f i)^2}+\frac {3 b e n \log \left (\frac {e (h+i x)}{e h-d i}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(e h-d i) (g h-f i)}-\frac {i (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(h+i x) (e h-d i) (g h-f i)}+\frac {g \log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(g h-f i)^2}-\frac {g \log \left (\frac {e (h+i x)}{e h-d i}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(g h-f i)^2}-\frac {6 b^3 e n^3 \text {Li}_3\left (-\frac {i (d+e x)}{e h-d i}\right )}{(e h-d i) (g h-f i)}+\frac {6 b^3 g n^3 \text {Li}_4\left (-\frac {g (d+e x)}{e f-d g}\right )}{(g h-f i)^2}-\frac {6 b^3 g n^3 \text {Li}_4\left (-\frac {i (d+e x)}{e h-d i}\right )}{(g h-f i)^2} \]
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Rubi [A] time = 0.73, antiderivative size = 602, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 7, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {2418, 2396, 2433, 2374, 2383, 6589, 2397} \[ \frac {6 b^2 e n^2 \text {PolyLog}\left (2,-\frac {i (d+e x)}{e h-d i}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(e h-d i) (g h-f i)}-\frac {6 b^2 g n^2 \text {PolyLog}\left (3,-\frac {g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(g h-f i)^2}+\frac {6 b^2 g n^2 \text {PolyLog}\left (3,-\frac {i (d+e x)}{e h-d i}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(g h-f i)^2}+\frac {3 b g n \text {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(g h-f i)^2}-\frac {3 b g n \text {PolyLog}\left (2,-\frac {i (d+e x)}{e h-d i}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(g h-f i)^2}-\frac {6 b^3 e n^3 \text {PolyLog}\left (3,-\frac {i (d+e x)}{e h-d i}\right )}{(e h-d i) (g h-f i)}+\frac {6 b^3 g n^3 \text {PolyLog}\left (4,-\frac {g (d+e x)}{e f-d g}\right )}{(g h-f i)^2}-\frac {6 b^3 g n^3 \text {PolyLog}\left (4,-\frac {i (d+e x)}{e h-d i}\right )}{(g h-f i)^2}+\frac {3 b e n \log \left (\frac {e (h+i x)}{e h-d i}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(e h-d i) (g h-f i)}-\frac {i (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(h+i x) (e h-d i) (g h-f i)}+\frac {g \log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(g h-f i)^2}-\frac {g \log \left (\frac {e (h+i x)}{e h-d i}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(g h-f i)^2} \]
Antiderivative was successfully verified.
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Rule 2374
Rule 2383
Rule 2396
Rule 2397
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+233 x)^2 (f+g x)} \, dx &=\int \left (\frac {233 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(233 f-g h) (h+233 x)^2}-\frac {233 g \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(233 f-g h)^2 (h+233 x)}+\frac {g^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(233 f-g h)^2 (f+g x)}\right ) \, dx\\ &=-\frac {(233 g) \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{h+233 x} \, dx}{(233 f-g h)^2}+\frac {g^2 \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{f+g x} \, dx}{(233 f-g h)^2}+\frac {233 \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(h+233 x)^2} \, dx}{233 f-g h}\\ &=-\frac {233 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(233 d-e h) (233 f-g h) (h+233 x)}-\frac {g \log \left (-\frac {e (h+233 x)}{233 d-e h}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(233 f-g h)^2}+\frac {g \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 )}{(233 f-g h)^2}+\frac {(3 b e g n) \int \frac {\log \left (\frac {e (h+233 x)}{-233 d+e h}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d+e x} \, dx}{(233 f-g h)^2}-\frac {(3 b e g 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}{(233 f-g h)^2}+\frac {(699 b e n) \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{h+233 x} \, dx}{(233 d-e h) (233 f-g h)}\\ &=\frac {3 b e n \log \left (-\frac {e (h+233 x)}{233 d-e h}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(233 d-e h) (233 f-g h)}-\frac {233 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(233 d-e h) (233 f-g h) (h+233 x)}-\frac {g \log \left (-\frac {e (h+233 x)}{233 d-e h}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(233 f-g h)^2}+\frac {g \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 )}{(233 f-g h)^2}+\frac {(3 b g n) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (\frac {e \left (\frac {-233 d+e h}{e}+\frac {233 x}{e}\right )}{-233 d+e h}\right )}{x} \, dx,x,d+e x\right )}{(233 f-g h)^2}-\frac {(3 b g 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 )}{(233 f-g h)^2}-\frac {\left (6 b^2 e^2 n^2\right ) \int \frac {\log \left (\frac {e (h+233 x)}{-233 d+e h}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x} \, dx}{(233 d-e h) (233 f-g h)}\\ &=\frac {3 b e n \log \left (-\frac {e (h+233 x)}{233 d-e h}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(233 d-e h) (233 f-g h)}-\frac {233 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(233 d-e h) (233 f-g h) (h+233 x)}-\frac {g \log \left (-\frac {e (h+233 x)}{233 d-e h}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(233 f-g h)^2}+\frac {g \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 )}{(233 f-g h)^2}+\frac {3 b g 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 )}{(233 f-g h)^2}-\frac {3 b g n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text {Li}_2\left (\frac {233 (d+e x)}{233 d-e h}\right )}{(233 f-g h)^2}-\frac {\left (6 b^2 g 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 )}{(233 f-g h)^2}+\frac {\left (6 b^2 g n^2\right ) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {233 x}{-233 d+e h}\right )}{x} \, dx,x,d+e x\right )}{(233 f-g h)^2}-\frac {\left (6 b^2 e n^2\right ) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (\frac {e \left (\frac {-233 d+e h}{e}+\frac {233 x}{e}\right )}{-233 d+e h}\right )}{x} \, dx,x,d+e x\right )}{(233 d-e h) (233 f-g h)}\\ &=\frac {3 b e n \log \left (-\frac {e (h+233 x)}{233 d-e h}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(233 d-e h) (233 f-g h)}-\frac {233 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(233 d-e h) (233 f-g h) (h+233 x)}-\frac {g \log \left (-\frac {e (h+233 x)}{233 d-e h}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(233 f-g h)^2}+\frac {g \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 )}{(233 f-g h)^2}+\frac {3 b g 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 )}{(233 f-g h)^2}+\frac {6 b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (\frac {233 (d+e x)}{233 d-e h}\right )}{(233 d-e h) (233 f-g h)}-\frac {3 b g n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text {Li}_2\left (\frac {233 (d+e x)}{233 d-e h}\right )}{(233 f-g h)^2}-\frac {6 b^2 g 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 )}{(233 f-g h)^2}+\frac {6 b^2 g n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_3\left (\frac {233 (d+e x)}{233 d-e h}\right )}{(233 f-g h)^2}+\frac {\left (6 b^3 g 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 )}{(233 f-g h)^2}-\frac {\left (6 b^3 g n^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3\left (-\frac {233 x}{-233 d+e h}\right )}{x} \, dx,x,d+e x\right )}{(233 f-g h)^2}-\frac {\left (6 b^3 e n^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {233 x}{-233 d+e h}\right )}{x} \, dx,x,d+e x\right )}{(233 d-e h) (233 f-g h)}\\ &=\frac {3 b e n \log \left (-\frac {e (h+233 x)}{233 d-e h}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(233 d-e h) (233 f-g h)}-\frac {233 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(233 d-e h) (233 f-g h) (h+233 x)}-\frac {g \log \left (-\frac {e (h+233 x)}{233 d-e h}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(233 f-g h)^2}+\frac {g \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 )}{(233 f-g h)^2}+\frac {3 b g 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 )}{(233 f-g h)^2}+\frac {6 b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (\frac {233 (d+e x)}{233 d-e h}\right )}{(233 d-e h) (233 f-g h)}-\frac {3 b g n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text {Li}_2\left (\frac {233 (d+e x)}{233 d-e h}\right )}{(233 f-g h)^2}-\frac {6 b^2 g 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 )}{(233 f-g h)^2}-\frac {6 b^3 e n^3 \text {Li}_3\left (\frac {233 (d+e x)}{233 d-e h}\right )}{(233 d-e h) (233 f-g h)}+\frac {6 b^2 g n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_3\left (\frac {233 (d+e x)}{233 d-e h}\right )}{(233 f-g h)^2}+\frac {6 b^3 g n^3 \text {Li}_4\left (-\frac {g (d+e x)}{e f-d g}\right )}{(233 f-g h)^2}-\frac {6 b^3 g n^3 \text {Li}_4\left (\frac {233 (d+e x)}{233 d-e h}\right )}{(233 f-g h)^2}\\ \end {align*}
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Mathematica [A] time = 1.33, size = 1025, normalized size = 1.70 \[ \frac {-b^3 \left ((g h-f i) \left (\left (i (d+e x) \log (d+e x)-3 e (h+i x) \log \left (\frac {e (h+i x)}{e h-d i}\right )\right ) \log ^2(d+e x)-6 e (h+i x) \text {Li}_2\left (\frac {i (d+e x)}{d i-e h}\right ) \log (d+e x)+6 e (h+i x) \text {Li}_3\left (\frac {i (d+e x)}{d i-e h}\right )\right )-g (e h-d i) (h+i x) \left (\log \left (\frac {e (f+g x)}{e f-d g}\right ) \log ^3(d+e x)+3 \text {Li}_2\left (\frac {g (d+e x)}{d g-e f}\right ) \log ^2(d+e x)-6 \text {Li}_3\left (\frac {g (d+e x)}{d g-e f}\right ) \log (d+e x)+6 \text {Li}_4\left (\frac {g (d+e x)}{d g-e f}\right )\right )+g (e h-d i) (h+i x) \left (\log \left (\frac {e (h+i x)}{e h-d i}\right ) \log ^3(d+e x)+3 \text {Li}_2\left (\frac {i (d+e x)}{d i-e h}\right ) \log ^2(d+e x)-6 \text {Li}_3\left (\frac {i (d+e x)}{d i-e h}\right ) \log (d+e x)+6 \text {Li}_4\left (\frac {i (d+e x)}{d i-e h}\right )\right )\right ) n^3-3 b^2 \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left ((g h-f i) \left (\log (d+e x) \left (i (d+e x) \log (d+e x)-2 e (h+i x) \log \left (\frac {e (h+i x)}{e h-d i}\right )\right )-2 e (h+i x) \text {Li}_2\left (\frac {i (d+e x)}{d i-e h}\right )\right )-g (e h-d i) (h+i x) \left (\log \left (\frac {e (f+g x)}{e f-d g}\right ) \log ^2(d+e x)+2 \text {Li}_2\left (\frac {g (d+e x)}{d g-e f}\right ) \log (d+e x)-2 \text {Li}_3\left (\frac {g (d+e x)}{d g-e f}\right )\right )+g (e h-d i) (h+i x) \left (\log \left (\frac {e (h+i x)}{e h-d i}\right ) \log ^2(d+e x)+2 \text {Li}_2\left (\frac {i (d+e x)}{d i-e h}\right ) \log (d+e x)-2 \text {Li}_3\left (\frac {i (d+e x)}{d i-e h}\right )\right )\right ) n^2-3 b \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2 \left ((g h-f i) (i (d+e x) \log (d+e x)-e (h+i x) \log (h+i x))-g (e h-d i) (h+i x) \left (\log (d+e x) \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 (e h-d i) (h+i x) \left (\log (d+e x) \log \left (\frac {e (h+i x)}{e h-d i}\right )+\text {Li}_2\left (\frac {i (d+e x)}{d i-e h}\right )\right )\right ) n+(e h-d i) (g h-f i) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^3+g (e h-d i) (h+i x) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^3 \log (f+g x)-g (e h-d i) (h+i x) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^3 \log (h+i x)}{(e h-d i) (g h-f i)^2 (h+i x)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{3} \log \left ({\left (e x + d\right )}^{n} c\right )^{3} + 3 \, a b^{2} \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + 3 \, a^{2} b \log \left ({\left (e x + d\right )}^{n} c\right ) + a^{3}}{g i^{2} x^{3} + f h^{2} + {\left (2 \, g h i + f i^{2}\right )} x^{2} + {\left (g h^{2} + 2 \, f h i\right )} x}, x\right ) \]
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 \[ \int \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{3}}{{\left (g x + f\right )} {\left (i x + h\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.53, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \left (e x +d \right )^{n}\right )+a \right )^{3}}{\left (g x +f \right ) \left (i x +h \right )^{2}}\, dx \]
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 \[ a^{3} {\left (\frac {g \log \left (g x + f\right )}{g^{2} h^{2} - 2 \, f g h i + f^{2} i^{2}} - \frac {g \log \left (i x + h\right )}{g^{2} h^{2} - 2 \, f g h i + f^{2} i^{2}} + \frac {1}{g h^{2} - f h i + {\left (g h i - f i^{2}\right )} x}\right )} + \int \frac {b^{3} \log \left ({\left (e x + d\right )}^{n}\right )^{3} + b^{3} \log \relax (c)^{3} + 3 \, a b^{2} \log \relax (c)^{2} + 3 \, a^{2} b \log \relax (c) + 3 \, {\left (b^{3} \log \relax (c) + a b^{2}\right )} \log \left ({\left (e x + d\right )}^{n}\right )^{2} + 3 \, {\left (b^{3} \log \relax (c)^{2} + 2 \, a b^{2} \log \relax (c) + a^{2} b\right )} \log \left ({\left (e x + d\right )}^{n}\right )}{g i^{2} x^{3} + f h^{2} + {\left (2 \, g h i + f i^{2}\right )} x^{2} + {\left (g h^{2} + 2 \, f h i\right )} x}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^3}{\left (f+g\,x\right )\,{\left (h+i\,x\right )}^2} \,d x \]
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 \[ \int \frac {\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{3}}{\left (f + g x\right ) \left (h + i x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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