Optimal. Leaf size=308 \[ -\frac{6 b^2 n^2 (g h-f i) \text{PolyLog}\left (3,-\frac{g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^2}+\frac{3 b n (g h-f i) \text{PolyLog}\left (2,-\frac{g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g^2}+\frac{6 b^3 n^3 (g h-f i) \text{PolyLog}\left (4,-\frac{g (d+e x)}{e f-d g}\right )}{g^2}+\frac{6 a b^2 i n^2 x}{g}+\frac{(g h-f i) \log \left (\frac{e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{g^2}-\frac{3 b i n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e g}+\frac{i (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e g}+\frac{6 b^3 i n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e g}-\frac{6 b^3 i n^3 x}{g} \]
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Rubi [A] time = 0.363952, antiderivative size = 308, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.31, Rules used = {2418, 2389, 2296, 2295, 2396, 2433, 2374, 2383, 6589} \[ -\frac{6 b^2 n^2 (g h-f i) \text{PolyLog}\left (3,-\frac{g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^2}+\frac{3 b n (g h-f i) \text{PolyLog}\left (2,-\frac{g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g^2}+\frac{6 b^3 n^3 (g h-f i) \text{PolyLog}\left (4,-\frac{g (d+e x)}{e f-d g}\right )}{g^2}+\frac{6 a b^2 i n^2 x}{g}+\frac{(g h-f i) \log \left (\frac{e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{g^2}-\frac{3 b i n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e g}+\frac{i (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e g}+\frac{6 b^3 i n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e g}-\frac{6 b^3 i n^3 x}{g} \]
Antiderivative was successfully verified.
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Rule 2418
Rule 2389
Rule 2296
Rule 2295
Rule 2396
Rule 2433
Rule 2374
Rule 2383
Rule 6589
Rubi steps
\begin{align*} \int \frac{(h+230 x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{f+g x} \, dx &=\int \left (\frac{230 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{g}+\frac{(-230 f+g h) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{g (f+g x)}\right ) \, dx\\ &=\frac{230 \int \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx}{g}+\frac{(-230 f+g h) \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{f+g x} \, dx}{g}\\ &=-\frac{(230 f-g h) \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 )}{g^2}+\frac{230 \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e x\right )}{e g}+\frac{(3 b e (230 f-g h) 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}{g^2}\\ &=\frac{230 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e g}-\frac{(230 f-g h) \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 )}{g^2}-\frac{(690 b n) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{e g}+\frac{(3 b (230 f-g h) 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 )}{g^2}\\ &=-\frac{690 b n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e g}+\frac{230 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e g}-\frac{(230 f-g h) \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 )}{g^2}-\frac{3 b (230 f-g h) 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 )}{g^2}+\frac{\left (1380 b^2 n^2\right ) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{e g}+\frac{\left (6 b^2 (230 f-g h) 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 )}{g^2}\\ &=\frac{1380 a b^2 n^2 x}{g}-\frac{690 b n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e g}+\frac{230 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e g}-\frac{(230 f-g h) \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 )}{g^2}-\frac{3 b (230 f-g h) 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 )}{g^2}+\frac{6 b^2 (230 f-g h) 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 )}{g^2}+\frac{\left (1380 b^3 n^2\right ) \operatorname{Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e g}-\frac{\left (6 b^3 (230 f-g h) 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 )}{g^2}\\ &=\frac{1380 a b^2 n^2 x}{g}-\frac{1380 b^3 n^3 x}{g}+\frac{1380 b^3 n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e g}-\frac{690 b n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e g}+\frac{230 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e g}-\frac{(230 f-g h) \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 )}{g^2}-\frac{3 b (230 f-g h) 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 )}{g^2}+\frac{6 b^2 (230 f-g h) 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 )}{g^2}-\frac{6 b^3 (230 f-g h) n^3 \text{Li}_4\left (-\frac{g (d+e x)}{e f-d g}\right )}{g^2}\\ \end{align*}
Mathematica [B] time = 0.378134, size = 799, normalized size = 2.59 \[ \frac{b^3 e g h \left (\log \left (\frac{e (f+g x)}{e f-d g}\right ) \log ^3(d+e x)+3 \text{PolyLog}\left (2,\frac{g (d+e x)}{d g-e f}\right ) \log ^2(d+e x)-6 \text{PolyLog}\left (3,\frac{g (d+e x)}{d g-e f}\right ) \log (d+e x)+6 \text{PolyLog}\left (4,\frac{g (d+e x)}{d g-e f}\right )\right ) n^3-b^3 i \left (g \left (-(d+e x) \log ^3(d+e x)+3 (d+e x) \log ^2(d+e x)-6 (d+e x) \log (d+e x)+6 e x\right )+e f \left (\log \left (\frac{e (f+g x)}{e f-d g}\right ) \log ^3(d+e x)+3 \text{PolyLog}\left (2,\frac{g (d+e x)}{d g-e f}\right ) \log ^2(d+e x)-6 \text{PolyLog}\left (3,\frac{g (d+e x)}{d g-e f}\right ) \log (d+e x)+6 \text{PolyLog}\left (4,\frac{g (d+e x)}{d g-e f}\right )\right )\right ) n^3+3 b^2 i \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left (g \left ((d+e x) \log ^2(d+e x)-2 (d+e x) \log (d+e x)+2 e x\right )-e f \left (\log \left (\frac{e (f+g x)}{e f-d g}\right ) \log ^2(d+e x)+2 \text{PolyLog}\left (2,\frac{g (d+e x)}{d g-e f}\right ) \log (d+e x)-2 \text{PolyLog}\left (3,\frac{g (d+e x)}{d g-e f}\right )\right )\right ) n^2+6 b^2 e g h \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left (\frac{1}{2} \log \left (\frac{e (f+g x)}{e f-d g}\right ) \log ^2(d+e x)+\text{PolyLog}\left (2,\frac{g (d+e x)}{d g-e f}\right ) \log (d+e x)-\text{PolyLog}\left (3,\frac{g (d+e x)}{d g-e f}\right )\right ) n^2+3 b e g h \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2 \left (\log (d+e x) \log \left (\frac{e (f+g x)}{e f-d g}\right )+\text{PolyLog}\left (2,\frac{g (d+e x)}{d g-e f}\right )\right ) n-3 b i \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2 \left (e f \left (\log (d+e x) \log \left (\frac{e (f+g x)}{e f-d g}\right )+\text{PolyLog}\left (2,\frac{g (d+e x)}{d g-e f}\right )\right )-g (d+e x) (\log (d+e x)-1)\right ) n+e g i x \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^3+e (g h-f i) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^3 \log (f+g x)}{e g^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 2.141, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ix+h \right ) \left ( a+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) \right ) ^{3}}{gx+f}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} a^{3} i{\left (\frac{x}{g} - \frac{f \log \left (g x + f\right )}{g^{2}}\right )} + \frac{a^{3} h \log \left (g x + f\right )}{g} + \int \frac{b^{3} h \log \left (c\right )^{3} + 3 \, a b^{2} h \log \left (c\right )^{2} + 3 \, a^{2} b h \log \left (c\right ) +{\left (b^{3} i x + b^{3} h\right )} \log \left ({\left (e x + d\right )}^{n}\right )^{3} + 3 \,{\left (b^{3} h \log \left (c\right ) + a b^{2} h +{\left (b^{3} i \log \left (c\right ) + a b^{2} i\right )} x\right )} \log \left ({\left (e x + d\right )}^{n}\right )^{2} +{\left (b^{3} i \log \left (c\right )^{3} + 3 \, a b^{2} i \log \left (c\right )^{2} + 3 \, a^{2} b i \log \left (c\right )\right )} x + 3 \,{\left (b^{3} h \log \left (c\right )^{2} + 2 \, a b^{2} h \log \left (c\right ) + a^{2} b h +{\left (b^{3} i \log \left (c\right )^{2} + 2 \, a b^{2} i \log \left (c\right ) + a^{2} b i\right )} x\right )} \log \left ({\left (e x + d\right )}^{n}\right )}{g x + f}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{a^{3} i x + a^{3} h +{\left (b^{3} i x + b^{3} h\right )} \log \left ({\left (e x + d\right )}^{n} c\right )^{3} + 3 \,{\left (a b^{2} i x + a b^{2} h\right )} \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + 3 \,{\left (a^{2} b i x + a^{2} b h\right )} \log \left ({\left (e x + d\right )}^{n} c\right )}{g x + f}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \log{\left (c \left (d + e x\right )^{n} \right )}\right )^{3} \left (h + i x\right )}{f + g x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (i x + h\right )}{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{3}}{g x + f}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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