Optimal. Leaf size=660 \[ -\frac{6 b^2 n^2 (g h-f i)^2 \text{PolyLog}\left (3,-\frac{g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^3}+\frac{3 b n (g h-f i)^2 \text{PolyLog}\left (2,-\frac{g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g^3}+\frac{6 b^3 n^3 (g h-f i)^2 \text{PolyLog}\left (4,-\frac{g (d+e x)}{e f-d g}\right )}{g^3}+\frac{3 b^2 i^2 n^2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 e^2 g}+\frac{6 a b^2 i n^2 x (e h-d i)}{e g}+\frac{6 a b^2 i n^2 x (g h-f i)}{g^2}-\frac{3 b i n (d+e x) (e h-d i) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2 g}+\frac{i (d+e x) (e h-d i) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^2 g}-\frac{3 b i^2 n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 e^2 g}+\frac{i^2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{2 e^2 g}-\frac{3 b i n (d+e x) (g h-f i) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e g^2}+\frac{i (d+e x) (g h-f i) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e g^2}+\frac{(g h-f i)^2 \log \left (\frac{e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{g^3}+\frac{6 b^3 i n^2 (d+e x) (e h-d i) \log \left (c (d+e x)^n\right )}{e^2 g}+\frac{6 b^3 i n^2 (d+e x) (g h-f i) \log \left (c (d+e x)^n\right )}{e g^2}-\frac{3 b^3 i^2 n^3 (d+e x)^2}{8 e^2 g}-\frac{6 b^3 i n^3 x (e h-d i)}{e g}-\frac{6 b^3 i n^3 x (g h-f i)}{g^2} \]
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Rubi [A] time = 0.733561, antiderivative size = 660, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 13, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.419, Rules used = {2418, 2389, 2296, 2295, 2396, 2433, 2374, 2383, 6589, 2401, 2390, 2305, 2304} \[ -\frac{6 b^2 n^2 (g h-f i)^2 \text{PolyLog}\left (3,-\frac{g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^3}+\frac{3 b n (g h-f i)^2 \text{PolyLog}\left (2,-\frac{g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g^3}+\frac{6 b^3 n^3 (g h-f i)^2 \text{PolyLog}\left (4,-\frac{g (d+e x)}{e f-d g}\right )}{g^3}+\frac{3 b^2 i^2 n^2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 e^2 g}+\frac{6 a b^2 i n^2 x (e h-d i)}{e g}+\frac{6 a b^2 i n^2 x (g h-f i)}{g^2}-\frac{3 b i n (d+e x) (e h-d i) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2 g}+\frac{i (d+e x) (e h-d i) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^2 g}-\frac{3 b i^2 n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 e^2 g}+\frac{i^2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{2 e^2 g}-\frac{3 b i n (d+e x) (g h-f i) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e g^2}+\frac{i (d+e x) (g h-f i) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e g^2}+\frac{(g h-f i)^2 \log \left (\frac{e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{g^3}+\frac{6 b^3 i n^2 (d+e x) (e h-d i) \log \left (c (d+e x)^n\right )}{e^2 g}+\frac{6 b^3 i n^2 (d+e x) (g h-f i) \log \left (c (d+e x)^n\right )}{e g^2}-\frac{3 b^3 i^2 n^3 (d+e x)^2}{8 e^2 g}-\frac{6 b^3 i n^3 x (e h-d i)}{e g}-\frac{6 b^3 i n^3 x (g h-f i)}{g^2} \]
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
Rule 2401
Rule 2390
Rule 2305
Rule 2304
Rubi steps
\begin{align*} \int \frac{(h+229 x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{f+g x} \, dx &=\int \left (\frac{229 (-229 f+g h) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{g^2}+\frac{229 (h+229 x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{g}+\frac{(-229 f+g h)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{g^2 (f+g x)}\right ) \, dx\\ &=\frac{229 \int (h+229 x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx}{g}-\frac{(229 (229 f-g h)) \int \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx}{g^2}+\frac{(229 f-g h)^2 \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{f+g x} \, dx}{g^2}\\ &=\frac{(229 f-g h)^2 \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^3}+\frac{229 \int \left (\frac{(-229 d+e h) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac{229 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}\right ) \, dx}{g}-\frac{(229 (229 f-g h)) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e x\right )}{e g^2}-\frac{\left (3 b e (229 f-g h)^2 n\right ) \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^3}\\ &=-\frac{229 (229 f-g h) (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e g^2}+\frac{(229 f-g h)^2 \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^3}+\frac{52441 \int (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx}{e g}-\frac{(229 (229 d-e h)) \int \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx}{e g}+\frac{(687 b (229 f-g h) n) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{e g^2}-\frac{\left (3 b (229 f-g h)^2 n\right ) \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^3}\\ &=\frac{687 b (229 f-g h) n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e g^2}-\frac{229 (229 f-g h) (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e g^2}+\frac{(229 f-g h)^2 \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^3}+\frac{3 b (229 f-g h)^2 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^3}+\frac{52441 \operatorname{Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e x\right )}{e^2 g}-\frac{(229 (229 d-e h)) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e x\right )}{e^2 g}-\frac{\left (1374 b^2 (229 f-g h) 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^2}-\frac{\left (6 b^2 (229 f-g h)^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 )}{g^3}\\ &=-\frac{1374 a b^2 (229 f-g h) n^2 x}{g^2}+\frac{687 b (229 f-g h) n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e g^2}-\frac{229 (229 d-e h) (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^2 g}-\frac{229 (229 f-g h) (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e g^2}+\frac{52441 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{2 e^2 g}+\frac{(229 f-g h)^2 \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^3}+\frac{3 b (229 f-g h)^2 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^3}-\frac{6 b^2 (229 f-g h)^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 )}{g^3}-\frac{(157323 b n) \operatorname{Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{2 e^2 g}+\frac{(687 b (229 d-e h) n) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{e^2 g}-\frac{\left (1374 b^3 (229 f-g h) n^2\right ) \operatorname{Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e g^2}+\frac{\left (6 b^3 (229 f-g h)^2 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^3}\\ &=-\frac{1374 a b^2 (229 f-g h) n^2 x}{g^2}+\frac{1374 b^3 (229 f-g h) n^3 x}{g^2}-\frac{1374 b^3 (229 f-g h) n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e g^2}+\frac{687 b (229 d-e h) n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2 g}+\frac{687 b (229 f-g h) n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e g^2}-\frac{157323 b n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 e^2 g}-\frac{229 (229 d-e h) (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^2 g}-\frac{229 (229 f-g h) (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e g^2}+\frac{52441 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{2 e^2 g}+\frac{(229 f-g h)^2 \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^3}+\frac{3 b (229 f-g h)^2 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^3}-\frac{6 b^2 (229 f-g h)^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 )}{g^3}+\frac{6 b^3 (229 f-g h)^2 n^3 \text{Li}_4\left (-\frac{g (d+e x)}{e f-d g}\right )}{g^3}+\frac{\left (157323 b^2 n^2\right ) \operatorname{Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{2 e^2 g}-\frac{\left (1374 b^2 (229 d-e h) n^2\right ) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{e^2 g}\\ &=-\frac{1374 a b^2 (229 d-e h) n^2 x}{e g}-\frac{1374 a b^2 (229 f-g h) n^2 x}{g^2}+\frac{1374 b^3 (229 f-g h) n^3 x}{g^2}-\frac{157323 b^3 n^3 (d+e x)^2}{8 e^2 g}-\frac{1374 b^3 (229 f-g h) n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e g^2}+\frac{157323 b^2 n^2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 e^2 g}+\frac{687 b (229 d-e h) n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2 g}+\frac{687 b (229 f-g h) n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e g^2}-\frac{157323 b n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 e^2 g}-\frac{229 (229 d-e h) (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^2 g}-\frac{229 (229 f-g h) (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e g^2}+\frac{52441 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{2 e^2 g}+\frac{(229 f-g h)^2 \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^3}+\frac{3 b (229 f-g h)^2 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^3}-\frac{6 b^2 (229 f-g h)^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 )}{g^3}+\frac{6 b^3 (229 f-g h)^2 n^3 \text{Li}_4\left (-\frac{g (d+e x)}{e f-d g}\right )}{g^3}-\frac{\left (1374 b^3 (229 d-e h) n^2\right ) \operatorname{Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e^2 g}\\ &=-\frac{1374 a b^2 (229 d-e h) n^2 x}{e g}-\frac{1374 a b^2 (229 f-g h) n^2 x}{g^2}+\frac{1374 b^3 (229 d-e h) n^3 x}{e g}+\frac{1374 b^3 (229 f-g h) n^3 x}{g^2}-\frac{157323 b^3 n^3 (d+e x)^2}{8 e^2 g}-\frac{1374 b^3 (229 d-e h) n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e^2 g}-\frac{1374 b^3 (229 f-g h) n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e g^2}+\frac{157323 b^2 n^2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 e^2 g}+\frac{687 b (229 d-e h) n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2 g}+\frac{687 b (229 f-g h) n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e g^2}-\frac{157323 b n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 e^2 g}-\frac{229 (229 d-e h) (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^2 g}-\frac{229 (229 f-g h) (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e g^2}+\frac{52441 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{2 e^2 g}+\frac{(229 f-g h)^2 \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^3}+\frac{3 b (229 f-g h)^2 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^3}-\frac{6 b^2 (229 f-g h)^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 )}{g^3}+\frac{6 b^3 (229 f-g h)^2 n^3 \text{Li}_4\left (-\frac{g (d+e x)}{e f-d g}\right )}{g^3}\\ \end{align*}
Mathematica [B] time = 0.862404, size = 1521, normalized size = 2.3 \[ \text{result too large to display} \]
Antiderivative was successfully verified.
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Maple [F] time = 2.272, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ix+h \right ) ^{2} \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*} 2 \, a^{3} h i{\left (\frac{x}{g} - \frac{f \log \left (g x + f\right )}{g^{2}}\right )} + \frac{1}{2} \, a^{3} i^{2}{\left (\frac{2 \, f^{2} \log \left (g x + f\right )}{g^{3}} + \frac{g x^{2} - 2 \, f x}{g^{2}}\right )} + \frac{a^{3} h^{2} \log \left (g x + f\right )}{g} + \int \frac{b^{3} h^{2} \log \left (c\right )^{3} + 3 \, a b^{2} h^{2} \log \left (c\right )^{2} + 3 \, a^{2} b h^{2} \log \left (c\right ) +{\left (b^{3} i^{2} x^{2} + 2 \, b^{3} h i x + b^{3} h^{2}\right )} \log \left ({\left (e x + d\right )}^{n}\right )^{3} +{\left (b^{3} i^{2} \log \left (c\right )^{3} + 3 \, a b^{2} i^{2} \log \left (c\right )^{2} + 3 \, a^{2} b i^{2} \log \left (c\right )\right )} x^{2} + 3 \,{\left (b^{3} h^{2} \log \left (c\right ) + a b^{2} h^{2} +{\left (b^{3} i^{2} \log \left (c\right ) + a b^{2} i^{2}\right )} x^{2} + 2 \,{\left (b^{3} h i \log \left (c\right ) + a b^{2} h i\right )} x\right )} \log \left ({\left (e x + d\right )}^{n}\right )^{2} + 2 \,{\left (b^{3} h i \log \left (c\right )^{3} + 3 \, a b^{2} h i \log \left (c\right )^{2} + 3 \, a^{2} b h i \log \left (c\right )\right )} x + 3 \,{\left (b^{3} h^{2} \log \left (c\right )^{2} + 2 \, a b^{2} h^{2} \log \left (c\right ) + a^{2} b h^{2} +{\left (b^{3} i^{2} \log \left (c\right )^{2} + 2 \, a b^{2} i^{2} \log \left (c\right ) + a^{2} b i^{2}\right )} x^{2} + 2 \,{\left (b^{3} h i \log \left (c\right )^{2} + 2 \, a b^{2} h i \log \left (c\right ) + a^{2} b h 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^{2} x^{2} + 2 \, a^{3} h i x + a^{3} h^{2} +{\left (b^{3} i^{2} x^{2} + 2 \, b^{3} h i x + b^{3} h^{2}\right )} \log \left ({\left (e x + d\right )}^{n} c\right )^{3} + 3 \,{\left (a b^{2} i^{2} x^{2} + 2 \, a b^{2} h i x + a b^{2} h^{2}\right )} \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + 3 \,{\left (a^{2} b i^{2} x^{2} + 2 \, a^{2} b h i x + a^{2} b h^{2}\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 )^{2}}{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 )}^{2}{\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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