Optimal. Leaf size=239 \[ \frac{b^2 e^3 n^2 \text{PolyLog}\left (2,\frac{d}{d+\frac{e}{x^{2/3}}}\right )}{d^3}-\frac{b e^3 n \log \left (1-\frac{d}{d+\frac{e}{x^{2/3}}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{d^3}-\frac{b e^2 n x^{2/3} \left (d+\frac{e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{d^3}+\frac{b e n x^{4/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{2 d}+\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2+\frac{b^2 e^2 n^2 x^{2/3}}{2 d^2}-\frac{b^2 e^3 n^2 \log \left (d+\frac{e}{x^{2/3}}\right )}{2 d^3}-\frac{b^2 e^3 n^2 \log (x)}{d^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.484285, antiderivative size = 264, normalized size of antiderivative = 1.1, number of steps used = 14, number of rules used = 12, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.546, Rules used = {2454, 2398, 2411, 2347, 2344, 2301, 2317, 2391, 2314, 31, 2319, 44} \[ -\frac{b^2 e^3 n^2 \text{PolyLog}\left (2,\frac{e}{d x^{2/3}}+1\right )}{d^3}+\frac{e^3 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{2 d^3}-\frac{b e^3 n \log \left (-\frac{e}{d x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{d^3}-\frac{b e^2 n x^{2/3} \left (d+\frac{e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{d^3}+\frac{b e n x^{4/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{2 d}+\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2+\frac{b^2 e^2 n^2 x^{2/3}}{2 d^2}-\frac{b^2 e^3 n^2 \log \left (d+\frac{e}{x^{2/3}}\right )}{2 d^3}-\frac{b^2 e^3 n^2 \log (x)}{d^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2454
Rule 2398
Rule 2411
Rule 2347
Rule 2344
Rule 2301
Rule 2317
Rule 2391
Rule 2314
Rule 31
Rule 2319
Rule 44
Rubi steps
\begin{align*} \int x \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2 \, dx &=-\left (\frac{3}{2} \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^4} \, dx,x,\frac{1}{x^{2/3}}\right )\right )\\ &=\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2-(b e n) \operatorname{Subst}\left (\int \frac{a+b \log \left (c (d+e x)^n\right )}{x^3 (d+e x)} \, dx,x,\frac{1}{x^{2/3}}\right )\\ &=\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2-(b n) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^3} \, dx,x,d+\frac{e}{x^{2/3}}\right )\\ &=\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2-\frac{(b n) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{\left (-\frac{d}{e}+\frac{x}{e}\right )^3} \, dx,x,d+\frac{e}{x^{2/3}}\right )}{d}+\frac{(b e n) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^2} \, dx,x,d+\frac{e}{x^{2/3}}\right )}{d}\\ &=\frac{b e n x^{4/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{2 d}+\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2+\frac{(b e n) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{\left (-\frac{d}{e}+\frac{x}{e}\right )^2} \, dx,x,d+\frac{e}{x^{2/3}}\right )}{d^2}-\frac{\left (b e^2 n\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x \left (-\frac{d}{e}+\frac{x}{e}\right )} \, dx,x,d+\frac{e}{x^{2/3}}\right )}{d^2}-\frac{\left (b^2 e n^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^2} \, dx,x,d+\frac{e}{x^{2/3}}\right )}{2 d}\\ &=-\frac{b e^2 n \left (d+\frac{e}{x^{2/3}}\right ) x^{2/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{d^3}+\frac{b e n x^{4/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{2 d}+\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2-\frac{\left (b e^2 n\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{-\frac{d}{e}+\frac{x}{e}} \, dx,x,d+\frac{e}{x^{2/3}}\right )}{d^3}+\frac{\left (b e^3 n\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x} \, dx,x,d+\frac{e}{x^{2/3}}\right )}{d^3}-\frac{\left (b^2 e n^2\right ) \operatorname{Subst}\left (\int \left (\frac{e^2}{d (d-x)^2}+\frac{e^2}{d^2 (d-x)}+\frac{e^2}{d^2 x}\right ) \, dx,x,d+\frac{e}{x^{2/3}}\right )}{2 d}+\frac{\left (b^2 e^2 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{d}{e}+\frac{x}{e}} \, dx,x,d+\frac{e}{x^{2/3}}\right )}{d^3}\\ &=\frac{b^2 e^2 n^2 x^{2/3}}{2 d^2}-\frac{b^2 e^3 n^2 \log \left (d+\frac{e}{x^{2/3}}\right )}{2 d^3}-\frac{b e^2 n \left (d+\frac{e}{x^{2/3}}\right ) x^{2/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{d^3}+\frac{b e n x^{4/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{2 d}+\frac{e^3 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{2 d^3}+\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2-\frac{b e^3 n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right ) \log \left (-\frac{e}{d x^{2/3}}\right )}{d^3}-\frac{b^2 e^3 n^2 \log (x)}{d^3}+\frac{\left (b^2 e^3 n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{d}\right )}{x} \, dx,x,d+\frac{e}{x^{2/3}}\right )}{d^3}\\ &=\frac{b^2 e^2 n^2 x^{2/3}}{2 d^2}-\frac{b^2 e^3 n^2 \log \left (d+\frac{e}{x^{2/3}}\right )}{2 d^3}-\frac{b e^2 n \left (d+\frac{e}{x^{2/3}}\right ) x^{2/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{d^3}+\frac{b e n x^{4/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{2 d}+\frac{e^3 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{2 d^3}+\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2-\frac{b e^3 n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right ) \log \left (-\frac{e}{d x^{2/3}}\right )}{d^3}-\frac{b^2 e^3 n^2 \log (x)}{d^3}-\frac{b^2 e^3 n^2 \text{Li}_2\left (1+\frac{e}{d x^{2/3}}\right )}{d^3}\\ \end{align*}
Mathematica [B] time = 0.44262, size = 542, normalized size = 2.27 \[ \frac{1}{2} x^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2-\frac{b e n \left (3 b e^2 n \left (-4 \text{PolyLog}\left (2,1-\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )+2 \text{PolyLog}\left (2,\frac{1}{2}-\frac{\sqrt{-d} \sqrt [3]{x}}{2 \sqrt{e}}\right )+\log \left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right ) \left (\log \left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right )+2 \log \left (\frac{1}{2} \left (\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}+1\right )\right )-4 \log \left (\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )\right )\right )+3 b e^2 n \left (2 \text{PolyLog}\left (2,\frac{1}{2} \left (\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}+1\right )\right )-4 \text{PolyLog}\left (2,\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}+1\right )+\log \left (\sqrt{-d} \sqrt [3]{x}+\sqrt{e}\right ) \left (\log \left (\sqrt{-d} \sqrt [3]{x}+\sqrt{e}\right )+2 \log \left (\frac{1}{2}-\frac{\sqrt{-d} \sqrt [3]{x}}{2 \sqrt{e}}\right )-4 \log \left (-\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )\right )\right )-3 d^2 x^{4/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )-6 e^2 \log \left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )-6 e^2 \log \left (\sqrt{-d} \sqrt [3]{x}+\sqrt{e}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )+6 d e x^{2/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )+2 b e^2 n \left (3 \log \left (d+\frac{e}{x^{2/3}}\right )+2 \log (x)\right )+b e n \left (3 e \log \left (d+\frac{e}{x^{2/3}}\right )-3 d x^{2/3}+2 e \log (x)\right )\right )}{6 d^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.35, size = 0, normalized size = 0. \begin{align*} \int x \left ( a+b\ln \left ( c \left ( d+{e{x}^{-{\frac{2}{3}}}} \right ) ^{n} \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \, b^{2} x^{2} \log \left ({\left (d x^{\frac{2}{3}} + e\right )}^{n}\right )^{2} - \int -\frac{3 \,{\left (b^{2} d \log \left (c\right )^{2} + 2 \, a b d \log \left (c\right ) + a^{2} d\right )} x^{2} + 12 \,{\left (b^{2} d x^{2} + b^{2} e x^{\frac{4}{3}}\right )} \log \left (x^{\frac{1}{3} \, n}\right )^{2} + 3 \,{\left (b^{2} e \log \left (c\right )^{2} + 2 \, a b e \log \left (c\right ) + a^{2} e\right )} x^{\frac{4}{3}} - 2 \,{\left (b^{2} d n x^{2} - 3 \,{\left (b^{2} d \log \left (c\right ) + a b d\right )} x^{2} - 3 \,{\left (b^{2} e \log \left (c\right ) + a b e\right )} x^{\frac{4}{3}} + 6 \,{\left (b^{2} d x^{2} + b^{2} e x^{\frac{4}{3}}\right )} \log \left (x^{\frac{1}{3} \, n}\right )\right )} \log \left ({\left (d x^{\frac{2}{3}} + e\right )}^{n}\right ) - 12 \,{\left ({\left (b^{2} d \log \left (c\right ) + a b d\right )} x^{2} +{\left (b^{2} e \log \left (c\right ) + a b e\right )} x^{\frac{4}{3}}\right )} \log \left (x^{\frac{1}{3} \, n}\right )}{3 \,{\left (d x + e x^{\frac{1}{3}}\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (b^{2} x \log \left (c \left (\frac{d x + e x^{\frac{1}{3}}}{x}\right )^{n}\right )^{2} + 2 \, a b x \log \left (c \left (\frac{d x + e x^{\frac{1}{3}}}{x}\right )^{n}\right ) + a^{2} x, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \log \left (c{\left (d + \frac{e}{x^{\frac{2}{3}}}\right )}^{n}\right ) + a\right )}^{2} x\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]