Optimal. Leaf size=364 \[ -\frac{4 i b^2 d^{3/2} n^2 \text{PolyLog}\left (2,1-\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} \sqrt [3]{x}}\right )}{e^{3/2}}-\frac{4 b d^{3/2} n \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{e^{3/2}}-\frac{4}{3} b n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )+x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2+\frac{4 a b d n \sqrt [3]{x}}{e}+\frac{4 b^2 d n \sqrt [3]{x} \log \left (c \left (d+e x^{2/3}\right )^n\right )}{e}-\frac{4 i b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )^2}{e^{3/2}}+\frac{32 b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )}{3 e^{3/2}}-\frac{8 b^2 d^{3/2} n^2 \log \left (\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} \sqrt [3]{x}}\right ) \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )}{e^{3/2}}-\frac{32 b^2 d n^2 \sqrt [3]{x}}{3 e}+\frac{8}{9} b^2 n^2 x \]
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Rubi [A] time = 0.446778, antiderivative size = 364, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 14, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.7, Rules used = {2451, 2457, 2476, 2448, 321, 205, 2455, 302, 2470, 12, 4920, 4854, 2402, 2315} \[ -\frac{4 i b^2 d^{3/2} n^2 \text{PolyLog}\left (2,1-\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} \sqrt [3]{x}}\right )}{e^{3/2}}-\frac{4 b d^{3/2} n \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{e^{3/2}}-\frac{4}{3} b n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )+x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2+\frac{4 a b d n \sqrt [3]{x}}{e}+\frac{4 b^2 d n \sqrt [3]{x} \log \left (c \left (d+e x^{2/3}\right )^n\right )}{e}-\frac{4 i b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )^2}{e^{3/2}}+\frac{32 b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )}{3 e^{3/2}}-\frac{8 b^2 d^{3/2} n^2 \log \left (\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} \sqrt [3]{x}}\right ) \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )}{e^{3/2}}-\frac{32 b^2 d n^2 \sqrt [3]{x}}{3 e}+\frac{8}{9} b^2 n^2 x \]
Antiderivative was successfully verified.
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Rule 2451
Rule 2457
Rule 2476
Rule 2448
Rule 321
Rule 205
Rule 2455
Rule 302
Rule 2470
Rule 12
Rule 4920
Rule 4854
Rule 2402
Rule 2315
Rubi steps
\begin{align*} \int \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \, dx &=3 \operatorname{Subst}\left (\int x^2 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2 \, dx,x,\sqrt [3]{x}\right )\\ &=x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-(4 b e n) \operatorname{Subst}\left (\int \frac{x^4 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )\\ &=x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-(4 b e n) \operatorname{Subst}\left (\int \left (-\frac{d \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^2}+\frac{x^2 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e}+\frac{d^2 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^2 \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-(4 b n) \operatorname{Subst}\left (\int x^2 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )+\frac{(4 b d n) \operatorname{Subst}\left (\int \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{e}-\frac{\left (4 b d^2 n\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+e x^2\right )^n\right )}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{e}\\ &=\frac{4 a b d n \sqrt [3]{x}}{e}-\frac{4}{3} b n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )-\frac{4 b d^{3/2} n \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{e^{3/2}}+x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2+\frac{\left (4 b^2 d n\right ) \operatorname{Subst}\left (\int \log \left (c \left (d+e x^2\right )^n\right ) \, dx,x,\sqrt [3]{x}\right )}{e}+\left (8 b^2 d^2 n^2\right ) \operatorname{Subst}\left (\int \frac{x \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} \sqrt{e} \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )+\frac{1}{3} \left (8 b^2 e n^2\right ) \operatorname{Subst}\left (\int \frac{x^4}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{4 a b d n \sqrt [3]{x}}{e}+\frac{4 b^2 d n \sqrt [3]{x} \log \left (c \left (d+e x^{2/3}\right )^n\right )}{e}-\frac{4}{3} b n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )-\frac{4 b d^{3/2} n \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{e^{3/2}}+x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-\left (8 b^2 d n^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )+\frac{\left (8 b^2 d^{3/2} n^2\right ) \operatorname{Subst}\left (\int \frac{x \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{\sqrt{e}}+\frac{1}{3} \left (8 b^2 e n^2\right ) \operatorname{Subst}\left (\int \left (-\frac{d}{e^2}+\frac{x^2}{e}+\frac{d^2}{e^2 \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{4 a b d n \sqrt [3]{x}}{e}-\frac{32 b^2 d n^2 \sqrt [3]{x}}{3 e}+\frac{8}{9} b^2 n^2 x-\frac{4 i b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )^2}{e^{3/2}}+\frac{4 b^2 d n \sqrt [3]{x} \log \left (c \left (d+e x^{2/3}\right )^n\right )}{e}-\frac{4}{3} b n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )-\frac{4 b d^{3/2} n \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{e^{3/2}}+x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-\frac{\left (8 b^2 d n^2\right ) \operatorname{Subst}\left (\int \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{i-\frac{\sqrt{e} x}{\sqrt{d}}} \, dx,x,\sqrt [3]{x}\right )}{e}+\frac{\left (8 b^2 d^2 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{3 e}+\frac{\left (8 b^2 d^2 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{e}\\ &=\frac{4 a b d n \sqrt [3]{x}}{e}-\frac{32 b^2 d n^2 \sqrt [3]{x}}{3 e}+\frac{8}{9} b^2 n^2 x+\frac{32 b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )}{3 e^{3/2}}-\frac{4 i b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )^2}{e^{3/2}}-\frac{8 b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right ) \log \left (\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} \sqrt [3]{x}}\right )}{e^{3/2}}+\frac{4 b^2 d n \sqrt [3]{x} \log \left (c \left (d+e x^{2/3}\right )^n\right )}{e}-\frac{4}{3} b n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )-\frac{4 b d^{3/2} n \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{e^{3/2}}+x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2+\frac{\left (8 b^2 d n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{2}{1+\frac{i \sqrt{e} x}{\sqrt{d}}}\right )}{1+\frac{e x^2}{d}} \, dx,x,\sqrt [3]{x}\right )}{e}\\ &=\frac{4 a b d n \sqrt [3]{x}}{e}-\frac{32 b^2 d n^2 \sqrt [3]{x}}{3 e}+\frac{8}{9} b^2 n^2 x+\frac{32 b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )}{3 e^{3/2}}-\frac{4 i b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )^2}{e^{3/2}}-\frac{8 b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right ) \log \left (\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} \sqrt [3]{x}}\right )}{e^{3/2}}+\frac{4 b^2 d n \sqrt [3]{x} \log \left (c \left (d+e x^{2/3}\right )^n\right )}{e}-\frac{4}{3} b n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )-\frac{4 b d^{3/2} n \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{e^{3/2}}+x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-\frac{\left (8 i b^2 d^{3/2} n^2\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+\frac{i \sqrt{e} \sqrt [3]{x}}{\sqrt{d}}}\right )}{e^{3/2}}\\ &=\frac{4 a b d n \sqrt [3]{x}}{e}-\frac{32 b^2 d n^2 \sqrt [3]{x}}{3 e}+\frac{8}{9} b^2 n^2 x+\frac{32 b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )}{3 e^{3/2}}-\frac{4 i b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )^2}{e^{3/2}}-\frac{8 b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right ) \log \left (\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} \sqrt [3]{x}}\right )}{e^{3/2}}+\frac{4 b^2 d n \sqrt [3]{x} \log \left (c \left (d+e x^{2/3}\right )^n\right )}{e}-\frac{4}{3} b n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )-\frac{4 b d^{3/2} n \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{e^{3/2}}+x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-\frac{4 i b^2 d^{3/2} n^2 \text{Li}_2\left (1-\frac{2}{1+\frac{i \sqrt{e} \sqrt [3]{x}}{\sqrt{d}}}\right )}{e^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.216445, size = 319, normalized size = 0.88 \[ \frac{-36 i b^2 d^{3/2} n^2 \text{PolyLog}\left (2,\frac{\sqrt{e} \sqrt [3]{x}+i \sqrt{d}}{\sqrt{e} \sqrt [3]{x}-i \sqrt{d}}\right )+\sqrt{e} \sqrt [3]{x} \left (9 a^2 e x^{2/3}+6 b \left (3 a e x^{2/3}+6 b d n-2 b e n x^{2/3}\right ) \log \left (c \left (d+e x^{2/3}\right )^n\right )+12 a b n \left (3 d-e x^{2/3}\right )+9 b^2 e x^{2/3} \log ^2\left (c \left (d+e x^{2/3}\right )^n\right )+8 b^2 n^2 \left (e x^{2/3}-12 d\right )\right )-12 b d^{3/2} n \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right ) \left (3 a+3 b \log \left (c \left (d+e x^{2/3}\right )^n\right )+6 b n \log \left (\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} \sqrt [3]{x}}\right )-8 b n\right )-36 i b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )^2}{9 e^{3/2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.343, size = 0, normalized size = 0. \begin{align*} \int \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.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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 (b^{2} \log \left ({\left (e x^{\frac{2}{3}} + d\right )}^{n} c\right )^{2} + 2 \, a b \log \left ({\left (e x^{\frac{2}{3}} + d\right )}^{n} c\right ) + a^{2}, 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 \left (a + b \log{\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.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \log \left ({\left (e x^{\frac{2}{3}} + d\right )}^{n} c\right ) + a\right )}^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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