Optimal. Leaf size=309 \[ -\frac {4 b e^{3/2} n \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{d^{3/2}}+x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2+\frac {4 a b e n \sqrt [3]{x}}{d}+\frac {4 b^2 e n \sqrt [3]{x} \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{d}+\frac {4 i b^2 e^{3/2} n^2 \text {Li}_2\left (\frac {2 \sqrt {e}}{\sqrt {e}-i \sqrt {d} \sqrt [3]{x}}-1\right )}{d^{3/2}}+\frac {4 i b^2 e^{3/2} n^2 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )^2}{d^{3/2}}+\frac {8 b^2 e^{3/2} n^2 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )}{d^{3/2}}-\frac {8 b^2 e^{3/2} n^2 \log \left (2-\frac {2 \sqrt {e}}{\sqrt {e}-i \sqrt {d} \sqrt [3]{x}}\right ) \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )}{d^{3/2}} \]
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Rubi [A] time = 0.44, antiderivative size = 309, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 13, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.650, Rules used = {2451, 2457, 2471, 2448, 263, 205, 2470, 12, 260, 6688, 4924, 4868, 2447} \[ \frac {4 i b^2 e^{3/2} n^2 \text {PolyLog}\left (2,-1+\frac {2 \sqrt {e}}{\sqrt {e}-i \sqrt {d} \sqrt [3]{x}}\right )}{d^{3/2}}-\frac {4 b e^{3/2} n \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{d^{3/2}}+x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2+\frac {4 a b e n \sqrt [3]{x}}{d}+\frac {4 b^2 e n \sqrt [3]{x} \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{d}+\frac {4 i b^2 e^{3/2} n^2 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )^2}{d^{3/2}}+\frac {8 b^2 e^{3/2} n^2 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )}{d^{3/2}}-\frac {8 b^2 e^{3/2} n^2 \log \left (2-\frac {2 \sqrt {e}}{\sqrt {e}-i \sqrt {d} \sqrt [3]{x}}\right ) \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )}{d^{3/2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 205
Rule 260
Rule 263
Rule 2447
Rule 2448
Rule 2451
Rule 2457
Rule 2470
Rule 2471
Rule 4868
Rule 4924
Rule 6688
Rubi steps
\begin {align*} \int \left (a+b \log \left (c \left (d+\frac {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+\frac {e}{x^2}\right )^n\right )\right )^2 \, dx,x,\sqrt [3]{x}\right )\\ &=x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2+(4 b e n) \operatorname {Subst}\left (\int \frac {a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )}{d+\frac {e}{x^2}} \, dx,x,\sqrt [3]{x}\right )\\ &=x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2+(4 b e n) \operatorname {Subst}\left (\int \left (\frac {a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )}{d}-\frac {e \left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )}{d \left (e+d x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2+\frac {(4 b e n) \operatorname {Subst}\left (\int \left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{d}-\frac {\left (4 b e^2 n\right ) \operatorname {Subst}\left (\int \frac {a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{d}\\ &=\frac {4 a b e n \sqrt [3]{x}}{d}-\frac {4 b e^{3/2} n \tan ^{-1}\left (\frac {\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 )}{d^{3/2}}+x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2+\frac {\left (4 b^2 e n\right ) \operatorname {Subst}\left (\int \log \left (c \left (d+\frac {e}{x^2}\right )^n\right ) \, dx,x,\sqrt [3]{x}\right )}{d}-\frac {\left (8 b^2 e^3 n^2\right ) \operatorname {Subst}\left (\int \frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {e}}\right )}{\sqrt {d} \sqrt {e} \left (d+\frac {e}{x^2}\right ) x^3} \, dx,x,\sqrt [3]{x}\right )}{d}\\ &=\frac {4 a b e n \sqrt [3]{x}}{d}+\frac {4 b^2 e n \sqrt [3]{x} \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{d}-\frac {4 b e^{3/2} n \tan ^{-1}\left (\frac {\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 )}{d^{3/2}}+x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2+\frac {\left (8 b^2 e^2 n^2\right ) \operatorname {Subst}\left (\int \frac {1}{\left (d+\frac {e}{x^2}\right ) x^2} \, dx,x,\sqrt [3]{x}\right )}{d}-\frac {\left (8 b^2 e^{5/2} n^2\right ) \operatorname {Subst}\left (\int \frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {e}}\right )}{\left (d+\frac {e}{x^2}\right ) x^3} \, dx,x,\sqrt [3]{x}\right )}{d^{3/2}}\\ &=\frac {4 a b e n \sqrt [3]{x}}{d}+\frac {4 b^2 e n \sqrt [3]{x} \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{d}-\frac {4 b e^{3/2} n \tan ^{-1}\left (\frac {\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 )}{d^{3/2}}+x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2+\frac {\left (8 b^2 e^2 n^2\right ) \operatorname {Subst}\left (\int \frac {1}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{d}-\frac {\left (8 b^2 e^{5/2} n^2\right ) \operatorname {Subst}\left (\int \frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {e}}\right )}{x \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{d^{3/2}}\\ &=\frac {4 a b e n \sqrt [3]{x}}{d}+\frac {8 b^2 e^{3/2} n^2 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )}{d^{3/2}}+\frac {4 i b^2 e^{3/2} n^2 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )^2}{d^{3/2}}+\frac {4 b^2 e n \sqrt [3]{x} \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{d}-\frac {4 b e^{3/2} n \tan ^{-1}\left (\frac {\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 )}{d^{3/2}}+x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2-\frac {\left (8 i b^2 e^{3/2} n^2\right ) \operatorname {Subst}\left (\int \frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {e}}\right )}{x \left (i+\frac {\sqrt {d} x}{\sqrt {e}}\right )} \, dx,x,\sqrt [3]{x}\right )}{d^{3/2}}\\ &=\frac {4 a b e n \sqrt [3]{x}}{d}+\frac {8 b^2 e^{3/2} n^2 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )}{d^{3/2}}+\frac {4 i b^2 e^{3/2} n^2 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )^2}{d^{3/2}}-\frac {8 b^2 e^{3/2} n^2 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right ) \log \left (2-\frac {2 \sqrt {e}}{\sqrt {e}-i \sqrt {d} \sqrt [3]{x}}\right )}{d^{3/2}}+\frac {4 b^2 e n \sqrt [3]{x} \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{d}-\frac {4 b e^{3/2} n \tan ^{-1}\left (\frac {\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 )}{d^{3/2}}+x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2+\frac {\left (8 b^2 e n^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (2-\frac {2}{1-\frac {i \sqrt {d} x}{\sqrt {e}}}\right )}{1+\frac {d x^2}{e}} \, dx,x,\sqrt [3]{x}\right )}{d}\\ &=\frac {4 a b e n \sqrt [3]{x}}{d}+\frac {8 b^2 e^{3/2} n^2 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )}{d^{3/2}}+\frac {4 i b^2 e^{3/2} n^2 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )^2}{d^{3/2}}-\frac {8 b^2 e^{3/2} n^2 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right ) \log \left (2-\frac {2 \sqrt {e}}{\sqrt {e}-i \sqrt {d} \sqrt [3]{x}}\right )}{d^{3/2}}+\frac {4 b^2 e n \sqrt [3]{x} \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{d}-\frac {4 b e^{3/2} n \tan ^{-1}\left (\frac {\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 )}{d^{3/2}}+x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2+\frac {4 i b^2 e^{3/2} n^2 \text {Li}_2\left (-1+\frac {2 \sqrt {e}}{\sqrt {e}-i \sqrt {d} \sqrt [3]{x}}\right )}{d^{3/2}}\\ \end {align*}
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Mathematica [A] time = 1.07, size = 523, normalized size = 1.69 \[ b e n \left (-\frac {2 \sqrt {e} \log \left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{(-d)^{3/2}}+\frac {2 \sqrt {e} \log \left (\sqrt {-d} \sqrt [3]{x}+\sqrt {e}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{(-d)^{3/2}}+\frac {4 a \sqrt [3]{x}}{d}+\frac {4 b \sqrt [3]{x} \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{d}-\frac {8 b \sqrt {e} n \tan ^{-1}\left (\frac {\sqrt {e}}{\sqrt {d} \sqrt [3]{x}}\right )}{d^{3/2}}+\frac {b \sqrt {e} n \left (-4 \text {Li}_2\left (1-\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )+2 \text {Li}_2\left (\frac {1}{2}-\frac {\sqrt {-d} \sqrt [3]{x}}{2 \sqrt {e}}\right )+\log \left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right ) \left (\log \left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right )+2 \log \left (\frac {1}{2} \left (\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}+1\right )\right )-4 \log \left (\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )\right )\right )}{(-d)^{3/2}}+\frac {b d \sqrt {e} n \left (2 \text {Li}_2\left (\frac {1}{2} \left (\frac {\sqrt [3]{x} \sqrt {-d}}{\sqrt {e}}+1\right )\right )-4 \text {Li}_2\left (\frac {\sqrt [3]{x} \sqrt {-d}}{\sqrt {e}}+1\right )+\log \left (\sqrt {-d} \sqrt [3]{x}+\sqrt {e}\right ) \left (\log \left (\sqrt {-d} \sqrt [3]{x}+\sqrt {e}\right )+2 \log \left (\frac {1}{2}-\frac {\sqrt {-d} \sqrt [3]{x}}{2 \sqrt {e}}\right )-4 \log \left (-\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )\right )\right )}{(-d)^{5/2}}\right )+x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2 \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b^{2} \log \left (c \left (\frac {d x + e x^{\frac {1}{3}}}{x}\right )^{n}\right )^{2} + 2 \, a b \log \left (c \left (\frac {d x + e x^{\frac {1}{3}}}{x}\right )^{n}\right ) + a^{2}, x\right ) \]
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 \[ \int {\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {2}{3}}}\right )}^{n}\right ) + a\right )}^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.10, size = 0, normalized size = 0.00 \[ \int \left (b \ln \left (c \left (d +\frac {e}{x^{\frac {2}{3}}}\right )^{n}\right )+a \right )^{2}\, dx \]
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 \[ -2 \, {\left (2 \, e n {\left (\frac {e \arctan \left (\frac {d x^{\frac {1}{3}}}{\sqrt {d e}}\right )}{\sqrt {d e} d} - \frac {x^{\frac {1}{3}}}{d}\right )} - x \log \left (c {\left (d + \frac {e}{x^{\frac {2}{3}}}\right )}^{n}\right )\right )} a b + {\left (n^{2} x \log \left (d x^{\frac {2}{3}} + e\right )^{2} - \int -\frac {3 \, d x \log \relax (c)^{2} + 3 \, e x^{\frac {1}{3}} \log \relax (c)^{2} - 2 \, {\left (2 \, d n x - 3 \, d x \log \relax (c) - 3 \, e x^{\frac {1}{3}} \log \relax (c) + 6 \, {\left (d x + e x^{\frac {1}{3}}\right )} \log \left (x^{\frac {1}{3} \, n}\right )\right )} n \log \left (d x^{\frac {2}{3}} + e\right ) + 12 \, {\left (d x + e x^{\frac {1}{3}}\right )} \log \left (x^{\frac {1}{3} \, n}\right )^{2} - 12 \, {\left (d x \log \relax (c) + e x^{\frac {1}{3}} \log \relax (c)\right )} \log \left (x^{\frac {1}{3} \, n}\right )}{3 \, {\left (d x + e x^{\frac {1}{3}}\right )}}\,{d x}\right )} b^{2} + a^{2} x \]
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 \[ \int {\left (a+b\,\ln \left (c\,{\left (d+\frac {e}{x^{2/3}}\right )}^n\right )\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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