Optimal. Leaf size=361 \[ -\frac {4 b d^{3/2} n \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{e^{3/2}}-\frac {4 b d n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{e \sqrt [3]{x}}+\frac {4 b n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{3 x}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x}+\frac {4 i b^2 d^{3/2} n^2 \text {Li}_2\left (\frac {2 \sqrt {e}}{\sqrt {e}-i \sqrt {d} \sqrt [3]{x}}-1\right )}{e^{3/2}}+\frac {4 i b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )^2}{e^{3/2}}+\frac {32 b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )}{3 e^{3/2}}-\frac {8 b^2 d^{3/2} n^2 \log \left (2-\frac {2 \sqrt {e}}{\sqrt {e}-i \sqrt {d} \sqrt [3]{x}}\right ) \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )}{e^{3/2}}+\frac {32 b^2 d n^2}{3 e \sqrt [3]{x}}-\frac {8 b^2 n^2}{9 x} \]
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Rubi [A] time = 0.59, antiderivative size = 361, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 14, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {2458, 2457, 2476, 2455, 263, 325, 205, 2470, 12, 260, 6688, 4924, 4868, 2447} \[ \frac {4 i b^2 d^{3/2} n^2 \text {PolyLog}\left (2,-1+\frac {2 \sqrt {e}}{\sqrt {e}-i \sqrt {d} \sqrt [3]{x}}\right )}{e^{3/2}}-\frac {4 b d^{3/2} n \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{e^{3/2}}-\frac {4 b d n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{e \sqrt [3]{x}}+\frac {4 b n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{3 x}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x}+\frac {4 i b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )^2}{e^{3/2}}+\frac {32 b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )}{3 e^{3/2}}-\frac {8 b^2 d^{3/2} n^2 \log \left (2-\frac {2 \sqrt {e}}{\sqrt {e}-i \sqrt {d} \sqrt [3]{x}}\right ) \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )}{e^{3/2}}+\frac {32 b^2 d n^2}{3 e \sqrt [3]{x}}-\frac {8 b^2 n^2}{9 x} \]
Antiderivative was successfully verified.
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Rule 12
Rule 205
Rule 260
Rule 263
Rule 325
Rule 2447
Rule 2455
Rule 2457
Rule 2458
Rule 2470
Rule 2476
Rule 4868
Rule 4924
Rule 6688
Rubi steps
\begin {align*} \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x^2} \, dx &=3 \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )^2}{x^4} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x}-(4 b e n) \operatorname {Subst}\left (\int \frac {a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )}{\left (d+\frac {e}{x^2}\right ) x^6} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x}-(4 b e n) \operatorname {Subst}\left (\int \left (\frac {a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )}{e x^4}-\frac {d \left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )}{e^2 x^2}+\frac {d^2 \left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )}{e^2 \left (e+d x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x}-(4 b n) \operatorname {Subst}\left (\int \frac {a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )}{x^4} \, dx,x,\sqrt [3]{x}\right )+\frac {(4 b d n) \operatorname {Subst}\left (\int \frac {a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )}{x^2} \, 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+\frac {e}{x^2}\right )^n\right )}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{e}\\ &=\frac {4 b n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{3 x}-\frac {4 b d n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{e \sqrt [3]{x}}-\frac {4 b d^{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 )}{e^{3/2}}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x}-\left (8 b^2 d n^2\right ) \operatorname {Subst}\left (\int \frac {1}{\left (d+\frac {e}{x^2}\right ) x^4} \, dx,x,\sqrt [3]{x}\right )-\left (8 b^2 d^2 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 )+\frac {1}{3} \left (8 b^2 e n^2\right ) \operatorname {Subst}\left (\int \frac {1}{\left (d+\frac {e}{x^2}\right ) x^6} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {4 b n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{3 x}-\frac {4 b d n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{e \sqrt [3]{x}}-\frac {4 b d^{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 )}{e^{3/2}}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x}-\left (8 b^2 d n^2\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )-\frac {\left (8 b^2 d^{3/2} 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 )}{\sqrt {e}}+\frac {1}{3} \left (8 b^2 e n^2\right ) \operatorname {Subst}\left (\int \frac {1}{x^4 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {8 b^2 n^2}{9 x}+\frac {8 b^2 d n^2}{e \sqrt [3]{x}}+\frac {4 b n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{3 x}-\frac {4 b d n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{e \sqrt [3]{x}}-\frac {4 b d^{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 )}{e^{3/2}}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x}-\frac {1}{3} \left (8 b^2 d n^2\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )+\frac {\left (8 b^2 d^2 n^2\right ) \operatorname {Subst}\left (\int \frac {1}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{e}-\frac {\left (8 b^2 d^{3/2} 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 )}{\sqrt {e}}\\ &=-\frac {8 b^2 n^2}{9 x}+\frac {32 b^2 d n^2}{3 e \sqrt [3]{x}}+\frac {8 b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )}{e^{3/2}}+\frac {4 i b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )^2}{e^{3/2}}+\frac {4 b n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{3 x}-\frac {4 b d n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{e \sqrt [3]{x}}-\frac {4 b d^{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 )}{e^{3/2}}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x}-\frac {\left (8 i b^2 d^{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 )}{e^{3/2}}+\frac {\left (8 b^2 d^2 n^2\right ) \operatorname {Subst}\left (\int \frac {1}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{3 e}\\ &=-\frac {8 b^2 n^2}{9 x}+\frac {32 b^2 d n^2}{3 e \sqrt [3]{x}}+\frac {32 b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )}{3 e^{3/2}}+\frac {4 i b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )^2}{e^{3/2}}-\frac {8 b^2 d^{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 )}{e^{3/2}}+\frac {4 b n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{3 x}-\frac {4 b d n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{e \sqrt [3]{x}}-\frac {4 b d^{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 )}{e^{3/2}}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x}+\frac {\left (8 b^2 d^2 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 )}{e^2}\\ &=-\frac {8 b^2 n^2}{9 x}+\frac {32 b^2 d n^2}{3 e \sqrt [3]{x}}+\frac {32 b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )}{3 e^{3/2}}+\frac {4 i b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )^2}{e^{3/2}}-\frac {8 b^2 d^{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 )}{e^{3/2}}+\frac {4 b n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{3 x}-\frac {4 b d n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{e \sqrt [3]{x}}-\frac {4 b d^{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 )}{e^{3/2}}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x}+\frac {4 i b^2 d^{3/2} n^2 \text {Li}_2\left (-1+\frac {2 \sqrt {e}}{\sqrt {e}-i \sqrt {d} \sqrt [3]{x}}\right )}{e^{3/2}}\\ \end {align*}
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Mathematica [A] time = 1.27, size = 598, normalized size = 1.66 \[ \frac {\frac {b n \left (12 e^{3/2} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )+18 (-d)^{3/2} x \log \left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )+18 d \sqrt {-d} x \log \left (\sqrt {-d} \sqrt [3]{x}+\sqrt {e}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )-36 d \sqrt {e} x^{2/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )-8 b n \left (3 d^{3/2} x \tan ^{-1}\left (\frac {\sqrt {e}}{\sqrt {d} \sqrt [3]{x}}\right )+\sqrt {e} \left (e-3 d x^{2/3}\right )\right )-72 b d^{3/2} n x \tan ^{-1}\left (\frac {\sqrt {e}}{\sqrt {d} \sqrt [3]{x}}\right )+9 b (-d)^{3/2} n x \left (2 \text {Li}_2\left (\frac {1}{2} \left (\frac {\sqrt [3]{x} \sqrt {-d}}{\sqrt {e}}+1\right )\right )-4 \text {Li}_2\left (\frac {\sqrt [3]{x} \sqrt {-d}}{\sqrt {e}}+1\right )+\log \left (\sqrt {-d} \sqrt [3]{x}+\sqrt {e}\right ) \left (\log \left (\sqrt {-d} \sqrt [3]{x}+\sqrt {e}\right )+2 \log \left (\frac {1}{2}-\frac {\sqrt {-d} \sqrt [3]{x}}{2 \sqrt {e}}\right )-4 \log \left (-\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )\right )\right )+9 b d \sqrt {-d} n x \left (-4 \text {Li}_2\left (1-\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )+2 \text {Li}_2\left (\frac {1}{2}-\frac {\sqrt {-d} \sqrt [3]{x}}{2 \sqrt {e}}\right )+\log \left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right ) \left (\log \left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right )+2 \log \left (\frac {1}{2} \left (\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}+1\right )\right )-4 \log \left (\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )\right )\right )+72 b d \sqrt {e} n x^{2/3}\right )}{e^{3/2}}-9 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{9 x} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \log \left (c \left (\frac {d x + e x^{\frac {1}{3}}}{x}\right )^{n}\right )^{2} + 2 \, a b \log \left (c \left (\frac {d x + e x^{\frac {1}{3}}}{x}\right )^{n}\right ) + a^{2}}{x^{2}}, x\right ) \]
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 \[ \int \frac {{\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {2}{3}}}\right )}^{n}\right ) + a\right )}^{2}}{x^{2}}\,{d x} \]
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 \[ \int \frac {\left (b \ln \left (c \left (d +\frac {e}{x^{\frac {2}{3}}}\right )^{n}\right )+a \right )^{2}}{x^{2}}\, dx \]
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 \[ -\frac {b^{2} n^{2} \log \left (d x^{\frac {2}{3}} + e\right )^{2}}{x} - \int -\frac {2 \, {\left (2 \, b^{2} d n x + 3 \, {\left (b^{2} d \log \relax (c) + a b d\right )} x - 6 \, {\left (b^{2} d x + b^{2} e x^{\frac {1}{3}}\right )} \log \left (x^{\frac {1}{3} \, n}\right ) + 3 \, {\left (b^{2} e \log \relax (c) + a b e\right )} x^{\frac {1}{3}}\right )} n \log \left (d x^{\frac {2}{3}} + e\right ) + 12 \, {\left (b^{2} d x + b^{2} e x^{\frac {1}{3}}\right )} \log \left (x^{\frac {1}{3} \, n}\right )^{2} + 3 \, {\left (b^{2} d \log \relax (c)^{2} + 2 \, a b d \log \relax (c) + a^{2} d\right )} x - 12 \, {\left ({\left (b^{2} d \log \relax (c) + a b d\right )} x + {\left (b^{2} e \log \relax (c) + a b e\right )} x^{\frac {1}{3}}\right )} \log \left (x^{\frac {1}{3} \, n}\right ) + 3 \, {\left (b^{2} e \log \relax (c)^{2} + 2 \, a b e \log \relax (c) + a^{2} e\right )} x^{\frac {1}{3}}}{3 \, {\left (d x^{3} + e x^{\frac {7}{3}}\right )}}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\ln \left (c\,{\left (d+\frac {e}{x^{2/3}}\right )}^n\right )\right )}^2}{x^2} \,d x \]
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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