Optimal. Leaf size=483 \[ -\frac {2 b d^2 n \text {Int}\left (\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x^{2/3} \left (d x^{2/3}+e\right )},x\right )}{e}+\frac {32 b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{e^{3/2}}+\frac {32 b^2 d n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{e \sqrt [3]{x}}-\frac {8 b^2 n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{3 x}-\frac {6 b d n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{e \sqrt [3]{x}}+\frac {2 b n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3}{x}-\frac {32 i b^3 d^{3/2} n^3 \text {Li}_2\left (\frac {2 \sqrt {e}}{\sqrt {e}-i \sqrt {d} \sqrt [3]{x}}-1\right )}{e^{3/2}}-\frac {32 i b^3 d^{3/2} n^3 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )^2}{e^{3/2}}-\frac {208 b^3 d^{3/2} n^3 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )}{3 e^{3/2}}+\frac {64 b^3 d^{3/2} n^3 \log \left (2-\frac {2 \sqrt {e}}{\sqrt {e}-i \sqrt {d} \sqrt [3]{x}}\right ) \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )}{e^{3/2}}-\frac {208 b^3 d n^3}{3 e \sqrt [3]{x}}+\frac {16 b^3 n^3}{9 x} \]
[Out]
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Rubi [A] time = 1.38, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3}{x^2} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3}{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 )^3}{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 )^3}{x}-(6 b e n) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )^2}{\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 )^3}{x}-(6 b e n) \operatorname {Subst}\left (\int \left (\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )^2}{e x^4}-\frac {d \left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )^2}{e^2 x^2}+\frac {d^2 \left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )^2}{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 )^3}{x}-(6 b n) \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 {(6 b d n) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )^2}{x^2} \, dx,x,\sqrt [3]{x}\right )}{e}-\frac {\left (6 b d^2 n\right ) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )^2}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{e}\\ &=\frac {2 b n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x}-\frac {6 b d n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{e \sqrt [3]{x}}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3}{x}-\frac {\left (6 b d^2 n\right ) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )^2}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{e}-\left (24 b^2 d n^2\right ) \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^4} \, dx,x,\sqrt [3]{x}\right )+\left (8 b^2 e n^2\right ) \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 {2 b n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x}-\frac {6 b d n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{e \sqrt [3]{x}}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3}{x}-\frac {\left (6 b d^2 n\right ) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )^2}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{e}-\left (24 b^2 d n^2\right ) \operatorname {Subst}\left (\int \left (\frac {a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )}{e x^2}-\frac {d \left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )}{e \left (e+d x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )+\left (8 b^2 e n^2\right ) \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 {2 b n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x}-\frac {6 b d n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{e \sqrt [3]{x}}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3}{x}-\frac {\left (6 b d^2 n\right ) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )^2}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{e}+\left (8 b^2 n^2\right ) \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 {\left (8 b^2 d n^2\right ) \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 (24 b^2 d n^2\right ) \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 (8 b^2 d^2 n^2\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 {\left (24 b^2 d^2 n^2\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 {8 b^2 n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{3 x}+\frac {32 b^2 d n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{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 ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{e^{3/2}}+\frac {2 b n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x}-\frac {6 b d n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{e \sqrt [3]{x}}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3}{x}-\frac {\left (6 b d^2 n\right ) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )^2}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{e}+\left (16 b^3 d n^3\right ) \operatorname {Subst}\left (\int \frac {1}{\left (d+\frac {e}{x^2}\right ) x^4} \, dx,x,\sqrt [3]{x}\right )+\left (48 b^3 d n^3\right ) \operatorname {Subst}\left (\int \frac {1}{\left (d+\frac {e}{x^2}\right ) x^4} \, dx,x,\sqrt [3]{x}\right )+\left (16 b^3 d^2 n^3\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 )+\left (48 b^3 d^2 n^3\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 (16 b^3 e n^3\right ) \operatorname {Subst}\left (\int \frac {1}{\left (d+\frac {e}{x^2}\right ) x^6} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {8 b^2 n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{3 x}+\frac {32 b^2 d n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{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 ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{e^{3/2}}+\frac {2 b n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x}-\frac {6 b d n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{e \sqrt [3]{x}}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3}{x}-\frac {\left (6 b d^2 n\right ) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )^2}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{e}+\left (16 b^3 d n^3\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )+\left (48 b^3 d n^3\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )+\frac {\left (16 b^3 d^{3/2} n^3\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 {\left (48 b^3 d^{3/2} n^3\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 (16 b^3 e n^3\right ) \operatorname {Subst}\left (\int \frac {1}{x^4 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {16 b^3 n^3}{9 x}-\frac {64 b^3 d n^3}{e \sqrt [3]{x}}-\frac {8 b^2 n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{3 x}+\frac {32 b^2 d n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{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 ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{e^{3/2}}+\frac {2 b n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x}-\frac {6 b d n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{e \sqrt [3]{x}}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3}{x}-\frac {\left (6 b d^2 n\right ) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )^2}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{e}+\frac {1}{3} \left (16 b^3 d n^3\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )-\frac {\left (16 b^3 d^2 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{e}-\frac {\left (48 b^3 d^2 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{e}+\frac {\left (16 b^3 d^{3/2} n^3\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 {\left (48 b^3 d^{3/2} n^3\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 {16 b^3 n^3}{9 x}-\frac {208 b^3 d n^3}{3 e \sqrt [3]{x}}-\frac {64 b^3 d^{3/2} n^3 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )}{e^{3/2}}-\frac {32 i b^3 d^{3/2} n^3 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )^2}{e^{3/2}}-\frac {8 b^2 n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{3 x}+\frac {32 b^2 d n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{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 ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{e^{3/2}}+\frac {2 b n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x}-\frac {6 b d n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{e \sqrt [3]{x}}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3}{x}-\frac {\left (6 b d^2 n\right ) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )^2}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{e}+\frac {\left (16 i b^3 d^{3/2} n^3\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 (48 i b^3 d^{3/2} n^3\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 (16 b^3 d^2 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{3 e}\\ &=\frac {16 b^3 n^3}{9 x}-\frac {208 b^3 d n^3}{3 e \sqrt [3]{x}}-\frac {208 b^3 d^{3/2} n^3 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )}{3 e^{3/2}}-\frac {32 i b^3 d^{3/2} n^3 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )^2}{e^{3/2}}+\frac {64 b^3 d^{3/2} n^3 \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 {8 b^2 n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{3 x}+\frac {32 b^2 d n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{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 ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{e^{3/2}}+\frac {2 b n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x}-\frac {6 b d n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{e \sqrt [3]{x}}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3}{x}-\frac {\left (6 b d^2 n\right ) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )^2}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{e}-\frac {\left (16 b^3 d^2 n^3\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 {\left (48 b^3 d^2 n^3\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 {16 b^3 n^3}{9 x}-\frac {208 b^3 d n^3}{3 e \sqrt [3]{x}}-\frac {208 b^3 d^{3/2} n^3 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )}{3 e^{3/2}}-\frac {32 i b^3 d^{3/2} n^3 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )^2}{e^{3/2}}+\frac {64 b^3 d^{3/2} n^3 \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 {8 b^2 n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{3 x}+\frac {32 b^2 d n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{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 ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{e^{3/2}}+\frac {2 b n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x}-\frac {6 b d n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{e \sqrt [3]{x}}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3}{x}-\frac {32 i b^3 d^{3/2} n^3 \text {Li}_2\left (-1+\frac {2 \sqrt {e}}{\sqrt {e}-i \sqrt {d} \sqrt [3]{x}}\right )}{e^{3/2}}-\frac {\left (6 b d^2 n\right ) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )^2}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{e}\\ \end {align*}
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Mathematica [A] time = 2.31, size = 1097, normalized size = 2.27 \[ \frac {b^3 \left (18 \left (x^{2/3} d+e\right ) \, _5F_4\left (-\frac {1}{2},1,1,1,1;2,2,2,2;\frac {e}{d x^{2/3}}+1\right )-\log \left (d+\frac {e}{x^{2/3}}\right ) \left (18 \left (x^{2/3} d+e\right ) \, _4F_3\left (-\frac {1}{2},1,1,1;2,2,2;\frac {e}{d x^{2/3}}+1\right )+\log \left (d+\frac {e}{x^{2/3}}\right ) \left (2 \left (x^{2/3} d+e \sqrt {-\frac {e}{d x^{2/3}}}\right ) \log \left (d+\frac {e}{x^{2/3}}\right )-9 \left (x^{2/3} d+e\right ) \, _3F_2\left (-\frac {1}{2},1,1;2,2;\frac {e}{d x^{2/3}}+1\right )\right )\right )\right ) n^3}{2 e \sqrt {-\frac {e}{d x^{2/3}}} x}+\frac {b^2 \left (-a+b n \log \left (d+\frac {e}{x^{2/3}}\right )-b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right ) \left (9 e^{3/2} \log ^2\left (d+\frac {e}{x^{2/3}}\right )-12 e^{3/2} \log \left (d+\frac {e}{x^{2/3}}\right )+18 \sqrt {-d} d x \log \left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right ) \log \left (d+\frac {e}{x^{2/3}}\right )+18 (-d)^{3/2} x \log \left (\sqrt [3]{x} \sqrt {-d}+\sqrt {e}\right ) \log \left (d+\frac {e}{x^{2/3}}\right )+36 d \sqrt {e} x^{2/3} \log \left (d+\frac {e}{x^{2/3}}\right )+9 (-d)^{3/2} x \log ^2\left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right )+9 \sqrt {-d} d x \log ^2\left (\sqrt [3]{x} \sqrt {-d}+\sqrt {e}\right )+8 e^{3/2}+96 d^{3/2} x \tan ^{-1}\left (\frac {\sqrt {e}}{\sqrt {d} \sqrt [3]{x}}\right )+18 \sqrt {-d} d x \log \left (\sqrt [3]{x} \sqrt {-d}+\sqrt {e}\right ) \log \left (\frac {1}{2}-\frac {\sqrt {-d} \sqrt [3]{x}}{2 \sqrt {e}}\right )+18 (-d)^{3/2} x \log \left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right ) \log \left (\frac {1}{2} \left (\frac {\sqrt [3]{x} \sqrt {-d}}{\sqrt {e}}+1\right )\right )+36 (-d)^{3/2} x \log \left (\sqrt [3]{x} \sqrt {-d}+\sqrt {e}\right ) \log \left (-\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )+36 \sqrt {-d} d x \log \left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right ) \log \left (\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )+36 \sqrt {-d} d x \text {Li}_2\left (1-\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )+18 (-d)^{3/2} x \text {Li}_2\left (\frac {1}{2}-\frac {\sqrt {-d} \sqrt [3]{x}}{2 \sqrt {e}}\right )+18 \sqrt {-d} d x \text {Li}_2\left (\frac {1}{2} \left (\frac {\sqrt [3]{x} \sqrt {-d}}{\sqrt {e}}+1\right )\right )+36 (-d)^{3/2} x \text {Li}_2\left (\frac {\sqrt [3]{x} \sqrt {-d}}{\sqrt {e}}+1\right )-96 d \sqrt {e} x^{2/3}\right ) n^2}{3 e^{3/2} x}-\frac {6 b d^{3/2} \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right ) \left (a-b n \log \left (d+\frac {e}{x^{2/3}}\right )+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2 n}{e^{3/2}}-\frac {3 b \log \left (d+\frac {e}{x^{2/3}}\right ) \left (a-b n \log \left (d+\frac {e}{x^{2/3}}\right )+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2 n}{x}-\frac {6 b d \left (a-b n \log \left (d+\frac {e}{x^{2/3}}\right )+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2 n}{e \sqrt [3]{x}}-\frac {\left (a-b n \log \left (d+\frac {e}{x^{2/3}}\right )+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2 \left (a-2 b n-b n \log \left (d+\frac {e}{x^{2/3}}\right )+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{3} \log \left (c \left (\frac {d x + e x^{\frac {1}{3}}}{x}\right )^{n}\right )^{3} + 3 \, a b^{2} \log \left (c \left (\frac {d x + e x^{\frac {1}{3}}}{x}\right )^{n}\right )^{2} + 3 \, a^{2} b \log \left (c \left (\frac {d x + e x^{\frac {1}{3}}}{x}\right )^{n}\right ) + a^{3}}{x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {2}{3}}}\right )}^{n}\right ) + a\right )}^{3}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \left (d +\frac {e}{x^{\frac {2}{3}}}\right )^{n}\right )+a \right )^{3}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {b^{3} n^{3} \log \left (d x^{\frac {2}{3}} + e\right )^{3}}{x} - \int -\frac {{\left (2 \, b^{3} d n x + 3 \, {\left (b^{3} d \log \relax (c) + a b^{2} d\right )} x - 6 \, {\left (b^{3} d x + b^{3} e x^{\frac {1}{3}}\right )} \log \left (x^{\frac {1}{3} \, n}\right ) + 3 \, {\left (b^{3} e \log \relax (c) + a b^{2} e\right )} x^{\frac {1}{3}}\right )} n^{2} \log \left (d x^{\frac {2}{3}} + e\right )^{2} - 8 \, {\left (b^{3} d x + b^{3} e x^{\frac {1}{3}}\right )} \log \left (x^{\frac {1}{3} \, n}\right )^{3} + 3 \, {\left (4 \, {\left (b^{3} d x + b^{3} e x^{\frac {1}{3}}\right )} \log \left (x^{\frac {1}{3} \, n}\right )^{2} + {\left (b^{3} d \log \relax (c)^{2} + 2 \, a b^{2} d \log \relax (c) + a^{2} b d\right )} x - 4 \, {\left ({\left (b^{3} d \log \relax (c) + a b^{2} d\right )} x + {\left (b^{3} e \log \relax (c) + a b^{2} e\right )} x^{\frac {1}{3}}\right )} \log \left (x^{\frac {1}{3} \, n}\right ) + {\left (b^{3} e \log \relax (c)^{2} + 2 \, a b^{2} e \log \relax (c) + a^{2} b e\right )} x^{\frac {1}{3}}\right )} n \log \left (d x^{\frac {2}{3}} + e\right ) + 12 \, {\left ({\left (b^{3} d \log \relax (c) + a b^{2} d\right )} x + {\left (b^{3} e \log \relax (c) + a b^{2} e\right )} x^{\frac {1}{3}}\right )} \log \left (x^{\frac {1}{3} \, n}\right )^{2} + {\left (b^{3} d \log \relax (c)^{3} + 3 \, a b^{2} d \log \relax (c)^{2} + 3 \, a^{2} b d \log \relax (c) + a^{3} d\right )} x - 6 \, {\left ({\left (b^{3} d \log \relax (c)^{2} + 2 \, a b^{2} d \log \relax (c) + a^{2} b d\right )} x + {\left (b^{3} e \log \relax (c)^{2} + 2 \, a b^{2} e \log \relax (c) + a^{2} b e\right )} x^{\frac {1}{3}}\right )} \log \left (x^{\frac {1}{3} \, n}\right ) + {\left (b^{3} e \log \relax (c)^{3} + 3 \, a b^{2} e \log \relax (c)^{2} + 3 \, a^{2} b e \log \relax (c) + a^{3} e\right )} x^{\frac {1}{3}}}{d x^{3} + e x^{\frac {7}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\ln \left (c\,{\left (d+\frac {e}{x^{2/3}}\right )}^n\right )\right )}^3}{x^2} \,d x \]
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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