Optimal. Leaf size=632 \[ \frac {2 b e^5 n \text {Int}\left (\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x^{2/3} \left (d+e x^{2/3}\right )},x\right )}{3 d^4}-\frac {1408 b^2 e^{9/2} n^2 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{105 d^{9/2}}-\frac {568 b^2 e^4 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{105 d^4 \sqrt [3]{x}}+\frac {32 b^2 e^3 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{35 d^3 x}-\frac {8 b^2 e^2 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{35 d^2 x^{5/3}}+\frac {2 b e^4 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{d^4 \sqrt [3]{x}}-\frac {2 b e^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{3 d^3 x}+\frac {2 b e^2 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{5 d^2 x^{5/3}}-\frac {2 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{7 d x^{7/3}}-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{3 x^3}-\frac {1408 i b^3 e^{9/2} n^3 \text {Li}_2\left (1-\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right )}{105 d^{9/2}}-\frac {1408 i b^3 e^{9/2} n^3 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2}{105 d^{9/2}}+\frac {1376 b^3 e^{9/2} n^3 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{105 d^{9/2}}-\frac {2816 b^3 e^{9/2} n^3 \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right ) \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{105 d^{9/2}}+\frac {16 b^3 e^4 n^3}{7 d^4 \sqrt [3]{x}}-\frac {16 b^3 e^3 n^3}{105 d^3 x} \]
[Out]
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Rubi [A] time = 1.87, 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 = {} \[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x^4} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x^4} \, dx &=3 \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^3}{x^{10}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{3 x^3}+(2 b e n) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{x^8 \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{3 x^3}+(2 b e n) \operatorname {Subst}\left (\int \left (\frac {\left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{d x^8}-\frac {e \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{d^2 x^6}+\frac {e^2 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{d^3 x^4}-\frac {e^3 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{d^4 x^2}+\frac {e^4 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{d^4 \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{3 x^3}+\frac {(2 b e n) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{x^8} \, dx,x,\sqrt [3]{x}\right )}{d}-\frac {\left (2 b e^2 n\right ) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{x^6} \, dx,x,\sqrt [3]{x}\right )}{d^2}+\frac {\left (2 b e^3 n\right ) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{x^4} \, dx,x,\sqrt [3]{x}\right )}{d^3}-\frac {\left (2 b e^4 n\right ) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{x^2} \, dx,x,\sqrt [3]{x}\right )}{d^4}+\frac {\left (2 b e^5 n\right ) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{d^4}\\ &=-\frac {2 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{7 d x^{7/3}}+\frac {2 b e^2 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{5 d^2 x^{5/3}}-\frac {2 b e^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{3 d^3 x}+\frac {2 b e^4 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{d^4 \sqrt [3]{x}}-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{3 x^3}+\frac {\left (2 b e^5 n\right ) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{d^4}+\frac {\left (8 b^2 e^2 n^2\right ) \operatorname {Subst}\left (\int \frac {a+b \log \left (c \left (d+e x^2\right )^n\right )}{x^6 \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{7 d}-\frac {\left (8 b^2 e^3 n^2\right ) \operatorname {Subst}\left (\int \frac {a+b \log \left (c \left (d+e x^2\right )^n\right )}{x^4 \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{5 d^2}+\frac {\left (8 b^2 e^4 n^2\right ) \operatorname {Subst}\left (\int \frac {a+b \log \left (c \left (d+e x^2\right )^n\right )}{x^2 \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{3 d^3}-\frac {\left (8 b^2 e^5 n^2\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 )}{d^4}\\ &=-\frac {8 b^2 e^{9/2} n^2 \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 )}{d^{9/2}}-\frac {2 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{7 d x^{7/3}}+\frac {2 b e^2 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{5 d^2 x^{5/3}}-\frac {2 b e^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{3 d^3 x}+\frac {2 b e^4 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{d^4 \sqrt [3]{x}}-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{3 x^3}+\frac {\left (2 b e^5 n\right ) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{d^4}+\frac {\left (8 b^2 e^2 n^2\right ) \operatorname {Subst}\left (\int \left (\frac {a+b \log \left (c \left (d+e x^2\right )^n\right )}{d x^6}-\frac {e \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{d^2 x^4}+\frac {e^2 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{d^3 x^2}-\frac {e^3 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{d^3 \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{7 d}-\frac {\left (8 b^2 e^3 n^2\right ) \operatorname {Subst}\left (\int \left (\frac {a+b \log \left (c \left (d+e x^2\right )^n\right )}{d x^4}-\frac {e \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{d^2 x^2}+\frac {e^2 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{d^2 \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{5 d^2}+\frac {\left (8 b^2 e^4 n^2\right ) \operatorname {Subst}\left (\int \left (\frac {a+b \log \left (c \left (d+e x^2\right )^n\right )}{d x^2}-\frac {e \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{d \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{3 d^3}+\frac {\left (16 b^3 e^6 n^3\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 )}{d^4}\\ &=-\frac {8 b^2 e^{9/2} n^2 \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 )}{d^{9/2}}-\frac {2 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{7 d x^{7/3}}+\frac {2 b e^2 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{5 d^2 x^{5/3}}-\frac {2 b e^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{3 d^3 x}+\frac {2 b e^4 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{d^4 \sqrt [3]{x}}-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{3 x^3}+\frac {\left (2 b e^5 n\right ) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{d^4}+\frac {\left (8 b^2 e^2 n^2\right ) \operatorname {Subst}\left (\int \frac {a+b \log \left (c \left (d+e x^2\right )^n\right )}{x^6} \, dx,x,\sqrt [3]{x}\right )}{7 d^2}-\frac {\left (8 b^2 e^3 n^2\right ) \operatorname {Subst}\left (\int \frac {a+b \log \left (c \left (d+e x^2\right )^n\right )}{x^4} \, dx,x,\sqrt [3]{x}\right )}{7 d^3}-\frac {\left (8 b^2 e^3 n^2\right ) \operatorname {Subst}\left (\int \frac {a+b \log \left (c \left (d+e x^2\right )^n\right )}{x^4} \, dx,x,\sqrt [3]{x}\right )}{5 d^3}+\frac {\left (8 b^2 e^4 n^2\right ) \operatorname {Subst}\left (\int \frac {a+b \log \left (c \left (d+e x^2\right )^n\right )}{x^2} \, dx,x,\sqrt [3]{x}\right )}{7 d^4}+\frac {\left (8 b^2 e^4 n^2\right ) \operatorname {Subst}\left (\int \frac {a+b \log \left (c \left (d+e x^2\right )^n\right )}{x^2} \, dx,x,\sqrt [3]{x}\right )}{5 d^4}+\frac {\left (8 b^2 e^4 n^2\right ) \operatorname {Subst}\left (\int \frac {a+b \log \left (c \left (d+e x^2\right )^n\right )}{x^2} \, dx,x,\sqrt [3]{x}\right )}{3 d^4}-\frac {\left (8 b^2 e^5 n^2\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 )}{7 d^4}-\frac {\left (8 b^2 e^5 n^2\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 )}{5 d^4}-\frac {\left (8 b^2 e^5 n^2\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 )}{3 d^4}+\frac {\left (16 b^3 e^{11/2} n^3\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 )}{d^{9/2}}\\ &=-\frac {8 i b^3 e^{9/2} n^3 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2}{d^{9/2}}-\frac {8 b^2 e^2 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{35 d^2 x^{5/3}}+\frac {32 b^2 e^3 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{35 d^3 x}-\frac {568 b^2 e^4 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{105 d^4 \sqrt [3]{x}}-\frac {1408 b^2 e^{9/2} n^2 \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 )}{105 d^{9/2}}-\frac {2 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{7 d x^{7/3}}+\frac {2 b e^2 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{5 d^2 x^{5/3}}-\frac {2 b e^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{3 d^3 x}+\frac {2 b e^4 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{d^4 \sqrt [3]{x}}-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{3 x^3}+\frac {\left (2 b e^5 n\right ) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{d^4}+\frac {\left (16 b^3 e^3 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{x^4 \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{35 d^2}-\frac {\left (16 b^3 e^4 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{21 d^3}-\frac {\left (16 b^3 e^4 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{15 d^3}-\frac {\left (16 b^3 e^5 n^3\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 )}{d^5}+\frac {\left (16 b^3 e^5 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{7 d^4}+\frac {\left (16 b^3 e^5 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{5 d^4}+\frac {\left (16 b^3 e^5 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{3 d^4}+\frac {\left (16 b^3 e^6 n^3\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 )}{7 d^4}+\frac {\left (16 b^3 e^6 n^3\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 )}{5 d^4}+\frac {\left (16 b^3 e^6 n^3\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 )}{3 d^4}\\ &=-\frac {16 b^3 e^3 n^3}{105 d^3 x}+\frac {64 b^3 e^4 n^3}{35 d^4 \sqrt [3]{x}}+\frac {1136 b^3 e^{9/2} n^3 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{105 d^{9/2}}-\frac {8 i b^3 e^{9/2} n^3 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2}{d^{9/2}}-\frac {16 b^3 e^{9/2} n^3 \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 )}{d^{9/2}}-\frac {8 b^2 e^2 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{35 d^2 x^{5/3}}+\frac {32 b^2 e^3 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{35 d^3 x}-\frac {568 b^2 e^4 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{105 d^4 \sqrt [3]{x}}-\frac {1408 b^2 e^{9/2} n^2 \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 )}{105 d^{9/2}}-\frac {2 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{7 d x^{7/3}}+\frac {2 b e^2 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{5 d^2 x^{5/3}}-\frac {2 b e^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{3 d^3 x}+\frac {2 b e^4 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{d^4 \sqrt [3]{x}}-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{3 x^3}+\frac {\left (2 b e^5 n\right ) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{d^4}-\frac {\left (16 b^3 e^4 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{35 d^3}+\frac {\left (16 b^3 e^5 n^3\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 )}{d^5}+\frac {\left (16 b^3 e^5 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{21 d^4}+\frac {\left (16 b^3 e^5 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{15 d^4}+\frac {\left (16 b^3 e^{11/2} n^3\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 )}{7 d^{9/2}}+\frac {\left (16 b^3 e^{11/2} n^3\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 )}{5 d^{9/2}}+\frac {\left (16 b^3 e^{11/2} n^3\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 )}{3 d^{9/2}}\\ &=-\frac {16 b^3 e^3 n^3}{105 d^3 x}+\frac {16 b^3 e^4 n^3}{7 d^4 \sqrt [3]{x}}+\frac {1328 b^3 e^{9/2} n^3 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{105 d^{9/2}}-\frac {1408 i b^3 e^{9/2} n^3 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2}{105 d^{9/2}}-\frac {16 b^3 e^{9/2} n^3 \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 )}{d^{9/2}}-\frac {8 b^2 e^2 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{35 d^2 x^{5/3}}+\frac {32 b^2 e^3 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{35 d^3 x}-\frac {568 b^2 e^4 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{105 d^4 \sqrt [3]{x}}-\frac {1408 b^2 e^{9/2} n^2 \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 )}{105 d^{9/2}}-\frac {2 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{7 d x^{7/3}}+\frac {2 b e^2 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{5 d^2 x^{5/3}}-\frac {2 b e^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{3 d^3 x}+\frac {2 b e^4 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{d^4 \sqrt [3]{x}}-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{3 x^3}+\frac {\left (2 b e^5 n\right ) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{d^4}-\frac {\left (16 i b^3 e^{9/2} n^3\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 )}{d^{9/2}}-\frac {\left (16 b^3 e^5 n^3\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 )}{7 d^5}-\frac {\left (16 b^3 e^5 n^3\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 )}{5 d^5}-\frac {\left (16 b^3 e^5 n^3\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 )}{3 d^5}+\frac {\left (16 b^3 e^5 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{35 d^4}\\ &=-\frac {16 b^3 e^3 n^3}{105 d^3 x}+\frac {16 b^3 e^4 n^3}{7 d^4 \sqrt [3]{x}}+\frac {1376 b^3 e^{9/2} n^3 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{105 d^{9/2}}-\frac {1408 i b^3 e^{9/2} n^3 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2}{105 d^{9/2}}-\frac {2816 b^3 e^{9/2} n^3 \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 )}{105 d^{9/2}}-\frac {8 b^2 e^2 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{35 d^2 x^{5/3}}+\frac {32 b^2 e^3 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{35 d^3 x}-\frac {568 b^2 e^4 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{105 d^4 \sqrt [3]{x}}-\frac {1408 b^2 e^{9/2} n^2 \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 )}{105 d^{9/2}}-\frac {2 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{7 d x^{7/3}}+\frac {2 b e^2 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{5 d^2 x^{5/3}}-\frac {2 b e^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{3 d^3 x}+\frac {2 b e^4 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{d^4 \sqrt [3]{x}}-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{3 x^3}-\frac {8 i b^3 e^{9/2} n^3 \text {Li}_2\left (1-\frac {2}{1+\frac {i \sqrt {e} \sqrt [3]{x}}{\sqrt {d}}}\right )}{d^{9/2}}+\frac {\left (2 b e^5 n\right ) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{d^4}+\frac {\left (16 b^3 e^5 n^3\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 )}{7 d^5}+\frac {\left (16 b^3 e^5 n^3\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 )}{5 d^5}+\frac {\left (16 b^3 e^5 n^3\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 )}{3 d^5}\\ &=-\frac {16 b^3 e^3 n^3}{105 d^3 x}+\frac {16 b^3 e^4 n^3}{7 d^4 \sqrt [3]{x}}+\frac {1376 b^3 e^{9/2} n^3 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{105 d^{9/2}}-\frac {1408 i b^3 e^{9/2} n^3 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2}{105 d^{9/2}}-\frac {2816 b^3 e^{9/2} n^3 \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 )}{105 d^{9/2}}-\frac {8 b^2 e^2 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{35 d^2 x^{5/3}}+\frac {32 b^2 e^3 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{35 d^3 x}-\frac {568 b^2 e^4 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{105 d^4 \sqrt [3]{x}}-\frac {1408 b^2 e^{9/2} n^2 \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 )}{105 d^{9/2}}-\frac {2 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{7 d x^{7/3}}+\frac {2 b e^2 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{5 d^2 x^{5/3}}-\frac {2 b e^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{3 d^3 x}+\frac {2 b e^4 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{d^4 \sqrt [3]{x}}-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{3 x^3}-\frac {8 i b^3 e^{9/2} n^3 \text {Li}_2\left (1-\frac {2}{1+\frac {i \sqrt {e} \sqrt [3]{x}}{\sqrt {d}}}\right )}{d^{9/2}}+\frac {\left (2 b e^5 n\right ) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{d^4}-\frac {\left (16 i b^3 e^{9/2} n^3\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 )}{7 d^{9/2}}-\frac {\left (16 i b^3 e^{9/2} n^3\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 )}{5 d^{9/2}}-\frac {\left (16 i b^3 e^{9/2} n^3\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 )}{3 d^{9/2}}\\ &=-\frac {16 b^3 e^3 n^3}{105 d^3 x}+\frac {16 b^3 e^4 n^3}{7 d^4 \sqrt [3]{x}}+\frac {1376 b^3 e^{9/2} n^3 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{105 d^{9/2}}-\frac {1408 i b^3 e^{9/2} n^3 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2}{105 d^{9/2}}-\frac {2816 b^3 e^{9/2} n^3 \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 )}{105 d^{9/2}}-\frac {8 b^2 e^2 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{35 d^2 x^{5/3}}+\frac {32 b^2 e^3 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{35 d^3 x}-\frac {568 b^2 e^4 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{105 d^4 \sqrt [3]{x}}-\frac {1408 b^2 e^{9/2} n^2 \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 )}{105 d^{9/2}}-\frac {2 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{7 d x^{7/3}}+\frac {2 b e^2 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{5 d^2 x^{5/3}}-\frac {2 b e^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{3 d^3 x}+\frac {2 b e^4 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{d^4 \sqrt [3]{x}}-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{3 x^3}-\frac {1408 i b^3 e^{9/2} n^3 \text {Li}_2\left (1-\frac {2}{1+\frac {i \sqrt {e} \sqrt [3]{x}}{\sqrt {d}}}\right )}{105 d^{9/2}}+\frac {\left (2 b e^5 n\right ) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{d^4}\\ \end {align*}
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Mathematica [A] time = 2.94, size = 803, normalized size = 1.27 \[ \frac {-70 \left (a-b n \log \left (d+e x^{2/3}\right )+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 d^5-210 b n \log \left (d+e x^{2/3}\right ) \left (a-b n \log \left (d+e x^{2/3}\right )+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 d^5-60 b e n x^{2/3} \left (a-b n \log \left (d+e x^{2/3}\right )+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 d^4+84 b e^2 n x^{4/3} \left (a-b n \log \left (d+e x^{2/3}\right )+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 d^3-140 b e^3 n x^2 \left (a-b n \log \left (d+e x^{2/3}\right )+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 d^2+420 b e^4 n x^{8/3} \left (a-b n \log \left (d+e x^{2/3}\right )+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 d+420 b e^{9/2} n x^3 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \left (a-b n \log \left (d+e x^{2/3}\right )+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \sqrt {d}+35 b^3 n^3 \left (54 \left (d+e x^{2/3}\right ) \sqrt {-\frac {e x^{2/3}}{d}} x^{8/3} \, _5F_4\left (1,1,1,1,\frac {11}{2};2,2,2,2;\frac {x^{2/3} e}{d}+1\right ) e^4+\log \left (d+e x^{2/3}\right ) \left (54 d \left (d+e x^{2/3}\right ) \left (-\frac {e x^{2/3}}{d}\right )^{3/2} x^2 \, _4F_3\left (1,1,1,\frac {11}{2};2,2,2;\frac {x^{2/3} e}{d}+1\right ) e^3+\log \left (d+e x^{2/3}\right ) \left (27 e^4 \left (d+e x^{2/3}\right ) \sqrt {-\frac {e x^{2/3}}{d}} x^{8/3} \, _3F_2\left (1,1,\frac {11}{2};2,2;\frac {x^{2/3} e}{d}+1\right )-2 d \left (d^4+e^3 \left (-\frac {e x^{2/3}}{d}\right )^{3/2} x^2 d\right ) \log \left (d+e x^{2/3}\right )\right )\right )\right )+\frac {210 b^2 n^2 \left (\log \left (d+e x^{2/3}\right ) \left (9 \left (d+e x^{2/3}\right ) x^{10/3} \, _3F_2\left (1,1,\frac {11}{2};2,2;\frac {x^{2/3} e}{d}+1\right ) e^5+d \left (\sqrt {-\frac {e x^{2/3}}{d}} d^5+e^5 x^{10/3}\right ) \log \left (d+e x^{2/3}\right )\right )-9 e^5 \left (d+e x^{2/3}\right ) x^{10/3} \, _4F_3\left (1,1,1,\frac {11}{2};2,2,2;\frac {x^{2/3} e}{d}+1\right )\right ) \left (-a+b n \log \left (d+e x^{2/3}\right )-b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{\sqrt {-\frac {e x^{2/3}}{d}} d}}{210 d^5 x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{3} \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right )^{3} + 3 \, a b^{2} \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right )^{2} + 3 \, a^{2} b \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right ) + a^{3}}{x^{4}}, x\right ) \]
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 \[ \int \frac {{\left (b \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right ) + a\right )}^{3}}{x^{4}}\,{d x} \]
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 \[ \int \frac {\left (b \ln \left (c \left (e \,x^{\frac {2}{3}}+d \right )^{n}\right )+a \right )^{3}}{x^{4}}\, dx \]
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 \[ -\frac {b^{3} n^{3} \log \left (e x^{\frac {2}{3}} + d\right )^{3}}{3 \, x^{3}} + \int \frac {{\left (2 \, b^{3} e n x + 9 \, {\left (b^{3} e \log \relax (c) + a b^{2} e\right )} x + 9 \, {\left (b^{3} d \log \relax (c) + a b^{2} d\right )} x^{\frac {1}{3}}\right )} n^{2} \log \left (e x^{\frac {2}{3}} + d\right )^{2} + 9 \, {\left ({\left (b^{3} e \log \relax (c)^{2} + 2 \, a b^{2} e \log \relax (c) + a^{2} b e\right )} x + {\left (b^{3} d \log \relax (c)^{2} + 2 \, a b^{2} d \log \relax (c) + a^{2} b d\right )} x^{\frac {1}{3}}\right )} n \log \left (e x^{\frac {2}{3}} + d\right ) + 3 \, {\left (b^{3} e \log \relax (c)^{3} + 3 \, a b^{2} e \log \relax (c)^{2} + 3 \, a^{2} b e \log \relax (c) + a^{3} e\right )} x + 3 \, {\left (b^{3} d \log \relax (c)^{3} + 3 \, a b^{2} d \log \relax (c)^{2} + 3 \, a^{2} b d \log \relax (c) + a^{3} d\right )} x^{\frac {1}{3}}}{3 \, {\left (e x^{5} + d x^{\frac {13}{3}}\right )}}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )\right )}^3}{x^4} \,d x \]
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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