Integrand size = 22, antiderivative size = 130 \[ \int x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \, dx=-\frac {2 b d^4 n \sqrt [3]{x}}{3 e^4}+\frac {2 b d^3 n x}{9 e^3}-\frac {2 b d^2 n x^{5/3}}{15 e^2}+\frac {2 b d n x^{7/3}}{21 e}-\frac {2}{27} b n x^3+\frac {2 b d^{9/2} n \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{3 e^{9/2}}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \]
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Time = 0.05 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2505, 348, 308, 211} \[ \int x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \, dx=\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )+\frac {2 b d^{9/2} n \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{3 e^{9/2}}-\frac {2 b d^4 n \sqrt [3]{x}}{3 e^4}+\frac {2 b d^3 n x}{9 e^3}-\frac {2 b d^2 n x^{5/3}}{15 e^2}+\frac {2 b d n x^{7/3}}{21 e}-\frac {2}{27} b n x^3 \]
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Rule 211
Rule 308
Rule 348
Rule 2505
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )-\frac {1}{9} (2 b e n) \int \frac {x^{8/3}}{d+e x^{2/3}} \, dx \\ & = \frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )-\frac {1}{3} (2 b e n) \text {Subst}\left (\int \frac {x^{10}}{d+e x^2} \, dx,x,\sqrt [3]{x}\right ) \\ & = \frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )-\frac {1}{3} (2 b e n) \text {Subst}\left (\int \left (\frac {d^4}{e^5}-\frac {d^3 x^2}{e^4}+\frac {d^2 x^4}{e^3}-\frac {d x^6}{e^2}+\frac {x^8}{e}-\frac {d^5}{e^5 \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right ) \\ & = -\frac {2 b d^4 n \sqrt [3]{x}}{3 e^4}+\frac {2 b d^3 n x}{9 e^3}-\frac {2 b d^2 n x^{5/3}}{15 e^2}+\frac {2 b d n x^{7/3}}{21 e}-\frac {2}{27} b n x^3+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )+\frac {\left (2 b d^5 n\right ) \text {Subst}\left (\int \frac {1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{3 e^4} \\ & = -\frac {2 b d^4 n \sqrt [3]{x}}{3 e^4}+\frac {2 b d^3 n x}{9 e^3}-\frac {2 b d^2 n x^{5/3}}{15 e^2}+\frac {2 b d n x^{7/3}}{21 e}-\frac {2}{27} b n x^3+\frac {2 b d^{9/2} n \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{3 e^{9/2}}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.98 \[ \int x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \, dx=\frac {a x^3}{3}-\frac {2}{9} b e n \left (\frac {3 d^4 \sqrt [3]{x}}{e^5}-\frac {d^3 x}{e^4}+\frac {3 d^2 x^{5/3}}{5 e^3}-\frac {3 d x^{7/3}}{7 e^2}+\frac {x^3}{3 e}-\frac {3 d^{9/2} \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{e^{11/2}}\right )+\frac {1}{3} b x^3 \log \left (c \left (d+e x^{2/3}\right )^n\right ) \]
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\[\int x^{2} \left (a +b \ln \left (c \left (d +e \,x^{\frac {2}{3}}\right )^{n}\right )\right )d x\]
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Time = 0.37 (sec) , antiderivative size = 337, normalized size of antiderivative = 2.59 \[ \int x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \, dx=\left [\frac {315 \, b e^{4} n x^{3} \log \left (e x^{\frac {2}{3}} + d\right ) + 315 \, b e^{4} x^{3} \log \left (c\right ) - 126 \, b d^{2} e^{2} n x^{\frac {5}{3}} + 315 \, b d^{4} n \sqrt {-\frac {d}{e}} \log \left (\frac {e^{3} x^{2} - 2 \, d e^{2} x \sqrt {-\frac {d}{e}} - d^{3} + 2 \, {\left (e^{3} x \sqrt {-\frac {d}{e}} + d^{2} e\right )} x^{\frac {2}{3}} - 2 \, {\left (d e^{2} x - d^{2} e \sqrt {-\frac {d}{e}}\right )} x^{\frac {1}{3}}}{e^{3} x^{2} + d^{3}}\right ) + 210 \, b d^{3} e n x - 35 \, {\left (2 \, b e^{4} n - 9 \, a e^{4}\right )} x^{3} + 90 \, {\left (b d e^{3} n x^{2} - 7 \, b d^{4} n\right )} x^{\frac {1}{3}}}{945 \, e^{4}}, \frac {315 \, b e^{4} n x^{3} \log \left (e x^{\frac {2}{3}} + d\right ) + 315 \, b e^{4} x^{3} \log \left (c\right ) - 126 \, b d^{2} e^{2} n x^{\frac {5}{3}} + 630 \, b d^{4} n \sqrt {\frac {d}{e}} \arctan \left (\frac {e x^{\frac {1}{3}} \sqrt {\frac {d}{e}}}{d}\right ) + 210 \, b d^{3} e n x - 35 \, {\left (2 \, b e^{4} n - 9 \, a e^{4}\right )} x^{3} + 90 \, {\left (b d e^{3} n x^{2} - 7 \, b d^{4} n\right )} x^{\frac {1}{3}}}{945 \, e^{4}}\right ] \]
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Timed out. \[ \int x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \, dx=\text {Timed out} \]
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Exception generated. \[ \int x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \, dx=\text {Exception raised: ValueError} \]
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Time = 0.35 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.88 \[ \int x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \, dx=\frac {1}{3} \, b x^{3} \log \left (c\right ) + \frac {1}{3} \, a x^{3} + \frac {1}{945} \, {\left (315 \, x^{3} \log \left (e x^{\frac {2}{3}} + d\right ) + 2 \, e {\left (\frac {315 \, d^{5} \arctan \left (\frac {e x^{\frac {1}{3}}}{\sqrt {d e}}\right )}{\sqrt {d e} e^{5}} - \frac {35 \, e^{8} x^{3} - 45 \, d e^{7} x^{\frac {7}{3}} + 63 \, d^{2} e^{6} x^{\frac {5}{3}} - 105 \, d^{3} e^{5} x + 315 \, d^{4} e^{4} x^{\frac {1}{3}}}{e^{9}}\right )}\right )} b n \]
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Timed out. \[ \int x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \, dx=\int x^2\,\left (a+b\,\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )\right ) \,d x \]
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