Optimal. Leaf size=130 \[ \frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )+\frac {2 b d^{9/2} n \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{3 e^{9/2}}-\frac {2 b d^4 n \sqrt [3]{x}}{3 e^4}+\frac {2 b d^3 n x}{9 e^3}-\frac {2 b d^2 n x^{5/3}}{15 e^2}+\frac {2 b d n x^{7/3}}{21 e}-\frac {2}{27} b n x^3 \]
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Rubi [A] time = 0.08, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2455, 341, 302, 205} \[ \frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )-\frac {2 b d^2 n x^{5/3}}{15 e^2}-\frac {2 b d^4 n \sqrt [3]{x}}{3 e^4}+\frac {2 b d^3 n x}{9 e^3}+\frac {2 b d^{9/2} n \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{3 e^{9/2}}+\frac {2 b d n x^{7/3}}{21 e}-\frac {2}{27} b n x^3 \]
Antiderivative was successfully verified.
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Rule 205
Rule 302
Rule 341
Rule 2455
Rubi steps
\begin {align*} \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 {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) \operatorname {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) \operatorname {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 ) \operatorname {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*}
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Mathematica [A] time = 0.10, size = 135, normalized size = 1.04 \[ \frac {a x^3}{3}+\frac {1}{3} b x^3 \log \left (c \left (d+e x^{2/3}\right )^n\right )+\frac {2 b d^{9/2} n \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{3 e^{9/2}}-\frac {2 b d^4 n \sqrt [3]{x}}{3 e^4}+\frac {2 b d^3 n x}{9 e^3}-\frac {2 b d^2 n x^{5/3}}{15 e^2}+\frac {2 b d n x^{7/3}}{21 e}-\frac {2}{27} b n x^3 \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 337, normalized size = 2.59 \[ \left [\frac {315 \, b e^{4} n x^{3} \log \left (e x^{\frac {2}{3}} + d\right ) + 315 \, b e^{4} x^{3} \log \relax (c) - 126 \, b d^{2} e^{2} n x^{\frac {5}{3}} + 315 \, b d^{4} n \sqrt {-\frac {d}{e}} \log \left (\frac {e^{3} x^{2} - 2 \, d e^{2} x \sqrt {-\frac {d}{e}} - d^{3} + 2 \, {\left (e^{3} x \sqrt {-\frac {d}{e}} + d^{2} e\right )} x^{\frac {2}{3}} - 2 \, {\left (d e^{2} x - d^{2} e \sqrt {-\frac {d}{e}}\right )} x^{\frac {1}{3}}}{e^{3} x^{2} + d^{3}}\right ) + 210 \, b d^{3} e n x - 35 \, {\left (2 \, b e^{4} n - 9 \, a e^{4}\right )} x^{3} + 90 \, {\left (b d e^{3} n x^{2} - 7 \, b d^{4} n\right )} x^{\frac {1}{3}}}{945 \, e^{4}}, \frac {315 \, b e^{4} n x^{3} \log \left (e x^{\frac {2}{3}} + d\right ) + 315 \, b e^{4} x^{3} \log \relax (c) - 126 \, b d^{2} e^{2} n x^{\frac {5}{3}} + 630 \, b d^{4} n \sqrt {\frac {d}{e}} \arctan \left (\frac {e x^{\frac {1}{3}} \sqrt {\frac {d}{e}}}{d}\right ) + 210 \, b d^{3} e n x - 35 \, {\left (2 \, b e^{4} n - 9 \, a e^{4}\right )} x^{3} + 90 \, {\left (b d e^{3} n x^{2} - 7 \, b d^{4} n\right )} x^{\frac {1}{3}}}{945 \, e^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 104, normalized size = 0.80 \[ \frac {1}{3} \, b x^{3} \log \relax (c) + \frac {1}{3} \, a x^{3} + \frac {1}{945} \, {\left (315 \, x^{3} \log \left (x^{\frac {2}{3}} e + d\right ) + 2 \, {\left (315 \, d^{\frac {9}{2}} \arctan \left (\frac {x^{\frac {1}{3}} e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\left (-\frac {11}{2}\right )} - {\left (315 \, d^{4} x^{\frac {1}{3}} e^{4} - 105 \, d^{3} x e^{5} + 63 \, d^{2} x^{\frac {5}{3}} e^{6} - 45 \, d x^{\frac {7}{3}} e^{7} + 35 \, x^{3} e^{8}\right )} e^{\left (-9\right )}\right )} e\right )} b n \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int \left (b \ln \left (c \left (e \,x^{\frac {2}{3}}+d \right )^{n}\right )+a \right ) x^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.00, size = 104, normalized size = 0.80 \[ \frac {1}{3} \, b x^{3} \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right ) + \frac {1}{3} \, a x^{3} + \frac {2}{945} \, b e n {\left (\frac {315 \, d^{5} \arctan \left (\frac {e x^{\frac {1}{3}}}{\sqrt {d e}}\right )}{\sqrt {d e} e^{5}} - \frac {35 \, e^{4} x^{3} - 45 \, d e^{3} x^{\frac {7}{3}} + 63 \, d^{2} e^{2} x^{\frac {5}{3}} - 105 \, d^{3} e x + 315 \, d^{4} x^{\frac {1}{3}}}{e^{5}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^2\,\left (a+b\,\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )\right ) \,d x \]
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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