Optimal. Leaf size=89 \[ -\frac {b d^2 n x^{2/3}}{2 e^2}+\frac {b d n x^{4/3}}{4 e}-\frac {1}{6} b n x^2+\frac {b d^3 n \log \left (d+e x^{2/3}\right )}{2 e^3}+\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \]
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Rubi [A]
time = 0.04, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2504, 2442, 45}
\begin {gather*} \frac {1}{2} x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )+\frac {b d^3 n \log \left (d+e x^{2/3}\right )}{2 e^3}-\frac {b d^2 n x^{2/3}}{2 e^2}+\frac {b d n x^{4/3}}{4 e}-\frac {1}{6} b n x^2 \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2442
Rule 2504
Rubi steps
\begin {align*} \int x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \, dx &=\frac {3}{2} \text {Subst}\left (\int x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx,x,x^{2/3}\right )\\ &=\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )-\frac {1}{2} (b e n) \text {Subst}\left (\int \frac {x^3}{d+e x} \, dx,x,x^{2/3}\right )\\ &=\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )-\frac {1}{2} (b e n) \text {Subst}\left (\int \left (\frac {d^2}{e^3}-\frac {d x}{e^2}+\frac {x^2}{e}-\frac {d^3}{e^3 (d+e x)}\right ) \, dx,x,x^{2/3}\right )\\ &=-\frac {b d^2 n x^{2/3}}{2 e^2}+\frac {b d n x^{4/3}}{4 e}-\frac {1}{6} b n x^2+\frac {b d^3 n \log \left (d+e x^{2/3}\right )}{2 e^3}+\frac {1}{2} x^2 \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 = 0.02, size = 94, normalized size = 1.06 \begin {gather*} -\frac {b d^2 n x^{2/3}}{2 e^2}+\frac {b d n x^{4/3}}{4 e}+\frac {a x^2}{2}-\frac {1}{6} b n x^2+\frac {b d^3 n \log \left (d+e x^{2/3}\right )}{2 e^3}+\frac {1}{2} b x^2 \log \left (c \left (d+e x^{2/3}\right )^n\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int x \left (a +b \ln \left (c \left (d +e \,x^{\frac {2}{3}}\right )^{n}\right )\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 76, normalized size = 0.85 \begin {gather*} \frac {1}{12} \, {\left (6 \, d^{3} e^{\left (-4\right )} \log \left (x^{\frac {2}{3}} e + d\right ) + {\left (3 \, d x^{\frac {4}{3}} e - 2 \, x^{2} e^{2} - 6 \, d^{2} x^{\frac {2}{3}}\right )} e^{\left (-3\right )}\right )} b n e + \frac {1}{2} \, b x^{2} \log \left ({\left (x^{\frac {2}{3}} e + d\right )}^{n} c\right ) + \frac {1}{2} \, a x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 77, normalized size = 0.87 \begin {gather*} -\frac {1}{12} \, {\left (6 \, b d^{2} n x^{\frac {2}{3}} e - 3 \, b d n x^{\frac {4}{3}} e^{2} - 6 \, b x^{2} e^{3} \log \left (c\right ) + 2 \, {\left (b n - 3 \, a\right )} x^{2} e^{3} - 6 \, {\left (b d^{3} n + b n x^{2} e^{3}\right )} \log \left (x^{\frac {2}{3}} e + d\right )\right )} e^{\left (-3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 54.92, size = 95, normalized size = 1.07 \begin {gather*} \frac {a x^{2}}{2} + b \left (- \frac {e n \left (- \frac {3 d^{3} \left (\begin {cases} \frac {x^{\frac {2}{3}}}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e x^{\frac {2}{3}} \right )}}{e} & \text {otherwise} \end {cases}\right )}{2 e^{3}} + \frac {3 d^{2} x^{\frac {2}{3}}}{2 e^{3}} - \frac {3 d x^{\frac {4}{3}}}{4 e^{2}} + \frac {x^{2}}{2 e}\right )}{3} + \frac {x^{2} \log {\left (c \left (d + e x^{\frac {2}{3}}\right )^{n} \right )}}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.97, size = 82, normalized size = 0.92 \begin {gather*} \frac {1}{2} \, b x^{2} \log \left (c\right ) + \frac {1}{12} \, {\left (6 \, x^{2} \log \left (x^{\frac {2}{3}} e + d\right ) + {\left (6 \, d^{3} e^{\left (-4\right )} \log \left ({\left | x^{\frac {2}{3}} e + d \right |}\right ) + {\left (3 \, d x^{\frac {4}{3}} e - 2 \, x^{2} e^{2} - 6 \, d^{2} x^{\frac {2}{3}}\right )} e^{\left (-3\right )}\right )} e\right )} b n + \frac {1}{2} \, a x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.39, size = 74, normalized size = 0.83 \begin {gather*} \frac {a\,x^2}{2}-\frac {b\,n\,x^2}{6}+\frac {b\,x^2\,\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )}{2}+\frac {b\,d\,n\,x^{4/3}}{4\,e}+\frac {b\,d^3\,n\,\ln \left (d+e\,x^{2/3}\right )}{2\,e^3}-\frac {b\,d^2\,n\,x^{2/3}}{2\,e^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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