Optimal. Leaf size=141 \[ \frac {b e^5 n \sqrt [3]{x}}{2 d^5}-\frac {b e^4 n x^{2/3}}{4 d^4}+\frac {b e^3 n x}{6 d^3}-\frac {b e^2 n x^{4/3}}{8 d^2}+\frac {b e n x^{5/3}}{10 d}-\frac {b e^6 n \log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{2 d^6}+\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )-\frac {b e^6 n \log (x)}{6 d^6} \]
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Rubi [A]
time = 0.06, antiderivative size = 141, 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, 46}
\begin {gather*} \frac {1}{2} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )-\frac {b e^6 n \log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{2 d^6}-\frac {b e^6 n \log (x)}{6 d^6}+\frac {b e^5 n \sqrt [3]{x}}{2 d^5}-\frac {b e^4 n x^{2/3}}{4 d^4}+\frac {b e^3 n x}{6 d^3}-\frac {b e^2 n x^{4/3}}{8 d^2}+\frac {b e n x^{5/3}}{10 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 2442
Rule 2504
Rubi steps
\begin {align*} \int x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right ) \, dx &=-\left (3 \text {Subst}\left (\int \frac {a+b \log \left (c (d+e x)^n\right )}{x^7} \, dx,x,\frac {1}{\sqrt [3]{x}}\right )\right )\\ &=\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )-\frac {1}{2} (b e n) \text {Subst}\left (\int \frac {1}{x^6 (d+e x)} \, dx,x,\frac {1}{\sqrt [3]{x}}\right )\\ &=\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )-\frac {1}{2} (b e n) \text {Subst}\left (\int \left (\frac {1}{d x^6}-\frac {e}{d^2 x^5}+\frac {e^2}{d^3 x^4}-\frac {e^3}{d^4 x^3}+\frac {e^4}{d^5 x^2}-\frac {e^5}{d^6 x}+\frac {e^6}{d^6 (d+e x)}\right ) \, dx,x,\frac {1}{\sqrt [3]{x}}\right )\\ &=\frac {b e^5 n \sqrt [3]{x}}{2 d^5}-\frac {b e^4 n x^{2/3}}{4 d^4}+\frac {b e^3 n x}{6 d^3}-\frac {b e^2 n x^{4/3}}{8 d^2}+\frac {b e n x^{5/3}}{10 d}-\frac {b e^6 n \log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{2 d^6}+\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )-\frac {b e^6 n \log (x)}{6 d^6}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 132, normalized size = 0.94 \begin {gather*} \frac {a x^2}{2}+\frac {1}{2} b x^2 \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )-\frac {1}{2} b e n \left (-\frac {e^4 \sqrt [3]{x}}{d^5}+\frac {e^3 x^{2/3}}{2 d^4}-\frac {e^2 x}{3 d^3}+\frac {e x^{4/3}}{4 d^2}-\frac {x^{5/3}}{5 d}+\frac {e^5 \log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^6}+\frac {e^5 \log (x)}{3 d^6}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int x \left (a +b \ln \left (c \left (d +\frac {e}{x^{\frac {1}{3}}}\right )^{n}\right )\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 95, normalized size = 0.67 \begin {gather*} \frac {1}{120} \, b n {\left (\frac {12 \, d^{4} x^{\frac {5}{3}} - 15 \, d^{3} x^{\frac {4}{3}} e + 20 \, d^{2} x e^{2} - 30 \, d x^{\frac {2}{3}} e^{3} + 60 \, x^{\frac {1}{3}} e^{4}}{d^{5}} - \frac {60 \, e^{5} \log \left (d x^{\frac {1}{3}} + e\right )}{d^{6}}\right )} e + \frac {1}{2} \, b x^{2} \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{n}\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 = 0.38, size = 152, normalized size = 1.08 \begin {gather*} \frac {60 \, b d^{6} x^{2} \log \left (c\right ) + 60 \, a d^{6} x^{2} - 60 \, b d^{6} n \log \left (x^{\frac {1}{3}}\right ) + 20 \, b d^{3} n x e^{3} + 60 \, {\left (b d^{6} n - b n e^{6}\right )} \log \left (d x^{\frac {1}{3}} + e\right ) + 60 \, {\left (b d^{6} n x^{2} - b d^{6} n\right )} \log \left (\frac {d x + x^{\frac {2}{3}} e}{x}\right ) + 6 \, {\left (2 \, b d^{5} n x e - 5 \, b d^{2} n e^{4}\right )} x^{\frac {2}{3}} - 15 \, {\left (b d^{4} n x e^{2} - 4 \, b d n e^{5}\right )} x^{\frac {1}{3}}}{120 \, d^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 13.52, size = 138, normalized size = 0.98 \begin {gather*} \frac {a x^{2}}{2} + b \left (\frac {e n \left (\frac {3 x^{\frac {5}{3}}}{5 d} - \frac {3 e x^{\frac {4}{3}}}{4 d^{2}} + \frac {e^{2} x}{d^{3}} - \frac {3 e^{3} x^{\frac {2}{3}}}{2 d^{4}} + \frac {3 e^{4} \sqrt [3]{x}}{d^{5}} - \frac {3 e^{6} \left (\begin {cases} \frac {1}{d \sqrt [3]{x}} & \text {for}\: e = 0 \\\frac {\log {\left (d + \frac {e}{\sqrt [3]{x}} \right )}}{e} & \text {otherwise} \end {cases}\right )}{d^{6}} + \frac {3 e^{5} \log {\left (\frac {1}{\sqrt [3]{x}} \right )}}{d^{6}}\right )}{6} + \frac {x^{2} \log {\left (c \left (d + \frac {e}{\sqrt [3]{x}}\right )^{n} \right )}}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.46, size = 101, normalized size = 0.72 \begin {gather*} \frac {1}{2} \, b x^{2} \log \left (c\right ) + \frac {1}{120} \, {\left (60 \, x^{2} \log \left (d + \frac {e}{x^{\frac {1}{3}}}\right ) + {\left (\frac {12 \, d^{4} x^{\frac {5}{3}} - 15 \, d^{3} x^{\frac {4}{3}} e + 20 \, d^{2} x e^{2} - 30 \, d x^{\frac {2}{3}} e^{3} + 60 \, x^{\frac {1}{3}} e^{4}}{d^{5}} - \frac {60 \, e^{5} \log \left ({\left | d x^{\frac {1}{3}} + e \right |}\right )}{d^{6}}\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 = 0.79, size = 112, normalized size = 0.79 \begin {gather*} \frac {x^{5/3}\,\left (\frac {b\,e\,n}{5\,d}-\frac {b\,e^2\,n}{4\,d^2\,x^{1/3}}-\frac {b\,e^4\,n}{2\,d^4\,x}+\frac {b\,e^3\,n}{3\,d^3\,x^{2/3}}+\frac {b\,e^5\,n}{d^5\,x^{4/3}}\right )}{2}+\frac {a\,x^2}{2}+\frac {b\,x^2\,\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )}{2}-\frac {b\,e^6\,n\,\mathrm {atanh}\left (\frac {2\,e}{d\,x^{1/3}}+1\right )}{d^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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