Optimal. Leaf size=136 \[ \frac {b d^5 n \sqrt [3]{x}}{2 e^5}-\frac {b d^4 n x^{2/3}}{4 e^4}+\frac {b d^3 n x}{6 e^3}-\frac {b d^2 n x^{4/3}}{8 e^2}+\frac {b d n x^{5/3}}{10 e}-\frac {1}{12} b n x^2-\frac {b d^6 n \log \left (d+e \sqrt [3]{x}\right )}{2 e^6}+\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \]
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Rubi [A]
time = 0.06, antiderivative size = 136, 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 \sqrt [3]{x}\right )^n\right )\right )-\frac {b d^6 n \log \left (d+e \sqrt [3]{x}\right )}{2 e^6}+\frac {b d^5 n \sqrt [3]{x}}{2 e^5}-\frac {b d^4 n x^{2/3}}{4 e^4}+\frac {b d^3 n x}{6 e^3}-\frac {b d^2 n x^{4/3}}{8 e^2}+\frac {b d n x^{5/3}}{10 e}-\frac {1}{12} 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 \sqrt [3]{x}\right )^n\right )\right ) \, dx &=3 \text {Subst}\left (\int x^5 \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )-\frac {1}{2} (b e n) \text {Subst}\left (\int \frac {x^6}{d+e x} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )-\frac {1}{2} (b e n) \text {Subst}\left (\int \left (-\frac {d^5}{e^6}+\frac {d^4 x}{e^5}-\frac {d^3 x^2}{e^4}+\frac {d^2 x^3}{e^3}-\frac {d x^4}{e^2}+\frac {x^5}{e}+\frac {d^6}{e^6 (d+e x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {b d^5 n \sqrt [3]{x}}{2 e^5}-\frac {b d^4 n x^{2/3}}{4 e^4}+\frac {b d^3 n x}{6 e^3}-\frac {b d^2 n x^{4/3}}{8 e^2}+\frac {b d n x^{5/3}}{10 e}-\frac {1}{12} b n x^2-\frac {b d^6 n \log \left (d+e \sqrt [3]{x}\right )}{2 e^6}+\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 133, normalized size = 0.98 \begin {gather*} \frac {a x^2}{2}-\frac {1}{2} b e n \left (-\frac {d^5 \sqrt [3]{x}}{e^6}+\frac {d^4 x^{2/3}}{2 e^5}-\frac {d^3 x}{3 e^4}+\frac {d^2 x^{4/3}}{4 e^3}-\frac {d x^{5/3}}{5 e^2}+\frac {x^2}{6 e}+\frac {d^6 \log \left (d+e \sqrt [3]{x}\right )}{e^7}\right )+\frac {1}{2} b x^2 \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int x \left (a +b \ln \left (c \left (d +e \,x^{\frac {1}{3}}\right )^{n}\right )\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 104, normalized size = 0.76 \begin {gather*} -\frac {1}{120} \, {\left (60 \, d^{6} e^{\left (-7\right )} \log \left (x^{\frac {1}{3}} e + d\right ) + {\left (30 \, d^{4} x^{\frac {2}{3}} e - 60 \, d^{5} x^{\frac {1}{3}} - 20 \, d^{3} x e^{2} + 15 \, d^{2} x^{\frac {4}{3}} e^{3} - 12 \, d x^{\frac {5}{3}} e^{4} + 10 \, x^{2} e^{5}\right )} e^{\left (-6\right )}\right )} b n e + \frac {1}{2} \, b x^{2} \log \left ({\left (x^{\frac {1}{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 = 114, normalized size = 0.84 \begin {gather*} \frac {1}{120} \, {\left (20 \, b d^{3} n x e^{3} + 60 \, b x^{2} e^{6} \log \left (c\right ) - 10 \, {\left (b n - 6 \, a\right )} x^{2} e^{6} - 60 \, {\left (b d^{6} n - b n x^{2} e^{6}\right )} \log \left (x^{\frac {1}{3}} e + d\right ) - 6 \, {\left (5 \, b d^{4} n e^{2} - 2 \, b d n x e^{5}\right )} x^{\frac {2}{3}} + 15 \, {\left (4 \, b d^{5} n e - b d^{2} n x e^{4}\right )} x^{\frac {1}{3}}\right )} e^{\left (-6\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 2.70, size = 131, normalized size = 0.96 \begin {gather*} \frac {a x^{2}}{2} + b \left (- \frac {e n \left (\frac {3 d^{6} \left (\begin {cases} \frac {\sqrt [3]{x}}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e \sqrt [3]{x} \right )}}{e} & \text {otherwise} \end {cases}\right )}{e^{6}} - \frac {3 d^{5} \sqrt [3]{x}}{e^{6}} + \frac {3 d^{4} x^{\frac {2}{3}}}{2 e^{5}} - \frac {d^{3} x}{e^{4}} + \frac {3 d^{2} x^{\frac {4}{3}}}{4 e^{3}} - \frac {3 d x^{\frac {5}{3}}}{5 e^{2}} + \frac {x^{2}}{2 e}\right )}{6} + \frac {x^{2} \log {\left (c \left (d + 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 [B] Leaf count of result is larger than twice the leaf count of optimal. 271 vs.
\(2 (104) = 208\).
time = 4.37, size = 271, normalized size = 1.99 \begin {gather*} \frac {1}{120} \, {\left (60 \, b x^{2} e \log \left (c\right ) + 60 \, a x^{2} e + {\left (60 \, {\left (x^{\frac {1}{3}} e + d\right )}^{6} e^{\left (-5\right )} \log \left (x^{\frac {1}{3}} e + d\right ) - 360 \, {\left (x^{\frac {1}{3}} e + d\right )}^{5} d e^{\left (-5\right )} \log \left (x^{\frac {1}{3}} e + d\right ) + 900 \, {\left (x^{\frac {1}{3}} e + d\right )}^{4} d^{2} e^{\left (-5\right )} \log \left (x^{\frac {1}{3}} e + d\right ) - 1200 \, {\left (x^{\frac {1}{3}} e + d\right )}^{3} d^{3} e^{\left (-5\right )} \log \left (x^{\frac {1}{3}} e + d\right ) + 900 \, {\left (x^{\frac {1}{3}} e + d\right )}^{2} d^{4} e^{\left (-5\right )} \log \left (x^{\frac {1}{3}} e + d\right ) - 360 \, {\left (x^{\frac {1}{3}} e + d\right )} d^{5} e^{\left (-5\right )} \log \left (x^{\frac {1}{3}} e + d\right ) - 10 \, {\left (x^{\frac {1}{3}} e + d\right )}^{6} e^{\left (-5\right )} + 72 \, {\left (x^{\frac {1}{3}} e + d\right )}^{5} d e^{\left (-5\right )} - 225 \, {\left (x^{\frac {1}{3}} e + d\right )}^{4} d^{2} e^{\left (-5\right )} + 400 \, {\left (x^{\frac {1}{3}} e + d\right )}^{3} d^{3} e^{\left (-5\right )} - 450 \, {\left (x^{\frac {1}{3}} e + d\right )}^{2} d^{4} e^{\left (-5\right )} + 360 \, {\left (x^{\frac {1}{3}} e + d\right )} d^{5} e^{\left (-5\right )}\right )} b n\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.42, size = 111, normalized size = 0.82 \begin {gather*} \frac {a\,x^2}{2}-\frac {b\,n\,x^2}{12}+\frac {b\,x^2\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{2}+\frac {b\,d^3\,n\,x}{6\,e^3}+\frac {b\,d\,n\,x^{5/3}}{10\,e}-\frac {b\,d^6\,n\,\ln \left (d+e\,x^{1/3}\right )}{2\,e^6}-\frac {b\,d^2\,n\,x^{4/3}}{8\,e^2}-\frac {b\,d^4\,n\,x^{2/3}}{4\,e^4}+\frac {b\,d^5\,n\,x^{1/3}}{2\,e^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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