Optimal. Leaf size=89 \[ \frac {b n}{6 x^2}-\frac {b d n}{4 e x^{4/3}}+\frac {b d^2 n}{2 e^2 x^{2/3}}-\frac {b d^3 n \log \left (d+\frac {e}{x^{2/3}}\right )}{2 e^3}-\frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{2 x^2} \]
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Rubi [A]
time = 0.05, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {2504, 2442, 45}
\begin {gather*} -\frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{2 x^2}-\frac {b d^3 n \log \left (d+\frac {e}{x^{2/3}}\right )}{2 e^3}+\frac {b d^2 n}{2 e^2 x^{2/3}}-\frac {b d n}{4 e x^{4/3}}+\frac {b n}{6 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2442
Rule 2504
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{x^3} \, dx &=-\left (\frac {3}{2} \text {Subst}\left (\int x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx,x,\frac {1}{x^{2/3}}\right )\right )\\ &=-\frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{2 x^2}+\frac {1}{2} (b e n) \text {Subst}\left (\int \frac {x^3}{d+e x} \, dx,x,\frac {1}{x^{2/3}}\right )\\ &=-\frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{2 x^2}+\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,\frac {1}{x^{2/3}}\right )\\ &=\frac {b n}{6 x^2}-\frac {b d n}{4 e x^{4/3}}+\frac {b d^2 n}{2 e^2 x^{2/3}}-\frac {b d^3 n \log \left (d+\frac {e}{x^{2/3}}\right )}{2 e^3}-\frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{2 x^2}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 94, normalized size = 1.06 \begin {gather*} -\frac {a}{2 x^2}+\frac {b n}{6 x^2}-\frac {b d n}{4 e x^{4/3}}+\frac {b d^2 n}{2 e^2 x^{2/3}}-\frac {b d^3 n \log \left (d+\frac {e}{x^{2/3}}\right )}{2 e^3}-\frac {b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{2 x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {a +b \ln \left (c \left (d +\frac {e}{x^{\frac {2}{3}}}\right )^{n}\right )}{x^{3}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 88, normalized size = 0.99 \begin {gather*} -\frac {1}{12} \, {\left (6 \, d^{3} e^{\left (-4\right )} \log \left (d x^{\frac {2}{3}} + e\right ) - 6 \, d^{3} e^{\left (-4\right )} \log \left (x^{\frac {2}{3}}\right ) - \frac {{\left (6 \, d^{2} x^{\frac {4}{3}} - 3 \, d x^{\frac {2}{3}} e + 2 \, e^{2}\right )} e^{\left (-3\right )}}{x^{2}}\right )} b n e - \frac {b \log \left (c {\left (d + \frac {e}{x^{\frac {2}{3}}}\right )}^{n}\right )}{2 \, x^{2}} - \frac {a}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 80, normalized size = 0.90 \begin {gather*} \frac {{\left (6 \, b d^{2} n x^{\frac {4}{3}} e - 3 \, b d n x^{\frac {2}{3}} e^{2} - 6 \, b e^{3} \log \left (c\right ) + 2 \, {\left (b n - 3 \, a\right )} e^{3} - 6 \, {\left (b d^{3} n x^{2} + b n e^{3}\right )} \log \left (\frac {d x + x^{\frac {1}{3}} e}{x}\right )\right )} e^{\left (-3\right )}}{12 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.33, size = 104, normalized size = 1.17 \begin {gather*} \frac {1}{12} \, {\left ({\left (12 \, d^{3} e^{\left (-4\right )} \log \left (x^{\frac {1}{3}}\right ) - 6 \, d^{3} e^{\left (-4\right )} \log \left ({\left | d x^{\frac {2}{3}} + e \right |}\right ) - \frac {{\left (11 \, d^{3} x^{2} - 6 \, d^{2} x^{\frac {4}{3}} e + 3 \, d x^{\frac {2}{3}} e^{2} - 2 \, e^{3}\right )} e^{\left (-4\right )}}{x^{2}}\right )} e - \frac {6 \, \log \left (d + \frac {e}{x^{\frac {2}{3}}}\right )}{x^{2}}\right )} b n - \frac {b \log \left (c\right )}{2 \, x^{2}} - \frac {a}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.44, size = 74, normalized size = 0.83 \begin {gather*} \frac {b\,n}{6\,x^2}-\frac {a}{2\,x^2}-\frac {b\,\ln \left (c\,{\left (d+\frac {e}{x^{2/3}}\right )}^n\right )}{2\,x^2}-\frac {b\,d\,n}{4\,e\,x^{4/3}}-\frac {b\,d^3\,n\,\ln \left (d+\frac {e}{x^{2/3}}\right )}{2\,e^3}+\frac {b\,d^2\,n}{2\,e^2\,x^{2/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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