Optimal. Leaf size=94 \[ -\frac {b e n}{4 d x^{4/3}}+\frac {b e^2 n}{2 d^2 x^{2/3}}-\frac {b e^3 n \log \left (d+e x^{2/3}\right )}{2 d^3}-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )}{2 x^2}+\frac {b e^3 n \log (x)}{3 d^3} \]
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Rubi [A]
time = 0.05, antiderivative size = 94, 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, 46}
\begin {gather*} -\frac {a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )}{2 x^2}-\frac {b e^3 n \log \left (d+e x^{2/3}\right )}{2 d^3}+\frac {b e^3 n \log (x)}{3 d^3}+\frac {b e^2 n}{2 d^2 x^{2/3}}-\frac {b e n}{4 d x^{4/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 2442
Rule 2504
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )}{x^3} \, dx &=\frac {3}{2} \text {Subst}\left (\int \frac {a+b \log \left (c (d+e x)^n\right )}{x^4} \, dx,x,x^{2/3}\right )\\ &=-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )}{2 x^2}+\frac {1}{2} (b e n) \text {Subst}\left (\int \frac {1}{x^3 (d+e x)} \, dx,x,x^{2/3}\right )\\ &=-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )}{2 x^2}+\frac {1}{2} (b e n) \text {Subst}\left (\int \left (\frac {1}{d x^3}-\frac {e}{d^2 x^2}+\frac {e^2}{d^3 x}-\frac {e^3}{d^3 (d+e x)}\right ) \, dx,x,x^{2/3}\right )\\ &=-\frac {b e n}{4 d x^{4/3}}+\frac {b e^2 n}{2 d^2 x^{2/3}}-\frac {b e^3 n \log \left (d+e x^{2/3}\right )}{2 d^3}-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )}{2 x^2}+\frac {b e^3 n \log (x)}{3 d^3}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 91, normalized size = 0.97 \begin {gather*} -\frac {a}{2 x^2}-\frac {b \log \left (c \left (d+e x^{2/3}\right )^n\right )}{2 x^2}+\frac {1}{2} b e n \left (-\frac {1}{2 d x^{4/3}}+\frac {e}{d^2 x^{2/3}}-\frac {e^2 \log \left (d+e x^{2/3}\right )}{d^3}+\frac {2 e^2 \log (x)}{3 d^3}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {a +b \ln \left (c \left (d +e \,x^{\frac {2}{3}}\right )^{n}\right )}{x^{3}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 79, normalized size = 0.84 \begin {gather*} -\frac {1}{4} \, b n {\left (\frac {2 \, e^{2} \log \left (x^{\frac {2}{3}} e + d\right )}{d^{3}} - \frac {2 \, e^{2} \log \left (x^{\frac {2}{3}}\right )}{d^{3}} - \frac {2 \, x^{\frac {2}{3}} e - d}{d^{2} x^{\frac {4}{3}}}\right )} e - \frac {b \log \left ({\left (x^{\frac {2}{3}} e + d\right )}^{n} 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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Fricas [A]
time = 0.37, size = 84, normalized size = 0.89 \begin {gather*} \frac {4 \, b n x^{2} e^{3} \log \left (x^{\frac {1}{3}}\right ) - b d^{2} n x^{\frac {2}{3}} e + 2 \, b d n x^{\frac {4}{3}} e^{2} - 2 \, b d^{3} \log \left (c\right ) - 2 \, a d^{3} - 2 \, {\left (b d^{3} n + b n x^{2} e^{3}\right )} \log \left (x^{\frac {2}{3}} e + d\right )}{4 \, d^{3} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.21, size = 95, normalized size = 1.01 \begin {gather*} \frac {1}{4} \, {\left ({\left (\frac {2 \, \log \left (x^{\frac {2}{3}} e\right )}{d^{3}} - \frac {2 \, \log \left ({\left | x^{\frac {2}{3}} e + d \right |}\right )}{d^{3}} + \frac {{\left (2 \, {\left (x^{\frac {2}{3}} e + d\right )} d - 3 \, d^{2}\right )} e^{\left (-2\right )}}{d^{3} x^{\frac {4}{3}}}\right )} e^{4} - \frac {2 \, e \log \left (x^{\frac {2}{3}} e + d\right )}{x^{2}}\right )} b n e^{\left (-1\right )} - \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 = 0.61, size = 74, normalized size = 0.79 \begin {gather*} -\frac {\frac {b\,e\,n}{2\,d}-\frac {b\,e^2\,n\,x^{2/3}}{d^2}}{2\,x^{4/3}}-\frac {a}{2\,x^2}-\frac {b\,\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )}{2\,x^2}-\frac {b\,e^3\,n\,\mathrm {atanh}\left (\frac {2\,e\,x^{2/3}}{d}+1\right )}{d^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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