Optimal. Leaf size=44 \[ \frac{3 x^{4 (1-n)} \left (a x^{-3 (1-n)}+b x^{4 n-3}\right )^{4/3}}{4 b n} \]
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Rubi [A] time = 0.0172021, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {1979, 2000} \[ \frac{3 x^{4 (1-n)} \left (a x^{-3 (1-n)}+b x^{4 n-3}\right )^{4/3}}{4 b n} \]
Antiderivative was successfully verified.
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Rule 1979
Rule 2000
Rubi steps
\begin{align*} \int \sqrt [3]{x^{3 (-1+n)} \left (a+b x^n\right )} \, dx &=\int \sqrt [3]{a x^{3 (-1+n)}+b x^{3 (-1+n)+n}} \, dx\\ &=\frac{3 x^{4 (1-n)} \left (a x^{-3 (1-n)}+b x^{-3+4 n}\right )^{4/3}}{4 b n}\\ \end{align*}
Mathematica [A] time = 0.0298387, size = 36, normalized size = 0.82 \[ \frac{3 x^{4-4 n} \left (x^{3 n-3} \left (a+b x^n\right )\right )^{4/3}}{4 b n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 40, normalized size = 0.9 \begin{align*}{\frac{3\,x \left ( a+b{x}^{n} \right ) }{4\,b{x}^{n}n}\sqrt [3]{{\frac{ \left ({x}^{n} \right ) ^{3} \left ( a+b{x}^{n} \right ) }{{x}^{3}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.19037, size = 23, normalized size = 0.52 \begin{align*} \frac{3 \,{\left (b x^{n} + a\right )}^{\frac{4}{3}}}{4 \, b n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.678657, size = 90, normalized size = 2.05 \begin{align*} \frac{3 \,{\left (b x x^{n} + a x\right )} \left (\frac{b x^{4 \, n} + a x^{3 \, n}}{x^{3}}\right )^{\frac{1}{3}}}{4 \, b n x^{n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left ({\left (b x^{n} + a\right )} x^{3 \, n - 3}\right )^{\frac{1}{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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