Optimal. Leaf size=19 \[ \frac{3 x \left (b x^n\right )^{2/3}}{2 n+3} \]
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Rubi [A] time = 0.0050974, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {15, 30} \[ \frac{3 x \left (b x^n\right )^{2/3}}{2 n+3} \]
Antiderivative was successfully verified.
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Rule 15
Rule 30
Rubi steps
\begin{align*} \int \left (b x^n\right )^{2/3} \, dx &=\left (x^{-2 n/3} \left (b x^n\right )^{2/3}\right ) \int x^{2 n/3} \, dx\\ &=\frac{3 x \left (b x^n\right )^{2/3}}{3+2 n}\\ \end{align*}
Mathematica [A] time = 0.002495, size = 20, normalized size = 1.05 \[ \frac{x \left (b x^n\right )^{2/3}}{\frac{2 n}{3}+1} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 18, normalized size = 1. \begin{align*} 3\,{\frac{x \left ( b{x}^{n} \right ) ^{2/3}}{3+2\,n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20755, size = 24, normalized size = 1.26 \begin{align*} \frac{3 \, b^{\frac{2}{3}} x x^{\frac{2}{3} \, n}}{2 \, n + 3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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