Optimal. Leaf size=24 \[ \frac{2 b x^{n+1} \sqrt{b x^n}}{3 n+2} \]
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Rubi [A] time = 0.0054126, antiderivative size = 24, 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} \[ \frac{2 b x^{n+1} \sqrt{b x^n}}{3 n+2} \]
Antiderivative was successfully verified.
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Rule 15
Rule 30
Rubi steps
\begin{align*} \int \left (b x^n\right )^{3/2} \, dx &=\left (b x^{-n/2} \sqrt{b x^n}\right ) \int x^{3 n/2} \, dx\\ &=\frac{2 b x^{1+n} \sqrt{b x^n}}{2+3 n}\\ \end{align*}
Mathematica [A] time = 0.0034921, size = 20, normalized size = 0.83 \[ \frac{x \left (b x^n\right )^{3/2}}{\frac{3 n}{2}+1} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 18, normalized size = 0.8 \begin{align*} 2\,{\frac{x \left ( b{x}^{n} \right ) ^{3/2}}{2+3\,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 = 1.18572, size = 24, normalized size = 1. \begin{align*} \frac{2 \, b^{\frac{3}{2}} x x^{\frac{3}{2} \, n}}{3 \, n + 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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