Optimal. Leaf size=20 \[ \frac{2 b x^n \sqrt{b x^n}}{3 n} \]
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Rubi [A] time = 0.0034291, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {15, 30} \[ \frac{2 b x^n \sqrt{b x^n}}{3 n} \]
Antiderivative was successfully verified.
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Rule 15
Rule 30
Rubi steps
\begin{align*} \int \frac{\left (b x^n\right )^{3/2}}{x} \, dx &=\left (b x^{-n/2} \sqrt{b x^n}\right ) \int x^{-1+\frac{3 n}{2}} \, dx\\ &=\frac{2 b x^n \sqrt{b x^n}}{3 n}\\ \end{align*}
Mathematica [A] time = 0.0020905, size = 16, normalized size = 0.8 \[ \frac{2 \left (b x^n\right )^{3/2}}{3 n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 13, normalized size = 0.7 \begin{align*}{\frac{2}{3\,n} \left ( b{x}^{n} \right ) ^{{\frac{3}{2}}}} \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 [A] time = 1.958, size = 34, normalized size = 1.7 \begin{align*} \frac{2 \, \sqrt{b x^{n}} b x^{n}}{3 \, n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.87608, size = 24, normalized size = 1.2 \begin{align*} \begin{cases} \frac{2 b^{\frac{3}{2}} \left (x^{n}\right )^{\frac{3}{2}}}{3 n} & \text{for}\: n \neq 0 \\b^{\frac{3}{2}} \log{\left (x \right )} & \text{otherwise} \end{cases} \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 \frac{\left (b x^{n}\right )^{\frac{3}{2}}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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