Optimal. Leaf size=24 \[ -\frac{2 b x^{n-2} \sqrt{b x^n}}{4-3 n} \]
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Rubi [A] time = 0.0067825, antiderivative size = 24, 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} \[ -\frac{2 b x^{n-2} \sqrt{b x^n}}{4-3 n} \]
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^3} \, dx &=\left (b x^{-n/2} \sqrt{b x^n}\right ) \int x^{-3+\frac{3 n}{2}} \, dx\\ &=-\frac{2 b x^{-2+n} \sqrt{b x^n}}{4-3 n}\\ \end{align*}
Mathematica [A] time = 0.0055559, size = 22, normalized size = 0.92 \[ \frac{\left (b x^n\right )^{3/2}}{\left (\frac{3 n}{2}-2\right ) x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 20, normalized size = 0.8 \begin{align*} 2\,{\frac{ \left ( b{x}^{n} \right ) ^{3/2}}{{x}^{2} \left ( 3\,n-4 \right ) }} \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (b x^{n}\right )^{\frac{3}{2}}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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