Optimal. Leaf size=28 \[ -\frac{2 x^{-n-2}}{b (3 n+4) \sqrt{b x^n}} \]
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Rubi [A] time = 0.0069323, antiderivative size = 28, 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} \[ -\frac{2 x^{-n-2}}{b (3 n+4) \sqrt{b x^n}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 30
Rubi steps
\begin{align*} \int \frac{1}{x^3 \left (b x^n\right )^{3/2}} \, dx &=\frac{x^{n/2} \int x^{-3-\frac{3 n}{2}} \, dx}{b \sqrt{b x^n}}\\ &=-\frac{2 x^{-2-n}}{b (4+3 n) \sqrt{b x^n}}\\ \end{align*}
Mathematica [A] time = 0.0066354, size = 22, normalized size = 0.79 \[ \frac{1}{\left (-\frac{3 n}{2}-2\right ) x^2 \left (b x^n\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 20, normalized size = 0.7 \begin{align*} -2\,{\frac{1}{{x}^{2} \left ( 4+3\,n \right ) \left ( b{x}^{n} \right ) ^{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 [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{1}{\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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