Optimal. Leaf size=28 \[ -\frac {2 x^{-n-2}}{b (3 n+4) \sqrt {b x^n}} \]
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Rubi [A] time = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.01, 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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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b x^{n}\right )^{\frac {3}{2}} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 20, normalized size = 0.71 \[ -\frac {2}{\left (3 n +4\right ) \left (b \,x^{n}\right )^{\frac {3}{2}} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 19, normalized size = 0.68 \[ -\frac {2}{\left (b x^{n}\right )^{\frac {3}{2}} {\left (3 \, n + 4\right )} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {1}{x^3\,{\left (b\,x^n\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \begin {cases} - \frac {2}{3 b^{\frac {3}{2}} n x^{2} \left (x^{n}\right )^{\frac {3}{2}} + 4 b^{\frac {3}{2}} x^{2} \left (x^{n}\right )^{\frac {3}{2}}} & \text {for}\: n \neq - \frac {4}{3} \\\int \frac {1}{x^{3} \left (\frac {b}{x^{\frac {4}{3}}}\right )^{\frac {3}{2}}}\, dx & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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