Optimal. Leaf size=32 \[ \frac {2 x^{m-n+1}}{a (2 m-3 n+2) \sqrt {a x^n}} \]
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Rubi [A] time = 0.01, antiderivative size = 32, 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} \[ \frac {2 x^{m-n+1}}{a (2 m-3 n+2) \sqrt {a x^n}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 30
Rubi steps
\begin {align*} \int \frac {x^m}{\left (a x^n\right )^{3/2}} \, dx &=\frac {x^{n/2} \int x^{m-\frac {3 n}{2}} \, dx}{a \sqrt {a x^n}}\\ &=\frac {2 x^{1+m-n}}{a (2+2 m-3 n) \sqrt {a x^n}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 25, normalized size = 0.78 \[ \frac {x^{m+1}}{\left (m-\frac {3 n}{2}+1\right ) \left (a x^n\right )^{3/2}} \]
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 \[ \int \frac {x^{m}}{\left (a x^{n}\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 25, normalized size = 0.78 \[ \frac {2 x^{m +1}}{\left (2 m -3 n +2\right ) \left (a \,x^{n}\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.49, size = 24, normalized size = 0.75 \[ \frac {2 \, x x^{m}}{a^{\frac {3}{2}} {\left (2 \, m - 3 \, n + 2\right )} {\left (x^{n}\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {x^m}{{\left (a\,x^n\right )}^{3/2}} \,d x \]
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 \[ \begin {cases} \frac {2 x x^{m}}{2 a^{\frac {3}{2}} m \left (x^{n}\right )^{\frac {3}{2}} - 3 a^{\frac {3}{2}} n \left (x^{n}\right )^{\frac {3}{2}} + 2 a^{\frac {3}{2}} \left (x^{n}\right )^{\frac {3}{2}}} & \text {for}\: m \neq \frac {3 n}{2} - 1 \\\int \frac {x^{\frac {3 n}{2} - 1}}{\left (a x^{n}\right )^{\frac {3}{2}}}\, dx & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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