Optimal. Leaf size=32 \[ \frac {2 x^{1+m-n}}{b (2+2 m-3 n) \sqrt {b x^n}} \]
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Rubi [A]
time = 0.00, 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}
\begin {gather*} \frac {2 x^{m-n+1}}{b (2 m-3 n+2) \sqrt {b x^n}} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 30
Rubi steps
\begin {align*} \int \frac {x^m}{\left (b x^n\right )^{3/2}} \, dx &=\frac {x^{n/2} \int x^{m-\frac {3 n}{2}} \, dx}{b \sqrt {b x^n}}\\ &=\frac {2 x^{1+m-n}}{b (2+2 m-3 n) \sqrt {b x^n}}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 25, normalized size = 0.78 \begin {gather*} \frac {x^{1+m}}{\left (1+m-\frac {3 n}{2}\right ) \left (b x^n\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 25, normalized size = 0.78
method | result | size |
gosper | \(\frac {2 x^{1+m}}{\left (2+2 m -3 n \right ) \left (b \,x^{n}\right )^{\frac {3}{2}}}\) | \(25\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 24, normalized size = 0.75 \begin {gather*} \frac {2 \, x x^{m}}{b^{\frac {3}{2}} {\left (2 \, m - 3 \, n + 2\right )} {\left (x^{n}\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 {gather*} \begin {cases} \frac {2 x x^{m}}{2 m \left (b x^{n}\right )^{\frac {3}{2}} - 3 n \left (b x^{n}\right )^{\frac {3}{2}} + 2 \left (b x^{n}\right )^{\frac {3}{2}}} & \text {for}\: m \neq \frac {3 n}{2} - 1 \\\int \frac {x^{\frac {3 n}{2} - 1}}{\left (b x^{n}\right )^{\frac {3}{2}}}\, dx & \text {otherwise} \end {cases} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {x^m}{{\left (b\,x^n\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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