Optimal. Leaf size=36 \[ \frac {2 x^{1-n} (c x)^m}{b (2 m-3 n+2) \sqrt {b x^n}} \]
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Rubi [A] time = 0.01, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {15, 20, 30} \[ \frac {2 x^{1-n} (c x)^m}{b (2 m-3 n+2) \sqrt {b x^n}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 20
Rule 30
Rubi steps
\begin {align*} \int \frac {(c x)^m}{\left (b x^n\right )^{3/2}} \, dx &=\frac {x^{n/2} \int x^{-3 n/2} (c x)^m \, dx}{b \sqrt {b x^n}}\\ &=\frac {\left (x^{-m+\frac {n}{2}} (c x)^m\right ) \int x^{m-\frac {3 n}{2}} \, dx}{b \sqrt {b x^n}}\\ &=\frac {2 x^{1-n} (c x)^m}{b (2+2 m-3 n) \sqrt {b x^n}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 26, normalized size = 0.72 \[ \frac {x (c x)^m}{\left (m-\frac {3 n}{2}+1\right ) \left (b x^n\right )^{3/2}} \]
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 \[ \int \frac {\left (c x\right )^{m}}{\left (b x^{n}\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 26, normalized size = 0.72 \[ \frac {2 x \left (c x \right )^{m}}{\left (2 m -3 n +2\right ) \left (b \,x^{n}\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.52, size = 27, normalized size = 0.75 \[ \frac {2 \, c^{m} x x^{m}}{b^{\frac {3}{2}} {\left (2 \, m - 3 \, n + 2\right )} {\left (x^{n}\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.03, size = 34, normalized size = 0.94 \[ \frac {2\,x^{1-2\,n}\,\sqrt {b\,x^n}\,{\left (c\,x\right )}^m}{b^2\,\left (2\,m-3\,n+2\right )} \]
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 \[ \begin {cases} \frac {2 c^{m} x x^{m}}{2 b^{\frac {3}{2}} m \left (x^{n}\right )^{\frac {3}{2}} - 3 b^{\frac {3}{2}} n \left (x^{n}\right )^{\frac {3}{2}} + 2 b^{\frac {3}{2}} \left (x^{n}\right )^{\frac {3}{2}}} & \text {for}\: m \neq \frac {3 n}{2} - 1 \\\int \frac {\left (c x\right )^{\frac {3 n}{2} - 1}}{\left (b x^{n}\right )^{\frac {3}{2}}}\, dx & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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