Optimal. Leaf size=32 \[ -\frac{c^2 x (c x)^{m-2}}{b (2-m) \sqrt{b x^2}} \]
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Rubi [A] time = 0.0119664, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {15, 16, 32} \[ -\frac{c^2 x (c x)^{m-2}}{b (2-m) \sqrt{b x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 16
Rule 32
Rubi steps
\begin{align*} \int \frac{(c x)^m}{\left (b x^2\right )^{3/2}} \, dx &=\frac{x \int \frac{(c x)^m}{x^3} \, dx}{b \sqrt{b x^2}}\\ &=\frac{\left (c^3 x\right ) \int (c x)^{-3+m} \, dx}{b \sqrt{b x^2}}\\ &=-\frac{c^2 x (c x)^{-2+m}}{b (2-m) \sqrt{b x^2}}\\ \end{align*}
Mathematica [A] time = 0.004051, size = 21, normalized size = 0.66 \[ \frac{x (c x)^m}{(m-2) \left (b x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 20, normalized size = 0.6 \begin{align*}{\frac{x \left ( cx \right ) ^{m}}{-2+m} \left ( b{x}^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05758, size = 24, normalized size = 0.75 \begin{align*} \frac{c^{m} x^{m}}{b^{\frac{3}{2}}{\left (m - 2\right )} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.79251, size = 58, normalized size = 1.81 \begin{align*} \frac{\sqrt{b x^{2}} \left (c x\right )^{m}}{{\left (b^{2} m - 2 \, b^{2}\right )} x^{3}} \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{\left (c x\right )^{m}}{\left (b x^{2}\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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