Optimal. Leaf size=29 \[ \frac{b \sqrt{b x^2} (c x)^{m+4}}{c^4 (m+4) x} \]
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Rubi [A] time = 0.0098853, antiderivative size = 29, 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} \[ \frac{b \sqrt{b x^2} (c x)^{m+4}}{c^4 (m+4) x} \]
Antiderivative was successfully verified.
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Rule 15
Rule 16
Rule 32
Rubi steps
\begin{align*} \int (c x)^m \left (b x^2\right )^{3/2} \, dx &=\frac{\left (b \sqrt{b x^2}\right ) \int x^3 (c x)^m \, dx}{x}\\ &=\frac{\left (b \sqrt{b x^2}\right ) \int (c x)^{3+m} \, dx}{c^3 x}\\ &=\frac{b (c x)^{4+m} \sqrt{b x^2}}{c^4 (4+m) x}\\ \end{align*}
Mathematica [A] time = 0.0044416, size = 21, normalized size = 0.72 \[ \frac{x \left (b x^2\right )^{3/2} (c x)^m}{m+4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 20, normalized size = 0.7 \begin{align*}{\frac{x \left ( cx \right ) ^{m}}{4+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 = 1.05888, size = 24, normalized size = 0.83 \begin{align*} \frac{b^{\frac{3}{2}} c^{m} x^{4} x^{m}}{m + 4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.50792, size = 47, normalized size = 1.62 \begin{align*} \frac{\sqrt{b x^{2}} \left (c x\right )^{m} b x^{3}}{m + 4} \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(-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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