Optimal. Leaf size=28 \[ \frac{\sqrt{b x^2} (c x)^{m+2}}{c^2 (m+2) x} \]
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Rubi [A] time = 0.0090569, antiderivative size = 28, 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} \[ \frac{\sqrt{b x^2} (c x)^{m+2}}{c^2 (m+2) x} \]
Antiderivative was successfully verified.
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Rule 15
Rule 16
Rule 32
Rubi steps
\begin{align*} \int (c x)^m \sqrt{b x^2} \, dx &=\frac{\sqrt{b x^2} \int x (c x)^m \, dx}{x}\\ &=\frac{\sqrt{b x^2} \int (c x)^{1+m} \, dx}{c x}\\ &=\frac{(c x)^{2+m} \sqrt{b x^2}}{c^2 (2+m) x}\\ \end{align*}
Mathematica [A] time = 0.0033506, size = 21, normalized size = 0.75 \[ \frac{x \sqrt{b x^2} (c x)^m}{m+2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 20, normalized size = 0.7 \begin{align*}{\frac{x \left ( cx \right ) ^{m}}{2+m}\sqrt{b{x}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03814, size = 24, normalized size = 0.86 \begin{align*} \frac{\sqrt{b} c^{m} x^{2} x^{m}}{m + 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.90814, size = 42, normalized size = 1.5 \begin{align*} \frac{\sqrt{b x^{2}} \left (c x\right )^{m} x}{m + 2} \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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