Optimal. Leaf size=19 \[ \frac{x (c x)^m}{m \sqrt{b x^2}} \]
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Rubi [A] time = 0.0049172, antiderivative size = 19, 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} \[ \frac{x (c x)^m}{m \sqrt{b x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 16
Rule 32
Rubi steps
\begin{align*} \int \frac{(c x)^m}{\sqrt{b x^2}} \, dx &=\frac{x \int \frac{(c x)^m}{x} \, dx}{\sqrt{b x^2}}\\ &=\frac{(c x) \int (c x)^{-1+m} \, dx}{\sqrt{b x^2}}\\ &=\frac{x (c x)^m}{m \sqrt{b x^2}}\\ \end{align*}
Mathematica [A] time = 0.0024654, size = 19, normalized size = 1. \[ \frac{x (c x)^m}{m \sqrt{b x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 18, normalized size = 1. \begin{align*}{\frac{x \left ( cx \right ) ^{m}}{m}{\frac{1}{\sqrt{b{x}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.997392, size = 18, normalized size = 0.95 \begin{align*} \frac{c^{m} x^{m}}{\sqrt{b} m} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.81149, size = 39, normalized size = 2.05 \begin{align*} \frac{\sqrt{b x^{2}} \left (c x\right )^{m}}{b m x} \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}}{\sqrt{b x^{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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