Optimal. Leaf size=32 \[ \frac{a x^2}{\sqrt{c x^2}}+\frac{b x^3}{2 \sqrt{c x^2}} \]
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Rubi [A] time = 0.0042455, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {15} \[ \frac{a x^2}{\sqrt{c x^2}}+\frac{b x^3}{2 \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rubi steps
\begin{align*} \int \frac{x (a+b x)}{\sqrt{c x^2}} \, dx &=\frac{x \int (a+b x) \, dx}{\sqrt{c x^2}}\\ &=\frac{a x^2}{\sqrt{c x^2}}+\frac{b x^3}{2 \sqrt{c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0014175, size = 23, normalized size = 0.72 \[ \frac{x^2 (2 a+b x)}{2 \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 20, normalized size = 0.6 \begin{align*}{\frac{{x}^{2} \left ( bx+2\,a \right ) }{2}{\frac{1}{\sqrt{c{x}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00629, size = 30, normalized size = 0.94 \begin{align*} \frac{b x^{2}}{2 \, \sqrt{c}} + \frac{\sqrt{c x^{2}} a}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.66548, size = 42, normalized size = 1.31 \begin{align*} \frac{\sqrt{c x^{2}}{\left (b x + 2 \, a\right )}}{2 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.437677, size = 34, normalized size = 1.06 \begin{align*} \frac{a x^{2}}{\sqrt{c} \sqrt{x^{2}}} + \frac{b x^{3}}{2 \sqrt{c} \sqrt{x^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07768, size = 30, normalized size = 0.94 \begin{align*} \frac{1}{2} \, \sqrt{c x^{2}}{\left (\frac{b x}{c} + \frac{2 \, a}{c}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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