Optimal. Leaf size=24 \[ \frac{x (a+b x)^3}{3 b \sqrt{c x^2}} \]
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Rubi [A] time = 0.0033883, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {15, 32} \[ \frac{x (a+b x)^3}{3 b \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 32
Rubi steps
\begin{align*} \int \frac{x (a+b x)^2}{\sqrt{c x^2}} \, dx &=\frac{x \int (a+b x)^2 \, dx}{\sqrt{c x^2}}\\ &=\frac{x (a+b x)^3}{3 b \sqrt{c x^2}}\\ \end{align*}
Mathematica [A] time = 0.001783, size = 24, normalized size = 1. \[ \frac{x (a+b x)^3}{3 b \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 31, normalized size = 1.3 \begin{align*}{\frac{{x}^{2} \left ({b}^{2}{x}^{2}+3\,abx+3\,{a}^{2} \right ) }{3}{\frac{1}{\sqrt{c{x}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.04516, size = 57, normalized size = 2.38 \begin{align*} \frac{\sqrt{c x^{2}} b^{2} x^{2}}{3 \, c} + \frac{a b x^{2}}{\sqrt{c}} + \frac{\sqrt{c x^{2}} a^{2}}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.45534, size = 63, normalized size = 2.62 \begin{align*} \frac{{\left (b^{2} x^{2} + 3 \, a b x + 3 \, a^{2}\right )} \sqrt{c x^{2}}}{3 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.507174, size = 56, normalized size = 2.33 \begin{align*} \frac{a^{2} x^{2}}{\sqrt{c} \sqrt{x^{2}}} + \frac{a b x^{3}}{\sqrt{c} \sqrt{x^{2}}} + \frac{b^{2} x^{4}}{3 \sqrt{c} \sqrt{x^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06458, size = 49, normalized size = 2.04 \begin{align*} \frac{1}{3} \, \sqrt{c x^{2}}{\left ({\left (\frac{b^{2} x}{c} + \frac{3 \, a b}{c}\right )} x + \frac{3 \, a^{2}}{c}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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