Optimal. Leaf size=26 \[ \frac{\sqrt{c x^2} (a+b x)^3}{3 b x} \]
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Rubi [A] time = 0.0036536, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {15, 32} \[ \frac{\sqrt{c x^2} (a+b x)^3}{3 b x} \]
Antiderivative was successfully verified.
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Rule 15
Rule 32
Rubi steps
\begin{align*} \int \frac{\sqrt{c x^2} (a+b x)^2}{x} \, dx &=\frac{\sqrt{c x^2} \int (a+b x)^2 \, dx}{x}\\ &=\frac{\sqrt{c x^2} (a+b x)^3}{3 b x}\\ \end{align*}
Mathematica [A] time = 0.005424, size = 25, normalized size = 0.96 \[ \frac{c x (a+b x)^3}{3 b \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 28, normalized size = 1.1 \begin{align*}{\frac{{b}^{2}{x}^{2}+3\,abx+3\,{a}^{2}}{3}\sqrt{c{x}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55154, size = 61, normalized size = 2.35 \begin{align*} \frac{1}{3} \,{\left (b^{2} x^{2} + 3 \, a b x + 3 \, a^{2}\right )} \sqrt{c x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.258034, size = 51, normalized size = 1.96 \begin{align*} a^{2} \sqrt{c} \sqrt{x^{2}} + a b \sqrt{c} x \sqrt{x^{2}} + \frac{b^{2} \sqrt{c} x^{2} \sqrt{x^{2}}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.04565, size = 39, normalized size = 1.5 \begin{align*} \frac{1}{3} \,{\left (\frac{{\left (b x + a\right )}^{3} \mathrm{sgn}\left (x\right )}{b} - \frac{a^{3} \mathrm{sgn}\left (x\right )}{b}\right )} \sqrt{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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