Optimal. Leaf size=35 \[ \frac{1}{2} a c x \sqrt{c x^2}+\frac{1}{3} b c x^2 \sqrt{c x^2} \]
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Rubi [A] time = 0.0084287, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {15, 43} \[ \frac{1}{2} a c x \sqrt{c x^2}+\frac{1}{3} b c x^2 \sqrt{c x^2} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (c x^2\right )^{3/2} (a+b x)}{x^2} \, dx &=\frac{\left (c \sqrt{c x^2}\right ) \int x (a+b x) \, dx}{x}\\ &=\frac{\left (c \sqrt{c x^2}\right ) \int \left (a x+b x^2\right ) \, dx}{x}\\ &=\frac{1}{2} a c x \sqrt{c x^2}+\frac{1}{3} b c x^2 \sqrt{c x^2}\\ \end{align*}
Mathematica [A] time = 0.0020262, size = 23, normalized size = 0.66 \[ \frac{1}{6} c x \sqrt{c x^2} (3 a+2 b x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 21, normalized size = 0.6 \begin{align*}{\frac{2\,bx+3\,a}{6\,x} \left ( c{x}^{2} \right ) ^{{\frac{3}{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 = 1.52435, size = 53, normalized size = 1.51 \begin{align*} \frac{1}{6} \,{\left (2 \, b c x^{2} + 3 \, a c x\right )} \sqrt{c x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.51171, size = 31, normalized size = 0.89 \begin{align*} \frac{a c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}}{2 x} + \frac{b c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07138, size = 30, normalized size = 0.86 \begin{align*} \frac{1}{6} \,{\left (2 \, b x^{3} \mathrm{sgn}\left (x\right ) + 3 \, a x^{2} \mathrm{sgn}\left (x\right )\right )} c^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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