Optimal. Leaf size=38 \[ \frac{a x^2}{c \sqrt{c x^2}}+\frac{b x^3}{2 c \sqrt{c x^2}} \]
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Rubi [A] time = 0.0056526, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {15} \[ \frac{a x^2}{c \sqrt{c x^2}}+\frac{b x^3}{2 c \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rubi steps
\begin{align*} \int \frac{x^3 (a+b x)}{\left (c x^2\right )^{3/2}} \, dx &=\frac{x \int (a+b x) \, dx}{c \sqrt{c x^2}}\\ &=\frac{a x^2}{c \sqrt{c x^2}}+\frac{b x^3}{2 c \sqrt{c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0040282, size = 23, normalized size = 0.61 \[ \frac{x^4 (2 a+b x)}{2 \left (c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 20, normalized size = 0.5 \begin{align*}{\frac{{x}^{4} \left ( bx+2\,a \right ) }{2} \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 [A] time = 1.0237, size = 43, normalized size = 1.13 \begin{align*} \frac{b x^{3}}{2 \, \sqrt{c x^{2}} c} + \frac{a x^{2}}{\sqrt{c x^{2}} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54675, size = 45, normalized size = 1.18 \begin{align*} \frac{\sqrt{c x^{2}}{\left (b x + 2 \, a\right )}}{2 \, c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.597934, size = 34, normalized size = 0.89 \begin{align*} \frac{a x^{4}}{c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}} + \frac{b x^{5}}{2 c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07361, size = 34, normalized size = 0.89 \begin{align*} \frac{\sqrt{c x^{2}}{\left (\frac{b x}{c} + \frac{2 \, a}{c}\right )}}{2 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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