Optimal. Leaf size=37 \[ \frac{1}{5} a c x^4 \sqrt{c x^2}+\frac{1}{6} b c x^5 \sqrt{c x^2} \]
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Rubi [A] time = 0.0108016, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {15, 43} \[ \frac{1}{5} a c x^4 \sqrt{c x^2}+\frac{1}{6} b c x^5 \sqrt{c x^2} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin{align*} \int x \left (c x^2\right )^{3/2} (a+b x) \, dx &=\frac{\left (c \sqrt{c x^2}\right ) \int x^4 (a+b x) \, dx}{x}\\ &=\frac{\left (c \sqrt{c x^2}\right ) \int \left (a x^4+b x^5\right ) \, dx}{x}\\ &=\frac{1}{5} a c x^4 \sqrt{c x^2}+\frac{1}{6} b c x^5 \sqrt{c x^2}\\ \end{align*}
Mathematica [A] time = 0.005597, size = 24, normalized size = 0.65 \[ \frac{1}{30} x^2 \left (c x^2\right )^{3/2} (6 a+5 b x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 21, normalized size = 0.6 \begin{align*}{\frac{{x}^{2} \left ( 5\,bx+6\,a \right ) }{30} \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 = 1.51571, size = 57, normalized size = 1.54 \begin{align*} \frac{1}{30} \,{\left (5 \, b c x^{5} + 6 \, a c x^{4}\right )} \sqrt{c x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.651173, size = 36, normalized size = 0.97 \begin{align*} \frac{a c^{\frac{3}{2}} x^{2} \left (x^{2}\right )^{\frac{3}{2}}}{5} + \frac{b c^{\frac{3}{2}} x^{3} \left (x^{2}\right )^{\frac{3}{2}}}{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06546, size = 30, normalized size = 0.81 \begin{align*} \frac{1}{30} \,{\left (5 \, b x^{6} \mathrm{sgn}\left (x\right ) + 6 \, a x^{5} \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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