Optimal. Leaf size=35 \[ \frac{1}{4} a x^3 \sqrt{c x^2}+\frac{1}{5} b x^4 \sqrt{c x^2} \]
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Rubi [A] time = 0.0099123, 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} \[ \frac{1}{4} a x^3 \sqrt{c x^2}+\frac{1}{5} b x^4 \sqrt{c x^2} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin{align*} \int x^2 \sqrt{c x^2} (a+b x) \, dx &=\frac{\sqrt{c x^2} \int x^3 (a+b x) \, dx}{x}\\ &=\frac{\sqrt{c x^2} \int \left (a x^3+b x^4\right ) \, dx}{x}\\ &=\frac{1}{4} a x^3 \sqrt{c x^2}+\frac{1}{5} b x^4 \sqrt{c x^2}\\ \end{align*}
Mathematica [A] time = 0.003507, size = 24, normalized size = 0.69 \[ \frac{1}{20} x^3 \sqrt{c x^2} (5 a+4 b x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 21, normalized size = 0.6 \begin{align*}{\frac{{x}^{3} \left ( 4\,bx+5\,a \right ) }{20}\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 = 1.56547, size = 51, normalized size = 1.46 \begin{align*} \frac{1}{20} \,{\left (4 \, b x^{4} + 5 \, a x^{3}\right )} \sqrt{c x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.305324, size = 36, normalized size = 1.03 \begin{align*} \frac{a \sqrt{c} x^{3} \sqrt{x^{2}}}{4} + \frac{b \sqrt{c} x^{4} \sqrt{x^{2}}}{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06644, size = 30, normalized size = 0.86 \begin{align*} \frac{1}{20} \,{\left (4 \, b x^{5} \mathrm{sgn}\left (x\right ) + 5 \, a x^{4} \mathrm{sgn}\left (x\right )\right )} \sqrt{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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