Optimal. Leaf size=35 \[ \frac{1}{3} a x^2 \sqrt{c x^2}+\frac{1}{4} b x^3 \sqrt{c x^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0086389, antiderivative size = 35, 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} \[ \frac{1}{3} a x^2 \sqrt{c x^2}+\frac{1}{4} b x^3 \sqrt{c x^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 15
Rule 43
Rubi steps
\begin{align*} \int x \sqrt{c x^2} (a+b x) \, dx &=\frac{\sqrt{c x^2} \int x^2 (a+b x) \, dx}{x}\\ &=\frac{\sqrt{c x^2} \int \left (a x^2+b x^3\right ) \, dx}{x}\\ &=\frac{1}{3} a x^2 \sqrt{c x^2}+\frac{1}{4} b x^3 \sqrt{c x^2}\\ \end{align*}
Mathematica [A] time = 0.003476, size = 24, normalized size = 0.69 \[ \frac{1}{12} x^2 \sqrt{c x^2} (4 a+3 b x) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.002, size = 21, normalized size = 0.6 \begin{align*}{\frac{{x}^{2} \left ( 3\,bx+4\,a \right ) }{12}\sqrt{c{x}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.46539, size = 51, normalized size = 1.46 \begin{align*} \frac{1}{12} \,{\left (3 \, b x^{3} + 4 \, a x^{2}\right )} \sqrt{c x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.23566, size = 36, normalized size = 1.03 \begin{align*} \frac{a \sqrt{c} x^{2} \sqrt{x^{2}}}{3} + \frac{b \sqrt{c} x^{3} \sqrt{x^{2}}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.07218, size = 30, normalized size = 0.86 \begin{align*} \frac{1}{12} \,{\left (3 \, b x^{4} \mathrm{sgn}\left (x\right ) + 4 \, a x^{3} \mathrm{sgn}\left (x\right )\right )} \sqrt{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]