Optimal. Leaf size=37 \[ \frac{1}{4} a c x^3 \sqrt{c x^2}+\frac{1}{5} b c x^4 \sqrt{c x^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0097555, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {15, 43} \[ \frac{1}{4} a c x^3 \sqrt{c x^2}+\frac{1}{5} b c x^4 \sqrt{c x^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 15
Rule 43
Rubi steps
\begin{align*} \int \left (c x^2\right )^{3/2} (a+b x) \, dx &=\frac{\left (c \sqrt{c x^2}\right ) \int x^3 (a+b x) \, dx}{x}\\ &=\frac{\left (c \sqrt{c x^2}\right ) \int \left (a x^3+b x^4\right ) \, dx}{x}\\ &=\frac{1}{4} a c x^3 \sqrt{c x^2}+\frac{1}{5} b c x^4 \sqrt{c x^2}\\ \end{align*}
Mathematica [A] time = 0.0052157, size = 22, normalized size = 0.59 \[ \frac{1}{20} x \left (c x^2\right )^{3/2} (5 a+4 b x) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.002, size = 19, normalized size = 0.5 \begin{align*}{\frac{x \left ( 4\,bx+5\,a \right ) }{20} \left ( c{x}^{2} \right ) ^{{\frac{3}{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 = 1.54806, size = 57, normalized size = 1.54 \begin{align*} \frac{1}{20} \,{\left (4 \, b c x^{4} + 5 \, a c x^{3}\right )} \sqrt{c x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.484755, size = 34, normalized size = 0.92 \begin{align*} \frac{a c^{\frac{3}{2}} x \left (x^{2}\right )^{\frac{3}{2}}}{4} + \frac{b c^{\frac{3}{2}} x^{2} \left (x^{2}\right )^{\frac{3}{2}}}{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.0628, size = 30, normalized size = 0.81 \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 )} c^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]