Optimal. Leaf size=27 \[ \frac{x (a+b x)^3}{3 b c \sqrt{c x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0042818, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {15, 32} \[ \frac{x (a+b x)^3}{3 b c \sqrt{c x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 15
Rule 32
Rubi steps
\begin{align*} \int \frac{x^3 (a+b x)^2}{\left (c x^2\right )^{3/2}} \, dx &=\frac{x \int (a+b x)^2 \, dx}{c \sqrt{c x^2}}\\ &=\frac{x (a+b x)^3}{3 b c \sqrt{c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0041706, size = 26, normalized size = 0.96 \[ \frac{x^3 (a+b x)^3}{3 b \left (c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.004, size = 31, normalized size = 1.2 \begin{align*}{\frac{{x}^{4} \left ({b}^{2}{x}^{2}+3\,abx+3\,{a}^{2} \right ) }{3} \left ( c{x}^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.06627, size = 70, normalized size = 2.59 \begin{align*} \frac{b^{2} x^{4}}{3 \, \sqrt{c x^{2}} c} + \frac{a b x^{3}}{\sqrt{c x^{2}} c} + \frac{a^{2} x^{2}}{\sqrt{c x^{2}} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.49582, size = 66, normalized size = 2.44 \begin{align*} \frac{{\left (b^{2} x^{2} + 3 \, a b x + 3 \, a^{2}\right )} \sqrt{c x^{2}}}{3 \, c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 0.758949, size = 56, normalized size = 2.07 \begin{align*} \frac{a^{2} x^{4}}{c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}} + \frac{a b x^{5}}{c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}} + \frac{b^{2} x^{6}}{3 c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.07006, size = 53, normalized size = 1.96 \begin{align*} \frac{\sqrt{c x^{2}}{\left ({\left (\frac{b^{2} x}{c} + \frac{3 \, a b}{c}\right )} x + \frac{3 \, a^{2}}{c}\right )}}{3 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]