Optimal. Leaf size=46 \[ \frac{2 B \sqrt{a+b x^3}}{3 b^2}-\frac{2 (A b-a B)}{3 b^2 \sqrt{a+b x^3}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0365055, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {444, 43} \[ \frac{2 B \sqrt{a+b x^3}}{3 b^2}-\frac{2 (A b-a B)}{3 b^2 \sqrt{a+b x^3}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 444
Rule 43
Rubi steps
\begin{align*} \int \frac{x^2 \left (A+B x^3\right )}{\left (a+b x^3\right )^{3/2}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{A+B x}{(a+b x)^{3/2}} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{A b-a B}{b (a+b x)^{3/2}}+\frac{B}{b \sqrt{a+b x}}\right ) \, dx,x,x^3\right )\\ &=-\frac{2 (A b-a B)}{3 b^2 \sqrt{a+b x^3}}+\frac{2 B \sqrt{a+b x^3}}{3 b^2}\\ \end{align*}
Mathematica [A] time = 0.0221462, size = 33, normalized size = 0.72 \[ \frac{2 \left (2 a B-A b+b B x^3\right )}{3 b^2 \sqrt{a+b x^3}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.006, size = 30, normalized size = 0.7 \begin{align*} -{\frac{-2\,bB{x}^{3}+2\,Ab-4\,Ba}{3\,{b}^{2}}{\frac{1}{\sqrt{b{x}^{3}+a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.942499, size = 63, normalized size = 1.37 \begin{align*} \frac{2}{3} \, B{\left (\frac{\sqrt{b x^{3} + a}}{b^{2}} + \frac{a}{\sqrt{b x^{3} + a} b^{2}}\right )} - \frac{2 \, A}{3 \, \sqrt{b x^{3} + a} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.75286, size = 85, normalized size = 1.85 \begin{align*} \frac{2 \,{\left (B b x^{3} + 2 \, B a - A b\right )} \sqrt{b x^{3} + a}}{3 \,{\left (b^{3} x^{3} + a b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.884662, size = 75, normalized size = 1.63 \begin{align*} \begin{cases} - \frac{2 A}{3 b \sqrt{a + b x^{3}}} + \frac{4 B a}{3 b^{2} \sqrt{a + b x^{3}}} + \frac{2 B x^{3}}{3 b \sqrt{a + b x^{3}}} & \text{for}\: b \neq 0 \\\frac{\frac{A x^{3}}{3} + \frac{B x^{6}}{6}}{a^{\frac{3}{2}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.1392, size = 47, normalized size = 1.02 \begin{align*} \frac{2 \,{\left (\sqrt{b x^{3} + a} B + \frac{B a - A b}{\sqrt{b x^{3} + a}}\right )}}{3 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]