Optimal. Leaf size=54 \[ -\frac{\left (1-a^2 x^2\right )^{3/2}}{9 a^3}+\frac{\sqrt{1-a^2 x^2}}{3 a^3}+\frac{1}{3} x^3 \sin ^{-1}(a x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.034606, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {4627, 266, 43} \[ -\frac{\left (1-a^2 x^2\right )^{3/2}}{9 a^3}+\frac{\sqrt{1-a^2 x^2}}{3 a^3}+\frac{1}{3} x^3 \sin ^{-1}(a x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4627
Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^2 \sin ^{-1}(a x) \, dx &=\frac{1}{3} x^3 \sin ^{-1}(a x)-\frac{1}{3} a \int \frac{x^3}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{1}{3} x^3 \sin ^{-1}(a x)-\frac{1}{6} a \operatorname{Subst}\left (\int \frac{x}{\sqrt{1-a^2 x}} \, dx,x,x^2\right )\\ &=\frac{1}{3} x^3 \sin ^{-1}(a x)-\frac{1}{6} a \operatorname{Subst}\left (\int \left (\frac{1}{a^2 \sqrt{1-a^2 x}}-\frac{\sqrt{1-a^2 x}}{a^2}\right ) \, dx,x,x^2\right )\\ &=\frac{\sqrt{1-a^2 x^2}}{3 a^3}-\frac{\left (1-a^2 x^2\right )^{3/2}}{9 a^3}+\frac{1}{3} x^3 \sin ^{-1}(a x)\\ \end{align*}
Mathematica [A] time = 0.0249748, size = 41, normalized size = 0.76 \[ \frac{1}{9} \left (\frac{\sqrt{1-a^2 x^2} \left (a^2 x^2+2\right )}{a^3}+3 x^3 \sin ^{-1}(a x)\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.003, size = 52, normalized size = 1. \begin{align*}{\frac{1}{{a}^{3}} \left ({\frac{{a}^{3}{x}^{3}\arcsin \left ( ax \right ) }{3}}+{\frac{{a}^{2}{x}^{2}}{9}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{2}{9}\sqrt{-{a}^{2}{x}^{2}+1}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.70877, size = 68, normalized size = 1.26 \begin{align*} \frac{1}{3} \, x^{3} \arcsin \left (a x\right ) + \frac{1}{9} \, a{\left (\frac{\sqrt{-a^{2} x^{2} + 1} x^{2}}{a^{2}} + \frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{a^{4}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.13791, size = 92, normalized size = 1.7 \begin{align*} \frac{3 \, a^{3} x^{3} \arcsin \left (a x\right ) +{\left (a^{2} x^{2} + 2\right )} \sqrt{-a^{2} x^{2} + 1}}{9 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.819754, size = 48, normalized size = 0.89 \begin{align*} \begin{cases} \frac{x^{3} \operatorname{asin}{\left (a x \right )}}{3} + \frac{x^{2} \sqrt{- a^{2} x^{2} + 1}}{9 a} + \frac{2 \sqrt{- a^{2} x^{2} + 1}}{9 a^{3}} & \text{for}\: a \neq 0 \\0 & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.30279, size = 86, normalized size = 1.59 \begin{align*} \frac{{\left (a^{2} x^{2} - 1\right )} x \arcsin \left (a x\right )}{3 \, a^{2}} + \frac{x \arcsin \left (a x\right )}{3 \, a^{2}} - \frac{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{9 \, a^{3}} + \frac{\sqrt{-a^{2} x^{2} + 1}}{3 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]