Optimal. Leaf size=78 \[ \frac{1}{18} \sqrt{1-x} x^{5/2}+\frac{5}{72} \sqrt{1-x} x^{3/2}+\frac{1}{3} x^3 \sin ^{-1}\left (\sqrt{x}\right )+\frac{5}{48} \sqrt{1-x} \sqrt{x}+\frac{5}{96} \sin ^{-1}(1-2 x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0302778, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {4842, 12, 50, 53, 619, 216} \[ \frac{1}{18} \sqrt{1-x} x^{5/2}+\frac{5}{72} \sqrt{1-x} x^{3/2}+\frac{1}{3} x^3 \sin ^{-1}\left (\sqrt{x}\right )+\frac{5}{48} \sqrt{1-x} \sqrt{x}+\frac{5}{96} \sin ^{-1}(1-2 x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4842
Rule 12
Rule 50
Rule 53
Rule 619
Rule 216
Rubi steps
\begin{align*} \int x^2 \sin ^{-1}\left (\sqrt{x}\right ) \, dx &=\frac{1}{3} x^3 \sin ^{-1}\left (\sqrt{x}\right )-\frac{1}{3} \int \frac{x^{5/2}}{2 \sqrt{1-x}} \, dx\\ &=\frac{1}{3} x^3 \sin ^{-1}\left (\sqrt{x}\right )-\frac{1}{6} \int \frac{x^{5/2}}{\sqrt{1-x}} \, dx\\ &=\frac{1}{18} \sqrt{1-x} x^{5/2}+\frac{1}{3} x^3 \sin ^{-1}\left (\sqrt{x}\right )-\frac{5}{36} \int \frac{x^{3/2}}{\sqrt{1-x}} \, dx\\ &=\frac{5}{72} \sqrt{1-x} x^{3/2}+\frac{1}{18} \sqrt{1-x} x^{5/2}+\frac{1}{3} x^3 \sin ^{-1}\left (\sqrt{x}\right )-\frac{5}{48} \int \frac{\sqrt{x}}{\sqrt{1-x}} \, dx\\ &=\frac{5}{48} \sqrt{1-x} \sqrt{x}+\frac{5}{72} \sqrt{1-x} x^{3/2}+\frac{1}{18} \sqrt{1-x} x^{5/2}+\frac{1}{3} x^3 \sin ^{-1}\left (\sqrt{x}\right )-\frac{5}{96} \int \frac{1}{\sqrt{1-x} \sqrt{x}} \, dx\\ &=\frac{5}{48} \sqrt{1-x} \sqrt{x}+\frac{5}{72} \sqrt{1-x} x^{3/2}+\frac{1}{18} \sqrt{1-x} x^{5/2}+\frac{1}{3} x^3 \sin ^{-1}\left (\sqrt{x}\right )-\frac{5}{96} \int \frac{1}{\sqrt{x-x^2}} \, dx\\ &=\frac{5}{48} \sqrt{1-x} \sqrt{x}+\frac{5}{72} \sqrt{1-x} x^{3/2}+\frac{1}{18} \sqrt{1-x} x^{5/2}+\frac{1}{3} x^3 \sin ^{-1}\left (\sqrt{x}\right )+\frac{5}{96} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2}} \, dx,x,1-2 x\right )\\ &=\frac{5}{48} \sqrt{1-x} \sqrt{x}+\frac{5}{72} \sqrt{1-x} x^{3/2}+\frac{1}{18} \sqrt{1-x} x^{5/2}+\frac{5}{96} \sin ^{-1}(1-2 x)+\frac{1}{3} x^3 \sin ^{-1}\left (\sqrt{x}\right )\\ \end{align*}
Mathematica [A] time = 0.0351987, size = 64, normalized size = 0.82 \[ \frac{1}{144} \left (8 \sqrt{1-x} x^{5/2}+10 \sqrt{1-x} x^{3/2}+3 \left (16 x^3-5\right ) \sin ^{-1}\left (\sqrt{x}\right )+15 \sqrt{-(x-1) x}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.011, size = 53, normalized size = 0.7 \begin{align*}{\frac{{x}^{3}}{3}\arcsin \left ( \sqrt{x} \right ) }+{\frac{1}{18}{x}^{{\frac{5}{2}}}\sqrt{1-x}}+{\frac{5}{72}{x}^{{\frac{3}{2}}}\sqrt{1-x}}+{\frac{5}{48}\sqrt{1-x}\sqrt{x}}-{\frac{5}{48}\arcsin \left ( \sqrt{x} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.42676, size = 70, normalized size = 0.9 \begin{align*} \frac{1}{3} \, x^{3} \arcsin \left (\sqrt{x}\right ) + \frac{1}{18} \, x^{\frac{5}{2}} \sqrt{-x + 1} + \frac{5}{72} \, x^{\frac{3}{2}} \sqrt{-x + 1} + \frac{5}{48} \, \sqrt{x} \sqrt{-x + 1} - \frac{5}{48} \, \arcsin \left (\sqrt{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.36802, size = 113, normalized size = 1.45 \begin{align*} \frac{1}{144} \,{\left (8 \, x^{2} + 10 \, x + 15\right )} \sqrt{x} \sqrt{-x + 1} + \frac{1}{48} \,{\left (16 \, x^{3} - 5\right )} \arcsin \left (\sqrt{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 10.5334, size = 73, normalized size = 0.94 \begin{align*} \frac{x^{3} \operatorname{asin}{\left (\sqrt{x} \right )}}{3} - \frac{\begin{cases} \frac{x^{\frac{3}{2}} \left (1 - x\right )^{\frac{3}{2}}}{6} + \frac{3 \sqrt{x} \left (1 - 2 x\right ) \sqrt{1 - x}}{16} - \frac{\sqrt{x} \sqrt{1 - x}}{2} + \frac{5 \operatorname{asin}{\left (\sqrt{x} \right )}}{16} & \text{for}\: x \geq 0 \wedge x < 1 \end{cases}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.17519, size = 104, normalized size = 1.33 \begin{align*} \frac{1}{3} \,{\left (x - 1\right )}^{3} \arcsin \left (\sqrt{x}\right ) + \frac{1}{18} \,{\left (x - 1\right )}^{2} \sqrt{x} \sqrt{-x + 1} +{\left (x - 1\right )}^{2} \arcsin \left (\sqrt{x}\right ) - \frac{13}{72} \, \sqrt{x}{\left (-x + 1\right )}^{\frac{3}{2}} +{\left (x - 1\right )} \arcsin \left (\sqrt{x}\right ) + \frac{11}{48} \, \sqrt{x} \sqrt{-x + 1} + \frac{11}{48} \, \arcsin \left (\sqrt{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]