3.362 \(\int x \sin ^{-1}(\sqrt{x}) \, dx\)

Optimal. Leaf size=60 \[ \frac{1}{8} \sqrt{1-x} x^{3/2}+\frac{1}{2} x^2 \sin ^{-1}\left (\sqrt{x}\right )+\frac{3}{16} \sqrt{1-x} \sqrt{x}+\frac{3}{32} \sin ^{-1}(1-2 x) \]

[Out]

(3*Sqrt[1 - x]*Sqrt[x])/16 + (Sqrt[1 - x]*x^(3/2))/8 + (3*ArcSin[1 - 2*x])/32 + (x^2*ArcSin[Sqrt[x]])/2

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Rubi [A]  time = 0.0208128, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.75, Rules used = {4842, 12, 50, 53, 619, 216} \[ \frac{1}{8} \sqrt{1-x} x^{3/2}+\frac{1}{2} x^2 \sin ^{-1}\left (\sqrt{x}\right )+\frac{3}{16} \sqrt{1-x} \sqrt{x}+\frac{3}{32} \sin ^{-1}(1-2 x) \]

Antiderivative was successfully verified.

[In]

Int[x*ArcSin[Sqrt[x]],x]

[Out]

(3*Sqrt[1 - x]*Sqrt[x])/16 + (Sqrt[1 - x]*x^(3/2))/8 + (3*ArcSin[1 - 2*x])/32 + (x^2*ArcSin[Sqrt[x]])/2

Rule 4842

Int[((a_.) + ArcSin[u_]*(b_.))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m + 1)*(a + b*ArcSin[
u]))/(d*(m + 1)), x] - Dist[b/(d*(m + 1)), Int[SimplifyIntegrand[((c + d*x)^(m + 1)*D[u, x])/Sqrt[1 - u^2], x]
, x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] &&  !FunctionOfQ[(c + d*x)^(
m + 1), u, x] &&  !FunctionOfExponentialQ[u, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 53

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Int[1/Sqrt[a*c - b*(a - c)*x - b^2*x^2]
, x] /; FreeQ[{a, b, c, d}, x] && EqQ[b + d, 0] && GtQ[a + c, 0]

Rule 619

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*((-4*c)/(b^2 - 4*a*c))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin{align*} \int x \sin ^{-1}\left (\sqrt{x}\right ) \, dx &=\frac{1}{2} x^2 \sin ^{-1}\left (\sqrt{x}\right )-\frac{1}{2} \int \frac{x^{3/2}}{2 \sqrt{1-x}} \, dx\\ &=\frac{1}{2} x^2 \sin ^{-1}\left (\sqrt{x}\right )-\frac{1}{4} \int \frac{x^{3/2}}{\sqrt{1-x}} \, dx\\ &=\frac{1}{8} \sqrt{1-x} x^{3/2}+\frac{1}{2} x^2 \sin ^{-1}\left (\sqrt{x}\right )-\frac{3}{16} \int \frac{\sqrt{x}}{\sqrt{1-x}} \, dx\\ &=\frac{3}{16} \sqrt{1-x} \sqrt{x}+\frac{1}{8} \sqrt{1-x} x^{3/2}+\frac{1}{2} x^2 \sin ^{-1}\left (\sqrt{x}\right )-\frac{3}{32} \int \frac{1}{\sqrt{1-x} \sqrt{x}} \, dx\\ &=\frac{3}{16} \sqrt{1-x} \sqrt{x}+\frac{1}{8} \sqrt{1-x} x^{3/2}+\frac{1}{2} x^2 \sin ^{-1}\left (\sqrt{x}\right )-\frac{3}{32} \int \frac{1}{\sqrt{x-x^2}} \, dx\\ &=\frac{3}{16} \sqrt{1-x} \sqrt{x}+\frac{1}{8} \sqrt{1-x} x^{3/2}+\frac{1}{2} x^2 \sin ^{-1}\left (\sqrt{x}\right )+\frac{3}{32} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2}} \, dx,x,1-2 x\right )\\ &=\frac{3}{16} \sqrt{1-x} \sqrt{x}+\frac{1}{8} \sqrt{1-x} x^{3/2}+\frac{3}{32} \sin ^{-1}(1-2 x)+\frac{1}{2} x^2 \sin ^{-1}\left (\sqrt{x}\right )\\ \end{align*}

Mathematica [A]  time = 0.024398, size = 47, normalized size = 0.78 \[ \frac{1}{16} \left (2 \sqrt{1-x} x^{3/2}+\left (8 x^2-3\right ) \sin ^{-1}\left (\sqrt{x}\right )+3 \sqrt{-(x-1) x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x*ArcSin[Sqrt[x]],x]

[Out]

(2*Sqrt[1 - x]*x^(3/2) + 3*Sqrt[-((-1 + x)*x)] + (-3 + 8*x^2)*ArcSin[Sqrt[x]])/16

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Maple [A]  time = 0.003, size = 41, normalized size = 0.7 \begin{align*}{\frac{{x}^{2}}{2}\arcsin \left ( \sqrt{x} \right ) }+{\frac{1}{8}{x}^{{\frac{3}{2}}}\sqrt{1-x}}+{\frac{3}{16}\sqrt{1-x}\sqrt{x}}-{\frac{3}{16}\arcsin \left ( \sqrt{x} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*arcsin(x^(1/2)),x)

[Out]

1/2*x^2*arcsin(x^(1/2))+1/8*x^(3/2)*(1-x)^(1/2)+3/16*(1-x)^(1/2)*x^(1/2)-3/16*arcsin(x^(1/2))

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Maxima [A]  time = 1.41846, size = 54, normalized size = 0.9 \begin{align*} \frac{1}{2} \, x^{2} \arcsin \left (\sqrt{x}\right ) + \frac{1}{8} \, x^{\frac{3}{2}} \sqrt{-x + 1} + \frac{3}{16} \, \sqrt{x} \sqrt{-x + 1} - \frac{3}{16} \, \arcsin \left (\sqrt{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*arcsin(x^(1/2)),x, algorithm="maxima")

[Out]

1/2*x^2*arcsin(sqrt(x)) + 1/8*x^(3/2)*sqrt(-x + 1) + 3/16*sqrt(x)*sqrt(-x + 1) - 3/16*arcsin(sqrt(x))

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Fricas [A]  time = 2.19271, size = 97, normalized size = 1.62 \begin{align*} \frac{1}{16} \,{\left (2 \, x + 3\right )} \sqrt{x} \sqrt{-x + 1} + \frac{1}{16} \,{\left (8 \, x^{2} - 3\right )} \arcsin \left (\sqrt{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*arcsin(x^(1/2)),x, algorithm="fricas")

[Out]

1/16*(2*x + 3)*sqrt(x)*sqrt(-x + 1) + 1/16*(8*x^2 - 3)*arcsin(sqrt(x))

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Sympy [A]  time = 4.15098, size = 58, normalized size = 0.97 \begin{align*} \frac{x^{2} \operatorname{asin}{\left (\sqrt{x} \right )}}{2} - \frac{\begin{cases} \frac{\sqrt{x} \left (1 - 2 x\right ) \sqrt{1 - x}}{8} - \frac{\sqrt{x} \sqrt{1 - x}}{2} + \frac{3 \operatorname{asin}{\left (\sqrt{x} \right )}}{8} & \text{for}\: x \geq 0 \wedge x < 1 \end{cases}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*asin(x**(1/2)),x)

[Out]

x**2*asin(sqrt(x))/2 - Piecewise((sqrt(x)*(1 - 2*x)*sqrt(1 - x)/8 - sqrt(x)*sqrt(1 - x)/2 + 3*asin(sqrt(x))/8,
(x >= 0) & (x < 1)))/2

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Giac [A]  time = 1.12907, size = 68, normalized size = 1.13 \begin{align*} \frac{1}{2} \,{\left (x - 1\right )}^{2} \arcsin \left (\sqrt{x}\right ) - \frac{1}{8} \, \sqrt{x}{\left (-x + 1\right )}^{\frac{3}{2}} +{\left (x - 1\right )} \arcsin \left (\sqrt{x}\right ) + \frac{5}{16} \, \sqrt{x} \sqrt{-x + 1} + \frac{5}{16} \, \arcsin \left (\sqrt{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*arcsin(x^(1/2)),x, algorithm="giac")

[Out]

1/2*(x - 1)^2*arcsin(sqrt(x)) - 1/8*sqrt(x)*(-x + 1)^(3/2) + (x - 1)*arcsin(sqrt(x)) + 5/16*sqrt(x)*sqrt(-x +
1) + 5/16*arcsin(sqrt(x))