3.4.61 \(\int x^2 \text {ArcSin}(\sqrt {x}) \, dx\) [361]

Optimal. Leaf size=78 \[ \frac {5}{48} \sqrt {1-x} \sqrt {x}+\frac {5}{72} \sqrt {1-x} x^{3/2}+\frac {1}{18} \sqrt {1-x} x^{5/2}+\frac {5}{96} \text {ArcSin}(1-2 x)+\frac {1}{3} x^3 \text {ArcSin}\left (\sqrt {x}\right ) \]

[Out]

-5/96*arcsin(-1+2*x)+1/3*x^3*arcsin(x^(1/2))+5/72*x^(3/2)*(1-x)^(1/2)+1/18*x^(5/2)*(1-x)^(1/2)+5/48*(1-x)^(1/2
)*x^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {4926, 12, 52, 55, 633, 222} \begin {gather*} \frac {1}{3} x^3 \text {ArcSin}\left (\sqrt {x}\right )+\frac {5}{96} \text {ArcSin}(1-2 x)+\frac {1}{18} \sqrt {1-x} x^{5/2}+\frac {5}{72} \sqrt {1-x} x^{3/2}+\frac {5}{48} \sqrt {1-x} \sqrt {x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*ArcSin[Sqrt[x]],x]

[Out]

(5*Sqrt[1 - x]*Sqrt[x])/48 + (5*Sqrt[1 - x]*x^(3/2))/72 + (Sqrt[1 - x]*x^(5/2))/18 + (5*ArcSin[1 - 2*x])/96 +
(x^3*ArcSin[Sqrt[x]])/3

Rule 12

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

Rule 52

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 55

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 222

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]

Rule 633

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 4926

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]

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} \text {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*}

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Mathematica [A]
time = 0.02, size = 64, normalized size = 0.82 \begin {gather*} \frac {1}{144} \left (10 \sqrt {1-x} x^{3/2}+8 \sqrt {1-x} x^{5/2}+15 \sqrt {-((-1+x) x)}+3 \left (-5+16 x^3\right ) \text {ArcSin}\left (\sqrt {x}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2*ArcSin[Sqrt[x]],x]

[Out]

(10*Sqrt[1 - x]*x^(3/2) + 8*Sqrt[1 - x]*x^(5/2) + 15*Sqrt[-((-1 + x)*x)] + 3*(-5 + 16*x^3)*ArcSin[Sqrt[x]])/14
4

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Maple [A]
time = 0.04, size = 53, normalized size = 0.68

method	result	size
derivativedivides	\(\frac {x^{3} \arcsin \left (\sqrt {x}\right )}{3}+\frac {x^{\frac {5}{2}} \sqrt {1-x}}{18}+\frac {5 x^{\frac {3}{2}} \sqrt {1-x}}{72}+\frac {5 \sqrt {1-x}\, \sqrt {x}}{48}-\frac {5 \arcsin \left (\sqrt {x}\right )}{48}\)	\(53\)
default	\(\frac {x^{3} \arcsin \left (\sqrt {x}\right )}{3}+\frac {x^{\frac {5}{2}} \sqrt {1-x}}{18}+\frac {5 x^{\frac {3}{2}} \sqrt {1-x}}{72}+\frac {5 \sqrt {1-x}\, \sqrt {x}}{48}-\frac {5 \arcsin \left (\sqrt {x}\right )}{48}\)	\(53\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*arcsin(x^(1/2)),x,method=_RETURNVERBOSE)

[Out]

1/3*x^3*arcsin(x^(1/2))+1/18*x^(5/2)*(1-x)^(1/2)+5/72*x^(3/2)*(1-x)^(1/2)+5/48*(1-x)^(1/2)*x^(1/2)-5/48*arcsin
(x^(1/2))

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Maxima [A]
time = 0.48, size = 52, normalized size = 0.67 \begin {gather*} \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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3*arcsin(sqrt(x)) + 1/18*x^(5/2)*sqrt(-x + 1) + 5/72*x^(3/2)*sqrt(-x + 1) + 5/48*sqrt(x)*sqrt(-x + 1) -
5/48*arcsin(sqrt(x))

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Fricas [A]
time = 2.41, size = 36, normalized size = 0.46 \begin {gather*} \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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/144*(8*x^2 + 10*x + 15)*sqrt(x)*sqrt(-x + 1) + 1/48*(16*x^3 - 5)*arcsin(sqrt(x))

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Sympy [A]
time = 4.93, size = 82, normalized size = 1.05 \begin {gather*} \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}\: \sqrt {x} > -1 \wedge \sqrt {x} < 1 \end {cases}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**3*asin(sqrt(x))/3 - Piecewise((x**(3/2)*(1 - x)**(3/2)/6 + 3*sqrt(x)*(1 - 2*x)*sqrt(1 - x)/16 - sqrt(x)*sqr
t(1 - x)/2 + 5*asin(sqrt(x))/16, (sqrt(x) > -1) & (sqrt(x) < 1)))/3

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Giac [A]
time = 0.39, size = 77, normalized size = 0.99 \begin {gather*} \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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(x - 1)^3*arcsin(sqrt(x)) + 1/18*(x - 1)^2*sqrt(x)*sqrt(-x + 1) + (x - 1)^2*arcsin(sqrt(x)) - 13/72*sqrt(x
)*(-x + 1)^(3/2) + (x - 1)*arcsin(sqrt(x)) + 11/48*sqrt(x)*sqrt(-x + 1) + 11/48*arcsin(sqrt(x))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^2\,\mathrm {asin}\left (\sqrt {x}\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^2*asin(x^(1/2)), x)

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