3.82 \(\int x^2 \sin ^{-1}(a x)^{3/2} \, dx\)

Optimal. Leaf size=147 \[ -\frac{3 \sqrt{\frac{\pi }{2}} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{8 a^3}+\frac{\sqrt{\frac{\pi }{6}} \text{FresnelC}\left (\sqrt{\frac{6}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{24 a^3}+\frac{x^2 \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{6 a}+\frac{\sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{3 a^3}+\frac{1}{3} x^3 \sin ^{-1}(a x)^{3/2} \]

[Out]

(Sqrt[1 - a^2*x^2]*Sqrt[ArcSin[a*x]])/(3*a^3) + (x^2*Sqrt[1 - a^2*x^2]*Sqrt[ArcSin[a*x]])/(6*a) + (x^3*ArcSin[
a*x]^(3/2))/3 - (3*Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(8*a^3) + (Sqrt[Pi/6]*FresnelC[Sqrt[6/Pi
]*Sqrt[ArcSin[a*x]]])/(24*a^3)

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Rubi [A]  time = 0.302717, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {4629, 4707, 4677, 4623, 3304, 3352, 4635, 4406} \[ -\frac{3 \sqrt{\frac{\pi }{2}} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{8 a^3}+\frac{\sqrt{\frac{\pi }{6}} \text{FresnelC}\left (\sqrt{\frac{6}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{24 a^3}+\frac{x^2 \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{6 a}+\frac{\sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{3 a^3}+\frac{1}{3} x^3 \sin ^{-1}(a x)^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[x^2*ArcSin[a*x]^(3/2),x]

[Out]

(Sqrt[1 - a^2*x^2]*Sqrt[ArcSin[a*x]])/(3*a^3) + (x^2*Sqrt[1 - a^2*x^2]*Sqrt[ArcSin[a*x]])/(6*a) + (x^3*ArcSin[
a*x]^(3/2))/3 - (3*Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(8*a^3) + (Sqrt[Pi/6]*FresnelC[Sqrt[6/Pi
]*Sqrt[ArcSin[a*x]]])/(24*a^3)

Rule 4629

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[(x^(m + 1)*(a + b*ArcSin[c*x])^n)/(m
+ 1), x] - Dist[(b*c*n)/(m + 1), Int[(x^(m + 1)*(a + b*ArcSin[c*x])^(n - 1))/Sqrt[1 - c^2*x^2], x], x] /; Fre
eQ[{a, b, c}, x] && IGtQ[m, 0] && GtQ[n, 0]

Rule 4707

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[
(f*(f*x)^(m - 1)*Sqrt[d + e*x^2]*(a + b*ArcSin[c*x])^n)/(e*m), x] + (Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)^(m
- 2)*(a + b*ArcSin[c*x])^n)/Sqrt[d + e*x^2], x], x] + Dist[(b*f*n*Sqrt[1 - c^2*x^2])/(c*m*Sqrt[d + e*x^2]), I
nt[(f*x)^(m - 1)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] &&
GtQ[n, 0] && GtQ[m, 1] && IntegerQ[m]

Rule 4677

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x^2)^
(p + 1)*(a + b*ArcSin[c*x])^n)/(2*e*(p + 1)), x] + Dist[(b*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*c*(p + 1
)*(1 - c^2*x^2)^FracPart[p]), Int[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x] /; FreeQ[{a, b,
c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]

Rule 4623

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[1/(b*c), Subst[Int[x^n*Cos[a/b - x/b], x], x, a
+ b*ArcSin[c*x]], x] /; FreeQ[{a, b, c, n}, x]

Rule 3304

Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Cos[(f*x^2)/d],
x], x, Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]

Rule 3352

Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)])/
(f*Rt[d, 2]), x] /; FreeQ[{d, e, f}, x]

Rule 4635

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[1/c^(m + 1), Subst[Int[(a + b*x)^n*S
in[x]^m*Cos[x], x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]

Rule 4406

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(c + d*x)^m, Sin[a + b*x]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0]
&& IGtQ[p, 0]

Rubi steps

\begin{align*} \int x^2 \sin ^{-1}(a x)^{3/2} \, dx &=\frac{1}{3} x^3 \sin ^{-1}(a x)^{3/2}-\frac{1}{2} a \int \frac{x^3 \sqrt{\sin ^{-1}(a x)}}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{x^2 \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{6 a}+\frac{1}{3} x^3 \sin ^{-1}(a x)^{3/2}-\frac{1}{12} \int \frac{x^2}{\sqrt{\sin ^{-1}(a x)}} \, dx-\frac{\int \frac{x \sqrt{\sin ^{-1}(a x)}}{\sqrt{1-a^2 x^2}} \, dx}{3 a}\\ &=\frac{\sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{3 a^3}+\frac{x^2 \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{6 a}+\frac{1}{3} x^3 \sin ^{-1}(a x)^{3/2}-\frac{\operatorname{Subst}\left (\int \frac{\cos (x) \sin ^2(x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{12 a^3}-\frac{\int \frac{1}{\sqrt{\sin ^{-1}(a x)}} \, dx}{6 a^2}\\ &=\frac{\sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{3 a^3}+\frac{x^2 \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{6 a}+\frac{1}{3} x^3 \sin ^{-1}(a x)^{3/2}-\frac{\operatorname{Subst}\left (\int \left (\frac{\cos (x)}{4 \sqrt{x}}-\frac{\cos (3 x)}{4 \sqrt{x}}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{12 a^3}-\frac{\operatorname{Subst}\left (\int \frac{\cos (x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{6 a^3}\\ &=\frac{\sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{3 a^3}+\frac{x^2 \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{6 a}+\frac{1}{3} x^3 \sin ^{-1}(a x)^{3/2}-\frac{\operatorname{Subst}\left (\int \frac{\cos (x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{48 a^3}+\frac{\operatorname{Subst}\left (\int \frac{\cos (3 x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{48 a^3}-\frac{\operatorname{Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{3 a^3}\\ &=\frac{\sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{3 a^3}+\frac{x^2 \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{6 a}+\frac{1}{3} x^3 \sin ^{-1}(a x)^{3/2}-\frac{\sqrt{\frac{\pi }{2}} C\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{3 a^3}-\frac{\operatorname{Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{24 a^3}+\frac{\operatorname{Subst}\left (\int \cos \left (3 x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{24 a^3}\\ &=\frac{\sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{3 a^3}+\frac{x^2 \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{6 a}+\frac{1}{3} x^3 \sin ^{-1}(a x)^{3/2}-\frac{3 \sqrt{\frac{\pi }{2}} C\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{8 a^3}+\frac{\sqrt{\frac{\pi }{6}} C\left (\sqrt{\frac{6}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{24 a^3}\\ \end{align*}

Mathematica [C]  time = 0.0591148, size = 136, normalized size = 0.93 \[ \frac{\sqrt{\sin ^{-1}(a x)} \left (27 \sqrt{i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{5}{2},-i \sin ^{-1}(a x)\right )+27 \sqrt{-i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{5}{2},i \sin ^{-1}(a x)\right )-\sqrt{3} \left (\sqrt{i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{5}{2},-3 i \sin ^{-1}(a x)\right )+\sqrt{-i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{5}{2},3 i \sin ^{-1}(a x)\right )\right )\right )}{216 a^3 \sqrt{\sin ^{-1}(a x)^2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^2*ArcSin[a*x]^(3/2),x]

[Out]

(Sqrt[ArcSin[a*x]]*(27*Sqrt[I*ArcSin[a*x]]*Gamma[5/2, (-I)*ArcSin[a*x]] + 27*Sqrt[(-I)*ArcSin[a*x]]*Gamma[5/2,
I*ArcSin[a*x]] - Sqrt[3]*(Sqrt[I*ArcSin[a*x]]*Gamma[5/2, (-3*I)*ArcSin[a*x]] + Sqrt[(-I)*ArcSin[a*x]]*Gamma[5
/2, (3*I)*ArcSin[a*x]])))/(216*a^3*Sqrt[ArcSin[a*x]^2])

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Maple [A]  time = 0.05, size = 131, normalized size = 0.9 \begin{align*} -{\frac{1}{144\,{a}^{3}} \left ( -36\,ax \left ( \arcsin \left ( ax \right ) \right ) ^{2}-\sqrt{3}\sqrt{2}\sqrt{\arcsin \left ( ax \right ) }\sqrt{\pi }{\it FresnelC} \left ({\frac{\sqrt{3}\sqrt{2}}{\sqrt{\pi }}\sqrt{\arcsin \left ( ax \right ) }} \right ) +12\, \left ( \arcsin \left ( ax \right ) \right ) ^{2}\sin \left ( 3\,\arcsin \left ( ax \right ) \right ) +27\,\sqrt{2}\sqrt{\arcsin \left ( ax \right ) }\sqrt{\pi }{\it FresnelC} \left ({\frac{\sqrt{2}\sqrt{\arcsin \left ( ax \right ) }}{\sqrt{\pi }}} \right ) -54\,\arcsin \left ( ax \right ) \sqrt{-{a}^{2}{x}^{2}+1}+6\,\arcsin \left ( ax \right ) \cos \left ( 3\,\arcsin \left ( ax \right ) \right ) \right ){\frac{1}{\sqrt{\arcsin \left ( ax \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*arcsin(a*x)^(3/2),x)

[Out]

-1/144/a^3/arcsin(a*x)^(1/2)*(-36*a*x*arcsin(a*x)^2-3^(1/2)*2^(1/2)*arcsin(a*x)^(1/2)*Pi^(1/2)*FresnelC(2^(1/2
)/Pi^(1/2)*3^(1/2)*arcsin(a*x)^(1/2))+12*arcsin(a*x)^2*sin(3*arcsin(a*x))+27*2^(1/2)*arcsin(a*x)^(1/2)*Pi^(1/2
)*FresnelC(2^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))-54*arcsin(a*x)*(-a^2*x^2+1)^(1/2)+6*arcsin(a*x)*cos(3*arcsin(a*
x)))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*arcsin(a*x)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*arcsin(a*x)^(3/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \operatorname{asin}^{\frac{3}{2}}{\left (a x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*asin(a*x)**(3/2),x)

[Out]

Integral(x**2*asin(a*x)**(3/2), x)

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Giac [C]  time = 1.39175, size = 320, normalized size = 2.18 \begin{align*} \frac{i \, \arcsin \left (a x\right )^{\frac{3}{2}} e^{\left (3 i \, \arcsin \left (a x\right )\right )}}{24 \, a^{3}} - \frac{i \, \arcsin \left (a x\right )^{\frac{3}{2}} e^{\left (i \, \arcsin \left (a x\right )\right )}}{8 \, a^{3}} + \frac{i \, \arcsin \left (a x\right )^{\frac{3}{2}} e^{\left (-i \, \arcsin \left (a x\right )\right )}}{8 \, a^{3}} - \frac{i \, \arcsin \left (a x\right )^{\frac{3}{2}} e^{\left (-3 i \, \arcsin \left (a x\right )\right )}}{24 \, a^{3}} - \frac{\left (i + 1\right ) \, \sqrt{6} \sqrt{\pi } \operatorname{erf}\left (\left (\frac{1}{2} i - \frac{1}{2}\right ) \, \sqrt{6} \sqrt{\arcsin \left (a x\right )}\right )}{576 \, a^{3}} + \frac{\left (i - 1\right ) \, \sqrt{6} \sqrt{\pi } \operatorname{erf}\left (-\left (\frac{1}{2} i + \frac{1}{2}\right ) \, \sqrt{6} \sqrt{\arcsin \left (a x\right )}\right )}{576 \, a^{3}} + \frac{\left (3 i + 3\right ) \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (\left (\frac{1}{2} i - \frac{1}{2}\right ) \, \sqrt{2} \sqrt{\arcsin \left (a x\right )}\right )}{64 \, a^{3}} - \frac{\left (3 i - 3\right ) \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (-\left (\frac{1}{2} i + \frac{1}{2}\right ) \, \sqrt{2} \sqrt{\arcsin \left (a x\right )}\right )}{64 \, a^{3}} - \frac{\sqrt{\arcsin \left (a x\right )} e^{\left (3 i \, \arcsin \left (a x\right )\right )}}{48 \, a^{3}} + \frac{3 \, \sqrt{\arcsin \left (a x\right )} e^{\left (i \, \arcsin \left (a x\right )\right )}}{16 \, a^{3}} + \frac{3 \, \sqrt{\arcsin \left (a x\right )} e^{\left (-i \, \arcsin \left (a x\right )\right )}}{16 \, a^{3}} - \frac{\sqrt{\arcsin \left (a x\right )} e^{\left (-3 i \, \arcsin \left (a x\right )\right )}}{48 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*arcsin(a*x)^(3/2),x, algorithm="giac")

[Out]

1/24*I*arcsin(a*x)^(3/2)*e^(3*I*arcsin(a*x))/a^3 - 1/8*I*arcsin(a*x)^(3/2)*e^(I*arcsin(a*x))/a^3 + 1/8*I*arcsi
n(a*x)^(3/2)*e^(-I*arcsin(a*x))/a^3 - 1/24*I*arcsin(a*x)^(3/2)*e^(-3*I*arcsin(a*x))/a^3 - (1/576*I + 1/576)*sq
rt(6)*sqrt(pi)*erf((1/2*I - 1/2)*sqrt(6)*sqrt(arcsin(a*x)))/a^3 + (1/576*I - 1/576)*sqrt(6)*sqrt(pi)*erf(-(1/2
*I + 1/2)*sqrt(6)*sqrt(arcsin(a*x)))/a^3 + (3/64*I + 3/64)*sqrt(2)*sqrt(pi)*erf((1/2*I - 1/2)*sqrt(2)*sqrt(arc
sin(a*x)))/a^3 - (3/64*I - 3/64)*sqrt(2)*sqrt(pi)*erf(-(1/2*I + 1/2)*sqrt(2)*sqrt(arcsin(a*x)))/a^3 - 1/48*sqr
t(arcsin(a*x))*e^(3*I*arcsin(a*x))/a^3 + 3/16*sqrt(arcsin(a*x))*e^(I*arcsin(a*x))/a^3 + 3/16*sqrt(arcsin(a*x))
*e^(-I*arcsin(a*x))/a^3 - 1/48*sqrt(arcsin(a*x))*e^(-3*I*arcsin(a*x))/a^3