\(\int x^2 \arcsin (a x)^{5/2} \, dx\) [88]

Optimal result

Integrand size = 12, antiderivative size = 178 \[ \int x^2 \arcsin (a x)^{5/2} \, dx=-\frac {5 x \sqrt {\arcsin (a x)}}{6 a^2}-\frac {5}{36} x^3 \sqrt {\arcsin (a x)}+\frac {5 \sqrt {1-a^2 x^2} \arcsin (a x)^{3/2}}{9 a^3}+\frac {5 x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^{3/2}}{18 a}+\frac {1}{3} x^3 \arcsin (a x)^{5/2}+\frac {15 \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{16 a^3}-\frac {5 \sqrt {\frac {\pi }{6}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arcsin (a x)}\right )}{144 a^3} \]

[Out]

1/3*x^3*arcsin(a*x)^(5/2)-5/864*FresnelS(6^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))*6^(1/2)*Pi^(1/2)/a^3+15/32*Fresne
lS(2^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))*2^(1/2)*Pi^(1/2)/a^3+5/9*arcsin(a*x)^(3/2)*(-a^2*x^2+1)^(1/2)/a^3+5/18*
x^2*arcsin(a*x)^(3/2)*(-a^2*x^2+1)^(1/2)/a-5/6*x*arcsin(a*x)^(1/2)/a^2-5/36*x^3*arcsin(a*x)^(1/2)

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {4725, 4795, 4767, 4715, 4809, 3386, 3432, 3393} \[ \int x^2 \arcsin (a x)^{5/2} \, dx=\frac {15 \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{16 a^3}-\frac {5 \sqrt {\frac {\pi }{6}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arcsin (a x)}\right )}{144 a^3}+\frac {5 x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^{3/2}}{18 a}-\frac {5 x \sqrt {\arcsin (a x)}}{6 a^2}+\frac {5 \sqrt {1-a^2 x^2} \arcsin (a x)^{3/2}}{9 a^3}+\frac {1}{3} x^3 \arcsin (a x)^{5/2}-\frac {5}{36} x^3 \sqrt {\arcsin (a x)} \]

[In]

Int[x^2*ArcSin[a*x]^(5/2),x]

[Out]

(-5*x*Sqrt[ArcSin[a*x]])/(6*a^2) - (5*x^3*Sqrt[ArcSin[a*x]])/36 + (5*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^(3/2))/(9*a
^3) + (5*x^2*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^(3/2))/(18*a) + (x^3*ArcSin[a*x]^(5/2))/3 + (15*Sqrt[Pi/2]*FresnelS
[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(16*a^3) - (5*Sqrt[Pi/6]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcSin[a*x]]])/(144*a^3)

Rule 3386

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Sin[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 3393

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

Rule 3432

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

Rule 4715

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

Rule 4725

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 4767

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/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^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 4795

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

Rule 4809

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

Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} x^3 \arcsin (a x)^{5/2}-\frac {1}{6} (5 a) \int \frac {x^3 \arcsin (a x)^{3/2}}{\sqrt {1-a^2 x^2}} \, dx \\ & = \frac {5 x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^{3/2}}{18 a}+\frac {1}{3} x^3 \arcsin (a x)^{5/2}-\frac {5}{12} \int x^2 \sqrt {\arcsin (a x)} \, dx-\frac {5 \int \frac {x \arcsin (a x)^{3/2}}{\sqrt {1-a^2 x^2}} \, dx}{9 a} \\ & = -\frac {5}{36} x^3 \sqrt {\arcsin (a x)}+\frac {5 \sqrt {1-a^2 x^2} \arcsin (a x)^{3/2}}{9 a^3}+\frac {5 x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^{3/2}}{18 a}+\frac {1}{3} x^3 \arcsin (a x)^{5/2}-\frac {5 \int \sqrt {\arcsin (a x)} \, dx}{6 a^2}+\frac {1}{72} (5 a) \int \frac {x^3}{\sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}} \, dx \\ & = -\frac {5 x \sqrt {\arcsin (a x)}}{6 a^2}-\frac {5}{36} x^3 \sqrt {\arcsin (a x)}+\frac {5 \sqrt {1-a^2 x^2} \arcsin (a x)^{3/2}}{9 a^3}+\frac {5 x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^{3/2}}{18 a}+\frac {1}{3} x^3 \arcsin (a x)^{5/2}+\frac {5 \text {Subst}\left (\int \frac {\sin ^3(x)}{\sqrt {x}} \, dx,x,\arcsin (a x)\right )}{72 a^3}+\frac {5 \int \frac {x}{\sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}} \, dx}{12 a} \\ & = -\frac {5 x \sqrt {\arcsin (a x)}}{6 a^2}-\frac {5}{36} x^3 \sqrt {\arcsin (a x)}+\frac {5 \sqrt {1-a^2 x^2} \arcsin (a x)^{3/2}}{9 a^3}+\frac {5 x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^{3/2}}{18 a}+\frac {1}{3} x^3 \arcsin (a x)^{5/2}+\frac {5 \text {Subst}\left (\int \left (\frac {3 \sin (x)}{4 \sqrt {x}}-\frac {\sin (3 x)}{4 \sqrt {x}}\right ) \, dx,x,\arcsin (a x)\right )}{72 a^3}+\frac {5 \text {Subst}\left (\int \frac {\sin (x)}{\sqrt {x}} \, dx,x,\arcsin (a x)\right )}{12 a^3} \\ & = -\frac {5 x \sqrt {\arcsin (a x)}}{6 a^2}-\frac {5}{36} x^3 \sqrt {\arcsin (a x)}+\frac {5 \sqrt {1-a^2 x^2} \arcsin (a x)^{3/2}}{9 a^3}+\frac {5 x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^{3/2}}{18 a}+\frac {1}{3} x^3 \arcsin (a x)^{5/2}-\frac {5 \text {Subst}\left (\int \frac {\sin (3 x)}{\sqrt {x}} \, dx,x,\arcsin (a x)\right )}{288 a^3}+\frac {5 \text {Subst}\left (\int \frac {\sin (x)}{\sqrt {x}} \, dx,x,\arcsin (a x)\right )}{96 a^3}+\frac {5 \text {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {\arcsin (a x)}\right )}{6 a^3} \\ & = -\frac {5 x \sqrt {\arcsin (a x)}}{6 a^2}-\frac {5}{36} x^3 \sqrt {\arcsin (a x)}+\frac {5 \sqrt {1-a^2 x^2} \arcsin (a x)^{3/2}}{9 a^3}+\frac {5 x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^{3/2}}{18 a}+\frac {1}{3} x^3 \arcsin (a x)^{5/2}+\frac {5 \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{6 a^3}-\frac {5 \text {Subst}\left (\int \sin \left (3 x^2\right ) \, dx,x,\sqrt {\arcsin (a x)}\right )}{144 a^3}+\frac {5 \text {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {\arcsin (a x)}\right )}{48 a^3} \\ & = -\frac {5 x \sqrt {\arcsin (a x)}}{6 a^2}-\frac {5}{36} x^3 \sqrt {\arcsin (a x)}+\frac {5 \sqrt {1-a^2 x^2} \arcsin (a x)^{3/2}}{9 a^3}+\frac {5 x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^{3/2}}{18 a}+\frac {1}{3} x^3 \arcsin (a x)^{5/2}+\frac {15 \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{16 a^3}-\frac {5 \sqrt {\frac {\pi }{6}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arcsin (a x)}\right )}{144 a^3} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.04 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.70 \[ \int x^2 \arcsin (a x)^{5/2} \, dx=\frac {-81 \sqrt {-i \arcsin (a x)} \Gamma \left (\frac {7}{2},-i \arcsin (a x)\right )-81 \sqrt {i \arcsin (a x)} \Gamma \left (\frac {7}{2},i \arcsin (a x)\right )+\sqrt {3} \left (\sqrt {-i \arcsin (a x)} \Gamma \left (\frac {7}{2},-3 i \arcsin (a x)\right )+\sqrt {i \arcsin (a x)} \Gamma \left (\frac {7}{2},3 i \arcsin (a x)\right )\right )}{648 a^3 \sqrt {\arcsin (a x)}} \]

[In]

Integrate[x^2*ArcSin[a*x]^(5/2),x]

[Out]

(-81*Sqrt[(-I)*ArcSin[a*x]]*Gamma[7/2, (-I)*ArcSin[a*x]] - 81*Sqrt[I*ArcSin[a*x]]*Gamma[7/2, I*ArcSin[a*x]] +
Sqrt[3]*(Sqrt[(-I)*ArcSin[a*x]]*Gamma[7/2, (-3*I)*ArcSin[a*x]] + Sqrt[I*ArcSin[a*x]]*Gamma[7/2, (3*I)*ArcSin[a
*x]]))/(648*a^3*Sqrt[ArcSin[a*x]])

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.88

method	result	size
default	\(-\frac {-216 a x \arcsin \left (a x \right )^{3}+72 \arcsin \left (a x \right )^{3} \sin \left (3 \arcsin \left (a x \right )\right )+5 \,\operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {3}\, \sqrt {2}\, \sqrt {\arcsin \left (a x \right )}\, \sqrt {\pi }+60 \arcsin \left (a x \right )^{2} \cos \left (3 \arcsin \left (a x \right )\right )-540 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}-405 \,\operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {2}\, \sqrt {\arcsin \left (a x \right )}\, \sqrt {\pi }+810 a x \arcsin \left (a x \right )-30 \arcsin \left (a x \right ) \sin \left (3 \arcsin \left (a x \right )\right )}{864 a^{3} \sqrt {\arcsin \left (a x \right )}}\)	\(156\)

[In]

int(x^2*arcsin(a*x)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-1/864/a^3/arcsin(a*x)^(1/2)*(-216*a*x*arcsin(a*x)^3+72*arcsin(a*x)^3*sin(3*arcsin(a*x))+5*FresnelS(2^(1/2)/Pi
^(1/2)*3^(1/2)*arcsin(a*x)^(1/2))*3^(1/2)*2^(1/2)*arcsin(a*x)^(1/2)*Pi^(1/2)+60*arcsin(a*x)^2*cos(3*arcsin(a*x
))-540*arcsin(a*x)^2*(-a^2*x^2+1)^(1/2)-405*FresnelS(2^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))*2^(1/2)*arcsin(a*x)^(
1/2)*Pi^(1/2)+810*a*x*arcsin(a*x)-30*arcsin(a*x)*sin(3*arcsin(a*x)))

Fricas [F(-2)]

Exception generated. \[ \int x^2 \arcsin (a x)^{5/2} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x^2*arcsin(a*x)^(5/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [F]

\[ \int x^2 \arcsin (a x)^{5/2} \, dx=\int x^{2} \operatorname {asin}^{\frac {5}{2}}{\left (a x \right )}\, dx \]

[In]

integrate(x**2*asin(a*x)**(5/2),x)

[Out]

Integral(x**2*asin(a*x)**(5/2), x)

Maxima [F(-2)]

Exception generated. \[ \int x^2 \arcsin (a x)^{5/2} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate(x^2*arcsin(a*x)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negative exponent.

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.34 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.74 \[ \int x^2 \arcsin (a x)^{5/2} \, dx=\frac {i \, \arcsin \left (a x\right )^{\frac {5}{2}} e^{\left (3 i \, \arcsin \left (a x\right )\right )}}{24 \, a^{3}} - \frac {i \, \arcsin \left (a x\right )^{\frac {5}{2}} e^{\left (i \, \arcsin \left (a x\right )\right )}}{8 \, a^{3}} + \frac {i \, \arcsin \left (a x\right )^{\frac {5}{2}} e^{\left (-i \, \arcsin \left (a x\right )\right )}}{8 \, a^{3}} - \frac {i \, \arcsin \left (a x\right )^{\frac {5}{2}} e^{\left (-3 i \, \arcsin \left (a x\right )\right )}}{24 \, a^{3}} - \frac {5 \, \arcsin \left (a x\right )^{\frac {3}{2}} e^{\left (3 i \, \arcsin \left (a x\right )\right )}}{144 \, a^{3}} + \frac {5 \, \arcsin \left (a x\right )^{\frac {3}{2}} e^{\left (i \, \arcsin \left (a x\right )\right )}}{16 \, a^{3}} + \frac {5 \, \arcsin \left (a x\right )^{\frac {3}{2}} e^{\left (-i \, \arcsin \left (a x\right )\right )}}{16 \, a^{3}} - \frac {5 \, \arcsin \left (a x\right )^{\frac {3}{2}} e^{\left (-3 i \, \arcsin \left (a x\right )\right )}}{144 \, a^{3}} - \frac {\left (5 i - 5\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 )}{3456 \, a^{3}} + \frac {\left (5 i + 5\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 )}{3456 \, a^{3}} + \frac {\left (15 i - 15\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 )}{128 \, a^{3}} - \frac {\left (15 i + 15\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 )}{128 \, a^{3}} - \frac {5 i \, \sqrt {\arcsin \left (a x\right )} e^{\left (3 i \, \arcsin \left (a x\right )\right )}}{288 \, a^{3}} + \frac {15 i \, \sqrt {\arcsin \left (a x\right )} e^{\left (i \, \arcsin \left (a x\right )\right )}}{32 \, a^{3}} - \frac {15 i \, \sqrt {\arcsin \left (a x\right )} e^{\left (-i \, \arcsin \left (a x\right )\right )}}{32 \, a^{3}} + \frac {5 i \, \sqrt {\arcsin \left (a x\right )} e^{\left (-3 i \, \arcsin \left (a x\right )\right )}}{288 \, a^{3}} \]

[In]

integrate(x^2*arcsin(a*x)^(5/2),x, algorithm="giac")

[Out]

1/24*I*arcsin(a*x)^(5/2)*e^(3*I*arcsin(a*x))/a^3 - 1/8*I*arcsin(a*x)^(5/2)*e^(I*arcsin(a*x))/a^3 + 1/8*I*arcsi
n(a*x)^(5/2)*e^(-I*arcsin(a*x))/a^3 - 1/24*I*arcsin(a*x)^(5/2)*e^(-3*I*arcsin(a*x))/a^3 - 5/144*arcsin(a*x)^(3
/2)*e^(3*I*arcsin(a*x))/a^3 + 5/16*arcsin(a*x)^(3/2)*e^(I*arcsin(a*x))/a^3 + 5/16*arcsin(a*x)^(3/2)*e^(-I*arcs
in(a*x))/a^3 - 5/144*arcsin(a*x)^(3/2)*e^(-3*I*arcsin(a*x))/a^3 - (5/3456*I - 5/3456)*sqrt(6)*sqrt(pi)*erf((1/
2*I - 1/2)*sqrt(6)*sqrt(arcsin(a*x)))/a^3 + (5/3456*I + 5/3456)*sqrt(6)*sqrt(pi)*erf(-(1/2*I + 1/2)*sqrt(6)*sq
rt(arcsin(a*x)))/a^3 + (15/128*I - 15/128)*sqrt(2)*sqrt(pi)*erf((1/2*I - 1/2)*sqrt(2)*sqrt(arcsin(a*x)))/a^3 -
(15/128*I + 15/128)*sqrt(2)*sqrt(pi)*erf(-(1/2*I + 1/2)*sqrt(2)*sqrt(arcsin(a*x)))/a^3 - 5/288*I*sqrt(arcsin(
a*x))*e^(3*I*arcsin(a*x))/a^3 + 15/32*I*sqrt(arcsin(a*x))*e^(I*arcsin(a*x))/a^3 - 15/32*I*sqrt(arcsin(a*x))*e^
(-I*arcsin(a*x))/a^3 + 5/288*I*sqrt(arcsin(a*x))*e^(-3*I*arcsin(a*x))/a^3

Mupad [F(-1)]

Timed out. \[ \int x^2 \arcsin (a x)^{5/2} \, dx=\int x^2\,{\mathrm {asin}\left (a\,x\right )}^{5/2} \,d x \]

[In]

int(x^2*asin(a*x)^(5/2),x)

[Out]

int(x^2*asin(a*x)^(5/2), x)