\(\int x^2 \arcsin (a x)^{5/2} \, dx\) [110]

Optimal result

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

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.03 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.70 \[ \int x^2 \arcsin (a x)^{5/2} \, dx=\frac {-81 \sqrt {-i \arcsin (a x)} \Gamma \left (\frac {7}{2},-i \arcsin (a x)\right )-81 \sqrt {i \arcsin (a x)} \Gamma \left (\frac {7}{2},i \arcsin (a x)\right )+\sqrt {3} \left (\sqrt {-i \arcsin (a x)} \Gamma \left (\frac {7}{2},-3 i \arcsin (a x)\right )+\sqrt {i \arcsin (a x)} \Gamma \left (\frac {7}{2},3 i \arcsin (a x)\right )\right )}{648 a^3 \sqrt {\arcsin (a x)}} \] Input:

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

Output:

(-81*Sqrt[(-I)*ArcSin[a*x]]*Gamma[7/2, (-I)*ArcSin[a*x]] - 81*Sqrt[I*ArcSi 
n[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]])
 

Rubi [A] (verified)

Time = 1.74 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.32, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.917, Rules used = {5140, 5210, 5140, 5182, 5130, 5224, 3042, 3786, 3793, 2009, 3832}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int x^2 \arcsin (a x)^{5/2} \, dx\)

\(\Big \downarrow \) 5140

\(\displaystyle \frac {1}{3} x^3 \arcsin (a x)^{5/2}-\frac {5}{6} a \int \frac {x^3 \arcsin (a x)^{3/2}}{\sqrt {1-a^2 x^2}}dx\)

\(\Big \downarrow \) 5210

\(\displaystyle \frac {1}{3} x^3 \arcsin (a x)^{5/2}-\frac {5}{6} a \left (\frac {2 \int \frac {x \arcsin (a x)^{3/2}}{\sqrt {1-a^2 x^2}}dx}{3 a^2}+\frac {\int x^2 \sqrt {\arcsin (a x)}dx}{2 a}-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^{3/2}}{3 a^2}\right )\)

\(\Big \downarrow \) 5140

\(\displaystyle \frac {1}{3} x^3 \arcsin (a x)^{5/2}-\frac {5}{6} a \left (\frac {2 \int \frac {x \arcsin (a x)^{3/2}}{\sqrt {1-a^2 x^2}}dx}{3 a^2}+\frac {\frac {1}{3} x^3 \sqrt {\arcsin (a x)}-\frac {1}{6} a \int \frac {x^3}{\sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}dx}{2 a}-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^{3/2}}{3 a^2}\right )\)

\(\Big \downarrow \) 5182

\(\displaystyle \frac {1}{3} x^3 \arcsin (a x)^{5/2}-\frac {5}{6} a \left (\frac {2 \left (\frac {3 \int \sqrt {\arcsin (a x)}dx}{2 a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^{3/2}}{a^2}\right )}{3 a^2}+\frac {\frac {1}{3} x^3 \sqrt {\arcsin (a x)}-\frac {1}{6} a \int \frac {x^3}{\sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}dx}{2 a}-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^{3/2}}{3 a^2}\right )\)

\(\Big \downarrow \) 5130

\(\displaystyle \frac {1}{3} x^3 \arcsin (a x)^{5/2}-\frac {5}{6} a \left (\frac {2 \left (\frac {3 \left (x \sqrt {\arcsin (a x)}-\frac {1}{2} a \int \frac {x}{\sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}dx\right )}{2 a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^{3/2}}{a^2}\right )}{3 a^2}+\frac {\frac {1}{3} x^3 \sqrt {\arcsin (a x)}-\frac {1}{6} a \int \frac {x^3}{\sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}dx}{2 a}-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^{3/2}}{3 a^2}\right )\)

\(\Big \downarrow \) 5224

\(\displaystyle \frac {1}{3} x^3 \arcsin (a x)^{5/2}-\frac {5}{6} a \left (\frac {\frac {1}{3} x^3 \sqrt {\arcsin (a x)}-\frac {\int \frac {a^3 x^3}{\sqrt {\arcsin (a x)}}d\arcsin (a x)}{6 a^3}}{2 a}+\frac {2 \left (\frac {3 \left (x \sqrt {\arcsin (a x)}-\frac {\int \frac {a x}{\sqrt {\arcsin (a x)}}d\arcsin (a x)}{2 a}\right )}{2 a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^{3/2}}{a^2}\right )}{3 a^2}-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^{3/2}}{3 a^2}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{3} x^3 \arcsin (a x)^{5/2}-\frac {5}{6} a \left (\frac {\frac {1}{3} x^3 \sqrt {\arcsin (a x)}-\frac {\int \frac {\sin (\arcsin (a x))^3}{\sqrt {\arcsin (a x)}}d\arcsin (a x)}{6 a^3}}{2 a}+\frac {2 \left (\frac {3 \left (x \sqrt {\arcsin (a x)}-\frac {\int \frac {\sin (\arcsin (a x))}{\sqrt {\arcsin (a x)}}d\arcsin (a x)}{2 a}\right )}{2 a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^{3/2}}{a^2}\right )}{3 a^2}-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^{3/2}}{3 a^2}\right )\)

\(\Big \downarrow \) 3786

\(\displaystyle \frac {1}{3} x^3 \arcsin (a x)^{5/2}-\frac {5}{6} a \left (\frac {\frac {1}{3} x^3 \sqrt {\arcsin (a x)}-\frac {\int \frac {\sin (\arcsin (a x))^3}{\sqrt {\arcsin (a x)}}d\arcsin (a x)}{6 a^3}}{2 a}+\frac {2 \left (\frac {3 \left (x \sqrt {\arcsin (a x)}-\frac {\int a xd\sqrt {\arcsin (a x)}}{a}\right )}{2 a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^{3/2}}{a^2}\right )}{3 a^2}-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^{3/2}}{3 a^2}\right )\)

\(\Big \downarrow \) 3793

\(\displaystyle \frac {1}{3} x^3 \arcsin (a x)^{5/2}-\frac {5}{6} a \left (\frac {\frac {1}{3} x^3 \sqrt {\arcsin (a x)}-\frac {\int \left (\frac {3 a x}{4 \sqrt {\arcsin (a x)}}-\frac {\sin (3 \arcsin (a x))}{4 \sqrt {\arcsin (a x)}}\right )d\arcsin (a x)}{6 a^3}}{2 a}+\frac {2 \left (\frac {3 \left (x \sqrt {\arcsin (a x)}-\frac {\int a xd\sqrt {\arcsin (a x)}}{a}\right )}{2 a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^{3/2}}{a^2}\right )}{3 a^2}-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^{3/2}}{3 a^2}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{3} x^3 \arcsin (a x)^{5/2}-\frac {5}{6} a \left (\frac {2 \left (\frac {3 \left (x \sqrt {\arcsin (a x)}-\frac {\int a xd\sqrt {\arcsin (a x)}}{a}\right )}{2 a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^{3/2}}{a^2}\right )}{3 a^2}+\frac {\frac {1}{3} x^3 \sqrt {\arcsin (a x)}-\frac {\frac {3}{2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )-\frac {1}{2} \sqrt {\frac {\pi }{6}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arcsin (a x)}\right )}{6 a^3}}{2 a}-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^{3/2}}{3 a^2}\right )\)

\(\Big \downarrow \) 3832

\(\displaystyle \frac {1}{3} x^3 \arcsin (a x)^{5/2}-\frac {5}{6} a \left (\frac {\frac {1}{3} x^3 \sqrt {\arcsin (a x)}-\frac {\frac {3}{2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )-\frac {1}{2} \sqrt {\frac {\pi }{6}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arcsin (a x)}\right )}{6 a^3}}{2 a}+\frac {2 \left (\frac {3 \left (x \sqrt {\arcsin (a x)}-\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{a}\right )}{2 a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^{3/2}}{a^2}\right )}{3 a^2}-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^{3/2}}{3 a^2}\right )\)

Input:

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

Output:

(x^3*ArcSin[a*x]^(5/2))/3 - (5*a*(-1/3*(x^2*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^ 
(3/2))/a^2 + (2*(-((Sqrt[1 - a^2*x^2]*ArcSin[a*x]^(3/2))/a^2) + (3*(x*Sqrt 
[ArcSin[a*x]] - (Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/a))/(2 
*a)))/(3*a^2) + ((x^3*Sqrt[ArcSin[a*x]])/3 - ((3*Sqrt[Pi/2]*FresnelS[Sqrt[ 
2/Pi]*Sqrt[ArcSin[a*x]]])/2 - (Sqrt[Pi/6]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcSin[ 
a*x]]])/2)/(6*a^3))/(2*a)))/6
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3786

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[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 3793

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In 
t[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 3832

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 5130

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*Ar 
cSin[c*x])^n, x] - Simp[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 5140

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

rule 5182

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] + Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]   I 
nt[(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 5210

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] + (Simp[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] + S 
imp[b*f*(n/(c*(m + 2*p + 1)))*Simp[(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 5224

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^ 
2)^(p_.), x_Symbol] :> Simp[(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]
 

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.88

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

Input:

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

Output:

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

Fricas [F(-2)]

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

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

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (constant residues)
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

Output:

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

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.20 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.74 \[ \int x^2 \arcsin (a x)^{5/2} \, dx =\text {Too large to display} \] Input:

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

Output:

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*arcsin(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*arcsin(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(p 
i)*erf(-(1/2*I + 1/2)*sqrt(6)*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 - (15/1 
28*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*arcs 
in(a*x))/a^3 + 5/288*I*sqrt(arcsin(a*x))*e^(-3*I*arcsin(a*x))/a^3
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int x^2 \arcsin (a x)^{5/2} \, dx=\int \sqrt {\mathit {asin} \left (a x \right )}\, \mathit {asin} \left (a x \right )^{2} x^{2}d x \] Input:

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

Output:

int(sqrt(asin(a*x))*asin(a*x)**2*x**2,x)