\(\int x^3 \arcsin (a x)^{3/2} \, dx\) [103]

Optimal result

Integrand size = 12, antiderivative size = 157 \[ \int x^3 \arcsin (a x)^{3/2} \, dx=\frac {9 x \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{64 a^3}+\frac {3 x^3 \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{32 a}-\frac {3 \arcsin (a x)^{3/2}}{32 a^4}+\frac {1}{4} x^4 \arcsin (a x)^{3/2}+\frac {3 \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{512 a^4}-\frac {3 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )}{64 a^4} \] Output:

9/64*x*(-a^2*x^2+1)^(1/2)*arcsin(a*x)^(1/2)/a^3+3/32*x^3*(-a^2*x^2+1)^(1/2 
)*arcsin(a*x)^(1/2)/a-3/32*arcsin(a*x)^(3/2)/a^4+1/4*x^4*arcsin(a*x)^(3/2) 
+3/1024*2^(1/2)*Pi^(1/2)*FresnelS(2*2^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))/a^ 
4-3/64*Pi^(1/2)*FresnelS(2*arcsin(a*x)^(1/2)/Pi^(1/2))/a^4
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.02 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.83 \[ \int x^3 \arcsin (a x)^{3/2} \, dx=\frac {8 \sqrt {2} \sqrt {-i \arcsin (a x)} \Gamma \left (\frac {5}{2},-2 i \arcsin (a x)\right )+8 \sqrt {2} \sqrt {i \arcsin (a x)} \Gamma \left (\frac {5}{2},2 i \arcsin (a x)\right )-\sqrt {-i \arcsin (a x)} \Gamma \left (\frac {5}{2},-4 i \arcsin (a x)\right )-\sqrt {i \arcsin (a x)} \Gamma \left (\frac {5}{2},4 i \arcsin (a x)\right )}{512 a^4 \sqrt {\arcsin (a x)}} \] Input:

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

Output:

(8*Sqrt[2]*Sqrt[(-I)*ArcSin[a*x]]*Gamma[5/2, (-2*I)*ArcSin[a*x]] + 8*Sqrt[ 
2]*Sqrt[I*ArcSin[a*x]]*Gamma[5/2, (2*I)*ArcSin[a*x]] - Sqrt[(-I)*ArcSin[a* 
x]]*Gamma[5/2, (-4*I)*ArcSin[a*x]] - Sqrt[I*ArcSin[a*x]]*Gamma[5/2, (4*I)* 
ArcSin[a*x]])/(512*a^4*Sqrt[ArcSin[a*x]])
 

Rubi [A] (verified)

Time = 1.77 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.28, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.083, Rules used = {5140, 5210, 5146, 4906, 2009, 5210, 5146, 4906, 27, 3042, 3786, 3832, 5152}

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.

\(\displaystyle \int x^3 \arcsin (a x)^{3/2} \, dx\)

\(\Big \downarrow \) 5140

\(\displaystyle \frac {1}{4} x^4 \arcsin (a x)^{3/2}-\frac {3}{8} a \int \frac {x^4 \sqrt {\arcsin (a x)}}{\sqrt {1-a^2 x^2}}dx\)

\(\Big \downarrow \) 5210

\(\displaystyle \frac {1}{4} x^4 \arcsin (a x)^{3/2}-\frac {3}{8} a \left (\frac {3 \int \frac {x^2 \sqrt {\arcsin (a x)}}{\sqrt {1-a^2 x^2}}dx}{4 a^2}+\frac {\int \frac {x^3}{\sqrt {\arcsin (a x)}}dx}{8 a}-\frac {x^3 \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{4 a^2}\right )\)

\(\Big \downarrow \) 5146

\(\displaystyle \frac {1}{4} x^4 \arcsin (a x)^{3/2}-\frac {3}{8} a \left (\frac {3 \int \frac {x^2 \sqrt {\arcsin (a x)}}{\sqrt {1-a^2 x^2}}dx}{4 a^2}+\frac {\int \frac {a^3 x^3 \sqrt {1-a^2 x^2}}{\sqrt {\arcsin (a x)}}d\arcsin (a x)}{8 a^5}-\frac {x^3 \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{4 a^2}\right )\)

\(\Big \downarrow \) 4906

\(\displaystyle \frac {1}{4} x^4 \arcsin (a x)^{3/2}-\frac {3}{8} a \left (\frac {\int \left (\frac {\sin (2 \arcsin (a x))}{4 \sqrt {\arcsin (a x)}}-\frac {\sin (4 \arcsin (a x))}{8 \sqrt {\arcsin (a x)}}\right )d\arcsin (a x)}{8 a^5}+\frac {3 \int \frac {x^2 \sqrt {\arcsin (a x)}}{\sqrt {1-a^2 x^2}}dx}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{4 a^2}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{4} x^4 \arcsin (a x)^{3/2}-\frac {3}{8} a \left (\frac {3 \int \frac {x^2 \sqrt {\arcsin (a x)}}{\sqrt {1-a^2 x^2}}dx}{4 a^2}+\frac {\frac {1}{4} \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )-\frac {1}{8} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{8 a^5}-\frac {x^3 \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{4 a^2}\right )\)

\(\Big \downarrow \) 5210

\(\displaystyle \frac {1}{4} x^4 \arcsin (a x)^{3/2}-\frac {3}{8} a \left (\frac {3 \left (\frac {\int \frac {\sqrt {\arcsin (a x)}}{\sqrt {1-a^2 x^2}}dx}{2 a^2}+\frac {\int \frac {x}{\sqrt {\arcsin (a x)}}dx}{4 a}-\frac {x \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{2 a^2}\right )}{4 a^2}+\frac {\frac {1}{4} \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )-\frac {1}{8} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{8 a^5}-\frac {x^3 \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{4 a^2}\right )\)

\(\Big \downarrow \) 5146

\(\displaystyle \frac {1}{4} x^4 \arcsin (a x)^{3/2}-\frac {3}{8} a \left (\frac {3 \left (\frac {\int \frac {\sqrt {\arcsin (a x)}}{\sqrt {1-a^2 x^2}}dx}{2 a^2}+\frac {\int \frac {a x \sqrt {1-a^2 x^2}}{\sqrt {\arcsin (a x)}}d\arcsin (a x)}{4 a^3}-\frac {x \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{2 a^2}\right )}{4 a^2}+\frac {\frac {1}{4} \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )-\frac {1}{8} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{8 a^5}-\frac {x^3 \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{4 a^2}\right )\)

\(\Big \downarrow \) 4906

\(\displaystyle \frac {1}{4} x^4 \arcsin (a x)^{3/2}-\frac {3}{8} a \left (\frac {3 \left (\frac {\int \frac {\sin (2 \arcsin (a x))}{2 \sqrt {\arcsin (a x)}}d\arcsin (a x)}{4 a^3}+\frac {\int \frac {\sqrt {\arcsin (a x)}}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{2 a^2}\right )}{4 a^2}+\frac {\frac {1}{4} \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )-\frac {1}{8} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{8 a^5}-\frac {x^3 \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{4 a^2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{4} x^4 \arcsin (a x)^{3/2}-\frac {3}{8} a \left (\frac {3 \left (\frac {\int \frac {\sin (2 \arcsin (a x))}{\sqrt {\arcsin (a x)}}d\arcsin (a x)}{8 a^3}+\frac {\int \frac {\sqrt {\arcsin (a x)}}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{2 a^2}\right )}{4 a^2}+\frac {\frac {1}{4} \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )-\frac {1}{8} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{8 a^5}-\frac {x^3 \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{4 a^2}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{4} x^4 \arcsin (a x)^{3/2}-\frac {3}{8} a \left (\frac {3 \left (\frac {\int \frac {\sin (2 \arcsin (a x))}{\sqrt {\arcsin (a x)}}d\arcsin (a x)}{8 a^3}+\frac {\int \frac {\sqrt {\arcsin (a x)}}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{2 a^2}\right )}{4 a^2}+\frac {\frac {1}{4} \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )-\frac {1}{8} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{8 a^5}-\frac {x^3 \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{4 a^2}\right )\)

\(\Big \downarrow \) 3786

\(\displaystyle \frac {1}{4} x^4 \arcsin (a x)^{3/2}-\frac {3}{8} a \left (\frac {3 \left (\frac {\int \sin (2 \arcsin (a x))d\sqrt {\arcsin (a x)}}{4 a^3}+\frac {\int \frac {\sqrt {\arcsin (a x)}}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{2 a^2}\right )}{4 a^2}+\frac {\frac {1}{4} \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )-\frac {1}{8} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{8 a^5}-\frac {x^3 \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{4 a^2}\right )\)

\(\Big \downarrow \) 3832

\(\displaystyle \frac {1}{4} x^4 \arcsin (a x)^{3/2}-\frac {3}{8} a \left (\frac {3 \left (\frac {\int \frac {\sqrt {\arcsin (a x)}}{\sqrt {1-a^2 x^2}}dx}{2 a^2}+\frac {\sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )}{8 a^3}-\frac {x \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{2 a^2}\right )}{4 a^2}+\frac {\frac {1}{4} \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )-\frac {1}{8} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{8 a^5}-\frac {x^3 \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{4 a^2}\right )\)

\(\Big \downarrow \) 5152

\(\displaystyle \frac {1}{4} x^4 \arcsin (a x)^{3/2}-\frac {3}{8} a \left (\frac {\frac {1}{4} \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )-\frac {1}{8} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{8 a^5}-\frac {x^3 \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{4 a^2}+\frac {3 \left (\frac {\sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )}{8 a^3}+\frac {\arcsin (a x)^{3/2}}{3 a^3}-\frac {x \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{2 a^2}\right )}{4 a^2}\right )\)

Input:

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

Output:

(x^4*ArcSin[a*x]^(3/2))/4 - (3*a*(-1/4*(x^3*Sqrt[1 - a^2*x^2]*Sqrt[ArcSin[ 
a*x]])/a^2 + (-1/8*(Sqrt[Pi/2]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]]) + 
 (Sqrt[Pi]*FresnelS[(2*Sqrt[ArcSin[a*x]])/Sqrt[Pi]])/4)/(8*a^5) + (3*(-1/2 
*(x*Sqrt[1 - a^2*x^2]*Sqrt[ArcSin[a*x]])/a^2 + ArcSin[a*x]^(3/2)/(3*a^3) + 
 (Sqrt[Pi]*FresnelS[(2*Sqrt[ArcSin[a*x]])/Sqrt[Pi]])/(8*a^3)))/(4*a^2)))/8
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 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 4906

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b 
_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(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] && IG 
tQ[p, 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 5146

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

rule 5152

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

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.77

method	result	size
default	\(-\frac {-3 \sqrt {2}\, \sqrt {\arcsin \left (a x \right )}\, \sqrt {\pi }\, \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right )+128 \arcsin \left (a x \right )^{2} \cos \left (2 \arcsin \left (a x \right )\right )-32 \arcsin \left (a x \right )^{2} \cos \left (4 \arcsin \left (a x \right )\right )+48 \sqrt {\arcsin \left (a x \right )}\, \sqrt {\pi }\, \operatorname {FresnelS}\left (\frac {2 \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right )-96 \arcsin \left (a x \right ) \sin \left (2 \arcsin \left (a x \right )\right )+12 \arcsin \left (a x \right ) \sin \left (4 \arcsin \left (a x \right )\right )}{1024 a^{4} \sqrt {\arcsin \left (a x \right )}}\)	\(121\)

Input:

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

Output:

-1/1024/a^4*(-3*2^(1/2)*arcsin(a*x)^(1/2)*Pi^(1/2)*FresnelS(2*2^(1/2)/Pi^( 
1/2)*arcsin(a*x)^(1/2))+128*arcsin(a*x)^2*cos(2*arcsin(a*x))-32*arcsin(a*x 
)^2*cos(4*arcsin(a*x))+48*arcsin(a*x)^(1/2)*Pi^(1/2)*FresnelS(2*arcsin(a*x 
)^(1/2)/Pi^(1/2))-96*arcsin(a*x)*sin(2*arcsin(a*x))+12*arcsin(a*x)*sin(4*a 
rcsin(a*x)))/arcsin(a*x)^(1/2)
 

Fricas [F(-2)]

Exception generated. \[ \int x^3 \arcsin (a x)^{3/2} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

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

Sympy [F]

\[ \int x^3 \arcsin (a x)^{3/2} \, dx=\int x^{3} \operatorname {asin}^{\frac {3}{2}}{\left (a x \right )}\, dx \] Input:

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

Output:

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

Maxima [F(-2)]

Exception generated. \[ \int x^3 \arcsin (a x)^{3/2} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate(x^3*arcsin(a*x)^(3/2),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 = 0.18 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.43 \[ \int x^3 \arcsin (a x)^{3/2} \, dx=\frac {\arcsin \left (a x\right )^{\frac {3}{2}} e^{\left (4 i \, \arcsin \left (a x\right )\right )}}{64 \, a^{4}} - \frac {\arcsin \left (a x\right )^{\frac {3}{2}} e^{\left (2 i \, \arcsin \left (a x\right )\right )}}{16 \, a^{4}} - \frac {\arcsin \left (a x\right )^{\frac {3}{2}} e^{\left (-2 i \, \arcsin \left (a x\right )\right )}}{16 \, a^{4}} + \frac {\arcsin \left (a x\right )^{\frac {3}{2}} e^{\left (-4 i \, \arcsin \left (a x\right )\right )}}{64 \, a^{4}} + \frac {\left (3 i - 3\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (i - 1\right ) \, \sqrt {2} \sqrt {\arcsin \left (a x\right )}\right )}{4096 \, a^{4}} - \frac {\left (3 i + 3\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (i + 1\right ) \, \sqrt {2} \sqrt {\arcsin \left (a x\right )}\right )}{4096 \, a^{4}} - \frac {\left (3 i - 3\right ) \, \sqrt {\pi } \operatorname {erf}\left (\left (i - 1\right ) \, \sqrt {\arcsin \left (a x\right )}\right )}{256 \, a^{4}} + \frac {\left (3 i + 3\right ) \, \sqrt {\pi } \operatorname {erf}\left (-\left (i + 1\right ) \, \sqrt {\arcsin \left (a x\right )}\right )}{256 \, a^{4}} + \frac {3 i \, \sqrt {\arcsin \left (a x\right )} e^{\left (4 i \, \arcsin \left (a x\right )\right )}}{512 \, a^{4}} - \frac {3 i \, \sqrt {\arcsin \left (a x\right )} e^{\left (2 i \, \arcsin \left (a x\right )\right )}}{64 \, a^{4}} + \frac {3 i \, \sqrt {\arcsin \left (a x\right )} e^{\left (-2 i \, \arcsin \left (a x\right )\right )}}{64 \, a^{4}} - \frac {3 i \, \sqrt {\arcsin \left (a x\right )} e^{\left (-4 i \, \arcsin \left (a x\right )\right )}}{512 \, a^{4}} \] Input:

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

Output:

1/64*arcsin(a*x)^(3/2)*e^(4*I*arcsin(a*x))/a^4 - 1/16*arcsin(a*x)^(3/2)*e^ 
(2*I*arcsin(a*x))/a^4 - 1/16*arcsin(a*x)^(3/2)*e^(-2*I*arcsin(a*x))/a^4 + 
1/64*arcsin(a*x)^(3/2)*e^(-4*I*arcsin(a*x))/a^4 + (3/4096*I - 3/4096)*sqrt 
(2)*sqrt(pi)*erf((I - 1)*sqrt(2)*sqrt(arcsin(a*x)))/a^4 - (3/4096*I + 3/40 
96)*sqrt(2)*sqrt(pi)*erf(-(I + 1)*sqrt(2)*sqrt(arcsin(a*x)))/a^4 - (3/256* 
I - 3/256)*sqrt(pi)*erf((I - 1)*sqrt(arcsin(a*x)))/a^4 + (3/256*I + 3/256) 
*sqrt(pi)*erf(-(I + 1)*sqrt(arcsin(a*x)))/a^4 + 3/512*I*sqrt(arcsin(a*x))* 
e^(4*I*arcsin(a*x))/a^4 - 3/64*I*sqrt(arcsin(a*x))*e^(2*I*arcsin(a*x))/a^4 
 + 3/64*I*sqrt(arcsin(a*x))*e^(-2*I*arcsin(a*x))/a^4 - 3/512*I*sqrt(arcsin 
(a*x))*e^(-4*I*arcsin(a*x))/a^4
 

Mupad [F(-1)]

Timed out. \[ \int x^3 \arcsin (a x)^{3/2} \, dx=\int x^3\,{\mathrm {asin}\left (a\,x\right )}^{3/2} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int x^3 \arcsin (a x)^{3/2} \, dx=\int \sqrt {\mathit {asin} \left (a x \right )}\, \mathit {asin} \left (a x \right ) x^{3}d x \] Input:

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

Output:

int(sqrt(asin(a*x))*asin(a*x)*x**3,x)