\(\int (a^2-x^2)^{3/2} \sqrt {\arcsin (\frac {x}{a})} \, dx\) [117]

Optimal result

Integrand size = 24, antiderivative size = 226 \[ \int \left (a^2-x^2\right )^{3/2} \sqrt {\arcsin \left (\frac {x}{a}\right )} \, dx=\frac {3}{8} a^2 x \sqrt {a^2-x^2} \sqrt {\arcsin \left (\frac {x}{a}\right )}+\frac {1}{4} x \left (a^2-x^2\right )^{3/2} \sqrt {\arcsin \left (\frac {x}{a}\right )}+\frac {a^3 \sqrt {a^2-x^2} \arcsin \left (\frac {x}{a}\right )^{3/2}}{4 \sqrt {1-\frac {x^2}{a^2}}}-\frac {a^3 \sqrt {\frac {\pi }{2}} \sqrt {a^2-x^2} \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arcsin \left (\frac {x}{a}\right )}\right )}{64 \sqrt {1-\frac {x^2}{a^2}}}-\frac {a^3 \sqrt {\pi } \sqrt {a^2-x^2} \operatorname {FresnelS}\left (\frac {2 \sqrt {\arcsin \left (\frac {x}{a}\right )}}{\sqrt {\pi }}\right )}{8 \sqrt {1-\frac {x^2}{a^2}}} \] Output:

3/8*a^2*x*(a^2-x^2)^(1/2)*arcsin(x/a)^(1/2)+1/4*x*(a^2-x^2)^(3/2)*arcsin(x 
/a)^(1/2)+1/4*a^3*(a^2-x^2)^(1/2)*arcsin(x/a)^(3/2)/(1-x^2/a^2)^(1/2)-1/12 
8*a^3*2^(1/2)*Pi^(1/2)*(a^2-x^2)^(1/2)*FresnelS(2*2^(1/2)/Pi^(1/2)*arcsin( 
x/a)^(1/2))/(1-x^2/a^2)^(1/2)-1/8*a^3*Pi^(1/2)*(a^2-x^2)^(1/2)*FresnelS(2* 
arcsin(x/a)^(1/2)/Pi^(1/2))/(1-x^2/a^2)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.14 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.81 \[ \int \left (a^2-x^2\right )^{3/2} \sqrt {\arcsin \left (\frac {x}{a}\right )} \, dx=\frac {a^3 \sqrt {a^2-x^2} \left (32 \arcsin \left (\frac {x}{a}\right )^2+8 \sqrt {2} \sqrt {-i \arcsin \left (\frac {x}{a}\right )} \Gamma \left (\frac {3}{2},-2 i \arcsin \left (\frac {x}{a}\right )\right )+8 \sqrt {2} \sqrt {i \arcsin \left (\frac {x}{a}\right )} \Gamma \left (\frac {3}{2},2 i \arcsin \left (\frac {x}{a}\right )\right )+\sqrt {-i \arcsin \left (\frac {x}{a}\right )} \Gamma \left (\frac {3}{2},-4 i \arcsin \left (\frac {x}{a}\right )\right )+\sqrt {i \arcsin \left (\frac {x}{a}\right )} \Gamma \left (\frac {3}{2},4 i \arcsin \left (\frac {x}{a}\right )\right )\right )}{128 \sqrt {1-\frac {x^2}{a^2}} \sqrt {\arcsin \left (\frac {x}{a}\right )}} \] Input:

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

Output:

(a^3*Sqrt[a^2 - x^2]*(32*ArcSin[x/a]^2 + 8*Sqrt[2]*Sqrt[(-I)*ArcSin[x/a]]* 
Gamma[3/2, (-2*I)*ArcSin[x/a]] + 8*Sqrt[2]*Sqrt[I*ArcSin[x/a]]*Gamma[3/2, 
(2*I)*ArcSin[x/a]] + Sqrt[(-I)*ArcSin[x/a]]*Gamma[3/2, (-4*I)*ArcSin[x/a]] 
 + Sqrt[I*ArcSin[x/a]]*Gamma[3/2, (4*I)*ArcSin[x/a]]))/(128*Sqrt[1 - x^2/a 
^2]*Sqrt[ArcSin[x/a]])
 

Rubi [A] (verified)

Time = 1.58 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.15, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.542, Rules used = {5158, 27, 5156, 5146, 4906, 27, 3042, 3786, 3832, 5152, 5224, 4906, 2009}

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.

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

\(\Big \downarrow \) 5158

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5156

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

\(\Big \downarrow \) 5146

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

\(\Big \downarrow \) 4906

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3786

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

\(\Big \downarrow \) 3832

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

\(\Big \downarrow \) 5152

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

\(\Big \downarrow \) 5224

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

\(\Big \downarrow \) 4906

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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 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 5156

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

rule 5158

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

\[\int \left (a^{2}-x^{2}\right )^{\frac {3}{2}} \sqrt {\arcsin \left (\frac {x}{a}\right )}d x\]

Input:

int((a^2-x^2)^(3/2)*arcsin(x/a)^(1/2),x)
 

Output:

int((a^2-x^2)^(3/2)*arcsin(x/a)^(1/2),x)
 

Fricas [F(-2)]

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

integrate((a^2-x^2)^(3/2)*arcsin(x/a)^(1/2),x, algorithm="fricas")
 

Output:

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

Sympy [F]

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

integrate((a**2-x**2)**(3/2)*asin(x/a)**(1/2),x)
 

Output:

Integral((-(-a + x)*(a + x))**(3/2)*sqrt(asin(x/a)), x)
 

Maxima [F(-2)]

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

integrate((a^2-x^2)^(3/2)*arcsin(x/a)^(1/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((a^2-x^2)^(3/2)*arcsin(x/a)^(1/2),x, algorithm="giac")
 

Output:

integrate((a^2 - x^2)^(3/2)*sqrt(arcsin(x/a)), x)
 

Mupad [F(-1)]

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

int(asin(x/a)^(1/2)*(a^2 - x^2)^(3/2),x)
 

Output:

int(asin(x/a)^(1/2)*(a^2 - x^2)^(3/2), x)
 

Reduce [F]

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

int((a^2-x^2)^(3/2)*asin(x/a)^(1/2),x)
 

Output:

 - int(sqrt(a**2 - x**2)*sqrt(asin(x/a))*x**2,x) + int(sqrt(a**2 - x**2)*s 
qrt(asin(x/a)),x)*a**2