\(\int \frac {1}{(a+b \arcsin (c x))^{5/2}} \, dx\) [168]

Optimal result

Integrand size = 12, antiderivative size = 163 \[ \int \frac {1}{(a+b \arcsin (c x))^{5/2}} \, dx=-\frac {2 \sqrt {1-c^2 x^2}}{3 b c (a+b \arcsin (c x))^{3/2}}+\frac {4 x}{3 b^2 \sqrt {a+b \arcsin (c x)}}-\frac {4 \sqrt {2 \pi } \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{3 b^{5/2} c}-\frac {4 \sqrt {2 \pi } \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{3 b^{5/2} c} \] Output:

-2/3*(-c^2*x^2+1)^(1/2)/b/c/(a+b*arcsin(c*x))^(3/2)+4/3*x/b^2/(a+b*arcsin( 
c*x))^(1/2)-4/3*2^(1/2)*Pi^(1/2)*cos(a/b)*FresnelC(2^(1/2)/Pi^(1/2)*(a+b*a 
rcsin(c*x))^(1/2)/b^(1/2))/b^(5/2)/c-4/3*2^(1/2)*Pi^(1/2)*FresnelS(2^(1/2) 
/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))*sin(a/b)/b^(5/2)/c
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.27 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.31 \[ \int \frac {1}{(a+b \arcsin (c x))^{5/2}} \, dx=\frac {e^{-\frac {i (a+b \arcsin (c x))}{b}} \left (-2 b e^{i \arcsin (c x)} \left (-\frac {i (a+b \arcsin (c x))}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {i (a+b \arcsin (c x))}{b}\right )-i e^{\frac {i a}{b}} \left (2 a \left (-1+e^{2 i \arcsin (c x)}\right )+b \left (-i-2 \arcsin (c x)+e^{2 i \arcsin (c x)} (-i+2 \arcsin (c x))\right )-2 i b e^{\frac {i (a+b \arcsin (c x))}{b}} \left (\frac {i (a+b \arcsin (c x))}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},\frac {i (a+b \arcsin (c x))}{b}\right )\right )\right )}{3 b^2 c (a+b \arcsin (c x))^{3/2}} \] Input:

Integrate[(a + b*ArcSin[c*x])^(-5/2),x]
 

Output:

(-2*b*E^(I*ArcSin[c*x])*(((-I)*(a + b*ArcSin[c*x]))/b)^(3/2)*Gamma[1/2, (( 
-I)*(a + b*ArcSin[c*x]))/b] - I*E^((I*a)/b)*(2*a*(-1 + E^((2*I)*ArcSin[c*x 
])) + b*(-I - 2*ArcSin[c*x] + E^((2*I)*ArcSin[c*x])*(-I + 2*ArcSin[c*x])) 
- (2*I)*b*E^((I*(a + b*ArcSin[c*x]))/b)*((I*(a + b*ArcSin[c*x]))/b)^(3/2)* 
Gamma[1/2, (I*(a + b*ArcSin[c*x]))/b]))/(3*b^2*c*E^((I*(a + b*ArcSin[c*x]) 
)/b)*(a + b*ArcSin[c*x])^(3/2))
 

Rubi [A] (verified)

Time = 1.02 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.04, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.917, Rules used = {5132, 5222, 5134, 3042, 3787, 25, 3042, 3785, 3786, 3832, 3833}

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.

\(\displaystyle \int \frac {1}{(a+b \arcsin (c x))^{5/2}} \, dx\)

\(\Big \downarrow \) 5132

\(\displaystyle -\frac {2 c \int \frac {x}{\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^{3/2}}dx}{3 b}-\frac {2 \sqrt {1-c^2 x^2}}{3 b c (a+b \arcsin (c x))^{3/2}}\)

\(\Big \downarrow \) 5222

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

\(\Big \downarrow \) 5134

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3787

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2 c \left (\frac {2 \left (\sin \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))+\cos \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arcsin (c x)}{b}+\frac {\pi }{2}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))\right )}{b^2 c^2}-\frac {2 x}{b c \sqrt {a+b \arcsin (c x)}}\right )}{3 b}-\frac {2 \sqrt {1-c^2 x^2}}{3 b c (a+b \arcsin (c x))^{3/2}}\)

\(\Big \downarrow \) 3785

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

\(\Big \downarrow \) 3786

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

\(\Big \downarrow \) 3832

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

\(\Big \downarrow \) 3833

\(\displaystyle -\frac {2 c \left (\frac {2 \left (\sqrt {2 \pi } \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )+\sqrt {2 \pi } \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )\right )}{b^2 c^2}-\frac {2 x}{b c \sqrt {a+b \arcsin (c x)}}\right )}{3 b}-\frac {2 \sqrt {1-c^2 x^2}}{3 b c (a+b \arcsin (c x))^{3/2}}\)

Input:

Int[(a + b*ArcSin[c*x])^(-5/2),x]
 

Output:

(-2*Sqrt[1 - c^2*x^2])/(3*b*c*(a + b*ArcSin[c*x])^(3/2)) - (2*c*((-2*x)/(b 
*c*Sqrt[a + b*ArcSin[c*x]]) + (2*(Sqrt[b]*Sqrt[2*Pi]*Cos[a/b]*FresnelC[(Sq 
rt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]] + Sqrt[b]*Sqrt[2*Pi]*FresnelS[( 
Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b]))/(b^2*c^2)))/(3*b)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 3042

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

rule 3785

Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> S 
imp[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 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 3787

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Cos 
[(d*e - c*f)/d]   Int[Sin[c*(f/d) + f*x]/Sqrt[c + d*x], x], x] + Simp[Sin[( 
d*e - c*f)/d]   Int[Cos[c*(f/d) + f*x]/Sqrt[c + d*x], x], x] /; FreeQ[{c, d 
, e, f}, x] && ComplexFreeQ[f] && NeQ[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 3833

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

rule 5132

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

rule 5134

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

rule 5222

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(339\) vs. \(2(129)=258\).

Time = 0.00 (sec) , antiderivative size = 340, normalized size of antiderivative = 2.09

method	result	size
default	\(-\frac {2 \left (2 \arcsin \left (c x \right ) \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \sqrt {a +b \arcsin \left (c x \right )}\, \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) b -2 \arcsin \left (c x \right ) \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \sqrt {a +b \arcsin \left (c x \right )}\, \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) b +2 \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \sqrt {a +b \arcsin \left (c x \right )}\, \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) a -2 \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \sqrt {a +b \arcsin \left (c x \right )}\, \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) a +2 \arcsin \left (c x \right ) \sin \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right ) b +\cos \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right ) b +2 \sin \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right ) a \right )}{3 c \,b^{2} \left (a +b \arcsin \left (c x \right )\right )^{\frac {3}{2}}}\)	\(340\)

Input:

int(1/(a+b*arcsin(c*x))^(5/2),x,method=_RETURNVERBOSE)
                                                                                    
                                                                                    
 

Output:

-2/3/c/b^2*(2*arcsin(c*x)*2^(1/2)*Pi^(1/2)*(-1/b)^(1/2)*(a+b*arcsin(c*x))^ 
(1/2)*cos(a/b)*FresnelC(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arcsin(c*x))^(1 
/2)/b)*b-2*arcsin(c*x)*2^(1/2)*Pi^(1/2)*(-1/b)^(1/2)*(a+b*arcsin(c*x))^(1/ 
2)*sin(a/b)*FresnelS(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2) 
/b)*b+2*2^(1/2)*Pi^(1/2)*(-1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)*cos(a/b)*Fre 
snelC(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*a-2*2^(1/2) 
*Pi^(1/2)*(-1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)*sin(a/b)*FresnelS(2^(1/2)/P 
i^(1/2)/(-1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*a+2*arcsin(c*x)*sin(-(a+b* 
arcsin(c*x))/b+a/b)*b+cos(-(a+b*arcsin(c*x))/b+a/b)*b+2*sin(-(a+b*arcsin(c 
*x))/b+a/b)*a)/(a+b*arcsin(c*x))^(3/2)
 

Fricas [F(-2)]

Exception generated. \[ \int \frac {1}{(a+b \arcsin (c x))^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(1/(a+b*arcsin(c*x))^(5/2),x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {1}{(a+b \arcsin (c x))^{5/2}} \, dx=\int \frac {1}{\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{\frac {5}{2}}}\, dx \] Input:

integrate(1/(a+b*asin(c*x))**(5/2),x)
 

Output:

Integral((a + b*asin(c*x))**(-5/2), x)
 

Maxima [F]

\[ \int \frac {1}{(a+b \arcsin (c x))^{5/2}} \, dx=\int { \frac {1}{{\left (b \arcsin \left (c x\right ) + a\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(1/(a+b*arcsin(c*x))^(5/2),x, algorithm="maxima")
 

Output:

integrate((b*arcsin(c*x) + a)^(-5/2), x)
 

Giac [F]

\[ \int \frac {1}{(a+b \arcsin (c x))^{5/2}} \, dx=\int { \frac {1}{{\left (b \arcsin \left (c x\right ) + a\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(1/(a+b*arcsin(c*x))^(5/2),x, algorithm="giac")
 

Output:

integrate((b*arcsin(c*x) + a)^(-5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(a+b \arcsin (c x))^{5/2}} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^{5/2}} \,d x \] Input:

int(1/(a + b*asin(c*x))^(5/2),x)
 

Output:

int(1/(a + b*asin(c*x))^(5/2), x)
 

Reduce [F]

\[ \int \frac {1}{(a+b \arcsin (c x))^{5/2}} \, dx=\frac {\frac {2 \mathit {asin} \left (c x \right )^{2} \left (\int \frac {\sqrt {\mathit {asin} \left (c x \right ) b +a}\, \sqrt {-c^{2} x^{2}+1}\, x}{\mathit {asin} \left (c x \right )^{2} b^{2} c^{2} x^{2}-\mathit {asin} \left (c x \right )^{2} b^{2}+2 \mathit {asin} \left (c x \right ) a b \,c^{2} x^{2}-2 \mathit {asin} \left (c x \right ) a b +a^{2} c^{2} x^{2}-a^{2}}d x \right ) b^{2} c^{2}}{3}+\frac {4 \mathit {asin} \left (c x \right ) \left (\int \frac {\sqrt {\mathit {asin} \left (c x \right ) b +a}\, \sqrt {-c^{2} x^{2}+1}\, x}{\mathit {asin} \left (c x \right )^{2} b^{2} c^{2} x^{2}-\mathit {asin} \left (c x \right )^{2} b^{2}+2 \mathit {asin} \left (c x \right ) a b \,c^{2} x^{2}-2 \mathit {asin} \left (c x \right ) a b +a^{2} c^{2} x^{2}-a^{2}}d x \right ) a b \,c^{2}}{3}-\frac {2 \sqrt {\mathit {asin} \left (c x \right ) b +a}\, \sqrt {-c^{2} x^{2}+1}}{3}+\frac {2 \left (\int \frac {\sqrt {\mathit {asin} \left (c x \right ) b +a}\, \sqrt {-c^{2} x^{2}+1}\, x}{\mathit {asin} \left (c x \right )^{2} b^{2} c^{2} x^{2}-\mathit {asin} \left (c x \right )^{2} b^{2}+2 \mathit {asin} \left (c x \right ) a b \,c^{2} x^{2}-2 \mathit {asin} \left (c x \right ) a b +a^{2} c^{2} x^{2}-a^{2}}d x \right ) a^{2} c^{2}}{3}}{b c \left (\mathit {asin} \left (c x \right )^{2} b^{2}+2 \mathit {asin} \left (c x \right ) a b +a^{2}\right )} \] Input:

int(1/(a+b*asin(c*x))^(5/2),x)
 

Output:

(2*(asin(c*x)**2*int((sqrt(asin(c*x)*b + a)*sqrt( - c**2*x**2 + 1)*x)/(asi 
n(c*x)**2*b**2*c**2*x**2 - asin(c*x)**2*b**2 + 2*asin(c*x)*a*b*c**2*x**2 - 
 2*asin(c*x)*a*b + a**2*c**2*x**2 - a**2),x)*b**2*c**2 + 2*asin(c*x)*int(( 
sqrt(asin(c*x)*b + a)*sqrt( - c**2*x**2 + 1)*x)/(asin(c*x)**2*b**2*c**2*x* 
*2 - asin(c*x)**2*b**2 + 2*asin(c*x)*a*b*c**2*x**2 - 2*asin(c*x)*a*b + a** 
2*c**2*x**2 - a**2),x)*a*b*c**2 - sqrt(asin(c*x)*b + a)*sqrt( - c**2*x**2 
+ 1) + int((sqrt(asin(c*x)*b + a)*sqrt( - c**2*x**2 + 1)*x)/(asin(c*x)**2* 
b**2*c**2*x**2 - asin(c*x)**2*b**2 + 2*asin(c*x)*a*b*c**2*x**2 - 2*asin(c* 
x)*a*b + a**2*c**2*x**2 - a**2),x)*a**2*c**2))/(3*b*c*(asin(c*x)**2*b**2 + 
 2*asin(c*x)*a*b + a**2))