3.3.48 \(\int (a+b \arcsin (c+d x))^{3/2} \, dx\) [248]

3.3.48.1 Optimal result

Integrand size = 14, antiderivative size = 175 \[ \int (a+b \arcsin (c+d x))^{3/2} \, dx=\frac {3 b \sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}}{2 d}+\frac {(c+d x) (a+b \arcsin (c+d x))^{3/2}}{d}-\frac {3 b^{3/2} \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{2 d}-\frac {3 b^{3/2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{2 d} \]

output

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

3.3.48.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.79 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.78 \[ \int (a+b \arcsin (c+d x))^{3/2} \, dx=\frac {a b e^{-\frac {i a}{b}} \left (\sqrt {-\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {3}{2},-\frac {i (a+b \arcsin (c+d x))}{b}\right )+e^{\frac {2 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {3}{2},\frac {i (a+b \arcsin (c+d x))}{b}\right )\right )}{2 d \sqrt {a+b \arcsin (c+d x)}}+\frac {\sqrt {b} \left (2 \sqrt {b} \sqrt {a+b \arcsin (c+d x)} \left (3 \sqrt {1-(c+d x)^2}+2 (c+d x) \arcsin (c+d x)\right )-\sqrt {2 \pi } \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right ) \left (3 b \cos \left (\frac {a}{b}\right )+2 a \sin \left (\frac {a}{b}\right )\right )+\sqrt {2 \pi } \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right ) \left (2 a \cos \left (\frac {a}{b}\right )-3 b \sin \left (\frac {a}{b}\right )\right )\right )}{4 d} \]

input

Integrate[(a + b*ArcSin[c + d*x])^(3/2),x]
 

output

(a*b*(Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[3/2, ((-I)*(a + b*ArcSi 
n[c + d*x]))/b] + E^(((2*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamm 
a[3/2, (I*(a + b*ArcSin[c + d*x]))/b]))/(2*d*E^((I*a)/b)*Sqrt[a + b*ArcSin 
[c + d*x]]) + (Sqrt[b]*(2*Sqrt[b]*Sqrt[a + b*ArcSin[c + d*x]]*(3*Sqrt[1 - 
(c + d*x)^2] + 2*(c + d*x)*ArcSin[c + d*x]) - Sqrt[2*Pi]*FresnelC[(Sqrt[2/ 
Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*(3*b*Cos[a/b] + 2*a*Sin[a/b]) + 
Sqrt[2*Pi]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*(2*a 
*Cos[a/b] - 3*b*Sin[a/b])))/(4*d)
 

3.3.48.3 Rubi [A] (verified)

Time = 0.78 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.94, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {5302, 5130, 5182, 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.

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

\(\Big \downarrow \) 5302

\(\displaystyle \frac {\int (a+b \arcsin (c+d x))^{3/2}d(c+d x)}{d}\)

\(\Big \downarrow \) 5130

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

\(\Big \downarrow \) 5182

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

\(\Big \downarrow \) 5134

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3787

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3785

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

\(\Big \downarrow \) 3786

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

\(\Big \downarrow \) 3832

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

\(\Big \downarrow \) 3833

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

input

Int[(a + b*ArcSin[c + d*x])^(3/2),x]
 

output

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

3.3.48.3.1 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 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 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 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 5302

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

3.3.48.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(303\) vs. \(2(139)=278\).

Time = 0.35 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.74

method	result	size
default	\(-\frac {3 \sqrt {a +b \arcsin \left (d x +c \right )}\, \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {2}\, b^{2}-3 \sqrt {a +b \arcsin \left (d x +c \right )}\, \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {2}\, b^{2}+4 \arcsin \left (d x +c \right )^{2} \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) b^{2}+8 \arcsin \left (d x +c \right ) \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) a b -6 \arcsin \left (d x +c \right ) \cos \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) b^{2}+4 \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) a^{2}-6 \cos \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) a b}{4 d \sqrt {a +b \arcsin \left (d x +c \right )}}\)	\(304\)

input

int((a+b*arcsin(d*x+c))^(3/2),x,method=_RETURNVERBOSE)
 

output

-1/4/d*(3*(a+b*arcsin(d*x+c))^(1/2)*cos(a/b)*FresnelC(2^(1/2)/Pi^(1/2)/(-1 
/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*(-1/b)^(1/2)*Pi^(1/2)*2^(1/2)*b^2-3 
*(a+b*arcsin(d*x+c))^(1/2)*sin(a/b)*FresnelS(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2) 
*(a+b*arcsin(d*x+c))^(1/2)/b)*(-1/b)^(1/2)*Pi^(1/2)*2^(1/2)*b^2+4*arcsin(d 
*x+c)^2*sin(-(a+b*arcsin(d*x+c))/b+a/b)*b^2+8*arcsin(d*x+c)*sin(-(a+b*arcs 
in(d*x+c))/b+a/b)*a*b-6*arcsin(d*x+c)*cos(-(a+b*arcsin(d*x+c))/b+a/b)*b^2+ 
4*sin(-(a+b*arcsin(d*x+c))/b+a/b)*a^2-6*cos(-(a+b*arcsin(d*x+c))/b+a/b)*a* 
b)/(a+b*arcsin(d*x+c))^(1/2)
 

3.3.48.5 Fricas [F(-2)]

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

input

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

output

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

3.3.48.6 Sympy [F]

\[ \int (a+b \arcsin (c+d x))^{3/2} \, dx=\int \left (a + b \operatorname {asin}{\left (c + d x \right )}\right )^{\frac {3}{2}}\, dx \]

input

integrate((a+b*asin(d*x+c))**(3/2),x)
 

output

Integral((a + b*asin(c + d*x))**(3/2), x)
 

3.3.48.7 Maxima [F]

\[ \int (a+b \arcsin (c+d x))^{3/2} \, dx=\int { {\left (b \arcsin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \,d x } \]

input

integrate((a+b*arcsin(d*x+c))^(3/2),x, algorithm="maxima")
 

output

integrate((b*arcsin(d*x + c) + a)^(3/2), x)
 

3.3.48.8 Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.90 (sec) , antiderivative size = 1061, normalized size of antiderivative = 6.06 \[ \int (a+b \arcsin (c+d x))^{3/2} \, dx=\text {Too large to display} \]

input

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

output

1/2*sqrt(2)*sqrt(pi)*a^2*b^2*erf(-1/2*I*sqrt(2)*sqrt(b*arcsin(d*x + c) + a 
)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e 
^(I*a/b)/((I*b^3/sqrt(abs(b)) + b^2*sqrt(abs(b)))*d) + 1/2*I*sqrt(2)*sqrt( 
pi)*a*b^3*erf(-1/2*I*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(abs(b)) - 1/ 
2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(I*a/b)/((I*b^3/sq 
rt(abs(b)) + b^2*sqrt(abs(b)))*d) + 1/2*sqrt(2)*sqrt(pi)*a^2*b^2*erf(1/2*I 
*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arc 
sin(d*x + c) + a)*sqrt(abs(b))/b)*e^(-I*a/b)/((-I*b^3/sqrt(abs(b)) + b^2*s 
qrt(abs(b)))*d) - 1/2*I*sqrt(2)*sqrt(pi)*a*b^3*erf(1/2*I*sqrt(2)*sqrt(b*ar 
csin(d*x + c) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)* 
sqrt(abs(b))/b)*e^(-I*a/b)/((-I*b^3/sqrt(abs(b)) + b^2*sqrt(abs(b)))*d) - 
1/2*I*sqrt(2)*sqrt(pi)*a*b^2*erf(-1/2*I*sqrt(2)*sqrt(b*arcsin(d*x + c) + a 
)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e 
^(I*a/b)/((I*b^2/sqrt(abs(b)) + b*sqrt(abs(b)))*d) + 3/8*sqrt(2)*sqrt(pi)* 
b^3*erf(-1/2*I*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(abs(b)) - 1/2*sqrt 
(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(I*a/b)/((I*b^2/sqrt(abs 
(b)) + b*sqrt(abs(b)))*d) + 1/2*I*sqrt(2)*sqrt(pi)*a*b^2*erf(1/2*I*sqrt(2) 
*sqrt(b*arcsin(d*x + c) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x 
+ c) + a)*sqrt(abs(b))/b)*e^(-I*a/b)/((-I*b^2/sqrt(abs(b)) + b*sqrt(abs(b) 
))*d) + 3/8*sqrt(2)*sqrt(pi)*b^3*erf(1/2*I*sqrt(2)*sqrt(b*arcsin(d*x + ...
 

3.3.48.9 Mupad [F(-1)]

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

input

int((a + b*asin(c + d*x))^(3/2),x)
 

output

int((a + b*asin(c + d*x))^(3/2), x)