3.7.91 \(\int (d+e x^2) (a+b \arcsin (c x))^{3/2} \, dx\) [691]

3.7.91.1 Optimal result

Integrand size = 20, antiderivative size = 482 \[ \int \left (d+e x^2\right ) (a+b \arcsin (c x))^{3/2} \, dx=\frac {3 b d \sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}{2 c}+\frac {b e \sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}{3 c^3}+\frac {b e x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}{6 c}+d x (a+b \arcsin (c x))^{3/2}+\frac {1}{3} e x^3 (a+b \arcsin (c x))^{3/2}-\frac {3 b^{3/2} d \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{2 c}-\frac {3 b^{3/2} e \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{8 c^3}+\frac {b^{3/2} e \sqrt {\frac {\pi }{6}} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{24 c^3}-\frac {3 b^{3/2} d \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{2 c}-\frac {3 b^{3/2} e \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{8 c^3}+\frac {b^{3/2} e \sqrt {\frac {\pi }{6}} \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{24 c^3} \]

output

d*x*(a+b*arcsin(c*x))^(3/2)+1/3*e*x^3*(a+b*arcsin(c*x))^(3/2)+1/144*b^(3/2 
)*e*cos(3*a/b)*FresnelC(6^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))* 
6^(1/2)*Pi^(1/2)/c^3+1/144*b^(3/2)*e*FresnelS(6^(1/2)/Pi^(1/2)*(a+b*arcsin 
(c*x))^(1/2)/b^(1/2))*sin(3*a/b)*6^(1/2)*Pi^(1/2)/c^3-3/4*b^(3/2)*d*cos(a/ 
b)*FresnelC(2^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))*2^(1/2)*Pi^( 
1/2)/c-3/16*b^(3/2)*e*cos(a/b)*FresnelC(2^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x)) 
^(1/2)/b^(1/2))*2^(1/2)*Pi^(1/2)/c^3-3/4*b^(3/2)*d*FresnelS(2^(1/2)/Pi^(1/ 
2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))*sin(a/b)*2^(1/2)*Pi^(1/2)/c-3/16*b^(3/ 
2)*e*FresnelS(2^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))*sin(a/b)*2 
^(1/2)*Pi^(1/2)/c^3+3/2*b*d*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))^(1/2)/c+1 
/3*b*e*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))^(1/2)/c^3+1/6*b*e*x^2*(-c^2*x^ 
2+1)^(1/2)*(a+b*arcsin(c*x))^(1/2)/c
 

3.7.91.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 8.61 (sec) , antiderivative size = 844, normalized size of antiderivative = 1.75 \[ \int \left (d+e x^2\right ) (a+b \arcsin (c x))^{3/2} \, dx=\frac {a b d e^{-\frac {i a}{b}} \left (\sqrt {-\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {3}{2},-\frac {i (a+b \arcsin (c x))}{b}\right )+e^{\frac {2 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {3}{2},\frac {i (a+b \arcsin (c x))}{b}\right )\right )}{2 c \sqrt {a+b \arcsin (c x)}}+\frac {a b e e^{-\frac {3 i a}{b}} \left (9 e^{\frac {2 i a}{b}} \sqrt {-\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {3}{2},-\frac {i (a+b \arcsin (c x))}{b}\right )+9 e^{\frac {4 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {3}{2},\frac {i (a+b \arcsin (c x))}{b}\right )-\sqrt {3} \left (\sqrt {-\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {3}{2},-\frac {3 i (a+b \arcsin (c x))}{b}\right )+e^{\frac {6 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {3}{2},\frac {3 i (a+b \arcsin (c x))}{b}\right )\right )\right )}{72 c^3 \sqrt {a+b \arcsin (c x)}}+\frac {\sqrt {b} d \left (2 \sqrt {b} \sqrt {a+b \arcsin (c x)} \left (3 \sqrt {1-c^2 x^2}+2 c x \arcsin (c x)\right )-\sqrt {2 \pi } \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c 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 x)}}{\sqrt {b}}\right ) \left (2 a \cos \left (\frac {a}{b}\right )-3 b \sin \left (\frac {a}{b}\right )\right )\right )}{4 c}+\frac {\sqrt {b} e \left (18 \sqrt {b} \sqrt {a+b \arcsin (c x)} \left (3 \sqrt {1-c^2 x^2}+2 c x \arcsin (c x)\right )-9 \sqrt {2 \pi } \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \left (3 b \cos \left (\frac {a}{b}\right )+2 a \sin \left (\frac {a}{b}\right )\right )+9 \sqrt {2 \pi } \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \left (2 a \cos \left (\frac {a}{b}\right )-3 b \sin \left (\frac {a}{b}\right )\right )+\sqrt {6 \pi } \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \left (b \cos \left (\frac {3 a}{b}\right )+2 a \sin \left (\frac {3 a}{b}\right )\right )-\sqrt {6 \pi } \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \left (2 a \cos \left (\frac {3 a}{b}\right )-b \sin \left (\frac {3 a}{b}\right )\right )-6 \sqrt {b} \sqrt {a+b \arcsin (c x)} (\cos (3 \arcsin (c x))+2 \arcsin (c x) \sin (3 \arcsin (c x)))\right )}{144 c^3} \]

input

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

output

(a*b*d*(Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[3/2, ((-I)*(a + b*ArcSin[ 
c*x]))/b] + E^(((2*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[3/2, (I* 
(a + b*ArcSin[c*x]))/b]))/(2*c*E^((I*a)/b)*Sqrt[a + b*ArcSin[c*x]]) + (a*b 
*e*(9*E^(((2*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[3/2, ((-I)* 
(a + b*ArcSin[c*x]))/b] + 9*E^(((4*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b 
]*Gamma[3/2, (I*(a + b*ArcSin[c*x]))/b] - Sqrt[3]*(Sqrt[((-I)*(a + b*ArcSi 
n[c*x]))/b]*Gamma[3/2, ((-3*I)*(a + b*ArcSin[c*x]))/b] + E^(((6*I)*a)/b)*S 
qrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[3/2, ((3*I)*(a + b*ArcSin[c*x]))/b])) 
)/(72*c^3*E^(((3*I)*a)/b)*Sqrt[a + b*ArcSin[c*x]]) + (Sqrt[b]*d*(2*Sqrt[b] 
*Sqrt[a + b*ArcSin[c*x]]*(3*Sqrt[1 - c^2*x^2] + 2*c*x*ArcSin[c*x]) - Sqrt[ 
2*Pi]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*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*x]] 
)/Sqrt[b]]*(2*a*Cos[a/b] - 3*b*Sin[a/b])))/(4*c) + (Sqrt[b]*e*(18*Sqrt[b]* 
Sqrt[a + b*ArcSin[c*x]]*(3*Sqrt[1 - c^2*x^2] + 2*c*x*ArcSin[c*x]) - 9*Sqrt 
[2*Pi]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*(3*b*Cos[a/b 
] + 2*a*Sin[a/b]) + 9*Sqrt[2*Pi]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c* 
x]])/Sqrt[b]]*(2*a*Cos[a/b] - 3*b*Sin[a/b]) + Sqrt[6*Pi]*FresnelC[(Sqrt[6/ 
Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*(b*Cos[(3*a)/b] + 2*a*Sin[(3*a)/b]) 
- Sqrt[6*Pi]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*(2*a*C 
os[(3*a)/b] - b*Sin[(3*a)/b]) - 6*Sqrt[b]*Sqrt[a + b*ArcSin[c*x]]*(Cos[...
 

3.7.91.3 Rubi [A] (verified)

Time = 1.49 (sec) , antiderivative size = 482, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {5172, 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.

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

\(\Big \downarrow \) 5172

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} e \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{8 c^3}+\frac {\sqrt {\frac {\pi }{6}} b^{3/2} e \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{24 c^3}-\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} e \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{8 c^3}+\frac {\sqrt {\frac {\pi }{6}} b^{3/2} e \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{24 c^3}-\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} d \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{2 c}-\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} d \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{2 c}+\frac {3 b d \sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}{2 c}+\frac {b e x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}{6 c}+\frac {b e \sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}{3 c^3}+d x (a+b \arcsin (c x))^{3/2}+\frac {1}{3} e x^3 (a+b \arcsin (c x))^{3/2}\)

input

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

output

(3*b*d*Sqrt[1 - c^2*x^2]*Sqrt[a + b*ArcSin[c*x]])/(2*c) + (b*e*Sqrt[1 - c^ 
2*x^2]*Sqrt[a + b*ArcSin[c*x]])/(3*c^3) + (b*e*x^2*Sqrt[1 - c^2*x^2]*Sqrt[ 
a + b*ArcSin[c*x]])/(6*c) + d*x*(a + b*ArcSin[c*x])^(3/2) + (e*x^3*(a + b* 
ArcSin[c*x])^(3/2))/3 - (3*b^(3/2)*d*Sqrt[Pi/2]*Cos[a/b]*FresnelC[(Sqrt[2/ 
Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(2*c) - (3*b^(3/2)*e*Sqrt[Pi/2]*Cos 
[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(8*c^3) + (b 
^(3/2)*e*Sqrt[Pi/6]*Cos[(3*a)/b]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c* 
x]])/Sqrt[b]])/(24*c^3) - (3*b^(3/2)*d*Sqrt[Pi/2]*FresnelS[(Sqrt[2/Pi]*Sqr 
t[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/(2*c) - (3*b^(3/2)*e*Sqrt[Pi/2]*F 
resnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/(8*c^3) + 
(b^(3/2)*e*Sqrt[Pi/6]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b 
]]*Sin[(3*a)/b])/(24*c^3)
 

3.7.91.3.1 Defintions of rubi rules used

rule 2009

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

rule 5172

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x 
_Symbol] :> Int[ExpandIntegrand[(a + b*ArcSin[c*x])^n, (d + e*x^2)^p, x], x 
] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] && (G 
tQ[p, 0] || IGtQ[n, 0])
 

3.7.91.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(849\) vs. \(2(374)=748\).

Time = 0.42 (sec) , antiderivative size = 850, normalized size of antiderivative = 1.76

method	result	size
default	\(\text {Expression too large to display}\)	\(850\)

input

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

output

1/144/c^3*(-108*(-1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)*Pi^(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)*2^(1/2)*b^2 
*c^2*d+108*(-1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)*Pi^(1/2)*sin(a/b)*FresnelS 
(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*2^(1/2)*b^2*c^2* 
d+(-3/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)*Pi^(1/2)*cos(3*a/b)*FresnelC(3*2^(1 
/2)/Pi^(1/2)/(-3/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*2^(1/2)*b^2*e-(-3/b)^ 
(1/2)*(a+b*arcsin(c*x))^(1/2)*Pi^(1/2)*sin(3*a/b)*FresnelS(3*2^(1/2)/Pi^(1 
/2)/(-3/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*2^(1/2)*b^2*e-27*(-1/b)^(1/2)* 
(a+b*arcsin(c*x))^(1/2)*Pi^(1/2)*cos(a/b)*FresnelC(2^(1/2)/Pi^(1/2)/(-1/b) 
^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*2^(1/2)*b^2*e+27*(-1/b)^(1/2)*(a+b*arcsi 
n(c*x))^(1/2)*Pi^(1/2)*sin(a/b)*FresnelS(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+ 
b*arcsin(c*x))^(1/2)/b)*2^(1/2)*b^2*e-144*arcsin(c*x)^2*sin(-(a+b*arcsin(c 
*x))/b+a/b)*b^2*c^2*d-288*arcsin(c*x)*sin(-(a+b*arcsin(c*x))/b+a/b)*a*b*c^ 
2*d+216*arcsin(c*x)*cos(-(a+b*arcsin(c*x))/b+a/b)*b^2*c^2*d+12*arcsin(c*x) 
^2*sin(-3*(a+b*arcsin(c*x))/b+3*a/b)*b^2*e-36*arcsin(c*x)^2*sin(-(a+b*arcs 
in(c*x))/b+a/b)*b^2*e-144*sin(-(a+b*arcsin(c*x))/b+a/b)*a^2*c^2*d+216*cos( 
-(a+b*arcsin(c*x))/b+a/b)*a*b*c^2*d+24*arcsin(c*x)*sin(-3*(a+b*arcsin(c*x) 
)/b+3*a/b)*a*b*e-6*arcsin(c*x)*cos(-3*(a+b*arcsin(c*x))/b+3*a/b)*b^2*e-72* 
arcsin(c*x)*sin(-(a+b*arcsin(c*x))/b+a/b)*a*b*e+54*arcsin(c*x)*cos(-(a+b*a 
rcsin(c*x))/b+a/b)*b^2*e+12*sin(-3*(a+b*arcsin(c*x))/b+3*a/b)*a^2*e-6*c...
 

3.7.91.5 Fricas [F(-2)]

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

input

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

output

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

3.7.91.6 Sympy [F]

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

input

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

output

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

3.7.91.7 Maxima [F]

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

input

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

output

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

3.7.91.8 Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 2.04 (sec) , antiderivative size = 2814, normalized size of antiderivative = 5.84 \[ \int \left (d+e x^2\right ) (a+b \arcsin (c x))^{3/2} \, dx=\text {Too large to display} \]

input

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

output

1/96*(48*sqrt(2)*sqrt(pi)*a^2*b*c^2*d*erf(-1/2*I*sqrt(2)*sqrt(b*arcsin(c*x 
) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*sqrt(abs(b))/b)* 
e^(I*a/b)/(I*b^2/sqrt(abs(b)) + b*sqrt(abs(b))) + 48*I*sqrt(2)*sqrt(pi)*a* 
b^2*c^2*d*erf(-1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sq 
rt(2)*sqrt(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(I*a/b)/(I*b^2/sqrt(abs(b) 
) + b*sqrt(abs(b))) + 48*sqrt(2)*sqrt(pi)*a^2*b*c^2*d*erf(1/2*I*sqrt(2)*sq 
rt(b*arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*s 
qrt(abs(b))/b)*e^(-I*a/b)/(-I*b^2/sqrt(abs(b)) + b*sqrt(abs(b))) - 48*I*sq 
rt(2)*sqrt(pi)*a*b^2*c^2*d*erf(1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqrt( 
abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(-I*a/b)/( 
-I*b^2/sqrt(abs(b)) + b*sqrt(abs(b))) - 48*I*sqrt(2)*sqrt(pi)*a*b*c^2*d*er 
f(-1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b 
*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(I*a/b)/(I*b/sqrt(abs(b)) + sqrt(abs(b 
))) + 36*sqrt(2)*sqrt(pi)*b^2*c^2*d*erf(-1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) 
+ a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^ 
(I*a/b)/(I*b/sqrt(abs(b)) + sqrt(abs(b))) + 48*I*sqrt(2)*sqrt(pi)*a*b*c^2* 
d*erf(1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqr 
t(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(-I*a/b)/(-I*b/sqrt(abs(b)) + sqrt( 
abs(b))) + 36*sqrt(2)*sqrt(pi)*b^2*c^2*d*erf(1/2*I*sqrt(2)*sqrt(b*arcsin(c 
*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*sqrt(abs(b)...
 

3.7.91.9 Mupad [F(-1)]

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

input

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

output

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