3.7.86 \(\int (d+e x^2)^2 \sqrt {a+b \arcsin (c x)} \, dx\) [686]

3.7.86.1 Optimal result

Integrand size = 22, antiderivative size = 754 \[ \int \left (d+e x^2\right )^2 \sqrt {a+b \arcsin (c x)} \, dx=d^2 x \sqrt {a+b \arcsin (c x)}+\frac {2}{3} d e x^3 \sqrt {a+b \arcsin (c x)}+\frac {1}{5} e^2 x^5 \sqrt {a+b \arcsin (c x)}-\frac {\sqrt {b} d^2 \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{c}-\frac {\sqrt {b} d e \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{2 c^3}-\frac {\sqrt {b} e^2 \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{8 c^5}+\frac {\sqrt {b} d e \sqrt {\frac {\pi }{6}} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{6 c^3}+\frac {\sqrt {b} e^2 \sqrt {\frac {\pi }{6}} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{16 c^5}-\frac {\sqrt {b} e^2 \sqrt {\frac {\pi }{10}} \cos \left (\frac {5 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{80 c^5}+\frac {\sqrt {b} d^2 \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{c}+\frac {\sqrt {b} d e \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{2 c^3}+\frac {\sqrt {b} e^2 \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{8 c^5}-\frac {\sqrt {b} d e \sqrt {\frac {\pi }{6}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{6 c^3}-\frac {\sqrt {b} e^2 \sqrt {\frac {\pi }{6}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{16 c^5}+\frac {\sqrt {b} e^2 \sqrt {\frac {\pi }{10}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {5 a}{b}\right )}{80 c^5} \]

output

-1/800*e^2*cos(5*a/b)*FresnelS(10^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b 
^(1/2))*b^(1/2)*10^(1/2)*Pi^(1/2)/c^5+1/800*e^2*FresnelC(10^(1/2)/Pi^(1/2) 
*(a+b*arcsin(c*x))^(1/2)/b^(1/2))*sin(5*a/b)*b^(1/2)*10^(1/2)*Pi^(1/2)/c^5 
+1/36*d*e*cos(3*a/b)*FresnelS(6^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^( 
1/2))*b^(1/2)*6^(1/2)*Pi^(1/2)/c^3+1/96*e^2*cos(3*a/b)*FresnelS(6^(1/2)/Pi 
^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))*b^(1/2)*6^(1/2)*Pi^(1/2)/c^5-1/36* 
d*e*FresnelC(6^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))*sin(3*a/b)* 
b^(1/2)*6^(1/2)*Pi^(1/2)/c^3-1/96*e^2*FresnelC(6^(1/2)/Pi^(1/2)*(a+b*arcsi 
n(c*x))^(1/2)/b^(1/2))*sin(3*a/b)*b^(1/2)*6^(1/2)*Pi^(1/2)/c^5-1/2*d^2*cos 
(a/b)*FresnelS(2^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))*b^(1/2)*2 
^(1/2)*Pi^(1/2)/c-1/4*d*e*cos(a/b)*FresnelS(2^(1/2)/Pi^(1/2)*(a+b*arcsin(c 
*x))^(1/2)/b^(1/2))*b^(1/2)*2^(1/2)*Pi^(1/2)/c^3-1/16*e^2*cos(a/b)*Fresnel 
S(2^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))*b^(1/2)*2^(1/2)*Pi^(1/ 
2)/c^5+1/2*d^2*FresnelC(2^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))* 
sin(a/b)*b^(1/2)*2^(1/2)*Pi^(1/2)/c+1/4*d*e*FresnelC(2^(1/2)/Pi^(1/2)*(a+b 
*arcsin(c*x))^(1/2)/b^(1/2))*sin(a/b)*b^(1/2)*2^(1/2)*Pi^(1/2)/c^3+1/16*e^ 
2*FresnelC(2^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))*sin(a/b)*b^(1 
/2)*2^(1/2)*Pi^(1/2)/c^5+d^2*x*(a+b*arcsin(c*x))^(1/2)+2/3*d*e*x^3*(a+b*ar 
csin(c*x))^(1/2)+1/5*e^2*x^5*(a+b*arcsin(c*x))^(1/2)
 

3.7.86.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.07 (sec) , antiderivative size = 400, normalized size of antiderivative = 0.53 \[ \int \left (d+e x^2\right )^2 \sqrt {a+b \arcsin (c x)} \, dx=\frac {b e^{-\frac {5 i a}{b}} \left (450 \left (8 c^4 d^2+4 c^2 d e+e^2\right ) e^{\frac {4 i a}{b}} \sqrt {-\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {3}{2},-\frac {i (a+b \arcsin (c x))}{b}\right )+450 \left (8 c^4 d^2+4 c^2 d e+e^2\right ) e^{\frac {6 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {3}{2},\frac {i (a+b \arcsin (c x))}{b}\right )-e \left (25 \sqrt {3} \left (8 c^2 d+3 e\right ) e^{\frac {2 i a}{b}} \sqrt {-\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {3}{2},-\frac {3 i (a+b \arcsin (c x))}{b}\right )+25 \sqrt {3} \left (8 c^2 d+3 e\right ) e^{\frac {8 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {3}{2},\frac {3 i (a+b \arcsin (c x))}{b}\right )-9 \sqrt {5} e \left (\sqrt {-\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {3}{2},-\frac {5 i (a+b \arcsin (c x))}{b}\right )+e^{\frac {10 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {3}{2},\frac {5 i (a+b \arcsin (c x))}{b}\right )\right )\right )\right )}{7200 c^5 \sqrt {a+b \arcsin (c x)}} \]

input

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

output

(b*(450*(8*c^4*d^2 + 4*c^2*d*e + e^2)*E^(((4*I)*a)/b)*Sqrt[((-I)*(a + b*Ar 
cSin[c*x]))/b]*Gamma[3/2, ((-I)*(a + b*ArcSin[c*x]))/b] + 450*(8*c^4*d^2 + 
 4*c^2*d*e + e^2)*E^(((6*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[3/ 
2, (I*(a + b*ArcSin[c*x]))/b] - e*(25*Sqrt[3]*(8*c^2*d + 3*e)*E^(((2*I)*a) 
/b)*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[3/2, ((-3*I)*(a + b*ArcSin[c* 
x]))/b] + 25*Sqrt[3]*(8*c^2*d + 3*e)*E^(((8*I)*a)/b)*Sqrt[(I*(a + b*ArcSin 
[c*x]))/b]*Gamma[3/2, ((3*I)*(a + b*ArcSin[c*x]))/b] - 9*Sqrt[5]*e*(Sqrt[( 
(-I)*(a + b*ArcSin[c*x]))/b]*Gamma[3/2, ((-5*I)*(a + b*ArcSin[c*x]))/b] + 
E^(((10*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[3/2, ((5*I)*(a + b* 
ArcSin[c*x]))/b]))))/(7200*c^5*E^(((5*I)*a)/b)*Sqrt[a + b*ArcSin[c*x]])
 

3.7.86.3 Rubi [A] (verified)

Time = 2.21 (sec) , antiderivative size = 754, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, 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.

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

\(\Big \downarrow \) 5172

\(\displaystyle \int \left (d^2 \sqrt {a+b \arcsin (c x)}+2 d e x^2 \sqrt {a+b \arcsin (c x)}+e^2 x^4 \sqrt {a+b \arcsin (c x)}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {\frac {\pi }{2}} \sqrt {b} e^2 \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{8 c^5}-\frac {\sqrt {\frac {\pi }{6}} \sqrt {b} e^2 \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{16 c^5}+\frac {\sqrt {\frac {\pi }{10}} \sqrt {b} e^2 \sin \left (\frac {5 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{80 c^5}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} e^2 \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{8 c^5}+\frac {\sqrt {\frac {\pi }{6}} \sqrt {b} e^2 \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{16 c^5}-\frac {\sqrt {\frac {\pi }{10}} \sqrt {b} e^2 \cos \left (\frac {5 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{80 c^5}+\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} d e \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{2 c^3}-\frac {\sqrt {\frac {\pi }{6}} \sqrt {b} d e \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{6 c^3}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} d e \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{2 c^3}+\frac {\sqrt {\frac {\pi }{6}} \sqrt {b} d e \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{6 c^3}+\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} d^2 \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{c}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} d^2 \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{c}+d^2 x \sqrt {a+b \arcsin (c x)}+\frac {2}{3} d e x^3 \sqrt {a+b \arcsin (c x)}+\frac {1}{5} e^2 x^5 \sqrt {a+b \arcsin (c x)}\)

input

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

output

d^2*x*Sqrt[a + b*ArcSin[c*x]] + (2*d*e*x^3*Sqrt[a + b*ArcSin[c*x]])/3 + (e 
^2*x^5*Sqrt[a + b*ArcSin[c*x]])/5 - (Sqrt[b]*d^2*Sqrt[Pi/2]*Cos[a/b]*Fresn 
elS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/c - (Sqrt[b]*d*e*Sqrt[P 
i/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(2*c 
^3) - (Sqrt[b]*e^2*Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*Arc 
Sin[c*x]])/Sqrt[b]])/(8*c^5) + (Sqrt[b]*d*e*Sqrt[Pi/6]*Cos[(3*a)/b]*Fresne 
lS[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(6*c^3) + (Sqrt[b]*e^2*S 
qrt[Pi/6]*Cos[(3*a)/b]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[ 
b]])/(16*c^5) - (Sqrt[b]*e^2*Sqrt[Pi/10]*Cos[(5*a)/b]*FresnelS[(Sqrt[10/Pi 
]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(80*c^5) + (Sqrt[b]*d^2*Sqrt[Pi/2]*Fr 
esnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/c + (Sqrt[b 
]*d*e*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Si 
n[a/b])/(2*c^3) + (Sqrt[b]*e^2*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b* 
ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/(8*c^5) - (Sqrt[b]*d*e*Sqrt[Pi/6]*Fresnel 
C[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[(3*a)/b])/(6*c^3) - (S 
qrt[b]*e^2*Sqrt[Pi/6]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b 
]]*Sin[(3*a)/b])/(16*c^5) + (Sqrt[b]*e^2*Sqrt[Pi/10]*FresnelC[(Sqrt[10/Pi] 
*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[(5*a)/b])/(80*c^5)
 

3.7.86.3.1 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])
 

3.7.86.4 Maple [A] (verified)

Time = 0.88 (sec) , antiderivative size = 1155, normalized size of antiderivative = 1.53

method	result	size
default	\(\text {Expression too large to display}\)	\(1155\)

input

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

output

-1/7200/c^5*(-9*cos(5*a/b)*Pi^(1/2)*(-5/b)^(1/2)*2^(1/2)*FresnelS(5*2^(1/2 
)/Pi^(1/2)/(-5/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*(a+b*arcsin(c*x))^(1/2) 
*b*e^2-9*Pi^(1/2)*(-5/b)^(1/2)*2^(1/2)*sin(5*a/b)*FresnelC(5*2^(1/2)/Pi^(1 
/2)/(-5/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*(a+b*arcsin(c*x))^(1/2)*b*e^2+ 
90*arcsin(c*x)*sin(-5*(a+b*arcsin(c*x))/b+5*a/b)*b*e^2+90*sin(-5*(a+b*arcs 
in(c*x))/b+5*a/b)*a*e^2-3600*Pi^(1/2)*2^(1/2)*cos(a/b)*FresnelS(2^(1/2)/Pi 
^(1/2)/(-1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*(a+b*arcsin(c*x))^(1/2)*(-1 
/b)^(1/2)*b*c^4*d^2-3600*Pi^(1/2)*2^(1/2)*sin(a/b)*FresnelC(2^(1/2)/Pi^(1/ 
2)/(-1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*(a+b*arcsin(c*x))^(1/2)*(-1/b)^ 
(1/2)*b*c^4*d^2-1800*Pi^(1/2)*2^(1/2)*cos(a/b)*FresnelS(2^(1/2)/Pi^(1/2)/( 
-1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*(a+b*arcsin(c*x))^(1/2)*(-1/b)^(1/2 
)*b*c^2*d*e-1800*Pi^(1/2)*2^(1/2)*sin(a/b)*FresnelC(2^(1/2)/Pi^(1/2)/(-1/b 
)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*(a+b*arcsin(c*x))^(1/2)*(-1/b)^(1/2)*b* 
c^2*d*e+7200*arcsin(c*x)*sin(-(a+b*arcsin(c*x))/b+a/b)*b*c^4*d^2-450*Pi^(1 
/2)*2^(1/2)*cos(a/b)*FresnelS(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arcsin(c* 
x))^(1/2)/b)*(a+b*arcsin(c*x))^(1/2)*(-1/b)^(1/2)*b*e^2-450*Pi^(1/2)*2^(1/ 
2)*sin(a/b)*FresnelC(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2) 
/b)*(a+b*arcsin(c*x))^(1/2)*(-1/b)^(1/2)*b*e^2+7200*sin(-(a+b*arcsin(c*x)) 
/b+a/b)*a*c^4*d^2+3600*arcsin(c*x)*sin(-(a+b*arcsin(c*x))/b+a/b)*b*c^2*d*e 
+3600*sin(-(a+b*arcsin(c*x))/b+a/b)*a*c^2*d*e+900*arcsin(c*x)*sin(-(a+b...
 

3.7.86.5 Fricas [F(-2)]

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

input

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

output

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

3.7.86.6 Sympy [F]

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

input

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

output

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

3.7.86.7 Maxima [F]

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

input

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

output

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

3.7.86.8 Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.86 (sec) , antiderivative size = 3216, normalized size of antiderivative = 4.27 \[ \int \left (d+e x^2\right )^2 \sqrt {a+b \arcsin (c x)} \, dx=\text {Too large to display} \]

input

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

output

1/480*(240*sqrt(2)*sqrt(pi)*a*b*c^4*d^2*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))) + 120*I*sqrt(2)*sqrt(pi) 
*b^2*c^4*d^2*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))) + 240*sqrt(2)*sqrt(pi)*a*b*c^4*d^2*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))) - 120 
*I*sqrt(2)*sqrt(pi)*b^2*c^4*d^2*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))) - 480*sqrt(pi)*a*c^4*d^2*erf(-1 
/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arc 
sin(c*x) + a)*sqrt(abs(b))/b)*e^(I*a/b)/(I*sqrt(2)*b/sqrt(abs(b)) + sqrt(2 
)*sqrt(abs(b))) - 480*sqrt(pi)*a*c^4*d^2*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*sqrt(2)*b/sqrt(abs(b)) + sqrt(2)*sqrt(abs(b))) + 120*sqrt 
(2)*sqrt(pi)*a*b*c^2*d*e*erf(-1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqrt(a 
bs(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))) + 60*I*sqrt(2)*sqrt(pi)*b^2*c^2*d*e*erf 
(-1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt...
 

3.7.86.9 Mupad [F(-1)]

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

input

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

output

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