\(\int \frac {(d-c^2 d x^2)^3}{\sqrt {a+b \arcsin (c x)}} \, dx\) [90]

Optimal result

Integrand size = 26, antiderivative size = 471 \[ \int \frac {\left (d-c^2 d x^2\right )^3}{\sqrt {a+b \arcsin (c x)}} \, dx=\frac {35 d^3 \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{32 \sqrt {b} c}+\frac {7 d^3 \sqrt {\frac {3 \pi }{2}} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{32 \sqrt {b} c}+\frac {7 d^3 \sqrt {\frac {\pi }{10}} \cos \left (\frac {5 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{32 \sqrt {b} c}+\frac {d^3 \sqrt {\frac {\pi }{14}} \cos \left (\frac {7 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {14}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{32 \sqrt {b} c}+\frac {35 d^3 \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{32 \sqrt {b} c}+\frac {7 d^3 \sqrt {\frac {3 \pi }{2}} \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{32 \sqrt {b} c}+\frac {7 d^3 \sqrt {\frac {\pi }{10}} \operatorname {FresnelS}\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {5 a}{b}\right )}{32 \sqrt {b} c}+\frac {d^3 \sqrt {\frac {\pi }{14}} \operatorname {FresnelS}\left (\frac {\sqrt {\frac {14}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {7 a}{b}\right )}{32 \sqrt {b} c} \] Output:

35/64*d^3*2^(1/2)*Pi^(1/2)*cos(a/b)*FresnelC(2^(1/2)/Pi^(1/2)*(a+b*arcsin( 
c*x))^(1/2)/b^(1/2))/b^(1/2)/c+7/64*d^3*6^(1/2)*Pi^(1/2)*cos(3*a/b)*Fresne 
lC(6^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))/b^(1/2)/c+7/320*d^3*1 
0^(1/2)*Pi^(1/2)*cos(5*a/b)*FresnelC(10^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^( 
1/2)/b^(1/2))/b^(1/2)/c+1/448*d^3*14^(1/2)*Pi^(1/2)*cos(7*a/b)*FresnelC(14 
^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))/b^(1/2)/c+35/64*d^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^(1/2)/c+7/64*d^3*6^(1/2)*Pi^(1/2)*FresnelS(6^(1/2)/Pi^(1/2)*(a+b*a 
rcsin(c*x))^(1/2)/b^(1/2))*sin(3*a/b)/b^(1/2)/c+7/320*d^3*10^(1/2)*Pi^(1/2 
)*FresnelS(10^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))*sin(5*a/b)/b 
^(1/2)/c+1/448*d^3*14^(1/2)*Pi^(1/2)*FresnelS(14^(1/2)/Pi^(1/2)*(a+b*arcsi 
n(c*x))^(1/2)/b^(1/2))*sin(7*a/b)/b^(1/2)/c
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.49 (sec) , antiderivative size = 455, normalized size of antiderivative = 0.97 \[ \int \frac {\left (d-c^2 d x^2\right )^3}{\sqrt {a+b \arcsin (c x)}} \, dx=\frac {i d^3 e^{-\frac {7 i a}{b}} \left (-1225 e^{\frac {6 i a}{b}} \sqrt {-\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},-\frac {i (a+b \arcsin (c x))}{b}\right )+1225 e^{\frac {8 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},\frac {i (a+b \arcsin (c x))}{b}\right )-245 \sqrt {3} e^{\frac {4 i a}{b}} \sqrt {-\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},-\frac {3 i (a+b \arcsin (c x))}{b}\right )+245 \sqrt {3} e^{\frac {10 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},\frac {3 i (a+b \arcsin (c x))}{b}\right )-49 \sqrt {5} e^{\frac {2 i a}{b}} \sqrt {-\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},-\frac {5 i (a+b \arcsin (c x))}{b}\right )+49 \sqrt {5} e^{\frac {12 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},\frac {5 i (a+b \arcsin (c x))}{b}\right )-5 \sqrt {7} \sqrt {-\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},-\frac {7 i (a+b \arcsin (c x))}{b}\right )+5 \sqrt {7} e^{\frac {14 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},\frac {7 i (a+b \arcsin (c x))}{b}\right )\right )}{4480 c \sqrt {a+b \arcsin (c x)}} \] Input:

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

Output:

((I/4480)*d^3*(-1225*E^(((6*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Ga 
mma[1/2, ((-I)*(a + b*ArcSin[c*x]))/b] + 1225*E^(((8*I)*a)/b)*Sqrt[(I*(a + 
 b*ArcSin[c*x]))/b]*Gamma[1/2, (I*(a + b*ArcSin[c*x]))/b] - 245*Sqrt[3]*E^ 
(((4*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((-3*I)*(a + b 
*ArcSin[c*x]))/b] + 245*Sqrt[3]*E^(((10*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c*x 
]))/b]*Gamma[1/2, ((3*I)*(a + b*ArcSin[c*x]))/b] - 49*Sqrt[5]*E^(((2*I)*a) 
/b)*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((-5*I)*(a + b*ArcSin[c* 
x]))/b] + 49*Sqrt[5]*E^(((12*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamm 
a[1/2, ((5*I)*(a + b*ArcSin[c*x]))/b] - 5*Sqrt[7]*Sqrt[((-I)*(a + b*ArcSin 
[c*x]))/b]*Gamma[1/2, ((-7*I)*(a + b*ArcSin[c*x]))/b] + 5*Sqrt[7]*E^(((14* 
I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((7*I)*(a + b*ArcSin[c 
*x]))/b]))/(c*E^(((7*I)*a)/b)*Sqrt[a + b*ArcSin[c*x]])
 

Rubi [A] (verified)

Time = 0.91 (sec) , antiderivative size = 433, normalized size of antiderivative = 0.92, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {5168, 3042, 3793, 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.

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

\(\Big \downarrow \) 5168

\(\displaystyle \frac {d^3 \int \frac {\cos ^7\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))}{b c}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {d^3 \int \frac {\sin \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}+\frac {\pi }{2}\right )^7}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))}{b c}\)

\(\Big \downarrow \) 3793

\(\displaystyle \frac {d^3 \int \left (\frac {\cos \left (\frac {7 a}{b}-\frac {7 (a+b \arcsin (c x))}{b}\right )}{64 \sqrt {a+b \arcsin (c x)}}+\frac {7 \cos \left (\frac {5 a}{b}-\frac {5 (a+b \arcsin (c x))}{b}\right )}{64 \sqrt {a+b \arcsin (c x)}}+\frac {21 \cos \left (\frac {3 a}{b}-\frac {3 (a+b \arcsin (c x))}{b}\right )}{64 \sqrt {a+b \arcsin (c x)}}+\frac {35 \cos \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{64 \sqrt {a+b \arcsin (c x)}}\right )d(a+b \arcsin (c x))}{b c}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {d^3 \left (\frac {35}{32} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )+\frac {7}{32} \sqrt {\frac {3 \pi }{2}} \sqrt {b} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )+\frac {7}{32} \sqrt {\frac {\pi }{10}} \sqrt {b} \cos \left (\frac {5 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )+\frac {1}{32} \sqrt {\frac {\pi }{14}} \sqrt {b} \cos \left (\frac {7 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {14}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )+\frac {35}{32} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )+\frac {7}{32} \sqrt {\frac {3 \pi }{2}} \sqrt {b} \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )+\frac {7}{32} \sqrt {\frac {\pi }{10}} \sqrt {b} \sin \left (\frac {5 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )+\frac {1}{32} \sqrt {\frac {\pi }{14}} \sqrt {b} \sin \left (\frac {7 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {14}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )\right )}{b c}\)

Input:

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

Output:

(d^3*((35*Sqrt[b]*Sqrt[Pi/2]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcS 
in[c*x]])/Sqrt[b]])/32 + (7*Sqrt[b]*Sqrt[(3*Pi)/2]*Cos[(3*a)/b]*FresnelC[( 
Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/32 + (7*Sqrt[b]*Sqrt[Pi/10]* 
Cos[(5*a)/b]*FresnelC[(Sqrt[10/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/32 + 
 (Sqrt[b]*Sqrt[Pi/14]*Cos[(7*a)/b]*FresnelC[(Sqrt[14/Pi]*Sqrt[a + b*ArcSin 
[c*x]])/Sqrt[b]])/32 + (35*Sqrt[b]*Sqrt[Pi/2]*FresnelS[(Sqrt[2/Pi]*Sqrt[a 
+ b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/32 + (7*Sqrt[b]*Sqrt[(3*Pi)/2]*Fresne 
lS[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[(3*a)/b])/32 + (7*Sqr 
t[b]*Sqrt[Pi/10]*FresnelS[(Sqrt[10/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*S 
in[(5*a)/b])/32 + (Sqrt[b]*Sqrt[Pi/14]*FresnelS[(Sqrt[14/Pi]*Sqrt[a + b*Ar 
cSin[c*x]])/Sqrt[b]]*Sin[(7*a)/b])/32))/(b*c)
 

Defintions of rubi rules used

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 3793

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In 
t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f 
, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
 

rule 5168

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x 
_Symbol] :> Simp[(1/(b*c))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]   Subst[Int[ 
x^n*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, 0]
 

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 413, normalized size of antiderivative = 0.88

method	result	size
default	\(\frac {d^{3} \sqrt {\pi }\, \sqrt {2}\, \sqrt {-\frac {1}{b}}\, \left (1225 \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right )-1225 \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right )-245 \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {3 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b}\right ) \sqrt {-\frac {1}{b}}\, \sqrt {-\frac {3}{b}}\, b +245 \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {3 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b}\right ) \sqrt {-\frac {1}{b}}\, \sqrt {-\frac {3}{b}}\, b -49 \cos \left (\frac {5 a}{b}\right ) \operatorname {FresnelC}\left (\frac {5 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {5}{b}}\, b}\right ) \sqrt {-\frac {1}{b}}\, \sqrt {-\frac {5}{b}}\, b +49 \sin \left (\frac {5 a}{b}\right ) \operatorname {FresnelS}\left (\frac {5 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {5}{b}}\, b}\right ) \sqrt {-\frac {1}{b}}\, \sqrt {-\frac {5}{b}}\, b -5 \cos \left (\frac {7 a}{b}\right ) \operatorname {FresnelC}\left (\frac {7 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {7}{b}}\, b}\right ) \sqrt {-\frac {1}{b}}\, \sqrt {-\frac {7}{b}}\, b +5 \sin \left (\frac {7 a}{b}\right ) \operatorname {FresnelS}\left (\frac {7 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {7}{b}}\, b}\right ) \sqrt {-\frac {1}{b}}\, \sqrt {-\frac {7}{b}}\, b \right )}{2240 c}\)	\(413\)

Input:

int((-c^2*d*x^2+d)^3/(a+b*arcsin(c*x))^(1/2),x,method=_RETURNVERBOSE)
 

Output:

1/2240/c*d^3*Pi^(1/2)*2^(1/2)*(-1/b)^(1/2)*(1225*cos(a/b)*FresnelC(2^(1/2) 
/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)-1225*sin(a/b)*FresnelS(2 
^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)-245*cos(3*a/b)*Fre 
snelC(3*2^(1/2)/Pi^(1/2)/(-3/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*(-1/b)^(1 
/2)*(-3/b)^(1/2)*b+245*sin(3*a/b)*FresnelS(3*2^(1/2)/Pi^(1/2)/(-3/b)^(1/2) 
*(a+b*arcsin(c*x))^(1/2)/b)*(-1/b)^(1/2)*(-3/b)^(1/2)*b-49*cos(5*a/b)*Fres 
nelC(5*2^(1/2)/Pi^(1/2)/(-5/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*(-1/b)^(1/ 
2)*(-5/b)^(1/2)*b+49*sin(5*a/b)*FresnelS(5*2^(1/2)/Pi^(1/2)/(-5/b)^(1/2)*( 
a+b*arcsin(c*x))^(1/2)/b)*(-1/b)^(1/2)*(-5/b)^(1/2)*b-5*cos(7*a/b)*Fresnel 
C(7*2^(1/2)/Pi^(1/2)/(-7/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*(-1/b)^(1/2)* 
(-7/b)^(1/2)*b+5*sin(7*a/b)*FresnelS(7*2^(1/2)/Pi^(1/2)/(-7/b)^(1/2)*(a+b* 
arcsin(c*x))^(1/2)/b)*(-1/b)^(1/2)*(-7/b)^(1/2)*b)
 

Fricas [F(-2)]

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

integrate((-c^2*d*x^2+d)^3/(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)
 

Sympy [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^3}{\sqrt {a+b \arcsin (c x)}} \, dx=- d^{3} \left (\int \frac {3 c^{2} x^{2}}{\sqrt {a + b \operatorname {asin}{\left (c x \right )}}}\, dx + \int \left (- \frac {3 c^{4} x^{4}}{\sqrt {a + b \operatorname {asin}{\left (c x \right )}}}\right )\, dx + \int \frac {c^{6} x^{6}}{\sqrt {a + b \operatorname {asin}{\left (c x \right )}}}\, dx + \int \left (- \frac {1}{\sqrt {a + b \operatorname {asin}{\left (c x \right )}}}\right )\, dx\right ) \] Input:

integrate((-c**2*d*x**2+d)**3/(a+b*asin(c*x))**(1/2),x)
 

Output:

-d**3*(Integral(3*c**2*x**2/sqrt(a + b*asin(c*x)), x) + Integral(-3*c**4*x 
**4/sqrt(a + b*asin(c*x)), x) + Integral(c**6*x**6/sqrt(a + b*asin(c*x)), 
x) + Integral(-1/sqrt(a + b*asin(c*x)), x))
 

Maxima [F]

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

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

Output:

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

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.72 (sec) , antiderivative size = 657, normalized size of antiderivative = 1.39 \[ \int \frac {\left (d-c^2 d x^2\right )^3}{\sqrt {a+b \arcsin (c x)}} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/64*sqrt(pi)*d^3*erf(-1/2*sqrt(14)*sqrt(b*arcsin(c*x) + a)/sqrt(b) - 1/2 
*I*sqrt(14)*sqrt(b*arcsin(c*x) + a)*sqrt(b)/abs(b))*e^(7*I*a/b)/((sqrt(14) 
*sqrt(b) + I*sqrt(14)*b^(3/2)/abs(b))*c) - 7/64*sqrt(pi)*d^3*erf(-1/2*sqrt 
(10)*sqrt(b*arcsin(c*x) + a)/sqrt(b) - 1/2*I*sqrt(10)*sqrt(b*arcsin(c*x) + 
 a)*sqrt(b)/abs(b))*e^(5*I*a/b)/((sqrt(10)*sqrt(b) + I*sqrt(10)*b^(3/2)/ab 
s(b))*c) - 21/64*sqrt(pi)*d^3*erf(-1/2*sqrt(6)*sqrt(b*arcsin(c*x) + a)/sqr 
t(b) - 1/2*I*sqrt(6)*sqrt(b*arcsin(c*x) + a)*sqrt(b)/abs(b))*e^(3*I*a/b)/( 
(sqrt(6)*sqrt(b) + I*sqrt(6)*b^(3/2)/abs(b))*c) - 35/64*sqrt(pi)*d^3*erf(- 
1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*ar 
csin(c*x) + a)*sqrt(abs(b))/b)*e^(I*a/b)/(c*(I*sqrt(2)*b/sqrt(abs(b)) + sq 
rt(2)*sqrt(abs(b)))) - 35/64*sqrt(pi)*d^3*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)/(c*(-I*sqrt(2)*b/sqrt(abs(b)) + sqrt(2)*sqrt(abs(b)))) - 21/ 
64*sqrt(pi)*d^3*erf(-1/2*sqrt(6)*sqrt(b*arcsin(c*x) + a)/sqrt(b) + 1/2*I*s 
qrt(6)*sqrt(b*arcsin(c*x) + a)*sqrt(b)/abs(b))*e^(-3*I*a/b)/((sqrt(6)*sqrt 
(b) - I*sqrt(6)*b^(3/2)/abs(b))*c) - 7/64*sqrt(pi)*d^3*erf(-1/2*sqrt(10)*s 
qrt(b*arcsin(c*x) + a)/sqrt(b) + 1/2*I*sqrt(10)*sqrt(b*arcsin(c*x) + a)*sq 
rt(b)/abs(b))*e^(-5*I*a/b)/((sqrt(10)*sqrt(b) - I*sqrt(10)*b^(3/2)/abs(b)) 
*c) - 1/64*sqrt(pi)*d^3*erf(-1/2*sqrt(14)*sqrt(b*arcsin(c*x) + a)/sqrt(b) 
+ 1/2*I*sqrt(14)*sqrt(b*arcsin(c*x) + a)*sqrt(b)/abs(b))*e^(-7*I*a/b)/(...
 

Mupad [F(-1)]

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

int((d - c^2*d*x^2)^3/(a + b*asin(c*x))^(1/2),x)
 

Output:

int((d - c^2*d*x^2)^3/(a + b*asin(c*x))^(1/2), x)
 

Reduce [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^3}{\sqrt {a+b \arcsin (c x)}} \, dx=d^{3} \left (\int \frac {\sqrt {\mathit {asin} \left (c x \right ) b +a}}{\mathit {asin} \left (c x \right ) b +a}d x -\left (\int \frac {\sqrt {\mathit {asin} \left (c x \right ) b +a}\, x^{6}}{\mathit {asin} \left (c x \right ) b +a}d x \right ) c^{6}+3 \left (\int \frac {\sqrt {\mathit {asin} \left (c x \right ) b +a}\, x^{4}}{\mathit {asin} \left (c x \right ) b +a}d x \right ) c^{4}-3 \left (\int \frac {\sqrt {\mathit {asin} \left (c x \right ) b +a}\, x^{2}}{\mathit {asin} \left (c x \right ) b +a}d x \right ) c^{2}\right ) \] Input:

int((-c^2*d*x^2+d)^3/(a+b*asin(c*x))^(1/2),x)
                                                                                    
                                                                                    
 

Output:

d**3*(int(sqrt(asin(c*x)*b + a)/(asin(c*x)*b + a),x) - int((sqrt(asin(c*x) 
*b + a)*x**6)/(asin(c*x)*b + a),x)*c**6 + 3*int((sqrt(asin(c*x)*b + a)*x** 
4)/(asin(c*x)*b + a),x)*c**4 - 3*int((sqrt(asin(c*x)*b + a)*x**2)/(asin(c* 
x)*b + a),x)*c**2)