\(\int x^3 (d-c^2 d x^2)^{3/2} (a+b \arcsin (c x)) \, dx\) [79]

Optimal result

Integrand size = 27, antiderivative size = 227 \[ \int x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x)) \, dx=\frac {2 b d x \sqrt {d-c^2 d x^2}}{35 c^3 \sqrt {1-c^2 x^2}}+\frac {b d x^3 \sqrt {d-c^2 d x^2}}{105 c \sqrt {1-c^2 x^2}}-\frac {8 b c d x^5 \sqrt {d-c^2 d x^2}}{175 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^4 d}+\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^4 d^2} \] Output:

2/35*b*d*x*(-c^2*d*x^2+d)^(1/2)/c^3/(-c^2*x^2+1)^(1/2)+1/105*b*d*x^3*(-c^2 
*d*x^2+d)^(1/2)/c/(-c^2*x^2+1)^(1/2)-8/175*b*c*d*x^5*(-c^2*d*x^2+d)^(1/2)/ 
(-c^2*x^2+1)^(1/2)+1/49*b*c^3*d*x^7*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2 
)-1/5*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))/c^4/d+1/7*(-c^2*d*x^2+d)^(7/2 
)*(a+b*arcsin(c*x))/c^4/d^2
 

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.56 \[ \int x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x)) \, dx=\frac {d \sqrt {d-c^2 d x^2} \left (-105 a \left (1-c^2 x^2\right )^{5/2} \left (2+5 c^2 x^2\right )+b c x \left (210+35 c^2 x^2-168 c^4 x^4+75 c^6 x^6\right )-105 b \left (1-c^2 x^2\right )^{5/2} \left (2+5 c^2 x^2\right ) \arcsin (c x)\right )}{3675 c^4 \sqrt {1-c^2 x^2}} \] Input:

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

Output:

(d*Sqrt[d - c^2*d*x^2]*(-105*a*(1 - c^2*x^2)^(5/2)*(2 + 5*c^2*x^2) + b*c*x 
*(210 + 35*c^2*x^2 - 168*c^4*x^4 + 75*c^6*x^6) - 105*b*(1 - c^2*x^2)^(5/2) 
*(2 + 5*c^2*x^2)*ArcSin[c*x]))/(3675*c^4*Sqrt[1 - c^2*x^2])
 

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.61, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {5194, 27, 290, 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.

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

\(\Big \downarrow \) 5194

\(\displaystyle -\frac {b c \sqrt {d-c^2 d x^2} \int -\frac {d \left (1-c^2 x^2\right )^2 \left (5 c^2 x^2+2\right )}{35 c^4}dx}{\sqrt {1-c^2 x^2}}+\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^4 d^2}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^4 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b d \sqrt {d-c^2 d x^2} \int \left (1-c^2 x^2\right )^2 \left (5 c^2 x^2+2\right )dx}{35 c^3 \sqrt {1-c^2 x^2}}+\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^4 d^2}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^4 d}\)

\(\Big \downarrow \) 290

\(\displaystyle \frac {b d \sqrt {d-c^2 d x^2} \int \left (5 c^6 x^6-8 c^4 x^4+c^2 x^2+2\right )dx}{35 c^3 \sqrt {1-c^2 x^2}}+\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^4 d^2}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^4 d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^4 d^2}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^4 d}+\frac {b d \left (\frac {5 c^6 x^7}{7}-\frac {8 c^4 x^5}{5}+\frac {c^2 x^3}{3}+2 x\right ) \sqrt {d-c^2 d x^2}}{35 c^3 \sqrt {1-c^2 x^2}}\)

Input:

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

Output:

(b*d*Sqrt[d - c^2*d*x^2]*(2*x + (c^2*x^3)/3 - (8*c^4*x^5)/5 + (5*c^6*x^7)/ 
7))/(35*c^3*Sqrt[1 - c^2*x^2]) - ((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x] 
))/(5*c^4*d) + ((d - c^2*d*x^2)^(7/2)*(a + b*ArcSin[c*x]))/(7*c^4*d^2)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 290

Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] :> I 
nt[ExpandIntegrand[(a + b*x^2)^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d 
}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && IGtQ[q, 0]
 

rule 2009

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

rule 5194

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_) 
, x_Symbol] :> With[{u = IntHide[x^m*(d + e*x^2)^p, x]}, Simp[(a + b*ArcSin 
[c*x])   u, x] - Simp[b*c*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]]   Int[Sim 
plifyIntegrand[u/Sqrt[d + e*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x] 
&& EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2] && NeQ[p, -2^(-1)] && (IGtQ[(m + 
1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])
 

Maple [A] (verified)

Time = 0.49 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.98

method	result	size
orering	\(\frac {\left (325 c^{8} x^{8}-866 c^{6} x^{6}+553 c^{4} x^{4}+420 c^{2} x^{2}-280\right ) \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \arcsin \left (c x \right )\right )}{1225 c^{4} \left (c x -1\right ) \left (c x +1\right ) \left (c^{2} x^{2}-1\right )}-\frac {\left (75 c^{6} x^{6}-168 c^{4} x^{4}+35 c^{2} x^{2}+210\right ) \left (3 x^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \arcsin \left (c x \right )\right )-3 x^{4} \sqrt {-c^{2} d \,x^{2}+d}\, \left (a +b \arcsin \left (c x \right )\right ) c^{2} d +\frac {x^{3} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} b c}{\sqrt {-c^{2} x^{2}+1}}\right )}{3675 x^{2} c^{4} \left (c x -1\right ) \left (c x +1\right )}\)	\(223\)
default	\(a \left (-\frac {x^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{7 c^{2} d}-\frac {2 \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{35 d \,c^{4}}\right )+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (64 c^{8} x^{8}-144 c^{6} x^{6}-64 i c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}+104 c^{4} x^{4}+112 i \sqrt {-c^{2} x^{2}+1}\, x^{5} c^{5}-25 c^{2} x^{2}-56 i \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}+7 i \sqrt {-c^{2} x^{2}+1}\, c x +1\right ) \left (i+7 \arcsin \left (c x \right )\right ) d}{6272 c^{4} \left (c^{2} x^{2}-1\right )}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i \sqrt {-c^{2} x^{2}+1}\, c x -1\right ) \left (\arcsin \left (c x \right )+i\right ) d}{128 c^{4} \left (c^{2} x^{2}-1\right )}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, c x +c^{2} x^{2}-1\right ) \left (\arcsin \left (c x \right )-i\right ) d}{128 c^{4} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 i \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}+4 c^{4} x^{4}-3 i \sqrt {-c^{2} x^{2}+1}\, c x -5 c^{2} x^{2}+1\right ) \left (-i+3 \arcsin \left (c x \right )\right ) d}{384 c^{4} \left (c^{2} x^{2}-1\right )}+\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, c x +c^{2} x^{2}-1\right ) \left (2 i+35 \arcsin \left (c x \right )\right ) \cos \left (6 \arcsin \left (c x \right )\right ) d}{39200 c^{4} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c^{2} x^{2}-c x \sqrt {-c^{2} x^{2}+1}-i\right ) \left (37 i+35 \arcsin \left (c x \right )\right ) \sin \left (6 \arcsin \left (c x \right )\right ) d}{78400 c^{4} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, c x +c^{2} x^{2}-1\right ) \left (7 i+15 \arcsin \left (c x \right )\right ) \cos \left (4 \arcsin \left (c x \right )\right ) d}{2400 c^{4} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c^{2} x^{2}-c x \sqrt {-c^{2} x^{2}+1}-i\right ) \left (11 i+45 \arcsin \left (c x \right )\right ) \sin \left (4 \arcsin \left (c x \right )\right ) d}{4800 c^{4} \left (c^{2} x^{2}-1\right )}\right )\)	\(727\)
parts	\(a \left (-\frac {x^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{7 c^{2} d}-\frac {2 \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{35 d \,c^{4}}\right )+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (64 c^{8} x^{8}-144 c^{6} x^{6}-64 i c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}+104 c^{4} x^{4}+112 i \sqrt {-c^{2} x^{2}+1}\, x^{5} c^{5}-25 c^{2} x^{2}-56 i \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}+7 i \sqrt {-c^{2} x^{2}+1}\, c x +1\right ) \left (i+7 \arcsin \left (c x \right )\right ) d}{6272 c^{4} \left (c^{2} x^{2}-1\right )}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i \sqrt {-c^{2} x^{2}+1}\, c x -1\right ) \left (\arcsin \left (c x \right )+i\right ) d}{128 c^{4} \left (c^{2} x^{2}-1\right )}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, c x +c^{2} x^{2}-1\right ) \left (\arcsin \left (c x \right )-i\right ) d}{128 c^{4} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 i \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}+4 c^{4} x^{4}-3 i \sqrt {-c^{2} x^{2}+1}\, c x -5 c^{2} x^{2}+1\right ) \left (-i+3 \arcsin \left (c x \right )\right ) d}{384 c^{4} \left (c^{2} x^{2}-1\right )}+\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, c x +c^{2} x^{2}-1\right ) \left (2 i+35 \arcsin \left (c x \right )\right ) \cos \left (6 \arcsin \left (c x \right )\right ) d}{39200 c^{4} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c^{2} x^{2}-c x \sqrt {-c^{2} x^{2}+1}-i\right ) \left (37 i+35 \arcsin \left (c x \right )\right ) \sin \left (6 \arcsin \left (c x \right )\right ) d}{78400 c^{4} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, c x +c^{2} x^{2}-1\right ) \left (7 i+15 \arcsin \left (c x \right )\right ) \cos \left (4 \arcsin \left (c x \right )\right ) d}{2400 c^{4} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c^{2} x^{2}-c x \sqrt {-c^{2} x^{2}+1}-i\right ) \left (11 i+45 \arcsin \left (c x \right )\right ) \sin \left (4 \arcsin \left (c x \right )\right ) d}{4800 c^{4} \left (c^{2} x^{2}-1\right )}\right )\)	\(727\)

Input:

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

Output:

1/1225*(325*c^8*x^8-866*c^6*x^6+553*c^4*x^4+420*c^2*x^2-280)/c^4/(c*x-1)/( 
c*x+1)/(c^2*x^2-1)*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))-1/3675/x^2*(75*c 
^6*x^6-168*c^4*x^4+35*c^2*x^2+210)/c^4/(c*x-1)/(c*x+1)*(3*x^2*(-c^2*d*x^2+ 
d)^(3/2)*(a+b*arcsin(c*x))-3*x^4*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))*c^ 
2*d+x^3*(-c^2*d*x^2+d)^(3/2)*b*c/(-c^2*x^2+1)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.83 \[ \int x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x)) \, dx=-\frac {{\left (75 \, b c^{7} d x^{7} - 168 \, b c^{5} d x^{5} + 35 \, b c^{3} d x^{3} + 210 \, b c d x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} + 105 \, {\left (5 \, a c^{8} d x^{8} - 13 \, a c^{6} d x^{6} + 9 \, a c^{4} d x^{4} + a c^{2} d x^{2} - 2 \, a d + {\left (5 \, b c^{8} d x^{8} - 13 \, b c^{6} d x^{6} + 9 \, b c^{4} d x^{4} + b c^{2} d x^{2} - 2 \, b d\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}}{3675 \, {\left (c^{6} x^{2} - c^{4}\right )}} \] Input:

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

Output:

-1/3675*((75*b*c^7*d*x^7 - 168*b*c^5*d*x^5 + 35*b*c^3*d*x^3 + 210*b*c*d*x) 
*sqrt(-c^2*d*x^2 + d)*sqrt(-c^2*x^2 + 1) + 105*(5*a*c^8*d*x^8 - 13*a*c^6*d 
*x^6 + 9*a*c^4*d*x^4 + a*c^2*d*x^2 - 2*a*d + (5*b*c^8*d*x^8 - 13*b*c^6*d*x 
^6 + 9*b*c^4*d*x^4 + b*c^2*d*x^2 - 2*b*d)*arcsin(c*x))*sqrt(-c^2*d*x^2 + d 
))/(c^6*x^2 - c^4)
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.66 \[ \int x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x)) \, dx=-\frac {1}{35} \, {\left (\frac {5 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{2}}{c^{2} d} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{c^{4} d}\right )} b \arcsin \left (c x\right ) - \frac {1}{35} \, {\left (\frac {5 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{2}}{c^{2} d} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{c^{4} d}\right )} a + \frac {{\left (75 \, c^{6} d^{\frac {3}{2}} x^{7} - 168 \, c^{4} d^{\frac {3}{2}} x^{5} + 35 \, c^{2} d^{\frac {3}{2}} x^{3} + 210 \, d^{\frac {3}{2}} x\right )} b}{3675 \, c^{3}} \] Input:

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

Output:

-1/35*(5*(-c^2*d*x^2 + d)^(5/2)*x^2/(c^2*d) + 2*(-c^2*d*x^2 + d)^(5/2)/(c^ 
4*d))*b*arcsin(c*x) - 1/35*(5*(-c^2*d*x^2 + d)^(5/2)*x^2/(c^2*d) + 2*(-c^2 
*d*x^2 + d)^(5/2)/(c^4*d))*a + 1/3675*(75*c^6*d^(3/2)*x^7 - 168*c^4*d^(3/2 
)*x^5 + 35*c^2*d^(3/2)*x^3 + 210*d^(3/2)*x)*b/c^3
 

Giac [F(-2)]

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x)) \, dx=\frac {\sqrt {d}\, d \left (-5 \sqrt {-c^{2} x^{2}+1}\, a \,c^{6} x^{6}+8 \sqrt {-c^{2} x^{2}+1}\, a \,c^{4} x^{4}-\sqrt {-c^{2} x^{2}+1}\, a \,c^{2} x^{2}-2 \sqrt {-c^{2} x^{2}+1}\, a -35 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) x^{5}d x \right ) b \,c^{6}+35 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) x^{3}d x \right ) b \,c^{4}\right )}{35 c^{4}} \] Input:

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

Output:

(sqrt(d)*d*( - 5*sqrt( - c**2*x**2 + 1)*a*c**6*x**6 + 8*sqrt( - c**2*x**2 
+ 1)*a*c**4*x**4 - sqrt( - c**2*x**2 + 1)*a*c**2*x**2 - 2*sqrt( - c**2*x** 
2 + 1)*a - 35*int(sqrt( - c**2*x**2 + 1)*asin(c*x)*x**5,x)*b*c**6 + 35*int 
(sqrt( - c**2*x**2 + 1)*asin(c*x)*x**3,x)*b*c**4))/(35*c**4)