\(\int x^2 (d-c^2 d x^2)^2 (a+b \arcsin (c x)) \, dx\) [12]

Optimal result

Integrand size = 25, antiderivative size = 161 \[ \int x^2 \left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x)) \, dx=\frac {8 b d^2 \sqrt {1-c^2 x^2}}{105 c^3}+\frac {4 b d^2 \left (1-c^2 x^2\right )^{3/2}}{315 c^3}+\frac {b d^2 \left (1-c^2 x^2\right )^{5/2}}{175 c^3}-\frac {b d^2 \left (1-c^2 x^2\right )^{7/2}}{49 c^3}+\frac {1}{3} d^2 x^3 (a+b \arcsin (c x))-\frac {2}{5} c^2 d^2 x^5 (a+b \arcsin (c x))+\frac {1}{7} c^4 d^2 x^7 (a+b \arcsin (c x)) \] Output:

8/105*b*d^2*(-c^2*x^2+1)^(1/2)/c^3+4/315*b*d^2*(-c^2*x^2+1)^(3/2)/c^3+1/17 
5*b*d^2*(-c^2*x^2+1)^(5/2)/c^3-1/49*b*d^2*(-c^2*x^2+1)^(7/2)/c^3+1/3*d^2*x 
^3*(a+b*arcsin(c*x))-2/5*c^2*d^2*x^5*(a+b*arcsin(c*x))+1/7*c^4*d^2*x^7*(a+ 
b*arcsin(c*x))
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.69 \[ \int x^2 \left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x)) \, dx=\frac {d^2 \left (105 a c^3 x^3 \left (35-42 c^2 x^2+15 c^4 x^4\right )+b \sqrt {1-c^2 x^2} \left (818+409 c^2 x^2-612 c^4 x^4+225 c^6 x^6\right )+105 b c^3 x^3 \left (35-42 c^2 x^2+15 c^4 x^4\right ) \arcsin (c x)\right )}{11025 c^3} \] Input:

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

Output:

(d^2*(105*a*c^3*x^3*(35 - 42*c^2*x^2 + 15*c^4*x^4) + b*Sqrt[1 - c^2*x^2]*( 
818 + 409*c^2*x^2 - 612*c^4*x^4 + 225*c^6*x^6) + 105*b*c^3*x^3*(35 - 42*c^ 
2*x^2 + 15*c^4*x^4)*ArcSin[c*x]))/(11025*c^3)
 

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.95, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5192, 27, 1578, 1195, 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.

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

\(\Big \downarrow \) 5192

\(\displaystyle -b c \int \frac {d^2 x^3 \left (15 c^4 x^4-42 c^2 x^2+35\right )}{105 \sqrt {1-c^2 x^2}}dx+\frac {1}{7} c^4 d^2 x^7 (a+b \arcsin (c x))-\frac {2}{5} c^2 d^2 x^5 (a+b \arcsin (c x))+\frac {1}{3} d^2 x^3 (a+b \arcsin (c x))\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {1}{105} b c d^2 \int \frac {x^3 \left (15 c^4 x^4-42 c^2 x^2+35\right )}{\sqrt {1-c^2 x^2}}dx+\frac {1}{7} c^4 d^2 x^7 (a+b \arcsin (c x))-\frac {2}{5} c^2 d^2 x^5 (a+b \arcsin (c x))+\frac {1}{3} d^2 x^3 (a+b \arcsin (c x))\)

\(\Big \downarrow \) 1578

\(\displaystyle -\frac {1}{210} b c d^2 \int \frac {x^2 \left (15 c^4 x^4-42 c^2 x^2+35\right )}{\sqrt {1-c^2 x^2}}dx^2+\frac {1}{7} c^4 d^2 x^7 (a+b \arcsin (c x))-\frac {2}{5} c^2 d^2 x^5 (a+b \arcsin (c x))+\frac {1}{3} d^2 x^3 (a+b \arcsin (c x))\)

\(\Big \downarrow \) 1195

\(\displaystyle -\frac {1}{210} b c d^2 \int \left (-\frac {15 \left (1-c^2 x^2\right )^{5/2}}{c^2}+\frac {3 \left (1-c^2 x^2\right )^{3/2}}{c^2}+\frac {4 \sqrt {1-c^2 x^2}}{c^2}+\frac {8}{c^2 \sqrt {1-c^2 x^2}}\right )dx^2+\frac {1}{7} c^4 d^2 x^7 (a+b \arcsin (c x))-\frac {2}{5} c^2 d^2 x^5 (a+b \arcsin (c x))+\frac {1}{3} d^2 x^3 (a+b \arcsin (c x))\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{7} c^4 d^2 x^7 (a+b \arcsin (c x))-\frac {2}{5} c^2 d^2 x^5 (a+b \arcsin (c x))+\frac {1}{3} d^2 x^3 (a+b \arcsin (c x))-\frac {1}{210} b c d^2 \left (\frac {30 \left (1-c^2 x^2\right )^{7/2}}{7 c^4}-\frac {6 \left (1-c^2 x^2\right )^{5/2}}{5 c^4}-\frac {8 \left (1-c^2 x^2\right )^{3/2}}{3 c^4}-\frac {16 \sqrt {1-c^2 x^2}}{c^4}\right )\)

Input:

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

Output:

-1/210*(b*c*d^2*((-16*Sqrt[1 - c^2*x^2])/c^4 - (8*(1 - c^2*x^2)^(3/2))/(3* 
c^4) - (6*(1 - c^2*x^2)^(5/2))/(5*c^4) + (30*(1 - c^2*x^2)^(7/2))/(7*c^4)) 
) + (d^2*x^3*(a + b*ArcSin[c*x]))/3 - (2*c^2*d^2*x^5*(a + b*ArcSin[c*x]))/ 
5 + (c^4*d^2*x^7*(a + b*ArcSin[c*x]))/7
 

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 1195

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x 
_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + 
 g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x 
] && IGtQ[p, 0]
 

rule 1578

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_ 
)^4)^(p_.), x_Symbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a 
+ b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && Int 
egerQ[(m - 1)/2]
 

rule 2009

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

rule 5192

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

Maple [A] (verified)

Time = 0.16 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.92

method	result	size
parts	\(a \,d^{2} \left (\frac {1}{7} c^{4} x^{7}-\frac {2}{5} c^{2} x^{5}+\frac {1}{3} x^{3}\right )+\frac {d^{2} b \left (\frac {\arcsin \left (c x \right ) c^{7} x^{7}}{7}-\frac {2 c^{5} x^{5} \arcsin \left (c x \right )}{5}+\frac {c^{3} x^{3} \arcsin \left (c x \right )}{3}+\frac {409 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{11025}+\frac {818 \sqrt {-c^{2} x^{2}+1}}{11025}-\frac {68 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{1225}+\frac {c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{49}\right )}{c^{3}}\)	\(148\)
derivativedivides	\(\frac {a \,d^{2} \left (\frac {1}{7} c^{7} x^{7}-\frac {2}{5} c^{5} x^{5}+\frac {1}{3} c^{3} x^{3}\right )+d^{2} b \left (\frac {\arcsin \left (c x \right ) c^{7} x^{7}}{7}-\frac {2 c^{5} x^{5} \arcsin \left (c x \right )}{5}+\frac {c^{3} x^{3} \arcsin \left (c x \right )}{3}+\frac {409 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{11025}+\frac {818 \sqrt {-c^{2} x^{2}+1}}{11025}-\frac {68 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{1225}+\frac {c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{49}\right )}{c^{3}}\)	\(152\)
default	\(\frac {a \,d^{2} \left (\frac {1}{7} c^{7} x^{7}-\frac {2}{5} c^{5} x^{5}+\frac {1}{3} c^{3} x^{3}\right )+d^{2} b \left (\frac {\arcsin \left (c x \right ) c^{7} x^{7}}{7}-\frac {2 c^{5} x^{5} \arcsin \left (c x \right )}{5}+\frac {c^{3} x^{3} \arcsin \left (c x \right )}{3}+\frac {409 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{11025}+\frac {818 \sqrt {-c^{2} x^{2}+1}}{11025}-\frac {68 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{1225}+\frac {c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{49}\right )}{c^{3}}\)	\(152\)
orering	\(\frac {\left (2925 c^{8} x^{8}-8532 c^{6} x^{6}+7353 c^{4} x^{4}+4090 c^{2} x^{2}-1636\right ) \left (-c^{2} d \,x^{2}+d \right )^{2} \left (a +b \arcsin \left (c x \right )\right )}{11025 c^{4} \left (c x -1\right ) x \left (c x +1\right ) \left (c^{2} x^{2}-1\right )}-\frac {\left (225 c^{6} x^{6}-612 c^{4} x^{4}+409 c^{2} x^{2}+818\right ) \left (2 x \left (-c^{2} d \,x^{2}+d \right )^{2} \left (a +b \arcsin \left (c x \right )\right )-4 x^{3} \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \arcsin \left (c x \right )\right ) c^{2} d +\frac {x^{2} \left (-c^{2} d \,x^{2}+d \right )^{2} b c}{\sqrt {-c^{2} x^{2}+1}}\right )}{11025 c^{4} \left (c x -1\right ) x^{2} \left (c x +1\right )}\)	\(222\)

Input:

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

Output:

a*d^2*(1/7*c^4*x^7-2/5*c^2*x^5+1/3*x^3)+d^2*b/c^3*(1/7*arcsin(c*x)*c^7*x^7 
-2/5*c^5*x^5*arcsin(c*x)+1/3*c^3*x^3*arcsin(c*x)+409/11025*c^2*x^2*(-c^2*x 
^2+1)^(1/2)+818/11025*(-c^2*x^2+1)^(1/2)-68/1225*c^4*x^4*(-c^2*x^2+1)^(1/2 
)+1/49*c^6*x^6*(-c^2*x^2+1)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.88 \[ \int x^2 \left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x)) \, dx=\frac {1575 \, a c^{7} d^{2} x^{7} - 4410 \, a c^{5} d^{2} x^{5} + 3675 \, a c^{3} d^{2} x^{3} + 105 \, {\left (15 \, b c^{7} d^{2} x^{7} - 42 \, b c^{5} d^{2} x^{5} + 35 \, b c^{3} d^{2} x^{3}\right )} \arcsin \left (c x\right ) + {\left (225 \, b c^{6} d^{2} x^{6} - 612 \, b c^{4} d^{2} x^{4} + 409 \, b c^{2} d^{2} x^{2} + 818 \, b d^{2}\right )} \sqrt {-c^{2} x^{2} + 1}}{11025 \, c^{3}} \] Input:

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

Output:

1/11025*(1575*a*c^7*d^2*x^7 - 4410*a*c^5*d^2*x^5 + 3675*a*c^3*d^2*x^3 + 10 
5*(15*b*c^7*d^2*x^7 - 42*b*c^5*d^2*x^5 + 35*b*c^3*d^2*x^3)*arcsin(c*x) + ( 
225*b*c^6*d^2*x^6 - 612*b*c^4*d^2*x^4 + 409*b*c^2*d^2*x^2 + 818*b*d^2)*sqr 
t(-c^2*x^2 + 1))/c^3
                                                                                    
                                                                                    
 

Sympy [A] (verification not implemented)

Time = 0.64 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.25 \[ \int x^2 \left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x)) \, dx=\begin {cases} \frac {a c^{4} d^{2} x^{7}}{7} - \frac {2 a c^{2} d^{2} x^{5}}{5} + \frac {a d^{2} x^{3}}{3} + \frac {b c^{4} d^{2} x^{7} \operatorname {asin}{\left (c x \right )}}{7} + \frac {b c^{3} d^{2} x^{6} \sqrt {- c^{2} x^{2} + 1}}{49} - \frac {2 b c^{2} d^{2} x^{5} \operatorname {asin}{\left (c x \right )}}{5} - \frac {68 b c d^{2} x^{4} \sqrt {- c^{2} x^{2} + 1}}{1225} + \frac {b d^{2} x^{3} \operatorname {asin}{\left (c x \right )}}{3} + \frac {409 b d^{2} x^{2} \sqrt {- c^{2} x^{2} + 1}}{11025 c} + \frac {818 b d^{2} \sqrt {- c^{2} x^{2} + 1}}{11025 c^{3}} & \text {for}\: c \neq 0 \\\frac {a d^{2} x^{3}}{3} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((a*c**4*d**2*x**7/7 - 2*a*c**2*d**2*x**5/5 + a*d**2*x**3/3 + b*c 
**4*d**2*x**7*asin(c*x)/7 + b*c**3*d**2*x**6*sqrt(-c**2*x**2 + 1)/49 - 2*b 
*c**2*d**2*x**5*asin(c*x)/5 - 68*b*c*d**2*x**4*sqrt(-c**2*x**2 + 1)/1225 + 
 b*d**2*x**3*asin(c*x)/3 + 409*b*d**2*x**2*sqrt(-c**2*x**2 + 1)/(11025*c) 
+ 818*b*d**2*sqrt(-c**2*x**2 + 1)/(11025*c**3), Ne(c, 0)), (a*d**2*x**3/3, 
 True))
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.66 \[ \int x^2 \left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x)) \, dx=\frac {1}{7} \, a c^{4} d^{2} x^{7} - \frac {2}{5} \, a c^{2} d^{2} x^{5} + \frac {1}{245} \, {\left (35 \, x^{7} \arcsin \left (c x\right ) + {\left (\frac {5 \, \sqrt {-c^{2} x^{2} + 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {-c^{2} x^{2} + 1}}{c^{8}}\right )} c\right )} b c^{4} d^{2} - \frac {2}{75} \, {\left (15 \, x^{5} \arcsin \left (c x\right ) + {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} b c^{2} d^{2} + \frac {1}{3} \, a d^{2} x^{3} + \frac {1}{9} \, {\left (3 \, x^{3} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b d^{2} \] Input:

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

Output:

1/7*a*c^4*d^2*x^7 - 2/5*a*c^2*d^2*x^5 + 1/245*(35*x^7*arcsin(c*x) + (5*sqr 
t(-c^2*x^2 + 1)*x^6/c^2 + 6*sqrt(-c^2*x^2 + 1)*x^4/c^4 + 8*sqrt(-c^2*x^2 + 
 1)*x^2/c^6 + 16*sqrt(-c^2*x^2 + 1)/c^8)*c)*b*c^4*d^2 - 2/75*(15*x^5*arcsi 
n(c*x) + (3*sqrt(-c^2*x^2 + 1)*x^4/c^2 + 4*sqrt(-c^2*x^2 + 1)*x^2/c^4 + 8* 
sqrt(-c^2*x^2 + 1)/c^6)*c)*b*c^2*d^2 + 1/3*a*d^2*x^3 + 1/9*(3*x^3*arcsin(c 
*x) + c*(sqrt(-c^2*x^2 + 1)*x^2/c^2 + 2*sqrt(-c^2*x^2 + 1)/c^4))*b*d^2
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.41 \[ \int x^2 \left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x)) \, dx=\frac {1}{7} \, a c^{4} d^{2} x^{7} - \frac {2}{5} \, a c^{2} d^{2} x^{5} + \frac {1}{3} \, a d^{2} x^{3} + \frac {{\left (c^{2} x^{2} - 1\right )}^{3} b d^{2} x \arcsin \left (c x\right )}{7 \, c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} b d^{2} x \arcsin \left (c x\right )}{35 \, c^{2}} - \frac {4 \, {\left (c^{2} x^{2} - 1\right )} b d^{2} x \arcsin \left (c x\right )}{105 \, c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{3} \sqrt {-c^{2} x^{2} + 1} b d^{2}}{49 \, c^{3}} + \frac {8 \, b d^{2} x \arcsin \left (c x\right )}{105 \, c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b d^{2}}{175 \, c^{3}} + \frac {4 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b d^{2}}{315 \, c^{3}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1} b d^{2}}{105 \, c^{3}} \] Input:

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

Output:

1/7*a*c^4*d^2*x^7 - 2/5*a*c^2*d^2*x^5 + 1/3*a*d^2*x^3 + 1/7*(c^2*x^2 - 1)^ 
3*b*d^2*x*arcsin(c*x)/c^2 + 1/35*(c^2*x^2 - 1)^2*b*d^2*x*arcsin(c*x)/c^2 - 
 4/105*(c^2*x^2 - 1)*b*d^2*x*arcsin(c*x)/c^2 + 1/49*(c^2*x^2 - 1)^3*sqrt(- 
c^2*x^2 + 1)*b*d^2/c^3 + 8/105*b*d^2*x*arcsin(c*x)/c^2 + 1/175*(c^2*x^2 - 
1)^2*sqrt(-c^2*x^2 + 1)*b*d^2/c^3 + 4/315*(-c^2*x^2 + 1)^(3/2)*b*d^2/c^3 + 
 8/105*sqrt(-c^2*x^2 + 1)*b*d^2/c^3
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.93 \[ \int x^2 \left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x)) \, dx=\frac {d^{2} \left (1575 \mathit {asin} \left (c x \right ) b \,c^{7} x^{7}-4410 \mathit {asin} \left (c x \right ) b \,c^{5} x^{5}+3675 \mathit {asin} \left (c x \right ) b \,c^{3} x^{3}+225 \sqrt {-c^{2} x^{2}+1}\, b \,c^{6} x^{6}-612 \sqrt {-c^{2} x^{2}+1}\, b \,c^{4} x^{4}+409 \sqrt {-c^{2} x^{2}+1}\, b \,c^{2} x^{2}+818 \sqrt {-c^{2} x^{2}+1}\, b +1575 a \,c^{7} x^{7}-4410 a \,c^{5} x^{5}+3675 a \,c^{3} x^{3}\right )}{11025 c^{3}} \] Input:

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

Output:

(d**2*(1575*asin(c*x)*b*c**7*x**7 - 4410*asin(c*x)*b*c**5*x**5 + 3675*asin 
(c*x)*b*c**3*x**3 + 225*sqrt( - c**2*x**2 + 1)*b*c**6*x**6 - 612*sqrt( - c 
**2*x**2 + 1)*b*c**4*x**4 + 409*sqrt( - c**2*x**2 + 1)*b*c**2*x**2 + 818*s 
qrt( - c**2*x**2 + 1)*b + 1575*a*c**7*x**7 - 4410*a*c**5*x**5 + 3675*a*c** 
3*x**3))/(11025*c**3)