\(\int (d-c^2 d x^2)^3 (a+b \arcsin (c x)) \, dx\) [1]

Optimal result

Integrand size = 22, antiderivative size = 175 \[ \int \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x)) \, dx=\frac {16 b d^3 \sqrt {1-c^2 x^2}}{35 c}+\frac {8 b d^3 \left (1-c^2 x^2\right )^{3/2}}{105 c}+\frac {6 b d^3 \left (1-c^2 x^2\right )^{5/2}}{175 c}+\frac {b d^3 \left (1-c^2 x^2\right )^{7/2}}{49 c}+d^3 x (a+b \arcsin (c x))-c^2 d^3 x^3 (a+b \arcsin (c x))+\frac {3}{5} c^4 d^3 x^5 (a+b \arcsin (c x))-\frac {1}{7} c^6 d^3 x^7 (a+b \arcsin (c x)) \] Output:

16/35*b*d^3*(-c^2*x^2+1)^(1/2)/c+8/105*b*d^3*(-c^2*x^2+1)^(3/2)/c+6/175*b* 
d^3*(-c^2*x^2+1)^(5/2)/c+1/49*b*d^3*(-c^2*x^2+1)^(7/2)/c+d^3*x*(a+b*arcsin 
(c*x))-c^2*d^3*x^3*(a+b*arcsin(c*x))+3/5*c^4*d^3*x^5*(a+b*arcsin(c*x))-1/7 
*c^6*d^3*x^7*(a+b*arcsin(c*x))
 

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.68 \[ \int \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x)) \, dx=-\frac {d^3 \left (105 a c x \left (-35+35 c^2 x^2-21 c^4 x^4+5 c^6 x^6\right )+b \sqrt {1-c^2 x^2} \left (-2161+757 c^2 x^2-351 c^4 x^4+75 c^6 x^6\right )+105 b c x \left (-35+35 c^2 x^2-21 c^4 x^4+5 c^6 x^6\right ) \arcsin (c x)\right )}{3675 c} \] Input:

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

Output:

-1/3675*(d^3*(105*a*c*x*(-35 + 35*c^2*x^2 - 21*c^4*x^4 + 5*c^6*x^6) + b*Sq 
rt[1 - c^2*x^2]*(-2161 + 757*c^2*x^2 - 351*c^4*x^4 + 75*c^6*x^6) + 105*b*c 
*x*(-35 + 35*c^2*x^2 - 21*c^4*x^4 + 5*c^6*x^6)*ArcSin[c*x]))/c
 

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.95, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {5154, 27, 2331, 2389, 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.

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

\(\Big \downarrow \) 5154

\(\displaystyle -b c \int \frac {d^3 x \left (-5 c^6 x^6+21 c^4 x^4-35 c^2 x^2+35\right )}{35 \sqrt {1-c^2 x^2}}dx-\frac {1}{7} c^6 d^3 x^7 (a+b \arcsin (c x))+\frac {3}{5} c^4 d^3 x^5 (a+b \arcsin (c x))-c^2 d^3 x^3 (a+b \arcsin (c x))+d^3 x (a+b \arcsin (c x))\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {1}{35} b c d^3 \int \frac {x \left (-5 c^6 x^6+21 c^4 x^4-35 c^2 x^2+35\right )}{\sqrt {1-c^2 x^2}}dx-\frac {1}{7} c^6 d^3 x^7 (a+b \arcsin (c x))+\frac {3}{5} c^4 d^3 x^5 (a+b \arcsin (c x))-c^2 d^3 x^3 (a+b \arcsin (c x))+d^3 x (a+b \arcsin (c x))\)

\(\Big \downarrow \) 2331

\(\displaystyle -\frac {1}{70} b c d^3 \int \frac {-5 c^6 x^6+21 c^4 x^4-35 c^2 x^2+35}{\sqrt {1-c^2 x^2}}dx^2-\frac {1}{7} c^6 d^3 x^7 (a+b \arcsin (c x))+\frac {3}{5} c^4 d^3 x^5 (a+b \arcsin (c x))-c^2 d^3 x^3 (a+b \arcsin (c x))+d^3 x (a+b \arcsin (c x))\)

\(\Big \downarrow \) 2389

\(\displaystyle -\frac {1}{70} b c d^3 \int \left (5 \left (1-c^2 x^2\right )^{5/2}+6 \left (1-c^2 x^2\right )^{3/2}+8 \sqrt {1-c^2 x^2}+\frac {16}{\sqrt {1-c^2 x^2}}\right )dx^2-\frac {1}{7} c^6 d^3 x^7 (a+b \arcsin (c x))+\frac {3}{5} c^4 d^3 x^5 (a+b \arcsin (c x))-c^2 d^3 x^3 (a+b \arcsin (c x))+d^3 x (a+b \arcsin (c x))\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {1}{7} c^6 d^3 x^7 (a+b \arcsin (c x))+\frac {3}{5} c^4 d^3 x^5 (a+b \arcsin (c x))-c^2 d^3 x^3 (a+b \arcsin (c x))+d^3 x (a+b \arcsin (c x))-\frac {1}{70} b c d^3 \left (-\frac {10 \left (1-c^2 x^2\right )^{7/2}}{7 c^2}-\frac {12 \left (1-c^2 x^2\right )^{5/2}}{5 c^2}-\frac {16 \left (1-c^2 x^2\right )^{3/2}}{3 c^2}-\frac {32 \sqrt {1-c^2 x^2}}{c^2}\right )\)

Input:

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

Output:

-1/70*(b*c*d^3*((-32*Sqrt[1 - c^2*x^2])/c^2 - (16*(1 - c^2*x^2)^(3/2))/(3* 
c^2) - (12*(1 - c^2*x^2)^(5/2))/(5*c^2) - (10*(1 - c^2*x^2)^(7/2))/(7*c^2) 
)) + d^3*x*(a + b*ArcSin[c*x]) - c^2*d^3*x^3*(a + b*ArcSin[c*x]) + (3*c^4* 
d^3*x^5*(a + b*ArcSin[c*x]))/5 - (c^6*d^3*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 2009

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

rule 2331

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/2   S 
ubst[Int[x^((m - 1)/2)*SubstFor[x^2, Pq, x]*(a + b*x)^p, x], x, x^2], x] /; 
 FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]
 

rule 2389

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand 
[Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p 
, 0] || EqQ[n, 1])
 

rule 5154

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbo 
l] :> With[{u = IntHide[(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]] /; Fr 
eeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
 

Maple [A] (verified)

Time = 0.13 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.93

method	result	size
parts	\(-d^{3} a \left (\frac {1}{7} c^{6} x^{7}-\frac {3}{5} c^{4} x^{5}+c^{2} x^{3}-x \right )-\frac {d^{3} b \left (\frac {\arcsin \left (c x \right ) c^{7} x^{7}}{7}-\frac {3 c^{5} x^{5} \arcsin \left (c x \right )}{5}+c^{3} x^{3} \arcsin \left (c x \right )-c x \arcsin \left (c x \right )-\frac {2161 \sqrt {-c^{2} x^{2}+1}}{3675}+\frac {757 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3675}-\frac {117 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{1225}+\frac {c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{49}\right )}{c}\)	\(162\)
derivativedivides	\(\frac {-d^{3} a \left (\frac {1}{7} c^{7} x^{7}-\frac {3}{5} c^{5} x^{5}+c^{3} x^{3}-c x \right )-d^{3} b \left (\frac {\arcsin \left (c x \right ) c^{7} x^{7}}{7}-\frac {3 c^{5} x^{5} \arcsin \left (c x \right )}{5}+c^{3} x^{3} \arcsin \left (c x \right )-c x \arcsin \left (c x \right )-\frac {2161 \sqrt {-c^{2} x^{2}+1}}{3675}+\frac {757 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3675}-\frac {117 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{1225}+\frac {c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{49}\right )}{c}\)	\(164\)
default	\(\frac {-d^{3} a \left (\frac {1}{7} c^{7} x^{7}-\frac {3}{5} c^{5} x^{5}+c^{3} x^{3}-c x \right )-d^{3} b \left (\frac {\arcsin \left (c x \right ) c^{7} x^{7}}{7}-\frac {3 c^{5} x^{5} \arcsin \left (c x \right )}{5}+c^{3} x^{3} \arcsin \left (c x \right )-c x \arcsin \left (c x \right )-\frac {2161 \sqrt {-c^{2} x^{2}+1}}{3675}+\frac {757 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3675}-\frac {117 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{1225}+\frac {c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{49}\right )}{c}\)	\(164\)
orering	\(\frac {x \left (325 c^{6} x^{6}-1437 c^{4} x^{4}+2739 c^{2} x^{2}-5547\right ) \left (-c^{2} d \,x^{2}+d \right )^{3} \left (a +b \arcsin \left (c x \right )\right )}{1225 \left (c x -1\right )^{2} \left (c x +1\right )^{2} \left (c^{2} x^{2}-1\right )}-\frac {\left (75 c^{6} x^{6}-351 c^{4} x^{4}+757 c^{2} x^{2}-2161\right ) \left (-6 \left (-c^{2} d \,x^{2}+d \right )^{2} \left (a +b \arcsin \left (c x \right )\right ) c^{2} d x +\frac {\left (-c^{2} d \,x^{2}+d \right )^{3} b c}{\sqrt {-c^{2} x^{2}+1}}\right )}{3675 c^{2} \left (c x -1\right )^{2} \left (c x +1\right )^{2}}\)	\(179\)

Input:

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

Output:

-d^3*a*(1/7*c^6*x^7-3/5*c^4*x^5+c^2*x^3-x)-d^3*b/c*(1/7*arcsin(c*x)*c^7*x^ 
7-3/5*c^5*x^5*arcsin(c*x)+c^3*x^3*arcsin(c*x)-c*x*arcsin(c*x)-2161/3675*(- 
c^2*x^2+1)^(1/2)+757/3675*c^2*x^2*(-c^2*x^2+1)^(1/2)-117/1225*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 = 0.09 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.90 \[ \int \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x)) \, dx=-\frac {525 \, a c^{7} d^{3} x^{7} - 2205 \, a c^{5} d^{3} x^{5} + 3675 \, a c^{3} d^{3} x^{3} - 3675 \, a c d^{3} x + 105 \, {\left (5 \, b c^{7} d^{3} x^{7} - 21 \, b c^{5} d^{3} x^{5} + 35 \, b c^{3} d^{3} x^{3} - 35 \, b c d^{3} x\right )} \arcsin \left (c x\right ) + {\left (75 \, b c^{6} d^{3} x^{6} - 351 \, b c^{4} d^{3} x^{4} + 757 \, b c^{2} d^{3} x^{2} - 2161 \, b d^{3}\right )} \sqrt {-c^{2} x^{2} + 1}}{3675 \, c} \] Input:

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

Output:

-1/3675*(525*a*c^7*d^3*x^7 - 2205*a*c^5*d^3*x^5 + 3675*a*c^3*d^3*x^3 - 367 
5*a*c*d^3*x + 105*(5*b*c^7*d^3*x^7 - 21*b*c^5*d^3*x^5 + 35*b*c^3*d^3*x^3 - 
 35*b*c*d^3*x)*arcsin(c*x) + (75*b*c^6*d^3*x^6 - 351*b*c^4*d^3*x^4 + 757*b 
*c^2*d^3*x^2 - 2161*b*d^3)*sqrt(-c^2*x^2 + 1))/c
                                                                                    
                                                                                    
 

Sympy [A] (verification not implemented)

Time = 0.72 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.26 \[ \int \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x)) \, dx=\begin {cases} - \frac {a c^{6} d^{3} x^{7}}{7} + \frac {3 a c^{4} d^{3} x^{5}}{5} - a c^{2} d^{3} x^{3} + a d^{3} x - \frac {b c^{6} d^{3} x^{7} \operatorname {asin}{\left (c x \right )}}{7} - \frac {b c^{5} d^{3} x^{6} \sqrt {- c^{2} x^{2} + 1}}{49} + \frac {3 b c^{4} d^{3} x^{5} \operatorname {asin}{\left (c x \right )}}{5} + \frac {117 b c^{3} d^{3} x^{4} \sqrt {- c^{2} x^{2} + 1}}{1225} - b c^{2} d^{3} x^{3} \operatorname {asin}{\left (c x \right )} - \frac {757 b c d^{3} x^{2} \sqrt {- c^{2} x^{2} + 1}}{3675} + b d^{3} x \operatorname {asin}{\left (c x \right )} + \frac {2161 b d^{3} \sqrt {- c^{2} x^{2} + 1}}{3675 c} & \text {for}\: c \neq 0 \\a d^{3} x & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((-a*c**6*d**3*x**7/7 + 3*a*c**4*d**3*x**5/5 - a*c**2*d**3*x**3 + 
 a*d**3*x - b*c**6*d**3*x**7*asin(c*x)/7 - b*c**5*d**3*x**6*sqrt(-c**2*x** 
2 + 1)/49 + 3*b*c**4*d**3*x**5*asin(c*x)/5 + 117*b*c**3*d**3*x**4*sqrt(-c* 
*2*x**2 + 1)/1225 - b*c**2*d**3*x**3*asin(c*x) - 757*b*c*d**3*x**2*sqrt(-c 
**2*x**2 + 1)/3675 + b*d**3*x*asin(c*x) + 2161*b*d**3*sqrt(-c**2*x**2 + 1) 
/(3675*c), Ne(c, 0)), (a*d**3*x, True))
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.75 \[ \int \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x)) \, dx=-\frac {1}{7} \, a c^{6} d^{3} x^{7} + \frac {3}{5} \, a c^{4} d^{3} x^{5} - \frac {1}{245} \, {\left (35 \, x^{7} \arcsin \left (c x\right ) + {\left (\frac {5 \, \sqrt {-c^{2} x^{2} + 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {-c^{2} x^{2} + 1}}{c^{8}}\right )} c\right )} b c^{6} d^{3} + \frac {1}{25} \, {\left (15 \, x^{5} \arcsin \left (c x\right ) + {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} b c^{4} d^{3} - a c^{2} d^{3} x^{3} - \frac {1}{3} \, {\left (3 \, x^{3} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b c^{2} d^{3} + a d^{3} x + \frac {{\left (c x \arcsin \left (c x\right ) + \sqrt {-c^{2} x^{2} + 1}\right )} b d^{3}}{c} \] Input:

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

Output:

-1/7*a*c^6*d^3*x^7 + 3/5*a*c^4*d^3*x^5 - 1/245*(35*x^7*arcsin(c*x) + (5*sq 
rt(-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^6*d^3 + 1/25*(15*x^5*arcs 
in(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^4*d^3 - a*c^2*d^3*x^3 - 1/3*(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*c^2*d^ 
3 + a*d^3*x + (c*x*arcsin(c*x) + sqrt(-c^2*x^2 + 1))*b*d^3/c
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.28 \[ \int \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x)) \, dx=-\frac {1}{7} \, a c^{6} d^{3} x^{7} + \frac {3}{5} \, a c^{4} d^{3} x^{5} - a c^{2} d^{3} x^{3} - \frac {1}{7} \, {\left (c^{2} x^{2} - 1\right )}^{3} b d^{3} x \arcsin \left (c x\right ) + \frac {6}{35} \, {\left (c^{2} x^{2} - 1\right )}^{2} b d^{3} x \arcsin \left (c x\right ) - \frac {8}{35} \, {\left (c^{2} x^{2} - 1\right )} b d^{3} x \arcsin \left (c x\right ) - \frac {{\left (c^{2} x^{2} - 1\right )}^{3} \sqrt {-c^{2} x^{2} + 1} b d^{3}}{49 \, c} + \frac {16}{35} \, b d^{3} x \arcsin \left (c x\right ) + \frac {6 \, {\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b d^{3}}{175 \, c} + a d^{3} x + \frac {8 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b d^{3}}{105 \, c} + \frac {16 \, \sqrt {-c^{2} x^{2} + 1} b d^{3}}{35 \, c} \] Input:

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

Output:

-1/7*a*c^6*d^3*x^7 + 3/5*a*c^4*d^3*x^5 - a*c^2*d^3*x^3 - 1/7*(c^2*x^2 - 1) 
^3*b*d^3*x*arcsin(c*x) + 6/35*(c^2*x^2 - 1)^2*b*d^3*x*arcsin(c*x) - 8/35*( 
c^2*x^2 - 1)*b*d^3*x*arcsin(c*x) - 1/49*(c^2*x^2 - 1)^3*sqrt(-c^2*x^2 + 1) 
*b*d^3/c + 16/35*b*d^3*x*arcsin(c*x) + 6/175*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 
 + 1)*b*d^3/c + a*d^3*x + 8/105*(-c^2*x^2 + 1)^(3/2)*b*d^3/c + 16/35*sqrt( 
-c^2*x^2 + 1)*b*d^3/c
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.93 \[ \int \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x)) \, dx=\frac {d^{3} \left (-525 \mathit {asin} \left (c x \right ) b \,c^{7} x^{7}+2205 \mathit {asin} \left (c x \right ) b \,c^{5} x^{5}-3675 \mathit {asin} \left (c x \right ) b \,c^{3} x^{3}+3675 \mathit {asin} \left (c x \right ) b c x -75 \sqrt {-c^{2} x^{2}+1}\, b \,c^{6} x^{6}+351 \sqrt {-c^{2} x^{2}+1}\, b \,c^{4} x^{4}-757 \sqrt {-c^{2} x^{2}+1}\, b \,c^{2} x^{2}+2161 \sqrt {-c^{2} x^{2}+1}\, b -525 a \,c^{7} x^{7}+2205 a \,c^{5} x^{5}-3675 a \,c^{3} x^{3}+3675 a c x \right )}{3675 c} \] Input:

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

Output:

(d**3*( - 525*asin(c*x)*b*c**7*x**7 + 2205*asin(c*x)*b*c**5*x**5 - 3675*as 
in(c*x)*b*c**3*x**3 + 3675*asin(c*x)*b*c*x - 75*sqrt( - c**2*x**2 + 1)*b*c 
**6*x**6 + 351*sqrt( - c**2*x**2 + 1)*b*c**4*x**4 - 757*sqrt( - c**2*x**2 
+ 1)*b*c**2*x**2 + 2161*sqrt( - c**2*x**2 + 1)*b - 525*a*c**7*x**7 + 2205* 
a*c**5*x**5 - 3675*a*c**3*x**3 + 3675*a*c*x))/(3675*c)