\(\int (d+c d x)^{5/2} \sqrt {e-c e x} (a+b \arcsin (c x))^2 \, dx\) [69]

Optimal result

Integrand size = 32, antiderivative size = 613 \[ \int (d+c d x)^{5/2} \sqrt {e-c e x} (a+b \arcsin (c x))^2 \, dx=\frac {8 b^2 d^2 \sqrt {d+c d x} \sqrt {e-c e x}}{9 c}-\frac {15}{64} b^2 d^2 x \sqrt {d+c d x} \sqrt {e-c e x}-\frac {1}{32} b^2 c^2 d^2 x^3 \sqrt {d+c d x} \sqrt {e-c e x}+\frac {4 b^2 d^2 \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right )}{27 c}+\frac {15 b^2 d^2 \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x)}{64 c \sqrt {1-c^2 x^2}}+\frac {4 b d^2 x \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{3 \sqrt {1-c^2 x^2}}-\frac {3 b c d^2 x^2 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{8 \sqrt {1-c^2 x^2}}-\frac {4 b c^2 d^2 x^3 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{9 \sqrt {1-c^2 x^2}}-\frac {b c^3 d^2 x^4 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{8 \sqrt {1-c^2 x^2}}+\frac {3}{8} d^2 x \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2+\frac {1}{4} c^2 d^2 x^3 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2-\frac {2 d^2 \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 c}+\frac {5 d^2 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^3}{24 b c \sqrt {1-c^2 x^2}} \] Output:

8/9*b^2*d^2*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)/c-15/64*b^2*d^2*x*(c*d*x+d)^( 
1/2)*(-c*e*x+e)^(1/2)-1/32*b^2*c^2*d^2*x^3*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2 
)+4/27*b^2*d^2*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)*(-c^2*x^2+1)/c+15/64*b^2*d 
^2*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)*arcsin(c*x)/c/(-c^2*x^2+1)^(1/2)+4/3*b 
*d^2*x*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)*(a+b*arcsin(c*x))/(-c^2*x^2+1)^(1/ 
2)-3/8*b*c*d^2*x^2*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)*(a+b*arcsin(c*x))/(-c^ 
2*x^2+1)^(1/2)-4/9*b*c^2*d^2*x^3*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)*(a+b*arc 
sin(c*x))/(-c^2*x^2+1)^(1/2)-1/8*b*c^3*d^2*x^4*(c*d*x+d)^(1/2)*(-c*e*x+e)^ 
(1/2)*(a+b*arcsin(c*x))/(-c^2*x^2+1)^(1/2)+3/8*d^2*x*(c*d*x+d)^(1/2)*(-c*e 
*x+e)^(1/2)*(a+b*arcsin(c*x))^2+1/4*c^2*d^2*x^3*(c*d*x+d)^(1/2)*(-c*e*x+e) 
^(1/2)*(a+b*arcsin(c*x))^2-2/3*d^2*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)*(-c^2* 
x^2+1)*(a+b*arcsin(c*x))^2/c+5/24*d^2*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)*(a+ 
b*arcsin(c*x))^3/b/c/(-c^2*x^2+1)^(1/2)
 

Mathematica [A] (verified)

Time = 3.09 (sec) , antiderivative size = 555, normalized size of antiderivative = 0.91 \[ \int (d+c d x)^{5/2} \sqrt {e-c e x} (a+b \arcsin (c x))^2 \, dx=\frac {1440 b^2 d^2 \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x)^3-4320 a^2 d^{5/2} \sqrt {e} \sqrt {1-c^2 x^2} \arctan \left (\frac {c x \sqrt {d+c d x} \sqrt {e-c e x}}{\sqrt {d} \sqrt {e} \left (-1+c^2 x^2\right )}\right )+12 b d^2 \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x) \left (576 b c x-768 a \sqrt {1-c^2 x^2}+768 a c^2 x^2 \sqrt {1-c^2 x^2}+144 b \cos (2 \arcsin (c x))-9 b \cos (4 \arcsin (c x))+288 a \sin (2 \arcsin (c x))+64 b \sin (3 \arcsin (c x))-36 a \sin (4 \arcsin (c x))\right )-72 b d^2 \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x)^2 \left (-60 a+48 b \sqrt {1-c^2 x^2}+16 b \cos (3 \arcsin (c x))-24 b \sin (2 \arcsin (c x))+3 b \sin (4 \arcsin (c x))\right )+d^2 \sqrt {d+c d x} \sqrt {e-c e x} \left (1728 a b \cos (2 \arcsin (c x))+256 b^2 \cos (3 \arcsin (c x))+3 \left (3072 a b c x-1024 a b c^3 x^3-1536 a^2 \sqrt {1-c^2 x^2}+2304 b^2 \sqrt {1-c^2 x^2}+864 a^2 c x \sqrt {1-c^2 x^2}+1536 a^2 c^2 x^2 \sqrt {1-c^2 x^2}+576 a^2 c^3 x^3 \sqrt {1-c^2 x^2}-36 a b \cos (4 \arcsin (c x))-288 b^2 \sin (2 \arcsin (c x))+9 b^2 \sin (4 \arcsin (c x))\right )\right )}{6912 c \sqrt {1-c^2 x^2}} \] Input:

Integrate[(d + c*d*x)^(5/2)*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x])^2,x]
 

Output:

(1440*b^2*d^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]^3 - 4320*a^2*d^( 
5/2)*Sqrt[e]*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x] 
)/(Sqrt[d]*Sqrt[e]*(-1 + c^2*x^2))] + 12*b*d^2*Sqrt[d + c*d*x]*Sqrt[e - c* 
e*x]*ArcSin[c*x]*(576*b*c*x - 768*a*Sqrt[1 - c^2*x^2] + 768*a*c^2*x^2*Sqrt 
[1 - c^2*x^2] + 144*b*Cos[2*ArcSin[c*x]] - 9*b*Cos[4*ArcSin[c*x]] + 288*a* 
Sin[2*ArcSin[c*x]] + 64*b*Sin[3*ArcSin[c*x]] - 36*a*Sin[4*ArcSin[c*x]]) - 
72*b*d^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]^2*(-60*a + 48*b*Sqrt[ 
1 - c^2*x^2] + 16*b*Cos[3*ArcSin[c*x]] - 24*b*Sin[2*ArcSin[c*x]] + 3*b*Sin 
[4*ArcSin[c*x]]) + d^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(1728*a*b*Cos[2*Arc 
Sin[c*x]] + 256*b^2*Cos[3*ArcSin[c*x]] + 3*(3072*a*b*c*x - 1024*a*b*c^3*x^ 
3 - 1536*a^2*Sqrt[1 - c^2*x^2] + 2304*b^2*Sqrt[1 - c^2*x^2] + 864*a^2*c*x* 
Sqrt[1 - c^2*x^2] + 1536*a^2*c^2*x^2*Sqrt[1 - c^2*x^2] + 576*a^2*c^3*x^3*S 
qrt[1 - c^2*x^2] - 36*a*b*Cos[4*ArcSin[c*x]] - 288*b^2*Sin[2*ArcSin[c*x]] 
+ 9*b^2*Sin[4*ArcSin[c*x]])))/(6912*c*Sqrt[1 - c^2*x^2])
 

Rubi [A] (verified)

Time = 1.40 (sec) , antiderivative size = 334, normalized size of antiderivative = 0.54, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5178, 27, 5262, 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.

\(\displaystyle \int (c d x+d)^{5/2} \sqrt {e-c e x} (a+b \arcsin (c x))^2 \, dx\)

\(\Big \downarrow \) 5178

\(\displaystyle \frac {\sqrt {c d x+d} \sqrt {e-c e x} \int d^2 (c x+1)^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2dx}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {d^2 \sqrt {c d x+d} \sqrt {e-c e x} \int (c x+1)^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2dx}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 5262

\(\displaystyle \frac {d^2 \sqrt {c d x+d} \sqrt {e-c e x} \int \left (c^2 x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2+2 c x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2+\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2\right )dx}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {d^2 \sqrt {c d x+d} \sqrt {e-c e x} \left (-\frac {1}{8} b c^3 x^4 (a+b \arcsin (c x))-\frac {4}{9} b c^2 x^3 (a+b \arcsin (c x))+\frac {3}{8} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2-\frac {2 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2}{3 c}+\frac {1}{4} c^2 x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2-\frac {3}{8} b c x^2 (a+b \arcsin (c x))+\frac {4}{3} b x (a+b \arcsin (c x))+\frac {5 (a+b \arcsin (c x))^3}{24 b c}+\frac {15 b^2 \arcsin (c x)}{64 c}-\frac {15}{64} b^2 x \sqrt {1-c^2 x^2}+\frac {4 b^2 \left (1-c^2 x^2\right )^{3/2}}{27 c}+\frac {8 b^2 \sqrt {1-c^2 x^2}}{9 c}-\frac {1}{32} b^2 c^2 x^3 \sqrt {1-c^2 x^2}\right )}{\sqrt {1-c^2 x^2}}\)

Input:

Int[(d + c*d*x)^(5/2)*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x])^2,x]
 

Output:

(d^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*((8*b^2*Sqrt[1 - c^2*x^2])/(9*c) - (1 
5*b^2*x*Sqrt[1 - c^2*x^2])/64 - (b^2*c^2*x^3*Sqrt[1 - c^2*x^2])/32 + (4*b^ 
2*(1 - c^2*x^2)^(3/2))/(27*c) + (15*b^2*ArcSin[c*x])/(64*c) + (4*b*x*(a + 
b*ArcSin[c*x]))/3 - (3*b*c*x^2*(a + b*ArcSin[c*x]))/8 - (4*b*c^2*x^3*(a + 
b*ArcSin[c*x]))/9 - (b*c^3*x^4*(a + b*ArcSin[c*x]))/8 + (3*x*Sqrt[1 - c^2* 
x^2]*(a + b*ArcSin[c*x])^2)/8 + (c^2*x^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c 
*x])^2)/4 - (2*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/(3*c) + (5*(a + 
b*ArcSin[c*x])^3)/(24*b*c)))/Sqrt[1 - c^2*x^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 2009

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

rule 5178

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_))^(p_)*((f_) 
 + (g_.)*(x_))^(q_), x_Symbol] :> Simp[(d + e*x)^q*((f + g*x)^q/(1 - c^2*x^ 
2)^q)   Int[(d + e*x)^(p - q)*(1 - c^2*x^2)^q*(a + b*ArcSin[c*x])^n, x], x] 
 /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^2 
- e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 0]
 

rule 5262

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_ 
) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^p*(a + 
 b*ArcSin[c*x])^n, (f + g*x)^m, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] & 
& EqQ[c^2*d + e, 0] && IGtQ[m, 0] && IntegerQ[p + 1/2] && GtQ[d, 0] && IGtQ 
[n, 0] && (m == 1 || p > 0 || (n == 1 && p > -1) || (m == 2 && p < -2))
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 4.67 (sec) , antiderivative size = 1819, normalized size of antiderivative = 2.97

method	result	size
default	\(\text {Expression too large to display}\)	\(1819\)
parts	\(\text {Expression too large to display}\)	\(1819\)

Input:

int((c*d*x+d)^(5/2)*(-c*e*x+e)^(1/2)*(a+b*arcsin(c*x))^2,x,method=_RETURNV 
ERBOSE)
 

Output:

-1/4*a^2/c/e*(c*d*x+d)^(5/2)*(-c*e*x+e)^(3/2)-5/12*a^2*d/c/e*(c*d*x+d)^(3/ 
2)*(-c*e*x+e)^(3/2)-5/8*a^2*d^2/c/e*(c*d*x+d)^(1/2)*(-c*e*x+e)^(3/2)+5/8*a 
^2*d^2/c*(-c*e*x+e)^(1/2)*(c*d*x+d)^(1/2)+5/8*a^2*d^3*e*((c*d*x+d)*(-c*e*x 
+e))^(1/2)/(-c*e*x+e)^(1/2)/(c*d*x+d)^(1/2)/(c^2*d*e)^(1/2)*arctan((c^2*d* 
e)^(1/2)*x/(-c^2*d*e*x^2+d*e)^(1/2))+b^2*(-5/24*(-e*(c*x-1))^(1/2)*(d*(c*x 
+1))^(1/2)*(-c^2*x^2+1)^(1/2)/c/(c^2*x^2-1)*arcsin(c*x)^3*d^2+1/512*(-e*(c 
*x-1))^(1/2)*(d*(c*x+1))^(1/2)*(-8*I*(-c^2*x^2+1)^(1/2)*x^4*c^4+8*c^5*x^5+ 
8*I*(-c^2*x^2+1)^(1/2)*x^2*c^2-12*c^3*x^3-I*(-c^2*x^2+1)^(1/2)+4*c*x)*(4*I 
*arcsin(c*x)+8*arcsin(c*x)^2-1)*d^2/c/(c^2*x^2-1)+1/108*(-e*(c*x-1))^(1/2) 
*(d*(c*x+1))^(1/2)*(4*c^4*x^4-5*c^2*x^2-4*I*(-c^2*x^2+1)^(1/2)*x^3*c^3+3*I 
*(-c^2*x^2+1)^(1/2)*x*c+1)*(6*I*arcsin(c*x)+9*arcsin(c*x)^2-2)*d^2/c/(c^2* 
x^2-1)-1/4*(-e*(c*x-1))^(1/2)*(d*(c*x+1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*c*x+ 
c^2*x^2-1)*(arcsin(c*x)^2-2-2*I*arcsin(c*x))*d^2/c/(c^2*x^2-1)+1/16*(-e*(c 
*x-1))^(1/2)*(d*(c*x+1))^(1/2)*(2*I*(-c^2*x^2+1)^(1/2)*x^2*c^2+2*c^3*x^3-I 
*(-c^2*x^2+1)^(1/2)-2*c*x)*(2*arcsin(c*x)^2-1-2*I*arcsin(c*x))*d^2/c/(c^2* 
x^2-1)-3/512*(-e*(c*x-1))^(1/2)*(d*(c*x+1))^(1/2)*(I*c^2*x^2-c*x*(-c^2*x^2 
+1)^(1/2)-I)*(20*I*arcsin(c*x)+24*arcsin(c*x)^2-11)*cos(3*arcsin(c*x))*d^2 
/c/(c^2*x^2-1)+1/512*(-e*(c*x-1))^(1/2)*(d*(c*x+1))^(1/2)*(I*(-c^2*x^2+1)^ 
(1/2)*c*x+c^2*x^2-1)*(68*I*arcsin(c*x)+56*arcsin(c*x)^2-31)*sin(3*arcsin(c 
*x))*d^2/c/(c^2*x^2-1)-1/27*(-e*(c*x-1))^(1/2)*(d*(c*x+1))^(1/2)*(I*(-c...
 

Fricas [F]

\[ \int (d+c d x)^{5/2} \sqrt {e-c e x} (a+b \arcsin (c x))^2 \, dx=\int { {\left (c d x + d\right )}^{\frac {5}{2}} \sqrt {-c e x + e} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} \,d x } \] Input:

integrate((c*d*x+d)^(5/2)*(-c*e*x+e)^(1/2)*(a+b*arcsin(c*x))^2,x, algorith 
m="fricas")
 

Output:

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

Sympy [F(-1)]

Timed out. \[ \int (d+c d x)^{5/2} \sqrt {e-c e x} (a+b \arcsin (c x))^2 \, dx=\text {Timed out} \] Input:

integrate((c*d*x+d)**(5/2)*(-c*e*x+e)**(1/2)*(a+b*asin(c*x))**2,x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int (d+c d x)^{5/2} \sqrt {e-c e x} (a+b \arcsin (c x))^2 \, dx=\text {Exception raised: ValueError} \] Input:

integrate((c*d*x+d)^(5/2)*(-c*e*x+e)^(1/2)*(a+b*arcsin(c*x))^2,x, algorith 
m="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [F]

\[ \int (d+c d x)^{5/2} \sqrt {e-c e x} (a+b \arcsin (c x))^2 \, dx=\int { {\left (c d x + d\right )}^{\frac {5}{2}} \sqrt {-c e x + e} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} \,d x } \] Input:

integrate((c*d*x+d)^(5/2)*(-c*e*x+e)^(1/2)*(a+b*arcsin(c*x))^2,x, algorith 
m="giac")
 

Output:

integrate((c*d*x + d)^(5/2)*sqrt(-c*e*x + e)*(b*arcsin(c*x) + a)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int (d+c d x)^{5/2} \sqrt {e-c e x} (a+b \arcsin (c x))^2 \, dx=\int {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\right )}^{5/2}\,\sqrt {e-c\,e\,x} \,d x \] Input:

int((a + b*asin(c*x))^2*(d + c*d*x)^(5/2)*(e - c*e*x)^(1/2),x)
 

Output:

int((a + b*asin(c*x))^2*(d + c*d*x)^(5/2)*(e - c*e*x)^(1/2), x)
 

Reduce [F]

\[ \int (d+c d x)^{5/2} \sqrt {e-c e x} (a+b \arcsin (c x))^2 \, dx=\frac {\sqrt {e}\, \sqrt {d}\, d^{2} \left (-30 \mathit {asin} \left (\frac {\sqrt {-c x +1}}{\sqrt {2}}\right ) a^{2}+6 \sqrt {c x +1}\, \sqrt {-c x +1}\, a^{2} c^{3} x^{3}+16 \sqrt {c x +1}\, \sqrt {-c x +1}\, a^{2} c^{2} x^{2}+9 \sqrt {c x +1}\, \sqrt {-c x +1}\, a^{2} c x -16 \sqrt {c x +1}\, \sqrt {-c x +1}\, a^{2}+48 \left (\int \sqrt {c x +1}\, \sqrt {-c x +1}\, \mathit {asin} \left (c x \right ) x^{2}d x \right ) a b \,c^{3}+96 \left (\int \sqrt {c x +1}\, \sqrt {-c x +1}\, \mathit {asin} \left (c x \right ) x d x \right ) a b \,c^{2}+48 \left (\int \sqrt {c x +1}\, \sqrt {-c x +1}\, \mathit {asin} \left (c x \right )d x \right ) a b c +24 \left (\int \sqrt {c x +1}\, \sqrt {-c x +1}\, \mathit {asin} \left (c x \right )^{2} x^{2}d x \right ) b^{2} c^{3}+48 \left (\int \sqrt {c x +1}\, \sqrt {-c x +1}\, \mathit {asin} \left (c x \right )^{2} x d x \right ) b^{2} c^{2}+24 \left (\int \sqrt {c x +1}\, \sqrt {-c x +1}\, \mathit {asin} \left (c x \right )^{2}d x \right ) b^{2} c \right )}{24 c} \] Input:

int((c*d*x+d)^(5/2)*(-c*e*x+e)^(1/2)*(a+b*asin(c*x))^2,x)
 

Output:

(sqrt(e)*sqrt(d)*d**2*( - 30*asin(sqrt( - c*x + 1)/sqrt(2))*a**2 + 6*sqrt( 
c*x + 1)*sqrt( - c*x + 1)*a**2*c**3*x**3 + 16*sqrt(c*x + 1)*sqrt( - c*x + 
1)*a**2*c**2*x**2 + 9*sqrt(c*x + 1)*sqrt( - c*x + 1)*a**2*c*x - 16*sqrt(c* 
x + 1)*sqrt( - c*x + 1)*a**2 + 48*int(sqrt(c*x + 1)*sqrt( - c*x + 1)*asin( 
c*x)*x**2,x)*a*b*c**3 + 96*int(sqrt(c*x + 1)*sqrt( - c*x + 1)*asin(c*x)*x, 
x)*a*b*c**2 + 48*int(sqrt(c*x + 1)*sqrt( - c*x + 1)*asin(c*x),x)*a*b*c + 2 
4*int(sqrt(c*x + 1)*sqrt( - c*x + 1)*asin(c*x)**2*x**2,x)*b**2*c**3 + 48*i 
nt(sqrt(c*x + 1)*sqrt( - c*x + 1)*asin(c*x)**2*x,x)*b**2*c**2 + 24*int(sqr 
t(c*x + 1)*sqrt( - c*x + 1)*asin(c*x)**2,x)*b**2*c))/(24*c)