\(\int (f+g x)^2 (d-c^2 d x^2)^{3/2} (a+b \arccos (c x)) \, dx\) [7]

Optimal result

Integrand size = 31, antiderivative size = 645 \[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x)) \, dx=-\frac {2 b d f g x \sqrt {d-c^2 d x^2}}{5 c \sqrt {1-c^2 x^2}}+\frac {3 b c d f^2 x^2 \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}-\frac {b d g^2 x^2 \sqrt {d-c^2 d x^2}}{32 c \sqrt {1-c^2 x^2}}+\frac {4 b c d f g x^3 \sqrt {d-c^2 d x^2}}{15 \sqrt {1-c^2 x^2}}+\frac {7 b c d g^2 x^4 \sqrt {d-c^2 d x^2}}{96 \sqrt {1-c^2 x^2}}-\frac {2 b c^3 d f g x^5 \sqrt {d-c^2 d x^2}}{25 \sqrt {1-c^2 x^2}}-\frac {b c^3 d g^2 x^6 \sqrt {d-c^2 d x^2}}{36 \sqrt {1-c^2 x^2}}-\frac {b d f^2 \left (1-c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2}}{16 c}+\frac {3}{8} d f^2 x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))-\frac {d g^2 x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{16 c^2}+\frac {1}{8} d g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {1}{4} f^2 x \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))+\frac {1}{6} g^2 x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))-\frac {2 f g \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))}{5 c^2 d}-\frac {3 d f^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{16 b c \sqrt {1-c^2 x^2}}-\frac {d g^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{32 b c^3 \sqrt {1-c^2 x^2}} \] Output:

-2/5*b*d*f*g*x*(-c^2*d*x^2+d)^(1/2)/c/(-c^2*x^2+1)^(1/2)+3/16*b*c*d*f^2*x^ 
2*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-1/32*b*d*g^2*x^2*(-c^2*d*x^2+d)^ 
(1/2)/c/(-c^2*x^2+1)^(1/2)+4/15*b*c*d*f*g*x^3*(-c^2*d*x^2+d)^(1/2)/(-c^2*x 
^2+1)^(1/2)+7/96*b*c*d*g^2*x^4*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-2/2 
5*b*c^3*d*f*g*x^5*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-1/36*b*c^3*d*g^2 
*x^6*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-1/16*b*d*f^2*(-c^2*x^2+1)^(3/ 
2)*(-c^2*d*x^2+d)^(1/2)/c+3/8*d*f^2*x*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x 
))-1/16*d*g^2*x*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))/c^2+1/8*d*g^2*x^3*( 
-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))+1/4*f^2*x*(-c^2*d*x^2+d)^(3/2)*(a+b* 
arccos(c*x))+1/6*g^2*x^3*(-c^2*d*x^2+d)^(3/2)*(a+b*arccos(c*x))-2/5*f*g*(- 
c^2*d*x^2+d)^(5/2)*(a+b*arccos(c*x))/c^2/d-3/16*d*f^2*(-c^2*d*x^2+d)^(1/2) 
*(a+b*arccos(c*x))^2/b/c/(-c^2*x^2+1)^(1/2)-1/32*d*g^2*(-c^2*d*x^2+d)^(1/2 
)*(a+b*arccos(c*x))^2/b/c^3/(-c^2*x^2+1)^(1/2)
 

Mathematica [A] (verified)

Time = 1.69 (sec) , antiderivative size = 591, normalized size of antiderivative = 0.92 \[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x)) \, dx=\frac {-1800 b d \left (6 c^2 f^2+g^2\right ) \sqrt {d-c^2 d x^2} \arccos (c x)^2-3600 a d^{3/2} \left (6 c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )-d \sqrt {d-c^2 d x^2} \left (14400 b c^2 f g x+23040 a c f g \sqrt {1-c^2 x^2}-36000 a c^3 f^2 x \sqrt {1-c^2 x^2}+3600 a c g^2 x \sqrt {1-c^2 x^2}-46080 a c^3 f g x^2 \sqrt {1-c^2 x^2}+14400 a c^5 f^2 x^3 \sqrt {1-c^2 x^2}-16800 a c^3 g^2 x^3 \sqrt {1-c^2 x^2}+23040 a c^5 f g x^4 \sqrt {1-c^2 x^2}+9600 a c^5 g^2 x^5 \sqrt {1-c^2 x^2}-450 b \left (16 c^2 f^2+g^2\right ) \cos (2 \arccos (c x))-2400 b c f g \cos (3 \arccos (c x))+450 b c^2 f^2 \cos (4 \arccos (c x))-225 b g^2 \cos (4 \arccos (c x))+288 b c f g \cos (5 \arccos (c x))+50 b g^2 \cos (6 \arccos (c x))\right )+60 b d \sqrt {d-c^2 d x^2} \arccos (c x) \left (-400 c f g \sqrt {1-c^2 x^2}+640 c^3 f g x^2 \sqrt {1-c^2 x^2}+15 \left (16 c^2 f^2+g^2\right ) \sin (2 \arccos (c x))-40 c f g \sin (3 \arccos (c x))-30 c^2 f^2 \sin (4 \arccos (c x))+15 g^2 \sin (4 \arccos (c x))-24 c f g \sin (5 \arccos (c x))-5 g^2 \sin (6 \arccos (c x))\right )}{57600 c^3 \sqrt {1-c^2 x^2}} \] Input:

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

Output:

(-1800*b*d*(6*c^2*f^2 + g^2)*Sqrt[d - c^2*d*x^2]*ArcCos[c*x]^2 - 3600*a*d^ 
(3/2)*(6*c^2*f^2 + g^2)*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2]) 
/(Sqrt[d]*(-1 + c^2*x^2))] - d*Sqrt[d - c^2*d*x^2]*(14400*b*c^2*f*g*x + 23 
040*a*c*f*g*Sqrt[1 - c^2*x^2] - 36000*a*c^3*f^2*x*Sqrt[1 - c^2*x^2] + 3600 
*a*c*g^2*x*Sqrt[1 - c^2*x^2] - 46080*a*c^3*f*g*x^2*Sqrt[1 - c^2*x^2] + 144 
00*a*c^5*f^2*x^3*Sqrt[1 - c^2*x^2] - 16800*a*c^3*g^2*x^3*Sqrt[1 - c^2*x^2] 
 + 23040*a*c^5*f*g*x^4*Sqrt[1 - c^2*x^2] + 9600*a*c^5*g^2*x^5*Sqrt[1 - c^2 
*x^2] - 450*b*(16*c^2*f^2 + g^2)*Cos[2*ArcCos[c*x]] - 2400*b*c*f*g*Cos[3*A 
rcCos[c*x]] + 450*b*c^2*f^2*Cos[4*ArcCos[c*x]] - 225*b*g^2*Cos[4*ArcCos[c* 
x]] + 288*b*c*f*g*Cos[5*ArcCos[c*x]] + 50*b*g^2*Cos[6*ArcCos[c*x]]) + 60*b 
*d*Sqrt[d - c^2*d*x^2]*ArcCos[c*x]*(-400*c*f*g*Sqrt[1 - c^2*x^2] + 640*c^3 
*f*g*x^2*Sqrt[1 - c^2*x^2] + 15*(16*c^2*f^2 + g^2)*Sin[2*ArcCos[c*x]] - 40 
*c*f*g*Sin[3*ArcCos[c*x]] - 30*c^2*f^2*Sin[4*ArcCos[c*x]] + 15*g^2*Sin[4*A 
rcCos[c*x]] - 24*c*f*g*Sin[5*ArcCos[c*x]] - 5*g^2*Sin[6*ArcCos[c*x]]))/(57 
600*c^3*Sqrt[1 - c^2*x^2])
 

Rubi [A] (verified)

Time = 1.01 (sec) , antiderivative size = 367, normalized size of antiderivative = 0.57, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {5277, 5263, 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.

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

\(\Big \downarrow \) 5277

\(\displaystyle \frac {d \sqrt {d-c^2 d x^2} \int (f+g x)^2 \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))dx}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 5263

\(\displaystyle \frac {d \sqrt {d-c^2 d x^2} \int \left (\left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x)) f^2+2 g x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x)) f+g^2 x^2 \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))\right )dx}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {d \sqrt {d-c^2 d x^2} \left (-\frac {g^2 (a+b \arccos (c x))^2}{32 b c^3}+\frac {1}{4} f^2 x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {3}{8} f^2 x \sqrt {1-c^2 x^2} (a+b \arccos (c x))-\frac {2 f g \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{5 c^2}-\frac {g^2 x \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{16 c^2}+\frac {1}{6} g^2 x^3 \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {1}{8} g^2 x^3 \sqrt {1-c^2 x^2} (a+b \arccos (c x))-\frac {3 f^2 (a+b \arccos (c x))^2}{16 b c}-\frac {1}{16} b c^3 f^2 x^4-\frac {2}{25} b c^3 f g x^5-\frac {1}{36} b c^3 g^2 x^6+\frac {5}{16} b c f^2 x^2+\frac {4}{15} b c f g x^3-\frac {2 b f g x}{5 c}+\frac {7}{96} b c g^2 x^4-\frac {b g^2 x^2}{32 c}\right )}{\sqrt {1-c^2 x^2}}\)

Input:

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

Output:

(d*Sqrt[d - c^2*d*x^2]*((-2*b*f*g*x)/(5*c) + (5*b*c*f^2*x^2)/16 - (b*g^2*x 
^2)/(32*c) + (4*b*c*f*g*x^3)/15 - (b*c^3*f^2*x^4)/16 + (7*b*c*g^2*x^4)/96 
- (2*b*c^3*f*g*x^5)/25 - (b*c^3*g^2*x^6)/36 + (3*f^2*x*Sqrt[1 - c^2*x^2]*( 
a + b*ArcCos[c*x]))/8 - (g^2*x*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x]))/(16* 
c^2) + (g^2*x^3*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x]))/8 + (f^2*x*(1 - c^2 
*x^2)^(3/2)*(a + b*ArcCos[c*x]))/4 + (g^2*x^3*(1 - c^2*x^2)^(3/2)*(a + b*A 
rcCos[c*x]))/6 - (2*f*g*(1 - c^2*x^2)^(5/2)*(a + b*ArcCos[c*x]))/(5*c^2) - 
 (3*f^2*(a + b*ArcCos[c*x])^2)/(16*b*c) - (g^2*(a + b*ArcCos[c*x])^2)/(32* 
b*c^3)))/Sqrt[1 - c^2*x^2]
 

Defintions of rubi rules used

rule 2009

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

rule 5263

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_ 
) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^p*(a + 
 b*ArcCos[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))
 

rule 5277

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

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.95 (sec) , antiderivative size = 1533, normalized size of antiderivative = 2.38

method	result	size
default	\(\text {Expression too large to display}\)	\(1533\)
parts	\(\text {Expression too large to display}\)	\(1533\)

Input:

int((g*x+f)^2*(-c^2*d*x^2+d)^(3/2)*(a+b*arccos(c*x)),x,method=_RETURNVERBO 
SE)
 

Output:

a*(f^2*(1/4*x*(-c^2*d*x^2+d)^(3/2)+3/4*d*(1/2*x*(-c^2*d*x^2+d)^(1/2)+1/2*d 
/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))))+g^2*(-1/6*x* 
(-c^2*d*x^2+d)^(5/2)/c^2/d+1/6/c^2*(1/4*x*(-c^2*d*x^2+d)^(3/2)+3/4*d*(1/2* 
x*(-c^2*d*x^2+d)^(1/2)+1/2*d/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d* 
x^2+d)^(1/2)))))-2/5*f*g*(-c^2*d*x^2+d)^(5/2)/c^2/d)+b*(1/32*(-d*(c^2*x^2- 
1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^3/(c^2*x^2-1)*arccos(c*x)^2*(6*c^2*f^2+g^2) 
*d-1/2304*(-d*(c^2*x^2-1))^(1/2)*(32*c^7*x^7-64*c^5*x^5+32*I*(-c^2*x^2+1)^ 
(1/2)*x^6*c^6+38*c^3*x^3-48*I*(-c^2*x^2+1)^(1/2)*x^4*c^4-6*c*x+18*I*(-c^2* 
x^2+1)^(1/2)*x^2*c^2-I*(-c^2*x^2+1)^(1/2))*g^2*(I+6*arccos(c*x))*d/c^3/(c^ 
2*x^2-1)-1/400*(-d*(c^2*x^2-1))^(1/2)*(16*c^6*x^6-28*c^4*x^4+16*I*(-c^2*x^ 
2+1)^(1/2)*x^5*c^5+13*c^2*x^2-20*I*(-c^2*x^2+1)^(1/2)*x^3*c^3+5*I*(-c^2*x^ 
2+1)^(1/2)*c*x-1)*f*g*(I+5*arccos(c*x))*d/c^2/(c^2*x^2-1)-1/512*(-d*(c^2*x 
^2-1))^(1/2)*(8*c^5*x^5-12*c^3*x^3+8*I*(-c^2*x^2+1)^(1/2)*x^4*c^4+4*c*x-8* 
I*(-c^2*x^2+1)^(1/2)*x^2*c^2+I*(-c^2*x^2+1)^(1/2))*(2*I*c^2*f^2+8*arccos(c 
*x)*c^2*f^2-I*g^2-4*arccos(c*x)*g^2)*d/c^3/(c^2*x^2-1)-1/8*(-d*(c^2*x^2-1) 
)^(1/2)*(I*(-c^2*x^2+1)^(1/2)*c*x+c^2*x^2-1)*f*g*(arccos(c*x)+I)*d/c^2/(c^ 
2*x^2-1)-1/8*(-d*(c^2*x^2-1))^(1/2)*(-I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)* 
f*g*(arccos(c*x)-I)*d/c^2/(c^2*x^2-1)+1/256*(-d*(c^2*x^2-1))^(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)*(-16*I*c^2 
*f^2+32*arccos(c*x)*c^2*f^2-I*g^2+2*arccos(c*x)*g^2)*d/c^3/(c^2*x^2-1)+...
 

Fricas [F]

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

integrate((g*x+f)^2*(-c^2*d*x^2+d)^(3/2)*(a+b*arccos(c*x)),x, algorithm="f 
ricas")
 

Output:

integral(-(a*c^2*d*g^2*x^4 + 2*a*c^2*d*f*g*x^3 - 2*a*d*f*g*x - a*d*f^2 + ( 
a*c^2*d*f^2 - a*d*g^2)*x^2 + (b*c^2*d*g^2*x^4 + 2*b*c^2*d*f*g*x^3 - 2*b*d* 
f*g*x - b*d*f^2 + (b*c^2*d*f^2 - b*d*g^2)*x^2)*arccos(c*x))*sqrt(-c^2*d*x^ 
2 + d), x)
 

Sympy [F(-1)]

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

integrate((g*x+f)**2*(-c**2*d*x**2+d)**(3/2)*(a+b*acos(c*x)),x)
 

Output:

Timed out
 

Maxima [F]

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

integrate((g*x+f)^2*(-c^2*d*x^2+d)^(3/2)*(a+b*arccos(c*x)),x, algorithm="m 
axima")
 

Output:

1/8*(2*(-c^2*d*x^2 + d)^(3/2)*x + 3*sqrt(-c^2*d*x^2 + d)*d*x + 3*d^(3/2)*a 
rcsin(c*x)/c)*a*f^2 + 1/48*a*g^2*(2*(-c^2*d*x^2 + d)^(3/2)*x/c^2 - 8*(-c^2 
*d*x^2 + d)^(5/2)*x/(c^2*d) + 3*sqrt(-c^2*d*x^2 + d)*d*x/c^2 + 3*d^(3/2)*a 
rcsin(c*x)/c^3) - 2/5*(-c^2*d*x^2 + d)^(5/2)*a*f*g/(c^2*d) + sqrt(d)*integ 
rate(-(b*c^2*d*g^2*x^4 + 2*b*c^2*d*f*g*x^3 - 2*b*d*f*g*x - b*d*f^2 + (b*c^ 
2*d*f^2 - b*d*g^2)*x^2)*sqrt(c*x + 1)*sqrt(-c*x + 1)*arctan2(sqrt(c*x + 1) 
*sqrt(-c*x + 1), c*x), x)
 

Giac [F(-2)]

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

integrate((g*x+f)^2*(-c^2*d*x^2+d)^(3/2)*(a+b*arccos(c*x)),x, algorithm="g 
iac")
 

Output:

Exception raised: RuntimeError >> an error occurred running a Giac command 
:INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve 
cteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x)) \, dx=\frac {\sqrt {d}\, d \left (90 \mathit {asin} \left (c x \right ) a \,c^{2} f^{2}+15 \mathit {asin} \left (c x \right ) a \,g^{2}-60 \sqrt {-c^{2} x^{2}+1}\, a \,c^{5} f^{2} x^{3}-96 \sqrt {-c^{2} x^{2}+1}\, a \,c^{5} f g \,x^{4}-40 \sqrt {-c^{2} x^{2}+1}\, a \,c^{5} g^{2} x^{5}+150 \sqrt {-c^{2} x^{2}+1}\, a \,c^{3} f^{2} x +192 \sqrt {-c^{2} x^{2}+1}\, a \,c^{3} f g \,x^{2}+70 \sqrt {-c^{2} x^{2}+1}\, a \,c^{3} g^{2} x^{3}-96 \sqrt {-c^{2} x^{2}+1}\, a c f g -15 \sqrt {-c^{2} x^{2}+1}\, a c \,g^{2} x -240 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) x^{4}d x \right ) b \,c^{5} g^{2}-480 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) x^{3}d x \right ) b \,c^{5} f g -240 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) x^{2}d x \right ) b \,c^{5} f^{2}+240 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) x^{2}d x \right ) b \,c^{3} g^{2}+480 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) x d x \right ) b \,c^{3} f g +240 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right )d x \right ) b \,c^{3} f^{2}+96 a c f g \right )}{240 c^{3}} \] Input:

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

Output:

(sqrt(d)*d*(90*asin(c*x)*a*c**2*f**2 + 15*asin(c*x)*a*g**2 - 60*sqrt( - c* 
*2*x**2 + 1)*a*c**5*f**2*x**3 - 96*sqrt( - c**2*x**2 + 1)*a*c**5*f*g*x**4 
- 40*sqrt( - c**2*x**2 + 1)*a*c**5*g**2*x**5 + 150*sqrt( - c**2*x**2 + 1)* 
a*c**3*f**2*x + 192*sqrt( - c**2*x**2 + 1)*a*c**3*f*g*x**2 + 70*sqrt( - c* 
*2*x**2 + 1)*a*c**3*g**2*x**3 - 96*sqrt( - c**2*x**2 + 1)*a*c*f*g - 15*sqr 
t( - c**2*x**2 + 1)*a*c*g**2*x - 240*int(sqrt( - c**2*x**2 + 1)*acos(c*x)* 
x**4,x)*b*c**5*g**2 - 480*int(sqrt( - c**2*x**2 + 1)*acos(c*x)*x**3,x)*b*c 
**5*f*g - 240*int(sqrt( - c**2*x**2 + 1)*acos(c*x)*x**2,x)*b*c**5*f**2 + 2 
40*int(sqrt( - c**2*x**2 + 1)*acos(c*x)*x**2,x)*b*c**3*g**2 + 480*int(sqrt 
( - c**2*x**2 + 1)*acos(c*x)*x,x)*b*c**3*f*g + 240*int(sqrt( - c**2*x**2 + 
 1)*acos(c*x),x)*b*c**3*f**2 + 96*a*c*f*g))/(240*c**3)