\(\int (f+g x)^3 (d-c^2 d x^2)^{3/2} (a+b \arccos (c x)) \, dx\) [6]

Optimal result

Integrand size = 31, antiderivative size = 904 \[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x)) \, dx=-\frac {3 b d f^2 g x \sqrt {d-c^2 d x^2}}{5 c \sqrt {1-c^2 x^2}}-\frac {2 b d g^3 x \sqrt {d-c^2 d x^2}}{35 c^3 \sqrt {1-c^2 x^2}}+\frac {3 b c d f^3 x^2 \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}-\frac {3 b d f g^2 x^2 \sqrt {d-c^2 d x^2}}{32 c \sqrt {1-c^2 x^2}}+\frac {2 b c d f^2 g x^3 \sqrt {d-c^2 d x^2}}{5 \sqrt {1-c^2 x^2}}-\frac {b d g^3 x^3 \sqrt {d-c^2 d x^2}}{105 c \sqrt {1-c^2 x^2}}+\frac {7 b c d f g^2 x^4 \sqrt {d-c^2 d x^2}}{32 \sqrt {1-c^2 x^2}}-\frac {3 b c^3 d f^2 g x^5 \sqrt {d-c^2 d x^2}}{25 \sqrt {1-c^2 x^2}}+\frac {8 b c d g^3 x^5 \sqrt {d-c^2 d x^2}}{175 \sqrt {1-c^2 x^2}}-\frac {b c^3 d f g^2 x^6 \sqrt {d-c^2 d x^2}}{12 \sqrt {1-c^2 x^2}}-\frac {b c^3 d g^3 x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {1-c^2 x^2}}-\frac {b d f^3 \left (1-c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2}}{16 c}+\frac {3}{8} d f^3 x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))-\frac {3 d f g^2 x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{16 c^2}+\frac {3}{8} d f g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {1}{4} f^3 x \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))+\frac {1}{2} f g^2 x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))-\frac {3 f^2 g \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))}{5 c^2 d}-\frac {g^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))}{5 c^4 d}+\frac {g^3 \left (d-c^2 d x^2\right )^{7/2} (a+b \arccos (c x))}{7 c^4 d^2}-\frac {3 d f^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{16 b c \sqrt {1-c^2 x^2}}-\frac {3 d f g^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{32 b c^3 \sqrt {1-c^2 x^2}} \] Output:

-3/5*b*d*f^2*g*x*(-c^2*d*x^2+d)^(1/2)/c/(-c^2*x^2+1)^(1/2)-2/35*b*d*g^3*x* 
(-c^2*d*x^2+d)^(1/2)/c^3/(-c^2*x^2+1)^(1/2)+3/16*b*c*d*f^3*x^2*(-c^2*d*x^2 
+d)^(1/2)/(-c^2*x^2+1)^(1/2)-3/32*b*d*f*g^2*x^2*(-c^2*d*x^2+d)^(1/2)/c/(-c 
^2*x^2+1)^(1/2)+2/5*b*c*d*f^2*g*x^3*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2 
)-1/105*b*d*g^3*x^3*(-c^2*d*x^2+d)^(1/2)/c/(-c^2*x^2+1)^(1/2)+7/32*b*c*d*f 
*g^2*x^4*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-3/25*b*c^3*d*f^2*g*x^5*(- 
c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+8/175*b*c*d*g^3*x^5*(-c^2*d*x^2+d)^( 
1/2)/(-c^2*x^2+1)^(1/2)-1/12*b*c^3*d*f*g^2*x^6*(-c^2*d*x^2+d)^(1/2)/(-c^2* 
x^2+1)^(1/2)-1/49*b*c^3*d*g^3*x^7*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)- 
1/16*b*d*f^3*(-c^2*x^2+1)^(3/2)*(-c^2*d*x^2+d)^(1/2)/c+3/8*d*f^3*x*(-c^2*d 
*x^2+d)^(1/2)*(a+b*arccos(c*x))-3/16*d*f*g^2*x*(-c^2*d*x^2+d)^(1/2)*(a+b*a 
rccos(c*x))/c^2+3/8*d*f*g^2*x^3*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))+1/4 
*f^3*x*(-c^2*d*x^2+d)^(3/2)*(a+b*arccos(c*x))+1/2*f*g^2*x^3*(-c^2*d*x^2+d) 
^(3/2)*(a+b*arccos(c*x))-3/5*f^2*g*(-c^2*d*x^2+d)^(5/2)*(a+b*arccos(c*x))/ 
c^2/d-1/5*g^3*(-c^2*d*x^2+d)^(5/2)*(a+b*arccos(c*x))/c^4/d+1/7*g^3*(-c^2*d 
*x^2+d)^(7/2)*(a+b*arccos(c*x))/c^4/d^2-3/16*d*f^3*(-c^2*d*x^2+d)^(1/2)*(a 
+b*arccos(c*x))^2/b/c/(-c^2*x^2+1)^(1/2)-3/32*d*f*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 = 3.12 (sec) , antiderivative size = 910, normalized size of antiderivative = 1.01 \[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x)) \, dx=\frac {-88200 b c d f \left (2 c^2 f^2+g^2\right ) \sqrt {d-c^2 d x^2} \arccos (c x)^2-176400 a c d^{3/2} f \left (2 c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )-d \sqrt {d-c^2 d x^2} \left (352800 b c^3 f^2 g x+44100 b c g^3 x+564480 a c^2 f^2 g \sqrt {1-c^2 x^2}+53760 a g^3 \sqrt {1-c^2 x^2}-588000 a c^4 f^3 x \sqrt {1-c^2 x^2}+176400 a c^2 f g^2 x \sqrt {1-c^2 x^2}-1128960 a c^4 f^2 g x^2 \sqrt {1-c^2 x^2}+26880 a c^2 g^3 x^2 \sqrt {1-c^2 x^2}+235200 a c^6 f^3 x^3 \sqrt {1-c^2 x^2}-823200 a c^4 f g^2 x^3 \sqrt {1-c^2 x^2}+564480 a c^6 f^2 g x^4 \sqrt {1-c^2 x^2}-215040 a c^4 g^3 x^4 \sqrt {1-c^2 x^2}+470400 a c^6 f g^2 x^5 \sqrt {1-c^2 x^2}+134400 a c^6 g^3 x^6 \sqrt {1-c^2 x^2}-7350 b c f \left (16 c^2 f^2+3 g^2\right ) \cos (2 \arccos (c x))-4900 b g \left (12 c^2 f^2+g^2\right ) \cos (3 \arccos (c x))+7350 b c^3 f^3 \cos (4 \arccos (c x))-11025 b c f g^2 \cos (4 \arccos (c x))+7056 b c^2 f^2 g \cos (5 \arccos (c x))-588 b g^3 \cos (5 \arccos (c x))+2450 b c f g^2 \cos (6 \arccos (c x))+300 b g^3 \cos (7 \arccos (c x))\right )+140 b d \sqrt {d-c^2 d x^2} \arccos (c x) \left (-4200 c^2 f^2 g \sqrt {1-c^2 x^2}+416 g^3 \sqrt {1-c^2 x^2}+6720 c^4 f^2 g x^2 \sqrt {1-c^2 x^2}-1256 c^2 g^3 x^2 \sqrt {1-c^2 x^2}+864 g^3 \left (1-c^2 x^2\right )^{3/2} \cos (2 \arccos (c x))+120 g^3 \left (1-c^2 x^2\right )^{3/2} \cos (4 \arccos (c x))+1680 c^3 f^3 \sin (2 \arccos (c x))+315 c f g^2 \sin (2 \arccos (c x))-420 c^2 f^2 g \sin (3 \arccos (c x))+140 g^3 \sin (3 \arccos (c x))-210 c^3 f^3 \sin (4 \arccos (c x))+315 c f g^2 \sin (4 \arccos (c x))-252 c^2 f^2 g \sin (5 \arccos (c x))+84 g^3 \sin (5 \arccos (c x))-105 c f g^2 \sin (6 \arccos (c x))\right )}{940800 c^4 \sqrt {1-c^2 x^2}} \] Input:

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

Output:

(-88200*b*c*d*f*(2*c^2*f^2 + g^2)*Sqrt[d - c^2*d*x^2]*ArcCos[c*x]^2 - 1764 
00*a*c*d^(3/2)*f*(2*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]*(352800*b*c^ 
3*f^2*g*x + 44100*b*c*g^3*x + 564480*a*c^2*f^2*g*Sqrt[1 - c^2*x^2] + 53760 
*a*g^3*Sqrt[1 - c^2*x^2] - 588000*a*c^4*f^3*x*Sqrt[1 - c^2*x^2] + 176400*a 
*c^2*f*g^2*x*Sqrt[1 - c^2*x^2] - 1128960*a*c^4*f^2*g*x^2*Sqrt[1 - c^2*x^2] 
 + 26880*a*c^2*g^3*x^2*Sqrt[1 - c^2*x^2] + 235200*a*c^6*f^3*x^3*Sqrt[1 - c 
^2*x^2] - 823200*a*c^4*f*g^2*x^3*Sqrt[1 - c^2*x^2] + 564480*a*c^6*f^2*g*x^ 
4*Sqrt[1 - c^2*x^2] - 215040*a*c^4*g^3*x^4*Sqrt[1 - c^2*x^2] + 470400*a*c^ 
6*f*g^2*x^5*Sqrt[1 - c^2*x^2] + 134400*a*c^6*g^3*x^6*Sqrt[1 - c^2*x^2] - 7 
350*b*c*f*(16*c^2*f^2 + 3*g^2)*Cos[2*ArcCos[c*x]] - 4900*b*g*(12*c^2*f^2 + 
 g^2)*Cos[3*ArcCos[c*x]] + 7350*b*c^3*f^3*Cos[4*ArcCos[c*x]] - 11025*b*c*f 
*g^2*Cos[4*ArcCos[c*x]] + 7056*b*c^2*f^2*g*Cos[5*ArcCos[c*x]] - 588*b*g^3* 
Cos[5*ArcCos[c*x]] + 2450*b*c*f*g^2*Cos[6*ArcCos[c*x]] + 300*b*g^3*Cos[7*A 
rcCos[c*x]]) + 140*b*d*Sqrt[d - c^2*d*x^2]*ArcCos[c*x]*(-4200*c^2*f^2*g*Sq 
rt[1 - c^2*x^2] + 416*g^3*Sqrt[1 - c^2*x^2] + 6720*c^4*f^2*g*x^2*Sqrt[1 - 
c^2*x^2] - 1256*c^2*g^3*x^2*Sqrt[1 - c^2*x^2] + 864*g^3*(1 - c^2*x^2)^(3/2 
)*Cos[2*ArcCos[c*x]] + 120*g^3*(1 - c^2*x^2)^(3/2)*Cos[4*ArcCos[c*x]] + 16 
80*c^3*f^3*Sin[2*ArcCos[c*x]] + 315*c*f*g^2*Sin[2*ArcCos[c*x]] - 420*c^2*f 
^2*g*Sin[3*ArcCos[c*x]] + 140*g^3*Sin[3*ArcCos[c*x]] - 210*c^3*f^3*Sin[...
 

Rubi [A] (verified)

Time = 1.25 (sec) , antiderivative size = 498, normalized size of antiderivative = 0.55, 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.

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

\(\Big \downarrow \) 5277

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

\(\Big \downarrow \) 5263

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {d \sqrt {d-c^2 d x^2} \left (-\frac {3 f g^2 (a+b \arccos (c x))^2}{32 b c^3}+\frac {1}{4} f^3 x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {3}{8} f^3 x \sqrt {1-c^2 x^2} (a+b \arccos (c x))-\frac {3 f^2 g \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{5 c^2}-\frac {3 f g^2 x \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{16 c^2}+\frac {1}{2} f g^2 x^3 \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {3}{8} f g^2 x^3 \sqrt {1-c^2 x^2} (a+b \arccos (c x))+\frac {g^3 \left (1-c^2 x^2\right )^{7/2} (a+b \arccos (c x))}{7 c^4}-\frac {g^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{5 c^4}-\frac {3 f^3 (a+b \arccos (c x))^2}{16 b c}-\frac {1}{16} b c^3 f^3 x^4-\frac {3}{25} b c^3 f^2 g x^5-\frac {1}{12} b c^3 f g^2 x^6-\frac {1}{49} b c^3 g^3 x^7-\frac {2 b g^3 x}{35 c^3}+\frac {5}{16} b c f^3 x^2+\frac {2}{5} b c f^2 g x^3-\frac {3 b f^2 g x}{5 c}+\frac {7}{32} b c f g^2 x^4-\frac {3 b f g^2 x^2}{32 c}+\frac {8}{175} b c g^3 x^5-\frac {b g^3 x^3}{105 c}\right )}{\sqrt {1-c^2 x^2}}\)

Input:

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

Output:

(d*Sqrt[d - c^2*d*x^2]*((-3*b*f^2*g*x)/(5*c) - (2*b*g^3*x)/(35*c^3) + (5*b 
*c*f^3*x^2)/16 - (3*b*f*g^2*x^2)/(32*c) + (2*b*c*f^2*g*x^3)/5 - (b*g^3*x^3 
)/(105*c) - (b*c^3*f^3*x^4)/16 + (7*b*c*f*g^2*x^4)/32 - (3*b*c^3*f^2*g*x^5 
)/25 + (8*b*c*g^3*x^5)/175 - (b*c^3*f*g^2*x^6)/12 - (b*c^3*g^3*x^7)/49 + ( 
3*f^3*x*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x]))/8 - (3*f*g^2*x*Sqrt[1 - c^2 
*x^2]*(a + b*ArcCos[c*x]))/(16*c^2) + (3*f*g^2*x^3*Sqrt[1 - c^2*x^2]*(a + 
b*ArcCos[c*x]))/8 + (f^3*x*(1 - c^2*x^2)^(3/2)*(a + b*ArcCos[c*x]))/4 + (f 
*g^2*x^3*(1 - c^2*x^2)^(3/2)*(a + b*ArcCos[c*x]))/2 - (3*f^2*g*(1 - c^2*x^ 
2)^(5/2)*(a + b*ArcCos[c*x]))/(5*c^2) - (g^3*(1 - c^2*x^2)^(5/2)*(a + b*Ar 
cCos[c*x]))/(5*c^4) + (g^3*(1 - c^2*x^2)^(7/2)*(a + b*ArcCos[c*x]))/(7*c^4 
) - (3*f^3*(a + b*ArcCos[c*x])^2)/(16*b*c) - (3*f*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 = 1.22 (sec) , antiderivative size = 2146, normalized size of antiderivative = 2.37

method	result	size
default	\(\text {Expression too large to display}\)	\(2146\)
parts	\(\text {Expression too large to display}\)	\(2146\)

Input:

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

Output:

a*(f^3*(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^3*(-1/7*x^ 
2*(-c^2*d*x^2+d)^(5/2)/c^2/d-2/35/d/c^4*(-c^2*d*x^2+d)^(5/2))+3*f*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)))))-3/5*f^2*g*(-c^2*d*x^2+d)^(5/2)/c^2/d)+b*(3/32*(-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*f*(2*c^2* 
f^2+g^2)*d-1/6272*(-d*(c^2*x^2-1))^(1/2)*(64*c^8*x^8-144*c^6*x^6+64*I*(-c^ 
2*x^2+1)^(1/2)*x^7*c^7+104*c^4*x^4-112*I*(-c^2*x^2+1)^(1/2)*x^5*c^5-25*c^2 
*x^2+56*I*(-c^2*x^2+1)^(1/2)*x^3*c^3-7*I*(-c^2*x^2+1)^(1/2)*c*x+1)*g^3*(I+ 
7*arccos(c*x))*d/c^4/(c^2*x^2-1)-1/768*(-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))*f* 
g^2*(I+6*arccos(c*x))*d/c^3/(c^2*x^2-1)-1/3200*(-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)*g*(12*I*c^2*f^2+60*arcco 
s(c*x)*c^2*f^2-I*g^2-5*arccos(c*x)*g^2)*d/c^4/(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))*f*(2*I*c^2*f^2+8*arccos 
(c*x)*c^2*f^2-3*I*g^2-12*arccos(c*x)*g^2)*d/c^3/(c^2*x^2-1)+1/384*(-d*(...
 

Fricas [F]

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

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

Output:

integral(-(a*c^2*d*g^3*x^5 + 3*a*c^2*d*f*g^2*x^4 - 3*a*d*f^2*g*x - a*d*f^3 
 + (3*a*c^2*d*f^2*g - a*d*g^3)*x^3 + (a*c^2*d*f^3 - 3*a*d*f*g^2)*x^2 + (b* 
c^2*d*g^3*x^5 + 3*b*c^2*d*f*g^2*x^4 - 3*b*d*f^2*g*x - b*d*f^3 + (3*b*c^2*d 
*f^2*g - b*d*g^3)*x^3 + (b*c^2*d*f^3 - 3*b*d*f*g^2)*x^2)*arccos(c*x))*sqrt 
(-c^2*d*x^2 + d), x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

integrate((g*x+f)^3*(-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^3 - 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*g^3 + 1/16*a*f*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)*arcsin(c*x)/c^3) - 3/5*(-c^2*d*x^2 + d)^(5/2)*a*f^2*g/(c^2*d 
) + sqrt(d)*integrate(-(b*c^2*d*g^3*x^5 + 3*b*c^2*d*f*g^2*x^4 - 3*b*d*f^2* 
g*x - b*d*f^3 + (3*b*c^2*d*f^2*g - b*d*g^3)*x^3 + (b*c^2*d*f^3 - 3*b*d*f*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. \[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x)) \, dx=\text {Exception raised: RuntimeError} \] Input:

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

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

Output:

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

Reduce [F]

\[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x)) \, dx=\frac {\sqrt {d}\, d \left (210 \mathit {asin} \left (c x \right ) a \,c^{3} f^{3}+105 \mathit {asin} \left (c x \right ) a c f \,g^{2}-140 \sqrt {-c^{2} x^{2}+1}\, a \,c^{6} f^{3} x^{3}-336 \sqrt {-c^{2} x^{2}+1}\, a \,c^{6} f^{2} g \,x^{4}-280 \sqrt {-c^{2} x^{2}+1}\, a \,c^{6} f \,g^{2} x^{5}-80 \sqrt {-c^{2} x^{2}+1}\, a \,c^{6} g^{3} x^{6}+350 \sqrt {-c^{2} x^{2}+1}\, a \,c^{4} f^{3} x +672 \sqrt {-c^{2} x^{2}+1}\, a \,c^{4} f^{2} g \,x^{2}+490 \sqrt {-c^{2} x^{2}+1}\, a \,c^{4} f \,g^{2} x^{3}+128 \sqrt {-c^{2} x^{2}+1}\, a \,c^{4} g^{3} x^{4}-336 \sqrt {-c^{2} x^{2}+1}\, a \,c^{2} f^{2} g -105 \sqrt {-c^{2} x^{2}+1}\, a \,c^{2} f \,g^{2} x -16 \sqrt {-c^{2} x^{2}+1}\, a \,c^{2} g^{3} x^{2}-32 \sqrt {-c^{2} x^{2}+1}\, a \,g^{3}-560 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) x^{5}d x \right ) b \,c^{6} g^{3}-1680 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) x^{4}d x \right ) b \,c^{6} f \,g^{2}-1680 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) x^{3}d x \right ) b \,c^{6} f^{2} g +560 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) x^{3}d x \right ) b \,c^{4} g^{3}-560 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) x^{2}d x \right ) b \,c^{6} f^{3}+1680 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) x^{2}d x \right ) b \,c^{4} f \,g^{2}+1680 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) x d x \right ) b \,c^{4} f^{2} g +560 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right )d x \right ) b \,c^{4} f^{3}+336 a \,c^{2} f^{2} g +32 a \,g^{3}\right )}{560 c^{4}} \] Input:

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

Output:

(sqrt(d)*d*(210*asin(c*x)*a*c**3*f**3 + 105*asin(c*x)*a*c*f*g**2 - 140*sqr 
t( - c**2*x**2 + 1)*a*c**6*f**3*x**3 - 336*sqrt( - c**2*x**2 + 1)*a*c**6*f 
**2*g*x**4 - 280*sqrt( - c**2*x**2 + 1)*a*c**6*f*g**2*x**5 - 80*sqrt( - c* 
*2*x**2 + 1)*a*c**6*g**3*x**6 + 350*sqrt( - c**2*x**2 + 1)*a*c**4*f**3*x + 
 672*sqrt( - c**2*x**2 + 1)*a*c**4*f**2*g*x**2 + 490*sqrt( - c**2*x**2 + 1 
)*a*c**4*f*g**2*x**3 + 128*sqrt( - c**2*x**2 + 1)*a*c**4*g**3*x**4 - 336*s 
qrt( - c**2*x**2 + 1)*a*c**2*f**2*g - 105*sqrt( - c**2*x**2 + 1)*a*c**2*f* 
g**2*x - 16*sqrt( - c**2*x**2 + 1)*a*c**2*g**3*x**2 - 32*sqrt( - c**2*x**2 
 + 1)*a*g**3 - 560*int(sqrt( - c**2*x**2 + 1)*acos(c*x)*x**5,x)*b*c**6*g** 
3 - 1680*int(sqrt( - c**2*x**2 + 1)*acos(c*x)*x**4,x)*b*c**6*f*g**2 - 1680 
*int(sqrt( - c**2*x**2 + 1)*acos(c*x)*x**3,x)*b*c**6*f**2*g + 560*int(sqrt 
( - c**2*x**2 + 1)*acos(c*x)*x**3,x)*b*c**4*g**3 - 560*int(sqrt( - c**2*x* 
*2 + 1)*acos(c*x)*x**2,x)*b*c**6*f**3 + 1680*int(sqrt( - c**2*x**2 + 1)*ac 
os(c*x)*x**2,x)*b*c**4*f*g**2 + 1680*int(sqrt( - c**2*x**2 + 1)*acos(c*x)* 
x,x)*b*c**4*f**2*g + 560*int(sqrt( - c**2*x**2 + 1)*acos(c*x),x)*b*c**4*f* 
*3 + 336*a*c**2*f**2*g + 32*a*g**3))/(560*c**4)