\(\int (f+g x) (d-c^2 d x^2)^{3/2} (a+b \arccos (c x)) \, dx\) [8]

Optimal result

Integrand size = 29, antiderivative size = 346 \[ \int (f+g x) \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x)) \, dx=-\frac {b d g x \sqrt {d-c^2 d x^2}}{5 c \sqrt {1-c^2 x^2}}+\frac {3 b c d f x^2 \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}+\frac {2 b c d g x^3 \sqrt {d-c^2 d x^2}}{15 \sqrt {1-c^2 x^2}}-\frac {b c^3 d g x^5 \sqrt {d-c^2 d x^2}}{25 \sqrt {1-c^2 x^2}}-\frac {b d f \left (1-c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2}}{16 c}+\frac {3}{8} d f x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {1}{4} f x \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))-\frac {g \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))}{5 c^2 d}-\frac {3 d f \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{16 b c \sqrt {1-c^2 x^2}} \] Output:

-1/5*b*d*g*x*(-c^2*d*x^2+d)^(1/2)/c/(-c^2*x^2+1)^(1/2)+3/16*b*c*d*f*x^2*(- 
c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+2/15*b*c*d*g*x^3*(-c^2*d*x^2+d)^(1/2 
)/(-c^2*x^2+1)^(1/2)-1/25*b*c^3*d*g*x^5*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^ 
(1/2)-1/16*b*d*f*(-c^2*x^2+1)^(3/2)*(-c^2*d*x^2+d)^(1/2)/c+3/8*d*f*x*(-c^2 
*d*x^2+d)^(1/2)*(a+b*arccos(c*x))+1/4*f*x*(-c^2*d*x^2+d)^(3/2)*(a+b*arccos 
(c*x))-1/5*g*(-c^2*d*x^2+d)^(5/2)*(a+b*arccos(c*x))/c^2/d-3/16*d*f*(-c^2*d 
*x^2+d)^(1/2)*(a+b*arccos(c*x))^2/b/c/(-c^2*x^2+1)^(1/2)
 

Mathematica [A] (verified)

Time = 1.19 (sec) , antiderivative size = 337, normalized size of antiderivative = 0.97 \[ \int (f+g x) \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x)) \, dx=\frac {-1800 b c d f \sqrt {d-c^2 d x^2} \arccos (c x)^2-3600 a c d^{3/2} f \sqrt {1-c^2 x^2} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )-d \sqrt {d-c^2 d x^2} \left (-1200 b c f \cos (2 \arccos (c x))-200 b g \cos (3 \arccos (c x))+3 \left (400 b c g x+80 a \sqrt {1-c^2 x^2} \left (8 g \left (-1+c^2 x^2\right )^2+5 c^2 f x \left (-5+2 c^2 x^2\right )\right )+25 b c f \cos (4 \arccos (c x))+8 b g \cos (5 \arccos (c x))\right )\right )+20 b d \sqrt {d-c^2 d x^2} \arccos (c x) \left (-100 g \sqrt {1-c^2 x^2}+160 c^2 g x^2 \sqrt {1-c^2 x^2}+120 c f \sin (2 \arccos (c x))-10 g \sin (3 \arccos (c x))-15 c f \sin (4 \arccos (c x))-6 g \sin (5 \arccos (c x))\right )}{9600 c^2 \sqrt {1-c^2 x^2}} \] Input:

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

Output:

(-1800*b*c*d*f*Sqrt[d - c^2*d*x^2]*ArcCos[c*x]^2 - 3600*a*c*d^(3/2)*f*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]*(-1200*b*c*f*Cos[2*ArcCos[c*x]] - 200*b*g*Cos[3*Arc 
Cos[c*x]] + 3*(400*b*c*g*x + 80*a*Sqrt[1 - c^2*x^2]*(8*g*(-1 + c^2*x^2)^2 
+ 5*c^2*f*x*(-5 + 2*c^2*x^2)) + 25*b*c*f*Cos[4*ArcCos[c*x]] + 8*b*g*Cos[5* 
ArcCos[c*x]])) + 20*b*d*Sqrt[d - c^2*d*x^2]*ArcCos[c*x]*(-100*g*Sqrt[1 - c 
^2*x^2] + 160*c^2*g*x^2*Sqrt[1 - c^2*x^2] + 120*c*f*Sin[2*ArcCos[c*x]] - 1 
0*g*Sin[3*ArcCos[c*x]] - 15*c*f*Sin[4*ArcCos[c*x]] - 6*g*Sin[5*ArcCos[c*x] 
]))/(9600*c^2*Sqrt[1 - c^2*x^2])
 

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.56, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, 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.

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

\(\Big \downarrow \) 5277

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

\(\Big \downarrow \) 5263

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(d*Sqrt[d - c^2*d*x^2]*(-1/5*(b*g*x)/c + (5*b*c*f*x^2)/16 + (2*b*c*g*x^3)/ 
15 - (b*c^3*f*x^4)/16 - (b*c^3*g*x^5)/25 + (3*f*x*Sqrt[1 - c^2*x^2]*(a + b 
*ArcCos[c*x]))/8 + (f*x*(1 - c^2*x^2)^(3/2)*(a + b*ArcCos[c*x]))/4 - (g*(1 
 - c^2*x^2)^(5/2)*(a + b*ArcCos[c*x]))/(5*c^2) - (3*f*(a + b*ArcCos[c*x])^ 
2)/(16*b*c)))/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 = 0.91 (sec) , antiderivative size = 1012, normalized size of antiderivative = 2.92

method	result	size
default	\(\frac {a f x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{4}+\frac {3 a f \sqrt {-c^{2} d \,x^{2}+d}\, x d}{8}+\frac {3 a f \,d^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{8 \sqrt {c^{2} d}}-\frac {a g \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{5 c^{2} d}+b \left (\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )^{2} f d}{16 c \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (16 c^{6} x^{6}-28 c^{4} x^{4}+16 i \sqrt {-c^{2} x^{2}+1}\, x^{5} c^{5}+13 c^{2} x^{2}-20 i \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}+5 i \sqrt {-c^{2} x^{2}+1}\, x c -1\right ) g \left (i+5 \arccos \left (c x \right )\right ) d}{800 c^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (8 c^{5} x^{5}-12 c^{3} x^{3}+8 i \sqrt {-c^{2} x^{2}+1}\, x^{4} c^{4}+4 c x -8 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+i \sqrt {-c^{2} x^{2}+1}\right ) f \left (i+4 \arccos \left (c x \right )\right ) d}{256 c \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) g \left (\arccos \left (c x \right )+i\right ) d}{16 c^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) g \left (\arccos \left (c x \right )-i\right ) d}{16 c^{2} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+2 c^{3} x^{3}+i \sqrt {-c^{2} x^{2}+1}-2 c x \right ) f \left (-i+2 \arccos \left (c x \right )\right ) d}{16 c \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-4 i \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}+4 c^{4} x^{4}+3 i \sqrt {-c^{2} x^{2}+1}\, x c -5 c^{2} x^{2}+1\right ) g \left (-i+3 \arccos \left (c x \right )\right ) d}{96 c^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) g \left (11 i+45 \arccos \left (c x \right )\right ) \cos \left (4 \arccos \left (c x \right )\right ) d}{1200 c^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c^{2} x^{2}+c x \sqrt {-c^{2} x^{2}+1}-i\right ) g \left (7 i+15 \arccos \left (c x \right )\right ) \sin \left (4 \arccos \left (c x \right )\right ) d}{600 c^{2} \left (c^{2} x^{2}-1\right )}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) f \left (5 i+12 \arccos \left (c x \right )\right ) \cos \left (3 \arccos \left (c x \right )\right ) d}{256 c \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c^{2} x^{2}+c x \sqrt {-c^{2} x^{2}+1}-i\right ) f \left (17 i+28 \arccos \left (c x \right )\right ) \sin \left (3 \arccos \left (c x \right )\right ) d}{256 c \left (c^{2} x^{2}-1\right )}\right )\)	\(1012\)
parts	\(\frac {a f x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{4}+\frac {3 a f \sqrt {-c^{2} d \,x^{2}+d}\, x d}{8}+\frac {3 a f \,d^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{8 \sqrt {c^{2} d}}-\frac {a g \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{5 c^{2} d}+b \left (\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )^{2} f d}{16 c \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (16 c^{6} x^{6}-28 c^{4} x^{4}+16 i \sqrt {-c^{2} x^{2}+1}\, x^{5} c^{5}+13 c^{2} x^{2}-20 i \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}+5 i \sqrt {-c^{2} x^{2}+1}\, x c -1\right ) g \left (i+5 \arccos \left (c x \right )\right ) d}{800 c^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (8 c^{5} x^{5}-12 c^{3} x^{3}+8 i \sqrt {-c^{2} x^{2}+1}\, x^{4} c^{4}+4 c x -8 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+i \sqrt {-c^{2} x^{2}+1}\right ) f \left (i+4 \arccos \left (c x \right )\right ) d}{256 c \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) g \left (\arccos \left (c x \right )+i\right ) d}{16 c^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) g \left (\arccos \left (c x \right )-i\right ) d}{16 c^{2} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+2 c^{3} x^{3}+i \sqrt {-c^{2} x^{2}+1}-2 c x \right ) f \left (-i+2 \arccos \left (c x \right )\right ) d}{16 c \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-4 i \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}+4 c^{4} x^{4}+3 i \sqrt {-c^{2} x^{2}+1}\, x c -5 c^{2} x^{2}+1\right ) g \left (-i+3 \arccos \left (c x \right )\right ) d}{96 c^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) g \left (11 i+45 \arccos \left (c x \right )\right ) \cos \left (4 \arccos \left (c x \right )\right ) d}{1200 c^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c^{2} x^{2}+c x \sqrt {-c^{2} x^{2}+1}-i\right ) g \left (7 i+15 \arccos \left (c x \right )\right ) \sin \left (4 \arccos \left (c x \right )\right ) d}{600 c^{2} \left (c^{2} x^{2}-1\right )}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) f \left (5 i+12 \arccos \left (c x \right )\right ) \cos \left (3 \arccos \left (c x \right )\right ) d}{256 c \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c^{2} x^{2}+c x \sqrt {-c^{2} x^{2}+1}-i\right ) f \left (17 i+28 \arccos \left (c x \right )\right ) \sin \left (3 \arccos \left (c x \right )\right ) d}{256 c \left (c^{2} x^{2}-1\right )}\right )\)	\(1012\)

Input:

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

Output:

1/4*a*f*x*(-c^2*d*x^2+d)^(3/2)+3/8*a*f*(-c^2*d*x^2+d)^(1/2)*x*d+3/8*a*f*d^ 
2/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))-1/5*a*g*(-c^2 
*d*x^2+d)^(5/2)/c^2/d+b*(3/16*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c/ 
(c^2*x^2-1)*arccos(c*x)^2*f*d-1/800*(-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*(I+5*arccos(c*x))*d/c^2/(c^2*x^2- 
1)-1/256*(-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*(I 
+4*arccos(c*x))*d/c/(c^2*x^2-1)-1/16*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1 
)^(1/2)*c*x+c^2*x^2-1)*g*(arccos(c*x)+I)*d/c^2/(c^2*x^2-1)-1/16*(-d*(c^2*x 
^2-1))^(1/2)*(-I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*g*(arccos(c*x)-I)*d/c^2 
/(c^2*x^2-1)+1/16*(-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)*f*(-I+2*arccos(c*x))*d/c/(c^2*x^2-1) 
+1/96*(-d*(c^2*x^2-1))^(1/2)*(-4*I*(-c^2*x^2+1)^(1/2)*x^3*c^3+4*c^4*x^4+3* 
I*(-c^2*x^2+1)^(1/2)*c*x-5*c^2*x^2+1)*g*(-I+3*arccos(c*x))*d/c^2/(c^2*x^2- 
1)-1/1200*(-d*(c^2*x^2-1))^(1/2)*(-I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*g*( 
11*I+45*arccos(c*x))*cos(4*arccos(c*x))*d/c^2/(c^2*x^2-1)-1/600*(-d*(c^2*x 
^2-1))^(1/2)*(I*c^2*x^2+c*x*(-c^2*x^2+1)^(1/2)-I)*g*(7*I+15*arccos(c*x))*s 
in(4*arccos(c*x))*d/c^2/(c^2*x^2-1)-3/256*(-d*(c^2*x^2-1))^(1/2)*(-I*(-c^2 
*x^2+1)^(1/2)*x*c+c^2*x^2-1)*f*(5*I+12*arccos(c*x))*cos(3*arccos(c*x))*...
 

Fricas [F]

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

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

Output:

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

Sympy [F]

\[ \int (f+g x) \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x)) \, dx=\int \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {acos}{\left (c x \right )}\right ) \left (f + g x\right )\, dx \] Input:

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

Output:

Integral((-d*(c*x - 1)*(c*x + 1))**(3/2)*(a + b*acos(c*x))*(f + g*x), x)
 

Maxima [F]

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

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

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

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

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

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

Output:

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

Reduce [F]

\[ \int (f+g x) \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x)) \, dx=\frac {\sqrt {d}\, d \left (15 \mathit {asin} \left (c x \right ) a c f -10 \sqrt {-c^{2} x^{2}+1}\, a \,c^{4} f \,x^{3}-8 \sqrt {-c^{2} x^{2}+1}\, a \,c^{4} g \,x^{4}+25 \sqrt {-c^{2} x^{2}+1}\, a \,c^{2} f x +16 \sqrt {-c^{2} x^{2}+1}\, a \,c^{2} g \,x^{2}-8 \sqrt {-c^{2} x^{2}+1}\, a g -40 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) x^{3}d x \right ) b \,c^{4} g -40 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) x^{2}d x \right ) b \,c^{4} f +40 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) x d x \right ) b \,c^{2} g +40 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right )d x \right ) b \,c^{2} f +8 a g \right )}{40 c^{2}} \] Input:

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

Output:

(sqrt(d)*d*(15*asin(c*x)*a*c*f - 10*sqrt( - c**2*x**2 + 1)*a*c**4*f*x**3 - 
 8*sqrt( - c**2*x**2 + 1)*a*c**4*g*x**4 + 25*sqrt( - c**2*x**2 + 1)*a*c**2 
*f*x + 16*sqrt( - c**2*x**2 + 1)*a*c**2*g*x**2 - 8*sqrt( - c**2*x**2 + 1)* 
a*g - 40*int(sqrt( - c**2*x**2 + 1)*acos(c*x)*x**3,x)*b*c**4*g - 40*int(sq 
rt( - c**2*x**2 + 1)*acos(c*x)*x**2,x)*b*c**4*f + 40*int(sqrt( - c**2*x**2 
 + 1)*acos(c*x)*x,x)*b*c**2*g + 40*int(sqrt( - c**2*x**2 + 1)*acos(c*x),x) 
*b*c**2*f + 8*a*g))/(40*c**2)