\(\int (f+g x) (d-c^2 d x^2)^{5/2} (a+b \arccos (c x)) \, dx\) [12]

Optimal result

Integrand size = 29, antiderivative size = 475 \[ \int (f+g x) \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x)) \, dx=-\frac {b d^2 g x \sqrt {d-c^2 d x^2}}{7 c \sqrt {1-c^2 x^2}}+\frac {5 b c d^2 f x^2 \sqrt {d-c^2 d x^2}}{32 \sqrt {1-c^2 x^2}}+\frac {b c d^2 g x^3 \sqrt {d-c^2 d x^2}}{7 \sqrt {1-c^2 x^2}}-\frac {3 b c^3 d^2 g x^5 \sqrt {d-c^2 d x^2}}{35 \sqrt {1-c^2 x^2}}+\frac {b c^5 d^2 g x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {1-c^2 x^2}}-\frac {5 b d^2 f \left (1-c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2}}{96 c}-\frac {b d^2 f \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2}}{36 c}+\frac {5}{16} d^2 f x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {5}{24} d f x \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))+\frac {1}{6} f x \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))-\frac {g \left (d-c^2 d x^2\right )^{7/2} (a+b \arccos (c x))}{7 c^2 d}-\frac {5 d^2 f \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{32 b c \sqrt {1-c^2 x^2}} \] Output:

-1/7*b*d^2*g*x*(-c^2*d*x^2+d)^(1/2)/c/(-c^2*x^2+1)^(1/2)+5/32*b*c*d^2*f*x^ 
2*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+1/7*b*c*d^2*g*x^3*(-c^2*d*x^2+d) 
^(1/2)/(-c^2*x^2+1)^(1/2)-3/35*b*c^3*d^2*g*x^5*(-c^2*d*x^2+d)^(1/2)/(-c^2* 
x^2+1)^(1/2)+1/49*b*c^5*d^2*g*x^7*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)- 
5/96*b*d^2*f*(-c^2*x^2+1)^(3/2)*(-c^2*d*x^2+d)^(1/2)/c-1/36*b*d^2*f*(-c^2* 
x^2+1)^(5/2)*(-c^2*d*x^2+d)^(1/2)/c+5/16*d^2*f*x*(-c^2*d*x^2+d)^(1/2)*(a+b 
*arccos(c*x))+5/24*d*f*x*(-c^2*d*x^2+d)^(3/2)*(a+b*arccos(c*x))+1/6*f*x*(- 
c^2*d*x^2+d)^(5/2)*(a+b*arccos(c*x))-1/7*g*(-c^2*d*x^2+d)^(7/2)*(a+b*arcco 
s(c*x))/c^2/d-5/32*d^2*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 = 2.00 (sec) , antiderivative size = 526, normalized size of antiderivative = 1.11 \[ \int (f+g x) \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x)) \, dx=\frac {d^2 \left (-88200 b c f \sqrt {d-c^2 d x^2} \arccos (c x)^2-176400 a c \sqrt {d} f \sqrt {1-c^2 x^2} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+\sqrt {d-c^2 d x^2} \left (-44100 b c g x-80640 a g \sqrt {1-c^2 x^2}+388080 a c^2 f x \sqrt {1-c^2 x^2}+241920 a c^2 g x^2 \sqrt {1-c^2 x^2}-305760 a c^4 f x^3 \sqrt {1-c^2 x^2}-241920 a c^4 g x^4 \sqrt {1-c^2 x^2}+94080 a c^6 f x^5 \sqrt {1-c^2 x^2}+80640 a c^6 g x^6 \sqrt {1-c^2 x^2}+66150 b c f \cos (2 \arccos (c x))+8820 b g \cos (3 \arccos (c x))-6615 b c f \cos (4 \arccos (c x))-1764 b g \cos (5 \arccos (c x))+490 b c f \cos (6 \arccos (c x))+180 b g \cos (7 \arccos (c x))\right )+84 b \sqrt {d-c^2 d x^2} \arccos (c x) \left (-1816 g \sqrt {1-c^2 x^2}+3496 c^2 g x^2 \sqrt {1-c^2 x^2}-864 g \left (1-c^2 x^2\right )^{3/2} \cos (2 \arccos (c x))-120 g \left (1-c^2 x^2\right )^{3/2} \cos (4 \arccos (c x))+1575 c f \sin (2 \arccos (c x))-280 g \sin (3 \arccos (c x))-315 c f \sin (4 \arccos (c x))-168 g \sin (5 \arccos (c x))+35 c f \sin (6 \arccos (c x))\right )\right )}{564480 c^2 \sqrt {1-c^2 x^2}} \] Input:

Integrate[(f + g*x)*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCos[c*x]),x]
 

Output:

(d^2*(-88200*b*c*f*Sqrt[d - c^2*d*x^2]*ArcCos[c*x]^2 - 176400*a*c*Sqrt[d]* 
f*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^ 
2))] + Sqrt[d - c^2*d*x^2]*(-44100*b*c*g*x - 80640*a*g*Sqrt[1 - c^2*x^2] + 
 388080*a*c^2*f*x*Sqrt[1 - c^2*x^2] + 241920*a*c^2*g*x^2*Sqrt[1 - c^2*x^2] 
 - 305760*a*c^4*f*x^3*Sqrt[1 - c^2*x^2] - 241920*a*c^4*g*x^4*Sqrt[1 - c^2* 
x^2] + 94080*a*c^6*f*x^5*Sqrt[1 - c^2*x^2] + 80640*a*c^6*g*x^6*Sqrt[1 - c^ 
2*x^2] + 66150*b*c*f*Cos[2*ArcCos[c*x]] + 8820*b*g*Cos[3*ArcCos[c*x]] - 66 
15*b*c*f*Cos[4*ArcCos[c*x]] - 1764*b*g*Cos[5*ArcCos[c*x]] + 490*b*c*f*Cos[ 
6*ArcCos[c*x]] + 180*b*g*Cos[7*ArcCos[c*x]]) + 84*b*Sqrt[d - c^2*d*x^2]*Ar 
cCos[c*x]*(-1816*g*Sqrt[1 - c^2*x^2] + 3496*c^2*g*x^2*Sqrt[1 - c^2*x^2] - 
864*g*(1 - c^2*x^2)^(3/2)*Cos[2*ArcCos[c*x]] - 120*g*(1 - c^2*x^2)^(3/2)*C 
os[4*ArcCos[c*x]] + 1575*c*f*Sin[2*ArcCos[c*x]] - 280*g*Sin[3*ArcCos[c*x]] 
 - 315*c*f*Sin[4*ArcCos[c*x]] - 168*g*Sin[5*ArcCos[c*x]] + 35*c*f*Sin[6*Ar 
cCos[c*x]])))/(564480*c^2*Sqrt[1 - c^2*x^2])
 

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 256, normalized size of antiderivative = 0.54, 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.

\(\displaystyle \int \left (d-c^2 d x^2\right )^{5/2} (f+g x) (a+b \arccos (c x)) \, dx\)

\(\Big \downarrow \) 5277

\(\displaystyle \frac {d^2 \sqrt {d-c^2 d x^2} \int (f+g x) \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))dx}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 5263

\(\displaystyle \frac {d^2 \sqrt {d-c^2 d x^2} \int \left (f (a+b \arccos (c x)) \left (1-c^2 x^2\right )^{5/2}+g x (a+b \arccos (c x)) \left (1-c^2 x^2\right )^{5/2}\right )dx}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{6} f x \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))+\frac {5}{24} f x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {5}{16} f x \sqrt {1-c^2 x^2} (a+b \arccos (c x))-\frac {g \left (1-c^2 x^2\right )^{7/2} (a+b \arccos (c x))}{7 c^2}-\frac {5 f (a+b \arccos (c x))^2}{32 b c}+\frac {1}{49} b c^5 g x^7-\frac {5}{96} b c^3 f x^4-\frac {3}{35} b c^3 g x^5-\frac {b f \left (1-c^2 x^2\right )^3}{36 c}+\frac {25}{96} b c f x^2+\frac {1}{7} b c g x^3-\frac {b g x}{7 c}\right )}{\sqrt {1-c^2 x^2}}\)

Input:

Int[(f + g*x)*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCos[c*x]),x]
 

Output:

(d^2*Sqrt[d - c^2*d*x^2]*(-1/7*(b*g*x)/c + (25*b*c*f*x^2)/96 + (b*c*g*x^3) 
/7 - (5*b*c^3*f*x^4)/96 - (3*b*c^3*g*x^5)/35 + (b*c^5*g*x^7)/49 - (b*f*(1 
- c^2*x^2)^3)/(36*c) + (5*f*x*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x]))/16 + 
(5*f*x*(1 - c^2*x^2)^(3/2)*(a + b*ArcCos[c*x]))/24 + (f*x*(1 - c^2*x^2)^(5 
/2)*(a + b*ArcCos[c*x]))/6 - (g*(1 - c^2*x^2)^(7/2)*(a + b*ArcCos[c*x]))/( 
7*c^2) - (5*f*(a + b*ArcCos[c*x])^2)/(32*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 = 1.14 (sec) , antiderivative size = 1419, normalized size of antiderivative = 2.99

method	result	size
default	\(\text {Expression too large to display}\)	\(1419\)
parts	\(\text {Expression too large to display}\)	\(1419\)

Input:

int((g*x+f)*(-c^2*d*x^2+d)^(5/2)*(a+b*arccos(c*x)),x,method=_RETURNVERBOSE 
)
 

Output:

1/6*a*f*x*(-c^2*d*x^2+d)^(5/2)+5/24*a*f*(-c^2*d*x^2+d)^(3/2)*x*d+5/16*a*f* 
(-c^2*d*x^2+d)^(1/2)*x*d^2+5/16*a*f*d^3/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2) 
*x/(-c^2*d*x^2+d)^(1/2))-1/7*a*g*(-c^2*d*x^2+d)^(7/2)/c^2/d+b*(5/32*(-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^2+1/62 
72*(-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*(I+7*arccos(c*x))*d^ 
2/c^2/(c^2*x^2-1)+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))*f*(I+6*arccos(c* 
x))*d^2/c/(c^2*x^2-1)-5/128*(-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^2/c^2/(c^2*x^2-1)-5/128*(-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^2/c^2/(c^ 
2*x^2-1)+15/256*(-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)*f*(-I+2*arccos(c*x))*d^2/c/(c^2*x^2-1) 
+1/128*(-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^2/c^2/(c^2*x 
^2-1)+1/7840*(-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+70*arccos(c*x))*cos(6*arccos(c*x))*d^2/c^2/(c^2*x^2-1)+3/15680*(-d 
*(c^2*x^2-1))^(1/2)*(I*c^2*x^2+c*x*(-c^2*x^2+1)^(1/2)-I)*g*(9*I+35*arcc...
 

Fricas [F]

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

integrate((g*x+f)*(-c^2*d*x^2+d)^(5/2)*(a+b*arccos(c*x)),x, algorithm="fri 
cas")
 

Output:

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

Sympy [F(-1)]

Timed out. \[ \int (f+g x) \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x)) \, dx=\text {Timed out} \] Input:

integrate((g*x+f)*(-c**2*d*x**2+d)**(5/2)*(a+b*acos(c*x)),x)
 

Output:

Timed out
 

Maxima [F]

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

integrate((g*x+f)*(-c^2*d*x^2+d)^(5/2)*(a+b*arccos(c*x)),x, algorithm="max 
ima")
 

Output:

1/48*(8*(-c^2*d*x^2 + d)^(5/2)*x + 10*(-c^2*d*x^2 + d)^(3/2)*d*x + 15*sqrt 
(-c^2*d*x^2 + d)*d^2*x + 15*d^(5/2)*arcsin(c*x)/c)*a*f - 1/7*(-c^2*d*x^2 + 
 d)^(7/2)*a*g/(c^2*d) + sqrt(d)*integrate((b*c^4*d^2*g*x^5 + b*c^4*d^2*f*x 
^4 - 2*b*c^2*d^2*g*x^3 - 2*b*c^2*d^2*f*x^2 + b*d^2*g*x + b*d^2*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. \[ \int (f+g x) \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x)) \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate((g*x+f)*(-c^2*d*x^2+d)^(5/2)*(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. \[ \int (f+g x) \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x)) \, dx=\int \left (f+g\,x\right )\,\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \] Input:

int((f + g*x)*(a + b*acos(c*x))*(d - c^2*d*x^2)^(5/2),x)
 

Output:

int((f + g*x)*(a + b*acos(c*x))*(d - c^2*d*x^2)^(5/2), x)
 

Reduce [F]

\[ \int (f+g x) \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x)) \, dx=\frac {\sqrt {d}\, d^{2} \left (105 \mathit {asin} \left (c x \right ) a c f +56 \sqrt {-c^{2} x^{2}+1}\, a \,c^{6} f \,x^{5}+48 \sqrt {-c^{2} x^{2}+1}\, a \,c^{6} g \,x^{6}-182 \sqrt {-c^{2} x^{2}+1}\, a \,c^{4} f \,x^{3}-144 \sqrt {-c^{2} x^{2}+1}\, a \,c^{4} g \,x^{4}+231 \sqrt {-c^{2} x^{2}+1}\, a \,c^{2} f x +144 \sqrt {-c^{2} x^{2}+1}\, a \,c^{2} g \,x^{2}-48 \sqrt {-c^{2} x^{2}+1}\, a g +336 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) x^{5}d x \right ) b \,c^{6} g +336 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) x^{4}d x \right ) b \,c^{6} f -672 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) x^{3}d x \right ) b \,c^{4} g -672 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) x^{2}d x \right ) b \,c^{4} f +336 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) x d x \right ) b \,c^{2} g +336 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right )d x \right ) b \,c^{2} f +48 a g \right )}{336 c^{2}} \] Input:

int((g*x+f)*(-c^2*d*x^2+d)^(5/2)*(a+b*acos(c*x)),x)
 

Output:

(sqrt(d)*d**2*(105*asin(c*x)*a*c*f + 56*sqrt( - c**2*x**2 + 1)*a*c**6*f*x* 
*5 + 48*sqrt( - c**2*x**2 + 1)*a*c**6*g*x**6 - 182*sqrt( - c**2*x**2 + 1)* 
a*c**4*f*x**3 - 144*sqrt( - c**2*x**2 + 1)*a*c**4*g*x**4 + 231*sqrt( - c** 
2*x**2 + 1)*a*c**2*f*x + 144*sqrt( - c**2*x**2 + 1)*a*c**2*g*x**2 - 48*sqr 
t( - c**2*x**2 + 1)*a*g + 336*int(sqrt( - c**2*x**2 + 1)*acos(c*x)*x**5,x) 
*b*c**6*g + 336*int(sqrt( - c**2*x**2 + 1)*acos(c*x)*x**4,x)*b*c**6*f - 67 
2*int(sqrt( - c**2*x**2 + 1)*acos(c*x)*x**3,x)*b*c**4*g - 672*int(sqrt( - 
c**2*x**2 + 1)*acos(c*x)*x**2,x)*b*c**4*f + 336*int(sqrt( - c**2*x**2 + 1) 
*acos(c*x)*x,x)*b*c**2*g + 336*int(sqrt( - c**2*x**2 + 1)*acos(c*x),x)*b*c 
**2*f + 48*a*g))/(336*c**2)