\(\int (d+c d x)^{5/2} (f-c f x)^{5/2} (a+b \arccos (c x)) \, dx\) [518]

Optimal result

Integrand size = 30, antiderivative size = 312 \[ \int (d+c d x)^{5/2} (f-c f x)^{5/2} (a+b \arccos (c x)) \, dx=-\frac {5 b c x^2 (d+c d x)^{5/2} (f-c f x)^{5/2}}{32 \left (1-c^2 x^2\right )^{5/2}}+\frac {5 b (d+c d x)^{5/2} (f-c f x)^{5/2}}{96 c \sqrt {1-c^2 x^2}}+\frac {b (d+c d x)^{5/2} (f-c f x)^{5/2} \sqrt {1-c^2 x^2}}{36 c}+\frac {1}{6} x (d+c d x)^{5/2} (f-c f x)^{5/2} (a+b \arccos (c x))+\frac {5 x (d+c d x)^{5/2} (f-c f x)^{5/2} (a+b \arccos (c x))}{16 \left (1-c^2 x^2\right )^2}+\frac {5 x (d+c d x)^{5/2} (f-c f x)^{5/2} (a+b \arccos (c x))}{24 \left (1-c^2 x^2\right )}+\frac {5 (d+c d x)^{5/2} (f-c f x)^{5/2} (a+b \arccos (c x))^2}{32 b c \left (1-c^2 x^2\right )^{5/2}} \] Output:

-5/32*b*c*x^2*(c*d*x+d)^(5/2)*(-c*f*x+f)^(5/2)/(-c^2*x^2+1)^(5/2)+5/96*b*( 
c*d*x+d)^(5/2)*(-c*f*x+f)^(5/2)/c/(-c^2*x^2+1)^(1/2)+1/36*b*(c*d*x+d)^(5/2 
)*(-c*f*x+f)^(5/2)*(-c^2*x^2+1)^(1/2)/c+1/6*x*(c*d*x+d)^(5/2)*(-c*f*x+f)^( 
5/2)*(a+b*arccos(c*x))+5/16*x*(c*d*x+d)^(5/2)*(-c*f*x+f)^(5/2)*(a+b*arccos 
(c*x))/(-c^2*x^2+1)^2+5*x*(c*d*x+d)^(5/2)*(-c*f*x+f)^(5/2)*(a+b*arccos(c*x 
))/(-24*c^2*x^2+24)+5/32*(c*d*x+d)^(5/2)*(-c*f*x+f)^(5/2)*(a+b*arccos(c*x) 
)^2/b/c/(-c^2*x^2+1)^(5/2)
 

Mathematica [A] (verified)

Time = 2.58 (sec) , antiderivative size = 303, normalized size of antiderivative = 0.97 \[ \int (d+c d x)^{5/2} (f-c f x)^{5/2} (a+b \arccos (c x)) \, dx=\frac {d^2 f^2 \left (-360 b \sqrt {d+c d x} \sqrt {f-c f x} \arccos (c x)^2-720 a \sqrt {d} \sqrt {f} \sqrt {1-c^2 x^2} \arctan \left (\frac {c x \sqrt {d+c d x} \sqrt {f-c f x}}{\sqrt {d} \sqrt {f} \left (-1+c^2 x^2\right )}\right )+\sqrt {d+c d x} \sqrt {f-c f x} \left (1584 a c x \sqrt {1-c^2 x^2}-1248 a c^3 x^3 \sqrt {1-c^2 x^2}+384 a c^5 x^5 \sqrt {1-c^2 x^2}+270 b \cos (2 \arccos (c x))-27 b \cos (4 \arccos (c x))+2 b \cos (6 \arccos (c x))\right )+12 b \sqrt {d+c d x} \sqrt {f-c f x} \arccos (c x) (45 \sin (2 \arccos (c x))-9 \sin (4 \arccos (c x))+\sin (6 \arccos (c x)))\right )}{2304 c \sqrt {1-c^2 x^2}} \] Input:

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

Output:

(d^2*f^2*(-360*b*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*ArcCos[c*x]^2 - 720*a*Sqr 
t[d]*Sqrt[f]*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[f - c*f*x] 
)/(Sqrt[d]*Sqrt[f]*(-1 + c^2*x^2))] + Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*(158 
4*a*c*x*Sqrt[1 - c^2*x^2] - 1248*a*c^3*x^3*Sqrt[1 - c^2*x^2] + 384*a*c^5*x 
^5*Sqrt[1 - c^2*x^2] + 270*b*Cos[2*ArcCos[c*x]] - 27*b*Cos[4*ArcCos[c*x]] 
+ 2*b*Cos[6*ArcCos[c*x]]) + 12*b*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*ArcCos[c* 
x]*(45*Sin[2*ArcCos[c*x]] - 9*Sin[4*ArcCos[c*x]] + Sin[6*ArcCos[c*x]])))/( 
2304*c*Sqrt[1 - c^2*x^2])
 

Rubi [A] (verified)

Time = 0.84 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.64, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5179, 5159, 241, 5159, 244, 2009, 5157, 15, 5153}

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.

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

\(\Big \downarrow \) 5179

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

\(\Big \downarrow \) 5159

\(\displaystyle \frac {(c d x+d)^{5/2} (f-c f x)^{5/2} \left (\frac {5}{6} \int \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))dx+\frac {1}{6} b c \int x \left (1-c^2 x^2\right )^2dx+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))\right )}{\left (1-c^2 x^2\right )^{5/2}}\)

\(\Big \downarrow \) 241

\(\displaystyle \frac {(c d x+d)^{5/2} (f-c f x)^{5/2} \left (\frac {5}{6} \int \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))dx+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))-\frac {b \left (1-c^2 x^2\right )^3}{36 c}\right )}{\left (1-c^2 x^2\right )^{5/2}}\)

\(\Big \downarrow \) 5159

\(\displaystyle \frac {(c d x+d)^{5/2} (f-c f x)^{5/2} \left (\frac {5}{6} \left (\frac {3}{4} \int \sqrt {1-c^2 x^2} (a+b \arccos (c x))dx+\frac {1}{4} b c \int x \left (1-c^2 x^2\right )dx+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))\right )+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))-\frac {b \left (1-c^2 x^2\right )^3}{36 c}\right )}{\left (1-c^2 x^2\right )^{5/2}}\)

\(\Big \downarrow \) 244

\(\displaystyle \frac {(c d x+d)^{5/2} (f-c f x)^{5/2} \left (\frac {5}{6} \left (\frac {3}{4} \int \sqrt {1-c^2 x^2} (a+b \arccos (c x))dx+\frac {1}{4} b c \int \left (x-c^2 x^3\right )dx+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))\right )+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))-\frac {b \left (1-c^2 x^2\right )^3}{36 c}\right )}{\left (1-c^2 x^2\right )^{5/2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {(c d x+d)^{5/2} (f-c f x)^{5/2} \left (\frac {5}{6} \left (\frac {3}{4} \int \sqrt {1-c^2 x^2} (a+b \arccos (c x))dx+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))-\frac {b \left (1-c^2 x^2\right )^3}{36 c}\right )}{\left (1-c^2 x^2\right )^{5/2}}\)

\(\Big \downarrow \) 5157

\(\displaystyle \frac {(c d x+d)^{5/2} (f-c f x)^{5/2} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {a+b \arccos (c x)}{\sqrt {1-c^2 x^2}}dx+\frac {1}{2} b c \int xdx+\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))-\frac {b \left (1-c^2 x^2\right )^3}{36 c}\right )}{\left (1-c^2 x^2\right )^{5/2}}\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {(c d x+d)^{5/2} (f-c f x)^{5/2} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {a+b \arccos (c x)}{\sqrt {1-c^2 x^2}}dx+\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))+\frac {1}{4} b c x^2\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))-\frac {b \left (1-c^2 x^2\right )^3}{36 c}\right )}{\left (1-c^2 x^2\right )^{5/2}}\)

\(\Big \downarrow \) 5153

\(\displaystyle \frac {(c d x+d)^{5/2} (f-c f x)^{5/2} \left (\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))+\frac {5}{6} \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))-\frac {(a+b \arccos (c x))^2}{4 b c}+\frac {1}{4} b c x^2\right )+\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )-\frac {b \left (1-c^2 x^2\right )^3}{36 c}\right )}{\left (1-c^2 x^2\right )^{5/2}}\)

Input:

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

Output:

((d + c*d*x)^(5/2)*(f - c*f*x)^(5/2)*(-1/36*(b*(1 - c^2*x^2)^3)/c + (x*(1 
- c^2*x^2)^(5/2)*(a + b*ArcCos[c*x]))/6 + (5*((b*c*(x^2/2 - (c^2*x^4)/4))/ 
4 + (x*(1 - c^2*x^2)^(3/2)*(a + b*ArcCos[c*x]))/4 + (3*((b*c*x^2)/4 + (x*S 
qrt[1 - c^2*x^2]*(a + b*ArcCos[c*x]))/2 - (a + b*ArcCos[c*x])^2/(4*b*c)))/ 
4))/6))/(1 - c^2*x^2)^(5/2)
 

Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

rule 241

Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ 
(2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -1]
 

rule 244

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand 
Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p 
, 0]
 

rule 2009

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

rule 5153

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S 
ymbol] :> Simp[(-(b*c*(n + 1))^(-1))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2] 
]*(a + b*ArcCos[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^ 
2*d + e, 0] && NeQ[n, -1]
 

rule 5157

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_S 
ymbol] :> Simp[x*Sqrt[d + e*x^2]*((a + b*ArcCos[c*x])^n/2), x] + (Simp[(1/2 
)*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]]   Int[(a + b*ArcCos[c*x])^n/Sqrt[ 
1 - c^2*x^2], x], x] + Simp[b*c*(n/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2 
]]   Int[x*(a + b*ArcCos[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x 
] && EqQ[c^2*d + e, 0] && GtQ[n, 0]
 

rule 5159

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x 
_Symbol] :> Simp[x*(d + e*x^2)^p*((a + b*ArcCos[c*x])^n/(2*p + 1)), x] + (S 
imp[2*d*(p/(2*p + 1))   Int[(d + e*x^2)^(p - 1)*(a + b*ArcCos[c*x])^n, x], 
x] + Simp[b*c*(n/(2*p + 1))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]   Int[x*(1 
- c^2*x^2)^(p - 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c 
, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0]
 

rule 5179

Int[((a_.) + ArcCos[(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*ArcCos[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]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 1.72 (sec) , antiderivative size = 895, normalized size of antiderivative = 2.87

method	result	size
default	\(-\frac {a \left (c d x +d \right )^{\frac {5}{2}} \left (-c f x +f \right )^{\frac {7}{2}}}{6 c f}-\frac {a d \left (c d x +d \right )^{\frac {3}{2}} \left (-c f x +f \right )^{\frac {7}{2}}}{6 c f}-\frac {a \,d^{2} \sqrt {c d x +d}\, \left (-c f x +f \right )^{\frac {7}{2}}}{8 c f}+\frac {a \,d^{2} \left (-c f x +f \right )^{\frac {5}{2}} \sqrt {c d x +d}}{24 c}+\frac {5 a \,d^{2} f \left (-c f x +f \right )^{\frac {3}{2}} \sqrt {c d x +d}}{48 c}+\frac {5 a \,d^{2} f^{2} \sqrt {-c f x +f}\, \sqrt {c d x +d}}{16 c}+\frac {5 a \,d^{3} f^{3} \sqrt {\left (-c f x +f \right ) \left (c d x +d \right )}\, \arctan \left (\frac {\sqrt {c^{2} d f}\, x}{\sqrt {-c^{2} d f \,x^{2}+d f}}\right )}{16 \sqrt {-c f x +f}\, \sqrt {c d x +d}\, \sqrt {c^{2} d f}}+b \left (\frac {5 \sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )^{2} d^{2} f^{2}}{32 \left (c^{2} x^{2}-1\right ) c}+\frac {\sqrt {-f \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \left (32 c^{7} x^{7}-64 c^{5} x^{5}+32 i \sqrt {-c^{2} x^{2}+1}\, x^{6} c^{6}+38 c^{3} x^{3}-48 i \sqrt {-c^{2} x^{2}+1}\, x^{4} c^{4}-6 c x +18 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}-i \sqrt {-c^{2} x^{2}+1}\right ) \left (i+6 \arccos \left (c x \right )\right ) d^{2} f^{2}}{2304 \left (c^{2} x^{2}-1\right ) c}+\frac {15 \sqrt {-f \left (c x -1\right )}\, \sqrt {d \left (c x +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 ) \left (-i+2 \arccos \left (c x \right )\right ) d^{2} f^{2}}{256 \left (c^{2} x^{2}-1\right ) c}+\frac {5 \sqrt {-f \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \left (-i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \left (5 i+24 \arccos \left (c x \right )\right ) \cos \left (5 \arccos \left (c x \right )\right ) d^{2} f^{2}}{4608 \left (c^{2} x^{2}-1\right ) c}+\frac {\sqrt {-f \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \left (i c^{2} x^{2}+c x \sqrt {-c^{2} x^{2}+1}-i\right ) \left (29 i+96 \arccos \left (c x \right )\right ) \sin \left (5 \arccos \left (c x \right )\right ) d^{2} f^{2}}{4608 \left (c^{2} x^{2}-1\right ) c}-\frac {9 \sqrt {-f \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \left (-i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \left (3 i+8 \arccos \left (c x \right )\right ) \cos \left (3 \arccos \left (c x \right )\right ) d^{2} f^{2}}{512 \left (c^{2} x^{2}-1\right ) c}-\frac {3 \sqrt {-f \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \left (i c^{2} x^{2}+c x \sqrt {-c^{2} x^{2}+1}-i\right ) \left (11 i+16 \arccos \left (c x \right )\right ) \sin \left (3 \arccos \left (c x \right )\right ) d^{2} f^{2}}{512 \left (c^{2} x^{2}-1\right ) c}\right )\)	\(895\)
parts	\(-\frac {a \left (c d x +d \right )^{\frac {5}{2}} \left (-c f x +f \right )^{\frac {7}{2}}}{6 c f}-\frac {a d \left (c d x +d \right )^{\frac {3}{2}} \left (-c f x +f \right )^{\frac {7}{2}}}{6 c f}-\frac {a \,d^{2} \sqrt {c d x +d}\, \left (-c f x +f \right )^{\frac {7}{2}}}{8 c f}+\frac {a \,d^{2} \left (-c f x +f \right )^{\frac {5}{2}} \sqrt {c d x +d}}{24 c}+\frac {5 a \,d^{2} f \left (-c f x +f \right )^{\frac {3}{2}} \sqrt {c d x +d}}{48 c}+\frac {5 a \,d^{2} f^{2} \sqrt {-c f x +f}\, \sqrt {c d x +d}}{16 c}+\frac {5 a \,d^{3} f^{3} \sqrt {\left (-c f x +f \right ) \left (c d x +d \right )}\, \arctan \left (\frac {\sqrt {c^{2} d f}\, x}{\sqrt {-c^{2} d f \,x^{2}+d f}}\right )}{16 \sqrt {-c f x +f}\, \sqrt {c d x +d}\, \sqrt {c^{2} d f}}+b \left (\frac {5 \sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )^{2} d^{2} f^{2}}{32 \left (c^{2} x^{2}-1\right ) c}+\frac {\sqrt {-f \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \left (32 c^{7} x^{7}-64 c^{5} x^{5}+32 i \sqrt {-c^{2} x^{2}+1}\, x^{6} c^{6}+38 c^{3} x^{3}-48 i \sqrt {-c^{2} x^{2}+1}\, x^{4} c^{4}-6 c x +18 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}-i \sqrt {-c^{2} x^{2}+1}\right ) \left (i+6 \arccos \left (c x \right )\right ) d^{2} f^{2}}{2304 \left (c^{2} x^{2}-1\right ) c}+\frac {15 \sqrt {-f \left (c x -1\right )}\, \sqrt {d \left (c x +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 ) \left (-i+2 \arccos \left (c x \right )\right ) d^{2} f^{2}}{256 \left (c^{2} x^{2}-1\right ) c}+\frac {5 \sqrt {-f \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \left (-i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \left (5 i+24 \arccos \left (c x \right )\right ) \cos \left (5 \arccos \left (c x \right )\right ) d^{2} f^{2}}{4608 \left (c^{2} x^{2}-1\right ) c}+\frac {\sqrt {-f \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \left (i c^{2} x^{2}+c x \sqrt {-c^{2} x^{2}+1}-i\right ) \left (29 i+96 \arccos \left (c x \right )\right ) \sin \left (5 \arccos \left (c x \right )\right ) d^{2} f^{2}}{4608 \left (c^{2} x^{2}-1\right ) c}-\frac {9 \sqrt {-f \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \left (-i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \left (3 i+8 \arccos \left (c x \right )\right ) \cos \left (3 \arccos \left (c x \right )\right ) d^{2} f^{2}}{512 \left (c^{2} x^{2}-1\right ) c}-\frac {3 \sqrt {-f \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \left (i c^{2} x^{2}+c x \sqrt {-c^{2} x^{2}+1}-i\right ) \left (11 i+16 \arccos \left (c x \right )\right ) \sin \left (3 \arccos \left (c x \right )\right ) d^{2} f^{2}}{512 \left (c^{2} x^{2}-1\right ) c}\right )\)	\(895\)

Input:

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

Output:

-1/6*a/c/f*(c*d*x+d)^(5/2)*(-c*f*x+f)^(7/2)-1/6*a*d/c/f*(c*d*x+d)^(3/2)*(- 
c*f*x+f)^(7/2)-1/8*a*d^2/c/f*(c*d*x+d)^(1/2)*(-c*f*x+f)^(7/2)+1/24*a*d^2/c 
*(-c*f*x+f)^(5/2)*(c*d*x+d)^(1/2)+5/48*a*d^2*f/c*(-c*f*x+f)^(3/2)*(c*d*x+d 
)^(1/2)+5/16*a*d^2*f^2/c*(-c*f*x+f)^(1/2)*(c*d*x+d)^(1/2)+5/16*a*d^3*f^3*( 
(-c*f*x+f)*(c*d*x+d))^(1/2)/(-c*f*x+f)^(1/2)/(c*d*x+d)^(1/2)/(c^2*d*f)^(1/ 
2)*arctan((c^2*d*f)^(1/2)*x/(-c^2*d*f*x^2+d*f)^(1/2))+b*(5/32*(d*(c*x+1))^ 
(1/2)*(-f*(c*x-1))^(1/2)*(-c^2*x^2+1)^(1/2)/(c^2*x^2-1)/c*arccos(c*x)^2*d^ 
2*f^2+1/2304*(-f*(c*x-1))^(1/2)*(d*(c*x+1))^(1/2)*(32*c^7*x^7-64*c^5*x^5+3 
2*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))*(I+6*arccos(c* 
x))*d^2*f^2/(c^2*x^2-1)/c+15/256*(-f*(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)*(-I+2*a 
rccos(c*x))*d^2*f^2/(c^2*x^2-1)/c+5/4608*(-f*(c*x-1))^(1/2)*(d*(c*x+1))^(1 
/2)*(-I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*(5*I+24*arccos(c*x))*cos(5*arcco 
s(c*x))*d^2*f^2/(c^2*x^2-1)/c+1/4608*(-f*(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)*(29*I+96*arccos(c*x))*sin(5*arccos(c* 
x))*d^2*f^2/(c^2*x^2-1)/c-9/512*(-f*(c*x-1))^(1/2)*(d*(c*x+1))^(1/2)*(-I*( 
-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*(3*I+8*arccos(c*x))*cos(3*arccos(c*x))*d^ 
2*f^2/(c^2*x^2-1)/c-3/512*(-f*(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)*(11*I+16*arccos(c*x))*sin(3*arccos(c*x))*d^2*...
 

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

b*sqrt(d)*sqrt(f)*integrate((c^4*d^2*f^2*x^4 - 2*c^2*d^2*f^2*x^2 + d^2*f^2 
)*sqrt(c*x + 1)*sqrt(-c*x + 1)*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x), 
 x) + 1/48*(15*sqrt(-c^2*d*f*x^2 + d*f)*d^2*f^2*x + 15*d^3*f^3*arcsin(c*x) 
/(sqrt(d*f)*c) + 10*(-c^2*d*f*x^2 + d*f)^(3/2)*d*f*x + 8*(-c^2*d*f*x^2 + d 
*f)^(5/2)*x)*a
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (d+c d x)^{5/2} (f-c f x)^{5/2} (a+b \arccos (c x)) \, dx=\frac {\sqrt {f}\, \sqrt {d}\, d^{2} f^{2} \left (-30 \mathit {asin} \left (\frac {\sqrt {-c x +1}}{\sqrt {2}}\right ) a +8 \sqrt {c x +1}\, \sqrt {-c x +1}\, a \,c^{5} x^{5}-26 \sqrt {c x +1}\, \sqrt {-c x +1}\, a \,c^{3} x^{3}+33 \sqrt {c x +1}\, \sqrt {-c x +1}\, a c x +48 \left (\int \sqrt {c x +1}\, \sqrt {-c x +1}\, \mathit {acos} \left (c x \right ) x^{4}d x \right ) b \,c^{5}-96 \left (\int \sqrt {c x +1}\, \sqrt {-c x +1}\, \mathit {acos} \left (c x \right ) x^{2}d x \right ) b \,c^{3}+48 \left (\int \sqrt {c x +1}\, \sqrt {-c x +1}\, \mathit {acos} \left (c x \right )d x \right ) b c \right )}{48 c} \] Input:

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

Output:

(sqrt(f)*sqrt(d)*d**2*f**2*( - 30*asin(sqrt( - c*x + 1)/sqrt(2))*a + 8*sqr 
t(c*x + 1)*sqrt( - c*x + 1)*a*c**5*x**5 - 26*sqrt(c*x + 1)*sqrt( - c*x + 1 
)*a*c**3*x**3 + 33*sqrt(c*x + 1)*sqrt( - c*x + 1)*a*c*x + 48*int(sqrt(c*x 
+ 1)*sqrt( - c*x + 1)*acos(c*x)*x**4,x)*b*c**5 - 96*int(sqrt(c*x + 1)*sqrt 
( - c*x + 1)*acos(c*x)*x**2,x)*b*c**3 + 48*int(sqrt(c*x + 1)*sqrt( - c*x + 
 1)*acos(c*x),x)*b*c))/(48*c)