\(\int \frac {(d+c d x)^{5/2} (a+b \arccos (c x))}{(f-c f x)^{3/2}} \, dx\) [530]

Optimal result

Integrand size = 30, antiderivative size = 400 \[ \int \frac {(d+c d x)^{5/2} (a+b \arccos (c x))}{(f-c f x)^{3/2}} \, dx=-\frac {4 b d^4 x \left (1-c^2 x^2\right )^{3/2}}{(d+c d x)^{3/2} (f-c f x)^{3/2}}-\frac {b c d^4 x^2 \left (1-c^2 x^2\right )^{3/2}}{4 (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac {15 b d^4 \left (1-c^2 x^2\right )^{3/2} \arccos (c x)^2}{4 c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac {8 d^4 (1+c x) \left (1-c^2 x^2\right ) (a+b \arccos (c x))}{c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac {4 d^4 \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))}{c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac {d^4 x \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))}{2 (d+c d x)^{3/2} (f-c f x)^{3/2}}-\frac {15 d^4 \left (1-c^2 x^2\right )^{3/2} \arccos (c x) (a+b \arccos (c x))}{2 c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac {8 b d^4 \left (1-c^2 x^2\right )^{3/2} \log (1-c x)}{c (d+c d x)^{3/2} (f-c f x)^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 4.15 (sec) , antiderivative size = 314, normalized size of antiderivative = 0.78 \[ \int \frac {(d+c d x)^{5/2} (a+b \arccos (c x))}{(f-c f x)^{3/2}} \, dx=\frac {d^2 (1+c x) \left (4 b \sqrt {d+c d x} \sqrt {f-c f x} \sqrt {1-c^2 x^2} \left (-24+7 c x+c^2 x^2\right ) \arccos (c x)+30 b (-1+c x) \sqrt {d+c d x} \sqrt {f-c f x} \arccos (c x)^2+60 a \sqrt {d} \sqrt {f} (-1+c x) \sqrt {1-c^2 x^2} \arctan \left (\frac {c x \sqrt {d+c d x} \sqrt {f-c f x}}{\sqrt {d} \sqrt {f} \left (-1+c^2 x^2\right )}\right )+\sqrt {d+c d x} \sqrt {f-c f x} \left (4 a \sqrt {1-c^2 x^2} \left (-24+7 c x+c^2 x^2\right )+b \left (1-33 c x+30 c^2 x^2+2 c^3 x^3\right )-128 b (-1+c x) \log \left (\sin \left (\frac {1}{2} \arccos (c x)\right )\right )\right )\right ) \sin ^2\left (\frac {1}{2} \arccos (c x)\right )}{4 c f^2 (-1+c x) \left (1-c^2 x^2\right )^{3/2}} \] Input:

Integrate[((d + c*d*x)^(5/2)*(a + b*ArcCos[c*x]))/(f - c*f*x)^(3/2),x]
 

Output:

(d^2*(1 + c*x)*(4*b*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*Sqrt[1 - c^2*x^2]*(-24 
 + 7*c*x + c^2*x^2)*ArcCos[c*x] + 30*b*(-1 + c*x)*Sqrt[d + c*d*x]*Sqrt[f - 
 c*f*x]*ArcCos[c*x]^2 + 60*a*Sqrt[d]*Sqrt[f]*(-1 + c*x)*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]*(4*a*Sqrt[1 - c^2*x^2]*(-24 + 7*c*x 
 + c^2*x^2) + b*(1 - 33*c*x + 30*c^2*x^2 + 2*c^3*x^3) - 128*b*(-1 + c*x)*L 
og[Sin[ArcCos[c*x]/2]]))*Sin[ArcCos[c*x]/2]^2)/(4*c*f^2*(-1 + c*x)*(1 - c^ 
2*x^2)^(3/2))
 

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.47, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {5179, 27, 5261, 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.

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

\(\Big \downarrow \) 5179

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5261

\(\displaystyle \frac {d^4 \left (1-c^2 x^2\right )^{3/2} \left (b c \int \left (\frac {x}{2}-\frac {15 \arcsin (c x)}{2 c \sqrt {1-c^2 x^2}}+\frac {4}{c}+\frac {8 (c x+1)}{c \left (1-c^2 x^2\right )}\right )dx-\frac {15 \arcsin (c x) (a+b \arccos (c x))}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))+\frac {4 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{c}+\frac {8 (c x+1) (a+b \arccos (c x))}{c \sqrt {1-c^2 x^2}}\right )}{(c d x+d)^{3/2} (f-c f x)^{3/2}}\)

\(\Big \downarrow \) 2009

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

Input:

Int[((d + c*d*x)^(5/2)*(a + b*ArcCos[c*x]))/(f - c*f*x)^(3/2),x]
 

Output:

(d^4*(1 - c^2*x^2)^(3/2)*((8*(1 + c*x)*(a + b*ArcCos[c*x]))/(c*Sqrt[1 - c^ 
2*x^2]) + (4*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x]))/c + (x*Sqrt[1 - c^2*x^ 
2]*(a + b*ArcCos[c*x]))/2 - (15*(a + b*ArcCos[c*x])*ArcSin[c*x])/(2*c) + b 
*c*((4*x)/c + x^2/4 - (15*ArcSin[c*x]^2)/(4*c^2) - (8*Log[1 - c*x])/c^2))) 
/((d + c*d*x)^(3/2)*(f - c*f*x)^(3/2))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

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

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]
 

rule 5261

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))*((f_) + (g_.)*(x_))^(m_.)*((d_) + (e 
_.)*(x_)^2)^(p_), x_Symbol] :> With[{u = IntHide[(f + g*x)^m*(d + e*x^2)^p, 
 x]}, Simp[(a + b*ArcCos[c*x])   u, x] + Simp[b*c   Int[1/Sqrt[1 - c^2*x^2] 
   u, x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[c^2*d + e, 0] && IG 
tQ[m, 0] && ILtQ[p + 1/2, 0] && GtQ[d, 0] && (LtQ[m, -2*p - 1] || GtQ[m, 3] 
)
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 36.72 (sec) , antiderivative size = 906, normalized size of antiderivative = 2.26

method	result	size
default	\(-\frac {15 \sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \sqrt {-c^{2} x^{2}+1}\, b \arccos \left (c x \right )^{2} d^{2}}{4 \left (c x -1\right ) f^{2} c \left (c x +1\right )}-\frac {15 \sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \sqrt {-c^{2} x^{2}+1}\, a \arccos \left (c x \right ) d^{2}}{2 \left (c x -1\right ) f^{2} c \left (c x +1\right )}+\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \left (2 c^{2} x^{2}+4 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+4 c^{3} x^{3}-1+2 i \sqrt {-c^{2} x^{2}+1}\, x c -i \sqrt {-c^{2} x^{2}+1}-3 c x \right ) \left (i b +2 b \arccos \left (c x \right )+2 a \right ) d^{2}}{32 \left (c x -1\right ) f^{2} c \left (c x +1\right )}-\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right ) \left (b \arccos \left (c x \right )+a +i b \right ) d^{2}}{\left (c x -1\right ) f^{2} c \left (c x +1\right )}+\frac {2 \sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \left (-i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \left (b \arccos \left (c x \right )+a -i b \right ) d^{2}}{\left (c x -1\right ) f^{2} c \left (c x +1\right )}-\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \left (-2 i \sqrt {-c^{2} x^{2}+1}\, x c +2 c^{2} x^{2}-i \sqrt {-c^{2} x^{2}+1}+c x -1\right ) \left (-i b +2 b \arccos \left (c x \right )+2 a \right ) d^{2}}{32 \left (c x -1\right ) f^{2} c \left (c x +1\right )}-\frac {16 i \sqrt {-c^{2} x^{2}+1}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, b \arccos \left (c x \right ) d^{2}}{\left (c x -1\right ) f^{2} c \left (c x +1\right )}-\frac {8 \sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \left (-i \sqrt {-c^{2} x^{2}+1}+c x +1\right ) \left (a +b \arccos \left (c x \right )\right ) d^{2}}{\left (c x -1\right ) f^{2} c \left (c x +1\right )}+\frac {16 \sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \sqrt {-c^{2} x^{2}+1}\, b \ln \left (i \sqrt {-c^{2} x^{2}+1}+c x -1\right ) d^{2}}{\left (c x -1\right ) f^{2} c \left (c x +1\right )}+\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \left (-i \sqrt {-c^{2} x^{2}+1}+c x +1\right ) \left (15 i b +16 b \arccos \left (c x \right )+16 a \right ) \cos \left (2 \arccos \left (c x \right )\right ) d^{2}}{16 \left (c x -1\right ) f^{2} c \left (c x +1\right )}+\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \left (i c x +\sqrt {-c^{2} x^{2}+1}+i\right ) \left (8 i b +7 b \arccos \left (c x \right )+7 a \right ) \sin \left (2 \arccos \left (c x \right )\right ) d^{2}}{8 \left (c x -1\right ) f^{2} c \left (c x +1\right )}\)	\(906\)

Input:

int((c*d*x+d)^(5/2)*(a+b*arccos(c*x))/(-c*f*x+f)^(3/2),x,method=_RETURNVER 
BOSE)
 

Output:

-15/4*(d*(c*x+1))^(1/2)*(-f*(c*x-1))^(1/2)*(-c^2*x^2+1)^(1/2)/(c*x-1)/f^2/ 
c/(c*x+1)*b*arccos(c*x)^2*d^2-15/2*(d*(c*x+1))^(1/2)*(-f*(c*x-1))^(1/2)*(- 
c^2*x^2+1)^(1/2)/(c*x-1)/f^2/c/(c*x+1)*a*arccos(c*x)*d^2+1/32*(d*(c*x+1))^ 
(1/2)*(-f*(c*x-1))^(1/2)*(2*c^2*x^2+4*I*(-c^2*x^2+1)^(1/2)*x^2*c^2+4*c^3*x 
^3-1+2*I*(-c^2*x^2+1)^(1/2)*c*x-I*(-c^2*x^2+1)^(1/2)-3*c*x)*(I*b+2*b*arcco 
s(c*x)+2*a)*d^2/(c*x-1)/f^2/c/(c*x+1)-(d*(c*x+1))^(1/2)*(-f*(c*x-1))^(1/2) 
*(1+c*x+I*(-c^2*x^2+1)^(1/2))*(b*arccos(c*x)+a+I*b)*d^2/(c*x-1)/f^2/c/(c*x 
+1)+2*(d*(c*x+1))^(1/2)*(-f*(c*x-1))^(1/2)*(-I*(-c^2*x^2+1)^(1/2)*x*c+c^2* 
x^2-1)*(b*arccos(c*x)+a-I*b)*d^2/(c*x-1)/f^2/c/(c*x+1)-1/32*(d*(c*x+1))^(1 
/2)*(-f*(c*x-1))^(1/2)*(-2*I*(-c^2*x^2+1)^(1/2)*c*x+2*c^2*x^2-I*(-c^2*x^2+ 
1)^(1/2)+c*x-1)*(-I*b+2*b*arccos(c*x)+2*a)*d^2/(c*x-1)/f^2/c/(c*x+1)-16*I* 
(-c^2*x^2+1)^(1/2)*(d*(c*x+1))^(1/2)*(-f*(c*x-1))^(1/2)/(c*x-1)/f^2/c/(c*x 
+1)*b*arccos(c*x)*d^2-8*(d*(c*x+1))^(1/2)*(-f*(c*x-1))^(1/2)*(-I*(-c^2*x^2 
+1)^(1/2)+c*x+1)*(a+b*arccos(c*x))*d^2/(c*x-1)/f^2/c/(c*x+1)+16*(d*(c*x+1) 
)^(1/2)*(-f*(c*x-1))^(1/2)*(-c^2*x^2+1)^(1/2)/(c*x-1)/f^2/c/(c*x+1)*b*ln(I 
*(-c^2*x^2+1)^(1/2)+c*x-1)*d^2+1/16*(d*(c*x+1))^(1/2)*(-f*(c*x-1))^(1/2)*( 
-I*(-c^2*x^2+1)^(1/2)+c*x+1)*(15*I*b+16*b*arccos(c*x)+16*a)*cos(2*arccos(c 
*x))*d^2/(c*x-1)/f^2/c/(c*x+1)+1/8*(d*(c*x+1))^(1/2)*(-f*(c*x-1))^(1/2)*(I 
*c*x+(-c^2*x^2+1)^(1/2)+I)*(8*I*b+7*b*arccos(c*x)+7*a)*sin(2*arccos(c*x))* 
d^2/(c*x-1)/f^2/c/(c*x+1)
 

Fricas [F]

\[ \int \frac {(d+c d x)^{5/2} (a+b \arccos (c x))}{(f-c f x)^{3/2}} \, dx=\int { \frac {{\left (c d x + d\right )}^{\frac {5}{2}} {\left (b \arccos \left (c x\right ) + a\right )}}{{\left (-c f x + f\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((c*d*x+d)^(5/2)*(a+b*arccos(c*x))/(-c*f*x+f)^(3/2),x, algorithm= 
"fricas")
 

Output:

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

Sympy [F(-1)]

Timed out. \[ \int \frac {(d+c d x)^{5/2} (a+b \arccos (c x))}{(f-c f x)^{3/2}} \, dx=\text {Timed out} \] Input:

integrate((c*d*x+d)**(5/2)*(a+b*acos(c*x))/(-c*f*x+f)**(3/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(d+c d x)^{5/2} (a+b \arccos (c x))}{(f-c f x)^{3/2}} \, dx=\int { \frac {{\left (c d x + d\right )}^{\frac {5}{2}} {\left (b \arccos \left (c x\right ) + a\right )}}{{\left (-c f x + f\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((c*d*x+d)^(5/2)*(a+b*arccos(c*x))/(-c*f*x+f)^(3/2),x, algorithm= 
"maxima")
 

Output:

-1/2*(c^2*d^3*x^3/(sqrt(-c^2*d*f*x^2 + d*f)*f) + 8*c*d^3*x^2/(sqrt(-c^2*d* 
f*x^2 + d*f)*f) - 17*d^3*x/(sqrt(-c^2*d*f*x^2 + d*f)*f) + 15*d^3*arcsin(c* 
x)/(sqrt(d*f)*c*f) - 24*d^3/(sqrt(-c^2*d*f*x^2 + d*f)*c*f))*a - b*sqrt(d)* 
integrate((c^2*d^2*x^2 + 2*c*d^2*x + d^2)*sqrt(c*x + 1)*arctan2(sqrt(c*x + 
 1)*sqrt(-c*x + 1), c*x)/((c*f*x - f)*sqrt(-c*x + 1)), x)/sqrt(f)
 

Giac [F]

\[ \int \frac {(d+c d x)^{5/2} (a+b \arccos (c x))}{(f-c f x)^{3/2}} \, dx=\int { \frac {{\left (c d x + d\right )}^{\frac {5}{2}} {\left (b \arccos \left (c x\right ) + a\right )}}{{\left (-c f x + f\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((c*d*x+d)^(5/2)*(a+b*arccos(c*x))/(-c*f*x+f)^(3/2),x, algorithm= 
"giac")
 

Output:

integrate((c*d*x + d)^(5/2)*(b*arccos(c*x) + a)/(-c*f*x + f)^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+c d x)^{5/2} (a+b \arccos (c x))}{(f-c f x)^{3/2}} \, dx=\int \frac {\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )\,{\left (d+c\,d\,x\right )}^{5/2}}{{\left (f-c\,f\,x\right )}^{3/2}} \,d x \] Input:

int(((a + b*acos(c*x))*(d + c*d*x)^(5/2))/(f - c*f*x)^(3/2),x)
 

Output:

int(((a + b*acos(c*x))*(d + c*d*x)^(5/2))/(f - c*f*x)^(3/2), x)
 

Reduce [F]

\[ \int \frac {(d+c d x)^{5/2} (a+b \arccos (c x))}{(f-c f x)^{3/2}} \, dx=\frac {\sqrt {d}\, d^{2} \left (30 \sqrt {-c x +1}\, \mathit {asin} \left (\frac {\sqrt {-c x +1}}{\sqrt {2}}\right ) a -2 \sqrt {-c x +1}\, \left (\int \frac {\sqrt {c x +1}\, \mathit {acos} \left (c x \right ) x^{2}}{\sqrt {-c x +1}\, c x -\sqrt {-c x +1}}d x \right ) b \,c^{3}-4 \sqrt {-c x +1}\, \left (\int \frac {\sqrt {c x +1}\, \mathit {acos} \left (c x \right ) x}{\sqrt {-c x +1}\, c x -\sqrt {-c x +1}}d x \right ) b \,c^{2}-2 \sqrt {-c x +1}\, \left (\int \frac {\sqrt {c x +1}\, \mathit {acos} \left (c x \right )}{\sqrt {-c x +1}\, c x -\sqrt {-c x +1}}d x \right ) b c -\sqrt {c x +1}\, a \,c^{2} x^{2}-7 \sqrt {c x +1}\, a c x +24 \sqrt {c x +1}\, a \right )}{2 \sqrt {f}\, \sqrt {-c x +1}\, c f} \] Input:

int((c*d*x+d)^(5/2)*(a+b*acos(c*x))/(-c*f*x+f)^(3/2),x)
 

Output:

(sqrt(d)*d**2*(30*sqrt( - c*x + 1)*asin(sqrt( - c*x + 1)/sqrt(2))*a - 2*sq 
rt( - c*x + 1)*int((sqrt(c*x + 1)*acos(c*x)*x**2)/(sqrt( - c*x + 1)*c*x - 
sqrt( - c*x + 1)),x)*b*c**3 - 4*sqrt( - c*x + 1)*int((sqrt(c*x + 1)*acos(c 
*x)*x)/(sqrt( - c*x + 1)*c*x - sqrt( - c*x + 1)),x)*b*c**2 - 2*sqrt( - c*x 
 + 1)*int((sqrt(c*x + 1)*acos(c*x))/(sqrt( - c*x + 1)*c*x - sqrt( - c*x + 
1)),x)*b*c - sqrt(c*x + 1)*a*c**2*x**2 - 7*sqrt(c*x + 1)*a*c*x + 24*sqrt(c 
*x + 1)*a))/(2*sqrt(f)*sqrt( - c*x + 1)*c*f)