\(\int \frac {(d+c d x)^{3/2} (a+b \arccos (c x))^2}{(e-c e x)^{5/2}} \, dx\) [573]

Optimal result

Integrand size = 32, antiderivative size = 544 \[ \int \frac {(d+c d x)^{3/2} (a+b \arccos (c x))^2}{(e-c e x)^{5/2}} \, dx=-\frac {8 i d^4 \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {d^4 \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))^3}{3 b c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {32 b d^4 \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x)) \log \left (1-i e^{-i \arccos (c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {32 i b^2 d^4 \left (1-c^2 x^2\right )^{5/2} \operatorname {PolyLog}\left (2,i e^{-i \arccos (c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {4 b d^4 \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x)) \sec ^2\left (\frac {\pi }{4}+\frac {1}{2} \arccos (c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {8 b^2 d^4 \left (1-c^2 x^2\right )^{5/2} \tan \left (\frac {\pi }{4}+\frac {1}{2} \arccos (c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {8 d^4 \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))^2 \tan \left (\frac {\pi }{4}+\frac {1}{2} \arccos (c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {2 d^4 \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))^2 \sec ^2\left (\frac {\pi }{4}+\frac {1}{2} \arccos (c x)\right ) \tan \left (\frac {\pi }{4}+\frac {1}{2} \arccos (c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}} \] Output:

-8/3*I*d^4*(-c^2*x^2+1)^(5/2)*(a+b*arccos(c*x))^2/c/(c*d*x+d)^(5/2)/(-c*e* 
x+e)^(5/2)+1/3*d^4*(-c^2*x^2+1)^(5/2)*(a+b*arccos(c*x))^3/b/c/(c*d*x+d)^(5 
/2)/(-c*e*x+e)^(5/2)-32/3*b*d^4*(-c^2*x^2+1)^(5/2)*(a+b*arccos(c*x))*ln(1- 
I/(c*x+I*(-c^2*x^2+1)^(1/2)))/c/(c*d*x+d)^(5/2)/(-c*e*x+e)^(5/2)-32/3*I*b^ 
2*d^4*(-c^2*x^2+1)^(5/2)*polylog(2,I/(c*x+I*(-c^2*x^2+1)^(1/2)))/c/(c*d*x+ 
d)^(5/2)/(-c*e*x+e)^(5/2)-4/3*b*d^4*(-c^2*x^2+1)^(5/2)*(a+b*arccos(c*x))*s 
ec(1/4*Pi+1/2*arccos(c*x))^2/c/(c*d*x+d)^(5/2)/(-c*e*x+e)^(5/2)+8/3*b^2*d^ 
4*(-c^2*x^2+1)^(5/2)*tan(1/4*Pi+1/2*arccos(c*x))/c/(c*d*x+d)^(5/2)/(-c*e*x 
+e)^(5/2)-8/3*d^4*(-c^2*x^2+1)^(5/2)*(a+b*arccos(c*x))^2*tan(1/4*Pi+1/2*ar 
ccos(c*x))/c/(c*d*x+d)^(5/2)/(-c*e*x+e)^(5/2)+2/3*d^4*(-c^2*x^2+1)^(5/2)*( 
a+b*arccos(c*x))^2*sec(1/4*Pi+1/2*arccos(c*x))^2*tan(1/4*Pi+1/2*arccos(c*x 
))/c/(c*d*x+d)^(5/2)/(-c*e*x+e)^(5/2)
 

Mathematica [A] (warning: unable to verify)

Time = 12.69 (sec) , antiderivative size = 544, normalized size of antiderivative = 1.00 \[ \int \frac {(d+c d x)^{3/2} (a+b \arccos (c x))^2}{(e-c e x)^{5/2}} \, dx=\frac {-3 a^2 d^{3/2} \sqrt {e} (-1+c x)^4 (1+c x) \arctan \left (\frac {c x \sqrt {d+c d x} \sqrt {e-c e x}}{\sqrt {d} \sqrt {e} \left (-1+c^2 x^2\right )}\right )-256 i b^2 d \sqrt {d+c d x} \sqrt {e-c e x} \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right ) \sin ^6\left (\frac {1}{2} \arccos (c x)\right )+4 d \sqrt {d+c d x} \sqrt {e-c e x} \left (-2 b^2 \sqrt {1-c^2 x^2} \arccos (c x)^3 \sin ^6\left (\frac {1}{2} \arccos (c x)\right )+b \arccos (c x)^2 \sin ^2\left (\frac {1}{2} \arccos (c x)\right ) \left (-2 b \left (-1+c^2 x^2\right )-16 i b \sqrt {1-c^2 x^2} \sin ^4\left (\frac {1}{2} \arccos (c x)\right )+\left (-1+c^2 x^2\right ) \sin \left (\frac {1}{2} \arccos (c x)\right ) \left (8 b \cos \left (\frac {1}{2} \arccos (c x)\right )+3 a \sin \left (\frac {1}{2} \arccos (c x)\right )\right ) \tan \left (\frac {1}{2} \arccos (c x)\right )\right )+4 b \arccos (c x) \sin ^2\left (\frac {1}{2} \arccos (c x)\right ) \left (4 a \left (-1+c^2 x^2\right ) \sin ^2\left (\frac {1}{2} \arccos (c x)\right )+16 b \sqrt {1-c^2 x^2} \log \left (1-e^{i \arccos (c x)}\right ) \sin ^4\left (\frac {1}{2} \arccos (c x)\right )-\left (-1+c^2 x^2\right ) \left (a+b \tan \left (\frac {1}{2} \arccos (c x)\right )\right )\right )+\left (-1+c^2 x^2\right ) \left (-8 b \sin ^4\left (\frac {1}{2} \arccos (c x)\right ) \left (b+4 a \log \left (\sin \left (\frac {1}{2} \arccos (c x)\right )\right ) \tan \left (\frac {1}{2} \arccos (c x)\right )\right )+a \left (a-3 a c x+2 a c^2 x^2-2 b \sqrt {1-c^2 x^2} \tan ^2\left (\frac {1}{2} \arccos (c x)\right )\right )\right )\right )}{3 c e^3 (-1+c x)^4 (1+c x)} \] Input:

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

Output:

(-3*a^2*d^(3/2)*Sqrt[e]*(-1 + c*x)^4*(1 + c*x)*ArcTan[(c*x*Sqrt[d + c*d*x] 
*Sqrt[e - c*e*x])/(Sqrt[d]*Sqrt[e]*(-1 + c^2*x^2))] - (256*I)*b^2*d*Sqrt[d 
 + c*d*x]*Sqrt[e - c*e*x]*Sqrt[1 - c^2*x^2]*PolyLog[2, E^(I*ArcCos[c*x])]* 
Sin[ArcCos[c*x]/2]^6 + 4*d*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(-2*b^2*Sqrt[1 
- c^2*x^2]*ArcCos[c*x]^3*Sin[ArcCos[c*x]/2]^6 + b*ArcCos[c*x]^2*Sin[ArcCos 
[c*x]/2]^2*(-2*b*(-1 + c^2*x^2) - (16*I)*b*Sqrt[1 - c^2*x^2]*Sin[ArcCos[c* 
x]/2]^4 + (-1 + c^2*x^2)*Sin[ArcCos[c*x]/2]*(8*b*Cos[ArcCos[c*x]/2] + 3*a* 
Sin[ArcCos[c*x]/2])*Tan[ArcCos[c*x]/2]) + 4*b*ArcCos[c*x]*Sin[ArcCos[c*x]/ 
2]^2*(4*a*(-1 + c^2*x^2)*Sin[ArcCos[c*x]/2]^2 + 16*b*Sqrt[1 - c^2*x^2]*Log 
[1 - E^(I*ArcCos[c*x])]*Sin[ArcCos[c*x]/2]^4 - (-1 + c^2*x^2)*(a + b*Tan[A 
rcCos[c*x]/2])) + (-1 + c^2*x^2)*(-8*b*Sin[ArcCos[c*x]/2]^4*(b + 4*a*Log[S 
in[ArcCos[c*x]/2]]*Tan[ArcCos[c*x]/2]) + a*(a - 3*a*c*x + 2*a*c^2*x^2 - 2* 
b*Sqrt[1 - c^2*x^2]*Tan[ArcCos[c*x]/2]^2))))/(3*c*e^3*(-1 + c*x)^4*(1 + c* 
x))
 

Rubi [A] (verified)

Time = 1.36 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.45, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5179, 27, 5275, 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.

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

\(\Big \downarrow \) 5179

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5275

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {d^4 \left (1-c^2 x^2\right )^{5/2} \left (-\frac {(a+b \arccos (c x))^3}{3 b c}-\frac {8 i (a+b \arccos (c x))^2}{3 c}+\frac {32 b \log \left (1-e^{i \arccos (c x)}\right ) (a+b \arccos (c x))}{3 c}-\frac {8 \cot \left (\frac {1}{2} \arccos (c x)\right ) (a+b \arccos (c x))^2}{3 c}+\frac {4 b \csc ^2\left (\frac {1}{2} \arccos (c x)\right ) (a+b \arccos (c x))}{3 c}+\frac {2 \cot \left (\frac {1}{2} \arccos (c x)\right ) \csc ^2\left (\frac {1}{2} \arccos (c x)\right ) (a+b \arccos (c x))^2}{3 c}-\frac {32 i b^2 \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )}{3 c}+\frac {8 b^2 \cot \left (\frac {1}{2} \arccos (c x)\right )}{3 c}\right )}{(c d x+d)^{5/2} (e-c e x)^{5/2}}\)

Input:

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

Output:

(d^4*(1 - c^2*x^2)^(5/2)*((((-8*I)/3)*(a + b*ArcCos[c*x])^2)/c - (a + b*Ar 
cCos[c*x])^3/(3*b*c) + (8*b^2*Cot[ArcCos[c*x]/2])/(3*c) - (8*(a + b*ArcCos 
[c*x])^2*Cot[ArcCos[c*x]/2])/(3*c) + (4*b*(a + b*ArcCos[c*x])*Csc[ArcCos[c 
*x]/2]^2)/(3*c) + (2*(a + b*ArcCos[c*x])^2*Cot[ArcCos[c*x]/2]*Csc[ArcCos[c 
*x]/2]^2)/(3*c) + (32*b*(a + b*ArcCos[c*x])*Log[1 - E^(I*ArcCos[c*x])])/(3 
*c) - (((32*I)/3)*b^2*PolyLog[2, E^(I*ArcCos[c*x])])/c))/((d + c*d*x)^(5/2 
)*(e - c*e*x)^(5/2))
 

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 5275

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_ 
) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcCos[c*x] 
)^n/Sqrt[d + e*x^2], (f + g*x)^m*(d + e*x^2)^(p + 1/2), x], x] /; FreeQ[{a, 
 b, c, d, e, f, g}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && ILtQ[p + 1/2, 
 0] && GtQ[d, 0] && IGtQ[n, 0]
 

Maple [A] (verified)

Time = 3.23 (sec) , antiderivative size = 743, normalized size of antiderivative = 1.37

method	result	size
default	\(\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (a +b \arccos \left (c x \right )\right )^{3} d}{3 \left (c x -1\right ) e^{3} c \left (c x +1\right ) b}+\frac {4 \sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \left (-2 i \sqrt {-c^{2} x^{2}+1}\, x c +2 c^{2} x^{2}+2 i \sqrt {-c^{2} x^{2}+1}+c x -1\right ) \left (12 b^{2} c^{2} x^{2} \arccos \left (c x \right )^{2}+8 i a b c x +24 a b \,c^{2} x^{2} \arccos \left (c x \right )+2 i \sqrt {-c^{2} x^{2}+1}\, b^{2} c x -15 \arccos \left (c x \right )^{2} b^{2} c x +4 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) b^{2} c x -4 i a b -4 i \arccos \left (c x \right ) b^{2}+12 a^{2} c^{2} x^{2}-10 x^{2} c^{2} b^{2}-30 \arccos \left (c x \right ) a b c x +4 \sqrt {-c^{2} x^{2}+1}\, a b c x +8 i \arccos \left (c x \right ) b^{2} c x +5 \arccos \left (c x \right )^{2} b^{2}-2 \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, b^{2}-4 i \arccos \left (c x \right ) b^{2} c^{2} x^{2}-4 i a b \,c^{2} x^{2}-15 a^{2} c x +16 c x \,b^{2}+10 \arccos \left (c x \right ) a b -2 \sqrt {-c^{2} x^{2}+1}\, a b -2 i \sqrt {-c^{2} x^{2}+1}\, b^{2}+5 a^{2}-6 b^{2}\right ) d}{3 \left (12 c^{4} x^{4}-39 c^{3} x^{3}+47 c^{2} x^{2}-25 c x +5\right ) e^{3} c \left (c x +1\right )}+\frac {16 i \sqrt {-c^{2} x^{2}+1}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, b \left (\arccos \left (c x \right )^{2} b +2 i \arccos \left (c x \right ) \ln \left (\sqrt {c x +i \sqrt {-c^{2} x^{2}+1}}+1\right ) b +2 i \arccos \left (c x \right ) \ln \left (1-\sqrt {c x +i \sqrt {-c^{2} x^{2}+1}}\right ) b +4 \operatorname {polylog}\left (2, \sqrt {c x +i \sqrt {-c^{2} x^{2}+1}}\right ) b +4 \operatorname {polylog}\left (2, -\sqrt {c x +i \sqrt {-c^{2} x^{2}+1}}\right ) b -2 i \ln \left (c x +i \sqrt {-c^{2} x^{2}+1}\right ) a +2 i \ln \left (\sqrt {c x +i \sqrt {-c^{2} x^{2}+1}}-1\right ) a +2 i \ln \left (\sqrt {c x +i \sqrt {-c^{2} x^{2}+1}}+1\right ) a \right ) d}{3 \left (c x -1\right ) e^{3} c \left (c x +1\right )}\)	\(743\)

Input:

int((c*d*x+d)^(3/2)*(a+b*arccos(c*x))^2/(-c*e*x+e)^(5/2),x,method=_RETURNV 
ERBOSE)
 

Output:

1/3*(d*(c*x+1))^(1/2)*(-e*(c*x-1))^(1/2)*(-c^2*x^2+1)^(1/2)/(c*x-1)/e^3/c/ 
(c*x+1)*(a+b*arccos(c*x))^3*d/b+4/3*(-e*(c*x-1))^(1/2)*(d*(c*x+1))^(1/2)*( 
-2*I*(-c^2*x^2+1)^(1/2)*c*x+2*c^2*x^2+2*I*(-c^2*x^2+1)^(1/2)+c*x-1)*(12*b^ 
2*c^2*x^2*arccos(c*x)^2+8*I*a*b*c*x+24*a*b*c^2*x^2*arccos(c*x)+2*I*(-c^2*x 
^2+1)^(1/2)*b^2*c*x-15*arccos(c*x)^2*b^2*c*x+4*(-c^2*x^2+1)^(1/2)*arccos(c 
*x)*b^2*c*x-4*I*a*b-4*I*arccos(c*x)*b^2+12*a^2*c^2*x^2-10*x^2*c^2*b^2-30*a 
rccos(c*x)*a*b*c*x+4*(-c^2*x^2+1)^(1/2)*a*b*c*x+8*I*arccos(c*x)*b^2*c*x+5* 
arccos(c*x)^2*b^2-2*arccos(c*x)*(-c^2*x^2+1)^(1/2)*b^2-4*I*arccos(c*x)*b^2 
*c^2*x^2-4*I*a*b*c^2*x^2-15*a^2*c*x+16*c*x*b^2+10*arccos(c*x)*a*b-2*(-c^2* 
x^2+1)^(1/2)*a*b-2*I*(-c^2*x^2+1)^(1/2)*b^2+5*a^2-6*b^2)*d/(12*c^4*x^4-39* 
c^3*x^3+47*c^2*x^2-25*c*x+5)/e^3/c/(c*x+1)+16/3*I*(-c^2*x^2+1)^(1/2)*(d*(c 
*x+1))^(1/2)*(-e*(c*x-1))^(1/2)/(c*x-1)/e^3/c/(c*x+1)*b*(arccos(c*x)^2*b+2 
*I*arccos(c*x)*ln((c*x+I*(-c^2*x^2+1)^(1/2))^(1/2)+1)*b+2*I*arccos(c*x)*ln 
(1-(c*x+I*(-c^2*x^2+1)^(1/2))^(1/2))*b+4*polylog(2,(c*x+I*(-c^2*x^2+1)^(1/ 
2))^(1/2))*b+4*polylog(2,-(c*x+I*(-c^2*x^2+1)^(1/2))^(1/2))*b-2*I*ln(c*x+I 
*(-c^2*x^2+1)^(1/2))*a+2*I*ln((c*x+I*(-c^2*x^2+1)^(1/2))^(1/2)-1)*a+2*I*ln 
((c*x+I*(-c^2*x^2+1)^(1/2))^(1/2)+1)*a)*d
 

Fricas [F]

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

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

Output:

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

Sympy [F]

\[ \int \frac {(d+c d x)^{3/2} (a+b \arccos (c x))^2}{(e-c e x)^{5/2}} \, dx=\int \frac {\left (d \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {acos}{\left (c x \right )}\right )^{2}}{\left (- e \left (c x - 1\right )\right )^{\frac {5}{2}}}\, dx \] Input:

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

Output:

Integral((d*(c*x + 1))**(3/2)*(a + b*acos(c*x))**2/(-e*(c*x - 1))**(5/2), 
x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(d+c d x)^{3/2} (a+b \arccos (c x))^2}{(e-c e x)^{5/2}} \, dx=\text {Exception raised: ValueError} \] Input:

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(d+c d x)^{3/2} (a+b \arccos (c x))^2}{(e-c e x)^{5/2}} \, dx=\frac {\sqrt {d}\, d \left (-6 \sqrt {-c x +1}\, \mathit {asin} \left (\frac {\sqrt {-c x +1}}{\sqrt {2}}\right ) a^{2} c x +6 \sqrt {-c x +1}\, \mathit {asin} \left (\frac {\sqrt {-c x +1}}{\sqrt {2}}\right ) a^{2}+6 \sqrt {-c x +1}\, \left (\int \frac {\sqrt {c x +1}\, \mathit {acos} \left (c x \right ) x}{\sqrt {-c x +1}\, c^{2} x^{2}-2 \sqrt {-c x +1}\, c x +\sqrt {-c x +1}}d x \right ) a b \,c^{3} x -6 \sqrt {-c x +1}\, \left (\int \frac {\sqrt {c x +1}\, \mathit {acos} \left (c x \right ) x}{\sqrt {-c x +1}\, c^{2} x^{2}-2 \sqrt {-c x +1}\, c x +\sqrt {-c x +1}}d x \right ) a b \,c^{2}+6 \sqrt {-c x +1}\, \left (\int \frac {\sqrt {c x +1}\, \mathit {acos} \left (c x \right )}{\sqrt {-c x +1}\, c^{2} x^{2}-2 \sqrt {-c x +1}\, c x +\sqrt {-c x +1}}d x \right ) a b \,c^{2} x -6 \sqrt {-c x +1}\, \left (\int \frac {\sqrt {c x +1}\, \mathit {acos} \left (c x \right )}{\sqrt {-c x +1}\, c^{2} x^{2}-2 \sqrt {-c x +1}\, c x +\sqrt {-c x +1}}d x \right ) a b c +3 \sqrt {-c x +1}\, \left (\int \frac {\sqrt {c x +1}\, \mathit {acos} \left (c x \right )^{2} x}{\sqrt {-c x +1}\, c^{2} x^{2}-2 \sqrt {-c x +1}\, c x +\sqrt {-c x +1}}d x \right ) b^{2} c^{3} x -3 \sqrt {-c x +1}\, \left (\int \frac {\sqrt {c x +1}\, \mathit {acos} \left (c x \right )^{2} x}{\sqrt {-c x +1}\, c^{2} x^{2}-2 \sqrt {-c x +1}\, c x +\sqrt {-c x +1}}d x \right ) b^{2} c^{2}+3 \sqrt {-c x +1}\, \left (\int \frac {\sqrt {c x +1}\, \mathit {acos} \left (c x \right )^{2}}{\sqrt {-c x +1}\, c^{2} x^{2}-2 \sqrt {-c x +1}\, c x +\sqrt {-c x +1}}d x \right ) b^{2} c^{2} x -3 \sqrt {-c x +1}\, \left (\int \frac {\sqrt {c x +1}\, \mathit {acos} \left (c x \right )^{2}}{\sqrt {-c x +1}\, c^{2} x^{2}-2 \sqrt {-c x +1}\, c x +\sqrt {-c x +1}}d x \right ) b^{2} c -8 \sqrt {c x +1}\, a^{2} c x +4 \sqrt {c x +1}\, a^{2}\right )}{3 \sqrt {e}\, \sqrt {-c x +1}\, c \,e^{2} \left (c x -1\right )} \] Input:

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

Output:

(sqrt(d)*d*( - 6*sqrt( - c*x + 1)*asin(sqrt( - c*x + 1)/sqrt(2))*a**2*c*x 
+ 6*sqrt( - c*x + 1)*asin(sqrt( - c*x + 1)/sqrt(2))*a**2 + 6*sqrt( - c*x + 
 1)*int((sqrt(c*x + 1)*acos(c*x)*x)/(sqrt( - c*x + 1)*c**2*x**2 - 2*sqrt( 
- c*x + 1)*c*x + sqrt( - c*x + 1)),x)*a*b*c**3*x - 6*sqrt( - c*x + 1)*int( 
(sqrt(c*x + 1)*acos(c*x)*x)/(sqrt( - c*x + 1)*c**2*x**2 - 2*sqrt( - c*x + 
1)*c*x + sqrt( - c*x + 1)),x)*a*b*c**2 + 6*sqrt( - c*x + 1)*int((sqrt(c*x 
+ 1)*acos(c*x))/(sqrt( - c*x + 1)*c**2*x**2 - 2*sqrt( - c*x + 1)*c*x + sqr 
t( - c*x + 1)),x)*a*b*c**2*x - 6*sqrt( - c*x + 1)*int((sqrt(c*x + 1)*acos( 
c*x))/(sqrt( - c*x + 1)*c**2*x**2 - 2*sqrt( - c*x + 1)*c*x + sqrt( - c*x + 
 1)),x)*a*b*c + 3*sqrt( - c*x + 1)*int((sqrt(c*x + 1)*acos(c*x)**2*x)/(sqr 
t( - c*x + 1)*c**2*x**2 - 2*sqrt( - c*x + 1)*c*x + sqrt( - c*x + 1)),x)*b* 
*2*c**3*x - 3*sqrt( - c*x + 1)*int((sqrt(c*x + 1)*acos(c*x)**2*x)/(sqrt( - 
 c*x + 1)*c**2*x**2 - 2*sqrt( - c*x + 1)*c*x + sqrt( - c*x + 1)),x)*b**2*c 
**2 + 3*sqrt( - c*x + 1)*int((sqrt(c*x + 1)*acos(c*x)**2)/(sqrt( - c*x + 1 
)*c**2*x**2 - 2*sqrt( - c*x + 1)*c*x + sqrt( - c*x + 1)),x)*b**2*c**2*x - 
3*sqrt( - c*x + 1)*int((sqrt(c*x + 1)*acos(c*x)**2)/(sqrt( - c*x + 1)*c**2 
*x**2 - 2*sqrt( - c*x + 1)*c*x + sqrt( - c*x + 1)),x)*b**2*c - 8*sqrt(c*x 
+ 1)*a**2*c*x + 4*sqrt(c*x + 1)*a**2))/(3*sqrt(e)*sqrt( - c*x + 1)*c*e**2* 
(c*x - 1))