\(\int \frac {(e-c e x)^{5/2} (a+b \arccos (c x))^2}{(d+c d x)^{3/2}} \, dx\) [558]

Optimal result

Integrand size = 32, antiderivative size = 918 \[ \int \frac {(e-c e x)^{5/2} (a+b \arccos (c x))^2}{(d+c d x)^{3/2}} \, dx=\frac {8 a b e^4 x \left (1-c^2 x^2\right )^{3/2}}{(d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {8 b^2 e^4 \left (1-c^2 x^2\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {b^2 e^4 x \left (1-c^2 x^2\right )^2}{4 (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {b^2 e^4 \left (1-c^2 x^2\right )^{3/2} \arccos (c x)}{4 c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {8 b^2 e^4 x \left (1-c^2 x^2\right )^{3/2} \arccos (c x)}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {b c e^4 x^2 \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))}{2 (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {8 e^4 \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {8 e^4 x \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {8 i e^4 \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {4 e^4 \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {e^4 x \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2}{2 (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {5 e^4 \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))^3}{2 b c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {32 i b e^4 \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x)) \arctan \left (e^{i \arccos (c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {16 b e^4 \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x)) \log \left (1+e^{2 i \arccos (c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {16 i b^2 e^4 \left (1-c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {16 i b^2 e^4 \left (1-c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {8 i b^2 e^4 \left (1-c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}} \] Output:

8*a*b*e^4*x*(-c^2*x^2+1)^(3/2)/(c*d*x+d)^(3/2)/(-c*e*x+e)^(3/2)+8*b^2*e^4* 
(-c^2*x^2+1)^2/c/(c*d*x+d)^(3/2)/(-c*e*x+e)^(3/2)-1/4*b^2*e^4*x*(-c^2*x^2+ 
1)^2/(c*d*x+d)^(3/2)/(-c*e*x+e)^(3/2)+1/4*b^2*e^4*(-c^2*x^2+1)^(3/2)*arcco 
s(c*x)/c/(c*d*x+d)^(3/2)/(-c*e*x+e)^(3/2)+8*b^2*e^4*x*(-c^2*x^2+1)^(3/2)*a 
rccos(c*x)/(c*d*x+d)^(3/2)/(-c*e*x+e)^(3/2)-1/2*b*c*e^4*x^2*(-c^2*x^2+1)^( 
3/2)*(a+b*arccos(c*x))/(c*d*x+d)^(3/2)/(-c*e*x+e)^(3/2)-8*e^4*(-c^2*x^2+1) 
*(a+b*arccos(c*x))^2/c/(c*d*x+d)^(3/2)/(-c*e*x+e)^(3/2)+8*e^4*x*(-c^2*x^2+ 
1)*(a+b*arccos(c*x))^2/(c*d*x+d)^(3/2)/(-c*e*x+e)^(3/2)-16*I*b^2*e^4*(-c^2 
*x^2+1)^(3/2)*polylog(2,I*(c*x+I*(-c^2*x^2+1)^(1/2)))/c/(c*d*x+d)^(3/2)/(- 
c*e*x+e)^(3/2)-4*e^4*(-c^2*x^2+1)^2*(a+b*arccos(c*x))^2/c/(c*d*x+d)^(3/2)/ 
(-c*e*x+e)^(3/2)+1/2*e^4*x*(-c^2*x^2+1)^2*(a+b*arccos(c*x))^2/(c*d*x+d)^(3 
/2)/(-c*e*x+e)^(3/2)-5/2*e^4*(-c^2*x^2+1)^(3/2)*(a+b*arccos(c*x))^3/b/c/(c 
*d*x+d)^(3/2)/(-c*e*x+e)^(3/2)-32*I*b*e^4*(-c^2*x^2+1)^(3/2)*(a+b*arccos(c 
*x))*arctan(c*x+I*(-c^2*x^2+1)^(1/2))/c/(c*d*x+d)^(3/2)/(-c*e*x+e)^(3/2)+1 
6*b*e^4*(-c^2*x^2+1)^(3/2)*(a+b*arccos(c*x))*ln(1+(c*x+I*(-c^2*x^2+1)^(1/2 
))^2)/c/(c*d*x+d)^(3/2)/(-c*e*x+e)^(3/2)-8*I*b^2*e^4*(-c^2*x^2+1)^(3/2)*po 
lylog(2,-(c*x+I*(-c^2*x^2+1)^(1/2))^2)/c/(c*d*x+d)^(3/2)/(-c*e*x+e)^(3/2)- 
8*I*e^4*(-c^2*x^2+1)^(3/2)*(a+b*arccos(c*x))^2/c/(c*d*x+d)^(3/2)/(-c*e*x+e 
)^(3/2)+16*I*b^2*e^4*(-c^2*x^2+1)^(3/2)*polylog(2,-I*(c*x+I*(-c^2*x^2+1)^( 
1/2)))/c/(c*d*x+d)^(3/2)/(-c*e*x+e)^(3/2)
 

Mathematica [A] (warning: unable to verify)

Time = 19.14 (sec) , antiderivative size = 999, normalized size of antiderivative = 1.09 \[ \int \frac {(e-c e x)^{5/2} (a+b \arccos (c x))^2}{(d+c d x)^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(e^2*(24*a^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[1 - c^2*x^2]*(-24 - 7*c* 
x + c^2*x^2) + 360*a^2*Sqrt[d]*Sqrt[e]*(1 + c*x)*Sqrt[1 - c^2*x^2]*ArcTan[ 
(c*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/(Sqrt[d]*Sqrt[e]*(-1 + c^2*x^2))] - 
48*a*b*(1 - c*x)*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Cot[ArcCos[c*x]/2]*(4*Arc 
Cos[c*x] - Cot[ArcCos[c*x]/2]*(ArcCos[c*x]^2 - 8*Log[Cos[ArcCos[c*x]/2]])) 
 - 192*a*b*(1 - c*x)*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Cot[ArcCos[c*x]/2]*(2 
*ArcCos[c*x] + Cot[ArcCos[c*x]/2]*(c*x + Sqrt[1 - c^2*x^2]*ArcCos[c*x] - A 
rcCos[c*x]^2 + 4*Log[Cos[ArcCos[c*x]/2]])) + 16*b^2*(1 - c*x)*Sqrt[d + c*d 
*x]*Sqrt[e - c*e*x]*Cot[ArcCos[c*x]/2]*Csc[ArcCos[c*x]/2]^2*(6 - 6*c^2*x^2 
 + 3*(-3 + 2*c*x + c^2*x^2 + (2*I)*Sqrt[1 - c^2*x^2])*ArcCos[c*x]^2 + 2*Sq 
rt[1 - c^2*x^2]*ArcCos[c*x]^3 - 6*Sqrt[1 - c^2*x^2]*ArcCos[c*x]*(c*x + 4*L 
og[1 + E^(I*ArcCos[c*x])]) + (24*I)*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^(I*Arc 
Cos[c*x])]) + 16*b^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[1 - c^2*x^2]*(Ar 
cCos[c*x]*(-6*ArcCos[c*x] + Cot[ArcCos[c*x]/2]*(ArcCos[c*x]*(6*I + ArcCos[ 
c*x]) - 24*Log[1 + E^(I*ArcCos[c*x])])) + (24*I)*Cot[ArcCos[c*x]/2]*PolyLo 
g[2, -E^(I*ArcCos[c*x])]) - b^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[1 - c 
^2*x^2]*Csc[ArcCos[c*x]/2]*(90*ArcCos[c*x]*Cos[(3*ArcCos[c*x])/2] - 6*ArcC 
os[c*x]*Cos[(5*ArcCos[c*x])/2] - 8*ArcCos[c*x]*Cos[ArcCos[c*x]/2]*(-12 + ( 
12*I)*ArcCos[c*x] + 5*ArcCos[c*x]^2 - 48*Log[1 + E^(I*ArcCos[c*x])]) - (38 
4*I)*Cos[ArcCos[c*x]/2]*PolyLog[2, -E^(I*ArcCos[c*x])] + 6*(-31 - 30*c*...
 

Rubi [A] (verified)

Time = 1.54 (sec) , antiderivative size = 403, normalized size of antiderivative = 0.44, 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.

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

\(\Big \downarrow \) 5179

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5275

\(\displaystyle \frac {e^4 \left (1-c^2 x^2\right )^{3/2} \int \left (-\frac {c^2 x^2 (a+b \arccos (c x))^2}{\sqrt {1-c^2 x^2}}+\frac {4 c x (a+b \arccos (c x))^2}{\sqrt {1-c^2 x^2}}-\frac {7 (a+b \arccos (c x))^2}{\sqrt {1-c^2 x^2}}+\frac {8 (1-c x) (a+b \arccos (c x))^2}{\left (1-c^2 x^2\right )^{3/2}}\right )dx}{(c d x+d)^{3/2} (e-c e x)^{3/2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {e^4 \left (1-c^2 x^2\right )^{3/2} \left (-\frac {32 b \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))}{c}+\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2-\frac {4 \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{c}+\frac {8 x (a+b \arccos (c x))^2}{\sqrt {1-c^2 x^2}}-\frac {8 (a+b \arccos (c x))^2}{c \sqrt {1-c^2 x^2}}+\frac {1}{2} b c x^2 (a+b \arccos (c x))+\frac {5 (a+b \arccos (c x))^3}{2 b c}+\frac {8 i (a+b \arccos (c x))^2}{c}-\frac {16 b \log \left (1-e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))}{c}-8 a b x+\frac {16 i b^2 \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )}{c}-\frac {16 i b^2 \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )}{c}+\frac {8 i b^2 \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right )}{c}-8 b^2 x \arccos (c x)+\frac {b^2 \arcsin (c x)}{4 c}-\frac {1}{4} b^2 x \sqrt {1-c^2 x^2}+\frac {8 b^2 \sqrt {1-c^2 x^2}}{c}\right )}{(c d x+d)^{3/2} (e-c e x)^{3/2}}\)

Input:

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

Output:

(e^4*(1 - c^2*x^2)^(3/2)*(-8*a*b*x + (8*b^2*Sqrt[1 - c^2*x^2])/c - (b^2*x* 
Sqrt[1 - c^2*x^2])/4 - 8*b^2*x*ArcCos[c*x] + (b*c*x^2*(a + b*ArcCos[c*x])) 
/2 + ((8*I)*(a + b*ArcCos[c*x])^2)/c - (8*(a + b*ArcCos[c*x])^2)/(c*Sqrt[1 
 - c^2*x^2]) + (8*x*(a + b*ArcCos[c*x])^2)/Sqrt[1 - c^2*x^2] - (4*Sqrt[1 - 
 c^2*x^2]*(a + b*ArcCos[c*x])^2)/c + (x*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c* 
x])^2)/2 + (5*(a + b*ArcCos[c*x])^3)/(2*b*c) + (b^2*ArcSin[c*x])/(4*c) - ( 
32*b*(a + b*ArcCos[c*x])*ArcTanh[E^(I*ArcCos[c*x])])/c - (16*b*(a + b*ArcC 
os[c*x])*Log[1 - E^((2*I)*ArcCos[c*x])])/c + ((16*I)*b^2*PolyLog[2, -E^(I* 
ArcCos[c*x])])/c - ((16*I)*b^2*PolyLog[2, E^(I*ArcCos[c*x])])/c + ((8*I)*b 
^2*PolyLog[2, E^((2*I)*ArcCos[c*x])])/c))/((d + c*d*x)^(3/2)*(e - c*e*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 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 = 3.98 (sec) , antiderivative size = 1076, normalized size of antiderivative = 1.17

method	result	size
default	\(\text {Expression too large to display}\)	\(1076\)

Input:

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

Output:

-5/2*(d*(c*x+1))^(1/2)*(-e*(c*x-1))^(1/2)*(-c^2*x^2+1)^(1/2)/(c*x+1)/d^2/( 
c*x-1)/c*(a+b*arccos(c*x))^3*e^2/b+1/32*(-e*(c*x-1))^(1/2)*(d*(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)*(2*I*arccos(c*x)*b^2+2*arccos(c*x)^ 
2*b^2+2*I*a*b+4*arccos(c*x)*a*b+2*a^2-b^2)*e^2/(c*x+1)/d^2/(c*x-1)/c-(-e*( 
c*x-1))^(1/2)*(d*(c*x+1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)+c*x-1)*(arccos(c*x)^ 
2*b^2+2*arccos(c*x)*a*b+a^2-2*b^2+2*I*arccos(c*x)*b^2+2*I*a*b)*e^2/(c*x+1) 
/d^2/(c*x-1)/c-2*(-e*(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)*(arccos(c*x)^2*b^2+2*arccos(c*x)*a*b+a^2-2*b^2-2*I*b^2*a 
rccos(c*x)-2*I*a*b)*e^2/(c*x+1)/d^2/(c*x-1)/c+1/32*(-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+I*(-c^2*x^2+1)^(1/2)- 
c*x-1)*(-2*I*b^2*arccos(c*x)+2*arccos(c*x)^2*b^2-2*I*a*b+4*arccos(c*x)*a*b 
+2*a^2-b^2)*e^2/(c*x+1)/d^2/(c*x-1)/c-8*(-e*(c*x-1))^(1/2)*(d*(c*x+1))^(1/ 
2)*(-I*(-c^2*x^2+1)^(1/2)+c*x-1)*(arccos(c*x)^2*b^2+2*arccos(c*x)*a*b+a^2) 
*e^2/(c*x+1)/d^2/(c*x-1)/c-16*I*(-c^2*x^2+1)^(1/2)*(d*(c*x+1))^(1/2)*(-e*( 
c*x-1))^(1/2)/(c*x+1)/d^2/(c*x-1)/c*b*(arccos(c*x)^2*b+2*I*arccos(c*x)*ln( 
1+c*x+I*(-c^2*x^2+1)^(1/2))*b+2*polylog(2,-c*x-I*(-c^2*x^2+1)^(1/2))*b+2*I 
*ln(1+c*x+I*(-c^2*x^2+1)^(1/2))*a-2*I*ln(c*x+I*(-c^2*x^2+1)^(1/2))*a)*e^2+ 
1/8*(-e*(c*x-1))^(1/2)*(d*(c*x+1))^(1/2)*(-I*(-c^2*x^2+1)^(1/2)+c*x-1)*(15 
*I*b^2*arccos(c*x)+8*arccos(c*x)^2*b^2+15*I*a*b+16*arccos(c*x)*a*b+8*a^...
 

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

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

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(e-c e x)^{5/2} (a+b \arccos (c x))^2}{(d+c d x)^{3/2}} \, dx=\frac {\sqrt {e}\, e^{2} \left (30 \sqrt {c x +1}\, \mathit {asin} \left (\frac {\sqrt {-c x +1}}{\sqrt {2}}\right ) a^{2}+\sqrt {-c x +1}\, a^{2} c^{2} x^{2}-7 \sqrt {-c x +1}\, a^{2} c x -24 \sqrt {-c x +1}\, a^{2}+4 \sqrt {c x +1}\, \left (\int \frac {\sqrt {-c x +1}\, \mathit {acos} \left (c x \right ) x^{2}}{\sqrt {c x +1}\, c x +\sqrt {c x +1}}d x \right ) a b \,c^{3}-8 \sqrt {c x +1}\, \left (\int \frac {\sqrt {-c x +1}\, \mathit {acos} \left (c x \right ) x}{\sqrt {c x +1}\, c x +\sqrt {c x +1}}d x \right ) a b \,c^{2}+4 \sqrt {c x +1}\, \left (\int \frac {\sqrt {-c x +1}\, \mathit {acos} \left (c x \right )}{\sqrt {c x +1}\, c x +\sqrt {c x +1}}d x \right ) a b c +2 \sqrt {c x +1}\, \left (\int \frac {\sqrt {-c x +1}\, \mathit {acos} \left (c x \right )^{2} x^{2}}{\sqrt {c x +1}\, c x +\sqrt {c x +1}}d x \right ) b^{2} c^{3}-4 \sqrt {c x +1}\, \left (\int \frac {\sqrt {-c x +1}\, \mathit {acos} \left (c x \right )^{2} x}{\sqrt {c x +1}\, c x +\sqrt {c x +1}}d x \right ) b^{2} c^{2}+2 \sqrt {c x +1}\, \left (\int \frac {\sqrt {-c x +1}\, \mathit {acos} \left (c x \right )^{2}}{\sqrt {c x +1}\, c x +\sqrt {c x +1}}d x \right ) b^{2} c \right )}{2 \sqrt {d}\, \sqrt {c x +1}\, c d} \] Input:

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

Output:

(sqrt(e)*e**2*(30*sqrt(c*x + 1)*asin(sqrt( - c*x + 1)/sqrt(2))*a**2 + sqrt 
( - c*x + 1)*a**2*c**2*x**2 - 7*sqrt( - c*x + 1)*a**2*c*x - 24*sqrt( - c*x 
 + 1)*a**2 + 4*sqrt(c*x + 1)*int((sqrt( - c*x + 1)*acos(c*x)*x**2)/(sqrt(c 
*x + 1)*c*x + sqrt(c*x + 1)),x)*a*b*c**3 - 8*sqrt(c*x + 1)*int((sqrt( - c* 
x + 1)*acos(c*x)*x)/(sqrt(c*x + 1)*c*x + sqrt(c*x + 1)),x)*a*b*c**2 + 4*sq 
rt(c*x + 1)*int((sqrt( - c*x + 1)*acos(c*x))/(sqrt(c*x + 1)*c*x + sqrt(c*x 
 + 1)),x)*a*b*c + 2*sqrt(c*x + 1)*int((sqrt( - c*x + 1)*acos(c*x)**2*x**2) 
/(sqrt(c*x + 1)*c*x + sqrt(c*x + 1)),x)*b**2*c**3 - 4*sqrt(c*x + 1)*int((s 
qrt( - c*x + 1)*acos(c*x)**2*x)/(sqrt(c*x + 1)*c*x + sqrt(c*x + 1)),x)*b** 
2*c**2 + 2*sqrt(c*x + 1)*int((sqrt( - c*x + 1)*acos(c*x)**2)/(sqrt(c*x + 1 
)*c*x + sqrt(c*x + 1)),x)*b**2*c))/(2*sqrt(d)*sqrt(c*x + 1)*c*d)