\(\int \frac {\sqrt {e-c e x} (a+b \arccos (c x))^2}{(d+c d x)^{5/2}} \, dx\) [547]

Optimal result

Integrand size = 32, antiderivative size = 486 \[ \int \frac {\sqrt {e-c e x} (a+b \arccos (c x))^2}{(d+c d x)^{5/2}} \, dx=\frac {i e^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {4 b^2 e^3 \left (1-c^2 x^2\right )^{5/2} \cot \left (\frac {\pi }{4}+\frac {1}{2} \arccos (c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {e^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} \arccos (c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {2 b e^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x)) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} \arccos (c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {e^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} \arccos (c x)\right ) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} \arccos (c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {4 b e^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x)) \log \left (1-i e^{i \arccos (c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {4 i b^2 e^3 \left (1-c^2 x^2\right )^{5/2} \operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}} \] Output:

1/3*I*e^3*(-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)-4/3*b^2*e^3*(-c^2*x^2+1)^(5/2)*cot(1/4*Pi+1/2*arccos(c*x))/c/(c* 
d*x+d)^(5/2)/(-c*e*x+e)^(5/2)+1/3*e^3*(-c^2*x^2+1)^(5/2)*(a+b*arccos(c*x)) 
^2*cot(1/4*Pi+1/2*arccos(c*x))/c/(c*d*x+d)^(5/2)/(-c*e*x+e)^(5/2)-2/3*b*e^ 
3*(-c^2*x^2+1)^(5/2)*(a+b*arccos(c*x))*csc(1/4*Pi+1/2*arccos(c*x))^2/c/(c* 
d*x+d)^(5/2)/(-c*e*x+e)^(5/2)-1/3*e^3*(-c^2*x^2+1)^(5/2)*(a+b*arccos(c*x)) 
^2*cot(1/4*Pi+1/2*arccos(c*x))*csc(1/4*Pi+1/2*arccos(c*x))^2/c/(c*d*x+d)^( 
5/2)/(-c*e*x+e)^(5/2)-4/3*b*e^3*(-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)+4/3*I*b^2 
*e^3*(-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)
 

Mathematica [A] (verified)

Time = 3.55 (sec) , antiderivative size = 315, normalized size of antiderivative = 0.65 \[ \int \frac {\sqrt {e-c e x} (a+b \arccos (c x))^2}{(d+c d x)^{5/2}} \, dx=\frac {(-1+c x) \sqrt {d+c d x} \sqrt {e-c e x} \left (a^2 \sqrt {1-c^2 x^2}+8 b^2 \cos ^2\left (\frac {1}{2} \arccos (c x)\right ) \cot \left (\frac {1}{2} \arccos (c x)\right )-4 a b \cot ^2\left (\frac {1}{2} \arccos (c x)\right )+2 b^2 \arccos (c x)^2 \cot \left (\frac {1}{2} \arccos (c x)\right ) \left (1+i \cos ^2\left (\frac {1}{2} \arccos (c x)\right ) \left (i+\cot \left (\frac {1}{2} \arccos (c x)\right )\right )\right )-4 b \arccos (c x) \cot \left (\frac {1}{2} \arccos (c x)\right ) \left (-a+b \cot \left (\frac {1}{2} \arccos (c x)\right )+\cos ^2\left (\frac {1}{2} \arccos (c x)\right ) \left (a+2 b \cot \left (\frac {1}{2} \arccos (c x)\right ) \log \left (1+e^{i \arccos (c x)}\right )\right )\right )-8 a b \cos ^2\left (\frac {1}{2} \arccos (c x)\right ) \cot ^2\left (\frac {1}{2} \arccos (c x)\right ) \log \left (\cos \left (\frac {1}{2} \arccos (c x)\right )\right )+8 i b^2 \cos ^2\left (\frac {1}{2} \arccos (c x)\right ) \cot ^2\left (\frac {1}{2} \arccos (c x)\right ) \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )\right )}{3 c d^3 (1+c x)^2 \sqrt {1-c^2 x^2}} \] Input:

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

Output:

((-1 + c*x)*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a^2*Sqrt[1 - c^2*x^2] + 8*b^2 
*Cos[ArcCos[c*x]/2]^2*Cot[ArcCos[c*x]/2] - 4*a*b*Cot[ArcCos[c*x]/2]^2 + 2* 
b^2*ArcCos[c*x]^2*Cot[ArcCos[c*x]/2]*(1 + I*Cos[ArcCos[c*x]/2]^2*(I + Cot[ 
ArcCos[c*x]/2])) - 4*b*ArcCos[c*x]*Cot[ArcCos[c*x]/2]*(-a + b*Cot[ArcCos[c 
*x]/2] + Cos[ArcCos[c*x]/2]^2*(a + 2*b*Cot[ArcCos[c*x]/2]*Log[1 + E^(I*Arc 
Cos[c*x])])) - 8*a*b*Cos[ArcCos[c*x]/2]^2*Cot[ArcCos[c*x]/2]^2*Log[Cos[Arc 
Cos[c*x]/2]] + (8*I)*b^2*Cos[ArcCos[c*x]/2]^2*Cot[ArcCos[c*x]/2]^2*PolyLog 
[2, -E^(I*ArcCos[c*x])]))/(3*c*d^3*(1 + c*x)^2*Sqrt[1 - c^2*x^2])
 

Rubi [A] (verified)

Time = 1.39 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.46, 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.

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

\(\Big \downarrow \) 5179

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5275

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(e^3*(1 - c^2*x^2)^(5/2)*(((-1/3*I)*(a + b*ArcCos[c*x])^2)/c + (4*b*(a + b 
*ArcCos[c*x])*Log[1 + E^(I*ArcCos[c*x])])/(3*c) - (((4*I)/3)*b^2*PolyLog[2 
, -E^(I*ArcCos[c*x])])/c + (2*b*(a + b*ArcCos[c*x])*Sec[ArcCos[c*x]/2]^2)/ 
(3*c) - (4*b^2*Tan[ArcCos[c*x]/2])/(3*c) + ((a + b*ArcCos[c*x])^2*Tan[ArcC 
os[c*x]/2])/(3*c) - ((a + b*ArcCos[c*x])^2*Sec[ArcCos[c*x]/2]^2*Tan[ArcCos 
[c*x]/2])/(3*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 [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2820 vs. \(2 (424 ) = 848\).

Time = 4.16 (sec) , antiderivative size = 2821, normalized size of antiderivative = 5.80

method	result	size
default	\(\text {Expression too large to display}\)	\(2821\)
parts	\(\text {Expression too large to display}\)	\(2821\)

Input:

int((-c*e*x+e)^(1/2)*(a+b*arccos(c*x))^2/(c*d*x+d)^(5/2),x,method=_RETURNV 
ERBOSE)
 

Output:

4*I*b^2*(d*(c*x+1))^(1/2)*(-e*(c*x-1))^(1/2)/(3*c^4*x^4+6*c^3*x^3+4*c^2*x^ 
2+2*c*x+1)/d^3*c/(c*x-1)*(-c^2*x^2+1)^(1/2)*x^2+4/3*I*b^2*(d*(c*x+1))^(1/2 
)*(-e*(c*x-1))^(1/2)/(3*c^4*x^4+6*c^3*x^3+4*c^2*x^2+2*c*x+1)/d^3*c/(c*x-1) 
*arccos(c*x)*x^2-4*b^2*(d*(c*x+1))^(1/2)*(-e*(c*x-1))^(1/2)/(3*c^4*x^4+6*c 
^3*x^3+4*c^2*x^2+2*c*x+1)/d^3*c/(c*x-1)*(-c^2*x^2+1)^(1/2)*arccos(c*x)*x^2 
-4/3*b^2*(-c^2*x^2+1)^(1/2)*(d*(c*x+1))^(1/2)*(-e*(c*x-1))^(1/2)/(c*x+1)/d 
^3/c/(c*x-1)*arccos(c*x)*ln(1+c*x+I*(-c^2*x^2+1)^(1/2))+4/3*I*b^2*(-c^2*x^ 
2+1)^(1/2)*(d*(c*x+1))^(1/2)*(-e*(c*x-1))^(1/2)/(c*x+1)/d^3/c/(c*x-1)*poly 
log(2,-c*x-I*(-c^2*x^2+1)^(1/2))-2/3*I*b^2*(d*(c*x+1))^(1/2)*(-e*(c*x-1))^ 
(1/2)/(3*c^4*x^4+6*c^3*x^3+4*c^2*x^2+2*c*x+1)/d^3*c^3/(c*x-1)*arccos(c*x)* 
x^4+4/3*I*b^2*(d*(c*x+1))^(1/2)*(-e*(c*x-1))^(1/2)/(3*c^4*x^4+6*c^3*x^3+4* 
c^2*x^2+2*c*x+1)/d^3/c/(c*x-1)*(-c^2*x^2+1)^(1/2)+4/3*b^2*(d*(c*x+1))^(1/2 
)*(-e*(c*x-1))^(1/2)/(3*c^4*x^4+6*c^3*x^3+4*c^2*x^2+2*c*x+1)/d^3*c/(c*x-1) 
*(-c^2*x^2+1)*x^2-4/3*b^2*(d*(c*x+1))^(1/2)*(-e*(c*x-1))^(1/2)/(3*c^4*x^4+ 
6*c^3*x^3+4*c^2*x^2+2*c*x+1)/d^3/c/(c*x-1)*(-c^2*x^2+1)^(1/2)*arccos(c*x)- 
2*b^2*(d*(c*x+1))^(1/2)*(-e*(c*x-1))^(1/2)/(3*c^4*x^4+6*c^3*x^3+4*c^2*x^2+ 
2*c*x+1)/d^3*c^2/(c*x-1)*arccos(c*x)^2*x^3+4/3*b^2*(d*(c*x+1))^(1/2)*(-e*( 
c*x-1))^(1/2)/(3*c^4*x^4+6*c^3*x^3+4*c^2*x^2+2*c*x+1)/d^3*c/(c*x-1)*arccos 
(c*x)^2*x^2-2/3*I*b^2*(d*(c*x+1))^(1/2)*(-e*(c*x-1))^(1/2)/(3*c^4*x^4+6*c^ 
3*x^3+4*c^2*x^2+2*c*x+1)/d^3/c/(c*x-1)*arccos(c*x)+4/3*I*b^2*(d*(c*x+1)...
 

Fricas [F]

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

integrate((-c*e*x+e)^(1/2)*(a+b*arccos(c*x))^2/(c*d*x+d)^(5/2),x, algorith 
m="fricas")
 

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((-c*e*x+e)^(1/2)*(a+b*arccos(c*x))^2/(c*d*x+d)^(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]

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

integrate((-c*e*x+e)^(1/2)*(a+b*arccos(c*x))^2/(c*d*x+d)^(5/2),x, algorith 
m="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(sqrt(e)*(sqrt( - c*x + 1)*a**2*c*x - sqrt( - c*x + 1)*a**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 + sqrt(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)/(sq 
rt(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))/(3*sqrt(d)*sqrt( 
c*x + 1)*c*d**2*(c*x + 1))