\(\int \frac {\sqrt {e-c e x} (a+b \arccos (c x))^2}{(d+c d x)^{3/2}} \, dx\) [546]

Optimal result

Integrand size = 32, antiderivative size = 530 \[ \int \frac {\sqrt {e-c e x} (a+b \arccos (c x))^2}{(d+c d x)^{3/2}} \, dx=-\frac {2 e^2 \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {2 e^2 x \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {2 i e^2 \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {e^2 \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))^3}{3 b c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {8 i b e^2 \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x)) \arctan \left (e^{i \arccos (c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {4 b e^2 \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x)) \log \left (1+e^{2 i \arccos (c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {4 i b^2 e^2 \left (1-c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {4 i b^2 e^2 \left (1-c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {2 i b^2 e^2 \left (1-c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}} \] Output:

-2*e^2*(-c^2*x^2+1)*(a+b*arccos(c*x))^2/c/(c*d*x+d)^(3/2)/(-c*e*x+e)^(3/2) 
+2*e^2*x*(-c^2*x^2+1)*(a+b*arccos(c*x))^2/(c*d*x+d)^(3/2)/(-c*e*x+e)^(3/2) 
-2*I*e^2*(-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)-1/3*e^2*(-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)-8*I*b*e^2*(-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)+4*b*e^2*(-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)+4*I*b^2*e^2*(-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*I*b^2*e^2*(- 
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)-2*I*b^2*e^2*(-c^2*x^2+1)^(3/2)*polylog(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)
 

Mathematica [A] (verified)

Time = 4.26 (sec) , antiderivative size = 410, normalized size of antiderivative = 0.77 \[ \int \frac {\sqrt {e-c e x} (a+b \arccos (c x))^2}{(d+c d x)^{3/2}} \, dx=-\frac {2 a^2 \sqrt {-e (-1+c x)} \sqrt {d (1+c x)}}{c d^2 (1+c x)}+\frac {a^2 \sqrt {e} \arctan \left (\frac {c x \sqrt {-e (-1+c x)} \sqrt {d (1+c x)}}{\sqrt {d} \sqrt {e} (-1+c x) (1+c x)}\right )}{c d^{3/2}}-\frac {a b (-1+c x) \sqrt {d+c d x} \sqrt {e-c e x} \sqrt {-d e \left (1-c^2 x^2\right )} \cot \left (\frac {1}{2} \arccos (c x)\right ) \left (-4 \arccos (c x)+\cot \left (\frac {1}{2} \arccos (c x)\right ) \left (\arccos (c x)^2-8 \log \left (\cos \left (\frac {1}{2} \arccos (c x)\right )\right )\right )\right )}{c d^2 (1+c x) \sqrt {(-d-c d x) (e-c e x)} \sqrt {1-c^2 x^2}}+\frac {b^2 \sqrt {d+c d x} \sqrt {e-c e x} \sqrt {-d e \left (1-c^2 x^2\right )} \left (\arccos (c x) \left (-6 \arccos (c x)+\cot \left (\frac {1}{2} \arccos (c x)\right ) \left (\arccos (c x) (6 i+\arccos (c x))-24 \log \left (1+e^{i \arccos (c x)}\right )\right )\right )+24 i \cot \left (\frac {1}{2} \arccos (c x)\right ) \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )\right )}{3 c d^2 (1+c x) \sqrt {(-d-c d x) (e-c e x)}} \] Input:

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

Output:

(-2*a^2*Sqrt[-(e*(-1 + c*x))]*Sqrt[d*(1 + c*x)])/(c*d^2*(1 + c*x)) + (a^2* 
Sqrt[e]*ArcTan[(c*x*Sqrt[-(e*(-1 + c*x))]*Sqrt[d*(1 + c*x)])/(Sqrt[d]*Sqrt 
[e]*(-1 + c*x)*(1 + c*x))])/(c*d^(3/2)) - (a*b*(-1 + c*x)*Sqrt[d + c*d*x]* 
Sqrt[e - c*e*x]*Sqrt[-(d*e*(1 - c^2*x^2))]*Cot[ArcCos[c*x]/2]*(-4*ArcCos[c 
*x] + Cot[ArcCos[c*x]/2]*(ArcCos[c*x]^2 - 8*Log[Cos[ArcCos[c*x]/2]])))/(c* 
d^2*(1 + c*x)*Sqrt[(-d - c*d*x)*(e - c*e*x)]*Sqrt[1 - c^2*x^2]) + (b^2*Sqr 
t[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[-(d*e*(1 - c^2*x^2))]*(ArcCos[c*x]*(-6*A 
rcCos[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]*PolyLog[2, -E^(I*ArcC 
os[c*x])]))/(3*c*d^2*(1 + c*x)*Sqrt[(-d - c*d*x)*(e - c*e*x)])
 

Rubi [A] (verified)

Time = 1.23 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.48, 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.

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

\(\Big \downarrow \) 5179

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5275

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(e^2*(1 - c^2*x^2)^(3/2)*(((2*I)*(a + b*ArcCos[c*x])^2)/c - (2*(a + b*ArcC 
os[c*x])^2)/(c*Sqrt[1 - c^2*x^2]) + (2*x*(a + b*ArcCos[c*x])^2)/Sqrt[1 - c 
^2*x^2] + (a + b*ArcCos[c*x])^3/(3*b*c) - (8*b*(a + b*ArcCos[c*x])*ArcTanh 
[E^(I*ArcCos[c*x])])/c - (4*b*(a + b*ArcCos[c*x])*Log[1 - E^((2*I)*ArcCos[ 
c*x])])/c + ((4*I)*b^2*PolyLog[2, -E^(I*ArcCos[c*x])])/c - ((4*I)*b^2*Poly 
Log[2, E^(I*ArcCos[c*x])])/c + ((2*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 = 2.85 (sec) , antiderivative size = 318, normalized size of antiderivative = 0.60

method	result	size
default	\(-\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (a +b \arccos \left (c x \right )\right )^{3}}{3 \left (c x +1\right ) d^{2} \left (c x -1\right ) c b}-\frac {2 \sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \left (-i \sqrt {-c^{2} x^{2}+1}+c x -1\right ) \left (\arccos \left (c x \right )^{2} b^{2}+2 \arccos \left (c x \right ) a b +a^{2}\right )}{\left (c x +1\right ) d^{2} \left (c x -1\right ) c}-\frac {4 i \sqrt {-c^{2} x^{2}+1}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, b \left (\arccos \left (c x \right )^{2} b +2 i \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right ) b +2 \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right ) b +2 i \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right ) a -2 i \ln \left (c x +i \sqrt {-c^{2} x^{2}+1}\right ) a \right )}{\left (c x +1\right ) d^{2} \left (c x -1\right ) c}\)	\(318\)

Input:

int((-c*e*x+e)^(1/2)*(a+b*arccos(c*x))^2/(c*d*x+d)^(3/2),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)/d^2/( 
c*x-1)/c*(a+b*arccos(c*x))^3/b-2*(-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)/(c*x+1 
)/d^2/(c*x-1)/c-4*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)
 

Fricas [F]

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

integrate((-c*e*x+e)^(1/2)*(a+b*arccos(c*x))^2/(c*d*x+d)^(3/2),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^2*d^2*x^2 + 2*c*d^2*x + d^2), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((-c*e*x+e)^(1/2)*(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]

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

integrate((-c*e*x+e)^(1/2)*(a+b*arccos(c*x))^2/(c*d*x+d)^(3/2),x, algorith 
m="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(sqrt(e)*(2*sqrt(c*x + 1)*asin(sqrt( - c*x + 1)/sqrt(2))*a**2 - 2*sqrt( - 
c*x + 1)*a**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)*a*b*c + 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))/(sqrt(d)*sqr 
t(c*x + 1)*c*d)