\(\int \frac {\sqrt {e-c e x} (a+b \arccos (c x))^2}{\sqrt {d+c d x}} \, dx\) [545]

Optimal result

Integrand size = 32, antiderivative size = 230 \[ \int \frac {\sqrt {e-c e x} (a+b \arccos (c x))^2}{\sqrt {d+c d x}} \, dx=-\frac {2 a b e x \sqrt {1-c^2 x^2}}{\sqrt {d+c d x} \sqrt {e-c e x}}-\frac {2 b^2 e \left (1-c^2 x^2\right )}{c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {2 b^2 e x \sqrt {1-c^2 x^2} \arccos (c x)}{\sqrt {d+c d x} \sqrt {e-c e x}}+\frac {e \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2}{c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {e \sqrt {1-c^2 x^2} (a+b \arccos (c x))^3}{3 b c \sqrt {d+c d x} \sqrt {e-c e x}} \] Output:

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

Mathematica [A] (verified)

Time = 2.32 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.36 \[ \int \frac {\sqrt {e-c e x} (a+b \arccos (c x))^2}{\sqrt {d+c d x}} \, dx=\frac {(-1+c x) \left (-3 \sqrt {d+c d x} \sqrt {e-c e x} \left (2 a b c x+a^2 \sqrt {1-c^2 x^2}-2 b^2 \sqrt {1-c^2 x^2}\right )-6 b \sqrt {d+c d x} \sqrt {e-c e x} \left (b c x+a \sqrt {1-c^2 x^2}\right ) \arccos (c x)+3 b \sqrt {d+c d x} \sqrt {e-c e x} \left (a-b \sqrt {1-c^2 x^2}\right ) \arccos (c x)^2+b^2 \sqrt {d+c d x} \sqrt {e-c e x} \arccos (c x)^3+3 a^2 \sqrt {d} \sqrt {e} \sqrt {1-c^2 x^2} \arctan \left (\frac {c x \sqrt {d+c d x} \sqrt {e-c e x}}{\sqrt {d} \sqrt {e} \left (-1+c^2 x^2\right )}\right )\right ) \csc ^2\left (\frac {1}{2} \arccos (c x)\right )}{6 c d \sqrt {1-c^2 x^2}} \] Input:

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

Output:

((-1 + c*x)*(-3*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(2*a*b*c*x + a^2*Sqrt[1 - 
c^2*x^2] - 2*b^2*Sqrt[1 - c^2*x^2]) - 6*b*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]* 
(b*c*x + a*Sqrt[1 - c^2*x^2])*ArcCos[c*x] + 3*b*Sqrt[d + c*d*x]*Sqrt[e - c 
*e*x]*(a - b*Sqrt[1 - c^2*x^2])*ArcCos[c*x]^2 + b^2*Sqrt[d + c*d*x]*Sqrt[e 
 - c*e*x]*ArcCos[c*x]^3 + 3*a^2*Sqrt[d]*Sqrt[e]*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))])*Cs 
c[ArcCos[c*x]/2]^2)/(6*c*d*Sqrt[1 - c^2*x^2])
 

Rubi [A] (verified)

Time = 0.74 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.53, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5179, 27, 5263, 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.

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

\(\Big \downarrow \) 5179

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5263

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(e*Sqrt[1 - c^2*x^2]*(2*a*b*x - (2*b^2*Sqrt[1 - c^2*x^2])/c + 2*b^2*x*ArcC 
os[c*x] + (Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x])^2)/c - (a + b*ArcCos[c*x] 
)^3/(3*b*c)))/(Sqrt[d + c*d*x]*Sqrt[e - c*e*x])
 

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 5263

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

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 2.14 (sec) , antiderivative size = 542, normalized size of antiderivative = 2.36

method	result	size
default	\(\frac {a^{2} \sqrt {-c e x +e}\, \sqrt {c d x +d}}{d c}+\frac {a^{2} e \sqrt {\left (-c e x +e \right ) \left (c d x +d \right )}\, \arctan \left (\frac {\sqrt {c^{2} d e}\, x}{\sqrt {-c^{2} d e \,x^{2}+d e}}\right )}{\sqrt {-c e x +e}\, \sqrt {c d x +d}\, \sqrt {c^{2} d e}}+b^{2} \left (\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )^{3}}{3 \left (c x +1\right ) d c \left (c x -1\right )}+\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \left (\arccos \left (c x \right )^{2}-2+2 i \arccos \left (c x \right )\right )}{2 \left (c x +1\right ) d c \left (c x -1\right )}+\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \left (-i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \left (\arccos \left (c x \right )^{2}-2-2 i \arccos \left (c x \right )\right )}{2 \left (c x +1\right ) d c \left (c x -1\right )}\right )+2 a b \left (\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )^{2}}{2 \left (c x +1\right ) d c \left (c x -1\right )}+\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \left (\arccos \left (c x \right )+i\right )}{2 \left (c x +1\right ) d c \left (c x -1\right )}+\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \left (-i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \left (\arccos \left (c x \right )-i\right )}{2 \left (c x +1\right ) d c \left (c x -1\right )}\right )\)	\(542\)
parts	\(\frac {a^{2} \sqrt {-c e x +e}\, \sqrt {c d x +d}}{d c}+\frac {a^{2} e \sqrt {\left (-c e x +e \right ) \left (c d x +d \right )}\, \arctan \left (\frac {\sqrt {c^{2} d e}\, x}{\sqrt {-c^{2} d e \,x^{2}+d e}}\right )}{\sqrt {-c e x +e}\, \sqrt {c d x +d}\, \sqrt {c^{2} d e}}+b^{2} \left (\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )^{3}}{3 \left (c x +1\right ) d c \left (c x -1\right )}+\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \left (\arccos \left (c x \right )^{2}-2+2 i \arccos \left (c x \right )\right )}{2 \left (c x +1\right ) d c \left (c x -1\right )}+\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \left (-i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \left (\arccos \left (c x \right )^{2}-2-2 i \arccos \left (c x \right )\right )}{2 \left (c x +1\right ) d c \left (c x -1\right )}\right )+2 a b \left (\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )^{2}}{2 \left (c x +1\right ) d c \left (c x -1\right )}+\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \left (\arccos \left (c x \right )+i\right )}{2 \left (c x +1\right ) d c \left (c x -1\right )}+\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \left (-i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \left (\arccos \left (c x \right )-i\right )}{2 \left (c x +1\right ) d c \left (c x -1\right )}\right )\)	\(542\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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