\(\int \frac {(e-c e x)^{3/2} (a+b \arccos (c x))^2}{\sqrt {d+c d x}} \, dx\) [551]

Optimal result

Integrand size = 32, antiderivative size = 398 \[ \int \frac {(e-c e x)^{3/2} (a+b \arccos (c x))^2}{\sqrt {d+c d x}} \, dx=-\frac {4 b^2 e^2 \left (1-c^2 x^2\right )}{c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {b^2 e^2 x \left (1-c^2 x^2\right )}{4 \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {b^2 e^2 \sqrt {1-c^2 x^2} \arccos (c x)}{4 c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {4 b e^2 x \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{\sqrt {d+c d x} \sqrt {e-c e x}}+\frac {b c e^2 x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{2 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {2 e^2 \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2}{c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {e^2 x \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2}{2 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {e^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))^3}{2 b c \sqrt {d+c d x} \sqrt {e-c e x}} \] Output:

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

Mathematica [A] (verified)

Time = 7.72 (sec) , antiderivative size = 522, normalized size of antiderivative = 1.31 \[ \int \frac {(e-c e x)^{3/2} (a+b \arccos (c x))^2}{\sqrt {d+c d x}} \, dx=\frac {24 a^2 e (4-c x) \sqrt {d+c d x} \sqrt {e-c e x} \sqrt {1-c^2 x^2}+48 a b e \sqrt {d+c d x} \sqrt {e-c e x} \left (2 c x+2 \sqrt {1-c^2 x^2} \arccos (c x)-\arccos (c x)^2\right )-72 a^2 \sqrt {d} e^{3/2} \sqrt {1-c^2 x^2} \arctan \left (\frac {c x \sqrt {d+c d x} \sqrt {e-c e x}}{\sqrt {d} \sqrt {e} \left (-1+c^2 x^2\right )}\right )-8 b^2 e (1-c x) \sqrt {d+c d x} \sqrt {e-c e x} \left (6 \sqrt {1-c^2 x^2}-6 c x \arccos (c x)-3 \sqrt {1-c^2 x^2} \arccos (c x)^2+\arccos (c x)^3\right ) \csc ^2\left (\frac {1}{2} \arccos (c x)\right )-b^2 e (1-c x) \sqrt {d+c d x} \sqrt {e-c e x} \csc ^2\left (\frac {1}{2} \arccos (c x)\right ) \left (48 \sqrt {1-c^2 x^2}+4 \arccos (c x)^3+6 \arccos (c x) (-8 c x+\cos (2 \arccos (c x)))-3 \sin (2 \arccos (c x))+6 \arccos (c x)^2 \left (-4 \sqrt {1-c^2 x^2}+\sin (2 \arccos (c x))\right )\right )+6 a b e (1-c x) \sqrt {d+c d x} \sqrt {e-c e x} \csc ^2\left (\frac {1}{2} \arccos (c x)\right ) \left (8 c x-\cos (2 \arccos (c x))-2 \arccos (c x) \left (-4 \sqrt {1-c^2 x^2}+\arccos (c x)+\sin (2 \arccos (c x))\right )\right )}{48 c d \sqrt {1-c^2 x^2}} \] Input:

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

Output:

(24*a^2*e*(4 - c*x)*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[1 - c^2*x^2] + 48 
*a*b*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(2*c*x + 2*Sqrt[1 - c^2*x^2]*ArcCos 
[c*x] - ArcCos[c*x]^2) - 72*a^2*Sqrt[d]*e^(3/2)*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))] - 8 
*b^2*e*(1 - c*x)*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(6*Sqrt[1 - c^2*x^2] - 6* 
c*x*ArcCos[c*x] - 3*Sqrt[1 - c^2*x^2]*ArcCos[c*x]^2 + ArcCos[c*x]^3)*Csc[A 
rcCos[c*x]/2]^2 - b^2*e*(1 - c*x)*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Csc[ArcC 
os[c*x]/2]^2*(48*Sqrt[1 - c^2*x^2] + 4*ArcCos[c*x]^3 + 6*ArcCos[c*x]*(-8*c 
*x + Cos[2*ArcCos[c*x]]) - 3*Sin[2*ArcCos[c*x]] + 6*ArcCos[c*x]^2*(-4*Sqrt 
[1 - c^2*x^2] + Sin[2*ArcCos[c*x]])) + 6*a*b*e*(1 - c*x)*Sqrt[d + c*d*x]*S 
qrt[e - c*e*x]*Csc[ArcCos[c*x]/2]^2*(8*c*x - Cos[2*ArcCos[c*x]] - 2*ArcCos 
[c*x]*(-4*Sqrt[1 - c^2*x^2] + ArcCos[c*x] + Sin[2*ArcCos[c*x]])))/(48*c*d* 
Sqrt[1 - c^2*x^2])
 

Rubi [A] (verified)

Time = 0.83 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.55, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {5179, 27, 5273, 3042, 3798, 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.

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

\(\Big \downarrow \) 5179

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5273

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3798

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-((e^2*Sqrt[1 - c^2*x^2]*(4*b^2*c^2*Sqrt[1 - c^2*x^2] - (b^2*c^3*x*Sqrt[1 
- c^2*x^2])/4 - (b^2*c^2*ArcCos[c*x])/4 - 4*b*c^3*x*(a + b*ArcCos[c*x]) + 
(b*c^4*x^2*(a + b*ArcCos[c*x]))/2 - 2*c^2*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[ 
c*x])^2 + (c^3*x*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x])^2)/2 + (c^2*(a + b* 
ArcCos[c*x])^3)/(2*b)))/(c^3*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 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3798

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.) 
, x_Symbol] :> Int[ExpandIntegrand[(c + d*x)^m, (a + b*Sin[e + f*x])^n, x], 
 x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[n, 0] && (EqQ[n, 1] || IGtQ[ 
m, 0] || NeQ[a^2 - b^2, 0])
 

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 5273

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

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 8.95 (sec) , antiderivative size = 1357, normalized size of antiderivative = 3.41

method	result	size
default	\(\text {Expression too large to display}\)	\(1357\)
parts	\(\text {Expression too large to display}\)	\(1357\)

Input:

int((-c*e*x+e)^(3/2)*(a+b*arccos(c*x))^2/(c*d*x+d)^(1/2),x,method=_RETURNV 
ERBOSE)
 

Output:

1/2*a^2/d/c*(-c*e*x+e)^(3/2)*(c*d*x+d)^(1/2)+3/2*a^2*e/d/c*(-c*e*x+e)^(1/2 
)*(c*d*x+d)^(1/2)+3/2*a^2*e^2*((-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*(1/2*(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)^3*e-1/32*(d*(c*x+1))^(1/2)*(-e*(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*arccos(c*x)^2-1+2*I*arccos(c* 
x))*e/(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)+c*x-1)*(arccos(c*x)^2-2+2*I*arccos(c*x))*e/(c*x+1)/d/c/(c*x 
-1)+(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))*e/(c*x+1)/d/c/(c*x-1)-1/32*(d*(c*x+ 
1))^(1/2)*(-e*(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*arccos(c*x)^2-1-2*I*arccos(c*x))*e/(c*x+1)/d/c/(c 
*x-1)-1/8*(d*(c*x+1))^(1/2)*(-e*(c*x-1))^(1/2)*(-I*(-c^2*x^2+1)^(1/2)+c*x- 
1)*(7*I*arccos(c*x)+4*arccos(c*x)^2-8)*cos(2*arccos(c*x))*e/(c*x+1)/d/c/(c 
*x-1)-1/16*(d*(c*x+1))^(1/2)*(-e*(c*x-1))^(1/2)*(I*c*x-I+(-c^2*x^2+1)^(1/2 
))*(16*I*arccos(c*x)+6*arccos(c*x)^2-15)*sin(2*arccos(c*x))*e/(c*x+1)/d/c/ 
(c*x-1))+2*a*b*(3/4*(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*e-1/32*(d*(c*x+1))^(1/2)*(-e*(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*...
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((-c*e*x+e)^(3/2)*(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]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(sqrt(e)*e*( - 6*asin(sqrt( - c*x + 1)/sqrt(2))*a**2 - sqrt(c*x + 1)*sqrt( 
 - c*x + 1)*a**2*c*x + 4*sqrt(c*x + 1)*sqrt( - c*x + 1)*a**2 - 4*int((sqrt 
( - c*x + 1)*acos(c*x)*x)/sqrt(c*x + 1),x)*a*b*c**2 + 4*int((sqrt( - c*x + 
 1)*acos(c*x))/sqrt(c*x + 1),x)*a*b*c - 2*int((sqrt( - c*x + 1)*acos(c*x)* 
*2*x)/sqrt(c*x + 1),x)*b**2*c**2 + 2*int((sqrt( - c*x + 1)*acos(c*x)**2)/s 
qrt(c*x + 1),x)*b**2*c))/(2*sqrt(d)*c)