\(\int (d+c d x)^{3/2} \sqrt {e-c e x} (a+b \arccos (c x))^2 \, dx\) [543]

Optimal result

Integrand size = 32, antiderivative size = 455 \[ \int (d+c d x)^{3/2} \sqrt {e-c e x} (a+b \arccos (c x))^2 \, dx=\frac {4 b^2 d \sqrt {d+c d x} \sqrt {e-c e x}}{9 c}-\frac {1}{4} b^2 d x \sqrt {d+c d x} \sqrt {e-c e x}+\frac {2 b^2 d \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right )}{27 c}+\frac {b^2 d \sqrt {d+c d x} \sqrt {e-c e x} \arccos (c x)}{4 c \sqrt {1-c^2 x^2}}+\frac {2 b d x \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arccos (c x))}{3 \sqrt {1-c^2 x^2}}-\frac {b c d x^2 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arccos (c x))}{2 \sqrt {1-c^2 x^2}}-\frac {2 b c^2 d x^3 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arccos (c x))}{9 \sqrt {1-c^2 x^2}}+\frac {1}{2} d x \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arccos (c x))^2-\frac {d \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2}{3 c}+\frac {d \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arccos (c x))^3}{6 b c \sqrt {1-c^2 x^2}} \] Output:

4/9*b^2*d*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)/c-1/4*b^2*d*x*(c*d*x+d)^(1/2)*( 
-c*e*x+e)^(1/2)+2/27*b^2*d*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)*(-c^2*x^2+1)/c 
+1/4*b^2*d*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)*arccos(c*x)/c/(-c^2*x^2+1)^(1/ 
2)+2/3*b*d*x*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)*(a+b*arccos(c*x))/(-c^2*x^2+ 
1)^(1/2)-1/2*b*c*d*x^2*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)*(a+b*arccos(c*x))/ 
(-c^2*x^2+1)^(1/2)-2/9*b*c^2*d*x^3*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)*(a+b*a 
rccos(c*x))/(-c^2*x^2+1)^(1/2)+1/2*d*x*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)*(a 
+b*arccos(c*x))^2-1/3*d*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)*(-c^2*x^2+1)*(a+b 
*arccos(c*x))^2/c+1/6*d*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)*(a+b*arccos(c*x)) 
^3/b/c/(-c^2*x^2+1)^(1/2)
 

Mathematica [A] (verified)

Time = 2.76 (sec) , antiderivative size = 457, normalized size of antiderivative = 1.00 \[ \int (d+c d x)^{3/2} \sqrt {e-c e x} (a+b \arccos (c x))^2 \, dx=\frac {-36 b^2 d \sqrt {d+c d x} \sqrt {e-c e x} \arccos (c x)^3-108 a^2 d^{3/2} \sqrt {e} \sqrt {1-c^2 x^2} \arctan \left (\frac {c x \sqrt {d+c d x} \sqrt {e-c e x}}{\sqrt {d} \sqrt {e} \left (-1+c^2 x^2\right )}\right )+18 b d \sqrt {d+c d x} \sqrt {e-c e x} \arccos (c x)^2 \left (-6 a-2 b \sqrt {1-c^2 x^2}+2 b \sqrt {1-c^2 x^2} \cos (2 \arccos (c x))+3 b \sin (2 \arccos (c x))\right )+d \sqrt {d+c d x} \sqrt {e-c e x} \left (104 b^2 \sqrt {1-c^2 x^2}+48 a b c x \left (-3+c^2 x^2\right )+36 a^2 \sqrt {1-c^2 x^2} \left (-2+3 c x+2 c^2 x^2\right )+2 b \left (27 a-4 b \sqrt {1-c^2 x^2}\right ) \cos (2 \arccos (c x))-27 b^2 \sin (2 \arccos (c x))\right )+6 b d \sqrt {d+c d x} \sqrt {e-c e x} \arccos (c x) \left (9 b \cos (2 \arccos (c x))+2 \left (-9 b c x-12 a \sqrt {1-c^2 x^2}+12 a c^2 x^2 \sqrt {1-c^2 x^2}+b \cos (3 \arccos (c x))+9 a \sin (2 \arccos (c x))\right )\right )}{216 c \sqrt {1-c^2 x^2}} \] Input:

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

Output:

(-36*b^2*d*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcCos[c*x]^3 - 108*a^2*d^(3/2) 
*Sqrt[e]*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/(S 
qrt[d]*Sqrt[e]*(-1 + c^2*x^2))] + 18*b*d*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*A 
rcCos[c*x]^2*(-6*a - 2*b*Sqrt[1 - c^2*x^2] + 2*b*Sqrt[1 - c^2*x^2]*Cos[2*A 
rcCos[c*x]] + 3*b*Sin[2*ArcCos[c*x]]) + d*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]* 
(104*b^2*Sqrt[1 - c^2*x^2] + 48*a*b*c*x*(-3 + c^2*x^2) + 36*a^2*Sqrt[1 - c 
^2*x^2]*(-2 + 3*c*x + 2*c^2*x^2) + 2*b*(27*a - 4*b*Sqrt[1 - c^2*x^2])*Cos[ 
2*ArcCos[c*x]] - 27*b^2*Sin[2*ArcCos[c*x]]) + 6*b*d*Sqrt[d + c*d*x]*Sqrt[e 
 - c*e*x]*ArcCos[c*x]*(9*b*Cos[2*ArcCos[c*x]] + 2*(-9*b*c*x - 12*a*Sqrt[1 
- c^2*x^2] + 12*a*c^2*x^2*Sqrt[1 - c^2*x^2] + b*Cos[3*ArcCos[c*x]] + 9*a*S 
in[2*ArcCos[c*x]])))/(216*c*Sqrt[1 - c^2*x^2])
 

Rubi [A] (verified)

Time = 0.88 (sec) , antiderivative size = 252, normalized size of antiderivative = 0.55, 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.

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

\(\Big \downarrow \) 5179

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5263

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {d \sqrt {c d x+d} \sqrt {e-c e x} \left (\frac {2}{9} b c^2 x^3 (a+b \arccos (c x))+\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))^2}{3 c}+\frac {1}{2} b c x^2 (a+b \arccos (c x))-\frac {2}{3} b x (a+b \arccos (c x))-\frac {(a+b \arccos (c x))^3}{6 b c}+\frac {b^2 \arcsin (c x)}{4 c}-\frac {1}{4} b^2 x \sqrt {1-c^2 x^2}+\frac {2 b^2 \left (1-c^2 x^2\right )^{3/2}}{27 c}+\frac {4 b^2 \sqrt {1-c^2 x^2}}{9 c}\right )}{\sqrt {1-c^2 x^2}}\)

Input:

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

Output:

(d*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*((4*b^2*Sqrt[1 - c^2*x^2])/(9*c) - (b^2 
*x*Sqrt[1 - c^2*x^2])/4 + (2*b^2*(1 - c^2*x^2)^(3/2))/(27*c) - (2*b*x*(a + 
 b*ArcCos[c*x]))/3 + (b*c*x^2*(a + b*ArcCos[c*x]))/2 + (2*b*c^2*x^3*(a + b 
*ArcCos[c*x]))/9 + (x*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x])^2)/2 - ((1 - c 
^2*x^2)^(3/2)*(a + b*ArcCos[c*x])^2)/(3*c) - (a + b*ArcCos[c*x])^3/(6*b*c) 
 + (b^2*ArcSin[c*x])/(4*c)))/Sqrt[1 - c^2*x^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 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 = 2.44 (sec) , antiderivative size = 1358, normalized size of antiderivative = 2.98

method	result	size
default	\(\text {Expression too large to display}\)	\(1358\)
parts	\(\text {Expression too large to display}\)	\(1358\)

Input:

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

Output:

-1/3*a^2/c/e*(c*d*x+d)^(3/2)*(-c*e*x+e)^(3/2)-1/2*a^2*d/c/e*(c*d*x+d)^(1/2 
)*(-c*e*x+e)^(3/2)+1/2*a^2*d/c*(-c*e*x+e)^(1/2)*(c*d*x+d)^(1/2)+1/2*a^2*d^ 
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*(1/6*(d*(c* 
x+1))^(1/2)*(-e*(c*x-1))^(1/2)*(-c^2*x^2+1)^(1/2)/(c^2*x^2-1)/c*arccos(c*x 
)^3*d+1/216*(d*(c*x+1))^(1/2)*(-e*(c*x-1))^(1/2)*(4*c^4*x^4-5*c^2*x^2+4*I* 
(-c^2*x^2+1)^(1/2)*x^3*c^3-3*I*(-c^2*x^2+1)^(1/2)*c*x+1)*(6*I*arccos(c*x)+ 
9*arccos(c*x)^2-2)*d/(c^2*x^2-1)/c+1/16*(d*(c*x+1))^(1/2)*(-e*(c*x-1))^(1/ 
2)*(2*c^3*x^3-2*c*x+2*I*(-c^2*x^2+1)^(1/2)*x^2*c^2-I*(-c^2*x^2+1)^(1/2))*( 
2*arccos(c*x)^2-1+2*I*arccos(c*x))*d/(c^2*x^2-1)/c-1/8*(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))*d/(c^2*x^2-1)/c+1/16*(d*(c*x+1))^(1/2)*(-e*(c*x-1))^(1/2)* 
(-2*I*(-c^2*x^2+1)^(1/2)*x^2*c^2+2*c^3*x^3+I*(-c^2*x^2+1)^(1/2)-2*c*x)*(2* 
arccos(c*x)^2-1-2*I*arccos(c*x))*d/(c^2*x^2-1)/c+1/54*(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)*(12*I*arccos(c*x)+9 
*arccos(c*x)^2-14)*cos(2*arccos(c*x))*d/(c^2*x^2-1)/c+1/108*(d*(c*x+1))^(1 
/2)*(-e*(c*x-1))^(1/2)*(I*c^2*x^2+c*x*(-c^2*x^2+1)^(1/2)-I)*(30*I*arccos(c 
*x)+9*arccos(c*x)^2-26)*sin(2*arccos(c*x))*d/(c^2*x^2-1)/c)+2*a*b*(1/4*(d* 
(c*x+1))^(1/2)*(-e*(c*x-1))^(1/2)*(-c^2*x^2+1)^(1/2)/(c^2*x^2-1)/c*arccos( 
c*x)^2*d+1/72*(d*(c*x+1))^(1/2)*(-e*(c*x-1))^(1/2)*(4*c^4*x^4-5*c^2*x^2...
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (d+c d x)^{3/2} \sqrt {e-c e x} (a+b \arccos (c x))^2 \, dx=\frac {\sqrt {e}\, \sqrt {d}\, d \left (-6 \mathit {asin} \left (\frac {\sqrt {-c x +1}}{\sqrt {2}}\right ) a^{2}+2 \sqrt {c x +1}\, \sqrt {-c x +1}\, a^{2} c^{2} x^{2}+3 \sqrt {c x +1}\, \sqrt {-c x +1}\, a^{2} c x -2 \sqrt {c x +1}\, \sqrt {-c x +1}\, a^{2}+12 \left (\int \sqrt {c x +1}\, \sqrt {-c x +1}\, \mathit {acos} \left (c x \right ) x d x \right ) a b \,c^{2}+12 \left (\int \sqrt {c x +1}\, \sqrt {-c x +1}\, \mathit {acos} \left (c x \right )d x \right ) a b c +6 \left (\int \sqrt {c x +1}\, \sqrt {-c x +1}\, \mathit {acos} \left (c x \right )^{2} x d x \right ) b^{2} c^{2}+6 \left (\int \sqrt {c x +1}\, \sqrt {-c x +1}\, \mathit {acos} \left (c x \right )^{2}d x \right ) b^{2} c \right )}{6 c} \] Input:

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

Output:

(sqrt(e)*sqrt(d)*d*( - 6*asin(sqrt( - c*x + 1)/sqrt(2))*a**2 + 2*sqrt(c*x 
+ 1)*sqrt( - c*x + 1)*a**2*c**2*x**2 + 3*sqrt(c*x + 1)*sqrt( - c*x + 1)*a* 
*2*c*x - 2*sqrt(c*x + 1)*sqrt( - c*x + 1)*a**2 + 12*int(sqrt(c*x + 1)*sqrt 
( - c*x + 1)*acos(c*x)*x,x)*a*b*c**2 + 12*int(sqrt(c*x + 1)*sqrt( - c*x + 
1)*acos(c*x),x)*a*b*c + 6*int(sqrt(c*x + 1)*sqrt( - c*x + 1)*acos(c*x)**2* 
x,x)*b**2*c**2 + 6*int(sqrt(c*x + 1)*sqrt( - c*x + 1)*acos(c*x)**2,x)*b**2 
*c))/(6*c)