3.643 \(\int \frac{1}{\left (1+4 x+4 x^2+4 x^4\right )^{3/2}} \, dx\)
Optimal. Leaf size=367 \[ -\frac{\left (3-\left (\frac{1}{x}+1\right )^2\right ) x^2}{\sqrt{4 x^4+4 x^2+4 x+1}}+\frac{\left (13-9 \left (\frac{1}{x}+1\right )^2\right ) \left (\frac{1}{x}+1\right ) x^2}{10 \sqrt{4 x^4+4 x^2+4 x+1}}+\frac{9 \left (\left (\frac{1}{x}+1\right )^4-2 \left (\frac{1}{x}+1\right )^2+5\right ) \left (\frac{1}{x}+1\right ) x^2}{10 \left (\left (\frac{1}{x}+1\right )^2+\sqrt{5}\right ) \sqrt{4 x^4+4 x^2+4 x+1}}+\frac{3 \left (3-\sqrt{5}\right ) \left (\left (\frac{1}{x}+1\right )^2+\sqrt{5}\right ) \sqrt{\frac{\left (\frac{1}{x}+1\right )^4-2 \left (\frac{1}{x}+1\right )^2+5}{\left (\left (\frac{1}{x}+1\right )^2+\sqrt{5}\right )^2}} x^2 F\left (2 \tan ^{-1}\left (\frac{1+\frac{1}{x}}{\sqrt [4]{5}}\right )|\frac{1}{10} \left (5+\sqrt{5}\right )\right )}{4\ 5^{3/4} \sqrt{4 x^4+4 x^2+4 x+1}}-\frac{9 \left (\left (\frac{1}{x}+1\right )^2+\sqrt{5}\right ) \sqrt{\frac{\left (\frac{1}{x}+1\right )^4-2 \left (\frac{1}{x}+1\right )^2+5}{\left (\left (\frac{1}{x}+1\right )^2+\sqrt{5}\right )^2}} x^2 E\left (2 \tan ^{-1}\left (\frac{1+\frac{1}{x}}{\sqrt [4]{5}}\right )|\frac{1}{10} \left (5+\sqrt{5}\right )\right )}{2\ 5^{3/4} \sqrt{4 x^4+4 x^2+4 x+1}} \]
[Out]
-(((3 - (1 + x^(-1))^2)*x^2)/Sqrt[1 + 4*x + 4*x^2 + 4*x^4]) + ((13 - 9*(1 + x^(-
1))^2)*(1 + x^(-1))*x^2)/(10*Sqrt[1 + 4*x + 4*x^2 + 4*x^4]) + (9*(5 - 2*(1 + x^(
-1))^2 + (1 + x^(-1))^4)*(1 + x^(-1))*x^2)/(10*(Sqrt[5] + (1 + x^(-1))^2)*Sqrt[1
+ 4*x + 4*x^2 + 4*x^4]) - (9*(Sqrt[5] + (1 + x^(-1))^2)*Sqrt[(5 - 2*(1 + x^(-1)
)^2 + (1 + x^(-1))^4)/(Sqrt[5] + (1 + x^(-1))^2)^2]*x^2*EllipticE[2*ArcTan[(1 +
x^(-1))/5^(1/4)], (5 + Sqrt[5])/10])/(2*5^(3/4)*Sqrt[1 + 4*x + 4*x^2 + 4*x^4]) +
(3*(3 - Sqrt[5])*(Sqrt[5] + (1 + x^(-1))^2)*Sqrt[(5 - 2*(1 + x^(-1))^2 + (1 + x
^(-1))^4)/(Sqrt[5] + (1 + x^(-1))^2)^2]*x^2*EllipticF[2*ArcTan[(1 + x^(-1))/5^(1
/4)], (5 + Sqrt[5])/10])/(4*5^(3/4)*Sqrt[1 + 4*x + 4*x^2 + 4*x^4])
_______________________________________________________________________________________
Rubi [A] time = 0.71022, antiderivative size = 367, normalized size of antiderivative = 1.,
number of steps used = 9, number of rules used = 9, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.474
\[ -\frac{\left (3-\left (\frac{1}{x}+1\right )^2\right ) x^2}{\sqrt{4 x^4+4 x^2+4 x+1}}+\frac{\left (13-9 \left (\frac{1}{x}+1\right )^2\right ) \left (\frac{1}{x}+1\right ) x^2}{10 \sqrt{4 x^4+4 x^2+4 x+1}}+\frac{9 \left (\left (\frac{1}{x}+1\right )^4-2 \left (\frac{1}{x}+1\right )^2+5\right ) \left (\frac{1}{x}+1\right ) x^2}{10 \left (\left (\frac{1}{x}+1\right )^2+\sqrt{5}\right ) \sqrt{4 x^4+4 x^2+4 x+1}}+\frac{3 \left (3-\sqrt{5}\right ) \left (\left (\frac{1}{x}+1\right )^2+\sqrt{5}\right ) \sqrt{\frac{\left (\frac{1}{x}+1\right )^4-2 \left (\frac{1}{x}+1\right )^2+5}{\left (\left (\frac{1}{x}+1\right )^2+\sqrt{5}\right )^2}} x^2 F\left (2 \tan ^{-1}\left (\frac{1+\frac{1}{x}}{\sqrt [4]{5}}\right )|\frac{1}{10} \left (5+\sqrt{5}\right )\right )}{4\ 5^{3/4} \sqrt{4 x^4+4 x^2+4 x+1}}-\frac{9 \left (\left (\frac{1}{x}+1\right )^2+\sqrt{5}\right ) \sqrt{\frac{\left (\frac{1}{x}+1\right )^4-2 \left (\frac{1}{x}+1\right )^2+5}{\left (\left (\frac{1}{x}+1\right )^2+\sqrt{5}\right )^2}} x^2 E\left (2 \tan ^{-1}\left (\frac{1+\frac{1}{x}}{\sqrt [4]{5}}\right )|\frac{1}{10} \left (5+\sqrt{5}\right )\right )}{2\ 5^{3/4} \sqrt{4 x^4+4 x^2+4 x+1}} \]
Antiderivative was successfully verified.
[In] Int[(1 + 4*x + 4*x^2 + 4*x^4)^(-3/2),x]
[Out]
-(((3 - (1 + x^(-1))^2)*x^2)/Sqrt[1 + 4*x + 4*x^2 + 4*x^4]) + ((13 - 9*(1 + x^(-
1))^2)*(1 + x^(-1))*x^2)/(10*Sqrt[1 + 4*x + 4*x^2 + 4*x^4]) + (9*(5 - 2*(1 + x^(
-1))^2 + (1 + x^(-1))^4)*(1 + x^(-1))*x^2)/(10*(Sqrt[5] + (1 + x^(-1))^2)*Sqrt[1
+ 4*x + 4*x^2 + 4*x^4]) - (9*(Sqrt[5] + (1 + x^(-1))^2)*Sqrt[(5 - 2*(1 + x^(-1)
)^2 + (1 + x^(-1))^4)/(Sqrt[5] + (1 + x^(-1))^2)^2]*x^2*EllipticE[2*ArcTan[(1 +
x^(-1))/5^(1/4)], (5 + Sqrt[5])/10])/(2*5^(3/4)*Sqrt[1 + 4*x + 4*x^2 + 4*x^4]) +
(3*(3 - Sqrt[5])*(Sqrt[5] + (1 + x^(-1))^2)*Sqrt[(5 - 2*(1 + x^(-1))^2 + (1 + x
^(-1))^4)/(Sqrt[5] + (1 + x^(-1))^2)^2]*x^2*EllipticF[2*ArcTan[(1 + x^(-1))/5^(1
/4)], (5 + Sqrt[5])/10])/(4*5^(3/4)*Sqrt[1 + 4*x + 4*x^2 + 4*x^4])
_______________________________________________________________________________________
Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - 16 \int ^{1 + \frac{1}{x}} \frac{1}{\left (\frac{256 x^{4} - 512 x^{2} + 1280}{\left (- 4 x + 4\right )^{4}}\right )^{\frac{3}{2}} \left (- 4 x + 4\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(4*x**4+4*x**2+4*x+1)**(3/2),x)
[Out]
-16*Integral(1/(((256*x**4 - 512*x**2 + 1280)/(-4*x + 4)**4)**(3/2)*(-4*x + 4)**
2), (x, 1 + 1/x))
_______________________________________________________________________________________
Mathematica [C] time = 6.05141, size = 3334, normalized size = 9.08 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(1 + 4*x + 4*x^2 + 4*x^4)^(-3/2),x]
[Out]
(19 + 42*x - 16*x^2 + 36*x^3)/(10*Sqrt[1 + 4*x + 4*x^2 + 4*x^4]) - (3*((-2*Ellip
ticF[ArcSin[Sqrt[((x - Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 1, 0])*(-Root[1 + 4*#
1 + 4*#1^2 + 4*#1^4 & , 2, 0] + Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 4, 0]))/((x
- Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 2, 0])*(-Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 &
, 1, 0] + Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 4, 0]))]], ((Root[1 + 4*#1 + 4*#1
^2 + 4*#1^4 & , 2, 0] - Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 3, 0])*(Root[1 + 4*#
1 + 4*#1^2 + 4*#1^4 & , 1, 0] - Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 4, 0]))/((Ro
ot[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 1, 0] - Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 3,
0])*(Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 2, 0] - Root[1 + 4*#1 + 4*#1^2 + 4*#1^
4 & , 4, 0]))]*(x - Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 2, 0])^2*Sqrt[((-Root[1
+ 4*#1 + 4*#1^2 + 4*#1^4 & , 1, 0] + Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 2, 0])*
(x - Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 3, 0]))/((x - Root[1 + 4*#1 + 4*#1^2 +
4*#1^4 & , 2, 0])*(-Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 1, 0] + Root[1 + 4*#1 +
4*#1^2 + 4*#1^4 & , 3, 0]))]*(Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 1, 0] - Root[1
+ 4*#1 + 4*#1^2 + 4*#1^4 & , 4, 0])*Sqrt[((-Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & ,
1, 0] + Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 2, 0])*(x - Root[1 + 4*#1 + 4*#1^2
+ 4*#1^4 & , 4, 0]))/((x - Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 2, 0])*(-Root[1 +
4*#1 + 4*#1^2 + 4*#1^4 & , 1, 0] + Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 4, 0]))]
*Sqrt[((x - Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 1, 0])*(-Root[1 + 4*#1 + 4*#1^2
+ 4*#1^4 & , 2, 0] + Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 4, 0]))/((x - Root[1 +
4*#1 + 4*#1^2 + 4*#1^4 & , 2, 0])*(-Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 1, 0] +
Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 4, 0]))])/(Sqrt[1 + 4*x + 4*x^2 + 4*x^4]*(-R
oot[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 1, 0] + Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 2
, 0])*(-Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 2, 0] + Root[1 + 4*#1 + 4*#1^2 + 4*#
1^4 & , 4, 0])) + (6*((x - Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 1, 0])*(x - Root[
1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 3, 0])*(x - Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 4
, 0]) + (x - Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 2, 0])^2*Sqrt[((-Root[1 + 4*#1
+ 4*#1^2 + 4*#1^4 & , 1, 0] + Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 2, 0])*(x - Ro
ot[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 3, 0]))/((x - Root[1 + 4*#1 + 4*#1^2 + 4*#1^4
& , 2, 0])*(-Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 1, 0] + Root[1 + 4*#1 + 4*#1^2
+ 4*#1^4 & , 3, 0]))]*Sqrt[((x - Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 1, 0])*(Roo
t[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 2, 0] - Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 4,
0]))/((x - Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 2, 0])*(Root[1 + 4*#1 + 4*#1^2 +
4*#1^4 & , 1, 0] - Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 4, 0]))]*Sqrt[((-Root[1 +
4*#1 + 4*#1^2 + 4*#1^4 & , 1, 0] + Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 2, 0])*(
x - Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 4, 0]))/((x - Root[1 + 4*#1 + 4*#1^2 + 4
*#1^4 & , 2, 0])*(-Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 1, 0] + Root[1 + 4*#1 + 4
*#1^2 + 4*#1^4 & , 4, 0]))]*(-Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 1, 0] + Root[1
+ 4*#1 + 4*#1^2 + 4*#1^4 & , 4, 0])*((EllipticE[ArcSin[Sqrt[((x - Root[1 + 4*#1
+ 4*#1^2 + 4*#1^4 & , 1, 0])*(Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 2, 0] - Root[
1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 4, 0]))/((x - Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & ,
2, 0])*(Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 1, 0] - Root[1 + 4*#1 + 4*#1^2 + 4*
#1^4 & , 4, 0]))]], -(((Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 2, 0] - Root[1 + 4*#
1 + 4*#1^2 + 4*#1^4 & , 3, 0])*(Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 1, 0] - Root
[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 4, 0]))/((-Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 1
, 0] + Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 3, 0])*(Root[1 + 4*#1 + 4*#1^2 + 4*#1
^4 & , 2, 0] - Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 4, 0])))]*(-Root[1 + 4*#1 + 4
*#1^2 + 4*#1^4 & , 1, 0] + Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 3, 0]))/(-Root[1
+ 4*#1 + 4*#1^2 + 4*#1^4 & , 1, 0] + Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 2, 0])
+ (EllipticF[ArcSin[Sqrt[((x - Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 1, 0])*(Root[
1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 2, 0] - Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 4, 0]
))/((x - Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 2, 0])*(Root[1 + 4*#1 + 4*#1^2 + 4*
#1^4 & , 1, 0] - Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 4, 0]))]], -(((Root[1 + 4*#
1 + 4*#1^2 + 4*#1^4 & , 2, 0] - Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 3, 0])*(Root
[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 1, 0] - Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 4, 0
]))/((-Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 1, 0] + Root[1 + 4*#1 + 4*#1^2 + 4*#1
^4 & , 3, 0])*(Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 2, 0] - Root[1 + 4*#1 + 4*#1^
2 + 4*#1^4 & , 4, 0])))]*(Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 2, 0]*(-Root[1 + 4
*#1 + 4*#1^2 + 4*#1^4 & , 2, 0] - Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 4, 0]) - R
oot[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 1, 0]*(Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 2,
0] - Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 4, 0])))/((-Root[1 + 4*#1 + 4*#1^2 + 4
*#1^4 & , 1, 0] + Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 2, 0])*(-Root[1 + 4*#1 + 4
*#1^2 + 4*#1^4 & , 2, 0] + Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 4, 0])) - (Ellipt
icPi[(-Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 1, 0] + Root[1 + 4*#1 + 4*#1^2 + 4*#1
^4 & , 4, 0])/(-Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 2, 0] + Root[1 + 4*#1 + 4*#1
^2 + 4*#1^4 & , 4, 0]), ArcSin[Sqrt[((x - Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 1,
0])*(Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 2, 0] - Root[1 + 4*#1 + 4*#1^2 + 4*#1^
4 & , 4, 0]))/((x - Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 2, 0])*(Root[1 + 4*#1 +
4*#1^2 + 4*#1^4 & , 1, 0] - Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 4, 0]))]], -(((R
oot[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 2, 0] - Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 3
, 0])*(Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 1, 0] - Root[1 + 4*#1 + 4*#1^2 + 4*#1
^4 & , 4, 0]))/((-Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 1, 0] + Root[1 + 4*#1 + 4*
#1^2 + 4*#1^4 & , 3, 0])*(Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 2, 0] - Root[1 + 4
*#1 + 4*#1^2 + 4*#1^4 & , 4, 0])))]*(-Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 1, 0]
- Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 2, 0] - Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 &
, 3, 0] - Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 4, 0]))/(-Root[1 + 4*#1 + 4*#1^2 +
4*#1^4 & , 2, 0] + Root[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , 4, 0]))))/Sqrt[1 + 4*x +
4*x^2 + 4*x^4]))/5
_______________________________________________________________________________________
Maple [C] time = 0.028, size = 2564, normalized size = 7. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(4*x^4+4*x^2+4*x+1)^(3/2),x)
[Out]
-8*(-9/20*x^3+1/5*x^2-21/40*x-19/80)/(4*x^4+4*x^2+4*x+1)^(1/2)+3/5*(RootOf(4*_Z^
4+4*_Z^2+4*_Z+1,index=1)-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=4))*((RootOf(4*_Z^4+4
*_Z^2+4*_Z+1,index=4)-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=2))*(x-RootOf(4*_Z^4+4*_
Z^2+4*_Z+1,index=1))/(RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=4)-RootOf(4*_Z^4+4*_Z^2+
4*_Z+1,index=1))/(x-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=2)))^(1/2)*(x-RootOf(4*_Z^
4+4*_Z^2+4*_Z+1,index=2))^2*((RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=2)-RootOf(4*_Z^4
+4*_Z^2+4*_Z+1,index=1))*(x-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=3))/(RootOf(4*_Z^4
+4*_Z^2+4*_Z+1,index=3)-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=1))/(x-RootOf(4*_Z^4+4
*_Z^2+4*_Z+1,index=2)))^(1/2)*((RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=2)-RootOf(4*_Z
^4+4*_Z^2+4*_Z+1,index=1))*(x-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=4))/(RootOf(4*_Z
^4+4*_Z^2+4*_Z+1,index=4)-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=1))/(x-RootOf(4*_Z^4
+4*_Z^2+4*_Z+1,index=2)))^(1/2)/(RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=4)-RootOf(4*_
Z^4+4*_Z^2+4*_Z+1,index=2))/(RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=2)-RootOf(4*_Z^4+
4*_Z^2+4*_Z+1,index=1))/((x-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=1))*(x-RootOf(4*_Z
^4+4*_Z^2+4*_Z+1,index=2))*(x-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=3))*(x-RootOf(4*
_Z^4+4*_Z^2+4*_Z+1,index=4)))^(1/2)*EllipticF(((RootOf(4*_Z^4+4*_Z^2+4*_Z+1,inde
x=4)-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=2))*(x-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=
1))/(RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=4)-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=1))/
(x-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=2)))^(1/2),((RootOf(4*_Z^4+4*_Z^2+4*_Z+1,in
dex=2)-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=3))*(RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=
1)-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=4))/(-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=3)+
RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=1))/(RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=2)-Root
Of(4*_Z^4+4*_Z^2+4*_Z+1,index=4)))^(1/2))-9/5*((x-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,in
dex=1))*(x-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=3))*(x-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,
index=4))+(RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=1)-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,inde
x=4))*((RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=4)-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=2
))*(x-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=1))/(RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=4
)-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=1))/(x-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=2))
)^(1/2)*(x-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=2))^2*((RootOf(4*_Z^4+4*_Z^2+4*_Z+1
,index=2)-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=1))*(x-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,i
ndex=3))/(RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=3)-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index
=1))/(x-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=2)))^(1/2)*((RootOf(4*_Z^4+4*_Z^2+4*_Z
+1,index=2)-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=1))*(x-RootOf(4*_Z^4+4*_Z^2+4*_Z+1
,index=4))/(RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=4)-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,ind
ex=1))/(x-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=2)))^(1/2)*((RootOf(4*_Z^4+4*_Z^2+4*
_Z+1,index=2)*RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=1)-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,i
ndex=1)*RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=4)+RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=2
)*RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=4)+RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=2)^2)/(
RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=4)-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=2))/(Root
Of(4*_Z^4+4*_Z^2+4*_Z+1,index=2)-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=1))*EllipticF
(((RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=4)-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=2))*(x
-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=1))/(RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=4)-Roo
tOf(4*_Z^4+4*_Z^2+4*_Z+1,index=1))/(x-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=2)))^(1/
2),((RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=2)-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=3))*
(RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=1)-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=4))/(-Ro
otOf(4*_Z^4+4*_Z^2+4*_Z+1,index=3)+RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=1))/(RootOf
(4*_Z^4+4*_Z^2+4*_Z+1,index=2)-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=4)))^(1/2))+(-R
ootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=3)+RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=1))*Ellipt
icE(((RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=4)-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=2))
*(x-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=1))/(RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=4)-
RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=1))/(x-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=2)))^
(1/2),((RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=2)-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=3
))*(RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=1)-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=4))/(
-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=3)+RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=1))/(Roo
tOf(4*_Z^4+4*_Z^2+4*_Z+1,index=2)-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=4)))^(1/2))/
(RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=2)-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=1))))/((
x-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=1))*(x-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=2))
*(x-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=3))*(x-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=4
)))^(1/2)
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (4 \, x^{4} + 4 \, x^{2} + 4 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((4*x^4 + 4*x^2 + 4*x + 1)^(-3/2),x, algorithm="maxima")
[Out]
integrate((4*x^4 + 4*x^2 + 4*x + 1)^(-3/2), x)
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (4 \, x^{4} + 4 \, x^{2} + 4 \, x + 1\right )}^{\frac{3}{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((4*x^4 + 4*x^2 + 4*x + 1)^(-3/2),x, algorithm="fricas")
[Out]
integral((4*x^4 + 4*x^2 + 4*x + 1)^(-3/2), x)
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (4 x^{4} + 4 x^{2} + 4 x + 1\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(4*x**4+4*x**2+4*x+1)**(3/2),x)
[Out]
Integral((4*x**4 + 4*x**2 + 4*x + 1)**(-3/2), x)
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (4 \, x^{4} + 4 \, x^{2} + 4 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((4*x^4 + 4*x^2 + 4*x + 1)^(-3/2),x, algorithm="giac")
[Out]
integrate((4*x^4 + 4*x^2 + 4*x + 1)^(-3/2), x)