3.639 \(\int \frac{x^2}{\left (a+8 x-8 x^2+4 x^3-x^4\right )^{5/2}} \, dx\)

Optimal. Leaf size=582 \[ \frac{(a+4) \left ((x-1)^2+2\right ) (x-1)}{6 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}}+\frac{\sqrt{\sqrt{a+4}+1} \left (\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right ) F\left (\tan ^{-1}\left (\frac{x-1}{\sqrt{\sqrt{a+4}+1}}\right )|-\frac{2 \sqrt{a+4}}{1-\sqrt{a+4}}\right )}{12 \left (a^2+7 a+12\right ) \sqrt{\frac{\frac{(x-1)^2}{1-\sqrt{a+4}}+1}{\frac{(x-1)^2}{\sqrt{a+4}+1}+1}} \sqrt{a-(x-1)^4-2 (x-1)^2+3}}+\frac{2 \left ((x-1)^2+1\right )}{3 (a+4)^2 \sqrt{a-(x-1)^4-2 (x-1)^2+3}}+\frac{(x-1)^2+1}{3 (a+4) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}}+\frac{(x-1) \left ((3 a+13) (x-1)^2+7 a+29\right )}{12 (a+3)^2 (a+4) \sqrt{a-(x-1)^4-2 (x-1)^2+3}}-\frac{(3 a+13) \left (1-\sqrt{a+4}\right ) (x-1) \left (\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right )}{12 (a+3)^2 (a+4) \sqrt{a-(x-1)^4-2 (x-1)^2+3}}+\frac{(3 a+13) \left (1-\sqrt{a+4}\right ) \sqrt{\sqrt{a+4}+1} \left (\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right ) E\left (\tan ^{-1}\left (\frac{x-1}{\sqrt{\sqrt{a+4}+1}}\right )|-\frac{2 \sqrt{a+4}}{1-\sqrt{a+4}}\right )}{12 (a+3)^2 (a+4) \sqrt{\frac{\frac{(x-1)^2}{1-\sqrt{a+4}}+1}{\frac{(x-1)^2}{\sqrt{a+4}+1}+1}} \sqrt{a-(x-1)^4-2 (x-1)^2+3}} \]

[Out]

(1 + (-1 + x)^2)/(3*(4 + a)*(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)^(3/2)) + (2*(1 +
(-1 + x)^2))/(3*(4 + a)^2*Sqrt[3 + a - 2*(-1 + x)^2 - (-1 + x)^4]) + ((4 + a)*(
2 + (-1 + x)^2)*(-1 + x))/(6*(12 + 7*a + a^2)*(3 + a - 2*(-1 + x)^2 - (-1 + x)^4
)^(3/2)) + ((29 + 7*a + (13 + 3*a)*(-1 + x)^2)*(-1 + x))/(12*(3 + a)^2*(4 + a)*S
qrt[3 + a - 2*(-1 + x)^2 - (-1 + x)^4]) - ((13 + 3*a)*(1 - Sqrt[4 + a])*(1 + (-1
+ x)^2/(1 - Sqrt[4 + a]))*(-1 + x))/(12*(3 + a)^2*(4 + a)*Sqrt[3 + a - 2*(-1 +
x)^2 - (-1 + x)^4]) + ((13 + 3*a)*(1 - Sqrt[4 + a])*Sqrt[1 + Sqrt[4 + a]]*(1 + (
-1 + x)^2/(1 - Sqrt[4 + a]))*EllipticE[ArcTan[(-1 + x)/Sqrt[1 + Sqrt[4 + a]]], (
-2*Sqrt[4 + a])/(1 - Sqrt[4 + a])])/(12*(3 + a)^2*(4 + a)*Sqrt[(1 + (-1 + x)^2/(
1 - Sqrt[4 + a]))/(1 + (-1 + x)^2/(1 + Sqrt[4 + a]))]*Sqrt[3 + a - 2*(-1 + x)^2
- (-1 + x)^4]) + (Sqrt[1 + Sqrt[4 + a]]*(1 + (-1 + x)^2/(1 - Sqrt[4 + a]))*Ellip
ticF[ArcTan[(-1 + x)/Sqrt[1 + Sqrt[4 + a]]], (-2*Sqrt[4 + a])/(1 - Sqrt[4 + a])]
)/(12*(12 + 7*a + a^2)*Sqrt[(1 + (-1 + x)^2/(1 - Sqrt[4 + a]))/(1 + (-1 + x)^2/(
1 + Sqrt[4 + a]))]*Sqrt[3 + a - 2*(-1 + x)^2 - (-1 + x)^4])

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Rubi [A]  time = 1.64818, antiderivative size = 582, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 12, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429 \[ -\frac{\sqrt{\sqrt{a+4}+1} \left (\frac{(1-x)^2}{1-\sqrt{a+4}}+1\right ) F\left (\tan ^{-1}\left (\frac{1-x}{\sqrt{\sqrt{a+4}+1}}\right )|-\frac{2 \sqrt{a+4}}{1-\sqrt{a+4}}\right )}{12 \left (a^2+7 a+12\right ) \sqrt{\frac{\frac{(1-x)^2}{1-\sqrt{a+4}}+1}{\frac{(1-x)^2}{\sqrt{a+4}+1}+1}} \sqrt{a-(1-x)^4-2 (1-x)^2+3}}+\frac{2 \left ((x-1)^2+1\right )}{3 (a+4)^2 \sqrt{a-(1-x)^4-2 (1-x)^2+3}}+\frac{(x-1)^2+1}{3 (a+4) \left (a-(1-x)^4-2 (1-x)^2+3\right )^{3/2}}-\frac{\left ((x-1)^2+2\right ) (1-x)}{6 (a+3) \left (a-(1-x)^4-2 (1-x)^2+3\right )^{3/2}}-\frac{(1-x) \left ((3 a+13) (1-x)^2+7 a+29\right )}{12 (a+3)^2 (a+4) \sqrt{a-(1-x)^4-2 (1-x)^2+3}}+\frac{(3 a+13) \left (1-\sqrt{a+4}\right ) (1-x) \left (\frac{(1-x)^2}{1-\sqrt{a+4}}+1\right )}{12 (a+3)^2 (a+4) \sqrt{a-(1-x)^4-2 (1-x)^2+3}}-\frac{(3 a+13) \left (1-\sqrt{a+4}\right ) \sqrt{\sqrt{a+4}+1} \left (\frac{(1-x)^2}{1-\sqrt{a+4}}+1\right ) E\left (\tan ^{-1}\left (\frac{1-x}{\sqrt{\sqrt{a+4}+1}}\right )|-\frac{2 \sqrt{a+4}}{1-\sqrt{a+4}}\right )}{12 (a+3)^2 (a+4) \sqrt{\frac{\frac{(1-x)^2}{1-\sqrt{a+4}}+1}{\frac{(1-x)^2}{\sqrt{a+4}+1}+1}} \sqrt{a-(1-x)^4-2 (1-x)^2+3}} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(a + 8*x - 8*x^2 + 4*x^3 - x^4)^(5/2),x]

[Out]

(1 + (-1 + x)^2)/(3*(4 + a)*(3 + a - 2*(1 - x)^2 - (1 - x)^4)^(3/2)) + (2*(1 + (
-1 + x)^2))/(3*(4 + a)^2*Sqrt[3 + a - 2*(1 - x)^2 - (1 - x)^4]) - ((29 + 7*a + (
13 + 3*a)*(1 - x)^2)*(1 - x))/(12*(3 + a)^2*(4 + a)*Sqrt[3 + a - 2*(1 - x)^2 - (
1 - x)^4]) + ((13 + 3*a)*(1 - Sqrt[4 + a])*(1 + (1 - x)^2/(1 - Sqrt[4 + a]))*(1
- x))/(12*(3 + a)^2*(4 + a)*Sqrt[3 + a - 2*(1 - x)^2 - (1 - x)^4]) - ((2 + (-1 +
x)^2)*(1 - x))/(6*(3 + a)*(3 + a - 2*(1 - x)^2 - (1 - x)^4)^(3/2)) - ((13 + 3*a
)*(1 - Sqrt[4 + a])*Sqrt[1 + Sqrt[4 + a]]*(1 + (1 - x)^2/(1 - Sqrt[4 + a]))*Elli
pticE[ArcTan[(1 - x)/Sqrt[1 + Sqrt[4 + a]]], (-2*Sqrt[4 + a])/(1 - Sqrt[4 + a])]
)/(12*(3 + a)^2*(4 + a)*Sqrt[(1 + (1 - x)^2/(1 - Sqrt[4 + a]))/(1 + (1 - x)^2/(1
+ Sqrt[4 + a]))]*Sqrt[3 + a - 2*(1 - x)^2 - (1 - x)^4]) - (Sqrt[1 + Sqrt[4 + a]
]*(1 + (1 - x)^2/(1 - Sqrt[4 + a]))*EllipticF[ArcTan[(1 - x)/Sqrt[1 + Sqrt[4 + a
]]], (-2*Sqrt[4 + a])/(1 - Sqrt[4 + a])])/(12*(12 + 7*a + a^2)*Sqrt[(1 + (1 - x)
^2/(1 - Sqrt[4 + a]))/(1 + (1 - x)^2/(1 + Sqrt[4 + a]))]*Sqrt[3 + a - 2*(1 - x)^
2 - (1 - x)^4])

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Rubi in Sympy [A]  time = 111.191, size = 500, normalized size = 0.86 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(-x**4+4*x**3-8*x**2+a+8*x)**(5/2),x)

[Out]

(x - 1)*(4*a + (2*a + 8)*(x - 1)**2 + (4*a + 20)*(x - 1) + 4*(x - 1)**3 + 16)/(1
2*(a + 3)*(a + 4)*(a - (x - 1)**4 - 2*(x - 1)**2 + 3)**(3/2)) - (3*a + 13)*(x -
1)*((x - 1)**2/(-sqrt(a + 4) + 1) + 1)*(-sqrt(a + 4) + 1)/(12*(a + 3)**2*(a + 4)
*sqrt(a - (x - 1)**4 - 2*(x - 1)**2 + 3)) + (3*a + 10)*sqrt(a - (x - 1)**4 - 2*(
x - 1)**2 + 3)/(3*(a + 3)**2*(a + 4)**2) + (x - 1)*(4*(a + 4)*(3*a + 13)*(x - 1)
**2 + (4*a + 16)*(7*a + 29) + (48*a + 160)*(x - 1)**3 + (x - 1)*(32*a**2 + 288*a
+ 608))/(48*(a + 3)**2*(a + 4)**2*sqrt(a - (x - 1)**4 - 2*(x - 1)**2 + 3)) + ((
x - 1)**2/(-sqrt(a + 4) + 1) + 1)*sqrt(sqrt(a + 4) + 1)*elliptic_f(atan((x - 1)/
sqrt(sqrt(a + 4) + 1)), 2*sqrt(a + 4)/(sqrt(a + 4) - 1))/(12*sqrt((-(x - 1)**2/(
sqrt(a + 4) - 1) + 1)/((x - 1)**2/(sqrt(a + 4) + 1) + 1))*(a + 3)*(a + 4)*sqrt(a
- (x - 1)**4 - 2*(x - 1)**2 + 3)) + (3*a + 13)*((x - 1)**2/(-sqrt(a + 4) + 1) +
1)*(-sqrt(a + 4) + 1)*sqrt(sqrt(a + 4) + 1)*elliptic_e(atan((x - 1)/sqrt(sqrt(a
+ 4) + 1)), 2*sqrt(a + 4)/(sqrt(a + 4) - 1))/(12*sqrt((-(x - 1)**2/(sqrt(a + 4)
- 1) + 1)/((x - 1)**2/(sqrt(a + 4) + 1) + 1))*(a + 3)**2*(a + 4)*sqrt(a - (x -
1)**4 - 2*(x - 1)**2 + 3))

_______________________________________________________________________________________

Mathematica [B]  time = 6.22607, size = 5812, normalized size = 9.99 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(a + 8*x - 8*x^2 + 4*x^3 - x^4)^(5/2),x]

[Out]

Result too large to show

_______________________________________________________________________________________

Maple [B]  time = 0.046, size = 2780, normalized size = 4.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(-x^4+4*x^3-8*x^2+a+8*x)^(5/2),x)

[Out]

(1/6/(3+a)*x^3-1/6*(6+a)/(a^2+7*a+12)*x^2+1/6*(8+a)/(a^2+7*a+12)*x+1/6*a/(a^2+7*
a+12))*(-x^4+4*x^3-8*x^2+a+8*x)^(1/2)/(x^4-4*x^3+8*x^2-a-8*x)^2+2*(1/24*(13+3*a)
/(3+a)/(a^2+7*a+12)*x^3-1/24*(a^2+27*a+84)/(a^2+7*a+12)^2*x^2+1/6*(9*a+32)/(a^2+
7*a+12)^2*x+1/12*(3*a^2+7*a-12)/(a^2+7*a+12)^2)/(-x^4+4*x^3-8*x^2+a+8*x)^(1/2)-(
-1/6*(a^2-9*a-44)/(a^2+7*a+12)^2-1/3*(9*a+32)/(a^2+7*a+12)^2)*((-1-(4+a)^(1/2))^
(1/2)+(-1+(4+a)^(1/2))^(1/2))*((-(-1-(4+a)^(1/2))^(1/2)+(-1+(4+a)^(1/2))^(1/2))*
(x-1-(-1+(4+a)^(1/2))^(1/2))/(-(-1-(4+a)^(1/2))^(1/2)-(-1+(4+a)^(1/2))^(1/2))/(x
-1+(-1+(4+a)^(1/2))^(1/2)))^(1/2)*(x-1+(-1+(4+a)^(1/2))^(1/2))^2*(-2*(-1+(4+a)^(
1/2))^(1/2)*(x-1-(-1-(4+a)^(1/2))^(1/2))/((-1-(4+a)^(1/2))^(1/2)-(-1+(4+a)^(1/2)
)^(1/2))/(x-1+(-1+(4+a)^(1/2))^(1/2)))^(1/2)*(-2*(-1+(4+a)^(1/2))^(1/2)*(x-1+(-1
-(4+a)^(1/2))^(1/2))/(-(-1-(4+a)^(1/2))^(1/2)-(-1+(4+a)^(1/2))^(1/2))/(x-1+(-1+(
4+a)^(1/2))^(1/2)))^(1/2)/(-(-1-(4+a)^(1/2))^(1/2)+(-1+(4+a)^(1/2))^(1/2))/(-1+(
4+a)^(1/2))^(1/2)/(-(x-1-(-1+(4+a)^(1/2))^(1/2))*(x-1+(-1+(4+a)^(1/2))^(1/2))*(x
-1-(-1-(4+a)^(1/2))^(1/2))*(x-1+(-1-(4+a)^(1/2))^(1/2)))^(1/2)*EllipticF(((-(-1-
(4+a)^(1/2))^(1/2)+(-1+(4+a)^(1/2))^(1/2))*(x-1-(-1+(4+a)^(1/2))^(1/2))/(-(-1-(4
+a)^(1/2))^(1/2)-(-1+(4+a)^(1/2))^(1/2))/(x-1+(-1+(4+a)^(1/2))^(1/2)))^(1/2),((-
(-1-(4+a)^(1/2))^(1/2)-(-1+(4+a)^(1/2))^(1/2))*((-1-(4+a)^(1/2))^(1/2)+(-1+(4+a)
^(1/2))^(1/2))/(-(-1-(4+a)^(1/2))^(1/2)+(-1+(4+a)^(1/2))^(1/2))/((-1-(4+a)^(1/2)
)^(1/2)-(-1+(4+a)^(1/2))^(1/2)))^(1/2))-(1/3*(a^2-a-16)/(a^2+7*a+12)^2+1/6*(a^2+
27*a+84)/(a^2+7*a+12)^2)*((-1-(4+a)^(1/2))^(1/2)+(-1+(4+a)^(1/2))^(1/2))*((-(-1-
(4+a)^(1/2))^(1/2)+(-1+(4+a)^(1/2))^(1/2))*(x-1-(-1+(4+a)^(1/2))^(1/2))/(-(-1-(4
+a)^(1/2))^(1/2)-(-1+(4+a)^(1/2))^(1/2))/(x-1+(-1+(4+a)^(1/2))^(1/2)))^(1/2)*(x-
1+(-1+(4+a)^(1/2))^(1/2))^2*(-2*(-1+(4+a)^(1/2))^(1/2)*(x-1-(-1-(4+a)^(1/2))^(1/
2))/((-1-(4+a)^(1/2))^(1/2)-(-1+(4+a)^(1/2))^(1/2))/(x-1+(-1+(4+a)^(1/2))^(1/2))
)^(1/2)*(-2*(-1+(4+a)^(1/2))^(1/2)*(x-1+(-1-(4+a)^(1/2))^(1/2))/(-(-1-(4+a)^(1/2
))^(1/2)-(-1+(4+a)^(1/2))^(1/2))/(x-1+(-1+(4+a)^(1/2))^(1/2)))^(1/2)/(-(-1-(4+a)
^(1/2))^(1/2)+(-1+(4+a)^(1/2))^(1/2))/(-1+(4+a)^(1/2))^(1/2)/(-(x-1-(-1+(4+a)^(1
/2))^(1/2))*(x-1+(-1+(4+a)^(1/2))^(1/2))*(x-1-(-1-(4+a)^(1/2))^(1/2))*(x-1+(-1-(
4+a)^(1/2))^(1/2)))^(1/2)*((1-(-1+(4+a)^(1/2))^(1/2))*EllipticF(((-(-1-(4+a)^(1/
2))^(1/2)+(-1+(4+a)^(1/2))^(1/2))*(x-1-(-1+(4+a)^(1/2))^(1/2))/(-(-1-(4+a)^(1/2)
)^(1/2)-(-1+(4+a)^(1/2))^(1/2))/(x-1+(-1+(4+a)^(1/2))^(1/2)))^(1/2),((-(-1-(4+a)
^(1/2))^(1/2)-(-1+(4+a)^(1/2))^(1/2))*((-1-(4+a)^(1/2))^(1/2)+(-1+(4+a)^(1/2))^(
1/2))/(-(-1-(4+a)^(1/2))^(1/2)+(-1+(4+a)^(1/2))^(1/2))/((-1-(4+a)^(1/2))^(1/2)-(
-1+(4+a)^(1/2))^(1/2)))^(1/2))+2*(-1+(4+a)^(1/2))^(1/2)*EllipticPi(((-(-1-(4+a)^
(1/2))^(1/2)+(-1+(4+a)^(1/2))^(1/2))*(x-1-(-1+(4+a)^(1/2))^(1/2))/(-(-1-(4+a)^(1
/2))^(1/2)-(-1+(4+a)^(1/2))^(1/2))/(x-1+(-1+(4+a)^(1/2))^(1/2)))^(1/2),(-(-1-(4+
a)^(1/2))^(1/2)-(-1+(4+a)^(1/2))^(1/2))/(-(-1-(4+a)^(1/2))^(1/2)+(-1+(4+a)^(1/2)
)^(1/2)),((-(-1-(4+a)^(1/2))^(1/2)-(-1+(4+a)^(1/2))^(1/2))*((-1-(4+a)^(1/2))^(1/
2)+(-1+(4+a)^(1/2))^(1/2))/(-(-1-(4+a)^(1/2))^(1/2)+(-1+(4+a)^(1/2))^(1/2))/((-1
-(4+a)^(1/2))^(1/2)-(-1+(4+a)^(1/2))^(1/2)))^(1/2)))-1/12*(13+3*a)/(3+a)/(a^2+7*
a+12)*((x-1-(-1+(4+a)^(1/2))^(1/2))*(x-1-(-1-(4+a)^(1/2))^(1/2))*(x-1+(-1-(4+a)^
(1/2))^(1/2))+((-1-(4+a)^(1/2))^(1/2)+(-1+(4+a)^(1/2))^(1/2))*((-(-1-(4+a)^(1/2)
)^(1/2)+(-1+(4+a)^(1/2))^(1/2))*(x-1-(-1+(4+a)^(1/2))^(1/2))/(-(-1-(4+a)^(1/2))^
(1/2)-(-1+(4+a)^(1/2))^(1/2))/(x-1+(-1+(4+a)^(1/2))^(1/2)))^(1/2)*(x-1+(-1+(4+a)
^(1/2))^(1/2))^2*(-2*(-1+(4+a)^(1/2))^(1/2)*(x-1-(-1-(4+a)^(1/2))^(1/2))/((-1-(4
+a)^(1/2))^(1/2)-(-1+(4+a)^(1/2))^(1/2))/(x-1+(-1+(4+a)^(1/2))^(1/2)))^(1/2)*(-2
*(-1+(4+a)^(1/2))^(1/2)*(x-1+(-1-(4+a)^(1/2))^(1/2))/(-(-1-(4+a)^(1/2))^(1/2)-(-
1+(4+a)^(1/2))^(1/2))/(x-1+(-1+(4+a)^(1/2))^(1/2)))^(1/2)*(-1/2*((1-(-1+(4+a)^(1
/2))^(1/2))*(1+(-1+(4+a)^(1/2))^(1/2))-(1-(-1-(4+a)^(1/2))^(1/2))*(1+(-1+(4+a)^(
1/2))^(1/2))+(1-(-1-(4+a)^(1/2))^(1/2))*(1-(-1+(4+a)^(1/2))^(1/2))+(1-(-1+(4+a)^
(1/2))^(1/2))^2)/(-(-1-(4+a)^(1/2))^(1/2)+(-1+(4+a)^(1/2))^(1/2))/(-1+(4+a)^(1/2
))^(1/2)*EllipticF(((-(-1-(4+a)^(1/2))^(1/2)+(-1+(4+a)^(1/2))^(1/2))*(x-1-(-1+(4
+a)^(1/2))^(1/2))/(-(-1-(4+a)^(1/2))^(1/2)-(-1+(4+a)^(1/2))^(1/2))/(x-1+(-1+(4+a
)^(1/2))^(1/2)))^(1/2),((-(-1-(4+a)^(1/2))^(1/2)-(-1+(4+a)^(1/2))^(1/2))*((-1-(4
+a)^(1/2))^(1/2)+(-1+(4+a)^(1/2))^(1/2))/(-(-1-(4+a)^(1/2))^(1/2)+(-1+(4+a)^(1/2
))^(1/2))/((-1-(4+a)^(1/2))^(1/2)-(-1+(4+a)^(1/2))^(1/2)))^(1/2))-1/2*(-(-1-(4+a
)^(1/2))^(1/2)+(-1+(4+a)^(1/2))^(1/2))*EllipticE(((-(-1-(4+a)^(1/2))^(1/2)+(-1+(
4+a)^(1/2))^(1/2))*(x-1-(-1+(4+a)^(1/2))^(1/2))/(-(-1-(4+a)^(1/2))^(1/2)-(-1+(4+
a)^(1/2))^(1/2))/(x-1+(-1+(4+a)^(1/2))^(1/2)))^(1/2),((-(-1-(4+a)^(1/2))^(1/2)-(
-1+(4+a)^(1/2))^(1/2))*((-1-(4+a)^(1/2))^(1/2)+(-1+(4+a)^(1/2))^(1/2))/(-(-1-(4+
a)^(1/2))^(1/2)+(-1+(4+a)^(1/2))^(1/2))/((-1-(4+a)^(1/2))^(1/2)-(-1+(4+a)^(1/2))
^(1/2)))^(1/2))/(-1+(4+a)^(1/2))^(1/2)-4/(-(-1-(4+a)^(1/2))^(1/2)+(-1+(4+a)^(1/2
))^(1/2))*EllipticPi(((-(-1-(4+a)^(1/2))^(1/2)+(-1+(4+a)^(1/2))^(1/2))*(x-1-(-1+
(4+a)^(1/2))^(1/2))/(-(-1-(4+a)^(1/2))^(1/2)-(-1+(4+a)^(1/2))^(1/2))/(x-1+(-1+(4
+a)^(1/2))^(1/2)))^(1/2),((-1-(4+a)^(1/2))^(1/2)+(-1+(4+a)^(1/2))^(1/2))/((-1-(4
+a)^(1/2))^(1/2)-(-1+(4+a)^(1/2))^(1/2)),((-(-1-(4+a)^(1/2))^(1/2)-(-1+(4+a)^(1/
2))^(1/2))*((-1-(4+a)^(1/2))^(1/2)+(-1+(4+a)^(1/2))^(1/2))/(-(-1-(4+a)^(1/2))^(1
/2)+(-1+(4+a)^(1/2))^(1/2))/((-1-(4+a)^(1/2))^(1/2)-(-1+(4+a)^(1/2))^(1/2)))^(1/
2))))/(-(x-1-(-1+(4+a)^(1/2))^(1/2))*(x-1+(-1+(4+a)^(1/2))^(1/2))*(x-1-(-1-(4+a)
^(1/2))^(1/2))*(x-1+(-1-(4+a)^(1/2))^(1/2)))^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{{\left (-x^{4} + 4 \, x^{3} - 8 \, x^{2} + a + 8 \, x\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(-x^4 + 4*x^3 - 8*x^2 + a + 8*x)^(5/2),x, algorithm="maxima")

[Out]

integrate(x^2/(-x^4 + 4*x^3 - 8*x^2 + a + 8*x)^(5/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{2}}{{\left (x^{8} - 8 \, x^{7} + 32 \, x^{6} - 2 \,{\left (a - 64\right )} x^{4} - 80 \, x^{5} + 8 \,{\left (a - 16\right )} x^{3} - 16 \,{\left (a - 4\right )} x^{2} + a^{2} + 16 \, a x\right )} \sqrt{-x^{4} + 4 \, x^{3} - 8 \, x^{2} + a + 8 \, x}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(-x^4 + 4*x^3 - 8*x^2 + a + 8*x)^(5/2),x, algorithm="fricas")

[Out]

integral(x^2/((x^8 - 8*x^7 + 32*x^6 - 2*(a - 64)*x^4 - 80*x^5 + 8*(a - 16)*x^3 -
16*(a - 4)*x^2 + a^2 + 16*a*x)*sqrt(-x^4 + 4*x^3 - 8*x^2 + a + 8*x)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\left (a - x^{4} + 4 x^{3} - 8 x^{2} + 8 x\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(-x**4+4*x**3-8*x**2+a+8*x)**(5/2),x)

[Out]

Integral(x**2/(a - x**4 + 4*x**3 - 8*x**2 + 8*x)**(5/2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{{\left (-x^{4} + 4 \, x^{3} - 8 \, x^{2} + a + 8 \, x\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(-x^4 + 4*x^3 - 8*x^2 + a + 8*x)^(5/2),x, algorithm="giac")

[Out]

integrate(x^2/(-x^4 + 4*x^3 - 8*x^2 + a + 8*x)^(5/2), x)