3.637 \(\int \frac{x^2}{\sqrt{a+8 x-8 x^2+4 x^3-x^4}} \, dx\)

Optimal. Leaf size=388 \[ \frac{\left (1-\sqrt{a+4}\right ) (x-1) \left (\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right )}{\sqrt{a-(x-1)^4-2 (x-1)^2+3}}+\tan ^{-1}\left (\frac{(x-1)^2+1}{\sqrt{a-(x-1)^4-2 (x-1)^2+3}}\right )+\frac{\sqrt{\sqrt{a+4}+1} \left (\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right ) F\left (\tan ^{-1}\left (\frac{x-1}{\sqrt{\sqrt{a+4}+1}}\right )|-\frac{2 \sqrt{a+4}}{1-\sqrt{a+4}}\right )}{\sqrt{\frac{\frac{(x-1)^2}{1-\sqrt{a+4}}+1}{\frac{(x-1)^2}{\sqrt{a+4}+1}+1}} \sqrt{a-(x-1)^4-2 (x-1)^2+3}}-\frac{\left (1-\sqrt{a+4}\right ) \sqrt{\sqrt{a+4}+1} \left (\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right ) E\left (\tan ^{-1}\left (\frac{x-1}{\sqrt{\sqrt{a+4}+1}}\right )|-\frac{2 \sqrt{a+4}}{1-\sqrt{a+4}}\right )}{\sqrt{\frac{\frac{(x-1)^2}{1-\sqrt{a+4}}+1}{\frac{(x-1)^2}{\sqrt{a+4}+1}+1}} \sqrt{a-(x-1)^4-2 (x-1)^2+3}} \]

[Out]

((1 - Sqrt[4 + a])*(1 + (-1 + x)^2/(1 - Sqrt[4 + a]))*(-1 + x))/Sqrt[3 + a - 2*(
-1 + x)^2 - (-1 + x)^4] + ArcTan[(1 + (-1 + x)^2)/Sqrt[3 + a - 2*(-1 + x)^2 - (-
1 + x)^4]] - ((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])])/(Sqrt[(1 + (-1 + x)^2/(1 - Sqrt[4 + a]))/(1 + (-1 + x)^2/(1 + Sq
rt[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])])/(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 = 1.03756, antiderivative size = 388, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 11, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.393 \[ -\frac{\left (1-\sqrt{a+4}\right ) (1-x) \left (\frac{(1-x)^2}{1-\sqrt{a+4}}+1\right )}{\sqrt{a-(1-x)^4-2 (1-x)^2+3}}+\tan ^{-1}\left (\frac{(x-1)^2+1}{\sqrt{a-(1-x)^4-2 (1-x)^2+3}}\right )-\frac{\sqrt{\sqrt{a+4}+1} \left (\frac{(1-x)^2}{1-\sqrt{a+4}}+1\right ) F\left (\tan ^{-1}\left (\frac{1-x}{\sqrt{\sqrt{a+4}+1}}\right )|-\frac{2 \sqrt{a+4}}{1-\sqrt{a+4}}\right )}{\sqrt{\frac{\frac{(1-x)^2}{1-\sqrt{a+4}}+1}{\frac{(1-x)^2}{\sqrt{a+4}+1}+1}} \sqrt{a-(1-x)^4-2 (1-x)^2+3}}+\frac{\left (1-\sqrt{a+4}\right ) \sqrt{\sqrt{a+4}+1} \left (\frac{(1-x)^2}{1-\sqrt{a+4}}+1\right ) E\left (\tan ^{-1}\left (\frac{1-x}{\sqrt{\sqrt{a+4}+1}}\right )|-\frac{2 \sqrt{a+4}}{1-\sqrt{a+4}}\right )}{\sqrt{\frac{\frac{(1-x)^2}{1-\sqrt{a+4}}+1}{\frac{(1-x)^2}{\sqrt{a+4}+1}+1}} \sqrt{a-(1-x)^4-2 (1-x)^2+3}} \]

Antiderivative was successfully verified.

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

[Out]

-(((1 - Sqrt[4 + a])*(1 + (1 - x)^2/(1 - Sqrt[4 + a]))*(1 - x))/Sqrt[3 + a - 2*(
1 - x)^2 - (1 - x)^4]) + ArcTan[(1 + (-1 + x)^2)/Sqrt[3 + a - 2*(1 - x)^2 - (1 -
x)^4]] + ((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])])/(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*Sqr
t[4 + a])/(1 - Sqrt[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])

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Rubi in Sympy [A]  time = 58.5489, size = 314, normalized size = 0.81 \[ \frac{\left (x - 1\right ) \left (\frac{\left (x - 1\right )^{2}}{- \sqrt{a + 4} + 1} + 1\right ) \left (- \sqrt{a + 4} + 1\right )}{\sqrt{a - \left (x - 1\right )^{4} - 2 \left (x - 1\right )^{2} + 3}} + \operatorname{atan}{\left (- \frac{- 2 \left (x - 1\right )^{2} - 2}{2 \sqrt{a - \left (x - 1\right )^{4} - 2 \left (x - 1\right )^{2} + 3}} \right )} - \frac{\left (\frac{\left (x - 1\right )^{2}}{- \sqrt{a + 4} + 1} + 1\right ) \left (- \sqrt{a + 4} + 1\right ) \sqrt{\sqrt{a + 4} + 1} E\left (\operatorname{atan}{\left (\frac{x - 1}{\sqrt{\sqrt{a + 4} + 1}} \right )}\middle | \frac{2 \sqrt{a + 4}}{\sqrt{a + 4} - 1}\right )}{\sqrt{\frac{- \frac{\left (x - 1\right )^{2}}{\sqrt{a + 4} - 1} + 1}{\frac{\left (x - 1\right )^{2}}{\sqrt{a + 4} + 1} + 1}} \sqrt{a - \left (x - 1\right )^{4} - 2 \left (x - 1\right )^{2} + 3}} + \frac{\left (\frac{\left (x - 1\right )^{2}}{- \sqrt{a + 4} + 1} + 1\right ) \sqrt{\sqrt{a + 4} + 1} F\left (\operatorname{atan}{\left (\frac{x - 1}{\sqrt{\sqrt{a + 4} + 1}} \right )}\middle | \frac{2 \sqrt{a + 4}}{\sqrt{a + 4} - 1}\right )}{\sqrt{\frac{- \frac{\left (x - 1\right )^{2}}{\sqrt{a + 4} - 1} + 1}{\frac{\left (x - 1\right )^{2}}{\sqrt{a + 4} + 1} + 1}} \sqrt{a - \left (x - 1\right )^{4} - 2 \left (x - 1\right )^{2} + 3}} \]

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)**(1/2),x)

[Out]

(x - 1)*((x - 1)**2/(-sqrt(a + 4) + 1) + 1)*(-sqrt(a + 4) + 1)/sqrt(a - (x - 1)*
*4 - 2*(x - 1)**2 + 3) + atan(-(-2*(x - 1)**2 - 2)/(2*sqrt(a - (x - 1)**4 - 2*(x
- 1)**2 + 3))) - ((x - 1)**2/(-sqrt(a + 4) + 1) + 1)*(-sqrt(a + 4) + 1)*sqrt(sq
rt(a + 4) + 1)*elliptic_e(atan((x - 1)/sqrt(sqrt(a + 4) + 1)), 2*sqrt(a + 4)/(sq
rt(a + 4) - 1))/(sqrt((-(x - 1)**2/(sqrt(a + 4) - 1) + 1)/((x - 1)**2/(sqrt(a +
4) + 1) + 1))*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))/(sqrt((-(x - 1)**2/(sqrt(a + 4) - 1) + 1)/((
x - 1)**2/(sqrt(a + 4) + 1) + 1))*sqrt(a - (x - 1)**4 - 2*(x - 1)**2 + 3))

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Mathematica [B]  time = 6.05408, size = 1247, normalized size = 3.21 \[ \frac{2 \left (\sqrt{-\sqrt{a+4}-1}+\sqrt{\sqrt{a+4}-1}\right ) \sqrt{\frac{\left (\sqrt{-\sqrt{a+4}-1}-\sqrt{\sqrt{a+4}-1}\right ) \left (x+\sqrt{-\sqrt{a+4}-1}-1\right )}{\left (\sqrt{-\sqrt{a+4}-1}+\sqrt{\sqrt{a+4}-1}\right ) \left (-x+\sqrt{-\sqrt{a+4}-1}+1\right )}} \sqrt{\frac{\sqrt{-\sqrt{a+4}-1} \left (x-\sqrt{\sqrt{a+4}-1}-1\right )}{\left (\sqrt{-\sqrt{a+4}-1}+\sqrt{\sqrt{a+4}-1}\right ) \left (x-\sqrt{-\sqrt{a+4}-1}-1\right )}} \sqrt{\frac{\sqrt{-\sqrt{a+4}-1} \left (x+\sqrt{\sqrt{a+4}-1}-1\right )}{\left (\sqrt{-\sqrt{a+4}-1}-\sqrt{\sqrt{a+4}-1}\right ) \left (x-\sqrt{-\sqrt{a+4}-1}-1\right )}} \left (\frac{\left (\sqrt{-\sqrt{a+4}-1}-\sqrt{\sqrt{a+4}-1}\right ) E\left (\sin ^{-1}\left (\sqrt{\frac{\left (\sqrt{-\sqrt{a+4}-1}-\sqrt{\sqrt{a+4}-1}\right ) \left (x+\sqrt{-\sqrt{a+4}-1}-1\right )}{\left (\sqrt{-\sqrt{a+4}-1}+\sqrt{\sqrt{a+4}-1}\right ) \left (-x+\sqrt{-\sqrt{a+4}-1}+1\right )}}\right )|\frac{\left (\sqrt{-\sqrt{a+4}-1}+\sqrt{\sqrt{a+4}-1}\right )^2}{\left (\sqrt{-\sqrt{a+4}-1}-\sqrt{\sqrt{a+4}-1}\right )^2}\right )}{2 \sqrt{-\sqrt{a+4}-1}}+\frac{\left (\left (\sqrt{-\sqrt{a+4}-1}-1\right ) \left (\sqrt{-\sqrt{a+4}-1}-\sqrt{\sqrt{a+4}-1}\right )-\left (-\sqrt{-\sqrt{a+4}-1}-1\right ) \left (-\sqrt{-\sqrt{a+4}-1}-\sqrt{\sqrt{a+4}-1}-2\right )\right ) F\left (\sin ^{-1}\left (\sqrt{\frac{\left (\sqrt{-\sqrt{a+4}-1}-\sqrt{\sqrt{a+4}-1}\right ) \left (x+\sqrt{-\sqrt{a+4}-1}-1\right )}{\left (\sqrt{-\sqrt{a+4}-1}+\sqrt{\sqrt{a+4}-1}\right ) \left (-x+\sqrt{-\sqrt{a+4}-1}+1\right )}}\right )|\frac{\left (\sqrt{-\sqrt{a+4}-1}+\sqrt{\sqrt{a+4}-1}\right )^2}{\left (\sqrt{-\sqrt{a+4}-1}-\sqrt{\sqrt{a+4}-1}\right )^2}\right )}{2 \sqrt{-\sqrt{a+4}-1} \left (\sqrt{\sqrt{a+4}-1}-\sqrt{-\sqrt{a+4}-1}\right )}+\frac{4 \Pi \left (\frac{\sqrt{-\sqrt{a+4}-1}+\sqrt{\sqrt{a+4}-1}}{\sqrt{\sqrt{a+4}-1}-\sqrt{-\sqrt{a+4}-1}};\sin ^{-1}\left (\sqrt{\frac{\left (\sqrt{-\sqrt{a+4}-1}-\sqrt{\sqrt{a+4}-1}\right ) \left (x+\sqrt{-\sqrt{a+4}-1}-1\right )}{\left (\sqrt{-\sqrt{a+4}-1}+\sqrt{\sqrt{a+4}-1}\right ) \left (-x+\sqrt{-\sqrt{a+4}-1}+1\right )}}\right )|\frac{\left (\sqrt{-\sqrt{a+4}-1}+\sqrt{\sqrt{a+4}-1}\right )^2}{\left (\sqrt{-\sqrt{a+4}-1}-\sqrt{\sqrt{a+4}-1}\right )^2}\right )}{\sqrt{\sqrt{a+4}-1}-\sqrt{-\sqrt{a+4}-1}}\right ) \left (x-\sqrt{-\sqrt{a+4}-1}-1\right )^2+\left (x+\sqrt{-\sqrt{a+4}-1}-1\right ) \left (x-\sqrt{\sqrt{a+4}-1}-1\right ) \left (x+\sqrt{\sqrt{a+4}-1}-1\right )}{\sqrt{a-x \left (x^3-4 x^2+8 x-8\right )}} \]

Warning: Unable to verify antiderivative.

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

[Out]

((-1 + Sqrt[-1 - Sqrt[4 + a]] + x)*(-1 - Sqrt[-1 + Sqrt[4 + a]] + x)*(-1 + Sqrt[
-1 + Sqrt[4 + a]] + x) + 2*(Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1
- Sqrt[-1 - Sqrt[4 + a]] + x)^2*Sqrt[((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[
4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 +
Sqrt[4 + a]])*(1 + Sqrt[-1 - Sqrt[4 + a]] - x))]*Sqrt[(Sqrt[-1 - Sqrt[4 + a]]*(
-1 - Sqrt[-1 + Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 +
a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]*Sqrt[(Sqrt[-1 - Sqrt[4 + a]]*(-1 + Sqrt
[-1 + Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1
- Sqrt[-1 - Sqrt[4 + a]] + x))]*(((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a
]])*EllipticE[ArcSin[Sqrt[((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1
+ Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]
])*(1 + Sqrt[-1 - Sqrt[4 + a]] - x))]], (Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt
[4 + a]])^2/(Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])^2])/(2*Sqrt[-1 - S
qrt[4 + a]]) + ((-((-1 - Sqrt[-1 - Sqrt[4 + a]])*(-2 - Sqrt[-1 - Sqrt[4 + a]] -
Sqrt[-1 + Sqrt[4 + a]])) + (-1 + Sqrt[-1 - Sqrt[4 + a]])*(Sqrt[-1 - Sqrt[4 + a]]
- Sqrt[-1 + Sqrt[4 + a]]))*EllipticF[ArcSin[Sqrt[((Sqrt[-1 - Sqrt[4 + a]] - Sqr
t[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]]
+ Sqrt[-1 + Sqrt[4 + a]])*(1 + Sqrt[-1 - Sqrt[4 + a]] - x))]], (Sqrt[-1 - Sqrt[
4 + a]] + Sqrt[-1 + Sqrt[4 + a]])^2/(Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 +
a]])^2])/(2*Sqrt[-1 - Sqrt[4 + a]]*(-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4
+ a]])) + (4*EllipticPi[(Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])/(-Sqrt
[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]]), ArcSin[Sqrt[((Sqrt[-1 - Sqrt[4 + a
]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt
[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(1 + Sqrt[-1 - Sqrt[4 + a]] - x))]], (Sqrt[-1
- Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])^2/(Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 +
Sqrt[4 + a]])^2])/(-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])))/Sqrt[a -
x*(-8 + 8*x - 4*x^2 + x^3)]

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Maple [B]  time = 0.029, size = 1147, normalized size = 3. \[ \text{result too large to display} \]

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)^(1/2),x)

[Out]

((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. \[ \int \frac{x^{2}}{\sqrt{-x^{4} + 4 \, x^{3} - 8 \, x^{2} + a + 8 \, x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(x^2/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. \[ \int \frac{x^{2}}{\sqrt{a - x^{4} + 4 x^{3} - 8 x^{2} + 8 x}}\, dx \]

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)**(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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