∖int x____________________ ∖left(a+8x−8x2+4x3−x4∖right)5∕2 ∖,dx

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

Optimal. Leaf size=591 \[ \frac{(x-1) \left (5 a^2+4 (2 a+7) (x-1)^2+47 a+104\right )}{12 (a+3)^2 (a+4)^2 \sqrt{a-(x-1)^4-2 (x-1)^2+3}}+\frac{(x-1) \left (a+(x-1)^2+5\right )}{6 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}}+\frac{(x-1)^2+1}{3 (a+4)^2 \sqrt{a-(x-1)^4-2 (x-1)^2+3}}+\frac{(x-1)^2+1}{6 (a+4) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}}-\frac{(2 a+7) \left (1-\sqrt{a+4}\right ) (x-1) \left (\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right )}{3 (a+3)^2 (a+4)^2 \sqrt{a-(x-1)^4-2 (x-1)^2+3}}+\frac{(5 a+16) \sqrt{\sqrt{a+4}+1} \left (\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right ) F\left (\tan ^{-1}\left (\frac{x-1}{\sqrt{\sqrt{a+4}+1}}\right )|-\frac{2 \sqrt{a+4}}{1-\sqrt{a+4}}\right )}{12 (a+3) (a+4)^2 \sqrt{\frac{\frac{(x-1)^2}{1-\sqrt{a+4}}+1}{\frac{(x-1)^2}{\sqrt{a+4}+1}+1}} \sqrt{a-(x-1)^4-2 (x-1)^2+3}}+\frac{(2 a+7) \left (1-\sqrt{a+4}\right ) \sqrt{\sqrt{a+4}+1} \left (\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right ) E\left (\tan ^{-1}\left (\frac{x-1}{\sqrt{\sqrt{a+4}+1}}\right )|-\frac{2 \sqrt{a+4}}{1-\sqrt{a+4}}\right )}{3 (a+3)^2 (a+4)^2 \sqrt{\frac{\frac{(x-1)^2}{1-\sqrt{a+4}}+1}{\frac{(x-1)^2}{\sqrt{a+4}+1}+1}} \sqrt{a-(x-1)^4-2 (x-1)^2+3}} \]

[Out]

(1 + (-1 + x)^2)/(6*(4 + a)*(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)^(3/2)) + (1 + (-

1 + x)^2)/(3*(4 + a)^2*Sqrt[3 + a - 2*(-1 + x)^2 - (-1 + x)^4]) + ((5 + a + (-1

+ x)^2)*(-1 + x))/(6*(12 + 7*a + a^2)*(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)^(3/2))

+ ((104 + 47*a + 5*a^2 + 4*(7 + 2*a)*(-1 + x)^2)*(-1 + x))/(12*(3 + a)^2*(4 + a

)^2*Sqrt[3 + a - 2*(-1 + x)^2 - (-1 + x)^4]) - ((7 + 2*a)*(1 - Sqrt[4 + a])*(1 +

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

-1 + x)^2 - (-1 + x)^4]) + ((7 + 2*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])])/(3*(3 + a)^2*(4 + 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]) + ((16 + 5*a)*Sqrt[1 + Sqrt[4 + a]]*(1 + (-1 + x)^2/(1 - Sqr

t[4 + a]))*EllipticF[ArcTan[(-1 + x)/Sqrt[1 + Sqrt[4 + a]]], (-2*Sqrt[4 + a])/(1

- Sqrt[4 + a])])/(12*(3 + a)*(4 + 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 = 1.52586, antiderivative size = 591, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 12, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.462 \[ -\frac{(1-x) \left (a+(x-1)^2+5\right )}{6 \left (a^2+7 a+12\right ) \left (a-(1-x)^4-2 (1-x)^2+3\right )^{3/2}}-\frac{(1-x) \left (5 a^2+4 (2 a+7) (1-x)^2+47 a+104\right )}{12 (a+3)^2 (a+4)^2 \sqrt{a-(1-x)^4-2 (1-x)^2+3}}+\frac{(x-1)^2+1}{3 (a+4)^2 \sqrt{a-(1-x)^4-2 (1-x)^2+3}}+\frac{(x-1)^2+1}{6 (a+4) \left (a-(1-x)^4-2 (1-x)^2+3\right )^{3/2}}+\frac{(2 a+7) \left (1-\sqrt{a+4}\right ) (1-x) \left (\frac{(1-x)^2}{1-\sqrt{a+4}}+1\right )}{3 (a+3)^2 (a+4)^2 \sqrt{a-(1-x)^4-2 (1-x)^2+3}}-\frac{(5 a+16) \sqrt{\sqrt{a+4}+1} \left (\frac{(1-x)^2}{1-\sqrt{a+4}}+1\right ) F\left (\tan ^{-1}\left (\frac{1-x}{\sqrt{\sqrt{a+4}+1}}\right )|-\frac{2 \sqrt{a+4}}{1-\sqrt{a+4}}\right )}{12 (a+3) (a+4)^2 \sqrt{\frac{\frac{(1-x)^2}{1-\sqrt{a+4}}+1}{\frac{(1-x)^2}{\sqrt{a+4}+1}+1}} \sqrt{a-(1-x)^4-2 (1-x)^2+3}}-\frac{(2 a+7) \left (1-\sqrt{a+4}\right ) \sqrt{\sqrt{a+4}+1} \left (\frac{(1-x)^2}{1-\sqrt{a+4}}+1\right ) E\left (\tan ^{-1}\left (\frac{1-x}{\sqrt{\sqrt{a+4}+1}}\right )|-\frac{2 \sqrt{a+4}}{1-\sqrt{a+4}}\right )}{3 (a+3)^2 (a+4)^2 \sqrt{\frac{\frac{(1-x)^2}{1-\sqrt{a+4}}+1}{\frac{(1-x)^2}{\sqrt{a+4}+1}+1}} \sqrt{a-(1-x)^4-2 (1-x)^2+3}} \]

Antiderivative was successfully veriﬁed.

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

[Out]

(1 + (-1 + x)^2)/(6*(4 + a)*(3 + a - 2*(1 - x)^2 - (1 - x)^4)^(3/2)) + (1 + (-1

+ x)^2)/(3*(4 + a)^2*Sqrt[3 + a - 2*(1 - x)^2 - (1 - x)^4]) - ((104 + 47*a + 5*a

^2 + 4*(7 + 2*a)*(1 - x)^2)*(1 - x))/(12*(3 + a)^2*(4 + a)^2*Sqrt[3 + a - 2*(1 -

x)^2 - (1 - x)^4]) + ((7 + 2*a)*(1 - Sqrt[4 + a])*(1 + (1 - x)^2/(1 - Sqrt[4 +

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

5 + a + (-1 + x)^2)*(1 - x))/(6*(12 + 7*a + a^2)*(3 + a - 2*(1 - x)^2 - (1 - x)^

4)^(3/2)) - ((7 + 2*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])])/(3*(3 + a)^2*(4 + 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]) -

((16 + 5*a)*Sqrt[1 + Sqrt[4 + a]]*(1 + (1 - x)^2/(1 - Sqrt[4 + a]))*EllipticF[A

rcTan[(1 - x)/Sqrt[1 + Sqrt[4 + a]]], (-2*Sqrt[4 + a])/(1 - Sqrt[4 + a])])/(12*(

3 + a)*(4 + 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 = 108.201, size = 502, normalized size = 0.85 \[ \text{result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

(x - 1)*(2*a + (2*a + 10)*(x - 1) + 2*(x - 1)**3 + 2*(x - 1)**2 + 10)/(12*(a + 3

)*(a + 4)*(a - (x - 1)**4 - 2*(x - 1)**2 + 3)**(3/2)) - (2*a + 7)*(x - 1)*((x -

1)**2/(-sqrt(a + 4) + 1) + 1)*(-sqrt(a + 4) + 1)/(3*(a + 3)**2*(a + 4)**2*sqrt(a

- (x - 1)**4 - 2*(x - 1)**2 + 3)) + (3*a + 10)*sqrt(a - (x - 1)**4 - 2*(x - 1)*

*2 + 3)/(6*(a + 3)**2*(a + 4)**2) + (x - 1)*(20*a**2 + 188*a + (24*a + 80)*(x -

1)**3 + (32*a + 112)*(x - 1)**2 + (x - 1)*(16*a**2 + 144*a + 304) + 416)/(48*(a

+ 3)**2*(a + 4)**2*sqrt(a - (x - 1)**4 - 2*(x - 1)**2 + 3)) + (5*a + 16)*((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)**2*sqrt(a -

(x - 1)**4 - 2*(x - 1)**2 + 3)) + (2*a + 7)*((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))/(3*sqrt((-(x - 1)**2/(sqrt(a + 4) - 1

) + 1)/((x - 1)**2/(sqrt(a + 4) + 1) + 1))*(a + 3)**2*(a + 4)**2*sqrt(a - (x - 1

)**4 - 2*(x - 1)**2 + 3))

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Mathematica [B] time = 6.16041, size = 6452, normalized size = 10.92 \[ \text{Result too large to show} \]

Antiderivative was successfully veriﬁed.

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

[Out]

Result too large to show

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Maple [B] time = 0.042, size = 2777, normalized size = 4.7 \[ \text{result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

(1/6/(a^2+7*a+12)*x^3+1/6*a/(a^2+7*a+12)*x^2-1/6*(a-2)/(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*(1/6*(7+2*a

)/(a^2+7*a+12)^2*x^3+1/6*(a^2-12)/(a^2+7*a+12)^2*x^2-1/24*(3*a^2-23*a-116)/(a^2+

7*a+12)^2*x+1/24*(3*a^2-7*a-60)/(a^2+7*a+12)^2)/(-x^4+4*x^3-8*x^2+a+8*x)^(1/2)-(

1/6*(a^2+23*a+68)/(a^2+7*a+12)^2+1/12*(3*a^2-23*a-116)/(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)*Elliptic

F(((-(-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))-(2/3*(a^2+2*a-5)/(a^2+7*a+12)^2

-2/3*(a^2-12)/(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)))-1/3*(7+2*a)/(a^2+7*a+1

2)^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-(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. \[ \int \frac{x}{{\left (-x^{4} + 4 \, x^{3} - 8 \, x^{2} + a + 8 \, x\right )}^{\frac{5}{2}}}\,{d x} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x/(-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. \[{\rm integral}\left (\frac{x}{{\left (x^{8} - 8 \, x^{7} + 32 \, x^{6} - 2 \,{\left (a - 64\right )} x^{4} - 80 \, x^{5} + 8 \,{\left (a - 16\right )} x^{3} - 16 \,{\left (a - 4\right )} x^{2} + a^{2} + 16 \, a x\right )} \sqrt{-x^{4} + 4 \, x^{3} - 8 \, x^{2} + a + 8 \, x}}, x\right ) \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(x/((x^8 - 8*x^7 + 32*x^6 - 2*(a - 64)*x^4 - 80*x^5 + 8*(a - 16)*x^3 - 1

6*(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. \[ \int \frac{x}{\left (a - x^{4} + 4 x^{3} - 8 x^{2} + 8 x\right )^{\frac{5}{2}}}\, dx \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x/(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. \[ \int \frac{x}{{\left (-x^{4} + 4 \, x^{3} - 8 \, x^{2} + a + 8 \, x\right )}^{\frac{5}{2}}}\,{d x} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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