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

Optimal. Leaf size=179 \[ \frac{1}{2} \tan ^{-1}\left (\frac{(x-1)^2+1}{\sqrt{a-(x-1)^4-2 (x-1)^2+3}}\right )+\frac{\sqrt{\sqrt{a+4}+1} \left (\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right ) F\left (\tan ^{-1}\left (\frac{x-1}{\sqrt{\sqrt{a+4}+1}}\right )|-\frac{2 \sqrt{a+4}}{1-\sqrt{a+4}}\right )}{\sqrt{\frac{\frac{(x-1)^2}{1-\sqrt{a+4}}+1}{\frac{(x-1)^2}{\sqrt{a+4}+1}+1}} \sqrt{a-(x-1)^4-2 (x-1)^2+3}} \]

[Out]

ArcTan[(1 + (-1 + x)^2)/Sqrt[3 + a - 2*(-1 + x)^2 - (-1 + x)^4]]/2 + (Sqrt[1 + S
qrt[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 = 0.334357, antiderivative size = 179, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ \frac{1}{2} \tan ^{-1}\left (\frac{(x-1)^2+1}{\sqrt{a-(1-x)^4-2 (1-x)^2+3}}\right )-\frac{\sqrt{\sqrt{a+4}+1} \left (\frac{(1-x)^2}{1-\sqrt{a+4}}+1\right ) F\left (\tan ^{-1}\left (\frac{1-x}{\sqrt{\sqrt{a+4}+1}}\right )|-\frac{2 \sqrt{a+4}}{1-\sqrt{a+4}}\right )}{\sqrt{\frac{\frac{(1-x)^2}{1-\sqrt{a+4}}+1}{\frac{(1-x)^2}{\sqrt{a+4}+1}+1}} \sqrt{a-(1-x)^4-2 (1-x)^2+3}} \]

Antiderivative was successfully verified.

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

[Out]

ArcTan[(1 + (-1 + x)^2)/Sqrt[3 + a - 2*(1 - x)^2 - (1 - x)^4]]/2 - (Sqrt[1 + Sqr
t[4 + a]]*(1 + (1 - x)^2/(1 - Sqrt[4 + a]))*EllipticF[ArcTan[(1 - x)/Sqrt[1 + Sq
rt[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 in Sympy [A]  time = 26.6278, size = 148, normalized size = 0.83 \[ \frac{\operatorname{atan}{\left (- \frac{- 2 \left (x - 1\right )^{2} - 2}{2 \sqrt{a - \left (x - 1\right )^{4} - 2 \left (x - 1\right )^{2} + 3}} \right )}}{2} + \frac{\left (\frac{\left (x - 1\right )^{2}}{- \sqrt{a + 4} + 1} + 1\right ) \sqrt{\sqrt{a + 4} + 1} F\left (\operatorname{atan}{\left (\frac{x - 1}{\sqrt{\sqrt{a + 4} + 1}} \right )}\middle | \frac{2 \sqrt{a + 4}}{\sqrt{a + 4} - 1}\right )}{\sqrt{\frac{- \frac{\left (x - 1\right )^{2}}{\sqrt{a + 4} - 1} + 1}{\frac{\left (x - 1\right )^{2}}{\sqrt{a + 4} + 1} + 1}} \sqrt{a - \left (x - 1\right )^{4} - 2 \left (x - 1\right )^{2} + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(-x**4+4*x**3-8*x**2+a+8*x)**(1/2),x)

[Out]

atan(-(-2*(x - 1)**2 - 2)/(2*sqrt(a - (x - 1)**4 - 2*(x - 1)**2 + 3)))/2 + ((x -
1)**2/(-sqrt(a + 4) + 1) + 1)*sqrt(sqrt(a + 4) + 1)*elliptic_f(atan((x - 1)/sqr
t(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 = 5.16382, size = 813, normalized size = 4.54 \[ \frac{2 \left (-x+\sqrt{-\sqrt{a+4}-1}+1\right ) \sqrt{\frac{\sqrt{-\sqrt{a+4}-1} \left (-x+\sqrt{\sqrt{a+4}-1}+1\right )}{\left (\sqrt{-\sqrt{a+4}-1}+\sqrt{\sqrt{a+4}-1}\right ) \left (-x+\sqrt{-\sqrt{a+4}-1}+1\right )}} \left (x+\sqrt{-\sqrt{a+4}-1}-1\right ) \sqrt{\frac{\sqrt{-\sqrt{a+4}-1} \left (x+\sqrt{\sqrt{a+4}-1}-1\right )}{\left (\sqrt{\sqrt{a+4}-1}-\sqrt{-\sqrt{a+4}-1}\right ) \left (-x+\sqrt{-\sqrt{a+4}-1}+1\right )}} \left (\left (\sqrt{-\sqrt{a+4}-1}+1\right ) F\left (\sin ^{-1}\left (\sqrt{\frac{\left (\sqrt{-\sqrt{a+4}-1}-\sqrt{\sqrt{a+4}-1}\right ) \left (x+\sqrt{-\sqrt{a+4}-1}-1\right )}{\left (\sqrt{-\sqrt{a+4}-1}+\sqrt{\sqrt{a+4}-1}\right ) \left (-x+\sqrt{-\sqrt{a+4}-1}+1\right )}}\right )|\frac{\left (\sqrt{-\sqrt{a+4}-1}+\sqrt{\sqrt{a+4}-1}\right )^2}{\left (\sqrt{-\sqrt{a+4}-1}-\sqrt{\sqrt{a+4}-1}\right )^2}\right )-2 \sqrt{-\sqrt{a+4}-1} \Pi \left (\frac{\sqrt{-\sqrt{a+4}-1}+\sqrt{\sqrt{a+4}-1}}{\sqrt{\sqrt{a+4}-1}-\sqrt{-\sqrt{a+4}-1}};\sin ^{-1}\left (\sqrt{\frac{\left (\sqrt{-\sqrt{a+4}-1}-\sqrt{\sqrt{a+4}-1}\right ) \left (x+\sqrt{-\sqrt{a+4}-1}-1\right )}{\left (\sqrt{-\sqrt{a+4}-1}+\sqrt{\sqrt{a+4}-1}\right ) \left (-x+\sqrt{-\sqrt{a+4}-1}+1\right )}}\right )|\frac{\left (\sqrt{-\sqrt{a+4}-1}+\sqrt{\sqrt{a+4}-1}\right )^2}{\left (\sqrt{-\sqrt{a+4}-1}-\sqrt{\sqrt{a+4}-1}\right )^2}\right )\right )}{\sqrt{-\sqrt{a+4}-1} \sqrt{\frac{\left (\sqrt{-\sqrt{a+4}-1}-\sqrt{\sqrt{a+4}-1}\right ) \left (x+\sqrt{-\sqrt{a+4}-1}-1\right )}{\left (\sqrt{-\sqrt{a+4}-1}+\sqrt{\sqrt{a+4}-1}\right ) \left (-x+\sqrt{-\sqrt{a+4}-1}+1\right )}} \sqrt{a-x \left (x^3-4 x^2+8 x-8\right )}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(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))]*(-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))]*((1 + Sqrt[-1 - Sqrt[4 + a]])*
EllipticF[ArcSin[Sqrt[((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 + S
qrt[-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]]*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 - Sq
rt[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[((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[a - x*(-8 + 8*x - 4*
x^2 + x^3)])

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Maple [B]  time = 0.026, size = 788, normalized size = 4.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(-x^4+4*x^3-8*x^2+a+8*x)^(1/2),x)

[Out]

-((-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)))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x/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. \[{\rm integral}\left (\frac{x}{\sqrt{-x^{4} + 4 \, x^{3} - 8 \, x^{2} + a + 8 \, x}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(-x**4+4*x**3-8*x**2+a+8*x)**(1/2),x)

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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