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

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 + 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 = 0.152437, 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, Rules used = {1680, 1673, 1104, 418, 1107, 621, 204} \[ \frac{1}{2} \tan ^{-1}\left (\frac{(x-1)^2+1}{\sqrt{a-(x-1)^4-2 (x-1)^2+3}}\right )+\frac{\sqrt{\sqrt{a+4}+1} \left (\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right ) F\left (\tan ^{-1}\left (\frac{x-1}{\sqrt{\sqrt{a+4}+1}}\right )|-\frac{2 \sqrt{a+4}}{1-\sqrt{a+4}}\right )}{\sqrt{\frac{\frac{(x-1)^2}{1-\sqrt{a+4}}+1}{\frac{(x-1)^2}{\sqrt{a+4}+1}+1}} \sqrt{a-(x-1)^4-2 (x-1)^2+3}} \]

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 + 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]
)

Rule 1680

Int[(Pq_)*(Q4_)^(p_), x_Symbol] :> With[{a = Coeff[Q4, x, 0], b = Coeff[Q4, x, 1], c = Coeff[Q4, x, 2], d = Co
eff[Q4, x, 3], e = Coeff[Q4, x, 4]}, Subst[Int[SimplifyIntegrand[(Pq /. x -> -(d/(4*e)) + x)*(a + d^4/(256*e^3
) - (b*d)/(8*e) + (c - (3*d^2)/(8*e))*x^2 + e*x^4)^p, x], x], x, d/(4*e) + x] /; EqQ[d^3 - 4*c*d*e + 8*b*e^2,
0] && NeQ[d, 0]] /; FreeQ[p, x] && PolyQ[Pq, x] && PolyQ[Q4, x, 4] &&  !IGtQ[p, 0]

Rule 1673

Int[(Pq_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], k}, Int[Sum[Coeff[
Pq, x, 2*k]*x^(2*k), {k, 0, q/2}]*(a + b*x^2 + c*x^4)^p, x] + Int[x*Sum[Coeff[Pq, x, 2*k + 1]*x^(2*k), {k, 0,
(q - 1)/2}]*(a + b*x^2 + c*x^4)^p, x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] &&  !PolyQ[Pq, x^2]

Rule 1104

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(Sqrt[1 + (2*
c*x^2)/(b - q)]*Sqrt[1 + (2*c*x^2)/(b + q)])/Sqrt[a + b*x^2 + c*x^4], Int[1/(Sqrt[1 + (2*c*x^2)/(b - q)]*Sqrt[
1 + (2*c*x^2)/(b + q)]), x], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && NegQ[c/a]

Rule 418

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(Sqrt[a + b*x^2]*EllipticF[ArcT
an[Rt[d/c, 2]*x], 1 - (b*c)/(a*d)])/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]), x] /
; FreeQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] &&  !SimplerSqrtQ[b/a, d/c]

Rule 1107

Int[(x_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[(a + b*x + c*x^2)^p, x],
x, x^2], x] /; FreeQ[{a, b, c, p}, x]

Rule 621

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{x}{\sqrt{a+8 x-8 x^2+4 x^3-x^4}} \, dx &=\operatorname{Subst}\left (\int \frac{1+x}{\sqrt{3+a-2 x^2-x^4}} \, dx,x,-1+x\right )\\ &=\operatorname{Subst}\left (\int \frac{1}{\sqrt{3+a-2 x^2-x^4}} \, dx,x,-1+x\right )+\operatorname{Subst}\left (\int \frac{x}{\sqrt{3+a-2 x^2-x^4}} \, dx,x,-1+x\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{3+a-2 x-x^2}} \, dx,x,(-1+x)^2\right )+\frac{\left (\sqrt{1-\frac{2 (-1+x)^2}{-2-2 \sqrt{4+a}}} \sqrt{1-\frac{2 (-1+x)^2}{-2+2 \sqrt{4+a}}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{2 x^2}{-2-2 \sqrt{4+a}}} \sqrt{1-\frac{2 x^2}{-2+2 \sqrt{4+a}}}} \, dx,x,-1+x\right )}{\sqrt{3+a-2 (-1+x)^2-(-1+x)^4}}\\ &=-\frac{\sqrt{1+\sqrt{4+a}} \left (1+\frac{(1-x)^2}{1-\sqrt{4+a}}\right ) F\left (\tan ^{-1}\left (\frac{1-x}{\sqrt{1+\sqrt{4+a}}}\right )|-\frac{2 \sqrt{4+a}}{1-\sqrt{4+a}}\right )}{\sqrt{\frac{1+\frac{(1-x)^2}{1-\sqrt{4+a}}}{1+\frac{(1-x)^2}{1+\sqrt{4+a}}}} \sqrt{3+a-2 (1-x)^2-(1-x)^4}}+\operatorname{Subst}\left (\int \frac{1}{-4-x^2} \, dx,x,-\frac{2 \left (1+(-1+x)^2\right )}{\sqrt{3+a-2 (-1+x)^2-(-1+x)^4}}\right )\\ &=\frac{1}{2} \tan ^{-1}\left (\frac{1+(-1+x)^2}{\sqrt{3+a-2 (-1+x)^2-(-1+x)^4}}\right )-\frac{\sqrt{1+\sqrt{4+a}} \left (1+\frac{(1-x)^2}{1-\sqrt{4+a}}\right ) F\left (\tan ^{-1}\left (\frac{1-x}{\sqrt{1+\sqrt{4+a}}}\right )|-\frac{2 \sqrt{4+a}}{1-\sqrt{4+a}}\right )}{\sqrt{\frac{1+\frac{(1-x)^2}{1-\sqrt{4+a}}}{1+\frac{(1-x)^2}{1+\sqrt{4+a}}}} \sqrt{3+a-2 (1-x)^2-(1-x)^4}}\\ \end{align*}

Mathematica [B]  time = 2.73358, 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)*S
qrt[(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 + 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 - S
qrt[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 - 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[((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 = 0.016, size = 788, normalized size = 4.4 \begin{align*} \text{result too large to display} \end{align*}

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. \begin{align*} \int \frac{x}{\sqrt{-x^{4} + 4 \, x^{3} - 8 \, x^{2} + a + 8 \, x}}\,{d x} \end{align*}

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, 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. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-x^{4} + 4 \, x^{3} - 8 \, x^{2} + a + 8 \, x} x}{x^{4} - 4 \, x^{3} + 8 \, x^{2} - a - 8 \, x}, x\right ) \end{align*}

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, algorithm="fricas")

[Out]

integral(-sqrt(-x^4 + 4*x^3 - 8*x^2 + a + 8*x)*x/(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. \begin{align*} \int \frac{x}{\sqrt{a - x^{4} + 4 x^{3} - 8 x^{2} + 8 x}}\, dx \end{align*}

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

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, algorithm="giac")

[Out]

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