3.767 \(\int \frac{1}{(8 x-8 x^2+4 x^3-x^4)^{3/2}} \, dx\)

Optimal. Leaf size=73 \[ \frac{\left ((x-1)^2+5\right ) (x-1)}{24 \sqrt{-(x-1)^4-2 (x-1)^2+3}}-\frac{F\left (\sin ^{-1}(1-x)|-\frac{1}{3}\right )}{4 \sqrt{3}}+\frac{E\left (\sin ^{-1}(1-x)|-\frac{1}{3}\right )}{8 \sqrt{3}} \]

[Out]

((5 + (-1 + x)^2)*(-1 + x))/(24*Sqrt[3 - 2*(-1 + x)^2 - (-1 + x)^4]) + EllipticE[ArcSin[1 - x], -1/3]/(8*Sqrt[
3]) - EllipticF[ArcSin[1 - x], -1/3]/(4*Sqrt[3])

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Rubi [A]  time = 0.0514859, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {1106, 1092, 1180, 524, 424, 419} \[ \frac{\left ((x-1)^2+5\right ) (x-1)}{24 \sqrt{-(x-1)^4-2 (x-1)^2+3}}-\frac{F\left (\sin ^{-1}(1-x)|-\frac{1}{3}\right )}{4 \sqrt{3}}+\frac{E\left (\sin ^{-1}(1-x)|-\frac{1}{3}\right )}{8 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((5 + (-1 + x)^2)*(-1 + x))/(24*Sqrt[3 - 2*(-1 + x)^2 - (-1 + x)^4]) + EllipticE[ArcSin[1 - x], -1/3]/(8*Sqrt[
3]) - EllipticF[ArcSin[1 - x], -1/3]/(4*Sqrt[3])

Rule 1106

Int[(P4_)^(p_), x_Symbol] :> With[{a = Coeff[P4, x, 0], b = Coeff[P4, x, 1], c = Coeff[P4, x, 2], d = Coeff[P4
, x, 3], e = Coeff[P4, x, 4]}, Subst[Int[SimplifyIntegrand[(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[P4, x, 4] && NeQ[p, 2] && NeQ[p, 3]

Rule 1092

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> -Simp[(x*(b^2 - 2*a*c + b*c*x^2)*(a + b*x^2 + c*x^
4)^(p + 1))/(2*a*(p + 1)*(b^2 - 4*a*c)), x] + Dist[1/(2*a*(p + 1)*(b^2 - 4*a*c)), Int[(b^2 - 2*a*c + 2*(p + 1)
*(b^2 - 4*a*c) + b*c*(4*p + 7)*x^2)*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*
a*c, 0] && LtQ[p, -1] && IntegerQ[2*p]

Rule 1180

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

Rule 524

Int[((e_) + (f_.)*(x_)^(n_))/(Sqrt[(a_) + (b_.)*(x_)^(n_)]*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/
b, Int[Sqrt[a + b*x^n]/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/(Sqrt[a + b*x^n]*Sqrt[c + d*x^n]),
x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] &&  !(EqQ[n, 2] && ((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && (PosQ[
d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-(b/a), -(d/c)]))))))

Rule 424

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[ArcSin[Rt[-(d/c)
, 2]*x], (b*c)/(a*d)])/(Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[
a, 0]

Rule 419

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1*EllipticF[ArcSin[Rt[-(d/c),
2]*x], (b*c)/(a*d)])/(Sqrt[a]*Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &
& GtQ[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-(b/a), -(d/c)])

Rubi steps

\begin{align*} \int \frac{1}{\left (8 x-8 x^2+4 x^3-x^4\right )^{3/2}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{\left (3-2 x^2-x^4\right )^{3/2}} \, dx,x,-1+x\right )\\ &=\frac{\left (5+(-1+x)^2\right ) (-1+x)}{24 \sqrt{3-2 (-1+x)^2-(-1+x)^4}}-\frac{1}{48} \operatorname{Subst}\left (\int \frac{-6+2 x^2}{\sqrt{3-2 x^2-x^4}} \, dx,x,-1+x\right )\\ &=\frac{\left (5+(-1+x)^2\right ) (-1+x)}{24 \sqrt{3-2 (-1+x)^2-(-1+x)^4}}-\frac{1}{24} \operatorname{Subst}\left (\int \frac{-6+2 x^2}{\sqrt{2-2 x^2} \sqrt{6+2 x^2}} \, dx,x,-1+x\right )\\ &=\frac{\left (5+(-1+x)^2\right ) (-1+x)}{24 \sqrt{3-2 (-1+x)^2-(-1+x)^4}}-\frac{1}{24} \operatorname{Subst}\left (\int \frac{\sqrt{6+2 x^2}}{\sqrt{2-2 x^2}} \, dx,x,-1+x\right )+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{2-2 x^2} \sqrt{6+2 x^2}} \, dx,x,-1+x\right )\\ &=\frac{\left (5+(-1+x)^2\right ) (-1+x)}{24 \sqrt{3-2 (-1+x)^2-(-1+x)^4}}+\frac{E\left (\sin ^{-1}(1-x)|-\frac{1}{3}\right )}{8 \sqrt{3}}-\frac{F\left (\sin ^{-1}(1-x)|-\frac{1}{3}\right )}{4 \sqrt{3}}\\ \end{align*}

Mathematica [C]  time = 0.869394, size = 261, normalized size = 3.58 \[ \frac{\sqrt{-x \left (x^3-4 x^2+8 x-8\right )} \left (\frac{\sqrt{2} \left (\sqrt{3}-i\right ) \sqrt{-\frac{i (x-2)}{\left (\sqrt{3}-i\right ) x}} E\left (\sin ^{-1}\left (\frac{\sqrt{\sqrt{3}+i-\frac{4 i}{x}}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{-i+\sqrt{3}}\right )}{\sqrt{\frac{x^2-2 x+4}{x^2}}}-\frac{x^2-4 i \sqrt{2} \sqrt{-\frac{i (x-2)}{\left (\sqrt{3}-i\right ) x}} \sqrt{\frac{x^2-2 x+4}{x^2}} x^2 F\left (\sin ^{-1}\left (\frac{\sqrt{\sqrt{3}+i-\frac{4 i}{x}}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{-i+\sqrt{3}}\right )+2}{x^2-2 x+4}\right )}{24 (x-2) x} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[-(x*(-8 + 8*x - 4*x^2 + x^3))]*((Sqrt[2]*(-I + Sqrt[3])*Sqrt[((-I)*(-2 + x))/((-I + Sqrt[3])*x)]*Ellipti
cE[ArcSin[Sqrt[I + Sqrt[3] - (4*I)/x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(-I + Sqrt[3])])/Sqrt[(4 - 2*x + x^2)/x^
2] - (2 + x^2 - (4*I)*Sqrt[2]*Sqrt[((-I)*(-2 + x))/((-I + Sqrt[3])*x)]*x^2*Sqrt[(4 - 2*x + x^2)/x^2]*EllipticF
[ArcSin[Sqrt[I + Sqrt[3] - (4*I)/x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(-I + Sqrt[3])])/(4 - 2*x + x^2)))/(24*(-2
+ x)*x)

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Maple [B]  time = 0.032, size = 963, normalized size = 13.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-x^4+4*x^3-8*x^2+8*x)^(3/2),x)

[Out]

-1/32*(-x^3+4*x^2-8*x+8)/(x*(-x^3+4*x^2-8*x+8))^(1/2)+2*x*(1/24+1/192*x^2)/(-x*(x^3-4*x^2+8*x-8))^(1/2)+1/6*(-
I*3^(1/2)-1)*((I*3^(1/2)-1)*x/(1+I*3^(1/2))/(-2+x))^(1/2)*(-2+x)^2*((x-1+I*3^(1/2))/(1-I*3^(1/2))/(-2+x))^(1/2
)*((x-1-I*3^(1/2))/(1+I*3^(1/2))/(-2+x))^(1/2)/(I*3^(1/2)-1)/(-x*(-2+x)*(x-1+I*3^(1/2))*(x-1-I*3^(1/2)))^(1/2)
*EllipticF(((I*3^(1/2)-1)*x/(1+I*3^(1/2))/(-2+x))^(1/2),((1+I*3^(1/2))*(-I*3^(1/2)-1)/(I*3^(1/2)-1)/(1-I*3^(1/
2)))^(1/2))+1/6*(-I*3^(1/2)-1)*((I*3^(1/2)-1)*x/(1+I*3^(1/2))/(-2+x))^(1/2)*(-2+x)^2*((x-1+I*3^(1/2))/(1-I*3^(
1/2))/(-2+x))^(1/2)*((x-1-I*3^(1/2))/(1+I*3^(1/2))/(-2+x))^(1/2)/(I*3^(1/2)-1)/(-x*(-2+x)*(x-1+I*3^(1/2))*(x-1
-I*3^(1/2)))^(1/2)*(2*EllipticF(((I*3^(1/2)-1)*x/(1+I*3^(1/2))/(-2+x))^(1/2),((1+I*3^(1/2))*(-I*3^(1/2)-1)/(I*
3^(1/2)-1)/(1-I*3^(1/2)))^(1/2))-2*EllipticPi(((I*3^(1/2)-1)*x/(1+I*3^(1/2))/(-2+x))^(1/2),(1+I*3^(1/2))/(I*3^
(1/2)-1),((1+I*3^(1/2))*(-I*3^(1/2)-1)/(I*3^(1/2)-1)/(1-I*3^(1/2)))^(1/2)))-1/24*(x*(x-1+I*3^(1/2))*(x-1-I*3^(
1/2))+2*(-I*3^(1/2)-1)*((I*3^(1/2)-1)*x/(1+I*3^(1/2))/(-2+x))^(1/2)*(-2+x)^2*((x-1+I*3^(1/2))/(1-I*3^(1/2))/(-
2+x))^(1/2)*((x-1-I*3^(1/2))/(1+I*3^(1/2))/(-2+x))^(1/2)*(1/2*(6+2*I*3^(1/2))/(I*3^(1/2)-1)*EllipticF(((I*3^(1
/2)-1)*x/(1+I*3^(1/2))/(-2+x))^(1/2),((1+I*3^(1/2))*(-I*3^(1/2)-1)/(I*3^(1/2)-1)/(1-I*3^(1/2)))^(1/2))+1/2*(I*
3^(1/2)-1)*EllipticE(((I*3^(1/2)-1)*x/(1+I*3^(1/2))/(-2+x))^(1/2),((1+I*3^(1/2))*(-I*3^(1/2)-1)/(I*3^(1/2)-1)/
(1-I*3^(1/2)))^(1/2))-4/(I*3^(1/2)-1)*EllipticPi(((I*3^(1/2)-1)*x/(1+I*3^(1/2))/(-2+x))^(1/2),(-I*3^(1/2)-1)/(
1-I*3^(1/2)),((1+I*3^(1/2))*(-I*3^(1/2)-1)/(I*3^(1/2)-1)/(1-I*3^(1/2)))^(1/2))))/(-x*(-2+x)*(x-1+I*3^(1/2))*(x
-1-I*3^(1/2)))^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (-x^{4} + 4 \, x^{3} - 8 \, x^{2} + 8 \, x\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x^4+4*x^3-8*x^2+8*x)^(3/2),x, algorithm="maxima")

[Out]

integrate((-x^4 + 4*x^3 - 8*x^2 + 8*x)^(-3/2), 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} + 8 \, x}}{x^{8} - 8 \, x^{7} + 32 \, x^{6} - 80 \, x^{5} + 128 \, x^{4} - 128 \, x^{3} + 64 \, x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x^4+4*x^3-8*x^2+8*x)^(3/2),x, algorithm="fricas")

[Out]

integral(sqrt(-x^4 + 4*x^3 - 8*x^2 + 8*x)/(x^8 - 8*x^7 + 32*x^6 - 80*x^5 + 128*x^4 - 128*x^3 + 64*x^2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (- x^{4} + 4 x^{3} - 8 x^{2} + 8 x\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x**4+4*x**3-8*x**2+8*x)**(3/2),x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (-x^{4} + 4 \, x^{3} - 8 \, x^{2} + 8 \, x\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x^4+4*x^3-8*x^2+8*x)^(3/2),x, algorithm="giac")

[Out]

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