3.765 \(\int \sqrt{8 x-8 x^2+4 x^3-x^4} \, dx\)
Optimal. Leaf size=62 \[ \frac{1}{3} \sqrt{-(x-1)^4-2 (x-1)^2+3} (x-1)-\frac{4 F\left (\sin ^{-1}(1-x)|-\frac{1}{3}\right )}{\sqrt{3}}+\frac{2 E\left (\sin ^{-1}(1-x)|-\frac{1}{3}\right )}{\sqrt{3}} \]
[Out]
(Sqrt[3 - 2*(-1 + x)^2 - (-1 + x)^4]*(-1 + x))/3 + (2*EllipticE[ArcSin[1 - x], -1/3])/Sqrt[3] - (4*EllipticF[A
rcSin[1 - x], -1/3])/Sqrt[3]
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Rubi [A] time = 0.0490289, antiderivative size = 62, 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, 1091, 1180, 524, 424, 419} \[ \frac{1}{3} \sqrt{-(x-1)^4-2 (x-1)^2+3} (x-1)-\frac{4 F\left (\sin ^{-1}(1-x)|-\frac{1}{3}\right )}{\sqrt{3}}+\frac{2 E\left (\sin ^{-1}(1-x)|-\frac{1}{3}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
[In]
Int[Sqrt[8*x - 8*x^2 + 4*x^3 - x^4],x]
[Out]
(Sqrt[3 - 2*(-1 + x)^2 - (-1 + x)^4]*(-1 + x))/3 + (2*EllipticE[ArcSin[1 - x], -1/3])/Sqrt[3] - (4*EllipticF[A
rcSin[1 - x], -1/3])/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 1091
Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(x*(a + b*x^2 + c*x^4)^p)/(4*p + 1), x] + Dis
t[(2*p)/(4*p + 1), Int[(2*a + b*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] && GtQ[p, 0] && 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 \sqrt{8 x-8 x^2+4 x^3-x^4} \, dx &=\operatorname{Subst}\left (\int \sqrt{3-2 x^2-x^4} \, dx,x,-1+x\right )\\ &=\frac{1}{3} \sqrt{3-2 (-1+x)^2-(-1+x)^4} (-1+x)+\frac{1}{3} \operatorname{Subst}\left (\int \frac{6-2 x^2}{\sqrt{3-2 x^2-x^4}} \, dx,x,-1+x\right )\\ &=\frac{1}{3} \sqrt{3-2 (-1+x)^2-(-1+x)^4} (-1+x)+\frac{2}{3} \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{1}{3} \sqrt{3-2 (-1+x)^2-(-1+x)^4} (-1+x)-\frac{2}{3} \operatorname{Subst}\left (\int \frac{\sqrt{6+2 x^2}}{\sqrt{2-2 x^2}} \, dx,x,-1+x\right )+8 \operatorname{Subst}\left (\int \frac{1}{\sqrt{2-2 x^2} \sqrt{6+2 x^2}} \, dx,x,-1+x\right )\\ &=\frac{1}{3} \sqrt{3-2 (-1+x)^2-(-1+x)^4} (-1+x)+\frac{2 E\left (\sin ^{-1}(1-x)|-\frac{1}{3}\right )}{\sqrt{3}}-\frac{4 F\left (\sin ^{-1}(1-x)|-\frac{1}{3}\right )}{\sqrt{3}}\\ \end{align*}
Mathematica [C] time = 0.597811, size = 256, normalized size = 4.13 \[ -\frac{x^5-5 x^4+14 x^3-24 x^2+8 i \sqrt{2} \sqrt{-\frac{i (x-2)}{\left (\sqrt{3}-i\right ) x}} \sqrt{\frac{x^2-2 x+4}{x^2}} x^2 F\left (\sin ^{-1}\left (\frac{\sqrt{\sqrt{3}+i-\frac{4 i}{x}}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{-i+\sqrt{3}}\right )-\frac{2 i \sqrt{2} (x-2) \sqrt{\frac{x^2-2 x+4}{x^2}} x E\left (\sin ^{-1}\left (\frac{\sqrt{\sqrt{3}+i-\frac{4 i}{x}}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{-i+\sqrt{3}}\right )}{\sqrt{-\frac{i (x-2)}{\left (\sqrt{3}-i\right ) x}}}+24 x-16}{3 \sqrt{-x \left (x^3-4 x^2+8 x-8\right )}} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[Sqrt[8*x - 8*x^2 + 4*x^3 - x^4],x]
[Out]
-(-16 + 24*x - 24*x^2 + 14*x^3 - 5*x^4 + x^5 - ((2*I)*Sqrt[2]*(-2 + x)*x*Sqrt[(4 - 2*x + x^2)/x^2]*EllipticE[A
rcSin[Sqrt[I + Sqrt[3] - (4*I)/x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(-I + Sqrt[3])])/Sqrt[((-I)*(-2 + x))/((-I +
Sqrt[3])*x)] + (8*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])])/(3*Sqrt[-(x*(-8 + 8*x - 4
*x^2 + x^3))])
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Maple [B] time = 0.024, size = 946, normalized size = 15.3 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((-x^4+4*x^3-8*x^2+8*x)^(1/2),x)
[Out]
1/3*x*(-x^4+4*x^3-8*x^2+8*x)^(1/2)-1/3*(-x^4+4*x^3-8*x^2+8*x)^(1/2)+8/3*(-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))+8/3*(-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*E
llipticPi(((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)))-2/3*(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 \sqrt{-x^{4} + 4 \, x^{3} - 8 \, x^{2} + 8 \, x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-x^4+4*x^3-8*x^2+8*x)^(1/2),x, algorithm="maxima")
[Out]
integrate(sqrt(-x^4 + 4*x^3 - 8*x^2 + 8*x), x)
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{-x^{4} + 4 \, x^{3} - 8 \, x^{2} + 8 \, x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-x^4+4*x^3-8*x^2+8*x)^(1/2),x, algorithm="fricas")
[Out]
integral(sqrt(-x^4 + 4*x^3 - 8*x^2 + 8*x), x)
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{- 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**4+4*x**3-8*x**2+8*x)**(1/2),x)
[Out]
Integral(sqrt(-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 \sqrt{-x^{4} + 4 \, x^{3} - 8 \, x^{2} + 8 \, x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-x^4+4*x^3-8*x^2+8*x)^(1/2),x, algorithm="giac")
[Out]
integrate(sqrt(-x^4 + 4*x^3 - 8*x^2 + 8*x), x)