3.769 \(\int ((2-x) x (4-2 x+x^2))^{3/2} \, dx\)

Optimal. Leaf size=102 \[ \frac{1}{7} (x-1) \left (-(x-1)^4-2 (x-1)^2+3\right )^{3/2}+\frac{2}{35} \left (13-3 (x-1)^2\right ) (x-1) \sqrt{-(x-1)^4-2 (x-1)^2+3}-\frac{176}{35} \sqrt{3} F\left (\sin ^{-1}(1-x)|-\frac{1}{3}\right )+\frac{16}{5} \sqrt{3} E\left (\sin ^{-1}(1-x)|-\frac{1}{3}\right ) \]

[Out]

(2*(13 - 3*(-1 + x)^2)*Sqrt[3 - 2*(-1 + x)^2 - (-1 + x)^4]*(-1 + x))/35 + ((3 - 2*(-1 + x)^2 - (-1 + x)^4)^(3/
2)*(-1 + x))/7 + (16*Sqrt[3]*EllipticE[ArcSin[1 - x], -1/3])/5 - (176*Sqrt[3]*EllipticF[ArcSin[1 - x], -1/3])/
35

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Rubi [A]  time = 0.069017, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368, Rules used = {1106, 1091, 1176, 1180, 524, 424, 419} \[ \frac{1}{7} (x-1) \left (-(x-1)^4-2 (x-1)^2+3\right )^{3/2}+\frac{2}{35} \left (13-3 (x-1)^2\right ) (x-1) \sqrt{-(x-1)^4-2 (x-1)^2+3}-\frac{176}{35} \sqrt{3} F\left (\sin ^{-1}(1-x)|-\frac{1}{3}\right )+\frac{16}{5} \sqrt{3} E\left (\sin ^{-1}(1-x)|-\frac{1}{3}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(13 - 3*(-1 + x)^2)*Sqrt[3 - 2*(-1 + x)^2 - (-1 + x)^4]*(-1 + x))/35 + ((3 - 2*(-1 + x)^2 - (-1 + x)^4)^(3/
2)*(-1 + x))/7 + (16*Sqrt[3]*EllipticE[ArcSin[1 - x], -1/3])/5 - (176*Sqrt[3]*EllipticF[ArcSin[1 - x], -1/3])/
35

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 1176

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(x*(2*b*e*p + c*d*(4*p
+ 3) + c*e*(4*p + 1)*x^2)*(a + b*x^2 + c*x^4)^p)/(c*(4*p + 1)*(4*p + 3)), x] + Dist[(2*p)/(c*(4*p + 1)*(4*p +
3)), Int[Simp[2*a*c*d*(4*p + 3) - a*b*e + (2*a*c*e*(4*p + 1) + b*c*d*(4*p + 3) - b^2*e*(2*p + 1))*x^2, x]*(a +
b*x^2 + c*x^4)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e
^2, 0] && GtQ[p, 0] && FractionQ[p] && 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 \left ((2-x) x \left (4-2 x+x^2\right )\right )^{3/2} \, dx &=\operatorname{Subst}\left (\int \left (3-2 x^2-x^4\right )^{3/2} \, dx,x,-1+x\right )\\ &=\frac{1}{7} \left (3-2 (-1+x)^2-(-1+x)^4\right )^{3/2} (-1+x)+\frac{3}{7} \operatorname{Subst}\left (\int \left (6-2 x^2\right ) \sqrt{3-2 x^2-x^4} \, dx,x,-1+x\right )\\ &=-\frac{2}{35} \left (13-3 (1-x)^2\right ) \sqrt{3-2 (1-x)^2-(1-x)^4} (1-x)+\frac{1}{7} \left (3-2 (-1+x)^2-(-1+x)^4\right )^{3/2} (-1+x)-\frac{1}{35} \operatorname{Subst}\left (\int \frac{-192+112 x^2}{\sqrt{3-2 x^2-x^4}} \, dx,x,-1+x\right )\\ &=-\frac{2}{35} \left (13-3 (1-x)^2\right ) \sqrt{3-2 (1-x)^2-(1-x)^4} (1-x)+\frac{1}{7} \left (3-2 (-1+x)^2-(-1+x)^4\right )^{3/2} (-1+x)-\frac{2}{35} \operatorname{Subst}\left (\int \frac{-192+112 x^2}{\sqrt{2-2 x^2} \sqrt{6+2 x^2}} \, dx,x,-1+x\right )\\ &=-\frac{2}{35} \left (13-3 (1-x)^2\right ) \sqrt{3-2 (1-x)^2-(1-x)^4} (1-x)+\frac{1}{7} \left (3-2 (-1+x)^2-(-1+x)^4\right )^{3/2} (-1+x)-\frac{16}{5} \operatorname{Subst}\left (\int \frac{\sqrt{6+2 x^2}}{\sqrt{2-2 x^2}} \, dx,x,-1+x\right )+\frac{1056}{35} \operatorname{Subst}\left (\int \frac{1}{\sqrt{2-2 x^2} \sqrt{6+2 x^2}} \, dx,x,-1+x\right )\\ &=-\frac{2}{35} \left (13-3 (1-x)^2\right ) \sqrt{3-2 (1-x)^2-(1-x)^4} (1-x)+\frac{1}{7} \left (3-2 (-1+x)^2-(-1+x)^4\right )^{3/2} (-1+x)+\frac{16}{5} \sqrt{3} E\left (\sin ^{-1}(1-x)|-\frac{1}{3}\right )-\frac{176}{35} \sqrt{3} F\left (\sin ^{-1}(1-x)|-\frac{1}{3}\right )\\ \end{align*}

Mathematica [C]  time = 0.988792, size = 278, normalized size = 2.73 \[ \frac{\sqrt{-x \left (x^3-4 x^2+8 x-8\right )} \left (\sqrt{\frac{x^2-2 x+4}{x^2}} \left (-5 x^7+35 x^6-116 x^5+230 x^4-228 x^3+44 x^2+152 x-224\right )+304 i \sqrt{2} \sqrt{-\frac{i (x-2)}{\left (\sqrt{3}-i\right ) x}} F\left (\sin ^{-1}\left (\frac{\sqrt{\sqrt{3}+i-\frac{4 i}{x}}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{-i+\sqrt{3}}\right )+112 \sqrt{2} \left (\sqrt{3}-i\right ) \sqrt{-\frac{i (x-2)}{\left (\sqrt{3}-i\right ) x}} E\left (\sin ^{-1}\left (\frac{\sqrt{\sqrt{3}+i-\frac{4 i}{x}}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{-i+\sqrt{3}}\right )\right )}{35 (x-2) x \sqrt{\frac{x^2-2 x+4}{x^2}}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[-(x*(-8 + 8*x - 4*x^2 + x^3))]*(Sqrt[(4 - 2*x + x^2)/x^2]*(-224 + 152*x + 44*x^2 - 228*x^3 + 230*x^4 - 1
16*x^5 + 35*x^6 - 5*x^7) + 112*Sqrt[2]*(-I + Sqrt[3])*Sqrt[((-I)*(-2 + x))/((-I + Sqrt[3])*x)]*EllipticE[ArcSi
n[Sqrt[I + Sqrt[3] - (4*I)/x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(-I + Sqrt[3])] + (304*I)*Sqrt[2]*Sqrt[((-I)*(-2
+ x))/((-I + Sqrt[3])*x)]*EllipticF[ArcSin[Sqrt[I + Sqrt[3] - (4*I)/x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(-I +
Sqrt[3])]))/(35*(-2 + x)*x*Sqrt[(4 - 2*x + x^2)/x^2])

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Maple [B]  time = 0.032, size = 1050, normalized size = 10.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7*x^5*(-x^4+4*x^3-8*x^2+8*x)^(1/2)+5/7*x^4*(-x^4+4*x^3-8*x^2+8*x)^(1/2)-66/35*x^3*(-x^4+4*x^3-8*x^2+8*x)^(1
/2)+14/5*x^2*(-x^4+4*x^3-8*x^2+8*x)^(1/2)-32/35*x*(-x^4+4*x^3-8*x^2+8*x)^(1/2)-4/7*(-x^4+4*x^3-8*x^2+8*x)^(1/2
)+32/7*(-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))+64/5*(-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)))-16/5*(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)*Elliptic
F(((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 \left (-{\left (x^{2} - 2 \, x + 4\right )}{\left (x - 2\right )} x\right )^{\frac{3}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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