3.773 \(\int \frac{1}{((2-x) x (4-2 x+x^2))^{5/2}} \, dx\)

Optimal. Leaf size=109 \[ \frac{\left (7 (x-1)^2+26\right ) (x-1)}{432 \sqrt{-(x-1)^4-2 (x-1)^2+3}}+\frac{\left ((x-1)^2+5\right ) (x-1)}{72 \left (-(x-1)^4-2 (x-1)^2+3\right )^{3/2}}-\frac{11 F\left (\sin ^{-1}(1-x)|-\frac{1}{3}\right )}{144 \sqrt{3}}+\frac{7 E\left (\sin ^{-1}(1-x)|-\frac{1}{3}\right )}{144 \sqrt{3}} \]

[Out]

((5 + (-1 + x)^2)*(-1 + x))/(72*(3 - 2*(-1 + x)^2 - (-1 + x)^4)^(3/2)) + ((26 + 7*(-1 + x)^2)*(-1 + x))/(432*S
qrt[3 - 2*(-1 + x)^2 - (-1 + x)^4]) + (7*EllipticE[ArcSin[1 - x], -1/3])/(144*Sqrt[3]) - (11*EllipticF[ArcSin[
1 - x], -1/3])/(144*Sqrt[3])

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Rubi [A]  time = 0.0704578, antiderivative size = 109, 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, 1092, 1178, 1180, 524, 424, 419} \[ \frac{\left (7 (x-1)^2+26\right ) (x-1)}{432 \sqrt{-(x-1)^4-2 (x-1)^2+3}}+\frac{\left ((x-1)^2+5\right ) (x-1)}{72 \left (-(x-1)^4-2 (x-1)^2+3\right )^{3/2}}-\frac{11 F\left (\sin ^{-1}(1-x)|-\frac{1}{3}\right )}{144 \sqrt{3}}+\frac{7 E\left (\sin ^{-1}(1-x)|-\frac{1}{3}\right )}{144 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[((2 - x)*x*(4 - 2*x + x^2))^(-5/2),x]

[Out]

((5 + (-1 + x)^2)*(-1 + x))/(72*(3 - 2*(-1 + x)^2 - (-1 + x)^4)^(3/2)) + ((26 + 7*(-1 + x)^2)*(-1 + x))/(432*S
qrt[3 - 2*(-1 + x)^2 - (-1 + x)^4]) + (7*EllipticE[ArcSin[1 - x], -1/3])/(144*Sqrt[3]) - (11*EllipticF[ArcSin[
1 - x], -1/3])/(144*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 1178

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(x*(a*b*e - d*(b^2 - 2*
a*c) - c*(b*d - 2*a*e)*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[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 7)*(d*b - 2*a*e)*c*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] && 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 ((2-x) x \left (4-2 x+x^2\right )\right )^{5/2}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{\left (3-2 x^2-x^4\right )^{5/2}} \, dx,x,-1+x\right )\\ &=\frac{\left (5+(-1+x)^2\right ) (-1+x)}{72 \left (3-2 (-1+x)^2-(-1+x)^4\right )^{3/2}}-\frac{1}{144} \operatorname{Subst}\left (\int \frac{-38-6 x^2}{\left (3-2 x^2-x^4\right )^{3/2}} \, dx,x,-1+x\right )\\ &=-\frac{\left (26+7 (1-x)^2\right ) (1-x)}{432 \sqrt{3-2 (1-x)^2-(1-x)^4}}+\frac{\left (5+(-1+x)^2\right ) (-1+x)}{72 \left (3-2 (-1+x)^2-(-1+x)^4\right )^{3/2}}+\frac{\operatorname{Subst}\left (\int \frac{192-112 x^2}{\sqrt{3-2 x^2-x^4}} \, dx,x,-1+x\right )}{6912}\\ &=-\frac{\left (26+7 (1-x)^2\right ) (1-x)}{432 \sqrt{3-2 (1-x)^2-(1-x)^4}}+\frac{\left (5+(-1+x)^2\right ) (-1+x)}{72 \left (3-2 (-1+x)^2-(-1+x)^4\right )^{3/2}}+\frac{\operatorname{Subst}\left (\int \frac{192-112 x^2}{\sqrt{2-2 x^2} \sqrt{6+2 x^2}} \, dx,x,-1+x\right )}{3456}\\ &=-\frac{\left (26+7 (1-x)^2\right ) (1-x)}{432 \sqrt{3-2 (1-x)^2-(1-x)^4}}+\frac{\left (5+(-1+x)^2\right ) (-1+x)}{72 \left (3-2 (-1+x)^2-(-1+x)^4\right )^{3/2}}-\frac{7}{432} \operatorname{Subst}\left (\int \frac{\sqrt{6+2 x^2}}{\sqrt{2-2 x^2}} \, dx,x,-1+x\right )+\frac{11}{72} \operatorname{Subst}\left (\int \frac{1}{\sqrt{2-2 x^2} \sqrt{6+2 x^2}} \, dx,x,-1+x\right )\\ &=-\frac{\left (26+7 (1-x)^2\right ) (1-x)}{432 \sqrt{3-2 (1-x)^2-(1-x)^4}}+\frac{\left (5+(-1+x)^2\right ) (-1+x)}{72 \left (3-2 (-1+x)^2-(-1+x)^4\right )^{3/2}}+\frac{7 E\left (\sin ^{-1}(1-x)|-\frac{1}{3}\right )}{144 \sqrt{3}}-\frac{11 F\left (\sin ^{-1}(1-x)|-\frac{1}{3}\right )}{144 \sqrt{3}}\\ \end{align*}

Mathematica [C]  time = 1.0615, size = 327, normalized size = 3. \[ \frac{(x-2)^3 x^2 \left (x^2-2 x+4\right )^2 \left (-\frac{7 x \left (x^2-2 x+4\right )}{x-2}+\frac{7 x^7-49 x^6+187 x^5-445 x^4+670 x^3-622 x^2+216 x+36}{(x-2)^2 x \left (x^2-2 x+4\right )}-19 i \sqrt{2} (x-2) \sqrt{\frac{i x}{\left (\sqrt{3}+i\right ) (x-2)}} \sqrt{\frac{x^2-2 x+4}{(x-2)^2}} F\left (\sin ^{-1}\left (\frac{\sqrt{\sqrt{3}-i-\frac{4 i}{x-2}}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{i+\sqrt{3}}\right )+\frac{7 i \sqrt{2} x \sqrt{\frac{x^2-2 x+4}{(x-2)^2}} E\left (\sin ^{-1}\left (\frac{\sqrt{\sqrt{3}-i-\frac{4 i}{x-2}}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{i+\sqrt{3}}\right )}{\sqrt{\frac{i x}{\left (\sqrt{3}+i\right ) (x-2)}}}\right )}{432 \left (-x \left (x^3-4 x^2+8 x-8\right )\right )^{5/2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[((2 - x)*x*(4 - 2*x + x^2))^(-5/2),x]

[Out]

((-2 + x)^3*x^2*(4 - 2*x + x^2)^2*((-7*x*(4 - 2*x + x^2))/(-2 + x) + (36 + 216*x - 622*x^2 + 670*x^3 - 445*x^4
+ 187*x^5 - 49*x^6 + 7*x^7)/((-2 + x)^2*x*(4 - 2*x + x^2)) + ((7*I)*Sqrt[2]*x*Sqrt[(4 - 2*x + x^2)/(-2 + x)^2
]*EllipticE[ArcSin[Sqrt[-I + Sqrt[3] - (4*I)/(-2 + x)]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(I + Sqrt[3])])/Sqrt[(I
*x)/((I + Sqrt[3])*(-2 + x))] - (19*I)*Sqrt[2]*(-2 + x)*Sqrt[(I*x)/((I + Sqrt[3])*(-2 + x))]*Sqrt[(4 - 2*x + x
^2)/(-2 + x)^2]*EllipticF[ArcSin[Sqrt[-I + Sqrt[3] - (4*I)/(-2 + x)]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(I + Sqrt
[3])]))/(432*(-(x*(-8 + 8*x - 4*x^2 + x^3)))^(5/2))

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Maple [B]  time = 0.032, size = 1039, normalized size = 9.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((2-x)*x*(x^2-2*x+4))^(5/2),x)

[Out]

(1/36+1/288*x^2-1/96*x)*(-x^4+4*x^3-8*x^2+8*x)^(1/2)/(x^3-4*x^2+8*x-8)^2+2*x*(53/3456+5/1728*x^2-19/4608*x)/(-
x*(x^3-4*x^2+8*x-8))^(1/2)-1/768*(-x^4+4*x^3-8*x^2+8*x)^(1/2)/x^2-1/96*(-x^3+4*x^2-8*x+8)/(x*(-x^3+4*x^2-8*x+8
))^(1/2)+5/216*(-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))+7/108*(-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)))-7/432*(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 (-{\left (x^{2} - 2 \, x + 4\right )}{\left (x - 2\right )} x\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((2-x)*x*(x^2-2*x+4))^(5/2),x, algorithm="maxima")

[Out]

integrate((-(x^2 - 2*x + 4)*(x - 2)*x)^(-5/2), 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^{12} - 12 \, x^{11} + 72 \, x^{10} - 280 \, x^{9} + 768 \, x^{8} - 1536 \, x^{7} + 2240 \, x^{6} - 2304 \, x^{5} + 1536 \, x^{4} - 512 \, x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((2-x)*x*(x^2-2*x+4))^(5/2),x, algorithm="fricas")

[Out]

integral(-sqrt(-x^4 + 4*x^3 - 8*x^2 + 8*x)/(x^12 - 12*x^11 + 72*x^10 - 280*x^9 + 768*x^8 - 1536*x^7 + 2240*x^6
- 2304*x^5 + 1536*x^4 - 512*x^3), 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(1/((2-x)*x*(x**2-2*x+4))**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (-{\left (x^{2} - 2 \, x + 4\right )}{\left (x - 2\right )} x\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((2-x)*x*(x^2-2*x+4))^(5/2),x, algorithm="giac")

[Out]

integrate((-(x^2 - 2*x + 4)*(x - 2)*x)^(-5/2), x)