3.6.15 \(\int \frac {2+x^2}{(-2+x^2) \sqrt {-2 x+2 x^2+x^3}} \, dx\)

Optimal. Leaf size=40 \[ -\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {x^3+2 x^2-2 x}}{x^2+2 x-2}\right ) \]

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Rubi [C]  time = 1.05, antiderivative size = 355, normalized size of antiderivative = 8.88, number of steps used = 15, number of rules used = 8, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2056, 6725, 716, 1098, 934, 168, 538, 537} \begin {gather*} \frac {\sqrt {x} \sqrt {\frac {2-\left (1-\sqrt {3}\right ) x}{2-\left (1+\sqrt {3}\right ) x}} \sqrt {\left (1+\sqrt {3}\right ) x-2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{3} \sqrt {x}}{\sqrt {\left (1+\sqrt {3}\right ) x-2}}\right )|\frac {1}{6} \left (3+\sqrt {3}\right )\right )}{\sqrt [4]{3} \sqrt {\frac {1}{2-\left (1+\sqrt {3}\right ) x}} \sqrt {x^3+2 x^2-2 x}}-\frac {2 \sqrt {2-\sqrt {3}} \sqrt {x} \sqrt {x+\sqrt {3}+1} \sqrt {\frac {x}{1-\sqrt {3}}+1} \Pi \left (-\frac {1-\sqrt {3}}{\sqrt {2}};\sin ^{-1}\left (\frac {\sqrt {x}}{\sqrt {-1+\sqrt {3}}}\right )|-2+\sqrt {3}\right )}{\sqrt {x^3+2 x^2-2 x}}-\frac {2 \sqrt {2-\sqrt {3}} \sqrt {x} \sqrt {x+\sqrt {3}+1} \sqrt {\frac {x}{1-\sqrt {3}}+1} \Pi \left (\frac {1-\sqrt {3}}{\sqrt {2}};\sin ^{-1}\left (\frac {\sqrt {x}}{\sqrt {-1+\sqrt {3}}}\right )|-2+\sqrt {3}\right )}{\sqrt {x^3+2 x^2-2 x}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[x]*Sqrt[(2 - (1 - Sqrt[3])*x)/(2 - (1 + Sqrt[3])*x)]*Sqrt[-2 + (1 + Sqrt[3])*x]*EllipticF[ArcSin[(Sqrt[2
]*3^(1/4)*Sqrt[x])/Sqrt[-2 + (1 + Sqrt[3])*x]], (3 + Sqrt[3])/6])/(3^(1/4)*Sqrt[(2 - (1 + Sqrt[3])*x)^(-1)]*Sq
rt[-2*x + 2*x^2 + x^3]) - (2*Sqrt[2 - Sqrt[3]]*Sqrt[x]*Sqrt[1 + Sqrt[3] + x]*Sqrt[1 + x/(1 - Sqrt[3])]*Ellipti
cPi[-((1 - Sqrt[3])/Sqrt[2]), ArcSin[Sqrt[x]/Sqrt[-1 + Sqrt[3]]], -2 + Sqrt[3]])/Sqrt[-2*x + 2*x^2 + x^3] - (2
*Sqrt[2 - Sqrt[3]]*Sqrt[x]*Sqrt[1 + Sqrt[3] + x]*Sqrt[1 + x/(1 - Sqrt[3])]*EllipticPi[(1 - Sqrt[3])/Sqrt[2], A
rcSin[Sqrt[x]/Sqrt[-1 + Sqrt[3]]], -2 + Sqrt[3]])/Sqrt[-2*x + 2*x^2 + x^3]

Rule 168

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Sym
bol] :> Dist[-2, Subst[Int[1/(Simp[b*c - a*d - b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + (f*x^2)/d, x]]*Sqrt[Simp[(d
*g - c*h)/d + (h*x^2)/d, x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && GtQ[(d*e - c
*f)/d, 0]

Rule 537

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

Rule 538

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

Rule 716

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

Rule 934

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Wi
th[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(Sqrt[b - q + 2*c*x]*Sqrt[b + q + 2*c*x])/Sqrt[a + b*x + c*x^2], Int[1/((d +
e*x)*Sqrt[f + g*x]*Sqrt[b - q + 2*c*x]*Sqrt[b + q + 2*c*x]), x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && Ne
Q[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 1098

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(Sqrt[(2*a +
(b - q)*x^2)/(2*a + (b + q)*x^2)]*Sqrt[(2*a + (b + q)*x^2)/q]*EllipticF[ArcSin[x/Sqrt[(2*a + (b + q)*x^2)/(2*q
)]], (b + q)/(2*q)])/(2*Sqrt[a + b*x^2 + c*x^4]*Sqrt[a/(2*a + (b + q)*x^2)]), x]] /; FreeQ[{a, b, c}, x] && Gt
Q[b^2 - 4*a*c, 0] && LtQ[a, 0] && GtQ[c, 0]

Rule 2056

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {2+x^2}{\left (-2+x^2\right ) \sqrt {-2 x+2 x^2+x^3}} \, dx &=\frac {\left (\sqrt {x} \sqrt {-2+2 x+x^2}\right ) \int \frac {2+x^2}{\sqrt {x} \left (-2+x^2\right ) \sqrt {-2+2 x+x^2}} \, dx}{\sqrt {-2 x+2 x^2+x^3}}\\ &=\frac {\left (\sqrt {x} \sqrt {-2+2 x+x^2}\right ) \int \left (\frac {1}{\sqrt {x} \sqrt {-2+2 x+x^2}}+\frac {4}{\sqrt {x} \left (-2+x^2\right ) \sqrt {-2+2 x+x^2}}\right ) \, dx}{\sqrt {-2 x+2 x^2+x^3}}\\ &=\frac {\left (\sqrt {x} \sqrt {-2+2 x+x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {-2+2 x+x^2}} \, dx}{\sqrt {-2 x+2 x^2+x^3}}+\frac {\left (4 \sqrt {x} \sqrt {-2+2 x+x^2}\right ) \int \frac {1}{\sqrt {x} \left (-2+x^2\right ) \sqrt {-2+2 x+x^2}} \, dx}{\sqrt {-2 x+2 x^2+x^3}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {-2+2 x+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-2+2 x^2+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {-2 x+2 x^2+x^3}}+\frac {\left (4 \sqrt {x} \sqrt {-2+2 x+x^2}\right ) \int \left (-\frac {1}{2 \sqrt {2} \left (\sqrt {2}-x\right ) \sqrt {x} \sqrt {-2+2 x+x^2}}-\frac {1}{2 \sqrt {2} \sqrt {x} \left (\sqrt {2}+x\right ) \sqrt {-2+2 x+x^2}}\right ) \, dx}{\sqrt {-2 x+2 x^2+x^3}}\\ &=\frac {\sqrt {x} \sqrt {\frac {2-\left (1-\sqrt {3}\right ) x}{2-\left (1+\sqrt {3}\right ) x}} \sqrt {-2+\left (1+\sqrt {3}\right ) x} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{3} \sqrt {x}}{\sqrt {-2+\left (1+\sqrt {3}\right ) x}}\right )|\frac {1}{6} \left (3+\sqrt {3}\right )\right )}{\sqrt [4]{3} \sqrt {\frac {1}{2-\left (1+\sqrt {3}\right ) x}} \sqrt {-2 x+2 x^2+x^3}}-\frac {\left (\sqrt {2} \sqrt {x} \sqrt {-2+2 x+x^2}\right ) \int \frac {1}{\left (\sqrt {2}-x\right ) \sqrt {x} \sqrt {-2+2 x+x^2}} \, dx}{\sqrt {-2 x+2 x^2+x^3}}-\frac {\left (\sqrt {2} \sqrt {x} \sqrt {-2+2 x+x^2}\right ) \int \frac {1}{\sqrt {x} \left (\sqrt {2}+x\right ) \sqrt {-2+2 x+x^2}} \, dx}{\sqrt {-2 x+2 x^2+x^3}}\\ &=\frac {\sqrt {x} \sqrt {\frac {2-\left (1-\sqrt {3}\right ) x}{2-\left (1+\sqrt {3}\right ) x}} \sqrt {-2+\left (1+\sqrt {3}\right ) x} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{3} \sqrt {x}}{\sqrt {-2+\left (1+\sqrt {3}\right ) x}}\right )|\frac {1}{6} \left (3+\sqrt {3}\right )\right )}{\sqrt [4]{3} \sqrt {\frac {1}{2-\left (1+\sqrt {3}\right ) x}} \sqrt {-2 x+2 x^2+x^3}}-\frac {\left (\sqrt {2} \sqrt {x} \sqrt {2-2 \sqrt {3}+2 x} \sqrt {2+2 \sqrt {3}+2 x}\right ) \int \frac {1}{\left (\sqrt {2}-x\right ) \sqrt {x} \sqrt {2-2 \sqrt {3}+2 x} \sqrt {2+2 \sqrt {3}+2 x}} \, dx}{\sqrt {-2 x+2 x^2+x^3}}-\frac {\left (\sqrt {2} \sqrt {x} \sqrt {2-2 \sqrt {3}+2 x} \sqrt {2+2 \sqrt {3}+2 x}\right ) \int \frac {1}{\sqrt {x} \left (\sqrt {2}+x\right ) \sqrt {2-2 \sqrt {3}+2 x} \sqrt {2+2 \sqrt {3}+2 x}} \, dx}{\sqrt {-2 x+2 x^2+x^3}}\\ &=\frac {\sqrt {x} \sqrt {\frac {2-\left (1-\sqrt {3}\right ) x}{2-\left (1+\sqrt {3}\right ) x}} \sqrt {-2+\left (1+\sqrt {3}\right ) x} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{3} \sqrt {x}}{\sqrt {-2+\left (1+\sqrt {3}\right ) x}}\right )|\frac {1}{6} \left (3+\sqrt {3}\right )\right )}{\sqrt [4]{3} \sqrt {\frac {1}{2-\left (1+\sqrt {3}\right ) x}} \sqrt {-2 x+2 x^2+x^3}}+\frac {\left (2 \sqrt {2} \sqrt {x} \sqrt {2-2 \sqrt {3}+2 x} \sqrt {2+2 \sqrt {3}+2 x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt {2}-x^2\right ) \sqrt {2 \left (1-\sqrt {3}\right )+2 x^2} \sqrt {2 \left (1+\sqrt {3}\right )+2 x^2}} \, dx,x,\sqrt {x}\right )}{\sqrt {-2 x+2 x^2+x^3}}+\frac {\left (2 \sqrt {2} \sqrt {x} \sqrt {2-2 \sqrt {3}+2 x} \sqrt {2+2 \sqrt {3}+2 x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt {2}+x^2\right ) \sqrt {2 \left (1-\sqrt {3}\right )+2 x^2} \sqrt {2 \left (1+\sqrt {3}\right )+2 x^2}} \, dx,x,\sqrt {x}\right )}{\sqrt {-2 x+2 x^2+x^3}}\\ &=\frac {\sqrt {x} \sqrt {\frac {2-\left (1-\sqrt {3}\right ) x}{2-\left (1+\sqrt {3}\right ) x}} \sqrt {-2+\left (1+\sqrt {3}\right ) x} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{3} \sqrt {x}}{\sqrt {-2+\left (1+\sqrt {3}\right ) x}}\right )|\frac {1}{6} \left (3+\sqrt {3}\right )\right )}{\sqrt [4]{3} \sqrt {\frac {1}{2-\left (1+\sqrt {3}\right ) x}} \sqrt {-2 x+2 x^2+x^3}}+\frac {\left (2 \sqrt {x} \sqrt {2-2 \sqrt {3}+2 x} \sqrt {2+2 \sqrt {3}+2 x} \sqrt {1+\frac {x}{1-\sqrt {3}}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt {2}-x^2\right ) \sqrt {2 \left (1+\sqrt {3}\right )+2 x^2} \sqrt {1+\frac {x^2}{1-\sqrt {3}}}} \, dx,x,\sqrt {x}\right )}{\sqrt {1-\sqrt {3}+x} \sqrt {-2 x+2 x^2+x^3}}+\frac {\left (2 \sqrt {x} \sqrt {2-2 \sqrt {3}+2 x} \sqrt {2+2 \sqrt {3}+2 x} \sqrt {1+\frac {x}{1-\sqrt {3}}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt {2}+x^2\right ) \sqrt {2 \left (1+\sqrt {3}\right )+2 x^2} \sqrt {1+\frac {x^2}{1-\sqrt {3}}}} \, dx,x,\sqrt {x}\right )}{\sqrt {1-\sqrt {3}+x} \sqrt {-2 x+2 x^2+x^3}}\\ &=\frac {\sqrt {x} \sqrt {\frac {2-\left (1-\sqrt {3}\right ) x}{2-\left (1+\sqrt {3}\right ) x}} \sqrt {-2+\left (1+\sqrt {3}\right ) x} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{3} \sqrt {x}}{\sqrt {-2+\left (1+\sqrt {3}\right ) x}}\right )|\frac {1}{6} \left (3+\sqrt {3}\right )\right )}{\sqrt [4]{3} \sqrt {\frac {1}{2-\left (1+\sqrt {3}\right ) x}} \sqrt {-2 x+2 x^2+x^3}}-\frac {2 \sqrt {2-\sqrt {3}} \sqrt {x} \sqrt {1+\sqrt {3}+x} \sqrt {1+\frac {x}{1-\sqrt {3}}} \Pi \left (-\frac {1-\sqrt {3}}{\sqrt {2}};\sin ^{-1}\left (\frac {\sqrt {x}}{\sqrt {-1+\sqrt {3}}}\right )|-2+\sqrt {3}\right )}{\sqrt {-2 x+2 x^2+x^3}}-\frac {2 \sqrt {2-\sqrt {3}} \sqrt {x} \sqrt {1+\sqrt {3}+x} \sqrt {1+\frac {x}{1-\sqrt {3}}} \Pi \left (\frac {1-\sqrt {3}}{\sqrt {2}};\sin ^{-1}\left (\frac {\sqrt {x}}{\sqrt {-1+\sqrt {3}}}\right )|-2+\sqrt {3}\right )}{\sqrt {-2 x+2 x^2+x^3}}\\ \end {align*}

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Mathematica [C]  time = 0.71, size = 173, normalized size = 4.32 \begin {gather*} -\frac {2 i \sqrt {-\frac {2}{x^2}+\frac {2}{x}+1} x^{3/2} \left (F\left (i \sinh ^{-1}\left (\frac {\sqrt {1+\sqrt {3}}}{\sqrt {x}}\right )|-2+\sqrt {3}\right )-\Pi \left (-\sqrt {\frac {3}{2}}+\frac {1}{\sqrt {2}};i \sinh ^{-1}\left (\frac {\sqrt {1+\sqrt {3}}}{\sqrt {x}}\right )|-2+\sqrt {3}\right )-\Pi \left (\frac {-1+\sqrt {3}}{\sqrt {2}};i \sinh ^{-1}\left (\frac {\sqrt {1+\sqrt {3}}}{\sqrt {x}}\right )|-2+\sqrt {3}\right )\right )}{\sqrt {1+\sqrt {3}} \sqrt {x \left (x^2+2 x-2\right )}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-2*I)*Sqrt[1 - 2/x^2 + 2/x]*x^(3/2)*(EllipticF[I*ArcSinh[Sqrt[1 + Sqrt[3]]/Sqrt[x]], -2 + Sqrt[3]] - Ellipti
cPi[-Sqrt[3/2] + 1/Sqrt[2], I*ArcSinh[Sqrt[1 + Sqrt[3]]/Sqrt[x]], -2 + Sqrt[3]] - EllipticPi[(-1 + Sqrt[3])/Sq
rt[2], I*ArcSinh[Sqrt[1 + Sqrt[3]]/Sqrt[x]], -2 + Sqrt[3]]))/(Sqrt[1 + Sqrt[3]]*Sqrt[x*(-2 + 2*x + x^2)])

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IntegrateAlgebraic [A]  time = 0.09, size = 40, normalized size = 1.00 \begin {gather*} -\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {-2 x+2 x^2+x^3}}{-2+2 x+x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(2 + x^2)/((-2 + x^2)*Sqrt[-2*x + 2*x^2 + x^3]),x]

[Out]

-(Sqrt[2]*ArcTanh[(Sqrt[2]*Sqrt[-2*x + 2*x^2 + x^3])/(-2 + 2*x + x^2)])

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fricas [A]  time = 0.47, size = 64, normalized size = 1.60 \begin {gather*} \frac {1}{4} \, \sqrt {2} \log \left (\frac {x^{4} + 16 \, x^{3} - 4 \, \sqrt {2} \sqrt {x^{3} + 2 \, x^{2} - 2 \, x} {\left (x^{2} + 4 \, x - 2\right )} + 28 \, x^{2} - 32 \, x + 4}{x^{4} - 4 \, x^{2} + 4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*sqrt(2)*log((x^4 + 16*x^3 - 4*sqrt(2)*sqrt(x^3 + 2*x^2 - 2*x)*(x^2 + 4*x - 2) + 28*x^2 - 32*x + 4)/(x^4 -
4*x^2 + 4))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} + 2}{\sqrt {x^{3} + 2 \, x^{2} - 2 \, x} {\left (x^{2} - 2\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.37, size = 63, normalized size = 1.58

method	result	size
trager	\(\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) x^{2}+4 \RootOf \left (\textit {\_Z}^{2}-2\right ) x -2 \RootOf \left (\textit {\_Z}^{2}-2\right )-4 \sqrt {x^{3}+2 x^{2}-2 x}}{x^{2}-2}\right )}{2}\)	\(63\)
default	\(\frac {\left (1+\sqrt {3}\right ) \sqrt {\frac {x +1+\sqrt {3}}{1+\sqrt {3}}}\, \sqrt {-6 \left (x -\sqrt {3}+1\right ) \sqrt {3}}\, \sqrt {-\frac {x}{1+\sqrt {3}}}\, \EllipticF \left (\sqrt {\frac {x +1+\sqrt {3}}{1+\sqrt {3}}}, \frac {\sqrt {6}\, \sqrt {\left (1+\sqrt {3}\right ) \sqrt {3}}}{6}\right )}{3 \sqrt {x^{3}+2 x^{2}-2 x}}+\frac {\sqrt {2}\, \sqrt {\frac {x}{1+\sqrt {3}}+\frac {1}{1+\sqrt {3}}+\frac {\sqrt {3}}{1+\sqrt {3}}}\, \sqrt {-6 x \sqrt {3}+18-6 \sqrt {3}}\, \sqrt {-\frac {x}{1+\sqrt {3}}}\, \EllipticPi \left (\sqrt {\frac {x +1+\sqrt {3}}{1+\sqrt {3}}}, \frac {-1-\sqrt {3}}{-1-\sqrt {3}-\sqrt {2}}, \frac {\sqrt {6}\, \sqrt {\left (1+\sqrt {3}\right ) \sqrt {3}}}{6}\right )}{3 \sqrt {x^{3}+2 x^{2}-2 x}\, \left (-1-\sqrt {3}-\sqrt {2}\right )}+\frac {\sqrt {2}\, \sqrt {\frac {x}{1+\sqrt {3}}+\frac {1}{1+\sqrt {3}}+\frac {\sqrt {3}}{1+\sqrt {3}}}\, \sqrt {-6 x \sqrt {3}+18-6 \sqrt {3}}\, \sqrt {-\frac {x}{1+\sqrt {3}}}\, \EllipticPi \left (\sqrt {\frac {x +1+\sqrt {3}}{1+\sqrt {3}}}, \frac {-1-\sqrt {3}}{-1-\sqrt {3}-\sqrt {2}}, \frac {\sqrt {6}\, \sqrt {\left (1+\sqrt {3}\right ) \sqrt {3}}}{6}\right ) \sqrt {3}}{3 \sqrt {x^{3}+2 x^{2}-2 x}\, \left (-1-\sqrt {3}-\sqrt {2}\right )}-\frac {\sqrt {2}\, \sqrt {\frac {x}{1+\sqrt {3}}+\frac {1}{1+\sqrt {3}}+\frac {\sqrt {3}}{1+\sqrt {3}}}\, \sqrt {-6 x \sqrt {3}+18-6 \sqrt {3}}\, \sqrt {-\frac {x}{1+\sqrt {3}}}\, \EllipticPi \left (\sqrt {\frac {x +1+\sqrt {3}}{1+\sqrt {3}}}, \frac {-1-\sqrt {3}}{-1-\sqrt {3}+\sqrt {2}}, \frac {\sqrt {6}\, \sqrt {\left (1+\sqrt {3}\right ) \sqrt {3}}}{6}\right )}{3 \sqrt {x^{3}+2 x^{2}-2 x}\, \left (-1-\sqrt {3}+\sqrt {2}\right )}-\frac {\sqrt {2}\, \sqrt {\frac {x}{1+\sqrt {3}}+\frac {1}{1+\sqrt {3}}+\frac {\sqrt {3}}{1+\sqrt {3}}}\, \sqrt {-6 x \sqrt {3}+18-6 \sqrt {3}}\, \sqrt {-\frac {x}{1+\sqrt {3}}}\, \EllipticPi \left (\sqrt {\frac {x +1+\sqrt {3}}{1+\sqrt {3}}}, \frac {-1-\sqrt {3}}{-1-\sqrt {3}+\sqrt {2}}, \frac {\sqrt {6}\, \sqrt {\left (1+\sqrt {3}\right ) \sqrt {3}}}{6}\right ) \sqrt {3}}{3 \sqrt {x^{3}+2 x^{2}-2 x}\, \left (-1-\sqrt {3}+\sqrt {2}\right )}\)	\(677\)
elliptic	\(\frac {\sqrt {\frac {x}{1+\sqrt {3}}+\frac {1}{1+\sqrt {3}}+\frac {\sqrt {3}}{1+\sqrt {3}}}\, \sqrt {-6 x \sqrt {3}+18-6 \sqrt {3}}\, \sqrt {-\frac {x}{1+\sqrt {3}}}\, \EllipticF \left (\sqrt {\frac {x +1+\sqrt {3}}{1+\sqrt {3}}}, \frac {\sqrt {6}\, \sqrt {\left (1+\sqrt {3}\right ) \sqrt {3}}}{6}\right )}{3 \sqrt {x^{3}+2 x^{2}-2 x}}+\frac {\sqrt {\frac {x}{1+\sqrt {3}}+\frac {1}{1+\sqrt {3}}+\frac {\sqrt {3}}{1+\sqrt {3}}}\, \sqrt {-6 x \sqrt {3}+18-6 \sqrt {3}}\, \sqrt {-\frac {x}{1+\sqrt {3}}}\, \EllipticF \left (\sqrt {\frac {x +1+\sqrt {3}}{1+\sqrt {3}}}, \frac {\sqrt {6}\, \sqrt {\left (1+\sqrt {3}\right ) \sqrt {3}}}{6}\right ) \sqrt {3}}{3 \sqrt {x^{3}+2 x^{2}-2 x}}+\frac {\sqrt {2}\, \sqrt {\frac {x}{1+\sqrt {3}}+\frac {1}{1+\sqrt {3}}+\frac {\sqrt {3}}{1+\sqrt {3}}}\, \sqrt {-6 x \sqrt {3}+18-6 \sqrt {3}}\, \sqrt {-\frac {x}{1+\sqrt {3}}}\, \EllipticPi \left (\sqrt {\frac {x +1+\sqrt {3}}{1+\sqrt {3}}}, \frac {-1-\sqrt {3}}{-1-\sqrt {3}-\sqrt {2}}, \frac {\sqrt {6}\, \sqrt {\left (1+\sqrt {3}\right ) \sqrt {3}}}{6}\right )}{3 \sqrt {x^{3}+2 x^{2}-2 x}\, \left (-1-\sqrt {3}-\sqrt {2}\right )}+\frac {\sqrt {2}\, \sqrt {\frac {x}{1+\sqrt {3}}+\frac {1}{1+\sqrt {3}}+\frac {\sqrt {3}}{1+\sqrt {3}}}\, \sqrt {-6 x \sqrt {3}+18-6 \sqrt {3}}\, \sqrt {-\frac {x}{1+\sqrt {3}}}\, \EllipticPi \left (\sqrt {\frac {x +1+\sqrt {3}}{1+\sqrt {3}}}, \frac {-1-\sqrt {3}}{-1-\sqrt {3}-\sqrt {2}}, \frac {\sqrt {6}\, \sqrt {\left (1+\sqrt {3}\right ) \sqrt {3}}}{6}\right ) \sqrt {3}}{3 \sqrt {x^{3}+2 x^{2}-2 x}\, \left (-1-\sqrt {3}-\sqrt {2}\right )}-\frac {\sqrt {2}\, \sqrt {\frac {x}{1+\sqrt {3}}+\frac {1}{1+\sqrt {3}}+\frac {\sqrt {3}}{1+\sqrt {3}}}\, \sqrt {-6 x \sqrt {3}+18-6 \sqrt {3}}\, \sqrt {-\frac {x}{1+\sqrt {3}}}\, \EllipticPi \left (\sqrt {\frac {x +1+\sqrt {3}}{1+\sqrt {3}}}, \frac {-1-\sqrt {3}}{-1-\sqrt {3}+\sqrt {2}}, \frac {\sqrt {6}\, \sqrt {\left (1+\sqrt {3}\right ) \sqrt {3}}}{6}\right )}{3 \sqrt {x^{3}+2 x^{2}-2 x}\, \left (-1-\sqrt {3}+\sqrt {2}\right )}-\frac {\sqrt {2}\, \sqrt {\frac {x}{1+\sqrt {3}}+\frac {1}{1+\sqrt {3}}+\frac {\sqrt {3}}{1+\sqrt {3}}}\, \sqrt {-6 x \sqrt {3}+18-6 \sqrt {3}}\, \sqrt {-\frac {x}{1+\sqrt {3}}}\, \EllipticPi \left (\sqrt {\frac {x +1+\sqrt {3}}{1+\sqrt {3}}}, \frac {-1-\sqrt {3}}{-1-\sqrt {3}+\sqrt {2}}, \frac {\sqrt {6}\, \sqrt {\left (1+\sqrt {3}\right ) \sqrt {3}}}{6}\right ) \sqrt {3}}{3 \sqrt {x^{3}+2 x^{2}-2 x}\, \left (-1-\sqrt {3}+\sqrt {2}\right )}\)	\(795\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2+2)/(x^2-2)/(x^3+2*x^2-2*x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/2*RootOf(_Z^2-2)*ln(-(RootOf(_Z^2-2)*x^2+4*RootOf(_Z^2-2)*x-2*RootOf(_Z^2-2)-4*(x^3+2*x^2-2*x)^(1/2))/(x^2-2
))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} + 2}{\sqrt {x^{3} + 2 \, x^{2} - 2 \, x} {\left (x^{2} - 2\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.15, size = 227, normalized size = 5.68 \begin {gather*} -\frac {2\,\sqrt {x}\,\sqrt {\frac {1}{\sqrt {3}+1}}\,\Pi \left (\sqrt {2}\,\left (\frac {\sqrt {3}}{2}-\frac {1}{2}\right );\mathrm {asin}\left (\sqrt {\frac {x}{\sqrt {3}-1}}\right )\middle |-\frac {\sqrt {3}-1}{\sqrt {3}+1}\right )\,\sqrt {x+\sqrt {3}+1}\,\sqrt {\sqrt {3}-x-1}+2\,\sqrt {x}\,\sqrt {\frac {1}{\sqrt {3}+1}}\,\Pi \left (-\sqrt {2}\,\left (\frac {\sqrt {3}}{2}-\frac {1}{2}\right );\mathrm {asin}\left (\sqrt {\frac {x}{\sqrt {3}-1}}\right )\middle |-\frac {\sqrt {3}-1}{\sqrt {3}+1}\right )\,\sqrt {x+\sqrt {3}+1}\,\sqrt {\sqrt {3}-x-1}-2\,\sqrt {x}\,\sqrt {\frac {1}{\sqrt {3}+1}}\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {\frac {x}{\sqrt {3}-1}}\right )\middle |-\frac {\sqrt {3}-1}{\sqrt {3}+1}\right )\,\sqrt {x+\sqrt {3}+1}\,\sqrt {\sqrt {3}-x-1}}{\sqrt {x^3+2\,x^2-\left (\sqrt {3}-1\right )\,\left (\sqrt {3}+1\right )\,x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*x^(1/2)*(1/(3^(1/2) + 1))^(1/2)*ellipticPi(2^(1/2)*(3^(1/2)/2 - 1/2), asin((x/(3^(1/2) - 1))^(1/2)), -(3^(
1/2) - 1)/(3^(1/2) + 1))*(x + 3^(1/2) + 1)^(1/2)*(3^(1/2) - x - 1)^(1/2) + 2*x^(1/2)*(1/(3^(1/2) + 1))^(1/2)*e
llipticPi(-2^(1/2)*(3^(1/2)/2 - 1/2), asin((x/(3^(1/2) - 1))^(1/2)), -(3^(1/2) - 1)/(3^(1/2) + 1))*(x + 3^(1/2
) + 1)^(1/2)*(3^(1/2) - x - 1)^(1/2) - 2*x^(1/2)*(1/(3^(1/2) + 1))^(1/2)*ellipticF(asin((x/(3^(1/2) - 1))^(1/2
)), -(3^(1/2) - 1)/(3^(1/2) + 1))*(x + 3^(1/2) + 1)^(1/2)*(3^(1/2) - x - 1)^(1/2))/(2*x^2 + x^3 - x*(3^(1/2) -
1)*(3^(1/2) + 1))^(1/2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} + 2}{\sqrt {x \left (x^{2} + 2 x - 2\right )} \left (x^{2} - 2\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((x**2 + 2)/(sqrt(x*(x**2 + 2*x - 2))*(x**2 - 2)), x)

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