3.10.57 \(\int \frac {-2+3 x+x^2}{(-1-2 x+x^2) \sqrt {-x+x^3}} \, dx\)

Optimal. Leaf size=73 \[ \frac {1}{4} \left (\sqrt {2}-6\right ) \tanh ^{-1}\left (\frac {x-1}{\left (\sqrt {2}-1\right ) \sqrt {x^3-x}}\right )+\frac {1}{4} \left (6+\sqrt {2}\right ) \tanh ^{-1}\left (\frac {x-1}{\left (1+\sqrt {2}\right ) \sqrt {x^3-x}}\right ) \]

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Rubi [C]  time = 0.82, antiderivative size = 180, normalized size of antiderivative = 2.47, number of steps used = 13, number of rules used = 7, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {2056, 6728, 329, 222, 933, 168, 537} \begin {gather*} \frac {\sqrt {2} \sqrt {x-1} \sqrt {x} \sqrt {x+1} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {x-1}}\right )|\frac {1}{2}\right )}{\sqrt {x^3-x}}+\frac {\left (5+2 \sqrt {2}\right ) \sqrt {x} \sqrt {1-x^2} \Pi \left (-\frac {1}{\sqrt {2}};\sin ^{-1}\left (\sqrt {1-x}\right )|\frac {1}{2}\right )}{2 \sqrt {x^3-x}}-\frac {\left (5-2 \sqrt {2}\right ) \sqrt {x} \sqrt {1-x^2} \Pi \left (\frac {1}{\sqrt {2}};\sin ^{-1}\left (\sqrt {1-x}\right )|\frac {1}{2}\right )}{2 \sqrt {x^3-x}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Int[(-2 + 3*x + x^2)/((-1 - 2*x + x^2)*Sqrt[-x + x^3]),x]

[Out]

(Sqrt[2]*Sqrt[-1 + x]*Sqrt[x]*Sqrt[1 + x]*EllipticF[ArcSin[(Sqrt[2]*Sqrt[x])/Sqrt[-1 + x]], 1/2])/Sqrt[-x + x^
3] + ((5 + 2*Sqrt[2])*Sqrt[x]*Sqrt[1 - x^2]*EllipticPi[-(1/Sqrt[2]), ArcSin[Sqrt[1 - x]], 1/2])/(2*Sqrt[-x + x
^3]) - ((5 - 2*Sqrt[2])*Sqrt[x]*Sqrt[1 - x^2]*EllipticPi[1/Sqrt[2], ArcSin[Sqrt[1 - x]], 1/2])/(2*Sqrt[-x + 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 222

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

Rule 329

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

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 933

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[-(c
/a), 2]}, Dist[Sqrt[1 + (c*x^2)/a]/Sqrt[a + c*x^2], Int[1/((d + e*x)*Sqrt[f + g*x]*Sqrt[1 - q*x]*Sqrt[1 + q*x]
), x], x]] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] &&  !GtQ[a, 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 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps

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

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Mathematica [C]  time = 0.66, size = 93, normalized size = 1.27 \begin {gather*} -\frac {\sqrt {1-\frac {1}{x^2}} x^{3/2} \left (4 F\left (\left .\sin ^{-1}\left (\frac {1}{\sqrt {x}}\right )\right |-1\right )-\left (1+3 \sqrt {2}\right ) \Pi \left (1-\sqrt {2};\left .\sin ^{-1}\left (\frac {1}{\sqrt {x}}\right )\right |-1\right )+\left (3 \sqrt {2}-1\right ) \Pi \left (1+\sqrt {2};\left .\sin ^{-1}\left (\frac {1}{\sqrt {x}}\right )\right |-1\right )\right )}{\sqrt {x \left (x^2-1\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2 + 3*x + x^2)/((-1 - 2*x + x^2)*Sqrt[-x + x^3]),x]

[Out]

-((Sqrt[1 - x^(-2)]*x^(3/2)*(4*EllipticF[ArcSin[1/Sqrt[x]], -1] - (1 + 3*Sqrt[2])*EllipticPi[1 - Sqrt[2], ArcS
in[1/Sqrt[x]], -1] + (-1 + 3*Sqrt[2])*EllipticPi[1 + Sqrt[2], ArcSin[1/Sqrt[x]], -1]))/Sqrt[x*(-1 + x^2)])

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IntegrateAlgebraic [A]  time = 0.74, size = 85, normalized size = 1.16 \begin {gather*} \frac {1}{4} \left (6+\sqrt {2}\right ) \tanh ^{-1}\left (\frac {1-\sqrt {2}+\left (-1+\sqrt {2}\right ) x}{\sqrt {-x+x^3}}\right )+\frac {1}{4} \left (-6+\sqrt {2}\right ) \tanh ^{-1}\left (\frac {-1-\sqrt {2}+\left (1+\sqrt {2}\right ) x}{\sqrt {-x+x^3}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-2 + 3*x + x^2)/((-1 - 2*x + x^2)*Sqrt[-x + x^3]),x]

[Out]

((6 + Sqrt[2])*ArcTanh[(1 - Sqrt[2] + (-1 + Sqrt[2])*x)/Sqrt[-x + x^3]])/4 + ((-6 + Sqrt[2])*ArcTanh[(-1 - Sqr
t[2] + (1 + Sqrt[2])*x)/Sqrt[-x + x^3]])/4

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fricas [B]  time = 0.66, size = 126, normalized size = 1.73 \begin {gather*} \frac {1}{8} \, \sqrt {2} \log \left (\frac {x^{4} + 12 \, x^{3} + 4 \, \sqrt {2} \sqrt {x^{3} - x} {\left (x^{2} + 2 \, x - 1\right )} + 2 \, x^{2} - 12 \, x + 1}{x^{4} - 4 \, x^{3} + 2 \, x^{2} + 4 \, x + 1}\right ) + \frac {3}{4} \, \log \left (\frac {x^{4} + 4 \, x^{3} + 2 \, x^{2} - 4 \, \sqrt {x^{3} - x} {\left (x^{2} + 1\right )} - 4 \, x + 1}{x^{4} - 4 \, x^{3} + 2 \, x^{2} + 4 \, x + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*sqrt(2)*log((x^4 + 12*x^3 + 4*sqrt(2)*sqrt(x^3 - x)*(x^2 + 2*x - 1) + 2*x^2 - 12*x + 1)/(x^4 - 4*x^3 + 2*x
^2 + 4*x + 1)) + 3/4*log((x^4 + 4*x^3 + 2*x^2 - 4*sqrt(x^3 - x)*(x^2 + 1) - 4*x + 1)/(x^4 - 4*x^3 + 2*x^2 + 4*
x + 1))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} + 3 \, x - 2}{\sqrt {x^{3} - x} {\left (x^{2} - 2 \, x - 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 1.30, size = 273, normalized size = 3.74

method	result	size
default	\(\frac {\sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \EllipticF \left (\sqrt {1+x}, \frac {\sqrt {2}}{2}\right )}{\sqrt {x^{3}-x}}+\frac {\sqrt {2}\, \sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \EllipticPi \left (\sqrt {1+x}, -\frac {1}{-2-\sqrt {2}}, \frac {\sqrt {2}}{2}\right )}{\sqrt {x^{3}-x}\, \left (-2-\sqrt {2}\right )}+\frac {5 \sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \EllipticPi \left (\sqrt {1+x}, -\frac {1}{-2-\sqrt {2}}, \frac {\sqrt {2}}{2}\right )}{2 \sqrt {x^{3}-x}\, \left (-2-\sqrt {2}\right )}-\frac {\sqrt {2}\, \sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \EllipticPi \left (\sqrt {1+x}, -\frac {1}{-2+\sqrt {2}}, \frac {\sqrt {2}}{2}\right )}{\sqrt {x^{3}-x}\, \left (-2+\sqrt {2}\right )}+\frac {5 \sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \EllipticPi \left (\sqrt {1+x}, -\frac {1}{-2+\sqrt {2}}, \frac {\sqrt {2}}{2}\right )}{2 \sqrt {x^{3}-x}\, \left (-2+\sqrt {2}\right )}\)	\(273\)
elliptic	\(\frac {\sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \EllipticF \left (\sqrt {1+x}, \frac {\sqrt {2}}{2}\right )}{\sqrt {x^{3}-x}}+\frac {\sqrt {2}\, \sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \EllipticPi \left (\sqrt {1+x}, -\frac {1}{-2-\sqrt {2}}, \frac {\sqrt {2}}{2}\right )}{\sqrt {x^{3}-x}\, \left (-2-\sqrt {2}\right )}+\frac {5 \sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \EllipticPi \left (\sqrt {1+x}, -\frac {1}{-2-\sqrt {2}}, \frac {\sqrt {2}}{2}\right )}{2 \sqrt {x^{3}-x}\, \left (-2-\sqrt {2}\right )}-\frac {\sqrt {2}\, \sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \EllipticPi \left (\sqrt {1+x}, -\frac {1}{-2+\sqrt {2}}, \frac {\sqrt {2}}{2}\right )}{\sqrt {x^{3}-x}\, \left (-2+\sqrt {2}\right )}+\frac {5 \sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \EllipticPi \left (\sqrt {1+x}, -\frac {1}{-2+\sqrt {2}}, \frac {\sqrt {2}}{2}\right )}{2 \sqrt {x^{3}-x}\, \left (-2+\sqrt {2}\right )}\)	\(273\)
trager	\(\frac {\RootOf \left (8 \textit {\_Z}^{2}-24 \textit {\_Z} +17\right ) \ln \left (-\frac {-278808 \RootOf \left (8 \textit {\_Z}^{2}-24 \textit {\_Z} +17\right )^{2} x^{2}+1394040 \RootOf \left (8 \textit {\_Z}^{2}-24 \textit {\_Z} +17\right )^{2} x +621337 \RootOf \left (8 \textit {\_Z}^{2}-24 \textit {\_Z} +17\right ) x^{2}+1038514 \sqrt {x^{3}-x}\, \RootOf \left (8 \textit {\_Z}^{2}-24 \textit {\_Z} +17\right )-1672848 \RootOf \left (8 \textit {\_Z}^{2}-24 \textit {\_Z} +17\right )^{2}-5650808 \RootOf \left (8 \textit {\_Z}^{2}-24 \textit {\_Z} +17\right ) x -193794 x^{2}-1924943 \sqrt {x^{3}-x}+6272145 \RootOf \left (8 \textit {\_Z}^{2}-24 \textit {\_Z} +17\right )+5684624 x -5878418}{\left (4 \RootOf \left (8 \textit {\_Z}^{2}-24 \textit {\_Z} +17\right ) x -8 \RootOf \left (8 \textit {\_Z}^{2}-24 \textit {\_Z} +17\right )-7 x +15\right )^{2}}\right )}{2}-\frac {\ln \left (\frac {278808 \RootOf \left (8 \textit {\_Z}^{2}-24 \textit {\_Z} +17\right )^{2} x^{2}-1394040 \RootOf \left (8 \textit {\_Z}^{2}-24 \textit {\_Z} +17\right )^{2} x -1051511 \RootOf \left (8 \textit {\_Z}^{2}-24 \textit {\_Z} +17\right ) x^{2}+1038514 \sqrt {x^{3}-x}\, \RootOf \left (8 \textit {\_Z}^{2}-24 \textit {\_Z} +17\right )+1672848 \RootOf \left (8 \textit {\_Z}^{2}-24 \textit {\_Z} +17\right )^{2}+2713432 \RootOf \left (8 \textit {\_Z}^{2}-24 \textit {\_Z} +17\right ) x +839055 x^{2}-1190599 \sqrt {x^{3}-x}-3764943 \RootOf \left (8 \textit {\_Z}^{2}-24 \textit {\_Z} +17\right )-1278560 x +2117615}{\left (4 \RootOf \left (8 \textit {\_Z}^{2}-24 \textit {\_Z} +17\right ) x -8 \RootOf \left (8 \textit {\_Z}^{2}-24 \textit {\_Z} +17\right )-5 x +9\right )^{2}}\right ) \RootOf \left (8 \textit {\_Z}^{2}-24 \textit {\_Z} +17\right )}{2}+\frac {3 \ln \left (\frac {278808 \RootOf \left (8 \textit {\_Z}^{2}-24 \textit {\_Z} +17\right )^{2} x^{2}-1394040 \RootOf \left (8 \textit {\_Z}^{2}-24 \textit {\_Z} +17\right )^{2} x -1051511 \RootOf \left (8 \textit {\_Z}^{2}-24 \textit {\_Z} +17\right ) x^{2}+1038514 \sqrt {x^{3}-x}\, \RootOf \left (8 \textit {\_Z}^{2}-24 \textit {\_Z} +17\right )+1672848 \RootOf \left (8 \textit {\_Z}^{2}-24 \textit {\_Z} +17\right )^{2}+2713432 \RootOf \left (8 \textit {\_Z}^{2}-24 \textit {\_Z} +17\right ) x +839055 x^{2}-1190599 \sqrt {x^{3}-x}-3764943 \RootOf \left (8 \textit {\_Z}^{2}-24 \textit {\_Z} +17\right )-1278560 x +2117615}{\left (4 \RootOf \left (8 \textit {\_Z}^{2}-24 \textit {\_Z} +17\right ) x -8 \RootOf \left (8 \textit {\_Z}^{2}-24 \textit {\_Z} +17\right )-5 x +9\right )^{2}}\right )}{2}\)	\(544\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1+x)^(1/2)*(2-2*x)^(1/2)*(-x)^(1/2)/(x^3-x)^(1/2)*EllipticF((1+x)^(1/2),1/2*2^(1/2))+2^(1/2)*(1+x)^(1/2)*(2-2
*x)^(1/2)*(-x)^(1/2)/(x^3-x)^(1/2)/(-2-2^(1/2))*EllipticPi((1+x)^(1/2),-1/(-2-2^(1/2)),1/2*2^(1/2))+5/2*(1+x)^
(1/2)*(2-2*x)^(1/2)*(-x)^(1/2)/(x^3-x)^(1/2)/(-2-2^(1/2))*EllipticPi((1+x)^(1/2),-1/(-2-2^(1/2)),1/2*2^(1/2))-
2^(1/2)*(1+x)^(1/2)*(2-2*x)^(1/2)*(-x)^(1/2)/(x^3-x)^(1/2)/(-2+2^(1/2))*EllipticPi((1+x)^(1/2),-1/(-2+2^(1/2))
,1/2*2^(1/2))+5/2*(1+x)^(1/2)*(2-2*x)^(1/2)*(-x)^(1/2)/(x^3-x)^(1/2)/(-2+2^(1/2))*EllipticPi((1+x)^(1/2),-1/(-
2+2^(1/2)),1/2*2^(1/2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} + 3 \, x - 2}{\sqrt {x^{3} - x} {\left (x^{2} - 2 \, x - 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.07, size = 159, normalized size = 2.18 \begin {gather*} \frac {\sqrt {2}\,\sqrt {-x}\,\left (5\,\sqrt {2}+4\right )\,\sqrt {1-x}\,\sqrt {x+1}\,\Pi \left (-\frac {1}{\sqrt {2}+1};\mathrm {asin}\left (\sqrt {-x}\right )\middle |-1\right )}{2\,\sqrt {x^3-x}\,\left (\sqrt {2}+1\right )}-\frac {\sqrt {2}\,\sqrt {-x}\,\left (5\,\sqrt {2}-4\right )\,\sqrt {1-x}\,\sqrt {x+1}\,\Pi \left (\frac {1}{\sqrt {2}-1};\mathrm {asin}\left (\sqrt {-x}\right )\middle |-1\right )}{2\,\sqrt {x^3-x}\,\left (\sqrt {2}-1\right )}-\frac {2\,\sqrt {-x}\,\sqrt {1-x}\,\sqrt {x+1}\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {-x}\right )\middle |-1\right )}{\sqrt {x^3-x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2^(1/2)*(-x)^(1/2)*(5*2^(1/2) + 4)*(1 - x)^(1/2)*(x + 1)^(1/2)*ellipticPi(-1/(2^(1/2) + 1), asin((-x)^(1/2)),
-1))/(2*(x^3 - x)^(1/2)*(2^(1/2) + 1)) - (2^(1/2)*(-x)^(1/2)*(5*2^(1/2) - 4)*(1 - x)^(1/2)*(x + 1)^(1/2)*elli
pticPi(1/(2^(1/2) - 1), asin((-x)^(1/2)), -1))/(2*(x^3 - x)^(1/2)*(2^(1/2) - 1)) - (2*(-x)^(1/2)*(1 - x)^(1/2)
*(x + 1)^(1/2)*ellipticF(asin((-x)^(1/2)), -1))/(x^3 - x)^(1/2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} + 3 x - 2}{\sqrt {x \left (x - 1\right ) \left (x + 1\right )} \left (x^{2} - 2 x - 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((x**2 + 3*x - 2)/(sqrt(x*(x - 1)*(x + 1))*(x**2 - 2*x - 1)), x)

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