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

Optimal. Leaf size=101 \[ -\frac {1}{2} \sqrt {\frac {11}{5}+\frac {2 i}{5}} \tan ^{-1}\left (\frac {\sqrt {1-2 i} \sqrt {x^3-x^2-x}}{x^2-x-1}\right )-\frac {1}{2} \sqrt {\frac {11}{5}-\frac {2 i}{5}} \tan ^{-1}\left (\frac {\sqrt {1+2 i} \sqrt {x^3-x^2-x}}{x^2-x-1}\right ) \]

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Rubi [C]  time = 1.20, antiderivative size = 383, normalized size of antiderivative = 3.79, number of steps used = 15, number of rules used = 8, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {2056, 6725, 716, 1098, 934, 168, 538, 537} \begin {gather*} \frac {\sqrt {x} \sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x+2}{\left (1-\sqrt {5}\right ) x+2}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x+2}} \sqrt {x^3-x^2-x}}-\frac {\left (1-\frac {i}{2}\right ) \sqrt {3+\sqrt {5}} \sqrt {x} \sqrt {2 x+\sqrt {5}-1} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \Pi \left (-\frac {1}{2} i \left (1+\sqrt {5}\right );\sin ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right )|\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{\sqrt {x^3-x^2-x}}-\frac {\left (1+\frac {i}{2}\right ) \sqrt {3+\sqrt {5}} \sqrt {x} \sqrt {2 x+\sqrt {5}-1} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \Pi \left (\frac {1}{2} i \left (1+\sqrt {5}\right );\sin ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right )|\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{\sqrt {x^3-x^2-x}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[x]*Sqrt[-2 - (1 - Sqrt[5])*x]*Sqrt[(2 + (1 + Sqrt[5])*x)/(2 + (1 - Sqrt[5])*x)]*EllipticF[ArcSin[(Sqrt[2
]*5^(1/4)*Sqrt[x])/Sqrt[-2 - (1 - Sqrt[5])*x]], (5 - Sqrt[5])/10])/(5^(1/4)*Sqrt[(2 + (1 - Sqrt[5])*x)^(-1)]*S
qrt[-x - x^2 + x^3]) - ((1 - I/2)*Sqrt[3 + Sqrt[5]]*Sqrt[x]*Sqrt[-1 + Sqrt[5] + 2*x]*Sqrt[1 - (2*x)/(1 + Sqrt[
5])]*EllipticPi[(-1/2*I)*(1 + Sqrt[5]), ArcSin[Sqrt[2/(1 + Sqrt[5])]*Sqrt[x]], (-3 - Sqrt[5])/2])/Sqrt[-x - x^
2 + x^3] - ((1 + I/2)*Sqrt[3 + Sqrt[5]]*Sqrt[x]*Sqrt[-1 + Sqrt[5] + 2*x]*Sqrt[1 - (2*x)/(1 + Sqrt[5])]*Ellipti
cPi[(I/2)*(1 + Sqrt[5]), ArcSin[Sqrt[2/(1 + Sqrt[5])]*Sqrt[x]], (-3 - Sqrt[5])/2])/Sqrt[-x - 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 {-1+x+x^2}{\left (1+x^2\right ) \sqrt {-x-x^2+x^3}} \, dx &=\frac {\left (\sqrt {x} \sqrt {-1-x+x^2}\right ) \int \frac {-1+x+x^2}{\sqrt {x} \left (1+x^2\right ) \sqrt {-1-x+x^2}} \, dx}{\sqrt {-x-x^2+x^3}}\\ &=\frac {\left (\sqrt {x} \sqrt {-1-x+x^2}\right ) \int \left (\frac {1}{\sqrt {x} \sqrt {-1-x+x^2}}-\frac {2-x}{\sqrt {x} \left (1+x^2\right ) \sqrt {-1-x+x^2}}\right ) \, dx}{\sqrt {-x-x^2+x^3}}\\ &=\frac {\left (\sqrt {x} \sqrt {-1-x+x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {-1-x+x^2}} \, dx}{\sqrt {-x-x^2+x^3}}-\frac {\left (\sqrt {x} \sqrt {-1-x+x^2}\right ) \int \frac {2-x}{\sqrt {x} \left (1+x^2\right ) \sqrt {-1-x+x^2}} \, dx}{\sqrt {-x-x^2+x^3}}\\ &=-\frac {\left (\sqrt {x} \sqrt {-1-x+x^2}\right ) \int \left (\frac {\frac {1}{2}+i}{(i-x) \sqrt {x} \sqrt {-1-x+x^2}}-\frac {\frac {1}{2}-i}{\sqrt {x} (i+x) \sqrt {-1-x+x^2}}\right ) \, dx}{\sqrt {-x-x^2+x^3}}+\frac {\left (2 \sqrt {x} \sqrt {-1-x+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1-x^2+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x-x^2+x^3}}\\ &=\frac {\sqrt {x} \sqrt {-2-\left (1-\sqrt {5}\right ) x} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x}{2+\left (1-\sqrt {5}\right ) x}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-2-\left (1-\sqrt {5}\right ) x}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x}} \sqrt {-x-x^2+x^3}}--\frac {\left (\left (\frac {1}{2}-i\right ) \sqrt {x} \sqrt {-1-x+x^2}\right ) \int \frac {1}{\sqrt {x} (i+x) \sqrt {-1-x+x^2}} \, dx}{\sqrt {-x-x^2+x^3}}-\frac {\left (\left (\frac {1}{2}+i\right ) \sqrt {x} \sqrt {-1-x+x^2}\right ) \int \frac {1}{(i-x) \sqrt {x} \sqrt {-1-x+x^2}} \, dx}{\sqrt {-x-x^2+x^3}}\\ &=\frac {\sqrt {x} \sqrt {-2-\left (1-\sqrt {5}\right ) x} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x}{2+\left (1-\sqrt {5}\right ) x}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-2-\left (1-\sqrt {5}\right ) x}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x}} \sqrt {-x-x^2+x^3}}--\frac {\left (\left (\frac {1}{2}-i\right ) \sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}\right ) \int \frac {1}{\sqrt {x} (i+x) \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}} \, dx}{\sqrt {-x-x^2+x^3}}-\frac {\left (\left (\frac {1}{2}+i\right ) \sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}\right ) \int \frac {1}{(i-x) \sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}} \, dx}{\sqrt {-x-x^2+x^3}}\\ &=\frac {\sqrt {x} \sqrt {-2-\left (1-\sqrt {5}\right ) x} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x}{2+\left (1-\sqrt {5}\right ) x}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-2-\left (1-\sqrt {5}\right ) x}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x}} \sqrt {-x-x^2+x^3}}--\frac {\left ((1+2 i) \sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-i+x^2\right ) \sqrt {-1-\sqrt {5}+2 x^2} \sqrt {-1+\sqrt {5}+2 x^2}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x-x^2+x^3}}-\frac {\left ((1-2 i) \sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-i-x^2\right ) \sqrt {-1-\sqrt {5}+2 x^2} \sqrt {-1+\sqrt {5}+2 x^2}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x-x^2+x^3}}\\ &=\frac {\sqrt {x} \sqrt {-2-\left (1-\sqrt {5}\right ) x} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x}{2+\left (1-\sqrt {5}\right ) x}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-2-\left (1-\sqrt {5}\right ) x}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x}} \sqrt {-x-x^2+x^3}}--\frac {\left ((1+2 i) \sqrt {x} \sqrt {-1+\sqrt {5}+2 x} \sqrt {1-\frac {2 x}{1+\sqrt {5}}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-i+x^2\right ) \sqrt {-1+\sqrt {5}+2 x^2} \sqrt {1+\frac {2 x^2}{-1-\sqrt {5}}}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x-x^2+x^3}}-\frac {\left ((1-2 i) \sqrt {x} \sqrt {-1+\sqrt {5}+2 x} \sqrt {1-\frac {2 x}{1+\sqrt {5}}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-i-x^2\right ) \sqrt {-1+\sqrt {5}+2 x^2} \sqrt {1+\frac {2 x^2}{-1-\sqrt {5}}}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x-x^2+x^3}}\\ &=\frac {\sqrt {x} \sqrt {-2-\left (1-\sqrt {5}\right ) x} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x}{2+\left (1-\sqrt {5}\right ) x}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-2-\left (1-\sqrt {5}\right ) x}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x}} \sqrt {-x-x^2+x^3}}-\frac {\left (1-\frac {i}{2}\right ) \sqrt {3+\sqrt {5}} \sqrt {x} \sqrt {-1+\sqrt {5}+2 x} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \Pi \left (-\frac {1}{2} i \left (1+\sqrt {5}\right );\sin ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right )|\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{\sqrt {-x-x^2+x^3}}-\frac {\left (1+\frac {i}{2}\right ) \sqrt {3+\sqrt {5}} \sqrt {x} \sqrt {-1+\sqrt {5}+2 x} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \Pi \left (\frac {1}{2} i \left (1+\sqrt {5}\right );\sin ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right )|\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{\sqrt {-x-x^2+x^3}}\\ \end {align*}

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Mathematica [C]  time = 0.63, size = 211, normalized size = 2.09 \begin {gather*} -\frac {i \sqrt {\frac {2}{\sqrt {5}-1}} \sqrt {-\frac {1}{x^2}-\frac {1}{x}+1} x^{3/2} \left (2 F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {2}{1+\sqrt {5}}}}{\sqrt {x}}\right )|-\frac {3}{2}-\frac {\sqrt {5}}{2}\right )-(2-i) \Pi \left (-\frac {1}{2} i \left (1+\sqrt {5}\right );i \sinh ^{-1}\left (\frac {\sqrt {\frac {2}{1+\sqrt {5}}}}{\sqrt {x}}\right )|\frac {1}{2} \left (-3-\sqrt {5}\right )\right )-(2+i) \Pi \left (\frac {1}{2} i \left (1+\sqrt {5}\right );i \sinh ^{-1}\left (\frac {\sqrt {\frac {2}{1+\sqrt {5}}}}{\sqrt {x}}\right )|\frac {1}{2} \left (-3-\sqrt {5}\right )\right )\right )}{\sqrt {x \left (x^2-x-1\right )}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-I)*Sqrt[2/(-1 + Sqrt[5])]*Sqrt[1 - x^(-2) - x^(-1)]*x^(3/2)*(2*EllipticF[I*ArcSinh[Sqrt[2/(1 + Sqrt[5])]/Sq
rt[x]], -3/2 - Sqrt[5]/2] - (2 - I)*EllipticPi[(-1/2*I)*(1 + Sqrt[5]), I*ArcSinh[Sqrt[2/(1 + Sqrt[5])]/Sqrt[x]
], (-3 - Sqrt[5])/2] - (2 + I)*EllipticPi[(I/2)*(1 + Sqrt[5]), I*ArcSinh[Sqrt[2/(1 + Sqrt[5])]/Sqrt[x]], (-3 -
Sqrt[5])/2]))/Sqrt[x*(-1 - x + x^2)]

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IntegrateAlgebraic [A]  time = 0.32, size = 101, normalized size = 1.00 \begin {gather*} -\frac {1}{2} \sqrt {\frac {11}{5}+\frac {2 i}{5}} \tan ^{-1}\left (\frac {\sqrt {1-2 i} \sqrt {-x-x^2+x^3}}{-1-x+x^2}\right )-\frac {1}{2} \sqrt {\frac {11}{5}-\frac {2 i}{5}} \tan ^{-1}\left (\frac {\sqrt {1+2 i} \sqrt {-x-x^2+x^3}}{-1-x+x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(Sqrt[11/5 + (2*I)/5]*ArcTan[(Sqrt[1 - 2*I]*Sqrt[-x - x^2 + x^3])/(-1 - x + x^2)]) - (Sqrt[11/5 - (2*I)/5
]*ArcTan[(Sqrt[1 + 2*I]*Sqrt[-x - x^2 + x^3])/(-1 - x + x^2)])/2

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fricas [B]  time = 1.02, size = 2504, normalized size = 24.79

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/800*125^(1/4)*(5*sqrt(5)*sqrt(2) - 11*sqrt(2))*sqrt(55*sqrt(5) + 125)*log(25*(25*x^4 - 100*x^3 + 2*125^(1/4)
*sqrt(x^3 - x^2 - x)*(sqrt(5)*sqrt(2)*(2*x^2 - 7*x - 2) - 5*sqrt(2)*(x^2 - 3*x - 1))*sqrt(55*sqrt(5) + 125) +
150*x^2 + 100*sqrt(5)*(x^3 - x^2 - x) + 100*x + 25)/(x^4 + 2*x^2 + 1)) - 1/800*125^(1/4)*(5*sqrt(5)*sqrt(2) -
11*sqrt(2))*sqrt(55*sqrt(5) + 125)*log(25*(25*x^4 - 100*x^3 - 2*125^(1/4)*sqrt(x^3 - x^2 - x)*(sqrt(5)*sqrt(2)
*(2*x^2 - 7*x - 2) - 5*sqrt(2)*(x^2 - 3*x - 1))*sqrt(55*sqrt(5) + 125) + 150*x^2 + 100*sqrt(5)*(x^3 - x^2 - x)
+ 100*x + 25)/(x^4 + 2*x^2 + 1)) - 1/100*125^(1/4)*sqrt(2)*sqrt(55*sqrt(5) + 125)*arctan(-1/2500*(1250*x^11 +
16250*x^10 - 83750*x^9 - 55000*x^8 + 355000*x^7 + 17500*x^6 - 355000*x^5 - 55000*x^4 + 83750*x^3 + 16250*x^2
+ 5*sqrt(x^3 - x^2 - x)*(125^(3/4)*(sqrt(5)*sqrt(2)*(x^10 - x^9 - 27*x^8 + 36*x^7 + 132*x^6 - 86*x^5 - 132*x^4
+ 36*x^3 + 27*x^2 - x - 1) - sqrt(2)*(2*x^10 + x^9 - 64*x^8 + 52*x^7 + 318*x^6 - 122*x^5 - 318*x^4 + 52*x^3 +
64*x^2 + x - 2)) + 10*125^(1/4)*(sqrt(5)*sqrt(2)*(7*x^9 + 33*x^8 - 24*x^7 - 159*x^6 + 50*x^5 + 159*x^4 - 24*x
^3 - 33*x^2 + 7*x) - 5*sqrt(2)*(3*x^9 + 17*x^8 - 20*x^7 - 63*x^6 + 34*x^5 + 63*x^4 - 20*x^3 - 17*x^2 + 3*x)))*
sqrt(55*sqrt(5) + 125) + 250*sqrt(5)*(5*x^11 - 15*x^10 - 15*x^9 + 20*x^8 - 20*x^7 + 70*x^6 + 20*x^5 + 20*x^4 +
15*x^3 - 15*x^2 - sqrt(5)*(x^11 + 13*x^10 - 67*x^9 - 44*x^8 + 284*x^7 + 14*x^6 - 284*x^5 - 44*x^4 + 67*x^3 +
13*x^2 - x) - 5*x) - 1250*sqrt(5)*(x^11 - 3*x^10 - 3*x^9 + 4*x^8 - 4*x^7 + 14*x^6 + 4*x^5 + 4*x^4 + 3*x^3 - 3*
x^2 - x) - (3000*x^10 + 2000*x^9 - 21000*x^8 - 6000*x^7 + 40000*x^6 + 6000*x^5 - 21000*x^4 - 2000*x^3 + 3000*x
^2 + sqrt(x^3 - x^2 - x)*(125^(3/4)*(sqrt(5)*sqrt(2)*(x^10 - 5*x^9 - 23*x^8 + 144*x^7 - 120*x^6 - 246*x^5 + 12
0*x^4 + 144*x^3 + 23*x^2 - 5*x - 1) - sqrt(2)*(2*x^10 - 9*x^9 - 44*x^8 + 312*x^7 - 302*x^6 - 542*x^5 + 302*x^4
+ 312*x^3 + 44*x^2 - 9*x - 2)) + 10*125^(1/4)*(sqrt(5)*sqrt(2)*(7*x^9 + 3*x^8 - 154*x^7 + 131*x^6 + 270*x^5 -
131*x^4 - 154*x^3 - 3*x^2 + 7*x) - 5*sqrt(2)*(3*x^9 + 3*x^8 - 70*x^7 + 51*x^6 + 126*x^5 - 51*x^4 - 70*x^3 - 3
*x^2 + 3*x)))*sqrt(55*sqrt(5) + 125) + 50*sqrt(5)*(5*x^11 - 25*x^10 - 105*x^9 + 440*x^8 + 50*x^7 - 830*x^6 - 5
0*x^5 + 440*x^4 + 105*x^3 - 25*x^2 - sqrt(5)*(x^11 + 3*x^10 - 37*x^9 + 96*x^8 - 6*x^7 - 166*x^6 + 6*x^5 + 96*x
^4 + 37*x^3 + 3*x^2 - x) - 5*x) - 1000*sqrt(5)*(x^10 + 6*x^9 - 23*x^8 - 2*x^7 + 40*x^6 + 2*x^5 - 23*x^4 - 6*x^
3 + x^2))*sqrt((25*x^4 - 100*x^3 + 2*125^(1/4)*sqrt(x^3 - x^2 - x)*(sqrt(5)*sqrt(2)*(2*x^2 - 7*x - 2) - 5*sqrt
(2)*(x^2 - 3*x - 1))*sqrt(55*sqrt(5) + 125) + 150*x^2 + 100*sqrt(5)*(x^3 - x^2 - x) + 100*x + 25)/(x^4 + 2*x^2
+ 1)) - 1250*x)/(x^11 - 9*x^10 - 45*x^9 + 180*x^8 + 18*x^7 - 326*x^6 - 18*x^5 + 180*x^4 + 45*x^3 - 9*x^2 - x)
) - 1/100*125^(1/4)*sqrt(2)*sqrt(55*sqrt(5) + 125)*arctan(1/2500*(1250*x^11 + 16250*x^10 - 83750*x^9 - 55000*x
^8 + 355000*x^7 + 17500*x^6 - 355000*x^5 - 55000*x^4 + 83750*x^3 + 16250*x^2 - 5*sqrt(x^3 - x^2 - x)*(125^(3/4
)*(sqrt(5)*sqrt(2)*(x^10 - x^9 - 27*x^8 + 36*x^7 + 132*x^6 - 86*x^5 - 132*x^4 + 36*x^3 + 27*x^2 - x - 1) - sqr
t(2)*(2*x^10 + x^9 - 64*x^8 + 52*x^7 + 318*x^6 - 122*x^5 - 318*x^4 + 52*x^3 + 64*x^2 + x - 2)) + 10*125^(1/4)*
(sqrt(5)*sqrt(2)*(7*x^9 + 33*x^8 - 24*x^7 - 159*x^6 + 50*x^5 + 159*x^4 - 24*x^3 - 33*x^2 + 7*x) - 5*sqrt(2)*(3
*x^9 + 17*x^8 - 20*x^7 - 63*x^6 + 34*x^5 + 63*x^4 - 20*x^3 - 17*x^2 + 3*x)))*sqrt(55*sqrt(5) + 125) + 250*sqrt
(5)*(5*x^11 - 15*x^10 - 15*x^9 + 20*x^8 - 20*x^7 + 70*x^6 + 20*x^5 + 20*x^4 + 15*x^3 - 15*x^2 - sqrt(5)*(x^11
+ 13*x^10 - 67*x^9 - 44*x^8 + 284*x^7 + 14*x^6 - 284*x^5 - 44*x^4 + 67*x^3 + 13*x^2 - x) - 5*x) - 1250*sqrt(5)
*(x^11 - 3*x^10 - 3*x^9 + 4*x^8 - 4*x^7 + 14*x^6 + 4*x^5 + 4*x^4 + 3*x^3 - 3*x^2 - x) - (3000*x^10 + 2000*x^9
- 21000*x^8 - 6000*x^7 + 40000*x^6 + 6000*x^5 - 21000*x^4 - 2000*x^3 + 3000*x^2 - sqrt(x^3 - x^2 - x)*(125^(3/
4)*(sqrt(5)*sqrt(2)*(x^10 - 5*x^9 - 23*x^8 + 144*x^7 - 120*x^6 - 246*x^5 + 120*x^4 + 144*x^3 + 23*x^2 - 5*x -
1) - sqrt(2)*(2*x^10 - 9*x^9 - 44*x^8 + 312*x^7 - 302*x^6 - 542*x^5 + 302*x^4 + 312*x^3 + 44*x^2 - 9*x - 2)) +
10*125^(1/4)*(sqrt(5)*sqrt(2)*(7*x^9 + 3*x^8 - 154*x^7 + 131*x^6 + 270*x^5 - 131*x^4 - 154*x^3 - 3*x^2 + 7*x)
- 5*sqrt(2)*(3*x^9 + 3*x^8 - 70*x^7 + 51*x^6 + 126*x^5 - 51*x^4 - 70*x^3 - 3*x^2 + 3*x)))*sqrt(55*sqrt(5) + 1
25) + 50*sqrt(5)*(5*x^11 - 25*x^10 - 105*x^9 + 440*x^8 + 50*x^7 - 830*x^6 - 50*x^5 + 440*x^4 + 105*x^3 - 25*x^
2 - sqrt(5)*(x^11 + 3*x^10 - 37*x^9 + 96*x^8 - 6*x^7 - 166*x^6 + 6*x^5 + 96*x^4 + 37*x^3 + 3*x^2 - x) - 5*x) -
1000*sqrt(5)*(x^10 + 6*x^9 - 23*x^8 - 2*x^7 + 40*x^6 + 2*x^5 - 23*x^4 - 6*x^3 + x^2))*sqrt((25*x^4 - 100*x^3
- 2*125^(1/4)*sqrt(x^3 - x^2 - x)*(sqrt(5)*sqrt(2)*(2*x^2 - 7*x - 2) - 5*sqrt(2)*(x^2 - 3*x - 1))*sqrt(55*sqrt
(5) + 125) + 150*x^2 + 100*sqrt(5)*(x^3 - x^2 - x) + 100*x + 25)/(x^4 + 2*x^2 + 1)) - 1250*x)/(x^11 - 9*x^10 -
45*x^9 + 180*x^8 + 18*x^7 - 326*x^6 - 18*x^5 + 180*x^4 + 45*x^3 - 9*x^2 - x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [C]  time = 1.23, size = 1353, normalized size = 13.40

method	result	size
default	\(\frac {2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \sqrt {-5 \left (x -\frac {1}{2}-\frac {\sqrt {5}}{2}\right ) \sqrt {5}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \EllipticF \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right )}{5 \sqrt {x^{3}-x^{2}-x}}+\frac {\sqrt {\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}-\frac {1}{2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}+\frac {\sqrt {5}}{\sqrt {5}-1}}\, \sqrt {-5 x \sqrt {5}+\frac {5 \sqrt {5}}{2}+\frac {25}{2}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \EllipticPi \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {\frac {1}{2}-\frac {\sqrt {5}}{2}}{\frac {1}{2}-i-\frac {\sqrt {5}}{2}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right ) \sqrt {5}}{10 \sqrt {x^{3}-x^{2}-x}\, \left (\frac {1}{2}-i-\frac {\sqrt {5}}{2}\right )}+\frac {i \sqrt {\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}-\frac {1}{2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}+\frac {\sqrt {5}}{\sqrt {5}-1}}\, \sqrt {-5 x \sqrt {5}+\frac {5 \sqrt {5}}{2}+\frac {25}{2}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \EllipticPi \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {\frac {1}{2}-\frac {\sqrt {5}}{2}}{\frac {1}{2}-i-\frac {\sqrt {5}}{2}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right ) \sqrt {5}}{5 \sqrt {x^{3}-x^{2}-x}\, \left (\frac {1}{2}-i-\frac {\sqrt {5}}{2}\right )}-\frac {\sqrt {\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}-\frac {1}{2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}+\frac {\sqrt {5}}{\sqrt {5}-1}}\, \sqrt {-5 x \sqrt {5}+\frac {5 \sqrt {5}}{2}+\frac {25}{2}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \EllipticPi \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {\frac {1}{2}-\frac {\sqrt {5}}{2}}{\frac {1}{2}-i-\frac {\sqrt {5}}{2}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right )}{10 \sqrt {x^{3}-x^{2}-x}\, \left (\frac {1}{2}-i-\frac {\sqrt {5}}{2}\right )}-\frac {i \sqrt {\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}-\frac {1}{2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}+\frac {\sqrt {5}}{\sqrt {5}-1}}\, \sqrt {-5 x \sqrt {5}+\frac {5 \sqrt {5}}{2}+\frac {25}{2}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \EllipticPi \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {\frac {1}{2}-\frac {\sqrt {5}}{2}}{\frac {1}{2}-i-\frac {\sqrt {5}}{2}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right )}{5 \sqrt {x^{3}-x^{2}-x}\, \left (\frac {1}{2}-i-\frac {\sqrt {5}}{2}\right )}+\frac {\sqrt {\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}-\frac {1}{2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}+\frac {\sqrt {5}}{\sqrt {5}-1}}\, \sqrt {-5 x \sqrt {5}+\frac {5 \sqrt {5}}{2}+\frac {25}{2}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \EllipticPi \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {\frac {1}{2}-\frac {\sqrt {5}}{2}}{\frac {1}{2}+i-\frac {\sqrt {5}}{2}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right ) \sqrt {5}}{10 \sqrt {x^{3}-x^{2}-x}\, \left (\frac {1}{2}+i-\frac {\sqrt {5}}{2}\right )}-\frac {i \sqrt {\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}-\frac {1}{2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}+\frac {\sqrt {5}}{\sqrt {5}-1}}\, \sqrt {-5 x \sqrt {5}+\frac {5 \sqrt {5}}{2}+\frac {25}{2}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \EllipticPi \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {\frac {1}{2}-\frac {\sqrt {5}}{2}}{\frac {1}{2}+i-\frac {\sqrt {5}}{2}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right ) \sqrt {5}}{5 \sqrt {x^{3}-x^{2}-x}\, \left (\frac {1}{2}+i-\frac {\sqrt {5}}{2}\right )}-\frac {\sqrt {\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}-\frac {1}{2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}+\frac {\sqrt {5}}{\sqrt {5}-1}}\, \sqrt {-5 x \sqrt {5}+\frac {5 \sqrt {5}}{2}+\frac {25}{2}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \EllipticPi \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {\frac {1}{2}-\frac {\sqrt {5}}{2}}{\frac {1}{2}+i-\frac {\sqrt {5}}{2}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right )}{10 \sqrt {x^{3}-x^{2}-x}\, \left (\frac {1}{2}+i-\frac {\sqrt {5}}{2}\right )}+\frac {i \sqrt {\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}-\frac {1}{2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}+\frac {\sqrt {5}}{\sqrt {5}-1}}\, \sqrt {-5 x \sqrt {5}+\frac {5 \sqrt {5}}{2}+\frac {25}{2}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \EllipticPi \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {\frac {1}{2}-\frac {\sqrt {5}}{2}}{\frac {1}{2}+i-\frac {\sqrt {5}}{2}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right )}{5 \sqrt {x^{3}-x^{2}-x}\, \left (\frac {1}{2}+i-\frac {\sqrt {5}}{2}\right )}\)	\(1353\)
elliptic	\(\frac {\sqrt {\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}-\frac {1}{2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}+\frac {\sqrt {5}}{\sqrt {5}-1}}\, \sqrt {-5 x \sqrt {5}+\frac {5 \sqrt {5}}{2}+\frac {25}{2}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \EllipticF \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right ) \sqrt {5}}{5 \sqrt {x^{3}-x^{2}-x}}-\frac {\sqrt {\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}-\frac {1}{2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}+\frac {\sqrt {5}}{\sqrt {5}-1}}\, \sqrt {-5 x \sqrt {5}+\frac {5 \sqrt {5}}{2}+\frac {25}{2}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \EllipticF \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right )}{5 \sqrt {x^{3}-x^{2}-x}}+\frac {\sqrt {\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}-\frac {1}{2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}+\frac {\sqrt {5}}{\sqrt {5}-1}}\, \sqrt {-5 x \sqrt {5}+\frac {5 \sqrt {5}}{2}+\frac {25}{2}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \EllipticPi \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {\frac {1}{2}-\frac {\sqrt {5}}{2}}{\frac {1}{2}-i-\frac {\sqrt {5}}{2}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right ) \sqrt {5}}{10 \sqrt {x^{3}-x^{2}-x}\, \left (\frac {1}{2}-i-\frac {\sqrt {5}}{2}\right )}+\frac {i \sqrt {\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}-\frac {1}{2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}+\frac {\sqrt {5}}{\sqrt {5}-1}}\, \sqrt {-5 x \sqrt {5}+\frac {5 \sqrt {5}}{2}+\frac {25}{2}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \EllipticPi \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {\frac {1}{2}-\frac {\sqrt {5}}{2}}{\frac {1}{2}-i-\frac {\sqrt {5}}{2}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right ) \sqrt {5}}{5 \sqrt {x^{3}-x^{2}-x}\, \left (\frac {1}{2}-i-\frac {\sqrt {5}}{2}\right )}-\frac {\sqrt {\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}-\frac {1}{2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}+\frac {\sqrt {5}}{\sqrt {5}-1}}\, \sqrt {-5 x \sqrt {5}+\frac {5 \sqrt {5}}{2}+\frac {25}{2}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \EllipticPi \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {\frac {1}{2}-\frac {\sqrt {5}}{2}}{\frac {1}{2}-i-\frac {\sqrt {5}}{2}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right )}{10 \sqrt {x^{3}-x^{2}-x}\, \left (\frac {1}{2}-i-\frac {\sqrt {5}}{2}\right )}-\frac {i \sqrt {\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}-\frac {1}{2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}+\frac {\sqrt {5}}{\sqrt {5}-1}}\, \sqrt {-5 x \sqrt {5}+\frac {5 \sqrt {5}}{2}+\frac {25}{2}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \EllipticPi \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {\frac {1}{2}-\frac {\sqrt {5}}{2}}{\frac {1}{2}-i-\frac {\sqrt {5}}{2}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right )}{5 \sqrt {x^{3}-x^{2}-x}\, \left (\frac {1}{2}-i-\frac {\sqrt {5}}{2}\right )}+\frac {\sqrt {\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}-\frac {1}{2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}+\frac {\sqrt {5}}{\sqrt {5}-1}}\, \sqrt {-5 x \sqrt {5}+\frac {5 \sqrt {5}}{2}+\frac {25}{2}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \EllipticPi \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {\frac {1}{2}-\frac {\sqrt {5}}{2}}{\frac {1}{2}+i-\frac {\sqrt {5}}{2}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right ) \sqrt {5}}{10 \sqrt {x^{3}-x^{2}-x}\, \left (\frac {1}{2}+i-\frac {\sqrt {5}}{2}\right )}-\frac {i \sqrt {\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}-\frac {1}{2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}+\frac {\sqrt {5}}{\sqrt {5}-1}}\, \sqrt {-5 x \sqrt {5}+\frac {5 \sqrt {5}}{2}+\frac {25}{2}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \EllipticPi \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {\frac {1}{2}-\frac {\sqrt {5}}{2}}{\frac {1}{2}+i-\frac {\sqrt {5}}{2}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right ) \sqrt {5}}{5 \sqrt {x^{3}-x^{2}-x}\, \left (\frac {1}{2}+i-\frac {\sqrt {5}}{2}\right )}-\frac {\sqrt {\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}-\frac {1}{2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}+\frac {\sqrt {5}}{\sqrt {5}-1}}\, \sqrt {-5 x \sqrt {5}+\frac {5 \sqrt {5}}{2}+\frac {25}{2}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \EllipticPi \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {\frac {1}{2}-\frac {\sqrt {5}}{2}}{\frac {1}{2}+i-\frac {\sqrt {5}}{2}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right )}{10 \sqrt {x^{3}-x^{2}-x}\, \left (\frac {1}{2}+i-\frac {\sqrt {5}}{2}\right )}+\frac {i \sqrt {\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}-\frac {1}{2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}+\frac {\sqrt {5}}{\sqrt {5}-1}}\, \sqrt {-5 x \sqrt {5}+\frac {5 \sqrt {5}}{2}+\frac {25}{2}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \EllipticPi \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {\frac {1}{2}-\frac {\sqrt {5}}{2}}{\frac {1}{2}+i-\frac {\sqrt {5}}{2}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right )}{5 \sqrt {x^{3}-x^{2}-x}\, \left (\frac {1}{2}+i-\frac {\sqrt {5}}{2}\right )}\)	\(1491\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*(1/2*5^(1/2)-1/2)*((x-1/2+1/2*5^(1/2))/(1/2*5^(1/2)-1/2))^(1/2)*(-5*(x-1/2-1/2*5^(1/2))*5^(1/2))^(1/2)*(-x
/(1/2*5^(1/2)-1/2))^(1/2)/(x^3-x^2-x)^(1/2)*EllipticF(((x-1/2+1/2*5^(1/2))/(1/2*5^(1/2)-1/2))^(1/2),1/5*5^(1/2
)*((1/2*5^(1/2)-1/2)*5^(1/2))^(1/2))+1/10*(x/(1/2*5^(1/2)-1/2)-1/2/(1/2*5^(1/2)-1/2)+1/2/(1/2*5^(1/2)-1/2)*5^(
1/2))^(1/2)*(-5*x*5^(1/2)+5/2*5^(1/2)+25/2)^(1/2)*(-x/(1/2*5^(1/2)-1/2))^(1/2)/(x^3-x^2-x)^(1/2)/(1/2-I-1/2*5^
(1/2))*EllipticPi(((x-1/2+1/2*5^(1/2))/(1/2*5^(1/2)-1/2))^(1/2),(1/2-1/2*5^(1/2))/(1/2-I-1/2*5^(1/2)),1/5*5^(1
/2)*((1/2*5^(1/2)-1/2)*5^(1/2))^(1/2))*5^(1/2)+1/5*I*(x/(1/2*5^(1/2)-1/2)-1/2/(1/2*5^(1/2)-1/2)+1/2/(1/2*5^(1/
2)-1/2)*5^(1/2))^(1/2)*(-5*x*5^(1/2)+5/2*5^(1/2)+25/2)^(1/2)*(-x/(1/2*5^(1/2)-1/2))^(1/2)/(x^3-x^2-x)^(1/2)/(1
/2-I-1/2*5^(1/2))*EllipticPi(((x-1/2+1/2*5^(1/2))/(1/2*5^(1/2)-1/2))^(1/2),(1/2-1/2*5^(1/2))/(1/2-I-1/2*5^(1/2
)),1/5*5^(1/2)*((1/2*5^(1/2)-1/2)*5^(1/2))^(1/2))*5^(1/2)-1/10*(x/(1/2*5^(1/2)-1/2)-1/2/(1/2*5^(1/2)-1/2)+1/2/
(1/2*5^(1/2)-1/2)*5^(1/2))^(1/2)*(-5*x*5^(1/2)+5/2*5^(1/2)+25/2)^(1/2)*(-x/(1/2*5^(1/2)-1/2))^(1/2)/(x^3-x^2-x
)^(1/2)/(1/2-I-1/2*5^(1/2))*EllipticPi(((x-1/2+1/2*5^(1/2))/(1/2*5^(1/2)-1/2))^(1/2),(1/2-1/2*5^(1/2))/(1/2-I-
1/2*5^(1/2)),1/5*5^(1/2)*((1/2*5^(1/2)-1/2)*5^(1/2))^(1/2))-1/5*I*(x/(1/2*5^(1/2)-1/2)-1/2/(1/2*5^(1/2)-1/2)+1
/2/(1/2*5^(1/2)-1/2)*5^(1/2))^(1/2)*(-5*x*5^(1/2)+5/2*5^(1/2)+25/2)^(1/2)*(-x/(1/2*5^(1/2)-1/2))^(1/2)/(x^3-x^
2-x)^(1/2)/(1/2-I-1/2*5^(1/2))*EllipticPi(((x-1/2+1/2*5^(1/2))/(1/2*5^(1/2)-1/2))^(1/2),(1/2-1/2*5^(1/2))/(1/2
-I-1/2*5^(1/2)),1/5*5^(1/2)*((1/2*5^(1/2)-1/2)*5^(1/2))^(1/2))+1/10*(x/(1/2*5^(1/2)-1/2)-1/2/(1/2*5^(1/2)-1/2)
+1/2/(1/2*5^(1/2)-1/2)*5^(1/2))^(1/2)*(-5*x*5^(1/2)+5/2*5^(1/2)+25/2)^(1/2)*(-x/(1/2*5^(1/2)-1/2))^(1/2)/(x^3-
x^2-x)^(1/2)/(1/2+I-1/2*5^(1/2))*EllipticPi(((x-1/2+1/2*5^(1/2))/(1/2*5^(1/2)-1/2))^(1/2),(1/2-1/2*5^(1/2))/(1
/2+I-1/2*5^(1/2)),1/5*5^(1/2)*((1/2*5^(1/2)-1/2)*5^(1/2))^(1/2))*5^(1/2)-1/5*I*(x/(1/2*5^(1/2)-1/2)-1/2/(1/2*5
^(1/2)-1/2)+1/2/(1/2*5^(1/2)-1/2)*5^(1/2))^(1/2)*(-5*x*5^(1/2)+5/2*5^(1/2)+25/2)^(1/2)*(-x/(1/2*5^(1/2)-1/2))^
(1/2)/(x^3-x^2-x)^(1/2)/(1/2+I-1/2*5^(1/2))*EllipticPi(((x-1/2+1/2*5^(1/2))/(1/2*5^(1/2)-1/2))^(1/2),(1/2-1/2*
5^(1/2))/(1/2+I-1/2*5^(1/2)),1/5*5^(1/2)*((1/2*5^(1/2)-1/2)*5^(1/2))^(1/2))*5^(1/2)-1/10*(x/(1/2*5^(1/2)-1/2)-
1/2/(1/2*5^(1/2)-1/2)+1/2/(1/2*5^(1/2)-1/2)*5^(1/2))^(1/2)*(-5*x*5^(1/2)+5/2*5^(1/2)+25/2)^(1/2)*(-x/(1/2*5^(1
/2)-1/2))^(1/2)/(x^3-x^2-x)^(1/2)/(1/2+I-1/2*5^(1/2))*EllipticPi(((x-1/2+1/2*5^(1/2))/(1/2*5^(1/2)-1/2))^(1/2)
,(1/2-1/2*5^(1/2))/(1/2+I-1/2*5^(1/2)),1/5*5^(1/2)*((1/2*5^(1/2)-1/2)*5^(1/2))^(1/2))+1/5*I*(x/(1/2*5^(1/2)-1/
2)-1/2/(1/2*5^(1/2)-1/2)+1/2/(1/2*5^(1/2)-1/2)*5^(1/2))^(1/2)*(-5*x*5^(1/2)+5/2*5^(1/2)+25/2)^(1/2)*(-x/(1/2*5
^(1/2)-1/2))^(1/2)/(x^3-x^2-x)^(1/2)/(1/2+I-1/2*5^(1/2))*EllipticPi(((x-1/2+1/2*5^(1/2))/(1/2*5^(1/2)-1/2))^(1
/2),(1/2-1/2*5^(1/2))/(1/2+I-1/2*5^(1/2)),1/5*5^(1/2)*((1/2*5^(1/2)-1/2)*5^(1/2))^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.84, size = 226, normalized size = 2.24 \begin {gather*} \frac {\left (\sqrt {5}\,\left (2+1{}\mathrm {i}\right )+2+1{}\mathrm {i}\right )\,\sqrt {\frac {x}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\,\sqrt {\frac {x+\frac {\sqrt {5}}{2}-\frac {1}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\,\sqrt {\frac {\frac {\sqrt {5}}{2}-x+\frac {1}{2}}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\,\left (-2\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {\frac {x}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\right )\middle |-\frac {\frac {\sqrt {5}}{2}+\frac {1}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}\right )+\Pi \left (-\frac {\sqrt {5}\,1{}\mathrm {i}}{2}-\frac {1}{2}{}\mathrm {i};\mathrm {asin}\left (\sqrt {\frac {x}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\right )\middle |-\frac {\frac {\sqrt {5}}{2}+\frac {1}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}\right )\,\left (2-\mathrm {i}\right )+\Pi \left (\frac {\sqrt {5}\,1{}\mathrm {i}}{2}+\frac {1}{2}{}\mathrm {i};\mathrm {asin}\left (\sqrt {\frac {x}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\right )\middle |-\frac {\frac {\sqrt {5}}{2}+\frac {1}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}\right )\,\left (2+1{}\mathrm {i}\right )\right )\,\left (-\frac {1}{5}+\frac {1}{10}{}\mathrm {i}\right )}{\sqrt {x^3-x^2-\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )\,\left (\frac {\sqrt {5}}{2}+\frac {1}{2}\right )\,x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((5^(1/2)*(2 + 1i) + (2 + 1i))*(x/(5^(1/2)/2 + 1/2))^(1/2)*((x + 5^(1/2)/2 - 1/2)/(5^(1/2)/2 - 1/2))^(1/2)*((5
^(1/2)/2 - x + 1/2)/(5^(1/2)/2 + 1/2))^(1/2)*(ellipticPi(- (5^(1/2)*1i)/2 - 1i/2, asin((x/(5^(1/2)/2 + 1/2))^(
1/2)), -(5^(1/2)/2 + 1/2)/(5^(1/2)/2 - 1/2))*(2 - 1i) - 2*ellipticF(asin((x/(5^(1/2)/2 + 1/2))^(1/2)), -(5^(1/
2)/2 + 1/2)/(5^(1/2)/2 - 1/2)) + ellipticPi((5^(1/2)*1i)/2 + 1i/2, asin((x/(5^(1/2)/2 + 1/2))^(1/2)), -(5^(1/2
)/2 + 1/2)/(5^(1/2)/2 - 1/2))*(2 + 1i))*(- 1/5 + 1i/10))/(x^3 - x^2 - x*(5^(1/2)/2 - 1/2)*(5^(1/2)/2 + 1/2))^(
1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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