\(\int \frac {-1+x+x^2}{(1+x^2) \sqrt {-x-x^2+x^3}} \, dx\) [1411]

Optimal result

Integrand size = 30, antiderivative size = 101 \[ \int \frac {-1+x+x^2}{\left (1+x^2\right ) \sqrt {-x-x^2+x^3}} \, dx=-\frac {1}{2} \sqrt {\frac {11}{5}+\frac {2 i}{5}} \arctan \left (\frac {\sqrt {1-2 i} \sqrt {-x-x^2+x^3}}{-1-x+x^2}\right )-\frac {1}{2} \sqrt {\frac {11}{5}-\frac {2 i}{5}} \arctan \left (\frac {\sqrt {1+2 i} \sqrt {-x-x^2+x^3}}{-1-x+x^2}\right ) \]

[Out]

-1/10*(55+10*I)^(1/2)*arctan((1-2*I)^(1/2)*(x^3-x^2-x)^(1/2)/(x^2-x-1))-1/10*(55-10*I)^(1/2)*arctan((1+2*I)^(1
/2)*(x^3-x^2-x)^(1/2)/(x^2-x-1))

Rubi [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 0.65 (sec) , antiderivative size = 383, normalized size of antiderivative = 3.79, number of steps used = 15, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {2081, 6857, 730, 1112, 948, 174, 552, 551} \[ \int \frac {-1+x+x^2}{\left (1+x^2\right ) \sqrt {-x-x^2+x^3}} \, dx=\frac {\sqrt {x} \sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x+2}{\left (1-\sqrt {5}\right ) x+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x+2}} \sqrt {x^3-x^2-x}}-\frac {\left (1-\frac {i}{2}\right ) \sqrt {3+\sqrt {5}} \sqrt {x} \sqrt {2 x+\sqrt {5}-1} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \operatorname {EllipticPi}\left (-\frac {1}{2} i \left (1+\sqrt {5}\right ),\arcsin \left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right ),\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{\sqrt {x^3-x^2-x}}-\frac {\left (1+\frac {i}{2}\right ) \sqrt {3+\sqrt {5}} \sqrt {x} \sqrt {2 x+\sqrt {5}-1} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \operatorname {EllipticPi}\left (\frac {1}{2} i \left (1+\sqrt {5}\right ),\arcsin \left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right ),\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{\sqrt {x^3-x^2-x}} \]

[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 174

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 551

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1/(a*Sqr
t[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b*(c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], 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 552

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

Rule 730

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 948

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 1112

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]/(2*Sqrt[a + b*x^2 + c*x^4]*Sqrt[a/(2*a + (b + q)
*x^2)]))*EllipticF[ArcSin[x/Sqrt[(2*a + (b + q)*x^2)/(2*q)]], (b + q)/(2*q)], x]] /; FreeQ[{a, b, c}, x] && Gt
Q[b^2 - 4*a*c, 0] && LtQ[a, 0] && GtQ[c, 0]

Rule 2081

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 6857

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*} \text {integral}& = \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 ) \text {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}} \operatorname {EllipticF}\left (\arcsin \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}} \operatorname {EllipticF}\left (\arcsin \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}} \operatorname {EllipticF}\left (\arcsin \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 ) \text {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 ) \text {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}} \operatorname {EllipticF}\left (\arcsin \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 ) \text {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 ) \text {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}} \operatorname {EllipticF}\left (\arcsin \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}}} \operatorname {EllipticPi}\left (-\frac {1}{2} i \left (1+\sqrt {5}\right ),\arcsin \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}}} \operatorname {EllipticPi}\left (\frac {1}{2} i \left (1+\sqrt {5}\right ),\arcsin \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*}

Mathematica [A] (verified)

Time = 0.45 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.08 \[ \int \frac {-1+x+x^2}{\left (1+x^2\right ) \sqrt {-x-x^2+x^3}} \, dx=-\frac {\sqrt {x} \sqrt {-1-x+x^2} \left (\sqrt {11+2 i} \arctan \left (\frac {\sqrt {1-2 i} \sqrt {x}}{\sqrt {-1-x+x^2}}\right )+\sqrt {11-2 i} \arctan \left (\frac {\sqrt {1+2 i} \sqrt {x}}{\sqrt {-1-x+x^2}}\right )\right )}{2 \sqrt {5} \sqrt {x \left (-1-x+x^2\right )}} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(197\) vs. \(2(77)=154\).

Time = 1.49 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.96

method	result	size
default	\(-\frac {3 \left (\left (-\sqrt {5}+\frac {5}{3}\right ) \ln \left (\frac {-\sqrt {x \left (x^{2}-x -1\right )}\, \sqrt {-2+2 \sqrt {5}}+x \sqrt {5}+x^{2}-x -1}{x}\right )+\left (\sqrt {5}-\frac {5}{3}\right ) \ln \left (\frac {\sqrt {x \left (x^{2}-x -1\right )}\, \sqrt {-2+2 \sqrt {5}}+x \sqrt {5}+x^{2}-x -1}{x}\right )+\frac {8 \left (\arctan \left (\frac {\sqrt {-2+2 \sqrt {5}}\, x -2 \sqrt {x \left (x^{2}-x -1\right )}}{x \sqrt {2+2 \sqrt {5}}}\right )-\arctan \left (\frac {\sqrt {-2+2 \sqrt {5}}\, x +2 \sqrt {x \left (x^{2}-x -1\right )}}{x \sqrt {2+2 \sqrt {5}}}\right )\right ) \left (\sqrt {5}+\frac {5}{2}\right )}{3}\right )}{20 \sqrt {2+2 \sqrt {5}}}\)	\(198\)
pseudoelliptic	\(-\frac {3 \left (\left (-\sqrt {5}+\frac {5}{3}\right ) \ln \left (\frac {-\sqrt {x \left (x^{2}-x -1\right )}\, \sqrt {-2+2 \sqrt {5}}+x \sqrt {5}+x^{2}-x -1}{x}\right )+\left (\sqrt {5}-\frac {5}{3}\right ) \ln \left (\frac {\sqrt {x \left (x^{2}-x -1\right )}\, \sqrt {-2+2 \sqrt {5}}+x \sqrt {5}+x^{2}-x -1}{x}\right )+\frac {8 \left (\arctan \left (\frac {\sqrt {-2+2 \sqrt {5}}\, x -2 \sqrt {x \left (x^{2}-x -1\right )}}{x \sqrt {2+2 \sqrt {5}}}\right )-\arctan \left (\frac {\sqrt {-2+2 \sqrt {5}}\, x +2 \sqrt {x \left (x^{2}-x -1\right )}}{x \sqrt {2+2 \sqrt {5}}}\right )\right ) \left (\sqrt {5}+\frac {5}{2}\right )}{3}\right )}{20 \sqrt {2+2 \sqrt {5}}}\)	\(198\)
elliptic	\(\text {Expression too large to display}\)	\(1491\)

[In]

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

[Out]

-3/20/(2+2*5^(1/2))^(1/2)*((-5^(1/2)+5/3)*ln((-(x*(x^2-x-1))^(1/2)*(-2+2*5^(1/2))^(1/2)+x*5^(1/2)+x^2-x-1)/x)+
(5^(1/2)-5/3)*ln(((x*(x^2-x-1))^(1/2)*(-2+2*5^(1/2))^(1/2)+x*5^(1/2)+x^2-x-1)/x)+8/3*(arctan(((-2+2*5^(1/2))^(
1/2)*x-2*(x*(x^2-x-1))^(1/2))/x/(2+2*5^(1/2))^(1/2))-arctan(((-2+2*5^(1/2))^(1/2)*x+2*(x*(x^2-x-1))^(1/2))/x/(
2+2*5^(1/2))^(1/2)))*(5^(1/2)+5/2))

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 297 vs. \(2 (69) = 138\).

Time = 0.30 (sec) , antiderivative size = 297, normalized size of antiderivative = 2.94 \[ \int \frac {-1+x+x^2}{\left (1+x^2\right ) \sqrt {-x-x^2+x^3}} \, dx=-\frac {1}{40} \, \sqrt {5} \sqrt {-2 i - 11} \log \left (\frac {25 \, x^{4} + \left (100 i - 50\right ) \, x^{3} - 2 \, \sqrt {5} \sqrt {-2 i - 11} \sqrt {x^{3} - x^{2} - x} {\left (-\left (3 i - 4\right ) \, x^{2} + \left (8 i + 6\right ) \, x + 3 i - 4\right )} - \left (100 i + 150\right ) \, x^{2} - \left (100 i - 50\right ) \, x + 25}{x^{4} + 2 \, x^{2} + 1}\right ) + \frac {1}{40} \, \sqrt {5} \sqrt {-2 i - 11} \log \left (\frac {25 \, x^{4} + \left (100 i - 50\right ) \, x^{3} - 2 \, \sqrt {5} \sqrt {-2 i - 11} \sqrt {x^{3} - x^{2} - x} {\left (\left (3 i - 4\right ) \, x^{2} - \left (8 i + 6\right ) \, x - 3 i + 4\right )} - \left (100 i + 150\right ) \, x^{2} - \left (100 i - 50\right ) \, x + 25}{x^{4} + 2 \, x^{2} + 1}\right ) - \frac {1}{40} \, \sqrt {5} \sqrt {2 i - 11} \log \left (\frac {25 \, x^{4} - \left (100 i + 50\right ) \, x^{3} - 2 \, \sqrt {5} \sqrt {2 i - 11} \sqrt {x^{3} - x^{2} - x} {\left (\left (3 i + 4\right ) \, x^{2} - \left (8 i - 6\right ) \, x - 3 i - 4\right )} + \left (100 i - 150\right ) \, x^{2} + \left (100 i + 50\right ) \, x + 25}{x^{4} + 2 \, x^{2} + 1}\right ) + \frac {1}{40} \, \sqrt {5} \sqrt {2 i - 11} \log \left (\frac {25 \, x^{4} - \left (100 i + 50\right ) \, x^{3} - 2 \, \sqrt {5} \sqrt {2 i - 11} \sqrt {x^{3} - x^{2} - x} {\left (-\left (3 i + 4\right ) \, x^{2} + \left (8 i - 6\right ) \, x + 3 i + 4\right )} + \left (100 i - 150\right ) \, x^{2} + \left (100 i + 50\right ) \, x + 25}{x^{4} + 2 \, x^{2} + 1}\right ) \]

[In]

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

[Out]

-1/40*sqrt(5)*sqrt(-2*I - 11)*log((25*x^4 + (100*I - 50)*x^3 - 2*sqrt(5)*sqrt(-2*I - 11)*sqrt(x^3 - x^2 - x)*(
-(3*I - 4)*x^2 + (8*I + 6)*x + 3*I - 4) - (100*I + 150)*x^2 - (100*I - 50)*x + 25)/(x^4 + 2*x^2 + 1)) + 1/40*s
qrt(5)*sqrt(-2*I - 11)*log((25*x^4 + (100*I - 50)*x^3 - 2*sqrt(5)*sqrt(-2*I - 11)*sqrt(x^3 - x^2 - x)*((3*I -
4)*x^2 - (8*I + 6)*x - 3*I + 4) - (100*I + 150)*x^2 - (100*I - 50)*x + 25)/(x^4 + 2*x^2 + 1)) - 1/40*sqrt(5)*s
qrt(2*I - 11)*log((25*x^4 - (100*I + 50)*x^3 - 2*sqrt(5)*sqrt(2*I - 11)*sqrt(x^3 - x^2 - x)*((3*I + 4)*x^2 - (
8*I - 6)*x - 3*I - 4) + (100*I - 150)*x^2 + (100*I + 50)*x + 25)/(x^4 + 2*x^2 + 1)) + 1/40*sqrt(5)*sqrt(2*I -
11)*log((25*x^4 - (100*I + 50)*x^3 - 2*sqrt(5)*sqrt(2*I - 11)*sqrt(x^3 - x^2 - x)*(-(3*I + 4)*x^2 + (8*I - 6)*
x + 3*I + 4) + (100*I - 150)*x^2 + (100*I + 50)*x + 25)/(x^4 + 2*x^2 + 1))

Sympy [F]

\[ \int \frac {-1+x+x^2}{\left (1+x^2\right ) \sqrt {-x-x^2+x^3}} \, dx=\int \frac {x^{2} + x - 1}{\sqrt {x \left (x^{2} - x - 1\right )} \left (x^{2} + 1\right )}\, dx \]

[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)

Maxima [F]

\[ \int \frac {-1+x+x^2}{\left (1+x^2\right ) \sqrt {-x-x^2+x^3}} \, dx=\int { \frac {x^{2} + x - 1}{\sqrt {x^{3} - x^{2} - x} {\left (x^{2} + 1\right )}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {-1+x+x^2}{\left (1+x^2\right ) \sqrt {-x-x^2+x^3}} \, dx=\int { \frac {x^{2} + x - 1}{\sqrt {x^{3} - x^{2} - x} {\left (x^{2} + 1\right )}} \,d x } \]

[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)

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 226, normalized size of antiderivative = 2.24 \[ \int \frac {-1+x+x^2}{\left (1+x^2\right ) \sqrt {-x-x^2+x^3}} \, dx=\frac {\left (\sqrt {5}\,\left (2+1{}\mathrm {i}\right )+2+1{}\mathrm {i}\right )\,\sqrt {\frac {x}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\,\sqrt {\frac {x+\frac {\sqrt {5}}{2}-\frac {1}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\,\sqrt {\frac {\frac {\sqrt {5}}{2}-x+\frac {1}{2}}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\,\left (-2\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {\frac {x}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\right )\middle |-\frac {\frac {\sqrt {5}}{2}+\frac {1}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}\right )+\Pi \left (-\frac {\sqrt {5}\,1{}\mathrm {i}}{2}-\frac {1}{2}{}\mathrm {i};\mathrm {asin}\left (\sqrt {\frac {x}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\right )\middle |-\frac {\frac {\sqrt {5}}{2}+\frac {1}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}\right )\,\left (2-\mathrm {i}\right )+\Pi \left (\frac {\sqrt {5}\,1{}\mathrm {i}}{2}+\frac {1}{2}{}\mathrm {i};\mathrm {asin}\left (\sqrt {\frac {x}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\right )\middle |-\frac {\frac {\sqrt {5}}{2}+\frac {1}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}\right )\,\left (2+1{}\mathrm {i}\right )\right )\,\left (-\frac {1}{5}+\frac {1}{10}{}\mathrm {i}\right )}{\sqrt {x^3-x^2-\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )\,\left (\frac {\sqrt {5}}{2}+\frac {1}{2}\right )\,x}} \]

[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)