3.5.36 \(\int \frac {1-2 x}{\sqrt {5+5 x-4 x^2-2 x^3+x^4}} \, dx\)

Optimal. Leaf size=35 \[ \log \left (-2 x^2+2 \sqrt {x^4-2 x^3-4 x^2+5 x+5}+2 x+5\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.05, antiderivative size = 37, normalized size of antiderivative = 1.06, number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {1680, 12, 1107, 621, 206} \begin {gather*} \tanh ^{-1}\left (\frac {11-4 \left (x-\frac {1}{2}\right )^2}{\sqrt {16 \left (x-\frac {1}{2}\right )^4-88 \left (x-\frac {1}{2}\right )^2+101}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)/Sqrt[5 + 5*x - 4*x^2 - 2*x^3 + x^4],x]

[Out]

ArcTanh[(11 - 4*(-1/2 + x)^2)/Sqrt[101 - 88*(-1/2 + x)^2 + 16*(-1/2 + x)^4]]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 621

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

Rule 1107

Int[(x_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[(a + b*x + c*x^2)^p, x],
x, x^2], x] /; FreeQ[{a, b, c, p}, x]

Rule 1680

Int[(Pq_)*(Q4_)^(p_), x_Symbol] :> With[{a = Coeff[Q4, x, 0], b = Coeff[Q4, x, 1], c = Coeff[Q4, x, 2], d = Co
eff[Q4, x, 3], e = Coeff[Q4, x, 4]}, Subst[Int[SimplifyIntegrand[(Pq /. x -> -(d/(4*e)) + x)*(a + d^4/(256*e^3
) - (b*d)/(8*e) + (c - (3*d^2)/(8*e))*x^2 + e*x^4)^p, x], x], x, d/(4*e) + x] /; EqQ[d^3 - 4*c*d*e + 8*b*e^2,
0] && NeQ[d, 0]] /; FreeQ[p, x] && PolyQ[Pq, x] && PolyQ[Q4, x, 4] &&  !IGtQ[p, 0]

Rubi steps

\begin {align*} \int \frac {1-2 x}{\sqrt {5+5 x-4 x^2-2 x^3+x^4}} \, dx &=\operatorname {Subst}\left (\int -\frac {8 x}{\sqrt {101-88 x^2+16 x^4}} \, dx,x,-\frac {1}{2}+x\right )\\ &=-\left (8 \operatorname {Subst}\left (\int \frac {x}{\sqrt {101-88 x^2+16 x^4}} \, dx,x,-\frac {1}{2}+x\right )\right )\\ &=-\left (4 \operatorname {Subst}\left (\int \frac {1}{\sqrt {101-88 x+16 x^2}} \, dx,x,\left (-\frac {1}{2}+x\right )^2\right )\right )\\ &=-\left (8 \operatorname {Subst}\left (\int \frac {1}{64-x^2} \, dx,x,\frac {8 \left (-11+4 \left (-\frac {1}{2}+x\right )^2\right )}{\sqrt {101+(1-2 x)^4-88 \left (-\frac {1}{2}+x\right )^2}}\right )\right )\\ &=\tanh ^{-1}\left (\frac {11-(-1+2 x)^2}{\sqrt {101-22 (1-2 x)^2+(1-2 x)^4}}\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 1.64, size = 789, normalized size = 22.54 \begin {gather*} \frac {\left (-2 x+\sqrt {11-2 \sqrt {5}}+1\right )^2 \sqrt {\frac {-2 x+\sqrt {11+2 \sqrt {5}}+1}{-2 x+\sqrt {11-2 \sqrt {5}}+1}} \sqrt {-\frac {2 x+\sqrt {11+2 \sqrt {5}}-1}{\left (\sqrt {11-2 \sqrt {5}}-\sqrt {11+2 \sqrt {5}}\right ) \left (-2 x+\sqrt {11-2 \sqrt {5}}+1\right )}} \left (\left (-\sqrt {11+2 \sqrt {5}} \sqrt {-\frac {2 x+\sqrt {11-2 \sqrt {5}}-1}{-2 x+\sqrt {11-2 \sqrt {5}}+1}}+\sqrt {\frac {\left (2 \sqrt {5}-11\right ) \left (2 x+\sqrt {11-2 \sqrt {5}}-1\right )}{-2 x+\sqrt {11-2 \sqrt {5}}+1}}+\sqrt {2} \sqrt {\frac {\left (\sqrt {101}-11\right ) \left (2 x+\sqrt {11-2 \sqrt {5}}-1\right )}{-2 x+\sqrt {11-2 \sqrt {5}}+1}}+\sqrt {2} \sqrt {-\frac {\left (2 \sqrt {5}-11\right ) \left (\sqrt {101}-11\right ) \left (2 x+\sqrt {11-2 \sqrt {5}}-1\right )}{-2 x+\sqrt {11-2 \sqrt {5}}+1}}\right ) F\left (\sin ^{-1}\left (\sqrt {\frac {\left (\sqrt {11-2 \sqrt {5}}-\sqrt {11+2 \sqrt {5}}\right ) \left (2 x+\sqrt {11-2 \sqrt {5}}-1\right )}{\left (\sqrt {11-2 \sqrt {5}}+\sqrt {11+2 \sqrt {5}}\right ) \left (-2 x+\sqrt {11-2 \sqrt {5}}+1\right )}}\right )|\frac {1}{20} \left (11+\sqrt {101}\right )^2\right )-2 \sqrt {2} \sqrt {-\frac {\left (2 \sqrt {5}-11\right ) \left (\sqrt {101}-11\right ) \left (2 x+\sqrt {11-2 \sqrt {5}}-1\right )}{-2 x+\sqrt {11-2 \sqrt {5}}+1}} \Pi \left (-\frac {\sqrt {11-2 \sqrt {5}}+\sqrt {11+2 \sqrt {5}}}{\sqrt {11-2 \sqrt {5}}-\sqrt {11+2 \sqrt {5}}};\sin ^{-1}\left (\sqrt {\frac {\left (\sqrt {11-2 \sqrt {5}}-\sqrt {11+2 \sqrt {5}}\right ) \left (2 x+\sqrt {11-2 \sqrt {5}}-1\right )}{\left (\sqrt {11-2 \sqrt {5}}+\sqrt {11+2 \sqrt {5}}\right ) \left (-2 x+\sqrt {11-2 \sqrt {5}}+1\right )}}\right )|\frac {1}{20} \left (11+\sqrt {101}\right )^2\right )\right )}{\left (\sqrt {11+2 \sqrt {5}}-\sqrt {11-2 \sqrt {5}}\right )^{3/2} \sqrt {x^4-2 x^3-4 x^2+5 x+5}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(1 - 2*x)/Sqrt[5 + 5*x - 4*x^2 - 2*x^3 + x^4],x]

[Out]

((1 + Sqrt[11 - 2*Sqrt[5]] - 2*x)^2*Sqrt[(1 + Sqrt[11 + 2*Sqrt[5]] - 2*x)/(1 + Sqrt[11 - 2*Sqrt[5]] - 2*x)]*Sq
rt[-((-1 + Sqrt[11 + 2*Sqrt[5]] + 2*x)/((Sqrt[11 - 2*Sqrt[5]] - Sqrt[11 + 2*Sqrt[5]])*(1 + Sqrt[11 - 2*Sqrt[5]
] - 2*x)))]*((-(Sqrt[11 + 2*Sqrt[5]]*Sqrt[-((-1 + Sqrt[11 - 2*Sqrt[5]] + 2*x)/(1 + Sqrt[11 - 2*Sqrt[5]] - 2*x)
)]) + Sqrt[((-11 + 2*Sqrt[5])*(-1 + Sqrt[11 - 2*Sqrt[5]] + 2*x))/(1 + Sqrt[11 - 2*Sqrt[5]] - 2*x)] + Sqrt[2]*S
qrt[((-11 + Sqrt[101])*(-1 + Sqrt[11 - 2*Sqrt[5]] + 2*x))/(1 + Sqrt[11 - 2*Sqrt[5]] - 2*x)] + Sqrt[2]*Sqrt[-((
(-11 + 2*Sqrt[5])*(-11 + Sqrt[101])*(-1 + Sqrt[11 - 2*Sqrt[5]] + 2*x))/(1 + Sqrt[11 - 2*Sqrt[5]] - 2*x))])*Ell
ipticF[ArcSin[Sqrt[((Sqrt[11 - 2*Sqrt[5]] - Sqrt[11 + 2*Sqrt[5]])*(-1 + Sqrt[11 - 2*Sqrt[5]] + 2*x))/((Sqrt[11
- 2*Sqrt[5]] + Sqrt[11 + 2*Sqrt[5]])*(1 + Sqrt[11 - 2*Sqrt[5]] - 2*x))]], (11 + Sqrt[101])^2/20] - 2*Sqrt[2]*
Sqrt[-(((-11 + 2*Sqrt[5])*(-11 + Sqrt[101])*(-1 + Sqrt[11 - 2*Sqrt[5]] + 2*x))/(1 + Sqrt[11 - 2*Sqrt[5]] - 2*x
))]*EllipticPi[-((Sqrt[11 - 2*Sqrt[5]] + Sqrt[11 + 2*Sqrt[5]])/(Sqrt[11 - 2*Sqrt[5]] - Sqrt[11 + 2*Sqrt[5]])),
ArcSin[Sqrt[((Sqrt[11 - 2*Sqrt[5]] - Sqrt[11 + 2*Sqrt[5]])*(-1 + Sqrt[11 - 2*Sqrt[5]] + 2*x))/((Sqrt[11 - 2*S
qrt[5]] + Sqrt[11 + 2*Sqrt[5]])*(1 + Sqrt[11 - 2*Sqrt[5]] - 2*x))]], (11 + Sqrt[101])^2/20]))/((-Sqrt[11 - 2*S
qrt[5]] + Sqrt[11 + 2*Sqrt[5]])^(3/2)*Sqrt[5 + 5*x - 4*x^2 - 2*x^3 + x^4])

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.23, size = 35, normalized size = 1.00 \begin {gather*} \log \left (5+2 x-2 x^2+2 \sqrt {5+5 x-4 x^2-2 x^3+x^4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(1 - 2*x)/Sqrt[5 + 5*x - 4*x^2 - 2*x^3 + x^4],x]

[Out]

Log[5 + 2*x - 2*x^2 + 2*Sqrt[5 + 5*x - 4*x^2 - 2*x^3 + x^4]]

________________________________________________________________________________________

fricas [A]  time = 0.51, size = 33, normalized size = 0.94 \begin {gather*} \log \left (-2 \, x^{2} + 2 \, x + 2 \, \sqrt {x^{4} - 2 \, x^{3} - 4 \, x^{2} + 5 \, x + 5} + 5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)/(x^4-2*x^3-4*x^2+5*x+5)^(1/2),x, algorithm="fricas")

[Out]

log(-2*x^2 + 2*x + 2*sqrt(x^4 - 2*x^3 - 4*x^2 + 5*x + 5) + 5)

________________________________________________________________________________________

giac [A]  time = 0.46, size = 35, normalized size = 1.00 \begin {gather*} \log \left ({\left | -2 \, x^{2} + 2 \, x + 2 \, \sqrt {{\left (x^{2} - x\right )}^{2} - 5 \, x^{2} + 5 \, x + 5} + 5 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)/(x^4-2*x^3-4*x^2+5*x+5)^(1/2),x, algorithm="giac")

[Out]

log(abs(-2*x^2 + 2*x + 2*sqrt((x^2 - x)^2 - 5*x^2 + 5*x + 5) + 5))

________________________________________________________________________________________

maple [A]  time = 0.49, size = 36, normalized size = 1.03

method	result	size
trager	\(-\ln \left (2 x^{2}+2 \sqrt {x^{4}-2 x^{3}-4 x^{2}+5 x +5}-2 x -5\right )\)	\(36\)
default	\(\frac {2 \left (-\frac {\sqrt {11-2 \sqrt {5}}}{2}-\frac {\sqrt {11+2 \sqrt {5}}}{2}\right ) \sqrt {\frac {\left (\frac {\sqrt {11+2 \sqrt {5}}}{2}-\frac {\sqrt {11-2 \sqrt {5}}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {11-2 \sqrt {5}}}{2}\right )}{\left (\frac {\sqrt {11+2 \sqrt {5}}}{2}+\frac {\sqrt {11-2 \sqrt {5}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {11-2 \sqrt {5}}}{2}\right )}}\, \left (x -\frac {1}{2}-\frac {\sqrt {11-2 \sqrt {5}}}{2}\right )^{2} \sqrt {-\frac {\left (x -\frac {1}{2}+\frac {\sqrt {11+2 \sqrt {5}}}{2}\right ) \sqrt {11-2 \sqrt {5}}}{\left (\frac {\sqrt {11+2 \sqrt {5}}}{2}-\frac {\sqrt {11-2 \sqrt {5}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {11-2 \sqrt {5}}}{2}\right )}}\, \sqrt {\frac {\sqrt {11-2 \sqrt {5}}\, \left (x -\frac {1}{2}-\frac {\sqrt {11+2 \sqrt {5}}}{2}\right )}{\left (\frac {\sqrt {11+2 \sqrt {5}}}{2}+\frac {\sqrt {11-2 \sqrt {5}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {11-2 \sqrt {5}}}{2}\right )}}\, \EllipticF \left (\sqrt {\frac {\left (\frac {\sqrt {11+2 \sqrt {5}}}{2}-\frac {\sqrt {11-2 \sqrt {5}}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {11-2 \sqrt {5}}}{2}\right )}{\left (\frac {\sqrt {11+2 \sqrt {5}}}{2}+\frac {\sqrt {11-2 \sqrt {5}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {11-2 \sqrt {5}}}{2}\right )}}, \sqrt {\frac {\left (\frac {\sqrt {11+2 \sqrt {5}}}{2}+\frac {\sqrt {11-2 \sqrt {5}}}{2}\right )^{2}}{\left (\frac {\sqrt {11+2 \sqrt {5}}}{2}-\frac {\sqrt {11-2 \sqrt {5}}}{2}\right )^{2}}}\right )}{\left (\frac {\sqrt {11+2 \sqrt {5}}}{2}-\frac {\sqrt {11-2 \sqrt {5}}}{2}\right ) \sqrt {11-2 \sqrt {5}}\, \sqrt {\left (x -\frac {1}{2}+\frac {\sqrt {11-2 \sqrt {5}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {11-2 \sqrt {5}}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {11+2 \sqrt {5}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {11+2 \sqrt {5}}}{2}\right )}}-\frac {4 \left (-\frac {\sqrt {11-2 \sqrt {5}}}{2}-\frac {\sqrt {11+2 \sqrt {5}}}{2}\right ) \sqrt {\frac {\left (\frac {\sqrt {11+2 \sqrt {5}}}{2}-\frac {\sqrt {11-2 \sqrt {5}}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {11-2 \sqrt {5}}}{2}\right )}{\left (\frac {\sqrt {11+2 \sqrt {5}}}{2}+\frac {\sqrt {11-2 \sqrt {5}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {11-2 \sqrt {5}}}{2}\right )}}\, \left (x -\frac {1}{2}-\frac {\sqrt {11-2 \sqrt {5}}}{2}\right )^{2} \sqrt {-\frac {\left (x -\frac {1}{2}+\frac {\sqrt {11+2 \sqrt {5}}}{2}\right ) \sqrt {11-2 \sqrt {5}}}{\left (\frac {\sqrt {11+2 \sqrt {5}}}{2}-\frac {\sqrt {11-2 \sqrt {5}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {11-2 \sqrt {5}}}{2}\right )}}\, \sqrt {\frac {\sqrt {11-2 \sqrt {5}}\, \left (x -\frac {1}{2}-\frac {\sqrt {11+2 \sqrt {5}}}{2}\right )}{\left (\frac {\sqrt {11+2 \sqrt {5}}}{2}+\frac {\sqrt {11-2 \sqrt {5}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {11-2 \sqrt {5}}}{2}\right )}}\, \left (\left (\frac {1}{2}+\frac {\sqrt {11-2 \sqrt {5}}}{2}\right ) \EllipticF \left (\sqrt {\frac {\left (\frac {\sqrt {11+2 \sqrt {5}}}{2}-\frac {\sqrt {11-2 \sqrt {5}}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {11-2 \sqrt {5}}}{2}\right )}{\left (\frac {\sqrt {11+2 \sqrt {5}}}{2}+\frac {\sqrt {11-2 \sqrt {5}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {11-2 \sqrt {5}}}{2}\right )}}, \sqrt {\frac {\left (\frac {\sqrt {11+2 \sqrt {5}}}{2}+\frac {\sqrt {11-2 \sqrt {5}}}{2}\right )^{2}}{\left (\frac {\sqrt {11+2 \sqrt {5}}}{2}-\frac {\sqrt {11-2 \sqrt {5}}}{2}\right )^{2}}}\right )-\sqrt {11-2 \sqrt {5}}\, \EllipticPi \left (\sqrt {\frac {\left (\frac {\sqrt {11+2 \sqrt {5}}}{2}-\frac {\sqrt {11-2 \sqrt {5}}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {11-2 \sqrt {5}}}{2}\right )}{\left (\frac {\sqrt {11+2 \sqrt {5}}}{2}+\frac {\sqrt {11-2 \sqrt {5}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {11-2 \sqrt {5}}}{2}\right )}}, \frac {\frac {\sqrt {11+2 \sqrt {5}}}{2}+\frac {\sqrt {11-2 \sqrt {5}}}{2}}{\frac {\sqrt {11+2 \sqrt {5}}}{2}-\frac {\sqrt {11-2 \sqrt {5}}}{2}}, \sqrt {\frac {\left (\frac {\sqrt {11+2 \sqrt {5}}}{2}+\frac {\sqrt {11-2 \sqrt {5}}}{2}\right )^{2}}{\left (\frac {\sqrt {11+2 \sqrt {5}}}{2}-\frac {\sqrt {11-2 \sqrt {5}}}{2}\right )^{2}}}\right )\right )}{\left (\frac {\sqrt {11+2 \sqrt {5}}}{2}-\frac {\sqrt {11-2 \sqrt {5}}}{2}\right ) \sqrt {11-2 \sqrt {5}}\, \sqrt {\left (x -\frac {1}{2}+\frac {\sqrt {11-2 \sqrt {5}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {11-2 \sqrt {5}}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {11+2 \sqrt {5}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {11+2 \sqrt {5}}}{2}\right )}}\)	\(1182\)
elliptic	\(\frac {2 \left (-\frac {\sqrt {11-2 \sqrt {5}}}{2}-\frac {\sqrt {11+2 \sqrt {5}}}{2}\right ) \sqrt {\frac {\left (\frac {\sqrt {11+2 \sqrt {5}}}{2}-\frac {\sqrt {11-2 \sqrt {5}}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {11-2 \sqrt {5}}}{2}\right )}{\left (\frac {\sqrt {11+2 \sqrt {5}}}{2}+\frac {\sqrt {11-2 \sqrt {5}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {11-2 \sqrt {5}}}{2}\right )}}\, \left (x -\frac {1}{2}-\frac {\sqrt {11-2 \sqrt {5}}}{2}\right )^{2} \sqrt {-\frac {\left (x -\frac {1}{2}+\frac {\sqrt {11+2 \sqrt {5}}}{2}\right ) \sqrt {11-2 \sqrt {5}}}{\left (\frac {\sqrt {11+2 \sqrt {5}}}{2}-\frac {\sqrt {11-2 \sqrt {5}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {11-2 \sqrt {5}}}{2}\right )}}\, \sqrt {\frac {\sqrt {11-2 \sqrt {5}}\, \left (x -\frac {1}{2}-\frac {\sqrt {11+2 \sqrt {5}}}{2}\right )}{\left (\frac {\sqrt {11+2 \sqrt {5}}}{2}+\frac {\sqrt {11-2 \sqrt {5}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {11-2 \sqrt {5}}}{2}\right )}}\, \EllipticF \left (\sqrt {\frac {\left (\frac {\sqrt {11+2 \sqrt {5}}}{2}-\frac {\sqrt {11-2 \sqrt {5}}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {11-2 \sqrt {5}}}{2}\right )}{\left (\frac {\sqrt {11+2 \sqrt {5}}}{2}+\frac {\sqrt {11-2 \sqrt {5}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {11-2 \sqrt {5}}}{2}\right )}}, \sqrt {\frac {\left (\frac {\sqrt {11+2 \sqrt {5}}}{2}+\frac {\sqrt {11-2 \sqrt {5}}}{2}\right )^{2}}{\left (\frac {\sqrt {11+2 \sqrt {5}}}{2}-\frac {\sqrt {11-2 \sqrt {5}}}{2}\right )^{2}}}\right )}{\left (\frac {\sqrt {11+2 \sqrt {5}}}{2}-\frac {\sqrt {11-2 \sqrt {5}}}{2}\right ) \sqrt {11-2 \sqrt {5}}\, \sqrt {\left (x -\frac {1}{2}+\frac {\sqrt {11-2 \sqrt {5}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {11-2 \sqrt {5}}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {11+2 \sqrt {5}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {11+2 \sqrt {5}}}{2}\right )}}-\frac {4 \left (-\frac {\sqrt {11-2 \sqrt {5}}}{2}-\frac {\sqrt {11+2 \sqrt {5}}}{2}\right ) \sqrt {\frac {\left (\frac {\sqrt {11+2 \sqrt {5}}}{2}-\frac {\sqrt {11-2 \sqrt {5}}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {11-2 \sqrt {5}}}{2}\right )}{\left (\frac {\sqrt {11+2 \sqrt {5}}}{2}+\frac {\sqrt {11-2 \sqrt {5}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {11-2 \sqrt {5}}}{2}\right )}}\, \left (x -\frac {1}{2}-\frac {\sqrt {11-2 \sqrt {5}}}{2}\right )^{2} \sqrt {-\frac {\left (x -\frac {1}{2}+\frac {\sqrt {11+2 \sqrt {5}}}{2}\right ) \sqrt {11-2 \sqrt {5}}}{\left (\frac {\sqrt {11+2 \sqrt {5}}}{2}-\frac {\sqrt {11-2 \sqrt {5}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {11-2 \sqrt {5}}}{2}\right )}}\, \sqrt {\frac {\sqrt {11-2 \sqrt {5}}\, \left (x -\frac {1}{2}-\frac {\sqrt {11+2 \sqrt {5}}}{2}\right )}{\left (\frac {\sqrt {11+2 \sqrt {5}}}{2}+\frac {\sqrt {11-2 \sqrt {5}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {11-2 \sqrt {5}}}{2}\right )}}\, \left (\left (\frac {1}{2}+\frac {\sqrt {11-2 \sqrt {5}}}{2}\right ) \EllipticF \left (\sqrt {\frac {\left (\frac {\sqrt {11+2 \sqrt {5}}}{2}-\frac {\sqrt {11-2 \sqrt {5}}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {11-2 \sqrt {5}}}{2}\right )}{\left (\frac {\sqrt {11+2 \sqrt {5}}}{2}+\frac {\sqrt {11-2 \sqrt {5}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {11-2 \sqrt {5}}}{2}\right )}}, \sqrt {\frac {\left (\frac {\sqrt {11+2 \sqrt {5}}}{2}+\frac {\sqrt {11-2 \sqrt {5}}}{2}\right )^{2}}{\left (\frac {\sqrt {11+2 \sqrt {5}}}{2}-\frac {\sqrt {11-2 \sqrt {5}}}{2}\right )^{2}}}\right )-\sqrt {11-2 \sqrt {5}}\, \EllipticPi \left (\sqrt {\frac {\left (\frac {\sqrt {11+2 \sqrt {5}}}{2}-\frac {\sqrt {11-2 \sqrt {5}}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {11-2 \sqrt {5}}}{2}\right )}{\left (\frac {\sqrt {11+2 \sqrt {5}}}{2}+\frac {\sqrt {11-2 \sqrt {5}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {11-2 \sqrt {5}}}{2}\right )}}, \frac {\frac {\sqrt {11+2 \sqrt {5}}}{2}+\frac {\sqrt {11-2 \sqrt {5}}}{2}}{\frac {\sqrt {11+2 \sqrt {5}}}{2}-\frac {\sqrt {11-2 \sqrt {5}}}{2}}, \sqrt {\frac {\left (\frac {\sqrt {11+2 \sqrt {5}}}{2}+\frac {\sqrt {11-2 \sqrt {5}}}{2}\right )^{2}}{\left (\frac {\sqrt {11+2 \sqrt {5}}}{2}-\frac {\sqrt {11-2 \sqrt {5}}}{2}\right )^{2}}}\right )\right )}{\left (\frac {\sqrt {11+2 \sqrt {5}}}{2}-\frac {\sqrt {11-2 \sqrt {5}}}{2}\right ) \sqrt {11-2 \sqrt {5}}\, \sqrt {\left (x -\frac {1}{2}+\frac {\sqrt {11-2 \sqrt {5}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {11-2 \sqrt {5}}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {11+2 \sqrt {5}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {11+2 \sqrt {5}}}{2}\right )}}\)	\(1182\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)/(x^4-2*x^3-4*x^2+5*x+5)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-ln(2*x^2+2*(x^4-2*x^3-4*x^2+5*x+5)^(1/2)-2*x-5)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {2 \, x - 1}{\sqrt {x^{4} - 2 \, x^{3} - 4 \, x^{2} + 5 \, x + 5}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)/(x^4-2*x^3-4*x^2+5*x+5)^(1/2),x, algorithm="maxima")

[Out]

-integrate((2*x - 1)/sqrt(x^4 - 2*x^3 - 4*x^2 + 5*x + 5), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int -\frac {2\,x-1}{\sqrt {x^4-2\,x^3-4\,x^2+5\,x+5}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*x - 1)/(5*x - 4*x^2 - 2*x^3 + x^4 + 5)^(1/2),x)

[Out]

int(-(2*x - 1)/(5*x - 4*x^2 - 2*x^3 + x^4 + 5)^(1/2), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {2 x}{\sqrt {x^{4} - 2 x^{3} - 4 x^{2} + 5 x + 5}}\, dx - \int \left (- \frac {1}{\sqrt {x^{4} - 2 x^{3} - 4 x^{2} + 5 x + 5}}\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)/(x**4-2*x**3-4*x**2+5*x+5)**(1/2),x)

[Out]

-Integral(2*x/sqrt(x**4 - 2*x**3 - 4*x**2 + 5*x + 5), x) - Integral(-1/sqrt(x**4 - 2*x**3 - 4*x**2 + 5*x + 5),
x)

________________________________________________________________________________________