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

Optimal. Leaf size=29 \[ \log \left (x^2+\sqrt {x^4+2 x^3-3 x^2-4 x-4}+x-2\right ) \]

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Rubi [A]  time = 0.05, antiderivative size = 39, normalized size of antiderivative = 1.34, 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 {9-4 \left (x+\frac {1}{2}\right )^2}{\sqrt {16 \left (x+\frac {1}{2}\right )^4-72 \left (x+\frac {1}{2}\right )^2-47}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-ArcTanh[(9 - 4*(1/2 + x)^2)/Sqrt[-47 - 72*(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 {-4-4 x-3 x^2+2 x^3+x^4}} \, dx &=\operatorname {Subst}\left (\int \frac {8 x}{\sqrt {-47-72 x^2+16 x^4}} \, dx,x,\frac {1}{2}+x\right )\\ &=8 \operatorname {Subst}\left (\int \frac {x}{\sqrt {-47-72 x^2+16 x^4}} \, dx,x,\frac {1}{2}+x\right )\\ &=4 \operatorname {Subst}\left (\int \frac {1}{\sqrt {-47-72 x+16 x^2}} \, dx,x,\left (\frac {1}{2}+x\right )^2\right )\\ &=8 \operatorname {Subst}\left (\int \frac {1}{64-x^2} \, dx,x,\frac {32 \left (-2+x+x^2\right )}{\sqrt {-47-72 \left (\frac {1}{2}+x\right )^2+(1+2 x)^4}}\right )\\ &=-\tanh ^{-1}\left (\frac {4 \left (2-x-x^2\right )}{\sqrt {-47-18 (1+2 x)^2+(1+2 x)^4}}\right )\\ \end {align*}

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Mathematica [C]  time = 2.48, size = 736, normalized size = 25.38 \begin {gather*} \frac {i \sqrt {8 \sqrt {2}-9} \left (-2 x+\sqrt {9+8 \sqrt {2}}-1\right ) \sqrt {\frac {\left (\sqrt {8 \sqrt {2}-9}+i \sqrt {9+8 \sqrt {2}}\right ) \left (-2 i x+\sqrt {8 \sqrt {2}-9}-i\right )}{\left (\sqrt {8 \sqrt {2}-9}-i \sqrt {9+8 \sqrt {2}}\right ) \left (2 i x+\sqrt {8 \sqrt {2}-9}+i\right )}} \left (2 i x+\sqrt {8 \sqrt {2}-9}+i\right ) \sqrt {\frac {2 x+\sqrt {9+8 \sqrt {2}}+1}{\left (\sqrt {9+8 \sqrt {2}}-i \sqrt {8 \sqrt {2}-9}\right ) \left (2 i x+\sqrt {8 \sqrt {2}-9}+i\right )}} \left (F\left (\sin ^{-1}\left (\sqrt {\frac {\left (\sqrt {-9+8 \sqrt {2}}+i \sqrt {9+8 \sqrt {2}}\right ) \left (-2 i x+\sqrt {-9+8 \sqrt {2}}-i\right )}{\left (\sqrt {-9+8 \sqrt {2}}-i \sqrt {9+8 \sqrt {2}}\right ) \left (2 i x+\sqrt {-9+8 \sqrt {2}}+i\right )}}\right )|\frac {9 i-\sqrt {47}}{9 i+\sqrt {47}}\right )-2 \Pi \left (-\frac {\sqrt {-9+8 \sqrt {2}}-i \sqrt {9+8 \sqrt {2}}}{\sqrt {-9+8 \sqrt {2}}+i \sqrt {9+8 \sqrt {2}}};\sin ^{-1}\left (\sqrt {\frac {\left (\sqrt {-9+8 \sqrt {2}}+i \sqrt {9+8 \sqrt {2}}\right ) \left (-2 i x+\sqrt {-9+8 \sqrt {2}}-i\right )}{\left (\sqrt {-9+8 \sqrt {2}}-i \sqrt {9+8 \sqrt {2}}\right ) \left (2 i x+\sqrt {-9+8 \sqrt {2}}+i\right )}}\right )|\frac {9 i-\sqrt {47}}{9 i+\sqrt {47}}\right )\right )}{\left (\sqrt {8 \sqrt {2}-9}+i \sqrt {9+8 \sqrt {2}}\right ) \sqrt {\frac {-2 x+\sqrt {9+8 \sqrt {2}}-1}{\left (\sqrt {9+8 \sqrt {2}}+i \sqrt {8 \sqrt {2}-9}\right ) \left (2 i x+\sqrt {8 \sqrt {2}-9}+i\right )}} \sqrt {x^4+2 x^3-3 x^2-4 x-4}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(I*Sqrt[-9 + 8*Sqrt[2]]*(-1 + Sqrt[9 + 8*Sqrt[2]] - 2*x)*Sqrt[((Sqrt[-9 + 8*Sqrt[2]] + I*Sqrt[9 + 8*Sqrt[2]])*
(-I + Sqrt[-9 + 8*Sqrt[2]] - (2*I)*x))/((Sqrt[-9 + 8*Sqrt[2]] - I*Sqrt[9 + 8*Sqrt[2]])*(I + Sqrt[-9 + 8*Sqrt[2
]] + (2*I)*x))]*(I + Sqrt[-9 + 8*Sqrt[2]] + (2*I)*x)*Sqrt[(1 + Sqrt[9 + 8*Sqrt[2]] + 2*x)/(((-I)*Sqrt[-9 + 8*S
qrt[2]] + Sqrt[9 + 8*Sqrt[2]])*(I + Sqrt[-9 + 8*Sqrt[2]] + (2*I)*x))]*(EllipticF[ArcSin[Sqrt[((Sqrt[-9 + 8*Sqr
t[2]] + I*Sqrt[9 + 8*Sqrt[2]])*(-I + Sqrt[-9 + 8*Sqrt[2]] - (2*I)*x))/((Sqrt[-9 + 8*Sqrt[2]] - I*Sqrt[9 + 8*Sq
rt[2]])*(I + Sqrt[-9 + 8*Sqrt[2]] + (2*I)*x))]], (9*I - Sqrt[47])/(9*I + Sqrt[47])] - 2*EllipticPi[-((Sqrt[-9
+ 8*Sqrt[2]] - I*Sqrt[9 + 8*Sqrt[2]])/(Sqrt[-9 + 8*Sqrt[2]] + I*Sqrt[9 + 8*Sqrt[2]])), ArcSin[Sqrt[((Sqrt[-9 +
8*Sqrt[2]] + I*Sqrt[9 + 8*Sqrt[2]])*(-I + Sqrt[-9 + 8*Sqrt[2]] - (2*I)*x))/((Sqrt[-9 + 8*Sqrt[2]] - I*Sqrt[9
+ 8*Sqrt[2]])*(I + Sqrt[-9 + 8*Sqrt[2]] + (2*I)*x))]], (9*I - Sqrt[47])/(9*I + Sqrt[47])]))/((Sqrt[-9 + 8*Sqrt
[2]] + I*Sqrt[9 + 8*Sqrt[2]])*Sqrt[(-1 + Sqrt[9 + 8*Sqrt[2]] - 2*x)/((I*Sqrt[-9 + 8*Sqrt[2]] + Sqrt[9 + 8*Sqrt
[2]])*(I + Sqrt[-9 + 8*Sqrt[2]] + (2*I)*x))]*Sqrt[-4 - 4*x - 3*x^2 + 2*x^3 + x^4])

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IntegrateAlgebraic [A]  time = 0.15, size = 29, normalized size = 1.00 \begin {gather*} \log \left (-2+x+x^2+\sqrt {-4-4 x-3 x^2+2 x^3+x^4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.50, size = 27, normalized size = 0.93 \begin {gather*} \log \left (x^{2} + x + \sqrt {x^{4} + 2 \, x^{3} - 3 \, x^{2} - 4 \, x - 4} - 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.38, size = 33, normalized size = 1.14 \begin {gather*} -\log \left ({\left | -x^{2} - x + \sqrt {{\left (x^{2} + x\right )}^{2} - 4 \, x^{2} - 4 \, x - 4} + 2 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(abs(-x^2 - x + sqrt((x^2 + x)^2 - 4*x^2 - 4*x - 4) + 2))

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maple [A]  time = 0.73, size = 34, normalized size = 1.17

method	result	size
trager	\(-\ln \left (-x^{2}+\sqrt {x^{4}+2 x^{3}-3 x^{2}-4 x -4}-x +2\right )\)	\(34\)
default	\(-\frac {2 i \left (-\frac {i \sqrt {-9+8 \sqrt {2}}}{2}-\frac {\sqrt {9+8 \sqrt {2}}}{2}\right ) \sqrt {\frac {\left (\frac {\sqrt {9+8 \sqrt {2}}}{2}-\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right ) \left (x +\frac {1}{2}+\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right )}{\left (\frac {\sqrt {9+8 \sqrt {2}}}{2}+\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right )}}\, \left (x +\frac {1}{2}-\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right )^{2} \sqrt {\frac {i \sqrt {-9+8 \sqrt {2}}\, \left (x +\frac {1}{2}+\frac {\sqrt {9+8 \sqrt {2}}}{2}\right )}{\left (-\frac {\sqrt {9+8 \sqrt {2}}}{2}+\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right )}}\, \sqrt {\frac {i \sqrt {-9+8 \sqrt {2}}\, \left (x +\frac {1}{2}-\frac {\sqrt {9+8 \sqrt {2}}}{2}\right )}{\left (\frac {\sqrt {9+8 \sqrt {2}}}{2}+\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right )}}\, \EllipticF \left (\sqrt {\frac {\left (\frac {\sqrt {9+8 \sqrt {2}}}{2}-\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right ) \left (x +\frac {1}{2}+\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right )}{\left (\frac {\sqrt {9+8 \sqrt {2}}}{2}+\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right )}}, \sqrt {\frac {\left (\frac {\sqrt {9+8 \sqrt {2}}}{2}+\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right ) \left (-\frac {i \sqrt {-9+8 \sqrt {2}}}{2}-\frac {\sqrt {9+8 \sqrt {2}}}{2}\right )}{\left (\frac {\sqrt {9+8 \sqrt {2}}}{2}-\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right ) \left (-\frac {\sqrt {9+8 \sqrt {2}}}{2}+\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right )}}\right )}{\left (\frac {\sqrt {9+8 \sqrt {2}}}{2}-\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right ) \sqrt {-9+8 \sqrt {2}}\, \sqrt {\left (x +\frac {1}{2}+\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right ) \left (x +\frac {1}{2}+\frac {\sqrt {9+8 \sqrt {2}}}{2}\right ) \left (x +\frac {1}{2}-\frac {\sqrt {9+8 \sqrt {2}}}{2}\right )}}-\frac {4 i \left (-\frac {i \sqrt {-9+8 \sqrt {2}}}{2}-\frac {\sqrt {9+8 \sqrt {2}}}{2}\right ) \sqrt {\frac {\left (\frac {\sqrt {9+8 \sqrt {2}}}{2}-\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right ) \left (x +\frac {1}{2}+\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right )}{\left (\frac {\sqrt {9+8 \sqrt {2}}}{2}+\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right )}}\, \left (x +\frac {1}{2}-\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right )^{2} \sqrt {\frac {i \sqrt {-9+8 \sqrt {2}}\, \left (x +\frac {1}{2}+\frac {\sqrt {9+8 \sqrt {2}}}{2}\right )}{\left (-\frac {\sqrt {9+8 \sqrt {2}}}{2}+\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right )}}\, \sqrt {\frac {i \sqrt {-9+8 \sqrt {2}}\, \left (x +\frac {1}{2}-\frac {\sqrt {9+8 \sqrt {2}}}{2}\right )}{\left (\frac {\sqrt {9+8 \sqrt {2}}}{2}+\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right )}}\, \left (\left (-\frac {1}{2}+\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right ) \EllipticF \left (\sqrt {\frac {\left (\frac {\sqrt {9+8 \sqrt {2}}}{2}-\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right ) \left (x +\frac {1}{2}+\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right )}{\left (\frac {\sqrt {9+8 \sqrt {2}}}{2}+\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right )}}, \sqrt {\frac {\left (\frac {\sqrt {9+8 \sqrt {2}}}{2}+\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right ) \left (-\frac {i \sqrt {-9+8 \sqrt {2}}}{2}-\frac {\sqrt {9+8 \sqrt {2}}}{2}\right )}{\left (\frac {\sqrt {9+8 \sqrt {2}}}{2}-\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right ) \left (-\frac {\sqrt {9+8 \sqrt {2}}}{2}+\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right )}}\right )-i \sqrt {-9+8 \sqrt {2}}\, \EllipticPi \left (\sqrt {\frac {\left (\frac {\sqrt {9+8 \sqrt {2}}}{2}-\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right ) \left (x +\frac {1}{2}+\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right )}{\left (\frac {\sqrt {9+8 \sqrt {2}}}{2}+\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right )}}, \frac {\frac {\sqrt {9+8 \sqrt {2}}}{2}+\frac {i \sqrt {-9+8 \sqrt {2}}}{2}}{\frac {\sqrt {9+8 \sqrt {2}}}{2}-\frac {i \sqrt {-9+8 \sqrt {2}}}{2}}, \sqrt {\frac {\left (\frac {\sqrt {9+8 \sqrt {2}}}{2}+\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right ) \left (-\frac {i \sqrt {-9+8 \sqrt {2}}}{2}-\frac {\sqrt {9+8 \sqrt {2}}}{2}\right )}{\left (\frac {\sqrt {9+8 \sqrt {2}}}{2}-\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right ) \left (-\frac {\sqrt {9+8 \sqrt {2}}}{2}+\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right )}}\right )\right )}{\left (\frac {\sqrt {9+8 \sqrt {2}}}{2}-\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right ) \sqrt {-9+8 \sqrt {2}}\, \sqrt {\left (x +\frac {1}{2}+\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right ) \left (x +\frac {1}{2}+\frac {\sqrt {9+8 \sqrt {2}}}{2}\right ) \left (x +\frac {1}{2}-\frac {\sqrt {9+8 \sqrt {2}}}{2}\right )}}\)	\(1382\)
elliptic	\(-\frac {2 i \left (-\frac {i \sqrt {-9+8 \sqrt {2}}}{2}-\frac {\sqrt {9+8 \sqrt {2}}}{2}\right ) \sqrt {\frac {\left (\frac {\sqrt {9+8 \sqrt {2}}}{2}-\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right ) \left (x +\frac {1}{2}+\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right )}{\left (\frac {\sqrt {9+8 \sqrt {2}}}{2}+\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right )}}\, \left (x +\frac {1}{2}-\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right )^{2} \sqrt {\frac {i \sqrt {-9+8 \sqrt {2}}\, \left (x +\frac {1}{2}+\frac {\sqrt {9+8 \sqrt {2}}}{2}\right )}{\left (-\frac {\sqrt {9+8 \sqrt {2}}}{2}+\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right )}}\, \sqrt {\frac {i \sqrt {-9+8 \sqrt {2}}\, \left (x +\frac {1}{2}-\frac {\sqrt {9+8 \sqrt {2}}}{2}\right )}{\left (\frac {\sqrt {9+8 \sqrt {2}}}{2}+\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right )}}\, \EllipticF \left (\sqrt {\frac {\left (\frac {\sqrt {9+8 \sqrt {2}}}{2}-\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right ) \left (x +\frac {1}{2}+\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right )}{\left (\frac {\sqrt {9+8 \sqrt {2}}}{2}+\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right )}}, \sqrt {\frac {\left (\frac {\sqrt {9+8 \sqrt {2}}}{2}+\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right ) \left (-\frac {i \sqrt {-9+8 \sqrt {2}}}{2}-\frac {\sqrt {9+8 \sqrt {2}}}{2}\right )}{\left (\frac {\sqrt {9+8 \sqrt {2}}}{2}-\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right ) \left (-\frac {\sqrt {9+8 \sqrt {2}}}{2}+\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right )}}\right )}{\left (\frac {\sqrt {9+8 \sqrt {2}}}{2}-\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right ) \sqrt {-9+8 \sqrt {2}}\, \sqrt {\left (x +\frac {1}{2}+\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right ) \left (x +\frac {1}{2}+\frac {\sqrt {9+8 \sqrt {2}}}{2}\right ) \left (x +\frac {1}{2}-\frac {\sqrt {9+8 \sqrt {2}}}{2}\right )}}-\frac {4 i \left (-\frac {i \sqrt {-9+8 \sqrt {2}}}{2}-\frac {\sqrt {9+8 \sqrt {2}}}{2}\right ) \sqrt {\frac {\left (\frac {\sqrt {9+8 \sqrt {2}}}{2}-\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right ) \left (x +\frac {1}{2}+\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right )}{\left (\frac {\sqrt {9+8 \sqrt {2}}}{2}+\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right )}}\, \left (x +\frac {1}{2}-\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right )^{2} \sqrt {\frac {i \sqrt {-9+8 \sqrt {2}}\, \left (x +\frac {1}{2}+\frac {\sqrt {9+8 \sqrt {2}}}{2}\right )}{\left (-\frac {\sqrt {9+8 \sqrt {2}}}{2}+\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right )}}\, \sqrt {\frac {i \sqrt {-9+8 \sqrt {2}}\, \left (x +\frac {1}{2}-\frac {\sqrt {9+8 \sqrt {2}}}{2}\right )}{\left (\frac {\sqrt {9+8 \sqrt {2}}}{2}+\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right )}}\, \left (\left (-\frac {1}{2}+\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right ) \EllipticF \left (\sqrt {\frac {\left (\frac {\sqrt {9+8 \sqrt {2}}}{2}-\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right ) \left (x +\frac {1}{2}+\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right )}{\left (\frac {\sqrt {9+8 \sqrt {2}}}{2}+\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right )}}, \sqrt {\frac {\left (\frac {\sqrt {9+8 \sqrt {2}}}{2}+\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right ) \left (-\frac {i \sqrt {-9+8 \sqrt {2}}}{2}-\frac {\sqrt {9+8 \sqrt {2}}}{2}\right )}{\left (\frac {\sqrt {9+8 \sqrt {2}}}{2}-\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right ) \left (-\frac {\sqrt {9+8 \sqrt {2}}}{2}+\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right )}}\right )-i \sqrt {-9+8 \sqrt {2}}\, \EllipticPi \left (\sqrt {\frac {\left (\frac {\sqrt {9+8 \sqrt {2}}}{2}-\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right ) \left (x +\frac {1}{2}+\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right )}{\left (\frac {\sqrt {9+8 \sqrt {2}}}{2}+\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right )}}, \frac {\frac {\sqrt {9+8 \sqrt {2}}}{2}+\frac {i \sqrt {-9+8 \sqrt {2}}}{2}}{\frac {\sqrt {9+8 \sqrt {2}}}{2}-\frac {i \sqrt {-9+8 \sqrt {2}}}{2}}, \sqrt {\frac {\left (\frac {\sqrt {9+8 \sqrt {2}}}{2}+\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right ) \left (-\frac {i \sqrt {-9+8 \sqrt {2}}}{2}-\frac {\sqrt {9+8 \sqrt {2}}}{2}\right )}{\left (\frac {\sqrt {9+8 \sqrt {2}}}{2}-\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right ) \left (-\frac {\sqrt {9+8 \sqrt {2}}}{2}+\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right )}}\right )\right )}{\left (\frac {\sqrt {9+8 \sqrt {2}}}{2}-\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right ) \sqrt {-9+8 \sqrt {2}}\, \sqrt {\left (x +\frac {1}{2}+\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {-9+8 \sqrt {2}}}{2}\right ) \left (x +\frac {1}{2}+\frac {\sqrt {9+8 \sqrt {2}}}{2}\right ) \left (x +\frac {1}{2}-\frac {\sqrt {9+8 \sqrt {2}}}{2}\right )}}\)	\(1382\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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