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

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

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Rubi [A]  time = 0.05, antiderivative size = 39, normalized size of antiderivative = 1.11, 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 {7-4 \left (x+\frac {1}{2}\right )^2}{\sqrt {16 \left (x+\frac {1}{2}\right )^4-56 \left (x+\frac {1}{2}\right )^2-51}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-ArcTanh[(7 - 4*(1/2 + x)^2)/Sqrt[-51 - 56*(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 {-4-3 x-2 x^2+2 x^3+x^4}} \, dx &=\operatorname {Subst}\left (\int \frac {8 x}{\sqrt {-51-56 x^2+16 x^4}} \, dx,x,\frac {1}{2}+x\right )\\ &=8 \operatorname {Subst}\left (\int \frac {x}{\sqrt {-51-56 x^2+16 x^4}} \, dx,x,\frac {1}{2}+x\right )\\ &=4 \operatorname {Subst}\left (\int \frac {1}{\sqrt {-51-56 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 {8 \left (-7+4 \left (\frac {1}{2}+x\right )^2\right )}{\sqrt {-51-56 \left (\frac {1}{2}+x\right )^2+(1+2 x)^4}}\right )\\ &=-\tanh ^{-1}\left (\frac {7-(1+2 x)^2}{\sqrt {-51-14 (1+2 x)^2+(1+2 x)^4}}\right )\\ \end {align*}

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Mathematica [C]  time = 0.98, size = 505, normalized size = 14.43 \begin {gather*} \frac {6 \sqrt [6]{-1} \left (-2 x+\sqrt {17}-1\right ) \left ((-1)^{2/3}-x\right ) \sqrt {\frac {i \left (2 x+\sqrt {17}+1\right )}{\left (\sqrt {17}-i \sqrt {3}\right ) \left ((-1)^{2/3}-x\right )}} \sqrt {\frac {-\sqrt {17} x+2 (-1)^{2/3} x+x-\sqrt [3]{-1} \sqrt {17}+\sqrt [3]{-1}-2}{\left (\sqrt {17}+i \sqrt {3}\right ) \left ((-1)^{2/3}-x\right )}} \left (F\left (\sin ^{-1}\left (\sqrt {\frac {2 i \sqrt {17} x+2 \sqrt {3} x-\sqrt {51}+i \sqrt {17}+\sqrt {3}+3 i}{2 i \sqrt {17} x-2 \sqrt {3} x+\sqrt {51}+i \sqrt {17}-\sqrt {3}+3 i}}\right )|\frac {7 i-\sqrt {51}}{7 i+\sqrt {51}}\right )-2 \Pi \left (-\frac {\sqrt {3}-i \sqrt {17}}{\sqrt {3}+i \sqrt {17}};\sin ^{-1}\left (\sqrt {\frac {2 i \sqrt {17} x+2 \sqrt {3} x-\sqrt {51}+i \sqrt {17}+\sqrt {3}+3 i}{2 i \sqrt {17} x-2 \sqrt {3} x+\sqrt {51}+i \sqrt {17}-\sqrt {3}+3 i}}\right )|\frac {7 i-\sqrt {51}}{7 i+\sqrt {51}}\right )\right )}{\left (1+\sqrt [3]{-1}\right ) \left (\sqrt {3}+i \sqrt {17}\right ) \sqrt {\frac {-2 x+\sqrt {17}-1}{\left (\sqrt {3}-i \sqrt {17}\right ) \left ((-1)^{2/3}-x\right )}} \sqrt {x^4+2 x^3-2 x^2-3 x-4}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(6*(-1)^(1/6)*(-1 + Sqrt[17] - 2*x)*((-1)^(2/3) - x)*Sqrt[(I*(1 + Sqrt[17] + 2*x))/(((-I)*Sqrt[3] + Sqrt[17])*
((-1)^(2/3) - x))]*Sqrt[(-2 + (-1)^(1/3) - (-1)^(1/3)*Sqrt[17] + x + 2*(-1)^(2/3)*x - Sqrt[17]*x)/((I*Sqrt[3]
+ Sqrt[17])*((-1)^(2/3) - x))]*(EllipticF[ArcSin[Sqrt[(3*I + Sqrt[3] + I*Sqrt[17] - Sqrt[51] + 2*Sqrt[3]*x + (
2*I)*Sqrt[17]*x)/(3*I - Sqrt[3] + I*Sqrt[17] + Sqrt[51] - 2*Sqrt[3]*x + (2*I)*Sqrt[17]*x)]], (7*I - Sqrt[51])/
(7*I + Sqrt[51])] - 2*EllipticPi[-((Sqrt[3] - I*Sqrt[17])/(Sqrt[3] + I*Sqrt[17])), ArcSin[Sqrt[(3*I + Sqrt[3]
+ I*Sqrt[17] - Sqrt[51] + 2*Sqrt[3]*x + (2*I)*Sqrt[17]*x)/(3*I - Sqrt[3] + I*Sqrt[17] + Sqrt[51] - 2*Sqrt[3]*x
+ (2*I)*Sqrt[17]*x)]], (7*I - Sqrt[51])/(7*I + Sqrt[51])]))/((1 + (-1)^(1/3))*(Sqrt[3] + I*Sqrt[17])*Sqrt[(-1
+ Sqrt[17] - 2*x)/((Sqrt[3] - I*Sqrt[17])*((-1)^(2/3) - x))]*Sqrt[-4 - 3*x - 2*x^2 + 2*x^3 + x^4])

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IntegrateAlgebraic [A]  time = 0.15, size = 35, normalized size = 1.00 \begin {gather*} \log \left (-3+2 x+2 x^2+2 \sqrt {-4-3 x-2 x^2+2 x^3+x^4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.49, size = 33, normalized size = 0.94 \begin {gather*} \log \left (2 \, x^{2} + 2 \, x + 2 \, \sqrt {x^{4} + 2 \, x^{3} - 2 \, x^{2} - 3 \, x - 4} - 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.73, size = 35, normalized size = 1.00 \begin {gather*} -\log \left ({\left | -2 \, x^{2} - 2 \, x + 2 \, \sqrt {{\left (x^{2} + x\right )}^{2} - 3 \, x^{2} - 3 \, x - 4} + 3 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.41, size = 36, normalized size = 1.03

method	result	size
trager	\(-\ln \left (-2 x^{2}+2 \sqrt {x^{4}+2 x^{3}-2 x^{2}-3 x -4}-2 x +3\right )\)	\(36\)
default	\(-\frac {2 i \left (-\frac {i \sqrt {3}}{2}-\frac {\sqrt {17}}{2}\right ) \sqrt {\frac {\left (\frac {\sqrt {17}}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (\frac {\sqrt {17}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )^{2} \sqrt {\frac {i \sqrt {3}\, \left (x +\frac {1}{2}+\frac {\sqrt {17}}{2}\right )}{\left (-\frac {\sqrt {17}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {i \sqrt {3}\, \left (x +\frac {1}{2}-\frac {\sqrt {17}}{2}\right )}{\left (\frac {\sqrt {17}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {3}\, \EllipticF \left (\sqrt {\frac {\left (\frac {\sqrt {17}}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (\frac {\sqrt {17}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}, \sqrt {\frac {\left (\frac {\sqrt {17}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {i \sqrt {3}}{2}-\frac {\sqrt {17}}{2}\right )}{\left (\frac {\sqrt {17}}{2}-\frac {i \sqrt {3}}{2}\right ) \left (-\frac {\sqrt {17}}{2}+\frac {i \sqrt {3}}{2}\right )}}\right )}{3 \left (\frac {\sqrt {17}}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}+\frac {\sqrt {17}}{2}\right ) \left (x +\frac {1}{2}-\frac {\sqrt {17}}{2}\right )}}-\frac {4 i \left (-\frac {i \sqrt {3}}{2}-\frac {\sqrt {17}}{2}\right ) \sqrt {\frac {\left (\frac {\sqrt {17}}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (\frac {\sqrt {17}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )^{2} \sqrt {\frac {i \sqrt {3}\, \left (x +\frac {1}{2}+\frac {\sqrt {17}}{2}\right )}{\left (-\frac {\sqrt {17}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {i \sqrt {3}\, \left (x +\frac {1}{2}-\frac {\sqrt {17}}{2}\right )}{\left (\frac {\sqrt {17}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {3}\, \left (\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \EllipticF \left (\sqrt {\frac {\left (\frac {\sqrt {17}}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (\frac {\sqrt {17}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}, \sqrt {\frac {\left (\frac {\sqrt {17}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {i \sqrt {3}}{2}-\frac {\sqrt {17}}{2}\right )}{\left (\frac {\sqrt {17}}{2}-\frac {i \sqrt {3}}{2}\right ) \left (-\frac {\sqrt {17}}{2}+\frac {i \sqrt {3}}{2}\right )}}\right )-i \sqrt {3}\, \EllipticPi \left (\sqrt {\frac {\left (\frac {\sqrt {17}}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (\frac {\sqrt {17}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}, \frac {\frac {\sqrt {17}}{2}+\frac {i \sqrt {3}}{2}}{\frac {\sqrt {17}}{2}-\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (\frac {\sqrt {17}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {i \sqrt {3}}{2}-\frac {\sqrt {17}}{2}\right )}{\left (\frac {\sqrt {17}}{2}-\frac {i \sqrt {3}}{2}\right ) \left (-\frac {\sqrt {17}}{2}+\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{3 \left (\frac {\sqrt {17}}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}+\frac {\sqrt {17}}{2}\right ) \left (x +\frac {1}{2}-\frac {\sqrt {17}}{2}\right )}}\)	\(782\)
elliptic	\(-\frac {2 i \left (-\frac {i \sqrt {3}}{2}-\frac {\sqrt {17}}{2}\right ) \sqrt {\frac {\left (\frac {\sqrt {17}}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (\frac {\sqrt {17}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )^{2} \sqrt {\frac {i \sqrt {3}\, \left (x +\frac {1}{2}+\frac {\sqrt {17}}{2}\right )}{\left (-\frac {\sqrt {17}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {i \sqrt {3}\, \left (x +\frac {1}{2}-\frac {\sqrt {17}}{2}\right )}{\left (\frac {\sqrt {17}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {3}\, \EllipticF \left (\sqrt {\frac {\left (\frac {\sqrt {17}}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (\frac {\sqrt {17}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}, \sqrt {\frac {\left (\frac {\sqrt {17}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {i \sqrt {3}}{2}-\frac {\sqrt {17}}{2}\right )}{\left (\frac {\sqrt {17}}{2}-\frac {i \sqrt {3}}{2}\right ) \left (-\frac {\sqrt {17}}{2}+\frac {i \sqrt {3}}{2}\right )}}\right )}{3 \left (\frac {\sqrt {17}}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}+\frac {\sqrt {17}}{2}\right ) \left (x +\frac {1}{2}-\frac {\sqrt {17}}{2}\right )}}-\frac {4 i \left (-\frac {i \sqrt {3}}{2}-\frac {\sqrt {17}}{2}\right ) \sqrt {\frac {\left (\frac {\sqrt {17}}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (\frac {\sqrt {17}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )^{2} \sqrt {\frac {i \sqrt {3}\, \left (x +\frac {1}{2}+\frac {\sqrt {17}}{2}\right )}{\left (-\frac {\sqrt {17}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {i \sqrt {3}\, \left (x +\frac {1}{2}-\frac {\sqrt {17}}{2}\right )}{\left (\frac {\sqrt {17}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {3}\, \left (\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \EllipticF \left (\sqrt {\frac {\left (\frac {\sqrt {17}}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (\frac {\sqrt {17}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}, \sqrt {\frac {\left (\frac {\sqrt {17}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {i \sqrt {3}}{2}-\frac {\sqrt {17}}{2}\right )}{\left (\frac {\sqrt {17}}{2}-\frac {i \sqrt {3}}{2}\right ) \left (-\frac {\sqrt {17}}{2}+\frac {i \sqrt {3}}{2}\right )}}\right )-i \sqrt {3}\, \EllipticPi \left (\sqrt {\frac {\left (\frac {\sqrt {17}}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (\frac {\sqrt {17}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}, \frac {\frac {\sqrt {17}}{2}+\frac {i \sqrt {3}}{2}}{\frac {\sqrt {17}}{2}-\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (\frac {\sqrt {17}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {i \sqrt {3}}{2}-\frac {\sqrt {17}}{2}\right )}{\left (\frac {\sqrt {17}}{2}-\frac {i \sqrt {3}}{2}\right ) \left (-\frac {\sqrt {17}}{2}+\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{3 \left (\frac {\sqrt {17}}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}+\frac {\sqrt {17}}{2}\right ) \left (x +\frac {1}{2}-\frac {\sqrt {17}}{2}\right )}}\)	\(782\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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} - 2 \, x^{2} - 3 \, 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^3-2*x^2-3*x-4)^(1/2),x, algorithm="maxima")

[Out]

integrate((2*x + 1)/sqrt(x^4 + 2*x^3 - 2*x^2 - 3*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^3-2\,x^2-3\,x-4}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((2*x + 1)/(2*x^3 - 2*x^2 - 3*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 {\left (x^{2} + x - 4\right ) \left (x^{2} + x + 1\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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