3.4.81 \(\int \frac {2+x}{\sqrt {13+16 x^2+8 x^3+x^4}} \, dx\)

Optimal. Leaf size=31 \[ \frac {1}{2} \log \left (x^2+\sqrt {x^4+8 x^3+16 x^2+13}+4 x\right ) \]

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Rubi [A]  time = 0.05, antiderivative size = 20, normalized size of antiderivative = 0.65, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {1680, 1107, 619, 215} \begin {gather*} -\frac {1}{2} \sinh ^{-1}\left (\frac {4-(x+2)^2}{\sqrt {13}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 + x)/Sqrt[13 + 16*x^2 + 8*x^3 + x^4],x]

[Out]

-1/2*ArcSinh[(4 - (2 + x)^2)/Sqrt[13]]

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rule 619

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*((-4*c)/(b^2 - 4*a*c))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/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 {2+x}{\sqrt {13+16 x^2+8 x^3+x^4}} \, dx &=\operatorname {Subst}\left (\int \frac {x}{\sqrt {29-8 x^2+x^4}} \, dx,x,2+x\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {29-8 x+x^2}} \, dx,x,(2+x)^2\right )\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{52}}} \, dx,x,2 x (4+x)\right )}{4 \sqrt {13}}\\ &=\frac {1}{2} \sinh ^{-1}\left (\frac {x (4+x)}{\sqrt {13}}\right )\\ \end {align*}

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Mathematica [C]  time = 2.28, size = 898, normalized size = 28.97 \begin {gather*} \frac {2 \left (\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right ) \sqrt {\frac {\sqrt {4-i \sqrt {13}} \left (-x+\sqrt {4+i \sqrt {13}}-2\right )}{\left (\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right ) \left (-x+\sqrt {4-i \sqrt {13}}-2\right )}} \left (x-\sqrt {4-i \sqrt {13}}+2\right )^2 \sqrt {\frac {\left (-\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right ) \left (x+\sqrt {4-i \sqrt {13}}+2\right )}{\left (\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right ) \left (x-\sqrt {4-i \sqrt {13}}+2\right )}} \sqrt {-\frac {\sqrt {4-i \sqrt {13}} \left (x+\sqrt {4+i \sqrt {13}}+2\right )}{\left (\sqrt {4-i \sqrt {13}}-\sqrt {4+i \sqrt {13}}\right ) \left (-x+\sqrt {4-i \sqrt {13}}-2\right )}} \left (F\left (\sin ^{-1}\left (\sqrt {\frac {i \left (\sqrt {4-i \sqrt {13}}-\sqrt {4+i \sqrt {13}}\right ) x-2 i \sqrt {4+i \sqrt {13}}+2 i \sqrt {4-i \sqrt {13}}-i \sqrt {29}+\sqrt {13}+4 i}{-i \left (\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right ) x-2 i \sqrt {4+i \sqrt {13}}-2 i \sqrt {4-i \sqrt {13}}+i \sqrt {29}+\sqrt {13}+4 i}}\right )|\frac {1}{13} \left (-45-8 \sqrt {29}\right )\right )-2 \Pi \left (-\frac {\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}}{\sqrt {4-i \sqrt {13}}-\sqrt {4+i \sqrt {13}}};\sin ^{-1}\left (\sqrt {\frac {i \left (\sqrt {4-i \sqrt {13}}-\sqrt {4+i \sqrt {13}}\right ) x-2 i \sqrt {4+i \sqrt {13}}+2 i \sqrt {4-i \sqrt {13}}-i \sqrt {29}+\sqrt {13}+4 i}{-i \left (\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right ) x-2 i \sqrt {4+i \sqrt {13}}-2 i \sqrt {4-i \sqrt {13}}+i \sqrt {29}+\sqrt {13}+4 i}}\right )|\frac {1}{13} \left (-45-8 \sqrt {29}\right )\right )\right )}{\left (\sqrt {4-i \sqrt {13}}-\sqrt {4+i \sqrt {13}}\right ) \sqrt {x^4+8 x^3+16 x^2+13}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(2 + x)/Sqrt[13 + 16*x^2 + 8*x^3 + x^4],x]

[Out]

(2*(Sqrt[4 - I*Sqrt[13]] + Sqrt[4 + I*Sqrt[13]])*Sqrt[(Sqrt[4 - I*Sqrt[13]]*(-2 + Sqrt[4 + I*Sqrt[13]] - x))/(
(Sqrt[4 - I*Sqrt[13]] + Sqrt[4 + I*Sqrt[13]])*(-2 + Sqrt[4 - I*Sqrt[13]] - x))]*(2 - Sqrt[4 - I*Sqrt[13]] + x)
^2*Sqrt[((-Sqrt[4 - I*Sqrt[13]] + Sqrt[4 + I*Sqrt[13]])*(2 + Sqrt[4 - I*Sqrt[13]] + x))/((Sqrt[4 - I*Sqrt[13]]
+ Sqrt[4 + I*Sqrt[13]])*(2 - Sqrt[4 - I*Sqrt[13]] + x))]*Sqrt[-((Sqrt[4 - I*Sqrt[13]]*(2 + Sqrt[4 + I*Sqrt[13
]] + x))/((Sqrt[4 - I*Sqrt[13]] - Sqrt[4 + I*Sqrt[13]])*(-2 + Sqrt[4 - I*Sqrt[13]] - x)))]*(EllipticF[ArcSin[S
qrt[(4*I + Sqrt[13] - I*Sqrt[29] + (2*I)*Sqrt[4 - I*Sqrt[13]] - (2*I)*Sqrt[4 + I*Sqrt[13]] + I*(Sqrt[4 - I*Sqr
t[13]] - Sqrt[4 + I*Sqrt[13]])*x)/(4*I + Sqrt[13] + I*Sqrt[29] - (2*I)*Sqrt[4 - I*Sqrt[13]] - (2*I)*Sqrt[4 + I
*Sqrt[13]] - I*(Sqrt[4 - I*Sqrt[13]] + Sqrt[4 + I*Sqrt[13]])*x)]], (-45 - 8*Sqrt[29])/13] - 2*EllipticPi[-((Sq
rt[4 - I*Sqrt[13]] + Sqrt[4 + I*Sqrt[13]])/(Sqrt[4 - I*Sqrt[13]] - Sqrt[4 + I*Sqrt[13]])), ArcSin[Sqrt[(4*I +
Sqrt[13] - I*Sqrt[29] + (2*I)*Sqrt[4 - I*Sqrt[13]] - (2*I)*Sqrt[4 + I*Sqrt[13]] + I*(Sqrt[4 - I*Sqrt[13]] - Sq
rt[4 + I*Sqrt[13]])*x)/(4*I + Sqrt[13] + I*Sqrt[29] - (2*I)*Sqrt[4 - I*Sqrt[13]] - (2*I)*Sqrt[4 + I*Sqrt[13]]
- I*(Sqrt[4 - I*Sqrt[13]] + Sqrt[4 + I*Sqrt[13]])*x)]], (-45 - 8*Sqrt[29])/13]))/((Sqrt[4 - I*Sqrt[13]] - Sqrt
[4 + I*Sqrt[13]])*Sqrt[13 + 16*x^2 + 8*x^3 + x^4])

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IntegrateAlgebraic [A]  time = 0.13, size = 31, normalized size = 1.00 \begin {gather*} \frac {1}{2} \log \left (4 x+x^2+\sqrt {13+16 x^2+8 x^3+x^4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(2 + x)/Sqrt[13 + 16*x^2 + 8*x^3 + x^4],x]

[Out]

Log[4*x + x^2 + Sqrt[13 + 16*x^2 + 8*x^3 + x^4]]/2

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fricas [A]  time = 0.50, size = 27, normalized size = 0.87 \begin {gather*} \frac {1}{2} \, \log \left (x^{2} + 4 \, x + \sqrt {x^{4} + 8 \, x^{3} + 16 \, x^{2} + 13}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+x)/(x^4+8*x^3+16*x^2+13)^(1/2),x, algorithm="fricas")

[Out]

1/2*log(x^2 + 4*x + sqrt(x^4 + 8*x^3 + 16*x^2 + 13))

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giac [A]  time = 0.66, size = 25, normalized size = 0.81 \begin {gather*} -\frac {1}{2} \, \log \left (-x^{2} - 4 \, x + \sqrt {{\left (x^{2} + 4 \, x\right )}^{2} + 13}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+x)/(x^4+8*x^3+16*x^2+13)^(1/2),x, algorithm="giac")

[Out]

-1/2*log(-x^2 - 4*x + sqrt((x^2 + 4*x)^2 + 13))

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maple [A]  time = 0.60, size = 30, normalized size = 0.97

method	result	size
trager	\(-\frac {\ln \left (-x^{2}+\sqrt {x^{4}+8 x^{3}+16 x^{2}+13}-4 x \right )}{2}\)	\(30\)
default	\(\frac {4 \left (-\sqrt {4+i \sqrt {13}}-\sqrt {4-i \sqrt {13}}\right ) \sqrt {\frac {\left (\sqrt {4-i \sqrt {13}}-\sqrt {4+i \sqrt {13}}\right ) \left (x +2+\sqrt {4+i \sqrt {13}}\right )}{\left (\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right ) \left (x +2-\sqrt {4+i \sqrt {13}}\right )}}\, \left (x +2-\sqrt {4+i \sqrt {13}}\right )^{2} \sqrt {\frac {\sqrt {4+i \sqrt {13}}\, \left (x +2+\sqrt {4-i \sqrt {13}}\right )}{\left (-\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right ) \left (x +2-\sqrt {4+i \sqrt {13}}\right )}}\, \sqrt {\frac {\sqrt {4+i \sqrt {13}}\, \left (x +2-\sqrt {4-i \sqrt {13}}\right )}{\left (\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right ) \left (x +2-\sqrt {4+i \sqrt {13}}\right )}}\, \EllipticF \left (\sqrt {\frac {\left (\sqrt {4-i \sqrt {13}}-\sqrt {4+i \sqrt {13}}\right ) \left (x +2+\sqrt {4+i \sqrt {13}}\right )}{\left (\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right ) \left (x +2-\sqrt {4+i \sqrt {13}}\right )}}, \sqrt {-\frac {\left (\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right )^{2}}{\left (\sqrt {4-i \sqrt {13}}-\sqrt {4+i \sqrt {13}}\right ) \left (-\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right )}}\right )}{\left (\sqrt {4-i \sqrt {13}}-\sqrt {4+i \sqrt {13}}\right ) \sqrt {4+i \sqrt {13}}\, \sqrt {\left (x +2+\sqrt {4+i \sqrt {13}}\right ) \left (x +2-\sqrt {4+i \sqrt {13}}\right ) \left (x +2+\sqrt {4-i \sqrt {13}}\right ) \left (x +2-\sqrt {4-i \sqrt {13}}\right )}}+\frac {2 \left (-\sqrt {4+i \sqrt {13}}-\sqrt {4-i \sqrt {13}}\right ) \sqrt {\frac {\left (\sqrt {4-i \sqrt {13}}-\sqrt {4+i \sqrt {13}}\right ) \left (x +2+\sqrt {4+i \sqrt {13}}\right )}{\left (\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right ) \left (x +2-\sqrt {4+i \sqrt {13}}\right )}}\, \left (x +2-\sqrt {4+i \sqrt {13}}\right )^{2} \sqrt {\frac {\sqrt {4+i \sqrt {13}}\, \left (x +2+\sqrt {4-i \sqrt {13}}\right )}{\left (-\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right ) \left (x +2-\sqrt {4+i \sqrt {13}}\right )}}\, \sqrt {\frac {\sqrt {4+i \sqrt {13}}\, \left (x +2-\sqrt {4-i \sqrt {13}}\right )}{\left (\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right ) \left (x +2-\sqrt {4+i \sqrt {13}}\right )}}\, \left (\left (-2+\sqrt {4+i \sqrt {13}}\right ) \EllipticF \left (\sqrt {\frac {\left (\sqrt {4-i \sqrt {13}}-\sqrt {4+i \sqrt {13}}\right ) \left (x +2+\sqrt {4+i \sqrt {13}}\right )}{\left (\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right ) \left (x +2-\sqrt {4+i \sqrt {13}}\right )}}, \sqrt {-\frac {\left (\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right )^{2}}{\left (\sqrt {4-i \sqrt {13}}-\sqrt {4+i \sqrt {13}}\right ) \left (-\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right )}}\right )-2 \sqrt {4+i \sqrt {13}}\, \EllipticPi \left (\sqrt {\frac {\left (\sqrt {4-i \sqrt {13}}-\sqrt {4+i \sqrt {13}}\right ) \left (x +2+\sqrt {4+i \sqrt {13}}\right )}{\left (\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right ) \left (x +2-\sqrt {4+i \sqrt {13}}\right )}}, \frac {\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}}{\sqrt {4-i \sqrt {13}}-\sqrt {4+i \sqrt {13}}}, \sqrt {-\frac {\left (\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right )^{2}}{\left (\sqrt {4-i \sqrt {13}}-\sqrt {4+i \sqrt {13}}\right ) \left (-\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right )}}\right )\right )}{\left (\sqrt {4-i \sqrt {13}}-\sqrt {4+i \sqrt {13}}\right ) \sqrt {4+i \sqrt {13}}\, \sqrt {\left (x +2+\sqrt {4+i \sqrt {13}}\right ) \left (x +2-\sqrt {4+i \sqrt {13}}\right ) \left (x +2+\sqrt {4-i \sqrt {13}}\right ) \left (x +2-\sqrt {4-i \sqrt {13}}\right )}}\)	\(1252\)
elliptic	\(\frac {4 \left (-\sqrt {4+i \sqrt {13}}-\sqrt {4-i \sqrt {13}}\right ) \sqrt {\frac {\left (\sqrt {4-i \sqrt {13}}-\sqrt {4+i \sqrt {13}}\right ) \left (x +2+\sqrt {4+i \sqrt {13}}\right )}{\left (\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right ) \left (x +2-\sqrt {4+i \sqrt {13}}\right )}}\, \left (x +2-\sqrt {4+i \sqrt {13}}\right )^{2} \sqrt {\frac {\sqrt {4+i \sqrt {13}}\, \left (x +2+\sqrt {4-i \sqrt {13}}\right )}{\left (-\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right ) \left (x +2-\sqrt {4+i \sqrt {13}}\right )}}\, \sqrt {\frac {\sqrt {4+i \sqrt {13}}\, \left (x +2-\sqrt {4-i \sqrt {13}}\right )}{\left (\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right ) \left (x +2-\sqrt {4+i \sqrt {13}}\right )}}\, \EllipticF \left (\sqrt {\frac {\left (\sqrt {4-i \sqrt {13}}-\sqrt {4+i \sqrt {13}}\right ) \left (x +2+\sqrt {4+i \sqrt {13}}\right )}{\left (\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right ) \left (x +2-\sqrt {4+i \sqrt {13}}\right )}}, \sqrt {-\frac {\left (\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right )^{2}}{\left (\sqrt {4-i \sqrt {13}}-\sqrt {4+i \sqrt {13}}\right ) \left (-\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right )}}\right )}{\left (\sqrt {4-i \sqrt {13}}-\sqrt {4+i \sqrt {13}}\right ) \sqrt {4+i \sqrt {13}}\, \sqrt {\left (x +2+\sqrt {4+i \sqrt {13}}\right ) \left (x +2-\sqrt {4+i \sqrt {13}}\right ) \left (x +2+\sqrt {4-i \sqrt {13}}\right ) \left (x +2-\sqrt {4-i \sqrt {13}}\right )}}+\frac {2 \left (-\sqrt {4+i \sqrt {13}}-\sqrt {4-i \sqrt {13}}\right ) \sqrt {\frac {\left (\sqrt {4-i \sqrt {13}}-\sqrt {4+i \sqrt {13}}\right ) \left (x +2+\sqrt {4+i \sqrt {13}}\right )}{\left (\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right ) \left (x +2-\sqrt {4+i \sqrt {13}}\right )}}\, \left (x +2-\sqrt {4+i \sqrt {13}}\right )^{2} \sqrt {\frac {\sqrt {4+i \sqrt {13}}\, \left (x +2+\sqrt {4-i \sqrt {13}}\right )}{\left (-\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right ) \left (x +2-\sqrt {4+i \sqrt {13}}\right )}}\, \sqrt {\frac {\sqrt {4+i \sqrt {13}}\, \left (x +2-\sqrt {4-i \sqrt {13}}\right )}{\left (\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right ) \left (x +2-\sqrt {4+i \sqrt {13}}\right )}}\, \left (\left (-2+\sqrt {4+i \sqrt {13}}\right ) \EllipticF \left (\sqrt {\frac {\left (\sqrt {4-i \sqrt {13}}-\sqrt {4+i \sqrt {13}}\right ) \left (x +2+\sqrt {4+i \sqrt {13}}\right )}{\left (\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right ) \left (x +2-\sqrt {4+i \sqrt {13}}\right )}}, \sqrt {-\frac {\left (\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right )^{2}}{\left (\sqrt {4-i \sqrt {13}}-\sqrt {4+i \sqrt {13}}\right ) \left (-\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right )}}\right )-2 \sqrt {4+i \sqrt {13}}\, \EllipticPi \left (\sqrt {\frac {\left (\sqrt {4-i \sqrt {13}}-\sqrt {4+i \sqrt {13}}\right ) \left (x +2+\sqrt {4+i \sqrt {13}}\right )}{\left (\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right ) \left (x +2-\sqrt {4+i \sqrt {13}}\right )}}, \frac {\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}}{\sqrt {4-i \sqrt {13}}-\sqrt {4+i \sqrt {13}}}, \sqrt {-\frac {\left (\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right )^{2}}{\left (\sqrt {4-i \sqrt {13}}-\sqrt {4+i \sqrt {13}}\right ) \left (-\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right )}}\right )\right )}{\left (\sqrt {4-i \sqrt {13}}-\sqrt {4+i \sqrt {13}}\right ) \sqrt {4+i \sqrt {13}}\, \sqrt {\left (x +2+\sqrt {4+i \sqrt {13}}\right ) \left (x +2-\sqrt {4+i \sqrt {13}}\right ) \left (x +2+\sqrt {4-i \sqrt {13}}\right ) \left (x +2-\sqrt {4-i \sqrt {13}}\right )}}\)	\(1252\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2+x)/(x^4+8*x^3+16*x^2+13)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/2*ln(-x^2+(x^4+8*x^3+16*x^2+13)^(1/2)-4*x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x + 2}{\sqrt {x^{4} + 8 \, x^{3} + 16 \, x^{2} + 13}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+x)/(x^4+8*x^3+16*x^2+13)^(1/2),x, algorithm="maxima")

[Out]

integrate((x + 2)/sqrt(x^4 + 8*x^3 + 16*x^2 + 13), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {x+2}{\sqrt {x^4+8\,x^3+16\,x^2+13}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + 2)/(16*x^2 + 8*x^3 + x^4 + 13)^(1/2),x)

[Out]

int((x + 2)/(16*x^2 + 8*x^3 + x^4 + 13)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x + 2}{\sqrt {x^{4} + 8 x^{3} + 16 x^{2} + 13}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+x)/(x**4+8*x**3+16*x**2+13)**(1/2),x)

[Out]

Integral((x + 2)/sqrt(x**4 + 8*x**3 + 16*x**2 + 13), x)

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