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

Optimal. Leaf size=24 \[ -\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{x^2+x+5}}{\sqrt{2}}\right ) \]

[Out]

-(Sqrt[2]*ArcTanh[Sqrt[5 + x + x^2]/Sqrt[2]])

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Rubi [A]  time = 0.0187828, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {1024, 206} \[ -\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{x^2+x+5}}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[2]*ArcTanh[Sqrt[5 + x + x^2]/Sqrt[2]])

Rule 1024

Int[((g_) + (h_.)*(x_))/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol
] :> Dist[-2*g, Subst[Int[1/(b*d - a*e - b*x^2), x], x, Sqrt[d + e*x + f*x^2]], x] /; FreeQ[{a, b, c, d, e, f,
g, h}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && EqQ[c*e - b*f, 0] && EqQ[h*e - 2*g*f, 0]

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])

Rubi steps

\begin{align*} \int \frac{1+2 x}{\left (3+x+x^2\right ) \sqrt{5+x+x^2}} \, dx &=-\left (2 \operatorname{Subst}\left (\int \frac{1}{2-x^2} \, dx,x,\sqrt{5+x+x^2}\right )\right )\\ &=-\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{5+x+x^2}}{\sqrt{2}}\right )\\ \end{align*}

Mathematica [C]  time = 0.0627908, size = 90, normalized size = 3.75 \[ -\frac{\tanh ^{-1}\left (\frac{-2 i \sqrt{11} x-i \sqrt{11}+19}{4 \sqrt{2} \sqrt{x^2+x+5}}\right )+\tanh ^{-1}\left (\frac{2 i \sqrt{11} x+i \sqrt{11}+19}{4 \sqrt{2} \sqrt{x^2+x+5}}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((ArcTanh[(19 - I*Sqrt[11] - (2*I)*Sqrt[11]*x)/(4*Sqrt[2]*Sqrt[5 + x + x^2])] + ArcTanh[(19 + I*Sqrt[11] + (2
*I)*Sqrt[11]*x)/(4*Sqrt[2]*Sqrt[5 + x + x^2])])/Sqrt[2])

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Maple [A]  time = 0.099, size = 20, normalized size = 0.8 \begin{align*} -{\it Artanh} \left ({\frac{\sqrt{2}}{2}\sqrt{{x}^{2}+x+5}} \right ) \sqrt{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-arctanh(1/2*(x^2+x+5)^(1/2)*2^(1/2))*2^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{2 \, x + 1}{\sqrt{x^{2} + x + 5}{\left (x^{2} + x + 3\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.26232, size = 103, normalized size = 4.29 \begin{align*} \frac{1}{2} \, \sqrt{2} \log \left (\frac{x^{2} - 2 \, \sqrt{2} \sqrt{x^{2} + x + 5} + x + 7}{x^{2} + x + 3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(2)*log((x^2 - 2*sqrt(2)*sqrt(x^2 + x + 5) + x + 7)/(x^2 + x + 3))

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Sympy [A]  time = 3.17531, size = 68, normalized size = 2.83 \begin{align*} 2 \left (\begin{cases} - \frac{\sqrt{2} \operatorname{acoth}{\left (\frac{\sqrt{2}}{\sqrt{x^{2} + x + 5}} \right )}}{2} & \text{for}\: \frac{1}{x^{2} + x + 5} > \frac{1}{2} \\- \frac{\sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2}}{\sqrt{x^{2} + x + 5}} \right )}}{2} & \text{for}\: \frac{1}{x^{2} + x + 5} < \frac{1}{2} \end{cases}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*Piecewise((-sqrt(2)*acoth(sqrt(2)/sqrt(x**2 + x + 5))/2, 1/(x**2 + x + 5) > 1/2), (-sqrt(2)*atanh(sqrt(2)/sq
rt(x**2 + x + 5))/2, 1/(x**2 + x + 5) < 1/2))

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Giac [B]  time = 1.25939, size = 127, normalized size = 5.29 \begin{align*} \frac{1}{2} \, \sqrt{2} \log \left ({\left (x - \sqrt{x^{2} + x + 5}\right )}^{2} +{\left (x - \sqrt{x^{2} + x + 5}\right )}{\left (2 \, \sqrt{2} + 1\right )} + \sqrt{2} + 5\right ) - \frac{1}{2} \, \sqrt{2} \log \left ({\left (x - \sqrt{x^{2} + x + 5}\right )}^{2} -{\left (x - \sqrt{x^{2} + x + 5}\right )}{\left (2 \, \sqrt{2} - 1\right )} - \sqrt{2} + 5\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(2)*log((x - sqrt(x^2 + x + 5))^2 + (x - sqrt(x^2 + x + 5))*(2*sqrt(2) + 1) + sqrt(2) + 5) - 1/2*sqrt(
2)*log((x - sqrt(x^2 + x + 5))^2 - (x - sqrt(x^2 + x + 5))*(2*sqrt(2) - 1) - sqrt(2) + 5)