3.1470 \(\int \frac{2+x}{\sqrt{3+4 x} (1+x^2)} \, dx\)

Optimal. Leaf size=29 \[ \tan ^{-1}\left (\sqrt{4 x+3}+2\right )-\tan ^{-1}\left (2-\sqrt{4 x+3}\right ) \]

[Out]

-ArcTan[2 - Sqrt[3 + 4*x]] + ArcTan[2 + Sqrt[3 + 4*x]]

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Rubi [A]  time = 0.0331659, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {827, 1161, 618, 204} \[ \tan ^{-1}\left (\sqrt{4 x+3}+2\right )-\tan ^{-1}\left (2-\sqrt{4 x+3}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-ArcTan[2 - Sqrt[3 + 4*x]] + ArcTan[2 + Sqrt[3 + 4*x]]

Rule 827

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

Rule 1161

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e - b/c, 2]},
Dist[e/(2*c), Int[1/Simp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /
; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - a*e^2, 0] && (GtQ[(2*d)/e - b/c, 0] || ( !Lt
Q[(2*d)/e - b/c, 0] && EqQ[d - e*Rt[a/c, 2], 0]))

Rule 618

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

Rule 204

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

Rubi steps

\begin{align*} \int \frac{2+x}{\sqrt{3+4 x} \left (1+x^2\right )} \, dx &=2 \operatorname{Subst}\left (\int \frac{5+x^2}{25-6 x^2+x^4} \, dx,x,\sqrt{3+4 x}\right )\\ &=\operatorname{Subst}\left (\int \frac{1}{5-4 x+x^2} \, dx,x,\sqrt{3+4 x}\right )+\operatorname{Subst}\left (\int \frac{1}{5+4 x+x^2} \, dx,x,\sqrt{3+4 x}\right )\\ &=-\left (2 \operatorname{Subst}\left (\int \frac{1}{-4-x^2} \, dx,x,-4+2 \sqrt{3+4 x}\right )\right )-2 \operatorname{Subst}\left (\int \frac{1}{-4-x^2} \, dx,x,4+2 \sqrt{3+4 x}\right )\\ &=-\tan ^{-1}\left (2-\sqrt{3+4 x}\right )+\tan ^{-1}\left (2+\sqrt{3+4 x}\right )\\ \end{align*}

Mathematica [C]  time = 0.0196804, size = 41, normalized size = 1.41 \[ \tan ^{-1}\left (\left (\frac{1}{5}+\frac{2 i}{5}\right ) \sqrt{4 x+3}\right )-i \tanh ^{-1}\left (\left (\frac{2}{5}+\frac{i}{5}\right ) \sqrt{4 x+3}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTan[(1/5 + (2*I)/5)*Sqrt[3 + 4*x]] - I*ArcTanh[(2/5 + I/5)*Sqrt[3 + 4*x]]

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Maple [A]  time = 0.009, size = 22, normalized size = 0.8 \begin{align*} \arctan \left ( -2+\sqrt{3+4\,x} \right ) +\arctan \left ( 2+\sqrt{3+4\,x} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.53419, size = 28, normalized size = 0.97 \begin{align*} \arctan \left (\sqrt{4 \, x + 3} + 2\right ) + \arctan \left (\sqrt{4 \, x + 3} - 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(sqrt(4*x + 3) + 2) + arctan(sqrt(4*x + 3) - 2)

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Fricas [A]  time = 1.67831, size = 45, normalized size = 1.55 \begin{align*} \arctan \left (\frac{2 \, x - 1}{\sqrt{4 \, x + 3}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan((2*x - 1)/sqrt(4*x + 3))

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Sympy [A]  time = 42.9989, size = 26, normalized size = 0.9 \begin{align*} \operatorname{atan}{\left (2 - \frac{5}{\sqrt{4 x + 3}} \right )} - \operatorname{atan}{\left (2 + \frac{5}{\sqrt{4 x + 3}} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

atan(2 - 5/sqrt(4*x + 3)) - atan(2 + 5/sqrt(4*x + 3))

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Giac [A]  time = 1.43077, size = 28, normalized size = 0.97 \begin{align*} \arctan \left (\sqrt{4 \, x + 3} + 2\right ) + \arctan \left (\sqrt{4 \, x + 3} - 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(sqrt(4*x + 3) + 2) + arctan(sqrt(4*x + 3) - 2)