\(\int \frac {3+2 x}{\sqrt {5+2 x+x^2}} \, dx\) [2402]

Optimal result

Integrand size = 18, antiderivative size = 23 \[ \int \frac {3+2 x}{\sqrt {5+2 x+x^2}} \, dx=2 \sqrt {5+2 x+x^2}+\text {arcsinh}\left (\frac {1+x}{2}\right ) \]

[Out]

arcsinh(1/2+1/2*x)+2*(x^2+2*x+5)^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {654, 633, 221} \[ \int \frac {3+2 x}{\sqrt {5+2 x+x^2}} \, dx=\text {arcsinh}\left (\frac {x+1}{2}\right )+2 \sqrt {x^2+2 x+5} \]

[In]

Int[(3 + 2*x)/Sqrt[5 + 2*x + x^2],x]

[Out]

2*Sqrt[5 + 2*x + x^2] + ArcSinh[(1 + x)/2]

Rule 221

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 633

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 654

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

Rubi steps \begin{align*} \text {integral}& = 2 \sqrt {5+2 x+x^2}+\int \frac {1}{\sqrt {5+2 x+x^2}} \, dx \\ & = 2 \sqrt {5+2 x+x^2}+\frac {1}{4} \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{16}}} \, dx,x,2+2 x\right ) \\ & = 2 \sqrt {5+2 x+x^2}+\sinh ^{-1}\left (\frac {1+x}{2}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.52 \[ \int \frac {3+2 x}{\sqrt {5+2 x+x^2}} \, dx=2 \sqrt {5+2 x+x^2}-\log \left (-1-x+\sqrt {5+2 x+x^2}\right ) \]

[In]

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

[Out]

2*Sqrt[5 + 2*x + x^2] - Log[-1 - x + Sqrt[5 + 2*x + x^2]]

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87

method	result	size
default	\(\operatorname {arcsinh}\left (\frac {1}{2}+\frac {x}{2}\right )+2 \sqrt {x^{2}+2 x +5}\)	\(20\)
risch	\(\operatorname {arcsinh}\left (\frac {1}{2}+\frac {x}{2}\right )+2 \sqrt {x^{2}+2 x +5}\)	\(20\)
trager	\(2 \sqrt {x^{2}+2 x +5}-\ln \left (\sqrt {x^{2}+2 x +5}-1-x \right )\)	\(32\)

[In]

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

[Out]

arcsinh(1/2+1/2*x)+2*(x^2+2*x+5)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.35 \[ \int \frac {3+2 x}{\sqrt {5+2 x+x^2}} \, dx=2 \, \sqrt {x^{2} + 2 \, x + 5} - \log \left (-x + \sqrt {x^{2} + 2 \, x + 5} - 1\right ) \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.37 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {3+2 x}{\sqrt {5+2 x+x^2}} \, dx=2 \sqrt {x^{2} + 2 x + 5} + \operatorname {asinh}{\left (\frac {x}{2} + \frac {1}{2} \right )} \]

[In]

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

[Out]

2*sqrt(x**2 + 2*x + 5) + asinh(x/2 + 1/2)

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \frac {3+2 x}{\sqrt {5+2 x+x^2}} \, dx=2 \, \sqrt {x^{2} + 2 \, x + 5} + \operatorname {arsinh}\left (\frac {1}{2} \, x + \frac {1}{2}\right ) \]

[In]

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

[Out]

2*sqrt(x^2 + 2*x + 5) + arcsinh(1/2*x + 1/2)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.35 \[ \int \frac {3+2 x}{\sqrt {5+2 x+x^2}} \, dx=2 \, \sqrt {x^{2} + 2 \, x + 5} - \log \left (-x + \sqrt {x^{2} + 2 \, x + 5} - 1\right ) \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 10.09 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.17 \[ \int \frac {3+2 x}{\sqrt {5+2 x+x^2}} \, dx=\ln \left (x+\sqrt {x^2+2\,x+5}+1\right )+2\,\sqrt {x^2+2\,x+5} \]

[In]

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

[Out]

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