\(\int \frac {1}{\sqrt {-2+5 x+3 x^2}} \, dx\) [123]

Optimal result

Integrand size = 14, antiderivative size = 35 \[ \int \frac {1}{\sqrt {-2+5 x+3 x^2}} \, dx=\frac {\text {arctanh}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {-2+5 x+3 x^2}}\right )}{\sqrt {3}} \]

[Out]

1/3*arctanh(1/6*(5+6*x)*3^(1/2)/(3*x^2+5*x-2)^(1/2))*3^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {635, 212} \[ \int \frac {1}{\sqrt {-2+5 x+3 x^2}} \, dx=\frac {\text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x-2}}\right )}{\sqrt {3}} \]

[In]

Int[1/Sqrt[-2 + 5*x + 3*x^2],x]

[Out]

ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[-2 + 5*x + 3*x^2])]/Sqrt[3]

Rule 212

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

Rule 635

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]

Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {-2+5 x+3 x^2}}\right ) \\ & = \frac {\tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {-2+5 x+3 x^2}}\right )}{\sqrt {3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.86 \[ \int \frac {1}{\sqrt {-2+5 x+3 x^2}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {-\frac {2}{3}+\frac {5 x}{3}+x^2}}{2+x}\right )}{\sqrt {3}} \]

[In]

Integrate[1/Sqrt[-2 + 5*x + 3*x^2],x]

[Out]

(2*ArcTanh[Sqrt[-2/3 + (5*x)/3 + x^2]/(2 + x)])/Sqrt[3]

Maple [A] (verified)

Time = 2.12 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.86

method	result	size
default	\(\frac {\ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x -2}\right ) \sqrt {3}}{3}\)	\(30\)
trager	\(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +6 \sqrt {3 x^{2}+5 x -2}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )\right )}{3}\)	\(42\)

[In]

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

[Out]

1/3*ln(1/3*(5/2+3*x)*3^(1/2)+(3*x^2+5*x-2)^(1/2))*3^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.09 \[ \int \frac {1}{\sqrt {-2+5 x+3 x^2}} \, dx=\frac {1}{6} \, \sqrt {3} \log \left (4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x - 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 1\right ) \]

[In]

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

[Out]

1/6*sqrt(3)*log(4*sqrt(3)*sqrt(3*x^2 + 5*x - 2)*(6*x + 5) + 72*x^2 + 120*x + 1)

Sympy [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\sqrt {-2+5 x+3 x^2}} \, dx=\frac {\sqrt {3} \log {\left (6 x + 2 \sqrt {3} \sqrt {3 x^{2} + 5 x - 2} + 5 \right )}}{3} \]

[In]

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

[Out]

sqrt(3)*log(6*x + 2*sqrt(3)*sqrt(3*x**2 + 5*x - 2) + 5)/3

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.80 \[ \int \frac {1}{\sqrt {-2+5 x+3 x^2}} \, dx=\frac {1}{3} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x - 2} + 6 \, x + 5\right ) \]

[In]

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

[Out]

1/3*sqrt(3)*log(2*sqrt(3)*sqrt(3*x^2 + 5*x - 2) + 6*x + 5)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.54 \[ \int \frac {1}{\sqrt {-2+5 x+3 x^2}} \, dx=\frac {1}{12} \, \sqrt {3 \, x^{2} + 5 \, x - 2} {\left (6 \, x + 5\right )} + \frac {49}{72} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x - 2}\right )} - 5 \right |}\right ) \]

[In]

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

[Out]

1/12*sqrt(3*x^2 + 5*x - 2)*(6*x + 5) + 49/72*sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x - 2)) -
5))

Mupad [B] (verification not implemented)

Time = 9.19 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.74 \[ \int \frac {1}{\sqrt {-2+5 x+3 x^2}} \, dx=\frac {\sqrt {3}\,\ln \left (\sqrt {3}\,\left (x+\frac {5}{6}\right )+\sqrt {3\,x^2+5\,x-2}\right )}{3} \]

[In]

int(1/(5*x + 3*x^2 - 2)^(1/2),x)

[Out]

(3^(1/2)*log(3^(1/2)*(x + 5/6) + (5*x + 3*x^2 - 2)^(1/2)))/3