\(\int \frac {1}{x \sqrt {-2+5 x+3 x^2}} \, dx\) [2436]

Optimal result

Integrand size = 18, antiderivative size = 36 \[ \int \frac {1}{x \sqrt {-2+5 x+3 x^2}} \, dx=-\frac {\arctan \left (\frac {4-5 x}{2 \sqrt {2} \sqrt {-2+5 x+3 x^2}}\right )}{\sqrt {2}} \]

[Out]

-1/2*arctan(1/4*(4-5*x)*2^(1/2)/(3*x^2+5*x-2)^(1/2))*2^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {738, 210} \[ \int \frac {1}{x \sqrt {-2+5 x+3 x^2}} \, dx=-\frac {\arctan \left (\frac {4-5 x}{2 \sqrt {2} \sqrt {3 x^2+5 x-2}}\right )}{\sqrt {2}} \]

[In]

Int[1/(x*Sqrt[-2 + 5*x + 3*x^2]),x]

[Out]

-(ArcTan[(4 - 5*x)/(2*Sqrt[2]*Sqrt[-2 + 5*x + 3*x^2])]/Sqrt[2])

Rule 210

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

Rule 738

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

Rubi steps \begin{align*} \text {integral}& = -\left (2 \text {Subst}\left (\int \frac {1}{-8-x^2} \, dx,x,\frac {-4+5 x}{\sqrt {-2+5 x+3 x^2}}\right )\right ) \\ & = -\frac {\tan ^{-1}\left (\frac {4-5 x}{2 \sqrt {2} \sqrt {-2+5 x+3 x^2}}\right )}{\sqrt {2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.97 \[ \int \frac {1}{x \sqrt {-2+5 x+3 x^2}} \, dx=-\sqrt {2} \arctan \left (\frac {\sqrt {-2+5 x+3 x^2}}{\sqrt {2} (-1+3 x)}\right ) \]

[In]

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

[Out]

-(Sqrt[2]*ArcTan[Sqrt[-2 + 5*x + 3*x^2]/(Sqrt[2]*(-1 + 3*x))])

Maple [A] (verified)

Time = 0.25 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.81

method	result	size
default	\(\frac {\sqrt {2}\, \arctan \left (\frac {\left (-4+5 x \right ) \sqrt {2}}{4 \sqrt {3 x^{2}+5 x -2}}\right )}{2}\)	\(29\)
trager	\(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x -4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right )+4 \sqrt {3 x^{2}+5 x -2}}{x}\right )}{2}\)	\(46\)

[In]

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

[Out]

1/2*2^(1/2)*arctan(1/4*(-4+5*x)*2^(1/2)/(3*x^2+5*x-2)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x \sqrt {-2+5 x+3 x^2}} \, dx=\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (5 \, x - 4\right )}}{4 \, \sqrt {3 \, x^{2} + 5 \, x - 2}}\right ) \]

[In]

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

[Out]

1/2*sqrt(2)*arctan(1/4*sqrt(2)*(5*x - 4)/sqrt(3*x^2 + 5*x - 2))

Sympy [F]

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

[In]

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

[Out]

Integral(1/(x*sqrt((x + 2)*(3*x - 1))), x)

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.56 \[ \int \frac {1}{x \sqrt {-2+5 x+3 x^2}} \, dx=\frac {1}{2} \, \sqrt {2} \arcsin \left (\frac {5 \, x}{7 \, {\left | x \right |}} - \frac {4}{7 \, {\left | x \right |}}\right ) \]

[In]

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

[Out]

1/2*sqrt(2)*arcsin(5/7*x/abs(x) - 4/7/abs(x))

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.83 \[ \int \frac {1}{x \sqrt {-2+5 x+3 x^2}} \, dx=\sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x - 2}\right )}\right ) \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.94 \[ \int \frac {1}{x \sqrt {-2+5 x+3 x^2}} \, dx=\frac {\sqrt {2}\,\ln \left (\frac {5\,x-4+\sqrt {2}\,\sqrt {3\,x^2+5\,x-2}\,2{}\mathrm {i}}{x}\right )\,1{}\mathrm {i}}{2} \]

[In]

int(1/(x*(5*x + 3*x^2 - 2)^(1/2)),x)

[Out]

(2^(1/2)*log((5*x + 2^(1/2)*(5*x + 3*x^2 - 2)^(1/2)*2i - 4)/x)*1i)/2