3.2419 \(\int \frac {1}{x \sqrt {4+12 x+9 x^2}} \, dx\)

Optimal. Leaf size=27 \[ -\frac {(3 x+2) \tanh ^{-1}(3 x+1)}{\sqrt {9 x^2+12 x+4}} \]

[Out]

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

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Rubi [B]  time = 0.01, antiderivative size = 55, normalized size of antiderivative = 2.04, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {646, 36, 29, 31} \[ \frac {(3 x+2) \log (x)}{2 \sqrt {9 x^2+12 x+4}}-\frac {(3 x+2) \log (3 x+2)}{2 \sqrt {9 x^2+12 x+4}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*Sqrt[4 + 12*x + 9*x^2]),x]

[Out]

((2 + 3*x)*Log[x])/(2*Sqrt[4 + 12*x + 9*x^2]) - ((2 + 3*x)*Log[2 + 3*x])/(2*Sqrt[4 + 12*x + 9*x^2])

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 36

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

Rule 646

Int[((d_.) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(a + b*x + c*x^2)^Fra
cPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b,
c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0] &&  !IntegerQ[p] && NeQ[2*c*d - b*e, 0]

Rubi steps

\begin {align*} \int \frac {1}{x \sqrt {4+12 x+9 x^2}} \, dx &=\frac {(6+9 x) \int \frac {1}{x (6+9 x)} \, dx}{\sqrt {4+12 x+9 x^2}}\\ &=\frac {(6+9 x) \int \frac {1}{x} \, dx}{6 \sqrt {4+12 x+9 x^2}}-\frac {(3 (6+9 x)) \int \frac {1}{6+9 x} \, dx}{2 \sqrt {4+12 x+9 x^2}}\\ &=\frac {(2+3 x) \log (x)}{2 \sqrt {4+12 x+9 x^2}}-\frac {(2+3 x) \log (2+3 x)}{2 \sqrt {4+12 x+9 x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 31, normalized size = 1.15 \[ \frac {(3 x+2) (\log (x)-\log (3 x+2))}{2 \sqrt {(3 x+2)^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*Sqrt[4 + 12*x + 9*x^2]),x]

[Out]

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

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fricas [A]  time = 1.32, size = 13, normalized size = 0.48 \[ -\frac {1}{2} \, \log \left (3 \, x + 2\right ) + \frac {1}{2} \, \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*log(3*x + 2) + 1/2*log(x)

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giac [A]  time = 0.20, size = 21, normalized size = 0.78 \[ -\frac {1}{2} \, {\left (\log \left ({\left | 3 \, x + 2 \right |}\right ) - \log \left ({\left | x \right |}\right )\right )} \mathrm {sgn}\left (3 \, x + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(log(abs(3*x + 2)) - log(abs(x)))*sgn(3*x + 2)

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maple [A]  time = 0.06, size = 28, normalized size = 1.04 \[ -\frac {\left (3 x +2\right ) \left (-\ln \relax (x )+\ln \left (3 x +2\right )\right )}{2 \sqrt {\left (3 x +2\right )^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.95, size = 24, normalized size = 0.89 \[ -\frac {1}{2} \, \left (-1\right )^{12 \, x + 8} \log \left (\frac {12 \, x}{{\left | x \right |}} + \frac {8}{{\left | x \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(-1)^(12*x + 8)*log(12*x/abs(x) + 8/abs(x))

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mupad [B]  time = 1.32, size = 23, normalized size = 0.85 \[ -\frac {\ln \left (\frac {6\,x+2\,\sqrt {{\left (3\,x+2\right )}^2}+4}{x}\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*((3*x + 2)^2)^(1/2)),x)

[Out]

-log((6*x + 2*((3*x + 2)^2)^(1/2) + 4)/x)/2

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sympy [A]  time = 0.12, size = 12, normalized size = 0.44 \[ \frac {\log {\relax (x )}}{2} - \frac {\log {\left (x + \frac {2}{3} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x)/2 - log(x + 2/3)/2

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