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

Optimal. Leaf size=48 \[ \frac {1}{9} \sqrt {9 x^2+12 x+4}-\frac {2 (3 x+2) \log (3 x+2)}{9 \sqrt {9 x^2+12 x+4}} \]

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Rubi [A]  time = 0.01, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {640, 608, 31} \begin {gather*} \frac {1}{9} \sqrt {9 x^2+12 x+4}-\frac {2 (3 x+2) \log (3 x+2)}{9 \sqrt {9 x^2+12 x+4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/Sqrt[4 + 12*x + 9*x^2],x]

[Out]

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

Rule 31

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

Rule 608

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

Rule 640

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

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Mathematica [A]  time = 0.01, size = 33, normalized size = 0.69 \begin {gather*} \frac {(3 x+2) (3 x-2 \log (3 x+2)+2)}{9 \sqrt {(3 x+2)^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/Sqrt[4 + 12*x + 9*x^2],x]

[Out]

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

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IntegrateAlgebraic [F]  time = 0.10, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\sqrt {4+12 x+9 x^2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x/Sqrt[4 + 12*x + 9*x^2],x]

[Out]

Defer[IntegrateAlgebraic][x/Sqrt[4 + 12*x + 9*x^2], x]

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fricas [A]  time = 0.43, size = 12, normalized size = 0.25 \begin {gather*} \frac {1}{3} \, x - \frac {2}{9} \, \log \left (3 \, x + 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x - 2/9*log(3*x + 2)

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giac [A]  time = 0.15, size = 25, normalized size = 0.52 \begin {gather*} \frac {1}{3} \, x \mathrm {sgn}\left (3 \, x + 2\right ) - \frac {2}{9} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \mathrm {sgn}\left (3 \, x + 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x*sgn(3*x + 2) - 2/9*log(abs(3*x + 2))*sgn(3*x + 2)

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maple [A]  time = 0.05, size = 29, normalized size = 0.60 \begin {gather*} -\frac {\left (3 x +2\right ) \left (-3 x +2 \ln \left (3 x +2\right )\right )}{9 \sqrt {\left (3 x +2\right )^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9*(3*x+2)*(-3*x+2*ln(3*x+2))/((3*x+2)^2)^(1/2)

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maxima [A]  time = 2.95, size = 21, normalized size = 0.44 \begin {gather*} \frac {1}{9} \, \sqrt {9 \, x^{2} + 12 \, x + 4} - \frac {2}{9} \, \log \left (x + \frac {2}{3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*sqrt(9*x^2 + 12*x + 4) - 2/9*log(x + 2/3)

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mupad [B]  time = 0.27, size = 32, normalized size = 0.67 \begin {gather*} \frac {\sqrt {9\,x^2+12\,x+4}}{9}-\frac {2\,\ln \left (x+\frac {\sqrt {{\left (3\,x+2\right )}^2}}{3}+\frac {2}{3}\right )}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/((3*x + 2)^2)^(1/2),x)

[Out]

(12*x + 9*x^2 + 4)^(1/2)/9 - (2*log(x + ((3*x + 2)^2)^(1/2)/3 + 2/3))/9

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sympy [A]  time = 0.10, size = 12, normalized size = 0.25 \begin {gather*} \frac {x}{3} - \frac {2 \log {\left (3 x + 2 \right )}}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/3 - 2*log(3*x + 2)/9

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