3.82 \(\int \frac{1}{2+5 x-3 x^2} \, dx\)

Optimal. Leaf size=21 \[ \frac{1}{7} \log (3 x+1)-\frac{1}{7} \log (2-x) \]

[Out]

-Log[2 - x]/7 + Log[1 + 3*x]/7

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Rubi [A]  time = 0.0055975, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {616, 31} \[ \frac{1}{7} \log (3 x+1)-\frac{1}{7} \log (2-x) \]

Antiderivative was successfully verified.

[In]

Int[(2 + 5*x - 3*x^2)^(-1),x]

[Out]

-Log[2 - x]/7 + Log[1 + 3*x]/7

Rule 616

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[c/q, Int[1/Simp
[b/2 - q/2 + c*x, x], x], x] - Dist[c/q, Int[1/Simp[b/2 + q/2 + c*x, x], x], x]] /; FreeQ[{a, b, c}, x] && NeQ
[b^2 - 4*a*c, 0] && PosQ[b^2 - 4*a*c] && PerfectSquareQ[b^2 - 4*a*c]

Rule 31

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

Rubi steps

\begin{align*} \int \frac{1}{2+5 x-3 x^2} \, dx &=-\left (\frac{3}{7} \int \frac{1}{-1-3 x} \, dx\right )+\frac{3}{7} \int \frac{1}{6-3 x} \, dx\\ &=-\frac{1}{7} \log (2-x)+\frac{1}{7} \log (1+3 x)\\ \end{align*}

Mathematica [A]  time = 0.0029986, size = 21, normalized size = 1. \[ \frac{1}{7} \log (3 x+1)-\frac{1}{7} \log (2-x) \]

Antiderivative was successfully verified.

[In]

Integrate[(2 + 5*x - 3*x^2)^(-1),x]

[Out]

-Log[2 - x]/7 + Log[1 + 3*x]/7

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Maple [A]  time = 0.049, size = 16, normalized size = 0.8 \begin{align*}{\frac{\ln \left ( 1+3\,x \right ) }{7}}-{\frac{\ln \left ( -2+x \right ) }{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*ln(1+3*x)-1/7*ln(-2+x)

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Maxima [A]  time = 1.09327, size = 20, normalized size = 0.95 \begin{align*} \frac{1}{7} \, \log \left (3 \, x + 1\right ) - \frac{1}{7} \, \log \left (x - 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*log(3*x + 1) - 1/7*log(x - 2)

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Fricas [A]  time = 2.14062, size = 47, normalized size = 2.24 \begin{align*} \frac{1}{7} \, \log \left (3 \, x + 1\right ) - \frac{1}{7} \, \log \left (x - 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*log(3*x + 1) - 1/7*log(x - 2)

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Sympy [A]  time = 0.16336, size = 14, normalized size = 0.67 \begin{align*} - \frac{\log{\left (x - 2 \right )}}{7} + \frac{\log{\left (x + \frac{1}{3} \right )}}{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x - 2)/7 + log(x + 1/3)/7

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Giac [A]  time = 1.20764, size = 23, normalized size = 1.1 \begin{align*} \frac{1}{7} \, \log \left ({\left | 3 \, x + 1 \right |}\right ) - \frac{1}{7} \, \log \left ({\left | x - 2 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*log(abs(3*x + 1)) - 1/7*log(abs(x - 2))