3.451 \(\int \frac{3+2 x}{(-2+x) (5+x)} \, dx\)

Optimal. Leaf size=11 \[ \log (2-x)+\log (x+5) \]

[Out]

Log[2 - x] + Log[5 + x]

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Rubi [A]  time = 0.0049503, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {72} \[ \log (2-x)+\log (x+5) \]

Antiderivative was successfully verified.

[In]

Int[(3 + 2*x)/((-2 + x)*(5 + x)),x]

[Out]

Log[2 - x] + Log[5 + x]

Rule 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rubi steps

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

Mathematica [A]  time = 0.0037906, size = 9, normalized size = 0.82 \[ \log (x-2)+\log (x+5) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[-2 + x] + Log[5 + x]

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Maple [A]  time = 0.001, size = 9, normalized size = 0.8 \begin{align*} \ln \left ( \left ( -2+x \right ) \left ( 5+x \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln((-2+x)*(5+x))

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Maxima [A]  time = 1.00099, size = 12, normalized size = 1.09 \begin{align*} \log \left (x + 5\right ) + \log \left (x - 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x + 5) + log(x - 2)

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Fricas [A]  time = 0.973725, size = 28, normalized size = 2.55 \begin{align*} \log \left (x^{2} + 3 \, x - 10\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x^2 + 3*x - 10)

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Sympy [A]  time = 0.08384, size = 8, normalized size = 0.73 \begin{align*} \log{\left (x^{2} + 3 x - 10 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x**2 + 3*x - 10)

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Giac [A]  time = 1.12288, size = 15, normalized size = 1.36 \begin{align*} \log \left ({\left | x + 5 \right |}\right ) + \log \left ({\left | x - 2 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(abs(x + 5)) + log(abs(x - 2))