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

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

[Out]

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

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Rubi [A]  time = 0.00, antiderivative size = 11, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, 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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fricas [A]  time = 0.58, size = 9, normalized size = 0.82 \[ \log \left (x^{2} + 3 \, x - 10\right ) \]

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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giac [A]  time = 0.29, size = 11, normalized size = 1.00 \[ \log \left ({\left | x + 5 \right |}\right ) + \log \left ({\left | x - 2 \right |}\right ) \]

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))

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maple [A]  time = 0.00, size = 9, normalized size = 0.82 \[ \ln \left (\left (x -2\right ) \left (x +5\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.45, size = 9, normalized size = 0.82 \[ \log \left (x + 5\right ) + \log \left (x - 2\right ) \]

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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mupad [B]  time = 2.22, size = 9, normalized size = 0.82 \[ \ln \left (x^2+3\,x-10\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x + 3)/((x - 2)*(x + 5)),x)

[Out]

log(3*x + x^2 - 10)

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sympy [A]  time = 0.09, size = 8, normalized size = 0.73 \[ \log {\left (x^{2} + 3 x - 10 \right )} \]

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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