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

Optimal. Leaf size=18 \[ -\frac{1}{x}+3 \log (2-x)+2 \log (x) \]

[Out]

-x^(-1) + 3*Log[2 - x] + 2*Log[x]

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Rubi [A]  time = 0.0143458, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {893} \[ -\frac{1}{x}+3 \log (2-x)+2 \log (x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-x^(-1) + 3*Log[2 - x] + 2*Log[x]

Rule 893

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && I
ntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rubi steps

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

Mathematica [A]  time = 0.0040948, size = 18, normalized size = 1. \[ -\frac{1}{x}+3 \log (2-x)+2 \log (x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-x^(-1) + 3*Log[2 - x] + 2*Log[x]

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Maple [A]  time = 0.007, size = 17, normalized size = 0.9 \begin{align*} -{x}^{-1}+2\,\ln \left ( x \right ) +3\,\ln \left ( -2+x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.938333, size = 22, normalized size = 1.22 \begin{align*} -\frac{1}{x} + 3 \, \log \left (x - 2\right ) + 2 \, \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/x + 3*log(x - 2) + 2*log(x)

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Fricas [A]  time = 1.5578, size = 50, normalized size = 2.78 \begin{align*} \frac{3 \, x \log \left (x - 2\right ) + 2 \, x \log \left (x\right ) - 1}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(3*x*log(x - 2) + 2*x*log(x) - 1)/x

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Sympy [A]  time = 0.103647, size = 14, normalized size = 0.78 \begin{align*} 2 \log{\left (x \right )} + 3 \log{\left (x - 2 \right )} - \frac{1}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*log(x) + 3*log(x - 2) - 1/x

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Giac [A]  time = 1.06285, size = 24, normalized size = 1.33 \begin{align*} -\frac{1}{x} + 3 \, \log \left ({\left | x - 2 \right |}\right ) + 2 \, \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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