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

Optimal. Leaf size=12 \[ \frac{1}{x}+3 \log (1-x) \]

[Out]

x^(-1) + 3*Log[1 - x]

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Rubi [A]  time = 0.0213736, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {1593, 893} \[ \frac{1}{x}+3 \log (1-x) \]

Antiderivative was successfully verified.

[In]

Int[(1 - x + 3*x^2)/(-x^2 + x^3),x]

[Out]

x^(-1) + 3*Log[1 - x]

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

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

Mathematica [A]  time = 0.0033556, size = 12, normalized size = 1. \[ \frac{1}{x}+3 \log (1-x) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - x + 3*x^2)/(-x^2 + x^3),x]

[Out]

x^(-1) + 3*Log[1 - x]

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Maple [A]  time = 0.006, size = 11, normalized size = 0.9 \begin{align*} 3\,\ln \left ( x-1 \right ) +{x}^{-1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.10043, size = 14, normalized size = 1.17 \begin{align*} \frac{1}{x} + 3 \, \log \left (x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/x + 3*log(x - 1)

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Fricas [A]  time = 1.45993, size = 32, normalized size = 2.67 \begin{align*} \frac{3 \, x \log \left (x - 1\right ) + 1}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(3*x*log(x - 1) + 1)/x

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Sympy [A]  time = 0.083881, size = 8, normalized size = 0.67 \begin{align*} 3 \log{\left (x - 1 \right )} + \frac{1}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*log(x - 1) + 1/x

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Giac [A]  time = 1.1625, size = 15, normalized size = 1.25 \begin{align*} \frac{1}{x} + 3 \, \log \left ({\left | x - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/x + 3*log(abs(x - 1))