3.326 \(\int \frac{3-4 x-5 x^2+3 x^3}{x^3 (-1+x+x^2)} \, dx\)

Optimal. Leaf size=65 \[ \frac{3}{2 x^2}-\frac{1}{x}+3 \log (x)-\frac{1}{10} \left (15-\sqrt{5}\right ) \log \left (2 x-\sqrt{5}+1\right )-\frac{1}{10} \left (15+\sqrt{5}\right ) \log \left (2 x+\sqrt{5}+1\right ) \]

[Out]

3/(2*x^2) - x^(-1) + 3*Log[x] - ((15 - Sqrt[5])*Log[1 - Sqrt[5] + 2*x])/10 - ((15 + Sqrt[5])*Log[1 + Sqrt[5] +
2*x])/10

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Rubi [A]  time = 0.0615783, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {1628, 632, 31} \[ \frac{3}{2 x^2}-\frac{1}{x}+3 \log (x)-\frac{1}{10} \left (15-\sqrt{5}\right ) \log \left (2 x-\sqrt{5}+1\right )-\frac{1}{10} \left (15+\sqrt{5}\right ) \log \left (2 x+\sqrt{5}+1\right ) \]

Antiderivative was successfully verified.

[In]

Int[(3 - 4*x - 5*x^2 + 3*x^3)/(x^3*(-1 + x + x^2)),x]

[Out]

3/(2*x^2) - x^(-1) + 3*Log[x] - ((15 - Sqrt[5])*Log[1 - Sqrt[5] + 2*x])/10 - ((15 + Sqrt[5])*Log[1 + Sqrt[5] +
2*x])/10

Rule 1628

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 632

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[
(c*d - e*(b/2 - q/2))/q, Int[1/(b/2 - q/2 + c*x), x], x] - Dist[(c*d - e*(b/2 + q/2))/q, Int[1/(b/2 + q/2 + c*
x), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] && NiceSqrtQ[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{3-4 x-5 x^2+3 x^3}{x^3 \left (-1+x+x^2\right )} \, dx &=\int \left (-\frac{3}{x^3}+\frac{1}{x^2}+\frac{3}{x}+\frac{-1-3 x}{-1+x+x^2}\right ) \, dx\\ &=\frac{3}{2 x^2}-\frac{1}{x}+3 \log (x)+\int \frac{-1-3 x}{-1+x+x^2} \, dx\\ &=\frac{3}{2 x^2}-\frac{1}{x}+3 \log (x)+\frac{1}{10} \left (-15+\sqrt{5}\right ) \int \frac{1}{\frac{1}{2}-\frac{\sqrt{5}}{2}+x} \, dx-\frac{1}{10} \left (15+\sqrt{5}\right ) \int \frac{1}{\frac{1}{2}+\frac{\sqrt{5}}{2}+x} \, dx\\ &=\frac{3}{2 x^2}-\frac{1}{x}+3 \log (x)-\frac{1}{10} \left (15-\sqrt{5}\right ) \log \left (1-\sqrt{5}+2 x\right )-\frac{1}{10} \left (15+\sqrt{5}\right ) \log \left (1+\sqrt{5}+2 x\right )\\ \end{align*}

Mathematica [A]  time = 0.0368658, size = 58, normalized size = 0.89 \[ \frac{1}{10} \left (\frac{15}{x^2}-\frac{10}{x}+\left (\sqrt{5}-15\right ) \log \left (-2 x+\sqrt{5}-1\right )+30 \log (x)-\left (15+\sqrt{5}\right ) \log \left (2 x+\sqrt{5}+1\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(3 - 4*x - 5*x^2 + 3*x^3)/(x^3*(-1 + x + x^2)),x]

[Out]

(15/x^2 - 10/x + (-15 + Sqrt[5])*Log[-1 + Sqrt[5] - 2*x] + 30*Log[x] - (15 + Sqrt[5])*Log[1 + Sqrt[5] + 2*x])/
10

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Maple [A]  time = 0.007, size = 41, normalized size = 0.6 \begin{align*} -{x}^{-1}+{\frac{3}{2\,{x}^{2}}}+3\,\ln \left ( x \right ) -{\frac{3\,\ln \left ({x}^{2}+x-1 \right ) }{2}}-{\frac{\sqrt{5}}{5}{\it Artanh} \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{5}}{5}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^3-5*x^2-4*x+3)/x^3/(x^2+x-1),x)

[Out]

-1/x+3/2/x^2+3*ln(x)-3/2*ln(x^2+x-1)-1/5*5^(1/2)*arctanh(1/5*(1+2*x)*5^(1/2))

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Maxima [A]  time = 1.67304, size = 69, normalized size = 1.06 \begin{align*} \frac{1}{10} \, \sqrt{5} \log \left (\frac{2 \, x - \sqrt{5} + 1}{2 \, x + \sqrt{5} + 1}\right ) - \frac{2 \, x - 3}{2 \, x^{2}} - \frac{3}{2} \, \log \left (x^{2} + x - 1\right ) + 3 \, \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^3-5*x^2-4*x+3)/x^3/(x^2+x-1),x, algorithm="maxima")

[Out]

1/10*sqrt(5)*log((2*x - sqrt(5) + 1)/(2*x + sqrt(5) + 1)) - 1/2*(2*x - 3)/x^2 - 3/2*log(x^2 + x - 1) + 3*log(x
)

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Fricas [A]  time = 1.49938, size = 182, normalized size = 2.8 \begin{align*} \frac{\sqrt{5} x^{2} \log \left (\frac{2 \, x^{2} - \sqrt{5}{\left (2 \, x + 1\right )} + 2 \, x + 3}{x^{2} + x - 1}\right ) - 15 \, x^{2} \log \left (x^{2} + x - 1\right ) + 30 \, x^{2} \log \left (x\right ) - 10 \, x + 15}{10 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^3-5*x^2-4*x+3)/x^3/(x^2+x-1),x, algorithm="fricas")

[Out]

1/10*(sqrt(5)*x^2*log((2*x^2 - sqrt(5)*(2*x + 1) + 2*x + 3)/(x^2 + x - 1)) - 15*x^2*log(x^2 + x - 1) + 30*x^2*
log(x) - 10*x + 15)/x^2

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Sympy [A]  time = 0.496657, size = 99, normalized size = 1.52 \begin{align*} 3 \log{\left (x \right )} + \left (- \frac{3}{2} + \frac{\sqrt{5}}{10}\right ) \log{\left (x - \frac{405}{202} - \frac{35 \sqrt{5}}{202} + \frac{110 \left (- \frac{3}{2} + \frac{\sqrt{5}}{10}\right )^{2}}{101} \right )} + \left (- \frac{3}{2} - \frac{\sqrt{5}}{10}\right ) \log{\left (x - \frac{405}{202} + \frac{35 \sqrt{5}}{202} + \frac{110 \left (- \frac{3}{2} - \frac{\sqrt{5}}{10}\right )^{2}}{101} \right )} - \frac{2 x - 3}{2 x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x**3-5*x**2-4*x+3)/x**3/(x**2+x-1),x)

[Out]

3*log(x) + (-3/2 + sqrt(5)/10)*log(x - 405/202 - 35*sqrt(5)/202 + 110*(-3/2 + sqrt(5)/10)**2/101) + (-3/2 - sq
rt(5)/10)*log(x - 405/202 + 35*sqrt(5)/202 + 110*(-3/2 - sqrt(5)/10)**2/101) - (2*x - 3)/(2*x**2)

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Giac [A]  time = 1.20338, size = 74, normalized size = 1.14 \begin{align*} \frac{1}{10} \, \sqrt{5} \log \left (\frac{{\left | 2 \, x - \sqrt{5} + 1 \right |}}{{\left | 2 \, x + \sqrt{5} + 1 \right |}}\right ) - \frac{2 \, x - 3}{2 \, x^{2}} - \frac{3}{2} \, \log \left ({\left | x^{2} + x - 1 \right |}\right ) + 3 \, \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^3-5*x^2-4*x+3)/x^3/(x^2+x-1),x, algorithm="giac")

[Out]

1/10*sqrt(5)*log(abs(2*x - sqrt(5) + 1)/abs(2*x + sqrt(5) + 1)) - 1/2*(2*x - 3)/x^2 - 3/2*log(abs(x^2 + x - 1)
) + 3*log(abs(x))