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^2-1/x+3*ln(x)-1/10*ln(1+2*x-5^(1/2))*(15-5^(1/2))-1/10*ln(1+2*x+5^(1/2))*(15+5^(1/2))

________________________________________________________________________________________

Rubi [A]  time = 0.06, antiderivative size = 65, normalized size of antiderivative = 1.00, 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 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

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 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]

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 = 0.04, 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

________________________________________________________________________________________

fricas [A]  time = 0.60, size = 66, normalized size = 1.02 \[ \frac {\sqrt {5} x^{2} \log \left (\frac {2 \, x^{2} - \sqrt {5} {\left (2 \, x + 1\right )} + 2 \, x + 3}{x^{2} + x - 1}\right ) - 15 \, x^{2} \log \left (x^{2} + x - 1\right ) + 30 \, x^{2} \log \relax (x) - 10 \, x + 15}{10 \, x^{2}} \]

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

________________________________________________________________________________________

giac [A]  time = 0.30, size = 55, normalized size = 0.85 \[ \frac {1}{10} \, \sqrt {5} \log \left (\frac {{\left | 2 \, x - \sqrt {5} + 1 \right |}}{{\left | 2 \, x + \sqrt {5} + 1 \right |}}\right ) - \frac {2 \, x - 3}{2 \, x^{2}} - \frac {3}{2} \, \log \left ({\left | x^{2} + x - 1 \right |}\right ) + 3 \, \log \left ({\left | x \right |}\right ) \]

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))

________________________________________________________________________________________

maple [A]  time = 0.01, size = 41, normalized size = 0.63 \[ -\frac {\sqrt {5}\, \arctanh \left (\frac {\left (2 x +1\right ) \sqrt {5}}{5}\right )}{5}+3 \ln \relax (x )-\frac {3 \ln \left (x^{2}+x -1\right )}{2}-\frac {1}{x}+\frac {3}{2 x^{2}} \]

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]

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

________________________________________________________________________________________

maxima [A]  time = 1.77, size = 51, normalized size = 0.78 \[ \frac {1}{10} \, \sqrt {5} \log \left (\frac {2 \, x - \sqrt {5} + 1}{2 \, x + \sqrt {5} + 1}\right ) - \frac {2 \, x - 3}{2 \, x^{2}} - \frac {3}{2} \, \log \left (x^{2} + x - 1\right ) + 3 \, \log \relax (x) \]

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
)

________________________________________________________________________________________

mupad [B]  time = 0.10, size = 48, normalized size = 0.74 \[ 3\,\ln \relax (x)-\frac {x-\frac {3}{2}}{x^2}+\ln \left (x-\frac {\sqrt {5}}{2}+\frac {1}{2}\right )\,\left (\frac {\sqrt {5}}{10}-\frac {3}{2}\right )-\ln \left (x+\frac {\sqrt {5}}{2}+\frac {1}{2}\right )\,\left (\frac {\sqrt {5}}{10}+\frac {3}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.45, size = 99, normalized size = 1.52 \[ 3 \log {\relax (x )} + \left (- \frac {3}{2} + \frac {\sqrt {5}}{10}\right ) \log {\left (x - \frac {405}{202} - \frac {35 \sqrt {5}}{202} + \frac {110 \left (- \frac {3}{2} + \frac {\sqrt {5}}{10}\right )^{2}}{101} \right )} + \left (- \frac {3}{2} - \frac {\sqrt {5}}{10}\right ) \log {\left (x - \frac {405}{202} + \frac {35 \sqrt {5}}{202} + \frac {110 \left (- \frac {3}{2} - \frac {\sqrt {5}}{10}\right )^{2}}{101} \right )} + \frac {3 - 2 x}{2 x^{2}} \]

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) + (3 - 2*x)/(2*x**2)

________________________________________________________________________________________