3.81.21 \(\int \frac {-99+6 x-10891 x^2+660 x^3-10 x^4+(33-2 x) \log (2)}{5445 x^2-330 x^3+5 x^4} \, dx\)

Optimal. Leaf size=30 \[ e^2-2 x+\frac {3-x-\log (2)}{5 (33-x) x} \]

________________________________________________________________________________________

Rubi [A]  time = 0.08, antiderivative size = 32, normalized size of antiderivative = 1.07, number of steps used = 5, number of rules used = 4, integrand size = 47, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.085, Rules used = {1594, 27, 12, 1620} \begin {gather*} -2 x-\frac {30+\log (2)}{165 (33-x)}+\frac {3-\log (2)}{165 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-99 + 6*x - 10891*x^2 + 660*x^3 - 10*x^4 + (33 - 2*x)*Log[2])/(5445*x^2 - 330*x^3 + 5*x^4),x]

[Out]

-2*x + (3 - Log[2])/(165*x) - (30 + Log[2])/(165*(33 - x))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 1594

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

Rule 1620

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-99+6 x-10891 x^2+660 x^3-10 x^4+(33-2 x) \log (2)}{x^2 \left (5445-330 x+5 x^2\right )} \, dx\\ &=\int \frac {-99+6 x-10891 x^2+660 x^3-10 x^4+(33-2 x) \log (2)}{5 (-33+x)^2 x^2} \, dx\\ &=\frac {1}{5} \int \frac {-99+6 x-10891 x^2+660 x^3-10 x^4+(33-2 x) \log (2)}{(-33+x)^2 x^2} \, dx\\ &=\frac {1}{5} \int \left (-10+\frac {-30-\log (2)}{33 (-33+x)^2}+\frac {-3+\log (2)}{33 x^2}\right ) \, dx\\ &=-2 x+\frac {3-\log (2)}{165 x}-\frac {30+\log (2)}{165 (33-x)}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 27, normalized size = 0.90 \begin {gather*} \frac {-3+x+330 x^2-10 x^3+\log (2)}{5 (-33+x) x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-99 + 6*x - 10891*x^2 + 660*x^3 - 10*x^4 + (33 - 2*x)*Log[2])/(5445*x^2 - 330*x^3 + 5*x^4),x]

[Out]

(-3 + x + 330*x^2 - 10*x^3 + Log[2])/(5*(-33 + x)*x)

________________________________________________________________________________________

fricas [A]  time = 0.81, size = 30, normalized size = 1.00 \begin {gather*} -\frac {10 \, x^{3} - 330 \, x^{2} - x - \log \relax (2) + 3}{5 \, {\left (x^{2} - 33 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x+33)*log(2)-10*x^4+660*x^3-10891*x^2+6*x-99)/(5*x^4-330*x^3+5445*x^2),x, algorithm="fricas")

[Out]

-1/5*(10*x^3 - 330*x^2 - x - log(2) + 3)/(x^2 - 33*x)

________________________________________________________________________________________

giac [A]  time = 0.20, size = 20, normalized size = 0.67 \begin {gather*} -2 \, x + \frac {x + \log \relax (2) - 3}{5 \, {\left (x^{2} - 33 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x+33)*log(2)-10*x^4+660*x^3-10891*x^2+6*x-99)/(5*x^4-330*x^3+5445*x^2),x, algorithm="giac")

[Out]

-2*x + 1/5*(x + log(2) - 3)/(x^2 - 33*x)

________________________________________________________________________________________

maple [A]  time = 0.05, size = 23, normalized size = 0.77

method	result	size
gosper	\(\frac {-10 x^{3}+\ln \relax (2)+10891 x -3}{5 x \left (x -33\right )}\)	\(23\)
risch	\(-2 x +\frac {\frac {x}{5}+\frac {\ln \relax (2)}{5}-\frac {3}{5}}{x \left (x -33\right )}\)	\(23\)
norman	\(\frac {\frac {10891 x}{5}-2 x^{3}+\frac {\ln \relax (2)}{5}-\frac {3}{5}}{x \left (x -33\right )}\)	\(24\)
default	\(-2 x -\frac {-\frac {\ln \relax (2)}{33}-\frac {10}{11}}{5 \left (x -33\right )}-\frac {\frac {\ln \relax (2)}{33}-\frac {1}{11}}{5 x}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x+33)*ln(2)-10*x^4+660*x^3-10891*x^2+6*x-99)/(5*x^4-330*x^3+5445*x^2),x,method=_RETURNVERBOSE)

[Out]

1/5/x*(-10*x^3+ln(2)+10891*x-3)/(x-33)

________________________________________________________________________________________

maxima [A]  time = 0.36, size = 20, normalized size = 0.67 \begin {gather*} -2 \, x + \frac {x + \log \relax (2) - 3}{5 \, {\left (x^{2} - 33 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x+33)*log(2)-10*x^4+660*x^3-10891*x^2+6*x-99)/(5*x^4-330*x^3+5445*x^2),x, algorithm="maxima")

[Out]

-2*x + 1/5*(x + log(2) - 3)/(x^2 - 33*x)

________________________________________________________________________________________

mupad [B]  time = 5.45, size = 34, normalized size = 1.13 \begin {gather*} -2\,x-\frac {\ln \relax (2)+x\,\left (\ln \left (\frac {2^{31/33}\,4^{1/33}}{2}\right )+1\right )-3}{165\,x-5\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(2)*(2*x - 33) - 6*x + 10891*x^2 - 660*x^3 + 10*x^4 + 99)/(5445*x^2 - 330*x^3 + 5*x^4),x)

[Out]

- 2*x - (log(2) + x*(log((2^(31/33)*4^(1/33))/2) + 1) - 3)/(165*x - 5*x^2)

________________________________________________________________________________________

sympy [A]  time = 0.32, size = 19, normalized size = 0.63 \begin {gather*} - 2 x - \frac {- x - \log {\relax (2 )} + 3}{5 x^{2} - 165 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x+33)*ln(2)-10*x**4+660*x**3-10891*x**2+6*x-99)/(5*x**4-330*x**3+5445*x**2),x)

[Out]

-2*x - (-x - log(2) + 3)/(5*x**2 - 165*x)

________________________________________________________________________________________