\(\int \frac {-72-2 e^{e^2}+48 x-12 x^2+x^3}{-64+48 x-12 x^2+x^3} \, dx\) [6712]

Optimal result

Integrand size = 36, antiderivative size = 24 \[ \int \frac {-72-2 e^{e^2}+48 x-12 x^2+x^3}{-64+48 x-12 x^2+x^3} \, dx=e^3+x+\frac {4+e^{e^2}}{-16 x+(4+x)^2} \]

[Out]

x+(exp(exp(2))+4)/((4+x)^2-16*x)+exp(3)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.71, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.028, Rules used = {2099} \[ \int \frac {-72-2 e^{e^2}+48 x-12 x^2+x^3}{-64+48 x-12 x^2+x^3} \, dx=x+\frac {4+e^{e^2}}{(4-x)^2} \]

[In]

Int[(-72 - 2*E^E^2 + 48*x - 12*x^2 + x^3)/(-64 + 48*x - 12*x^2 + x^3),x]

[Out]

(4 + E^E^2)/(4 - x)^2 + x

Rule 2099

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps \begin{align*} \text {integral}& = \int \left (1-\frac {2 \left (4+e^{e^2}\right )}{(-4+x)^3}\right ) \, dx \\ & = \frac {4+e^{e^2}}{(4-x)^2}+x \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.62 \[ \int \frac {-72-2 e^{e^2}+48 x-12 x^2+x^3}{-64+48 x-12 x^2+x^3} \, dx=\frac {4+e^{e^2}}{(-4+x)^2}+x \]

[In]

Integrate[(-72 - 2*E^E^2 + 48*x - 12*x^2 + x^3)/(-64 + 48*x - 12*x^2 + x^3),x]

[Out]

(4 + E^E^2)/(-4 + x)^2 + x

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.71

method	result	size
default	\(x -\frac {-8-2 \,{\mathrm e}^{{\mathrm e}^{2}}}{2 \left (x -4\right )^{2}}\)	\(17\)
norman	\(\frac {x^{3}+{\mathrm e}^{{\mathrm e}^{2}}-48 x +132}{\left (x -4\right )^{2}}\)	\(18\)
gosper	\(\frac {x^{3}+{\mathrm e}^{{\mathrm e}^{2}}-48 x +132}{x^{2}-8 x +16}\)	\(23\)
parallelrisch	\(\frac {x^{3}+{\mathrm e}^{{\mathrm e}^{2}}-48 x +132}{x^{2}-8 x +16}\)	\(23\)
risch	\(x +\frac {{\mathrm e}^{{\mathrm e}^{2}}}{x^{2}-8 x +16}+\frac {4}{x^{2}-8 x +16}\)	\(29\)

[In]

int((-2*exp(exp(2))+x^3-12*x^2+48*x-72)/(x^3-12*x^2+48*x-64),x,method=_RETURNVERBOSE)

[Out]

x-1/2*(-8-2*exp(exp(2)))/(x-4)^2

Fricas [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.12 \[ \int \frac {-72-2 e^{e^2}+48 x-12 x^2+x^3}{-64+48 x-12 x^2+x^3} \, dx=\frac {x^{3} - 8 \, x^{2} + 16 \, x + e^{\left (e^{2}\right )} + 4}{x^{2} - 8 \, x + 16} \]

[In]

integrate((-2*exp(exp(2))+x^3-12*x^2+48*x-72)/(x^3-12*x^2+48*x-64),x, algorithm="fricas")

[Out]

(x^3 - 8*x^2 + 16*x + e^(e^2) + 4)/(x^2 - 8*x + 16)

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.62 \[ \int \frac {-72-2 e^{e^2}+48 x-12 x^2+x^3}{-64+48 x-12 x^2+x^3} \, dx=x + \frac {4 + e^{e^{2}}}{x^{2} - 8 x + 16} \]

[In]

integrate((-2*exp(exp(2))+x**3-12*x**2+48*x-72)/(x**3-12*x**2+48*x-64),x)

[Out]

x + (4 + exp(exp(2)))/(x**2 - 8*x + 16)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.75 \[ \int \frac {-72-2 e^{e^2}+48 x-12 x^2+x^3}{-64+48 x-12 x^2+x^3} \, dx=x + \frac {e^{\left (e^{2}\right )} + 4}{x^{2} - 8 \, x + 16} \]

[In]

integrate((-2*exp(exp(2))+x^3-12*x^2+48*x-72)/(x^3-12*x^2+48*x-64),x, algorithm="maxima")

[Out]

x + (e^(e^2) + 4)/(x^2 - 8*x + 16)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.54 \[ \int \frac {-72-2 e^{e^2}+48 x-12 x^2+x^3}{-64+48 x-12 x^2+x^3} \, dx=x + \frac {e^{\left (e^{2}\right )} + 4}{{\left (x - 4\right )}^{2}} \]

[In]

integrate((-2*exp(exp(2))+x^3-12*x^2+48*x-72)/(x^3-12*x^2+48*x-64),x, algorithm="giac")

[Out]

x + (e^(e^2) + 4)/(x - 4)^2

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.54 \[ \int \frac {-72-2 e^{e^2}+48 x-12 x^2+x^3}{-64+48 x-12 x^2+x^3} \, dx=x+\frac {{\mathrm {e}}^{{\mathrm {e}}^2}+4}{{\left (x-4\right )}^2} \]

[In]

int(-(2*exp(exp(2)) - 48*x + 12*x^2 - x^3 + 72)/(48*x - 12*x^2 + x^3 - 64),x)

[Out]

x + (exp(exp(2)) + 4)/(x - 4)^2