\(\int -\frac {84 e^2}{6480+80 e^2+360 x+5 x^2+e (1440+40 x)} \, dx\) [5042]

Optimal result

Integrand size = 29, antiderivative size = 19 \[ \int -\frac {84 e^2}{6480+80 e^2+360 x+5 x^2+e (1440+40 x)} \, dx=e^2 \left (10+\frac {84}{5 (4 (9+e)+x)}\right ) \]

[Out]

exp(2)*(10+4/(5/21*x+60/7+20/21*exp(1)))

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {12, 2006, 27, 32} \[ \int -\frac {84 e^2}{6480+80 e^2+360 x+5 x^2+e (1440+40 x)} \, dx=\frac {84 e^2}{5 (x+4 (9+e))} \]

[In]

Int[(-84*E^2)/(6480 + 80*E^2 + 360*x + 5*x^2 + E*(1440 + 40*x)),x]

[Out]

(84*E^2)/(5*(4*(9 + E) + 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 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 2006

Int[(u_)^(p_), x_Symbol] :> Int[ExpandToSum[u, x]^p, x] /; FreeQ[p, x] && QuadraticQ[u, x] &&  !QuadraticMatch
Q[u, x]

Rubi steps \begin{align*} \text {integral}& = -\left (\left (84 e^2\right ) \int \frac {1}{6480+80 e^2+360 x+5 x^2+e (1440+40 x)} \, dx\right ) \\ & = -\left (\left (84 e^2\right ) \int \frac {1}{80 (9+e)^2+40 (9+e) x+5 x^2} \, dx\right ) \\ & = -\left (\left (84 e^2\right ) \int \frac {1}{5 (36+4 e+x)^2} \, dx\right ) \\ & = -\left (\frac {1}{5} \left (84 e^2\right ) \int \frac {1}{(36+4 e+x)^2} \, dx\right ) \\ & = \frac {84 e^2}{5 (4 (9+e)+x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int -\frac {84 e^2}{6480+80 e^2+360 x+5 x^2+e (1440+40 x)} \, dx=\frac {84 e^2}{5 (36+4 e+x)} \]

[In]

Integrate[(-84*E^2)/(6480 + 80*E^2 + 360*x + 5*x^2 + E*(1440 + 40*x)),x]

[Out]

(84*E^2)/(5*(36 + 4*E + x))

Maple [A] (verified)

Time = 0.44 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.74

method	result	size
gosper	\(\frac {84 \,{\mathrm e}^{2}}{5 \left (4 \,{\mathrm e}+x +36\right )}\)	\(14\)
norman	\(\frac {84 \,{\mathrm e}^{2}}{5 \left (4 \,{\mathrm e}+x +36\right )}\)	\(14\)
risch	\(\frac {21 \,{\mathrm e}^{2}}{5 \left ({\mathrm e}+\frac {x}{4}+9\right )}\)	\(14\)
parallelrisch	\(\frac {84 \,{\mathrm e}^{2}}{5 \left (4 \,{\mathrm e}+x +36\right )}\)	\(14\)
meijerg	\(-\frac {21 \,{\mathrm e}^{2} x}{5 \left (4 \,{\mathrm e}+36\right ) \left ({\mathrm e}+9\right ) \left (1+\frac {x}{4 \,{\mathrm e}+36}\right )}\)	\(33\)

[In]

int(-84*exp(2)/(80*exp(1)^2+(40*x+1440)*exp(1)+5*x^2+360*x+6480),x,method=_RETURNVERBOSE)

[Out]

84/5*exp(2)/(4*exp(1)+x+36)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.68 \[ \int -\frac {84 e^2}{6480+80 e^2+360 x+5 x^2+e (1440+40 x)} \, dx=\frac {84 \, e^{2}}{5 \, {\left (x + 4 \, e + 36\right )}} \]

[In]

integrate(-84*exp(2)/(80*exp(1)^2+(40*x+1440)*exp(1)+5*x^2+360*x+6480),x, algorithm="fricas")

[Out]

84/5*e^2/(x + 4*e + 36)

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.74 \[ \int -\frac {84 e^2}{6480+80 e^2+360 x+5 x^2+e (1440+40 x)} \, dx=\frac {84 e^{2}}{5 x + 20 e + 180} \]

[In]

integrate(-84*exp(2)/(80*exp(1)**2+(40*x+1440)*exp(1)+5*x**2+360*x+6480),x)

[Out]

84*exp(2)/(5*x + 20*E + 180)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.68 \[ \int -\frac {84 e^2}{6480+80 e^2+360 x+5 x^2+e (1440+40 x)} \, dx=\frac {84 \, e^{2}}{5 \, {\left (x + 4 \, e + 36\right )}} \]

[In]

integrate(-84*exp(2)/(80*exp(1)^2+(40*x+1440)*exp(1)+5*x^2+360*x+6480),x, algorithm="maxima")

[Out]

84/5*e^2/(x + 4*e + 36)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.68 \[ \int -\frac {84 e^2}{6480+80 e^2+360 x+5 x^2+e (1440+40 x)} \, dx=\frac {84 \, e^{2}}{5 \, {\left (x + 4 \, e + 36\right )}} \]

[In]

integrate(-84*exp(2)/(80*exp(1)^2+(40*x+1440)*exp(1)+5*x^2+360*x+6480),x, algorithm="giac")

[Out]

84/5*e^2/(x + 4*e + 36)

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int -\frac {84 e^2}{6480+80 e^2+360 x+5 x^2+e (1440+40 x)} \, dx=\frac {84\,{\mathrm {e}}^2}{5\,\left (x+4\,\mathrm {e}+36\right )} \]

[In]

int(-(84*exp(2))/(360*x + 80*exp(2) + 5*x^2 + exp(1)*(40*x + 1440) + 6480),x)

[Out]

(84*exp(2))/(5*(x + 4*exp(1) + 36))