\(\int -\frac {2 e^{14}}{144 x^2+216 e^5 x^2+81 e^{10} x^2} \, dx\) [2119]

Optimal result

Integrand size = 29, antiderivative size = 19 \[ \int -\frac {2 e^{14}}{144 x^2+216 e^5 x^2+81 e^{10} x^2} \, dx=\frac {2 e^4}{9 \left (3+\frac {4}{e^5}\right )^2 x} \]

[Out]

2/9/x*exp(4)/(3+4/exp(5))^2

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {6, 12, 30} \[ \int -\frac {2 e^{14}}{144 x^2+216 e^5 x^2+81 e^{10} x^2} \, dx=\frac {2 e^{14}}{9 \left (4+3 e^5\right )^2 x} \]

[In]

Int[(-2*E^14)/(144*x^2 + 216*E^5*x^2 + 81*E^10*x^2),x]

[Out]

(2*E^14)/(9*(4 + 3*E^5)^2*x)

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, 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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = \int -\frac {2 e^{14}}{81 e^{10} x^2+\left (144+216 e^5\right ) x^2} \, dx \\ & = \int -\frac {2 e^{14}}{\left (144+216 e^5+81 e^{10}\right ) x^2} \, dx \\ & = -\frac {\left (2 e^{14}\right ) \int \frac {1}{x^2} \, dx}{9 \left (4+3 e^5\right )^2} \\ & = \frac {2 e^{14}}{9 \left (4+3 e^5\right )^2 x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int -\frac {2 e^{14}}{144 x^2+216 e^5 x^2+81 e^{10} x^2} \, dx=\frac {2 e^{14}}{9 \left (4+3 e^5\right )^2 x} \]

[In]

Integrate[(-2*E^14)/(144*x^2 + 216*E^5*x^2 + 81*E^10*x^2),x]

[Out]

(2*E^14)/(9*(4 + 3*E^5)^2*x)

Maple [A] (verified)

Time = 0.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05

method	result	size
norman	\(\frac {2 \,{\mathrm e}^{4} {\mathrm e}^{10}}{9 \left (3 \,{\mathrm e}^{5}+4\right )^{2} x}\)	\(20\)
risch	\(\frac {2 \,{\mathrm e}^{14}}{9 x \left (9 \,{\mathrm e}^{10}+24 \,{\mathrm e}^{5}+16\right )}\)	\(20\)
gosper	\(\frac {2 \,{\mathrm e}^{4} {\mathrm e}^{10}}{9 x \left (9 \,{\mathrm e}^{10}+24 \,{\mathrm e}^{5}+16\right )}\)	\(26\)
default	\(\frac {2 \,{\mathrm e}^{4} {\mathrm e}^{10}}{9 x \left (9 \,{\mathrm e}^{10}+24 \,{\mathrm e}^{5}+16\right )}\)	\(26\)
parallelrisch	\(\frac {2 \,{\mathrm e}^{4} {\mathrm e}^{10}}{9 x \left (9 \,{\mathrm e}^{10}+24 \,{\mathrm e}^{5}+16\right )}\)	\(26\)

[In]

int(-2*exp(4)*exp(5)^2/(81*x^2*exp(5)^2+216*x^2*exp(5)+144*x^2),x,method=_RETURNVERBOSE)

[Out]

2/9*exp(4)*exp(5)^2/(3*exp(5)+4)^2/x

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int -\frac {2 e^{14}}{144 x^2+216 e^5 x^2+81 e^{10} x^2} \, dx=\frac {2 \, e^{14}}{9 \, {\left (9 \, x e^{10} + 24 \, x e^{5} + 16 \, x\right )}} \]

[In]

integrate(-2*exp(4)*exp(5)^2/(81*x^2*exp(5)^2+216*x^2*exp(5)+144*x^2),x, algorithm="fricas")

[Out]

2/9*e^14/(9*x*e^10 + 24*x*e^5 + 16*x)

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int -\frac {2 e^{14}}{144 x^2+216 e^5 x^2+81 e^{10} x^2} \, dx=\frac {2 e^{14}}{x \left (144 + 216 e^{5} + 81 e^{10}\right )} \]

[In]

integrate(-2*exp(4)*exp(5)**2/(81*x**2*exp(5)**2+216*x**2*exp(5)+144*x**2),x)

[Out]

2*exp(14)/(x*(144 + 216*exp(5) + 81*exp(10)))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int -\frac {2 e^{14}}{144 x^2+216 e^5 x^2+81 e^{10} x^2} \, dx=\frac {2 \, e^{14}}{9 \, x {\left (9 \, e^{10} + 24 \, e^{5} + 16\right )}} \]

[In]

integrate(-2*exp(4)*exp(5)^2/(81*x^2*exp(5)^2+216*x^2*exp(5)+144*x^2),x, algorithm="maxima")

[Out]

2/9*e^14/(x*(9*e^10 + 24*e^5 + 16))

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int -\frac {2 e^{14}}{144 x^2+216 e^5 x^2+81 e^{10} x^2} \, dx=\frac {2 \, e^{14}}{9 \, x {\left (9 \, e^{10} + 24 \, e^{5} + 16\right )}} \]

[In]

integrate(-2*exp(4)*exp(5)^2/(81*x^2*exp(5)^2+216*x^2*exp(5)+144*x^2),x, algorithm="giac")

[Out]

2/9*e^14/(x*(9*e^10 + 24*e^5 + 16))

Mupad [B] (verification not implemented)

Time = 10.18 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int -\frac {2 e^{14}}{144 x^2+216 e^5 x^2+81 e^{10} x^2} \, dx=\frac {2\,{\mathrm {e}}^{14}}{9\,x\,{\left (3\,{\mathrm {e}}^5+4\right )}^2} \]

[In]

int(-(2*exp(14))/(216*x^2*exp(5) + 81*x^2*exp(10) + 144*x^2),x)

[Out]

(2*exp(14))/(9*x*(3*exp(5) + 4)^2)