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

Optimal. Leaf size=19 \[ \frac {2 e^4}{9 \left (3+\frac {4}{e^5}\right )^2 x} \]

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Rubi [A]  time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {6, 12, 30} \begin {gather*} \frac {2 e^{14}}{9 \left (4+3 e^5\right )^2 x} \end {gather*}

Antiderivative was successfully verified.

[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 {gather*} \begin {aligned} \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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 19, normalized size = 1.00 \begin {gather*} \frac {2 e^{14}}{9 \left (4+3 e^5\right )^2 x} \end {gather*}

Antiderivative was successfully verified.

[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)

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fricas [A]  time = 0.50, size = 20, normalized size = 1.05 \begin {gather*} \frac {2 \, e^{14}}{9 \, {\left (9 \, x e^{10} + 24 \, x e^{5} + 16 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [A]  time = 1.16, size = 19, normalized size = 1.00 \begin {gather*} \frac {2 \, e^{14}}{9 \, x {\left (9 \, e^{10} + 24 \, e^{5} + 16\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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maple [A]  time = 0.38, size = 20, normalized size = 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	\(\frac {2 \,{\mathrm e}^{4} {\mathrm e}^{10}}{9 x \left (9 \,{\mathrm e}^{10}+24 \,{\mathrm e}^{5}+16\right )}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[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

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maxima [A]  time = 0.38, size = 19, normalized size = 1.00 \begin {gather*} \frac {2 \, e^{14}}{9 \, x {\left (9 \, e^{10} + 24 \, e^{5} + 16\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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mupad [B]  time = 1.17, size = 15, normalized size = 0.79 \begin {gather*} \frac {2\,{\mathrm {e}}^{14}}{9\,x\,{\left (3\,{\mathrm {e}}^5+4\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [A]  time = 0.07, size = 17, normalized size = 0.89 \begin {gather*} \frac {2 e^{14}}{x \left (144 + 216 e^{5} + 81 e^{10}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)))

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