\(\int \frac {-2916-3240 e^4 x^2}{e^8 x^5} \, dx\) [4397]

Optimal result

Integrand size = 17, antiderivative size = 14 \[ \int \frac {-2916-3240 e^4 x^2}{e^8 x^5} \, dx=9 \left (10+\frac {9}{e^4 x^2}\right )^2 \]

[Out]

3*(10+9*exp(-2)^2/x^2)*(30+27*exp(-2)^2/x^2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.21, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {12, 14} \[ \int \frac {-2916-3240 e^4 x^2}{e^8 x^5} \, dx=\frac {729}{e^8 x^4}+\frac {1620}{e^4 x^2} \]

[In]

Int[(-2916 - 3240*E^4*x^2)/(E^8*x^5),x]

[Out]

729/(E^8*x^4) + 1620/(E^4*x^2)

Rule 12

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {-2916-3240 e^4 x^2}{x^5} \, dx}{e^8} \\ & = \frac {\int \left (-\frac {2916}{x^5}-\frac {3240 e^4}{x^3}\right ) \, dx}{e^8} \\ & = \frac {729}{e^8 x^4}+\frac {1620}{e^4 x^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.50 \[ \int \frac {-2916-3240 e^4 x^2}{e^8 x^5} \, dx=-\frac {324 \left (-\frac {9}{4 x^4}-\frac {5 e^4}{x^2}\right )}{e^8} \]

[In]

Integrate[(-2916 - 3240*E^4*x^2)/(E^8*x^5),x]

[Out]

(-324*(-9/(4*x^4) - (5*E^4)/x^2))/E^8

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14

method	result	size
risch	\(\frac {{\mathrm e}^{-8} \left (1620 x^{2} {\mathrm e}^{4}+729\right )}{x^{4}}\)	\(16\)
default	\(324 \,{\mathrm e}^{-8} \left (\frac {9}{4 x^{4}}+\frac {5 \,{\mathrm e}^{4}}{x^{2}}\right )\)	\(20\)
parallelrisch	\(\frac {{\mathrm e}^{-8} \left (1620 x^{2} {\mathrm e}^{4}+729\right )}{x^{4}}\)	\(20\)
gosper	\(\frac {81 \left (20 x^{2} {\mathrm e}^{4}+9\right ) {\mathrm e}^{-8}}{x^{4}}\)	\(21\)
norman	\(\frac {\left (729 \,{\mathrm e}^{-2}+1620 x^{2} {\mathrm e}^{2}\right ) {\mathrm e}^{-6}}{x^{4}}\)	\(23\)

[In]

int((-3240*x^2*exp(2)^2-2916)/x^5/exp(2)^4,x,method=_RETURNVERBOSE)

[Out]

exp(-8)*(1620*x^2*exp(4)+729)/x^4

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {-2916-3240 e^4 x^2}{e^8 x^5} \, dx=\frac {81 \, {\left (20 \, x^{2} e^{4} + 9\right )} e^{\left (-8\right )}}{x^{4}} \]

[In]

integrate((-3240*x^2*exp(2)^2-2916)/x^5/exp(2)^4,x, algorithm="fricas")

[Out]

81*(20*x^2*e^4 + 9)*e^(-8)/x^4

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.36 \[ \int \frac {-2916-3240 e^4 x^2}{e^8 x^5} \, dx=- \frac {- 1620 x^{2} e^{4} - 729}{x^{4} e^{8}} \]

[In]

integrate((-3240*x**2*exp(2)**2-2916)/x**5/exp(2)**4,x)

[Out]

-(-1620*x**2*exp(4) - 729)*exp(-8)/x**4

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {-2916-3240 e^4 x^2}{e^8 x^5} \, dx=\frac {81 \, {\left (20 \, x^{2} e^{4} + 9\right )} e^{\left (-8\right )}}{x^{4}} \]

[In]

integrate((-3240*x^2*exp(2)^2-2916)/x^5/exp(2)^4,x, algorithm="maxima")

[Out]

81*(20*x^2*e^4 + 9)*e^(-8)/x^4

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {-2916-3240 e^4 x^2}{e^8 x^5} \, dx=\frac {81 \, {\left (20 \, x^{2} e^{4} + 9\right )} e^{\left (-8\right )}}{x^{4}} \]

[In]

integrate((-3240*x^2*exp(2)^2-2916)/x^5/exp(2)^4,x, algorithm="giac")

[Out]

81*(20*x^2*e^4 + 9)*e^(-8)/x^4

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.07 \[ \int \frac {-2916-3240 e^4 x^2}{e^8 x^5} \, dx=\frac {{\mathrm {e}}^{-8}\,\left (1620\,{\mathrm {e}}^4\,x^2+729\right )}{x^4} \]

[In]

int(-(exp(-8)*(3240*x^2*exp(4) + 2916))/x^5,x)

[Out]

(exp(-8)*(1620*x^2*exp(4) + 729))/x^4