\(\int \frac {e^{\frac {49+126 x^2+81 x^4-9 x^5}{9 x^4}} (-196-252 x^2-9 x^5)}{9 x^5} \, dx\) [5737]

Optimal result

Integrand size = 45, antiderivative size = 24 \[ \int \frac {e^{\frac {49+126 x^2+81 x^4-9 x^5}{9 x^4}} \left (-196-252 x^2-9 x^5\right )}{9 x^5} \, dx=e^{-x+\frac {1}{9} \left (2+\frac {7 \left (\frac {1}{x}+x\right )}{x}\right )^2} \]

[Out]

exp(1/3*(2+7*(x+1/x)/x)*(2/3+7/3*(x+1/x)/x)-x)

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.79, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {12, 6820, 6838} \[ \int \frac {e^{\frac {49+126 x^2+81 x^4-9 x^5}{9 x^4}} \left (-196-252 x^2-9 x^5\right )}{9 x^5} \, dx=e^{\frac {49}{9 x^4}+\frac {14}{x^2}-x+9} \]

[In]

Int[(E^((49 + 126*x^2 + 81*x^4 - 9*x^5)/(9*x^4))*(-196 - 252*x^2 - 9*x^5))/(9*x^5),x]

[Out]

E^(9 + 49/(9*x^4) + 14/x^2 - 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 6820

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6838

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[q*(F^v/Log[F]), x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{9} \int \frac {e^{\frac {49+126 x^2+81 x^4-9 x^5}{9 x^4}} \left (-196-252 x^2-9 x^5\right )}{x^5} \, dx \\ & = \frac {1}{9} \int \frac {e^{9+\frac {49}{9 x^4}+\frac {14}{x^2}-x} \left (-196-252 x^2-9 x^5\right )}{x^5} \, dx \\ & = e^{9+\frac {49}{9 x^4}+\frac {14}{x^2}-x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.79 \[ \int \frac {e^{\frac {49+126 x^2+81 x^4-9 x^5}{9 x^4}} \left (-196-252 x^2-9 x^5\right )}{9 x^5} \, dx=e^{9+\frac {49}{9 x^4}+\frac {14}{x^2}-x} \]

[In]

Integrate[(E^((49 + 126*x^2 + 81*x^4 - 9*x^5)/(9*x^4))*(-196 - 252*x^2 - 9*x^5))/(9*x^5),x]

[Out]

E^(9 + 49/(9*x^4) + 14/x^2 - x)

Maple [A] (verified)

Time = 0.11 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00

method	result	size
gosper	\({\mathrm e}^{-\frac {9 x^{5}-81 x^{4}-126 x^{2}-49}{9 x^{4}}}\)	\(24\)
norman	\({\mathrm e}^{\frac {-9 x^{5}+81 x^{4}+126 x^{2}+49}{9 x^{4}}}\)	\(24\)
risch	\({\mathrm e}^{-\frac {9 x^{5}-81 x^{4}-126 x^{2}-49}{9 x^{4}}}\)	\(24\)
parallelrisch	\({\mathrm e}^{-\frac {9 x^{5}-81 x^{4}-126 x^{2}-49}{9 x^{4}}}\)	\(24\)

[In]

int(1/9*(-9*x^5-252*x^2-196)*exp(1/9*(-9*x^5+81*x^4+126*x^2+49)/x^4)/x^5,x,method=_RETURNVERBOSE)

[Out]

exp(-1/9*(9*x^5-81*x^4-126*x^2-49)/x^4)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96 \[ \int \frac {e^{\frac {49+126 x^2+81 x^4-9 x^5}{9 x^4}} \left (-196-252 x^2-9 x^5\right )}{9 x^5} \, dx=e^{\left (-\frac {9 \, x^{5} - 81 \, x^{4} - 126 \, x^{2} - 49}{9 \, x^{4}}\right )} \]

[In]

integrate(1/9*(-9*x^5-252*x^2-196)*exp(1/9*(-9*x^5+81*x^4+126*x^2+49)/x^4)/x^5,x, algorithm="fricas")

[Out]

e^(-1/9*(9*x^5 - 81*x^4 - 126*x^2 - 49)/x^4)

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {e^{\frac {49+126 x^2+81 x^4-9 x^5}{9 x^4}} \left (-196-252 x^2-9 x^5\right )}{9 x^5} \, dx=e^{\frac {- x^{5} + 9 x^{4} + 14 x^{2} + \frac {49}{9}}{x^{4}}} \]

[In]

integrate(1/9*(-9*x**5-252*x**2-196)*exp(1/9*(-9*x**5+81*x**4+126*x**2+49)/x**4)/x**5,x)

[Out]

exp((-x**5 + 9*x**4 + 14*x**2 + 49/9)/x**4)

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.67 \[ \int \frac {e^{\frac {49+126 x^2+81 x^4-9 x^5}{9 x^4}} \left (-196-252 x^2-9 x^5\right )}{9 x^5} \, dx=e^{\left (-x + \frac {14}{x^{2}} + \frac {49}{9 \, x^{4}} + 9\right )} \]

[In]

integrate(1/9*(-9*x^5-252*x^2-196)*exp(1/9*(-9*x^5+81*x^4+126*x^2+49)/x^4)/x^5,x, algorithm="maxima")

[Out]

e^(-x + 14/x^2 + 49/9/x^4 + 9)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.67 \[ \int \frac {e^{\frac {49+126 x^2+81 x^4-9 x^5}{9 x^4}} \left (-196-252 x^2-9 x^5\right )}{9 x^5} \, dx=e^{\left (-x + \frac {14}{x^{2}} + \frac {49}{9 \, x^{4}} + 9\right )} \]

[In]

integrate(1/9*(-9*x^5-252*x^2-196)*exp(1/9*(-9*x^5+81*x^4+126*x^2+49)/x^4)/x^5,x, algorithm="giac")

[Out]

e^(-x + 14/x^2 + 49/9/x^4 + 9)

Mupad [B] (verification not implemented)

Time = 11.17 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.79 \[ \int \frac {e^{\frac {49+126 x^2+81 x^4-9 x^5}{9 x^4}} \left (-196-252 x^2-9 x^5\right )}{9 x^5} \, dx={\mathrm {e}}^{-x}\,{\mathrm {e}}^9\,{\mathrm {e}}^{\frac {14}{x^2}}\,{\mathrm {e}}^{\frac {49}{9\,x^4}} \]

[In]

int(-(exp((14*x^2 + 9*x^4 - x^5 + 49/9)/x^4)*(252*x^2 + 9*x^5 + 196))/(9*x^5),x)

[Out]

exp(-x)*exp(9)*exp(14/x^2)*exp(49/(9*x^4))