3.62.63 \(\int \frac {-81 e^{4+x}+675 e^9 x^2}{81 e^{2 x}+625 e^{10} x^6+e^x (-450 e^5 x^3-162 \log (8))+450 e^5 x^3 \log (8)+81 \log ^2(8)} \, dx\)

Optimal. Leaf size=24 \[ \frac {e^4}{e^x-\frac {25 e^5 x^3}{9}-\log (8)} \]

________________________________________________________________________________________

Rubi [A]  time = 0.18, antiderivative size = 25, normalized size of antiderivative = 1.04, number of steps used = 2, number of rules used = 2, integrand size = 68, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {6688, 6686} \begin {gather*} \frac {9 e^4}{-25 e^5 x^3+9 e^x-9 \log (8)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-81*E^(4 + x) + 675*E^9*x^2)/(81*E^(2*x) + 625*E^10*x^6 + E^x*(-450*E^5*x^3 - 162*Log[8]) + 450*E^5*x^3*L
og[8] + 81*Log[8]^2),x]

[Out]

(9*E^4)/(9*E^x - 25*E^5*x^3 - 9*Log[8])

Rule 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rule 6688

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-81 e^{4+x}+675 e^9 x^2}{\left (9 e^x-25 e^5 x^3-9 \log (8)\right )^2} \, dx\\ &=\frac {9 e^4}{9 e^x-25 e^5 x^3-9 \log (8)}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.04, size = 25, normalized size = 1.04 \begin {gather*} -\frac {9 e^4}{-9 e^x+25 e^5 x^3+9 \log (8)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-81*E^(4 + x) + 675*E^9*x^2)/(81*E^(2*x) + 625*E^10*x^6 + E^x*(-450*E^5*x^3 - 162*Log[8]) + 450*E^5
*x^3*Log[8] + 81*Log[8]^2),x]

[Out]

(-9*E^4)/(-9*E^x + 25*E^5*x^3 + 9*Log[8])

________________________________________________________________________________________

fricas [A]  time = 0.62, size = 26, normalized size = 1.08 \begin {gather*} -\frac {9 \, e^{8}}{25 \, x^{3} e^{9} + 27 \, e^{4} \log \relax (2) - 9 \, e^{\left (x + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-81*exp(4)*exp(x)+675*x^2*exp(4)*exp(5))/(81*exp(x)^2+(-486*log(2)-450*x^3*exp(5))*exp(x)+729*log(2
)^2+1350*x^3*exp(5)*log(2)+625*x^6*exp(5)^2),x, algorithm="fricas")

[Out]

-9*e^8/(25*x^3*e^9 + 27*e^4*log(2) - 9*e^(x + 4))

________________________________________________________________________________________

giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-81*exp(4)*exp(x)+675*x^2*exp(4)*exp(5))/(81*exp(x)^2+(-486*log(2)-450*x^3*exp(5))*exp(x)+729*log(2
)^2+1350*x^3*exp(5)*log(2)+625*x^6*exp(5)^2),x, algorithm="giac")

[Out]

Timed out

________________________________________________________________________________________

maple [A]  time = 0.32, size = 23, normalized size = 0.96

method	result	size
norman	\(-\frac {9 \,{\mathrm e}^{4}}{25 x^{3} {\mathrm e}^{5}-9 \,{\mathrm e}^{x}+27 \ln \relax (2)}\)	\(23\)
risch	\(-\frac {9 \,{\mathrm e}^{4}}{25 x^{3} {\mathrm e}^{5}-9 \,{\mathrm e}^{x}+27 \ln \relax (2)}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-81*exp(4)*exp(x)+675*x^2*exp(4)*exp(5))/(81*exp(x)^2+(-486*ln(2)-450*x^3*exp(5))*exp(x)+729*ln(2)^2+1350
*x^3*exp(5)*ln(2)+625*x^6*exp(5)^2),x,method=_RETURNVERBOSE)

[Out]

-9*exp(4)/(25*x^3*exp(5)-9*exp(x)+27*ln(2))

________________________________________________________________________________________

maxima [A]  time = 0.47, size = 22, normalized size = 0.92 \begin {gather*} -\frac {9 \, e^{4}}{25 \, x^{3} e^{5} - 9 \, e^{x} + 27 \, \log \relax (2)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-81*exp(4)*exp(x)+675*x^2*exp(4)*exp(5))/(81*exp(x)^2+(-486*log(2)-450*x^3*exp(5))*exp(x)+729*log(2
)^2+1350*x^3*exp(5)*log(2)+625*x^6*exp(5)^2),x, algorithm="maxima")

[Out]

-9*e^4/(25*x^3*e^5 - 9*e^x + 27*log(2))

________________________________________________________________________________________

mupad [B]  time = 5.18, size = 22, normalized size = 0.92 \begin {gather*} -\frac {9\,{\mathrm {e}}^4}{27\,\ln \relax (2)-9\,{\mathrm {e}}^x+25\,x^3\,{\mathrm {e}}^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((675*x^2*exp(9) - 81*exp(4)*exp(x))/(81*exp(2*x) - exp(x)*(486*log(2) + 450*x^3*exp(5)) + 625*x^6*exp(10)
+ 729*log(2)^2 + 1350*x^3*exp(5)*log(2)),x)

[Out]

-(9*exp(4))/(27*log(2) - 9*exp(x) + 25*x^3*exp(5))

________________________________________________________________________________________

sympy [A]  time = 0.13, size = 22, normalized size = 0.92 \begin {gather*} \frac {9 e^{4}}{- 25 x^{3} e^{5} + 9 e^{x} - 27 \log {\relax (2 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-81*exp(4)*exp(x)+675*x**2*exp(4)*exp(5))/(81*exp(x)**2+(-486*ln(2)-450*x**3*exp(5))*exp(x)+729*ln(
2)**2+1350*x**3*exp(5)*ln(2)+625*x**6*exp(5)**2),x)

[Out]

9*exp(4)/(-25*x**3*exp(5) + 9*exp(x) - 27*log(2))

________________________________________________________________________________________