3.21.36 \(\int \frac {-250 e^{3 x} x-150 e^{2 x} x^2-2 x^4+e^x (-30 x^3+e^{\frac {1+4 e^2}{e^2}} (250 x-250 x^2) \log (2))}{125 e^{\frac {1+4 e^2}{e^2}+3 x}+75 e^{\frac {1+4 e^2}{e^2}+2 x} x+15 e^{\frac {1+4 e^2}{e^2}+x} x^2+e^{\frac {1+4 e^2}{e^2}} x^3} \, dx\)

Optimal. Leaf size=30 \[ x \left (-e^{-4-\frac {1}{e^2}} x+\frac {x \log (2)}{\left (e^x+\frac {x}{5}\right )^2}\right ) \]

________________________________________________________________________________________

Rubi [F]  time = 1.41, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-250 e^{3 x} x-150 e^{2 x} x^2-2 x^4+e^x \left (-30 x^3+e^{\frac {1+4 e^2}{e^2}} \left (250 x-250 x^2\right ) \log (2)\right )}{125 e^{\frac {1+4 e^2}{e^2}+3 x}+75 e^{\frac {1+4 e^2}{e^2}+2 x} x+15 e^{\frac {1+4 e^2}{e^2}+x} x^2+e^{\frac {1+4 e^2}{e^2}} x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-250*E^(3*x)*x - 150*E^(2*x)*x^2 - 2*x^4 + E^x*(-30*x^3 + E^((1 + 4*E^2)/E^2)*(250*x - 250*x^2)*Log[2]))/
(125*E^((1 + 4*E^2)/E^2 + 3*x) + 75*E^((1 + 4*E^2)/E^2 + 2*x)*x + 15*E^((1 + 4*E^2)/E^2 + x)*x^2 + E^((1 + 4*E
^2)/E^2)*x^3),x]

[Out]

-(E^(-4 - E^(-2))*x^2) - 50*Log[2]*Defer[Int][x^2/(5*E^x + x)^3, x] + 50*Log[2]*Defer[Int][x^3/(5*E^x + x)^3,
x] + 50*Log[2]*Defer[Int][x/(5*E^x + x)^2, x] - 50*Log[2]*Defer[Int][x^2/(5*E^x + x)^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 e^{-4-\frac {1}{e^2}} x \left (-125 e^{3 x}-75 e^{2 x} x-15 e^x x^2-x^3-125 e^{4+\frac {1}{e^2}+x} (-1+x) \log (2)\right )}{\left (5 e^x+x\right )^3} \, dx\\ &=\left (2 e^{-4-\frac {1}{e^2}}\right ) \int \frac {x \left (-125 e^{3 x}-75 e^{2 x} x-15 e^x x^2-x^3-125 e^{4+\frac {1}{e^2}+x} (-1+x) \log (2)\right )}{\left (5 e^x+x\right )^3} \, dx\\ &=\left (2 e^{-4-\frac {1}{e^2}}\right ) \int \left (-x+\frac {25 e^{4+\frac {1}{e^2}} (-1+x) x^2 \log (2)}{\left (5 e^x+x\right )^3}-\frac {25 e^{4+\frac {1}{e^2}} (-1+x) x \log (2)}{\left (5 e^x+x\right )^2}\right ) \, dx\\ &=-e^{-4-\frac {1}{e^2}} x^2+(50 \log (2)) \int \frac {(-1+x) x^2}{\left (5 e^x+x\right )^3} \, dx-(50 \log (2)) \int \frac {(-1+x) x}{\left (5 e^x+x\right )^2} \, dx\\ &=-e^{-4-\frac {1}{e^2}} x^2+(50 \log (2)) \int \left (-\frac {x^2}{\left (5 e^x+x\right )^3}+\frac {x^3}{\left (5 e^x+x\right )^3}\right ) \, dx-(50 \log (2)) \int \left (-\frac {x}{\left (5 e^x+x\right )^2}+\frac {x^2}{\left (5 e^x+x\right )^2}\right ) \, dx\\ &=-e^{-4-\frac {1}{e^2}} x^2-(50 \log (2)) \int \frac {x^2}{\left (5 e^x+x\right )^3} \, dx+(50 \log (2)) \int \frac {x^3}{\left (5 e^x+x\right )^3} \, dx+(50 \log (2)) \int \frac {x}{\left (5 e^x+x\right )^2} \, dx-(50 \log (2)) \int \frac {x^2}{\left (5 e^x+x\right )^2} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.24, size = 36, normalized size = 1.20 \begin {gather*} -e^{-4-\frac {1}{e^2}} x^2 \left (1-\frac {25 e^{4+\frac {1}{e^2}} \log (2)}{\left (5 e^x+x\right )^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-250*E^(3*x)*x - 150*E^(2*x)*x^2 - 2*x^4 + E^x*(-30*x^3 + E^((1 + 4*E^2)/E^2)*(250*x - 250*x^2)*Log
[2]))/(125*E^((1 + 4*E^2)/E^2 + 3*x) + 75*E^((1 + 4*E^2)/E^2 + 2*x)*x + 15*E^((1 + 4*E^2)/E^2 + x)*x^2 + E^((1
+ 4*E^2)/E^2)*x^3),x]

[Out]

-(E^(-4 - E^(-2))*x^2*(1 - (25*E^(4 + E^(-2))*Log[2])/(5*E^x + x)^2))

________________________________________________________________________________________

fricas [B]  time = 0.72, size = 150, normalized size = 5.00 \begin {gather*} -\frac {x^{4} e^{\left (2 \, {\left (4 \, e^{2} + 1\right )} e^{\left (-2\right )}\right )} + 10 \, x^{3} e^{\left ({\left ({\left (x + 4\right )} e^{2} + 1\right )} e^{\left (-2\right )} + {\left (4 \, e^{2} + 1\right )} e^{\left (-2\right )}\right )} - 25 \, x^{2} e^{\left (3 \, {\left (4 \, e^{2} + 1\right )} e^{\left (-2\right )}\right )} \log \relax (2) + 25 \, x^{2} e^{\left (2 \, {\left ({\left (x + 4\right )} e^{2} + 1\right )} e^{\left (-2\right )}\right )}}{x^{2} e^{\left (3 \, {\left (4 \, e^{2} + 1\right )} e^{\left (-2\right )}\right )} + 10 \, x e^{\left ({\left ({\left (x + 4\right )} e^{2} + 1\right )} e^{\left (-2\right )} + 2 \, {\left (4 \, e^{2} + 1\right )} e^{\left (-2\right )}\right )} + 25 \, e^{\left (2 \, {\left ({\left (x + 4\right )} e^{2} + 1\right )} e^{\left (-2\right )} + {\left (4 \, e^{2} + 1\right )} e^{\left (-2\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-250*x*exp(x)^3-150*exp(x)^2*x^2+((-250*x^2+250*x)*log(2)*exp((4*exp(2)+1)/exp(2))-30*x^3)*exp(x)-2
*x^4)/(125*exp((4*exp(2)+1)/exp(2))*exp(x)^3+75*x*exp((4*exp(2)+1)/exp(2))*exp(x)^2+15*x^2*exp((4*exp(2)+1)/ex
p(2))*exp(x)+x^3*exp((4*exp(2)+1)/exp(2))),x, algorithm="fricas")

[Out]

-(x^4*e^(2*(4*e^2 + 1)*e^(-2)) + 10*x^3*e^(((x + 4)*e^2 + 1)*e^(-2) + (4*e^2 + 1)*e^(-2)) - 25*x^2*e^(3*(4*e^2
+ 1)*e^(-2))*log(2) + 25*x^2*e^(2*((x + 4)*e^2 + 1)*e^(-2)))/(x^2*e^(3*(4*e^2 + 1)*e^(-2)) + 10*x*e^(((x + 4)
*e^2 + 1)*e^(-2) + 2*(4*e^2 + 1)*e^(-2)) + 25*e^(2*((x + 4)*e^2 + 1)*e^(-2) + (4*e^2 + 1)*e^(-2)))

________________________________________________________________________________________

giac [B]  time = 0.30, size = 87, normalized size = 2.90 \begin {gather*} -\frac {x^{4} + 10 \, x^{3} e^{x} - 25 \, x^{2} e^{\left ({\left (4 \, e^{2} + 1\right )} e^{\left (-2\right )}\right )} \log \relax (2) + 25 \, x^{2} e^{\left (2 \, x\right )}}{x^{2} e^{\left ({\left (4 \, e^{2} + 1\right )} e^{\left (-2\right )}\right )} + 10 \, x e^{\left ({\left (4 \, e^{2} + 1\right )} e^{\left (-2\right )} + x\right )} + 25 \, e^{\left ({\left (4 \, e^{2} + 1\right )} e^{\left (-2\right )} + 2 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-250*x*exp(x)^3-150*exp(x)^2*x^2+((-250*x^2+250*x)*log(2)*exp((4*exp(2)+1)/exp(2))-30*x^3)*exp(x)-2
*x^4)/(125*exp((4*exp(2)+1)/exp(2))*exp(x)^3+75*x*exp((4*exp(2)+1)/exp(2))*exp(x)^2+15*x^2*exp((4*exp(2)+1)/ex
p(2))*exp(x)+x^3*exp((4*exp(2)+1)/exp(2))),x, algorithm="giac")

[Out]

-(x^4 + 10*x^3*e^x - 25*x^2*e^((4*e^2 + 1)*e^(-2))*log(2) + 25*x^2*e^(2*x))/(x^2*e^((4*e^2 + 1)*e^(-2)) + 10*x
*e^((4*e^2 + 1)*e^(-2) + x) + 25*e^((4*e^2 + 1)*e^(-2) + 2*x))

________________________________________________________________________________________

maple [A]  time = 0.22, size = 29, normalized size = 0.97

method	result	size
risch	\(-x^{2} {\mathrm e}^{-4-{\mathrm e}^{-2}}+\frac {25 \ln \relax (2) x^{2}}{\left (5 \,{\mathrm e}^{x}+x \right )^{2}}\)	\(29\)
norman	\(\frac {-625 \ln \relax (2) {\mathrm e}^{2 x}-250 x \ln \relax (2) {\mathrm e}^{x}-{\mathrm e}^{-{\mathrm e}^{-2}} {\mathrm e}^{-4} x^{4}-25 \,{\mathrm e}^{-{\mathrm e}^{-2}} {\mathrm e}^{-4} x^{2} {\mathrm e}^{2 x}-10 \,{\mathrm e}^{-{\mathrm e}^{-2}} {\mathrm e}^{-4} x^{3} {\mathrm e}^{x}}{\left (5 \,{\mathrm e}^{x}+x \right )^{2}}\)	\(80\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-250*x*exp(x)^3-150*exp(x)^2*x^2+((-250*x^2+250*x)*ln(2)*exp((4*exp(2)+1)/exp(2))-30*x^3)*exp(x)-2*x^4)/(
125*exp((4*exp(2)+1)/exp(2))*exp(x)^3+75*x*exp((4*exp(2)+1)/exp(2))*exp(x)^2+15*x^2*exp((4*exp(2)+1)/exp(2))*e
xp(x)+x^3*exp((4*exp(2)+1)/exp(2))),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

maxima [B]  time = 0.97, size = 65, normalized size = 2.17 \begin {gather*} -\frac {x^{4} + 10 \, x^{3} e^{x} - 25 \, x^{2} e^{\left (e^{\left (-2\right )} + 4\right )} \log \relax (2) + 25 \, x^{2} e^{\left (2 \, x\right )}}{x^{2} e^{\left (e^{\left (-2\right )} + 4\right )} + 10 \, x e^{\left (x + e^{\left (-2\right )} + 4\right )} + 25 \, e^{\left (2 \, x + e^{\left (-2\right )} + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-250*x*exp(x)^3-150*exp(x)^2*x^2+((-250*x^2+250*x)*log(2)*exp((4*exp(2)+1)/exp(2))-30*x^3)*exp(x)-2
*x^4)/(125*exp((4*exp(2)+1)/exp(2))*exp(x)^3+75*x*exp((4*exp(2)+1)/exp(2))*exp(x)^2+15*x^2*exp((4*exp(2)+1)/ex
p(2))*exp(x)+x^3*exp((4*exp(2)+1)/exp(2))),x, algorithm="maxima")

[Out]

-(x^4 + 10*x^3*e^x - 25*x^2*e^(e^(-2) + 4)*log(2) + 25*x^2*e^(2*x))/(x^2*e^(e^(-2) + 4) + 10*x*e^(x + e^(-2) +
4) + 25*e^(2*x + e^(-2) + 4))

________________________________________________________________________________________

mupad [B]  time = 0.33, size = 65, normalized size = 2.17 \begin {gather*} -\frac {x^2\,\left (25\,{\mathrm {e}}^{2\,x}-25\,{\mathrm {e}}^{{\mathrm {e}}^{-2}\,\left (4\,{\mathrm {e}}^2+1\right )}\,\ln \relax (2)+10\,x\,{\mathrm {e}}^x+x^2\right )}{25\,{\mathrm {e}}^{2\,x+{\mathrm {e}}^{-2}+4}+x^2\,{\mathrm {e}}^{{\mathrm {e}}^{-2}+4}+10\,x\,{\mathrm {e}}^{x+{\mathrm {e}}^{-2}+4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(250*x*exp(3*x) + exp(x)*(30*x^3 - exp(exp(-2)*(4*exp(2) + 1))*log(2)*(250*x - 250*x^2)) + 150*x^2*exp(2*
x) + 2*x^4)/(x^3*exp(exp(-2)*(4*exp(2) + 1)) + 125*exp(3*x)*exp(exp(-2)*(4*exp(2) + 1)) + 75*x*exp(2*x)*exp(ex
p(-2)*(4*exp(2) + 1)) + 15*x^2*exp(exp(-2)*(4*exp(2) + 1))*exp(x)),x)

[Out]

-(x^2*(25*exp(2*x) - 25*exp(exp(-2)*(4*exp(2) + 1))*log(2) + 10*x*exp(x) + x^2))/(25*exp(2*x + exp(-2) + 4) +
x^2*exp(exp(-2) + 4) + 10*x*exp(x + exp(-2) + 4))

________________________________________________________________________________________

sympy [A]  time = 0.18, size = 39, normalized size = 1.30 \begin {gather*} - \frac {x^{2}}{e^{4} e^{e^{-2}}} + \frac {5 x^{2} \log {\relax (2 )}}{\frac {x^{2}}{5} + 2 x e^{x} + 5 e^{2 x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-250*x*exp(x)**3-150*exp(x)**2*x**2+((-250*x**2+250*x)*ln(2)*exp((4*exp(2)+1)/exp(2))-30*x**3)*exp(
x)-2*x**4)/(125*exp((4*exp(2)+1)/exp(2))*exp(x)**3+75*x*exp((4*exp(2)+1)/exp(2))*exp(x)**2+15*x**2*exp((4*exp(
2)+1)/exp(2))*exp(x)+x**3*exp((4*exp(2)+1)/exp(2))),x)

[Out]

-x**2*exp(-4)*exp(-exp(-2)) + 5*x**2*log(2)/(x**2/5 + 2*x*exp(x) + 5*exp(2*x))

________________________________________________________________________________________