3.27.41 \(\int \frac {10 e^3+e^{25-x^2} (x^2-2 x^4)}{-10 e^3 x+5 x^2+e^{25-x^2} x^3} \, dx\)

Optimal. Leaf size=25 \[ \log \left (2-\frac {4 e^3}{x}+\frac {2}{5} e^{25-x^2} x\right ) \]

________________________________________________________________________________________

Rubi [F]  time = 1.02, 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 {10 e^3+e^{25-x^2} \left (x^2-2 x^4\right )}{-10 e^3 x+5 x^2+e^{25-x^2} x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(10*E^3 + E^(25 - x^2)*(x^2 - 2*x^4))/(-10*E^3*x + 5*x^2 + E^(25 - x^2)*x^3),x]

[Out]

Log[2*E^3 - x] - Log[x] - 2*E^28*Defer[Int][(-10*E^(3 + x^2) + 5*E^x^2*x + E^25*x^2)^(-1), x] + 4*E^31*Defer[I
nt][1/((2*E^3 - x)*(-10*E^(3 + x^2) + 5*E^x^2*x + E^25*x^2)), x] + E^25*Defer[Int][x/(-10*E^(3 + x^2) + 5*E^x^
2*x + E^25*x^2), x] - 2*E^25*Defer[Int][x^3/(-10*E^(3 + x^2) + 5*E^x^2*x + E^25*x^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {2 e^3}{\left (2 e^3-x\right ) x}-\frac {e^{25} x \left (-4 e^3+x+4 e^3 x^2-2 x^3\right )}{\left (2 e^3-x\right ) \left (-10 e^{3+x^2}+5 e^{x^2} x+e^{25} x^2\right )}\right ) \, dx\\ &=-\left (\left (2 e^3\right ) \int \frac {1}{\left (2 e^3-x\right ) x} \, dx\right )-e^{25} \int \frac {x \left (-4 e^3+x+4 e^3 x^2-2 x^3\right )}{\left (2 e^3-x\right ) \left (-10 e^{3+x^2}+5 e^{x^2} x+e^{25} x^2\right )} \, dx\\ &=-\left (e^{25} \int \left (\frac {2 e^3}{-10 e^{3+x^2}+5 e^{x^2} x+e^{25} x^2}-\frac {4 e^6}{\left (2 e^3-x\right ) \left (-10 e^{3+x^2}+5 e^{x^2} x+e^{25} x^2\right )}-\frac {x}{-10 e^{3+x^2}+5 e^{x^2} x+e^{25} x^2}+\frac {2 x^3}{-10 e^{3+x^2}+5 e^{x^2} x+e^{25} x^2}\right ) \, dx\right )-\int \frac {1}{2 e^3-x} \, dx-\int \frac {1}{x} \, dx\\ &=\log \left (2 e^3-x\right )-\log (x)+e^{25} \int \frac {x}{-10 e^{3+x^2}+5 e^{x^2} x+e^{25} x^2} \, dx-\left (2 e^{25}\right ) \int \frac {x^3}{-10 e^{3+x^2}+5 e^{x^2} x+e^{25} x^2} \, dx-\left (2 e^{28}\right ) \int \frac {1}{-10 e^{3+x^2}+5 e^{x^2} x+e^{25} x^2} \, dx+\left (4 e^{31}\right ) \int \frac {1}{\left (2 e^3-x\right ) \left (-10 e^{3+x^2}+5 e^{x^2} x+e^{25} x^2\right )} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.29, size = 37, normalized size = 1.48 \begin {gather*} -x^2-\log (x)+\log \left (10 e^{3+x^2}-5 e^{x^2} x-e^{25} x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(10*E^3 + E^(25 - x^2)*(x^2 - 2*x^4))/(-10*E^3*x + 5*x^2 + E^(25 - x^2)*x^3),x]

[Out]

-x^2 - Log[x] + Log[10*E^(3 + x^2) - 5*E^x^2*x - E^25*x^2]

________________________________________________________________________________________

fricas [A]  time = 0.61, size = 28, normalized size = 1.12 \begin {gather*} \log \relax (x) + \log \left (\frac {x^{2} e^{\left (-x^{2} + 25\right )} + 5 \, x - 10 \, e^{3}}{x^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^4+x^2)*exp(-x^2+25)+10*exp(3))/(x^3*exp(-x^2+25)-10*x*exp(3)+5*x^2),x, algorithm="fricas")

[Out]

log(x) + log((x^2*e^(-x^2 + 25) + 5*x - 10*e^3)/x^2)

________________________________________________________________________________________

giac [A]  time = 0.21, size = 26, normalized size = 1.04 \begin {gather*} \log \left (x^{2} e^{\left (-x^{2} + 25\right )} + 5 \, x - 10 \, e^{3}\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^4+x^2)*exp(-x^2+25)+10*exp(3))/(x^3*exp(-x^2+25)-10*x*exp(3)+5*x^2),x, algorithm="giac")

[Out]

log(x^2*e^(-x^2 + 25) + 5*x - 10*e^3) - log(x)

________________________________________________________________________________________

maple [A]  time = 0.06, size = 28, normalized size = 1.12

method	result	size
norman	\(-\ln \relax (x )+\ln \left (-x^{2} {\mathrm e}^{-x^{2}+25}+10 \,{\mathrm e}^{3}-5 x \right )\)	\(28\)
risch	\(\ln \relax (x )-25+\ln \left ({\mathrm e}^{-\left (x -5\right ) \left (5+x \right )}-\frac {5 \left (2 \,{\mathrm e}^{3}-x \right )}{x^{2}}\right )\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x^4+x^2)*exp(-x^2+25)+10*exp(3))/(x^3*exp(-x^2+25)-10*x*exp(3)+5*x^2),x,method=_RETURNVERBOSE)

[Out]

-ln(x)+ln(-x^2*exp(-x^2+25)+10*exp(3)-5*x)

________________________________________________________________________________________

maxima [B]  time = 0.49, size = 47, normalized size = 1.88 \begin {gather*} -x^{2} + \log \left (x - 2 \, e^{3}\right ) - \log \relax (x) + \log \left (\frac {x^{2} e^{25} + 5 \, {\left (x - 2 \, e^{3}\right )} e^{\left (x^{2}\right )}}{5 \, {\left (x - 2 \, e^{3}\right )}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^4+x^2)*exp(-x^2+25)+10*exp(3))/(x^3*exp(-x^2+25)-10*x*exp(3)+5*x^2),x, algorithm="maxima")

[Out]

-x^2 + log(x - 2*e^3) - log(x) + log(1/5*(x^2*e^25 + 5*(x - 2*e^3)*e^(x^2))/(x - 2*e^3))

________________________________________________________________________________________

mupad [B]  time = 0.25, size = 26, normalized size = 1.04 \begin {gather*} \ln \left (5\,x-10\,{\mathrm {e}}^3+x^2\,{\mathrm {e}}^{25}\,{\mathrm {e}}^{-x^2}\right )-\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((10*exp(3) + exp(25 - x^2)*(x^2 - 2*x^4))/(x^3*exp(25 - x^2) - 10*x*exp(3) + 5*x^2),x)

[Out]

log(5*x - 10*exp(3) + x^2*exp(25)*exp(-x^2)) - log(x)

________________________________________________________________________________________

sympy [A]  time = 0.20, size = 22, normalized size = 0.88 \begin {gather*} \log {\relax (x )} + \log {\left (e^{25 - x^{2}} + \frac {5 x - 10 e^{3}}{x^{2}} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x**4+x**2)*exp(-x**2+25)+10*exp(3))/(x**3*exp(-x**2+25)-10*x*exp(3)+5*x**2),x)

[Out]

log(x) + log(exp(25 - x**2) + (5*x - 10*exp(3))/x**2)

________________________________________________________________________________________