3.57.20 \(\int \frac {-5-e^x x^2}{-5 x-5 x^2+e^x x^2} \, dx\)

Optimal. Leaf size=24 \[ \frac {1}{2} \log \left (\frac {256 e^{20}}{\left (5-e^x+\frac {5}{x}\right )^2}\right ) \]

________________________________________________________________________________________

Rubi [F]  time = 0.40, 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 {-5-e^x x^2}{-5 x-5 x^2+e^x x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-5 - E^x*x^2)/(-5*x - 5*x^2 + E^x*x^2),x]

[Out]

-x - 5*Defer[Int][(-5 - 5*x + E^x*x)^(-1), x] - 5*Defer[Int][1/(x*(-5 - 5*x + E^x*x)), x] - 5*Defer[Int][x/(-5
- 5*x + E^x*x), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-1-\frac {5 \left (1+x+x^2\right )}{x \left (-5-5 x+e^x x\right )}\right ) \, dx\\ &=-x-5 \int \frac {1+x+x^2}{x \left (-5-5 x+e^x x\right )} \, dx\\ &=-x-5 \int \left (\frac {1}{-5-5 x+e^x x}+\frac {1}{x \left (-5-5 x+e^x x\right )}+\frac {x}{-5-5 x+e^x x}\right ) \, dx\\ &=-x-5 \int \frac {1}{-5-5 x+e^x x} \, dx-5 \int \frac {1}{x \left (-5-5 x+e^x x\right )} \, dx-5 \int \frac {x}{-5-5 x+e^x x} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.12, size = 17, normalized size = 0.71 \begin {gather*} \log (x)-\log \left (5+5 x-e^x x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-5 - E^x*x^2)/(-5*x - 5*x^2 + E^x*x^2),x]

[Out]

Log[x] - Log[5 + 5*x - E^x*x]

________________________________________________________________________________________

fricas [A]  time = 0.87, size = 16, normalized size = 0.67 \begin {gather*} -\log \left (\frac {x e^{x} - 5 \, x - 5}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log((x*e^x - 5*x - 5)/x)

________________________________________________________________________________________

giac [A]  time = 0.21, size = 15, normalized size = 0.62 \begin {gather*} -\log \left (x e^{x} - 5 \, x - 5\right ) + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x*e^x - 5*x - 5) + log(x)

________________________________________________________________________________________

maple [A]  time = 0.03, size = 15, normalized size = 0.62

method	result	size
risch	\(-\ln \left ({\mathrm e}^{x}-\frac {5 \left (x +1\right )}{x}\right )\)	\(15\)
norman	\(-\ln \left ({\mathrm e}^{x} x -5 x -5\right )+\ln \relax (x )\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-ln(exp(x)-5*(x+1)/x)

________________________________________________________________________________________

maxima [A]  time = 0.44, size = 16, normalized size = 0.67 \begin {gather*} -\log \left (\frac {x e^{x} - 5 \, x - 5}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log((x*e^x - 5*x - 5)/x)

________________________________________________________________________________________

mupad [B]  time = 0.09, size = 16, normalized size = 0.67 \begin {gather*} \ln \relax (x)-\ln \left (5\,x-x\,{\mathrm {e}}^x+5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2*exp(x) + 5)/(5*x - x^2*exp(x) + 5*x^2),x)

[Out]

log(x) - log(5*x - x*exp(x) + 5)

________________________________________________________________________________________

sympy [A]  time = 0.15, size = 14, normalized size = 0.58 \begin {gather*} - \log {\left (e^{x} + \frac {- 5 x - 5}{x} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(x)*x**2-5)/(exp(x)*x**2-5*x**2-5*x),x)

[Out]

-log(exp(x) + (-5*x - 5)/x)

________________________________________________________________________________________