\(\int \frac {2-4 x-2 x^2}{-e x-4 x^2-x^3+x \log (25 x^2)} \, dx\) [4770]

Optimal result

Integrand size = 36, antiderivative size = 19 \[ \int \frac {2-4 x-2 x^2}{-e x-4 x^2-x^3+x \log \left (25 x^2\right )} \, dx=4+\log \left (e+4 x+x^2-\log \left (25 x^2\right )\right ) \]

[Out]

ln(4*x+exp(1)+x^2-ln(25*x^2))+4

Rubi [F]

\[ \int \frac {2-4 x-2 x^2}{-e x-4 x^2-x^3+x \log \left (25 x^2\right )} \, dx=\int \frac {2-4 x-2 x^2}{-e x-4 x^2-x^3+x \log \left (25 x^2\right )} \, dx \]

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {4}{e+4 x+x^2-\log \left (25 x^2\right )}-\frac {2}{x \left (e+4 x+x^2-\log \left (25 x^2\right )\right )}+\frac {2 x}{e+4 x+x^2-\log \left (25 x^2\right )}\right ) \, dx \\ & = -\left (2 \int \frac {1}{x \left (e+4 x+x^2-\log \left (25 x^2\right )\right )} \, dx\right )+2 \int \frac {x}{e+4 x+x^2-\log \left (25 x^2\right )} \, dx+4 \int \frac {1}{e+4 x+x^2-\log \left (25 x^2\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84 \[ \int \frac {2-4 x-2 x^2}{-e x-4 x^2-x^3+x \log \left (25 x^2\right )} \, dx=\log \left (e+x (4+x)-\log \left (25 x^2\right )\right ) \]

[In]

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

[Out]

Log[E + x*(4 + x) - Log[25*x^2]]

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00

method	result	size
norman	\(\ln \left (4 x +{\mathrm e}+x^{2}-\ln \left (25 x^{2}\right )\right )\)	\(19\)
parallelrisch	\(\ln \left (4 x +{\mathrm e}+x^{2}-\ln \left (25 x^{2}\right )\right )\)	\(19\)
risch	\(\ln \left (-x^{2}+\ln \left (25 x^{2}\right )-{\mathrm e}-4 x \right )\)	\(21\)

[In]

int((-2*x^2-4*x+2)/(x*ln(25*x^2)-x*exp(1)-x^3-4*x^2),x,method=_RETURNVERBOSE)

[Out]

ln(4*x+exp(1)+x^2-ln(25*x^2))

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {2-4 x-2 x^2}{-e x-4 x^2-x^3+x \log \left (25 x^2\right )} \, dx=\log \left (-x^{2} - 4 \, x - e + \log \left (25 \, x^{2}\right )\right ) \]

[In]

integrate((-2*x^2-4*x+2)/(x*log(25*x^2)-x*exp(1)-x^3-4*x^2),x, algorithm="fricas")

[Out]

log(-x^2 - 4*x - e + log(25*x^2))

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int \frac {2-4 x-2 x^2}{-e x-4 x^2-x^3+x \log \left (25 x^2\right )} \, dx=\log {\left (- x^{2} - 4 x + \log {\left (25 x^{2} \right )} - e \right )} \]

[In]

integrate((-2*x**2-4*x+2)/(x*ln(25*x**2)-x*exp(1)-x**3-4*x**2),x)

[Out]

log(-x**2 - 4*x + log(25*x**2) - E)

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95 \[ \int \frac {2-4 x-2 x^2}{-e x-4 x^2-x^3+x \log \left (25 x^2\right )} \, dx=\log \left (-\frac {1}{2} \, x^{2} - 2 \, x - \frac {1}{2} \, e + \log \left (5\right ) + \log \left (x\right )\right ) \]

[In]

integrate((-2*x^2-4*x+2)/(x*log(25*x^2)-x*exp(1)-x^3-4*x^2),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {2-4 x-2 x^2}{-e x-4 x^2-x^3+x \log \left (25 x^2\right )} \, dx=\log \left (-x^{2} - 4 \, x - e + \log \left (25 \, x^{2}\right )\right ) \]

[In]

integrate((-2*x^2-4*x+2)/(x*log(25*x^2)-x*exp(1)-x^3-4*x^2),x, algorithm="giac")

[Out]

log(-x^2 - 4*x - e + log(25*x^2))

Mupad [B] (verification not implemented)

Time = 11.10 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95 \[ \int \frac {2-4 x-2 x^2}{-e x-4 x^2-x^3+x \log \left (25 x^2\right )} \, dx=\ln \left (4\,x+\mathrm {e}-\ln \left (25\,x^2\right )+x^2\right ) \]

[In]

int((4*x + 2*x^2 - 2)/(x*exp(1) - x*log(25*x^2) + 4*x^2 + x^3),x)

[Out]

log(4*x + exp(1) - log(25*x^2) + x^2)