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

Optimal. Leaf size=19 \[ 4+\log \left (e+4 x+x^2-\log \left (25 x^2\right )\right ) \]

________________________________________________________________________________________

Rubi [F]  time = 0.22, 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 {2-4 x-2 x^2}{-e x-4 x^2-x^3+x \log \left (25 x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[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 {gather*} \begin {aligned} \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 {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.23, size = 16, normalized size = 0.84 \begin {gather*} \log \left (e+x (4+x)-\log \left (25 x^2\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[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]]

________________________________________________________________________________________

fricas [A]  time = 1.08, size = 20, normalized size = 1.05 \begin {gather*} \log \left (-x^{2} - 4 \, x - e + \log \left (25 \, x^{2}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

________________________________________________________________________________________

giac [A]  time = 0.14, size = 20, normalized size = 1.05 \begin {gather*} \log \left (-x^{2} - 4 \, x - e + \log \left (25 \, x^{2}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

________________________________________________________________________________________

maple [A]  time = 0.05, size = 19, normalized size = 1.00

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

Verification of antiderivative is not currently implemented for this CAS.

[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))

________________________________________________________________________________________

maxima [A]  time = 0.49, size = 18, normalized size = 0.95 \begin {gather*} \log \left (-\frac {1}{2} \, x^{2} - 2 \, x - \frac {1}{2} \, e + \log \relax (5) + \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

________________________________________________________________________________________

mupad [B]  time = 3.36, size = 18, normalized size = 0.95 \begin {gather*} \ln \left (4\,x+\mathrm {e}-\ln \left (25\,x^2\right )+x^2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

sympy [A]  time = 0.15, size = 17, normalized size = 0.89 \begin {gather*} \log {\left (- x^{2} - 4 x + \log {\left (25 x^{2} \right )} - e \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________