3.22.71 \(\int \frac {5-4 e^{4-x}+8 x}{-3+e^5+4 e^{4-x}+5 x+4 x^2} \, dx\)

Optimal. Leaf size=27 \[ \log \left (3-e^5-x-4 x^2-4 \left (e^{4-x}+x\right )\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.08, antiderivative size = 25, normalized size of antiderivative = 0.93, number of steps used = 1, number of rules used = 1, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {6684} \begin {gather*} \log \left (-4 x^2-5 x-4 e^{4-x}-e^5+3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(5 - 4*E^(4 - x) + 8*x)/(-3 + E^5 + 4*E^(4 - x) + 5*x + 4*x^2),x]

[Out]

Log[3 - E^5 - 4*E^(4 - x) - 5*x - 4*x^2]

Rule 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\log \left (3-e^5-4 e^{4-x}-5 x-4 x^2\right )\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.78, size = 30, normalized size = 1.11 \begin {gather*} -x+\log \left (4 e^4+e^{5+x}+e^x \left (-3+5 x+4 x^2\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(5 - 4*E^(4 - x) + 8*x)/(-3 + E^5 + 4*E^(4 - x) + 5*x + 4*x^2),x]

[Out]

-x + Log[4*E^4 + E^(5 + x) + E^x*(-3 + 5*x + 4*x^2)]

________________________________________________________________________________________

fricas [A]  time = 0.83, size = 21, normalized size = 0.78 \begin {gather*} \log \left (4 \, x^{2} + 5 \, x + e^{5} + 4 \, e^{\left (-x + 4\right )} - 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*exp(-x+4)+8*x+5)/(4*exp(-x+4)+exp(5)+4*x^2+5*x-3),x, algorithm="fricas")

[Out]

log(4*x^2 + 5*x + e^5 + 4*e^(-x + 4) - 3)

________________________________________________________________________________________

giac [A]  time = 0.21, size = 21, normalized size = 0.78 \begin {gather*} \log \left (4 \, x^{2} + 5 \, x + e^{5} + 4 \, e^{\left (-x + 4\right )} - 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*exp(-x+4)+8*x+5)/(4*exp(-x+4)+exp(5)+4*x^2+5*x-3),x, algorithm="giac")

[Out]

log(4*x^2 + 5*x + e^5 + 4*e^(-x + 4) - 3)

________________________________________________________________________________________

maple [A]  time = 0.06, size = 22, normalized size = 0.81

method	result	size
derivativedivides	\(\ln \left (4 \,{\mathrm e}^{-x +4}+{\mathrm e}^{5}+4 x^{2}+5 x -3\right )\)	\(22\)
default	\(\ln \left (4 \,{\mathrm e}^{-x +4}+{\mathrm e}^{5}+4 x^{2}+5 x -3\right )\)	\(22\)
norman	\(\ln \left (4 \,{\mathrm e}^{-x +4}+{\mathrm e}^{5}+4 x^{2}+5 x -3\right )\)	\(22\)
risch	\(-4+\ln \left (x^{2}+\frac {{\mathrm e}^{5}}{4}+\frac {5 x}{4}+{\mathrm e}^{-x +4}-\frac {3}{4}\right )\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-4*exp(-x+4)+8*x+5)/(4*exp(-x+4)+exp(5)+4*x^2+5*x-3),x,method=_RETURNVERBOSE)

[Out]

ln(4*exp(-x+4)+exp(5)+4*x^2+5*x-3)

________________________________________________________________________________________

maxima [A]  time = 0.44, size = 21, normalized size = 0.78 \begin {gather*} \log \left (4 \, x^{2} + 5 \, x + e^{5} + 4 \, e^{\left (-x + 4\right )} - 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*exp(-x+4)+8*x+5)/(4*exp(-x+4)+exp(5)+4*x^2+5*x-3),x, algorithm="maxima")

[Out]

log(4*x^2 + 5*x + e^5 + 4*e^(-x + 4) - 3)

________________________________________________________________________________________

mupad [B]  time = 0.14, size = 19, normalized size = 0.70 \begin {gather*} \ln \left (\frac {5\,x}{4}+\frac {{\mathrm {e}}^5}{4}+{\mathrm {e}}^{4-x}+x^2-\frac {3}{4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((8*x - 4*exp(4 - x) + 5)/(5*x + exp(5) + 4*exp(4 - x) + 4*x^2 - 3),x)

[Out]

log((5*x)/4 + exp(5)/4 + exp(4 - x) + x^2 - 3/4)

________________________________________________________________________________________

sympy [A]  time = 0.17, size = 22, normalized size = 0.81 \begin {gather*} \log {\left (x^{2} + \frac {5 x}{4} + e^{4 - x} - \frac {3}{4} + \frac {e^{5}}{4} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*exp(-x+4)+8*x+5)/(4*exp(-x+4)+exp(5)+4*x**2+5*x-3),x)

[Out]

log(x**2 + 5*x/4 + exp(4 - x) - 3/4 + exp(5)/4)

________________________________________________________________________________________