\(\int \frac {e^{4 e+x}-3750 x}{e^{4 e} (-27+e^x)-1875 x^2} \, dx\) [4835]

Optimal result

Integrand size = 31, antiderivative size = 20 \[ \int \frac {e^{4 e+x}-3750 x}{e^{4 e} \left (-27+e^x\right )-1875 x^2} \, dx=\log \left (9-\frac {e^x}{3}+625 e^{-4 e} x^2\right ) \]

[Out]

ln(9-1/3*exp(x)+625/exp(exp(1))^4*x^2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.032, Rules used = {6816} \[ \int \frac {e^{4 e+x}-3750 x}{e^{4 e} \left (-27+e^x\right )-1875 x^2} \, dx=\log \left (1875 x^2+e^{4 e} \left (27-e^x\right )\right ) \]

[In]

Int[(E^(4*E + x) - 3750*x)/(E^(4*E)*(-27 + E^x) - 1875*x^2),x]

[Out]

Log[E^(4*E)*(27 - E^x) + 1875*x^2]

Rule 6816

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

Rubi steps \begin{align*} \text {integral}& = \log \left (e^{4 e} \left (27-e^x\right )+1875 x^2\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {e^{4 e+x}-3750 x}{e^{4 e} \left (-27+e^x\right )-1875 x^2} \, dx=\log \left (e^{4 e} \left (-27+e^x\right )-1875 x^2\right ) \]

[In]

Integrate[(E^(4*E + x) - 3750*x)/(E^(4*E)*(-27 + E^x) - 1875*x^2),x]

[Out]

Log[E^(4*E)*(-27 + E^x) - 1875*x^2]

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80

method	result	size
risch	\(\ln \left (-1875 \,{\mathrm e}^{-4 \,{\mathrm e}} x^{2}-27+{\mathrm e}^{x}\right )\)	\(16\)
derivativedivides	\(\ln \left (\left ({\mathrm e}^{x}-27\right ) {\mathrm e}^{4 \,{\mathrm e}}-1875 x^{2}\right )\)	\(18\)
default	\(\ln \left (\left ({\mathrm e}^{x}-27\right ) {\mathrm e}^{4 \,{\mathrm e}}-1875 x^{2}\right )\)	\(18\)
parallelrisch	\(\ln \left (-\frac {{\mathrm e}^{x} {\mathrm e}^{4 \,{\mathrm e}}}{1875}+\frac {9 \,{\mathrm e}^{4 \,{\mathrm e}}}{625}+x^{2}\right )\)	\(22\)
norman	\(\ln \left ({\mathrm e}^{x} {\mathrm e}^{4 \,{\mathrm e}}-27 \,{\mathrm e}^{4 \,{\mathrm e}}-1875 x^{2}\right )\)	\(23\)

[In]

int((exp(x)*exp(exp(1))^4-3750*x)/((exp(x)-27)*exp(exp(1))^4-1875*x^2),x,method=_RETURNVERBOSE)

[Out]

ln(-1875*exp(-4*exp(1))*x^2-27+exp(x))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.05 \[ \int \frac {e^{4 e+x}-3750 x}{e^{4 e} \left (-27+e^x\right )-1875 x^2} \, dx=\log \left (-1875 \, x^{2} + e^{\left (x + 4 \, e\right )} - 27 \, e^{\left (4 \, e\right )}\right ) \]

[In]

integrate((exp(x)*exp(exp(1))^4-3750*x)/((exp(x)-27)*exp(exp(1))^4-1875*x^2),x, algorithm="fricas")

[Out]

log(-1875*x^2 + e^(x + 4*e) - 27*e^(4*e))

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.30 \[ \int \frac {e^{4 e+x}-3750 x}{e^{4 e} \left (-27+e^x\right )-1875 x^2} \, dx=\log {\left (\frac {- 1875 x^{2} - 27 e^{4 e}}{e^{4 e}} + e^{x} \right )} \]

[In]

integrate((exp(x)*exp(exp(1))**4-3750*x)/((exp(x)-27)*exp(exp(1))**4-1875*x**2),x)

[Out]

log((-1875*x**2 - 27*exp(4*E))*exp(-4*E) + exp(x))

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {e^{4 e+x}-3750 x}{e^{4 e} \left (-27+e^x\right )-1875 x^2} \, dx=\log \left (1875 \, x^{2} - {\left (e^{x} - 27\right )} e^{\left (4 \, e\right )}\right ) \]

[In]

integrate((exp(x)*exp(exp(1))^4-3750*x)/((exp(x)-27)*exp(exp(1))^4-1875*x^2),x, algorithm="maxima")

[Out]

log(1875*x^2 - (e^x - 27)*e^(4*e))

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.05 \[ \int \frac {e^{4 e+x}-3750 x}{e^{4 e} \left (-27+e^x\right )-1875 x^2} \, dx=\log \left (-1875 \, x^{2} + e^{\left (x + 4 \, e\right )} - 27 \, e^{\left (4 \, e\right )}\right ) \]

[In]

integrate((exp(x)*exp(exp(1))^4-3750*x)/((exp(x)-27)*exp(exp(1))^4-1875*x^2),x, algorithm="giac")

[Out]

log(-1875*x^2 + e^(x + 4*e) - 27*e^(4*e))

Mupad [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.15 \[ \int \frac {e^{4 e+x}-3750 x}{e^{4 e} \left (-27+e^x\right )-1875 x^2} \, dx=\ln \left (27\,{\mathrm {e}}^{4\,\mathrm {e}}-{\mathrm {e}}^{x+4\,\mathrm {e}}+1875\,x^2\right ) \]

[In]

int(-(3750*x - exp(4*exp(1))*exp(x))/(exp(4*exp(1))*(exp(x) - 27) - 1875*x^2),x)

[Out]

log(27*exp(4*exp(1)) - exp(x + 4*exp(1)) + 1875*x^2)