3.55.33 \(\int \frac {3 x^3+e^{3/x} (15+e (-48-16 x)+5 x)}{-60 x^3+15 x^4+e^{3/x} (-25 x^2+80 e x^2)} \, dx\)

Optimal. Leaf size=27 \[ \frac {1}{5} \log \left (-4+\frac {e^{3/x} (-5+16 e)}{3 x}+x\right ) \]

________________________________________________________________________________________

Rubi [F]  time = 0.80, 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 {3 x^3+e^{3/x} (15+e (-48-16 x)+5 x)}{-60 x^3+15 x^4+e^{3/x} \left (-25 x^2+80 e x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(3*x^3 + E^(3/x)*(15 + E*(-48 - 16*x) + 5*x))/(-60*x^3 + 15*x^4 + E^(3/x)*(-25*x^2 + 80*E*x^2)),x]

[Out]

3/(5*x) - Log[x]/5 + (3*Defer[Int][(-16*(1 - 5/(16*E))*E^(1 + 3/x) + 12*x - 3*x^2)^(-1), x])/5 + (36*Defer[Int
][1/(x*(-16*(1 - 5/(16*E))*E^(1 + 3/x) + 12*x - 3*x^2)), x])/5 + (6*Defer[Int][x/(16*(1 - 5/(16*E))*E^(1 + 3/x
) - 12*x + 3*x^2), x])/5

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {-3-x}{5 x^2}+\frac {3 \left (-12-x+2 x^2\right )}{5 x \left (16 \left (1-\frac {5}{16 e}\right ) e^{1+\frac {3}{x}}-12 x+3 x^2\right )}\right ) \, dx\\ &=\frac {1}{5} \int \frac {-3-x}{x^2} \, dx+\frac {3}{5} \int \frac {-12-x+2 x^2}{x \left (16 \left (1-\frac {5}{16 e}\right ) e^{1+\frac {3}{x}}-12 x+3 x^2\right )} \, dx\\ &=\frac {1}{5} \int \left (-\frac {3}{x^2}-\frac {1}{x}\right ) \, dx+\frac {3}{5} \int \left (\frac {1}{-16 \left (1-\frac {5}{16 e}\right ) e^{1+\frac {3}{x}}+12 x-3 x^2}+\frac {12}{x \left (-16 \left (1-\frac {5}{16 e}\right ) e^{1+\frac {3}{x}}+12 x-3 x^2\right )}+\frac {2 x}{16 \left (1-\frac {5}{16 e}\right ) e^{1+\frac {3}{x}}-12 x+3 x^2}\right ) \, dx\\ &=\frac {3}{5 x}-\frac {\log (x)}{5}+\frac {3}{5} \int \frac {1}{-16 \left (1-\frac {5}{16 e}\right ) e^{1+\frac {3}{x}}+12 x-3 x^2} \, dx+\frac {6}{5} \int \frac {x}{16 \left (1-\frac {5}{16 e}\right ) e^{1+\frac {3}{x}}-12 x+3 x^2} \, dx+\frac {36}{5} \int \frac {1}{x \left (-16 \left (1-\frac {5}{16 e}\right ) e^{1+\frac {3}{x}}+12 x-3 x^2\right )} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.34, size = 39, normalized size = 1.44 \begin {gather*} \frac {1}{5} \left (-\log (x)+\log \left (16 e^{1+\frac {3}{x}}-5 e^{3/x}-12 x+3 x^2\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3*x^3 + E^(3/x)*(15 + E*(-48 - 16*x) + 5*x))/(-60*x^3 + 15*x^4 + E^(3/x)*(-25*x^2 + 80*E*x^2)),x]

[Out]

(-Log[x] + Log[16*E^(1 + 3/x) - 5*E^(3/x) - 12*x + 3*x^2])/5

________________________________________________________________________________________

fricas [A]  time = 1.11, size = 30, normalized size = 1.11 \begin {gather*} \frac {1}{5} \, \log \left (3 \, x^{2} + {\left (16 \, e - 5\right )} e^{\frac {3}{x}} - 12 \, x\right ) - \frac {1}{5} \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-16*x-48)*exp(1)+5*x+15)*exp(3/x)+3*x^3)/((80*x^2*exp(1)-25*x^2)*exp(3/x)+15*x^4-60*x^3),x, algor
ithm="fricas")

[Out]

1/5*log(3*x^2 + (16*e - 5)*e^(3/x) - 12*x) - 1/5*log(x)

________________________________________________________________________________________

giac [A]  time = 0.28, size = 43, normalized size = 1.59 \begin {gather*} -\frac {1}{5} \, \log \left (\frac {3}{x}\right ) + \frac {1}{5} \, \log \left (-\frac {108}{x} - \frac {45 \, e^{\frac {3}{x}}}{x^{2}} + \frac {144 \, e^{\left (\frac {3}{x} + 1\right )}}{x^{2}} + 27\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-16*x-48)*exp(1)+5*x+15)*exp(3/x)+3*x^3)/((80*x^2*exp(1)-25*x^2)*exp(3/x)+15*x^4-60*x^3),x, algor
ithm="giac")

[Out]

-1/5*log(3/x) + 1/5*log(-108/x - 45*e^(3/x)/x^2 + 144*e^(3/x + 1)/x^2 + 27)

________________________________________________________________________________________

maple [A]  time = 0.42, size = 30, normalized size = 1.11

method	result	size
risch	\(-\frac {\ln \relax (x )}{5}+\frac {\ln \left ({\mathrm e}^{\frac {3}{x}}+\frac {3 \left (x -4\right ) x}{16 \,{\mathrm e}-5}\right )}{5}\)	\(30\)
norman	\(-\frac {\ln \relax (x )}{5}+\frac {\ln \left (16 \,{\mathrm e} \,{\mathrm e}^{\frac {3}{x}}+3 x^{2}-5 \,{\mathrm e}^{\frac {3}{x}}-12 x \right )}{5}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-16*x-48)*exp(1)+5*x+15)*exp(3/x)+3*x^3)/((80*x^2*exp(1)-25*x^2)*exp(3/x)+15*x^4-60*x^3),x,method=_RETU
RNVERBOSE)

[Out]

-1/5*ln(x)+1/5*ln(exp(3/x)+3*(x-4)*x/(16*exp(1)-5))

________________________________________________________________________________________

maxima [A]  time = 0.41, size = 39, normalized size = 1.44 \begin {gather*} -\frac {1}{5} \, \log \relax (x) + \frac {1}{5} \, \log \left (\frac {3 \, x^{2} + {\left (16 \, e - 5\right )} e^{\frac {3}{x}} - 12 \, x}{16 \, e - 5}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-16*x-48)*exp(1)+5*x+15)*exp(3/x)+3*x^3)/((80*x^2*exp(1)-25*x^2)*exp(3/x)+15*x^4-60*x^3),x, algor
ithm="maxima")

[Out]

-1/5*log(x) + 1/5*log((3*x^2 + (16*e - 5)*e^(3/x) - 12*x)/(16*e - 5))

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {{\mathrm {e}}^{3/x}\,\left (5\,x-\mathrm {e}\,\left (16\,x+48\right )+15\right )+3\,x^3}{{\mathrm {e}}^{3/x}\,\left (80\,x^2\,\mathrm {e}-25\,x^2\right )-60\,x^3+15\,x^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(3/x)*(5*x - exp(1)*(16*x + 48) + 15) + 3*x^3)/(exp(3/x)*(80*x^2*exp(1) - 25*x^2) - 60*x^3 + 15*x^4),x
)

[Out]

int((exp(3/x)*(5*x - exp(1)*(16*x + 48) + 15) + 3*x^3)/(exp(3/x)*(80*x^2*exp(1) - 25*x^2) - 60*x^3 + 15*x^4),
x)

________________________________________________________________________________________

sympy [A]  time = 0.34, size = 27, normalized size = 1.00 \begin {gather*} - \frac {\log {\relax (x )}}{5} + \frac {\log {\left (\frac {3 x^{2} - 12 x}{-5 + 16 e} + e^{\frac {3}{x}} \right )}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-16*x-48)*exp(1)+5*x+15)*exp(3/x)+3*x**3)/((80*x**2*exp(1)-25*x**2)*exp(3/x)+15*x**4-60*x**3),x)

[Out]

-log(x)/5 + log((3*x**2 - 12*x)/(-5 + 16*E) + exp(3/x))/5

________________________________________________________________________________________