3.9.30 \(\int \frac {6 x^2+2 x^3+e^{\frac {e^{9/4}}{x^3}} (3 e^{9/4}+x^3)}{3 x^3+2 x^4+e^{\frac {e^{9/4}}{x^3}} x^4} \, dx\)

Optimal. Leaf size=25 \[ \log \left (\frac {x^2}{3+2 x+e^{\frac {e^{9/4}}{x^3}} x}\right ) \]

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Rubi [F]  time = 1.00, 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 {6 x^2+2 x^3+e^{\frac {e^{9/4}}{x^3}} \left (3 e^{9/4}+x^3\right )}{3 x^3+2 x^4+e^{\frac {e^{9/4}}{x^3}} x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(6*x^2 + 2*x^3 + E^(E^(9/4)/x^3)*(3*E^(9/4) + x^3))/(3*x^3 + 2*x^4 + E^(E^(9/4)/x^3)*x^4),x]

[Out]

-(E^(9/4)/x^3) + Log[x] - 9*E^(9/4)*Defer[Int][1/(x^4*(3 + 2*x + E^(E^(9/4)/x^3)*x)), x] - 6*E^(9/4)*Defer[Int
][1/(x^3*(3 + 2*x + E^(E^(9/4)/x^3)*x)), x] + 3*Defer[Int][1/(x*(3 + 2*x + E^(E^(9/4)/x^3)*x)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {3 e^{9/4}+x^3}{x^4}+\frac {3 \left (-3 e^{9/4}-2 e^{9/4} x+x^3\right )}{x^4 \left (3+2 x+e^{\frac {e^{9/4}}{x^3}} x\right )}\right ) \, dx\\ &=3 \int \frac {-3 e^{9/4}-2 e^{9/4} x+x^3}{x^4 \left (3+2 x+e^{\frac {e^{9/4}}{x^3}} x\right )} \, dx+\int \frac {3 e^{9/4}+x^3}{x^4} \, dx\\ &=3 \int \left (-\frac {3 e^{9/4}}{x^4 \left (3+2 x+e^{\frac {e^{9/4}}{x^3}} x\right )}-\frac {2 e^{9/4}}{x^3 \left (3+2 x+e^{\frac {e^{9/4}}{x^3}} x\right )}+\frac {1}{x \left (3+2 x+e^{\frac {e^{9/4}}{x^3}} x\right )}\right ) \, dx+\int \left (\frac {3 e^{9/4}}{x^4}+\frac {1}{x}\right ) \, dx\\ &=-\frac {e^{9/4}}{x^3}+\log (x)+3 \int \frac {1}{x \left (3+2 x+e^{\frac {e^{9/4}}{x^3}} x\right )} \, dx-\left (6 e^{9/4}\right ) \int \frac {1}{x^3 \left (3+2 x+e^{\frac {e^{9/4}}{x^3}} x\right )} \, dx-\left (9 e^{9/4}\right ) \int \frac {1}{x^4 \left (3+2 x+e^{\frac {e^{9/4}}{x^3}} x\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.28, size = 26, normalized size = 1.04 \begin {gather*} 2 \log (x)-\log \left (3+2 x+e^{\frac {e^{9/4}}{x^3}} x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(6*x^2 + 2*x^3 + E^(E^(9/4)/x^3)*(3*E^(9/4) + x^3))/(3*x^3 + 2*x^4 + E^(E^(9/4)/x^3)*x^4),x]

[Out]

2*Log[x] - Log[3 + 2*x + E^(E^(9/4)/x^3)*x]

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fricas [A]  time = 0.58, size = 24, normalized size = 0.96 \begin {gather*} \log \relax (x) - \log \left (\frac {x e^{\left (\frac {e^{\frac {9}{4}}}{x^{3}}\right )} + 2 \, x + 3}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*exp(9/4)+x^3)*exp(exp(9/4)/x^3)+2*x^3+6*x^2)/(x^4*exp(exp(9/4)/x^3)+2*x^4+3*x^3),x, algorithm="f
ricas")

[Out]

log(x) - log((x*e^(e^(9/4)/x^3) + 2*x + 3)/x)

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giac [B]  time = 0.96, size = 46, normalized size = 1.84 \begin {gather*} -\frac {x^{3} \log \left (x e^{\left (\frac {e^{\frac {9}{4}}}{x^{3}}\right )} + 2 \, x + 3\right ) - x^{3} \log \relax (x) - e^{\frac {9}{4}}}{x^{3}} - \frac {e^{\frac {9}{4}}}{x^{3}} + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*exp(9/4)+x^3)*exp(exp(9/4)/x^3)+2*x^3+6*x^2)/(x^4*exp(exp(9/4)/x^3)+2*x^4+3*x^3),x, algorithm="g
iac")

[Out]

-(x^3*log(x*e^(e^(9/4)/x^3) + 2*x + 3) - x^3*log(x) - e^(9/4))/x^3 - e^(9/4)/x^3 + log(x)

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maple [A]  time = 0.16, size = 23, normalized size = 0.92

method	result	size
norman	\(2 \ln \relax (x )-\ln \left (3+2 x +{\mathrm e}^{\frac {{\mathrm e}^{\frac {9}{4}}}{x^{3}}} x \right )\)	\(23\)
risch	\(\ln \relax (x )-\ln \left ({\mathrm e}^{\frac {{\mathrm e}^{\frac {9}{4}}}{x^{3}}}+\frac {2 x +3}{x}\right )\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3*exp(9/4)+x^3)*exp(exp(9/4)/x^3)+2*x^3+6*x^2)/(x^4*exp(exp(9/4)/x^3)+2*x^4+3*x^3),x,method=_RETURNVERBO
SE)

[Out]

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

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maxima [A]  time = 0.53, size = 24, normalized size = 0.96 \begin {gather*} \log \relax (x) - \log \left (\frac {x e^{\left (\frac {e^{\frac {9}{4}}}{x^{3}}\right )} + 2 \, x + 3}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*exp(9/4)+x^3)*exp(exp(9/4)/x^3)+2*x^3+6*x^2)/(x^4*exp(exp(9/4)/x^3)+2*x^4+3*x^3),x, algorithm="m
axima")

[Out]

log(x) - log((x*e^(e^(9/4)/x^3) + 2*x + 3)/x)

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mupad [B]  time = 0.74, size = 24, normalized size = 0.96 \begin {gather*} \ln \relax (x)-\ln \left (\frac {2\,x+x\,{\mathrm {e}}^{\frac {{\mathrm {e}}^{9/4}}{x^3}}+3}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(exp(9/4)/x^3)*(3*exp(9/4) + x^3) + 6*x^2 + 2*x^3)/(x^4*exp(exp(9/4)/x^3) + 3*x^3 + 2*x^4),x)

[Out]

log(x) - log((2*x + x*exp(exp(9/4)/x^3) + 3)/x)

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sympy [A]  time = 0.19, size = 20, normalized size = 0.80 \begin {gather*} \log {\relax (x )} - \log {\left (e^{\frac {e^{\frac {9}{4}}}{x^{3}}} + \frac {2 x + 3}{x} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*exp(9/4)+x**3)*exp(exp(9/4)/x**3)+2*x**3+6*x**2)/(x**4*exp(exp(9/4)/x**3)+2*x**4+3*x**3),x)

[Out]

log(x) - log(exp(exp(9/4)/x**3) + (2*x + 3)/x)

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