3.69.94 $\int \frac{-60e^{\frac{19}{9}+\frac{20e^2}{x}}+x^2}{3\sqrt[9]{e}{x}^2+3e^{\frac{1}{9}+\frac{20e^2}{x}}{x}^2+x^3}dx$

Optimal. Leaf size=23 $$\log (\sqrt[9]{e}(3+3e^{\frac{20e^2}{x}})+x)$$

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Rubi [F]  time = 1.37, 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 = {} $$\begin{array}{c}\int \frac{-60e^{\frac{19}{9}+\frac{20e^2}{x}}+x^2}{3\sqrt[9]{e}{x}^2+3e^{\frac{1}{9}+\frac{20e^2}{x}}{x}^2+x^3}dx\end{array}$$

Verification is not applicable to the result.

[In]

Int[(-60*E^(19/9 + (20*E^2)/x) + x^2)/(3*E^(1/9)*x^2 + 3*E^(1/9 + (20*E^2)/x)*x^2 + x^3),x]

[Out]

(20*E^2)/x + Defer[Int][(3*E^(1/9) + 3*E^(1/9 + (20*E^2)/x) + x)^(-1), x] + 60*E^(19/9)*Defer[Int][1/(x^2*(3*E
^(1/9) + 3*E^(1/9 + (20*E^2)/x) + x)), x] + 20*E^2*Defer[Int][1/(x*(3*E^(1/9) + 3*E^(1/9 + (20*E^2)/x) + x)),
x]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int (-\frac{20e^2}{x^2}+\frac{60e^{19/9}+20e^{2}x+x^2}{x^2(3\sqrt[9]{e}+3e^{\frac{1}{9}+\frac{20e^2}{x}}+x)})dx\\ & =\frac{20e^2}{x}+\int \frac{60e^{19/9}+20e^{2}x+x^2}{x^2(3\sqrt[9]{e}+3e^{\frac{1}{9}+\frac{20e^2}{x}}+x)}dx\\ & =\frac{20e^2}{x}+\int (\frac{1}{3\sqrt[9]{e}+3e^{\frac{1}{9}+\frac{20e^2}{x}}+x}+\frac{60e^{19/9}}{x^2(3\sqrt[9]{e}+3e^{\frac{1}{9}+\frac{20e^2}{x}}+x)}+\frac{20e^2}{x(3\sqrt[9]{e}+3e^{\frac{1}{9}+\frac{20e^2}{x}}+x)})dx\\ & =\frac{20e^2}{x}+\left(20e^2\right)\int \frac{1}{x(3\sqrt[9]{e}+3e^{\frac{1}{9}+\frac{20e^2}{x}}+x)}dx+\left(60e^{19/9}\right)\int \frac{1}{x^2(3\sqrt[9]{e}+3e^{\frac{1}{9}+\frac{20e^2}{x}}+x)}dx+\int \frac{1}{3\sqrt[9]{e}+3e^{\frac{1}{9}+\frac{20e^2}{x}}+x}dx\end{array}\end{array}$$

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Mathematica [A]  time = 0.21, size = 26, normalized size = 1.13 $$\begin{array}{c}\log (3\sqrt[9]{e}+3e^{\frac{1}{9}+\frac{20e^2}{x}}+x)\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(-60*E^(19/9 + (20*E^2)/x) + x^2)/(3*E^(1/9)*x^2 + 3*E^(1/9 + (20*E^2)/x)*x^2 + x^3),x]

[Out]

Log[3*E^(1/9) + 3*E^(1/9 + (20*E^2)/x) + x]

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fricas [A]  time = 0.66, size = 26, normalized size = 1.13 $$\begin{array}{c}\log (x{e}^2+3e^{\frac{19}{9}}+3e^{\left(\frac{19x+180e^2}{9x}\right)})\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-60*exp(1/9)*exp(2)*exp(20*exp(2)/x)+x^2)/(3*x^2*exp(1/9)*exp(20*exp(2)/x)+3*x^2*exp(1/9)+x^3),x, a
lgorithm="fricas")

[Out]

log(x*e^2 + 3*e^(19/9) + 3*e^(1/9*(19*x + 180*e^2)/x))

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giac [B]  time = 0.19, size = 44, normalized size = 1.91 $$\begin{array}{c}(e^2\log (\frac{3e^2}{x}+\frac{3e^{(\frac{20e^2}{x}+2)}}{x}+e^{\frac{17}{9}})-e^2\log \left(\frac{e^2}{x}\right))e^{(-2)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-60*exp(1/9)*exp(2)*exp(20*exp(2)/x)+x^2)/(3*x^2*exp(1/9)*exp(20*exp(2)/x)+3*x^2*exp(1/9)+x^3),x, a
lgorithm="giac")

[Out]

(e^2*log(3*e^2/x + 3*e^(20*e^2/x + 2)/x + e^(17/9)) - e^2*log(e^2/x))*e^(-2)

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maple [A]  time = 0.24, size = 20, normalized size = 0.87

method	result	size
norman	$\ln (3{\mathrm{e}}^{\frac{1}{9}}{\mathrm{e}}^{\frac{20{\mathrm{e}}^2}{x}}+3{\mathrm{e}}^{\frac{1}{9}}+x)$	$20$
risch	$\ln ({\mathrm{e}}^{\frac{20{\mathrm{e}}^2}{x}}+\frac{(3{\mathrm{e}}^{\frac{1}{9}}+x){\mathrm{e}}^{-\frac{1}{9}}}{3})$	$21$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-60*exp(1/9)*exp(2)*exp(20*exp(2)/x)+x^2)/(3*x^2*exp(1/9)*exp(20*exp(2)/x)+3*x^2*exp(1/9)+x^3),x,method=_
RETURNVERBOSE)

[Out]

ln(3*exp(1/9)*exp(20*exp(2)/x)+3*exp(1/9)+x)

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maxima [A]  time = 0.38, size = 23, normalized size = 1.00 $$\begin{array}{c}\log \left(\frac{1}{3}(x+3e^{\frac{1}{9}}+3e^{(\frac{20e^2}{x}+\frac{1}{9})})e^{(-\frac{1}{9})}\right)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-60*exp(1/9)*exp(2)*exp(20*exp(2)/x)+x^2)/(3*x^2*exp(1/9)*exp(20*exp(2)/x)+3*x^2*exp(1/9)+x^3),x, a
lgorithm="maxima")

[Out]

log(1/3*(x + 3*e^(1/9) + 3*e^(20*e^2/x + 1/9))*e^(-1/9))

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mupad [B]  time = 4.69, size = 30, normalized size = 1.30 $$\begin{array}{c}\ln \left(\frac{x+3{\mathrm{e}}^{1/9}+3{\mathrm{e}}^{\frac{20{\mathrm{e}}^2}{x}}{\mathrm{e}}^{1/9}}{x}\right)-\ln \left(\frac{1}{x}\right)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(60*exp((20*exp(2))/x)*exp(19/9) - x^2)/(3*x^2*exp(1/9) + x^3 + 3*x^2*exp((20*exp(2))/x)*exp(1/9)),x)

[Out]

log((x + 3*exp(1/9) + 3*exp((20*exp(2))/x)*exp(1/9))/x) - log(1/x)

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sympy [A]  time = 0.20, size = 24, normalized size = 1.04 $$\begin{array}{c}\log (\frac{x+3e^{\frac{1}{9}}}{3e^{\frac{1}{9}}}+e^{\frac{20e^2}{x}})\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-60*exp(1/9)*exp(2)*exp(20*exp(2)/x)+x**2)/(3*x**2*exp(1/9)*exp(20*exp(2)/x)+3*x**2*exp(1/9)+x**3),
x)

[Out]

log((x + 3*exp(1/9))*exp(-1/9)/3 + exp(20*exp(2)/x))

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