3.5.42 \(\int \frac {-162 x^2+270 x^3-18 x^4+e^{2 x} (-18+30 x-2 x^2)+e^x (108 x-180 x^2+3 x^3+9 x^4)}{e^{2 x} x^3-6 e^x x^4+9 x^5} \, dx\) [442]

Optimal. Leaf size=30 \[ \left (-5+\frac {3}{x}\right )^2+\frac {9}{3-\frac {e^x}{x}}-\log \left (x^2\right ) \]

[Out]

3/(1-1/3*exp(x)/x)+(3/x-5)^2-ln(x^2)

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Rubi [F]
time = 0.62, 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 {-162 x^2+270 x^3-18 x^4+e^{2 x} \left (-18+30 x-2 x^2\right )+e^x \left (108 x-180 x^2+3 x^3+9 x^4\right )}{e^{2 x} x^3-6 e^x x^4+9 x^5} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-162*x^2 + 270*x^3 - 18*x^4 + E^(2*x)*(-18 + 30*x - 2*x^2) + E^x*(108*x - 180*x^2 + 3*x^3 + 9*x^4))/(E^(2
*x)*x^3 - 6*E^x*x^4 + 9*x^5),x]

[Out]

9/x^2 - 30/x - 2*Log[x] - 9*Defer[Int][(E^x - 3*x)^(-1), x] - 27*Defer[Int][x/(E^x - 3*x)^2, x] + 9*Defer[Int]
[x/(E^x - 3*x), x] + 27*Defer[Int][x^2/(E^x - 3*x)^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-162 x^2+270 x^3-18 x^4+e^{2 x} \left (-18+30 x-2 x^2\right )+e^x \left (108 x-180 x^2+3 x^3+9 x^4\right )}{\left (e^x-3 x\right )^2 x^3} \, dx\\ &=\int \left (\frac {9 (-1+x)}{e^x-3 x}+\frac {27 (-1+x) x}{\left (e^x-3 x\right )^2}-\frac {2 \left (9-15 x+x^2\right )}{x^3}\right ) \, dx\\ &=-\left (2 \int \frac {9-15 x+x^2}{x^3} \, dx\right )+9 \int \frac {-1+x}{e^x-3 x} \, dx+27 \int \frac {(-1+x) x}{\left (e^x-3 x\right )^2} \, dx\\ &=-\left (2 \int \left (\frac {9}{x^3}-\frac {15}{x^2}+\frac {1}{x}\right ) \, dx\right )+9 \int \left (-\frac {1}{e^x-3 x}+\frac {x}{e^x-3 x}\right ) \, dx+27 \int \left (-\frac {x}{\left (e^x-3 x\right )^2}+\frac {x^2}{\left (e^x-3 x\right )^2}\right ) \, dx\\ &=\frac {9}{x^2}-\frac {30}{x}-2 \log (x)-9 \int \frac {1}{e^x-3 x} \, dx+9 \int \frac {x}{e^x-3 x} \, dx-27 \int \frac {x}{\left (e^x-3 x\right )^2} \, dx+27 \int \frac {x^2}{\left (e^x-3 x\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 1.39, size = 27, normalized size = 0.90 \begin {gather*} \frac {9}{x^2}-\frac {30}{x}-\frac {9 x}{e^x-3 x}-2 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-162*x^2 + 270*x^3 - 18*x^4 + E^(2*x)*(-18 + 30*x - 2*x^2) + E^x*(108*x - 180*x^2 + 3*x^3 + 9*x^4))
/(E^(2*x)*x^3 - 6*E^x*x^4 + 9*x^5),x]

[Out]

9/x^2 - 30/x - (9*x)/(E^x - 3*x) - 2*Log[x]

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Maple [A]
time = 0.08, size = 28, normalized size = 0.93

method	result	size
risch	\(\frac {-30 x +9}{x^{2}}-2 \ln \left (x \right )+\frac {9 x}{3 x -{\mathrm e}^{x}}\)	\(28\)
norman	\(\frac {9 x^{3}+27 x -90 x^{2}+30 \,{\mathrm e}^{x} x -9 \,{\mathrm e}^{x}}{x^{2} \left (3 x -{\mathrm e}^{x}\right )}-2 \ln \left (x \right )\)	\(43\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x^2+30*x-18)*exp(x)^2+(9*x^4+3*x^3-180*x^2+108*x)*exp(x)-18*x^4+270*x^3-162*x^2)/(exp(x)^2*x^3-6*exp(
x)*x^4+9*x^5),x,method=_RETURNVERBOSE)

[Out]

(-30*x+9)/x^2-2*ln(x)+9*x/(3*x-exp(x))

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Maxima [A]
time = 0.55, size = 44, normalized size = 1.47 \begin {gather*} \frac {3 \, {\left (3 \, x^{3} - 30 \, x^{2} + {\left (10 \, x - 3\right )} e^{x} + 9 \, x\right )}}{3 \, x^{3} - x^{2} e^{x}} - 2 \, \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^2+30*x-18)*exp(x)^2+(9*x^4+3*x^3-180*x^2+108*x)*exp(x)-18*x^4+270*x^3-162*x^2)/(exp(x)^2*x^3-
6*exp(x)*x^4+9*x^5),x, algorithm="maxima")

[Out]

3*(3*x^3 - 30*x^2 + (10*x - 3)*e^x + 9*x)/(3*x^3 - x^2*e^x) - 2*log(x)

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Fricas [A]
time = 0.32, size = 56, normalized size = 1.87 \begin {gather*} \frac {9 \, x^{3} - 90 \, x^{2} + 3 \, {\left (10 \, x - 3\right )} e^{x} - 2 \, {\left (3 \, x^{3} - x^{2} e^{x}\right )} \log \left (x\right ) + 27 \, x}{3 \, x^{3} - x^{2} e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^2+30*x-18)*exp(x)^2+(9*x^4+3*x^3-180*x^2+108*x)*exp(x)-18*x^4+270*x^3-162*x^2)/(exp(x)^2*x^3-
6*exp(x)*x^4+9*x^5),x, algorithm="fricas")

[Out]

(9*x^3 - 90*x^2 + 3*(10*x - 3)*e^x - 2*(3*x^3 - x^2*e^x)*log(x) + 27*x)/(3*x^3 - x^2*e^x)

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Sympy [A]
time = 0.06, size = 24, normalized size = 0.80 \begin {gather*} - \frac {9 x}{- 3 x + e^{x}} - 2 \log {\left (x \right )} - \frac {30 x - 9}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x**2+30*x-18)*exp(x)**2+(9*x**4+3*x**3-180*x**2+108*x)*exp(x)-18*x**4+270*x**3-162*x**2)/(exp(x
)**2*x**3-6*exp(x)*x**4+9*x**5),x)

[Out]

-9*x/(-3*x + exp(x)) - 2*log(x) - (30*x - 9)/x**2

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Giac [A]
time = 0.40, size = 56, normalized size = 1.87 \begin {gather*} -\frac {6 \, x^{3} \log \left (x\right ) - 2 \, x^{2} e^{x} \log \left (x\right ) - 9 \, x^{3} + 90 \, x^{2} - 30 \, x e^{x} - 27 \, x + 9 \, e^{x}}{3 \, x^{3} - x^{2} e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^2+30*x-18)*exp(x)^2+(9*x^4+3*x^3-180*x^2+108*x)*exp(x)-18*x^4+270*x^3-162*x^2)/(exp(x)^2*x^3-
6*exp(x)*x^4+9*x^5),x, algorithm="giac")

[Out]

-(6*x^3*log(x) - 2*x^2*e^x*log(x) - 9*x^3 + 90*x^2 - 30*x*e^x - 27*x + 9*e^x)/(3*x^3 - x^2*e^x)

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Mupad [B]
time = 0.54, size = 34, normalized size = 1.13 \begin {gather*} \frac {9}{x^2}-\frac {9\,x^3}{x^2\,{\mathrm {e}}^x-3\,x^3}-\frac {30}{x}-2\,\ln \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(2*x)*(2*x^2 - 30*x + 18) - exp(x)*(108*x - 180*x^2 + 3*x^3 + 9*x^4) + 162*x^2 - 270*x^3 + 18*x^4)/(x
^3*exp(2*x) - 6*x^4*exp(x) + 9*x^5),x)

[Out]

9/x^2 - (9*x^3)/(x^2*exp(x) - 3*x^3) - 30/x - 2*log(x)

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