3.5.13 \(\int \frac {e^{\frac {3 x-x^2+x^{-\frac {x}{-10+5 x}}+9 x^2 \log (x)}{x}} (160 x^2-160 x^3+40 x^4+(180 x^2-180 x^3+45 x^4) \log (x)+x^{-\frac {x}{-10+5 x}} (-20+22 x-6 x^2+2 x \log (x)))}{20 x^2-20 x^3+5 x^4} \, dx\)

Optimal. Leaf size=28 \[ e^{3-x+x^{-1+\frac {x}{5 (2-x)}}+9 x \log (x)} \]

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Rubi [F]  time = 17.03, 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 {e^{\frac {3 x-x^2+x^{-\frac {x}{-10+5 x}}+9 x^2 \log (x)}{x}} \left (160 x^2-160 x^3+40 x^4+\left (180 x^2-180 x^3+45 x^4\right ) \log (x)+x^{-\frac {x}{-10+5 x}} \left (-20+22 x-6 x^2+2 x \log (x)\right )\right )}{20 x^2-20 x^3+5 x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((3*x - x^2 + x^(-(x/(-10 + 5*x))) + 9*x^2*Log[x])/x)*(160*x^2 - 160*x^3 + 40*x^4 + (180*x^2 - 180*x^3
+ 45*x^4)*Log[x] + (-20 + 22*x - 6*x^2 + 2*x*Log[x])/x^(x/(-10 + 5*x))))/(20*x^2 - 20*x^3 + 5*x^4),x]

[Out]

8*Defer[Int][E^(3 - x + x^(-1 - x/(-10 + 5*x)) + 9*x*Log[x]), x] - (6*Defer[Int][E^(3 - x + x^(-1 - x/(-10 + 5
*x)) + 9*x*Log[x])*x^(-2 + x/(10 - 5*x)), x])/5 - (2*Defer[Int][(E^(3 - x + x^(-1 - x/(-10 + 5*x)) + 9*x*Log[x
])*x^(-2 + x/(10 - 5*x)))/(-2 + x), x])/5 + 9*Defer[Int][E^(3 - x + x^(-1 - x/(-10 + 5*x)) + 9*x*Log[x])*Log[x
], x] + (2*Defer[Int][(E^(3 - x + x^(-1 - x/(-10 + 5*x)) + 9*x*Log[x])*x^(-1 + x/(10 - 5*x))*Log[x])/(-2 + x)^
2, x])/5

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{\frac {3 x-x^2+x^{-\frac {x}{-10+5 x}}+9 x^2 \log (x)}{x}} \left (160 x^2-160 x^3+40 x^4+\left (180 x^2-180 x^3+45 x^4\right ) \log (x)+x^{-\frac {x}{-10+5 x}} \left (-20+22 x-6 x^2+2 x \log (x)\right )\right )}{x^2 \left (20-20 x+5 x^2\right )} \, dx\\ &=\int \frac {e^{\frac {3 x-x^2+x^{-\frac {x}{-10+5 x}}+9 x^2 \log (x)}{x}} \left (160 x^2-160 x^3+40 x^4+\left (180 x^2-180 x^3+45 x^4\right ) \log (x)+x^{-\frac {x}{-10+5 x}} \left (-20+22 x-6 x^2+2 x \log (x)\right )\right )}{5 (-2+x)^2 x^2} \, dx\\ &=\frac {1}{5} \int \frac {e^{\frac {3 x-x^2+x^{-\frac {x}{-10+5 x}}+9 x^2 \log (x)}{x}} \left (160 x^2-160 x^3+40 x^4+\left (180 x^2-180 x^3+45 x^4\right ) \log (x)+x^{-\frac {x}{-10+5 x}} \left (-20+22 x-6 x^2+2 x \log (x)\right )\right )}{(-2+x)^2 x^2} \, dx\\ &=\frac {1}{5} \int \frac {e^{3-x+x^{-1-\frac {x}{-10+5 x}}+9 x \log (x)} \left (160 x^2-160 x^3+40 x^4+\left (180 x^2-180 x^3+45 x^4\right ) \log (x)+x^{-\frac {x}{-10+5 x}} \left (-20+22 x-6 x^2+2 x \log (x)\right )\right )}{(2-x)^2 x^2} \, dx\\ &=\frac {1}{5} \int \left (5 e^{3-x+x^{-1-\frac {x}{-10+5 x}}+9 x \log (x)} (8+9 \log (x))-\frac {2 e^{3-x+x^{-1-\frac {x}{-10+5 x}}+9 x \log (x)} x^{-2+\frac {x}{10-5 x}} \left (10-11 x+3 x^2-x \log (x)\right )}{(-2+x)^2}\right ) \, dx\\ &=-\left (\frac {2}{5} \int \frac {e^{3-x+x^{-1-\frac {x}{-10+5 x}}+9 x \log (x)} x^{-2+\frac {x}{10-5 x}} \left (10-11 x+3 x^2-x \log (x)\right )}{(-2+x)^2} \, dx\right )+\int e^{3-x+x^{-1-\frac {x}{-10+5 x}}+9 x \log (x)} (8+9 \log (x)) \, dx\\ &=-\left (\frac {2}{5} \int \left (\frac {e^{3-x+x^{-1-\frac {x}{-10+5 x}}+9 x \log (x)} x^{-2+\frac {x}{10-5 x}} (-5+3 x)}{-2+x}-\frac {e^{3-x+x^{-1-\frac {x}{-10+5 x}}+9 x \log (x)} x^{-1+\frac {x}{10-5 x}} \log (x)}{(-2+x)^2}\right ) \, dx\right )+\int \left (8 e^{3-x+x^{-1-\frac {x}{-10+5 x}}+9 x \log (x)}+9 e^{3-x+x^{-1-\frac {x}{-10+5 x}}+9 x \log (x)} \log (x)\right ) \, dx\\ &=-\left (\frac {2}{5} \int \frac {e^{3-x+x^{-1-\frac {x}{-10+5 x}}+9 x \log (x)} x^{-2+\frac {x}{10-5 x}} (-5+3 x)}{-2+x} \, dx\right )+\frac {2}{5} \int \frac {e^{3-x+x^{-1-\frac {x}{-10+5 x}}+9 x \log (x)} x^{-1+\frac {x}{10-5 x}} \log (x)}{(-2+x)^2} \, dx+8 \int e^{3-x+x^{-1-\frac {x}{-10+5 x}}+9 x \log (x)} \, dx+9 \int e^{3-x+x^{-1-\frac {x}{-10+5 x}}+9 x \log (x)} \log (x) \, dx\\ &=-\left (\frac {2}{5} \int \left (3 e^{3-x+x^{-1-\frac {x}{-10+5 x}}+9 x \log (x)} x^{-2+\frac {x}{10-5 x}}+\frac {e^{3-x+x^{-1-\frac {x}{-10+5 x}}+9 x \log (x)} x^{-2+\frac {x}{10-5 x}}}{-2+x}\right ) \, dx\right )+\frac {2}{5} \int \frac {e^{3-x+x^{-1-\frac {x}{-10+5 x}}+9 x \log (x)} x^{-1+\frac {x}{10-5 x}} \log (x)}{(-2+x)^2} \, dx+8 \int e^{3-x+x^{-1-\frac {x}{-10+5 x}}+9 x \log (x)} \, dx+9 \int e^{3-x+x^{-1-\frac {x}{-10+5 x}}+9 x \log (x)} \log (x) \, dx\\ &=-\left (\frac {2}{5} \int \frac {e^{3-x+x^{-1-\frac {x}{-10+5 x}}+9 x \log (x)} x^{-2+\frac {x}{10-5 x}}}{-2+x} \, dx\right )+\frac {2}{5} \int \frac {e^{3-x+x^{-1-\frac {x}{-10+5 x}}+9 x \log (x)} x^{-1+\frac {x}{10-5 x}} \log (x)}{(-2+x)^2} \, dx-\frac {6}{5} \int e^{3-x+x^{-1-\frac {x}{-10+5 x}}+9 x \log (x)} x^{-2+\frac {x}{10-5 x}} \, dx+8 \int e^{3-x+x^{-1-\frac {x}{-10+5 x}}+9 x \log (x)} \, dx+9 \int e^{3-x+x^{-1-\frac {x}{-10+5 x}}+9 x \log (x)} \log (x) \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.29, size = 27, normalized size = 0.96 \begin {gather*} e^{3-x+x^{-1-\frac {x}{5 (-2+x)}}} x^{9 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((3*x - x^2 + x^(-(x/(-10 + 5*x))) + 9*x^2*Log[x])/x)*(160*x^2 - 160*x^3 + 40*x^4 + (180*x^2 - 18
0*x^3 + 45*x^4)*Log[x] + (-20 + 22*x - 6*x^2 + 2*x*Log[x])/x^(x/(-10 + 5*x))))/(20*x^2 - 20*x^3 + 5*x^4),x]

[Out]

E^(3 - x + x^(-1 - x/(5*(-2 + x))))*x^(9*x)

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fricas [A]  time = 0.62, size = 46, normalized size = 1.64 \begin {gather*} e^{\left (\frac {{\left (9 \, x^{2} \log \relax (x) - x^{2} + 3 \, x\right )} x^{\frac {x}{5 \, {\left (x - 2\right )}}} + 1}{x x^{\frac {x}{5 \, {\left (x - 2\right )}}}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x*log(x)-6*x^2+22*x-20)*exp(-x*log(x)/(5*x-10))+(45*x^4-180*x^3+180*x^2)*log(x)+40*x^4-160*x^3+1
60*x^2)*exp((exp(-x*log(x)/(5*x-10))+9*x^2*log(x)-x^2+3*x)/x)/(5*x^4-20*x^3+20*x^2),x, algorithm="fricas")

[Out]

e^(((9*x^2*log(x) - x^2 + 3*x)*x^(1/5*x/(x - 2)) + 1)/(x*x^(1/5*x/(x - 2))))

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x*log(x)-6*x^2+22*x-20)*exp(-x*log(x)/(5*x-10))+(45*x^4-180*x^3+180*x^2)*log(x)+40*x^4-160*x^3+1
60*x^2)*exp((exp(-x*log(x)/(5*x-10))+9*x^2*log(x)-x^2+3*x)/x)/(5*x^4-20*x^3+20*x^2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Unable to divide, perhaps due to rounding error%%%{-2880000,[1,8]%%%}+%%%{25920000,[1,7]%%%} / %%%{1600000,
[0,6]%%%} E

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maple [A]  time = 0.08, size = 32, normalized size = 1.14

method	result	size
risch	\(x^{9 x} {\mathrm e}^{-\frac {x^{2}-x^{-\frac {x}{5 \left (x -2\right )}}-3 x}{x}}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x*ln(x)-6*x^2+22*x-20)*exp(-x*ln(x)/(5*x-10))+(45*x^4-180*x^3+180*x^2)*ln(x)+40*x^4-160*x^3+160*x^2)*e
xp((exp(-x*ln(x)/(5*x-10))+9*x^2*ln(x)-x^2+3*x)/x)/(5*x^4-20*x^3+20*x^2),x,method=_RETURNVERBOSE)

[Out]

x^(9*x)*exp(-(x^2-x^(-1/5*x/(x-2))-3*x)/x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {1}{5} \, \int \frac {{\left (40 \, x^{4} - 160 \, x^{3} + 160 \, x^{2} + 45 \, {\left (x^{4} - 4 \, x^{3} + 4 \, x^{2}\right )} \log \relax (x) - \frac {2 \, {\left (3 \, x^{2} - x \log \relax (x) - 11 \, x + 10\right )}}{x^{\frac {x}{5 \, {\left (x - 2\right )}}}}\right )} e^{\left (\frac {9 \, x^{2} \log \relax (x) - x^{2} + 3 \, x + \frac {1}{x^{\frac {x}{5 \, {\left (x - 2\right )}}}}}{x}\right )}}{x^{4} - 4 \, x^{3} + 4 \, x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x*log(x)-6*x^2+22*x-20)*exp(-x*log(x)/(5*x-10))+(45*x^4-180*x^3+180*x^2)*log(x)+40*x^4-160*x^3+1
60*x^2)*exp((exp(-x*log(x)/(5*x-10))+9*x^2*log(x)-x^2+3*x)/x)/(5*x^4-20*x^3+20*x^2),x, algorithm="maxima")

[Out]

1/5*integrate((40*x^4 - 160*x^3 + 160*x^2 + 45*(x^4 - 4*x^3 + 4*x^2)*log(x) - 2*(3*x^2 - x*log(x) - 11*x + 10)
/x^(1/5*x/(x - 2)))*e^((9*x^2*log(x) - x^2 + 3*x + 1/x^(1/5*x/(x - 2)))/x)/(x^4 - 4*x^3 + 4*x^2), x)

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mupad [B]  time = 0.69, size = 28, normalized size = 1.00 \begin {gather*} x^{9\,x}\,{\mathrm {e}}^{\frac {1}{x^{\frac {x}{5\,x-10}+1}}}\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^3 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((3*x + exp(-(x*log(x))/(5*x - 10)) + 9*x^2*log(x) - x^2)/x)*(exp(-(x*log(x))/(5*x - 10))*(22*x + 2*x*
log(x) - 6*x^2 - 20) + log(x)*(180*x^2 - 180*x^3 + 45*x^4) + 160*x^2 - 160*x^3 + 40*x^4))/(20*x^2 - 20*x^3 + 5
*x^4),x)

[Out]

x^(9*x)*exp(1/x^(x/(5*x - 10) + 1))*exp(-x)*exp(3)

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sympy [A]  time = 1.44, size = 31, normalized size = 1.11 \begin {gather*} e^{\frac {9 x^{2} \log {\relax (x )} - x^{2} + 3 x + e^{- \frac {x \log {\relax (x )}}{5 x - 10}}}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x*ln(x)-6*x**2+22*x-20)*exp(-x*ln(x)/(5*x-10))+(45*x**4-180*x**3+180*x**2)*ln(x)+40*x**4-160*x**
3+160*x**2)*exp((exp(-x*ln(x)/(5*x-10))+9*x**2*ln(x)-x**2+3*x)/x)/(5*x**4-20*x**3+20*x**2),x)

[Out]

exp((9*x**2*log(x) - x**2 + 3*x + exp(-x*log(x)/(5*x - 10)))/x)

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