3.72.54 \(\int \frac {e^{\frac {36+48 x-7 x^2-7 x^3+x^4+(-3 x^2+x^3) \log (5)}{-12-20 x-4 x^2+x^3+x^2 \log (5)}} (144+456 x+476 x^2+136 x^3-25 x^4-8 x^5+x^6+(-24 x^2-40 x^3-8 x^4+2 x^5) \log (5)+x^4 \log ^2(5))}{144+480 x+496 x^2+136 x^3-24 x^4-8 x^5+x^6+(-24 x^2-40 x^3-8 x^4+2 x^5) \log (5)+x^4 \log ^2(5)} \, dx\)

Optimal. Leaf size=24 \[ e^{-3+x+\frac {1}{x-\frac {4 \left (5+\frac {3}{x}+x\right )}{x}+\log (5)}} \]

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Rubi [F]  time = 16.68, 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 {\exp \left (\frac {36+48 x-7 x^2-7 x^3+x^4+\left (-3 x^2+x^3\right ) \log (5)}{-12-20 x-4 x^2+x^3+x^2 \log (5)}\right ) \left (144+456 x+476 x^2+136 x^3-25 x^4-8 x^5+x^6+\left (-24 x^2-40 x^3-8 x^4+2 x^5\right ) \log (5)+x^4 \log ^2(5)\right )}{144+480 x+496 x^2+136 x^3-24 x^4-8 x^5+x^6+\left (-24 x^2-40 x^3-8 x^4+2 x^5\right ) \log (5)+x^4 \log ^2(5)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((36 + 48*x - 7*x^2 - 7*x^3 + x^4 + (-3*x^2 + x^3)*Log[5])/(-12 - 20*x - 4*x^2 + x^3 + x^2*Log[5]))*(14
4 + 456*x + 476*x^2 + 136*x^3 - 25*x^4 - 8*x^5 + x^6 + (-24*x^2 - 40*x^3 - 8*x^4 + 2*x^5)*Log[5] + x^4*Log[5]^
2))/(144 + 480*x + 496*x^2 + 136*x^3 - 24*x^4 - 8*x^5 + x^6 + (-24*x^2 - 40*x^3 - 8*x^4 + 2*x^5)*Log[5] + x^4*
Log[5]^2),x]

[Out]

Defer[Int][E^((36 + 48*x + x^4 - x^3*(7 - Log[5]) - x^2*(7 + Log[125]))/(-12 - 20*x + x^3 - x^2*(4 - Log[5])))
, x] - 12*(4 - Log[5])*Defer[Int][E^((36 + 48*x + x^4 - x^3*(7 - Log[5]) - x^2*(7 + Log[125]))/(-12 - 20*x + x
^3 - x^2*(4 - Log[5])))/(12 + 20*x - x^3 + x^2*(4 - Log[5]))^2, x] - 4*(29 - 5*Log[5])*Defer[Int][(E^((36 + 48
*x + x^4 - x^3*(7 - Log[5]) - x^2*(7 + Log[125]))/(-12 - 20*x + x^3 - x^2*(4 - Log[5])))*x)/(12 + 20*x - x^3 +
x^2*(4 - Log[5]))^2, x] - (56 - 8*Log[5] + Log[5]^2)*Defer[Int][(E^((36 + 48*x + x^4 - x^3*(7 - Log[5]) - x^2
*(7 + Log[125]))/(-12 - 20*x + x^3 - x^2*(4 - Log[5])))*x^2)/(12 + 20*x - x^3 + x^2*(4 - Log[5]))^2, x] + (4 -
Log[5])*Defer[Int][E^((36 + 48*x + x^4 - x^3*(7 - Log[5]) - x^2*(7 + Log[125]))/(-12 - 20*x + x^3 - x^2*(4 -
Log[5])))/(12 + 20*x - x^3 + x^2*(4 - Log[5])), x] + Defer[Int][(E^((36 + 48*x + x^4 - x^3*(7 - Log[5]) - x^2*
(7 + Log[125]))/(-12 - 20*x + x^3 - x^2*(4 - Log[5])))*x)/(12 + 20*x - x^3 + x^2*(4 - Log[5])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (\frac {36+48 x-7 x^2-7 x^3+x^4+\left (-3 x^2+x^3\right ) \log (5)}{-12-20 x-4 x^2+x^3+x^2 \log (5)}\right ) \left (144+456 x+476 x^2+136 x^3-8 x^5+x^6+\left (-24 x^2-40 x^3-8 x^4+2 x^5\right ) \log (5)+x^4 \left (-25+\log ^2(5)\right )\right )}{144+480 x+496 x^2+136 x^3-24 x^4-8 x^5+x^6+\left (-24 x^2-40 x^3-8 x^4+2 x^5\right ) \log (5)+x^4 \log ^2(5)} \, dx\\ &=\int \frac {\exp \left (\frac {36+48 x-7 x^2-7 x^3+x^4+\left (-3 x^2+x^3\right ) \log (5)}{-12-20 x-4 x^2+x^3+x^2 \log (5)}\right ) \left (144+456 x+476 x^2+136 x^3-8 x^5+x^6+\left (-24 x^2-40 x^3-8 x^4+2 x^5\right ) \log (5)+x^4 \left (-25+\log ^2(5)\right )\right )}{144+480 x+496 x^2+136 x^3-8 x^5+x^6+\left (-24 x^2-40 x^3-8 x^4+2 x^5\right ) \log (5)+x^4 \left (-24+\log ^2(5)\right )} \, dx\\ &=\int \frac {\exp \left (\frac {36+48 x+x^4-x^3 (7-\log (5))-x^2 (7+\log (125))}{-12-20 x+x^3-x^2 (4-\log (5))}\right ) \left (144+456 x+476 x^2+136 x^3-8 x^5+x^6+\left (-24 x^2-40 x^3-8 x^4+2 x^5\right ) \log (5)+x^4 \left (-25+\log ^2(5)\right )\right )}{\left (12+20 x-x^3+x^2 (4-\log (5))\right )^2} \, dx\\ &=\int \left (\exp \left (\frac {36+48 x+x^4-x^3 (7-\log (5))-x^2 (7+\log (125))}{-12-20 x+x^3-x^2 (4-\log (5))}\right )+\frac {\exp \left (\frac {36+48 x+x^4-x^3 (7-\log (5))-x^2 (7+\log (125))}{-12-20 x+x^3-x^2 (4-\log (5))}\right ) (4+x-\log (5))}{12+20 x-x^3+x^2 (4-\log (5))}+\frac {\exp \left (\frac {36+48 x+x^4-x^3 (7-\log (5))-x^2 (7+\log (125))}{-12-20 x+x^3-x^2 (4-\log (5))}\right ) \left (-4 x (29-5 \log (5))-12 (4-\log (5))-x^2 \left (56-8 \log (5)+\log ^2(5)\right )\right )}{\left (12+20 x-x^3+x^2 (4-\log (5))\right )^2}\right ) \, dx\\ &=\int \exp \left (\frac {36+48 x+x^4-x^3 (7-\log (5))-x^2 (7+\log (125))}{-12-20 x+x^3-x^2 (4-\log (5))}\right ) \, dx+\int \frac {\exp \left (\frac {36+48 x+x^4-x^3 (7-\log (5))-x^2 (7+\log (125))}{-12-20 x+x^3-x^2 (4-\log (5))}\right ) (4+x-\log (5))}{12+20 x-x^3+x^2 (4-\log (5))} \, dx+\int \frac {\exp \left (\frac {36+48 x+x^4-x^3 (7-\log (5))-x^2 (7+\log (125))}{-12-20 x+x^3-x^2 (4-\log (5))}\right ) \left (-4 x (29-5 \log (5))-12 (4-\log (5))-x^2 \left (56-8 \log (5)+\log ^2(5)\right )\right )}{\left (12+20 x-x^3+x^2 (4-\log (5))\right )^2} \, dx\\ &=\int \exp \left (\frac {36+48 x+x^4-x^3 (7-\log (5))-x^2 (7+\log (125))}{-12-20 x+x^3-x^2 (4-\log (5))}\right ) \, dx+\int \left (\frac {\exp \left (\frac {36+48 x+x^4-x^3 (7-\log (5))-x^2 (7+\log (125))}{-12-20 x+x^3-x^2 (4-\log (5))}\right ) x}{12+20 x-x^3+x^2 (4-\log (5))}+\frac {4 \exp \left (\frac {36+48 x+x^4-x^3 (7-\log (5))-x^2 (7+\log (125))}{-12-20 x+x^3-x^2 (4-\log (5))}\right ) \left (1-\frac {\log (5)}{4}\right )}{12+20 x-x^3+x^2 (4-\log (5))}\right ) \, dx+\int \left (\frac {12 \exp \left (\frac {36+48 x+x^4-x^3 (7-\log (5))-x^2 (7+\log (125))}{-12-20 x+x^3-x^2 (4-\log (5))}\right ) (-4+\log (5))}{\left (12+20 x-x^3+x^2 (4-\log (5))\right )^2}+\frac {4 \exp \left (\frac {36+48 x+x^4-x^3 (7-\log (5))-x^2 (7+\log (125))}{-12-20 x+x^3-x^2 (4-\log (5))}\right ) x (-29+5 \log (5))}{\left (12+20 x-x^3+x^2 (4-\log (5))\right )^2}+\frac {\exp \left (\frac {36+48 x+x^4-x^3 (7-\log (5))-x^2 (7+\log (125))}{-12-20 x+x^3-x^2 (4-\log (5))}\right ) x^2 \left (-56+8 \log (5)-\log ^2(5)\right )}{\left (12+20 x-x^3+x^2 (4-\log (5))\right )^2}\right ) \, dx\\ &=-\left ((4 (29-5 \log (5))) \int \frac {\exp \left (\frac {36+48 x+x^4-x^3 (7-\log (5))-x^2 (7+\log (125))}{-12-20 x+x^3-x^2 (4-\log (5))}\right ) x}{\left (12+20 x-x^3+x^2 (4-\log (5))\right )^2} \, dx\right )+(4-\log (5)) \int \frac {\exp \left (\frac {36+48 x+x^4-x^3 (7-\log (5))-x^2 (7+\log (125))}{-12-20 x+x^3-x^2 (4-\log (5))}\right )}{12+20 x-x^3+x^2 (4-\log (5))} \, dx-(12 (4-\log (5))) \int \frac {\exp \left (\frac {36+48 x+x^4-x^3 (7-\log (5))-x^2 (7+\log (125))}{-12-20 x+x^3-x^2 (4-\log (5))}\right )}{\left (12+20 x-x^3+x^2 (4-\log (5))\right )^2} \, dx+\left (-56+8 \log (5)-\log ^2(5)\right ) \int \frac {\exp \left (\frac {36+48 x+x^4-x^3 (7-\log (5))-x^2 (7+\log (125))}{-12-20 x+x^3-x^2 (4-\log (5))}\right ) x^2}{\left (12+20 x-x^3+x^2 (4-\log (5))\right )^2} \, dx+\int \exp \left (\frac {36+48 x+x^4-x^3 (7-\log (5))-x^2 (7+\log (125))}{-12-20 x+x^3-x^2 (4-\log (5))}\right ) \, dx+\int \frac {\exp \left (\frac {36+48 x+x^4-x^3 (7-\log (5))-x^2 (7+\log (125))}{-12-20 x+x^3-x^2 (4-\log (5))}\right ) x}{12+20 x-x^3+x^2 (4-\log (5))} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.20, size = 79, normalized size = 3.29 \begin {gather*} 5^{-\frac {x^2 \log (5)}{-12-20 x+x^3+x^2 (-4+\log (5))}} e^{\frac {36+48 x+x^4+x^3 (-7+\log (5))+x^2 \left (-7-3 \log (5)+\log ^2(5)\right )}{-12-20 x+x^3+x^2 (-4+\log (5))}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((36 + 48*x - 7*x^2 - 7*x^3 + x^4 + (-3*x^2 + x^3)*Log[5])/(-12 - 20*x - 4*x^2 + x^3 + x^2*Log[5]
))*(144 + 456*x + 476*x^2 + 136*x^3 - 25*x^4 - 8*x^5 + x^6 + (-24*x^2 - 40*x^3 - 8*x^4 + 2*x^5)*Log[5] + x^4*L
og[5]^2))/(144 + 480*x + 496*x^2 + 136*x^3 - 24*x^4 - 8*x^5 + x^6 + (-24*x^2 - 40*x^3 - 8*x^4 + 2*x^5)*Log[5]
+ x^4*Log[5]^2),x]

[Out]

E^((36 + 48*x + x^4 + x^3*(-7 + Log[5]) + x^2*(-7 - 3*Log[5] + Log[5]^2))/(-12 - 20*x + x^3 + x^2*(-4 + Log[5]
)))/5^((x^2*Log[5])/(-12 - 20*x + x^3 + x^2*(-4 + Log[5])))

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fricas [B]  time = 0.55, size = 53, normalized size = 2.21 \begin {gather*} e^{\left (\frac {x^{4} - 7 \, x^{3} - 7 \, x^{2} + {\left (x^{3} - 3 \, x^{2}\right )} \log \relax (5) + 48 \, x + 36}{x^{3} + x^{2} \log \relax (5) - 4 \, x^{2} - 20 \, x - 12}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4*log(5)^2+(2*x^5-8*x^4-40*x^3-24*x^2)*log(5)+x^6-8*x^5-25*x^4+136*x^3+476*x^2+456*x+144)*exp(((x
^3-3*x^2)*log(5)+x^4-7*x^3-7*x^2+48*x+36)/(x^2*log(5)+x^3-4*x^2-20*x-12))/(x^4*log(5)^2+(2*x^5-8*x^4-40*x^3-24
*x^2)*log(5)+x^6-8*x^5-24*x^4+136*x^3+496*x^2+480*x+144),x, algorithm="fricas")

[Out]

e^((x^4 - 7*x^3 - 7*x^2 + (x^3 - 3*x^2)*log(5) + 48*x + 36)/(x^3 + x^2*log(5) - 4*x^2 - 20*x - 12))

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giac [B]  time = 0.27, size = 181, normalized size = 7.54 \begin {gather*} e^{\left (\frac {x^{4}}{x^{3} + x^{2} \log \relax (5) - 4 \, x^{2} - 20 \, x - 12} + \frac {x^{3} \log \relax (5)}{x^{3} + x^{2} \log \relax (5) - 4 \, x^{2} - 20 \, x - 12} - \frac {7 \, x^{3}}{x^{3} + x^{2} \log \relax (5) - 4 \, x^{2} - 20 \, x - 12} - \frac {3 \, x^{2} \log \relax (5)}{x^{3} + x^{2} \log \relax (5) - 4 \, x^{2} - 20 \, x - 12} - \frac {7 \, x^{2}}{x^{3} + x^{2} \log \relax (5) - 4 \, x^{2} - 20 \, x - 12} + \frac {48 \, x}{x^{3} + x^{2} \log \relax (5) - 4 \, x^{2} - 20 \, x - 12} + \frac {36}{x^{3} + x^{2} \log \relax (5) - 4 \, x^{2} - 20 \, x - 12}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4*log(5)^2+(2*x^5-8*x^4-40*x^3-24*x^2)*log(5)+x^6-8*x^5-25*x^4+136*x^3+476*x^2+456*x+144)*exp(((x
^3-3*x^2)*log(5)+x^4-7*x^3-7*x^2+48*x+36)/(x^2*log(5)+x^3-4*x^2-20*x-12))/(x^4*log(5)^2+(2*x^5-8*x^4-40*x^3-24
*x^2)*log(5)+x^6-8*x^5-24*x^4+136*x^3+496*x^2+480*x+144),x, algorithm="giac")

[Out]

e^(x^4/(x^3 + x^2*log(5) - 4*x^2 - 20*x - 12) + x^3*log(5)/(x^3 + x^2*log(5) - 4*x^2 - 20*x - 12) - 7*x^3/(x^3
+ x^2*log(5) - 4*x^2 - 20*x - 12) - 3*x^2*log(5)/(x^3 + x^2*log(5) - 4*x^2 - 20*x - 12) - 7*x^2/(x^3 + x^2*lo
g(5) - 4*x^2 - 20*x - 12) + 48*x/(x^3 + x^2*log(5) - 4*x^2 - 20*x - 12) + 36/(x^3 + x^2*log(5) - 4*x^2 - 20*x
- 12))

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maple [B]  time = 0.37, size = 55, normalized size = 2.29

method	result	size
gosper	\({\mathrm e}^{\frac {x^{3} \ln \relax (5)+x^{4}-3 x^{2} \ln \relax (5)-7 x^{3}-7 x^{2}+48 x +36}{x^{2} \ln \relax (5)+x^{3}-4 x^{2}-20 x -12}}\)	\(55\)
risch	\({\mathrm e}^{\frac {x^{3} \ln \relax (5)+x^{4}-3 x^{2} \ln \relax (5)-7 x^{3}-7 x^{2}+48 x +36}{x^{2} \ln \relax (5)+x^{3}-4 x^{2}-20 x -12}}\)	\(55\)
norman	\(\frac {x^{3} {\mathrm e}^{\frac {\left (x^{3}-3 x^{2}\right ) \ln \relax (5)+x^{4}-7 x^{3}-7 x^{2}+48 x +36}{x^{2} \ln \relax (5)+x^{3}-4 x^{2}-20 x -12}}+\left (\ln \relax (5)-4\right ) x^{2} {\mathrm e}^{\frac {\left (x^{3}-3 x^{2}\right ) \ln \relax (5)+x^{4}-7 x^{3}-7 x^{2}+48 x +36}{x^{2} \ln \relax (5)+x^{3}-4 x^{2}-20 x -12}}-20 x \,{\mathrm e}^{\frac {\left (x^{3}-3 x^{2}\right ) \ln \relax (5)+x^{4}-7 x^{3}-7 x^{2}+48 x +36}{x^{2} \ln \relax (5)+x^{3}-4 x^{2}-20 x -12}}-12 \,{\mathrm e}^{\frac {\left (x^{3}-3 x^{2}\right ) \ln \relax (5)+x^{4}-7 x^{3}-7 x^{2}+48 x +36}{x^{2} \ln \relax (5)+x^{3}-4 x^{2}-20 x -12}}}{x^{2} \ln \relax (5)+x^{3}-4 x^{2}-20 x -12}\)	\(253\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4*ln(5)^2+(2*x^5-8*x^4-40*x^3-24*x^2)*ln(5)+x^6-8*x^5-25*x^4+136*x^3+476*x^2+456*x+144)*exp(((x^3-3*x^2
)*ln(5)+x^4-7*x^3-7*x^2+48*x+36)/(x^2*ln(5)+x^3-4*x^2-20*x-12))/(x^4*ln(5)^2+(2*x^5-8*x^4-40*x^3-24*x^2)*ln(5)
+x^6-8*x^5-24*x^4+136*x^3+496*x^2+480*x+144),x,method=_RETURNVERBOSE)

[Out]

exp((x^3*ln(5)+x^4-3*x^2*ln(5)-7*x^3-7*x^2+48*x+36)/(x^2*ln(5)+x^3-4*x^2-20*x-12))

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maxima [A]  time = 1.03, size = 26, normalized size = 1.08 \begin {gather*} e^{\left (x + \frac {x^{2}}{x^{3} + x^{2} {\left (\log \relax (5) - 4\right )} - 20 \, x - 12} - 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4*log(5)^2+(2*x^5-8*x^4-40*x^3-24*x^2)*log(5)+x^6-8*x^5-25*x^4+136*x^3+476*x^2+456*x+144)*exp(((x
^3-3*x^2)*log(5)+x^4-7*x^3-7*x^2+48*x+36)/(x^2*log(5)+x^3-4*x^2-20*x-12))/(x^4*log(5)^2+(2*x^5-8*x^4-40*x^3-24
*x^2)*log(5)+x^6-8*x^5-24*x^4+136*x^3+496*x^2+480*x+144),x, algorithm="maxima")

[Out]

e^(x + x^2/(x^3 + x^2*(log(5) - 4) - 20*x - 12) - 3)

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mupad [B]  time = 5.31, size = 184, normalized size = 7.67 \begin {gather*} 5^{\frac {3\,x^2-x^3}{20\,x-x^2\,\ln \relax (5)+4\,x^2-x^3+12}}\,{\mathrm {e}}^{-\frac {x^4}{20\,x-x^2\,\ln \relax (5)+4\,x^2-x^3+12}}\,{\mathrm {e}}^{\frac {7\,x^2}{20\,x-x^2\,\ln \relax (5)+4\,x^2-x^3+12}}\,{\mathrm {e}}^{\frac {7\,x^3}{20\,x-x^2\,\ln \relax (5)+4\,x^2-x^3+12}}\,{\mathrm {e}}^{-\frac {36}{20\,x-x^2\,\ln \relax (5)+4\,x^2-x^3+12}}\,{\mathrm {e}}^{-\frac {48\,x}{20\,x-x^2\,\ln \relax (5)+4\,x^2-x^3+12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-(48*x - log(5)*(3*x^2 - x^3) - 7*x^2 - 7*x^3 + x^4 + 36)/(20*x - x^2*log(5) + 4*x^2 - x^3 + 12))*(45
6*x + x^4*log(5)^2 - log(5)*(24*x^2 + 40*x^3 + 8*x^4 - 2*x^5) + 476*x^2 + 136*x^3 - 25*x^4 - 8*x^5 + x^6 + 144
))/(480*x + x^4*log(5)^2 - log(5)*(24*x^2 + 40*x^3 + 8*x^4 - 2*x^5) + 496*x^2 + 136*x^3 - 24*x^4 - 8*x^5 + x^6
+ 144),x)

[Out]

5^((3*x^2 - x^3)/(20*x - x^2*log(5) + 4*x^2 - x^3 + 12))*exp(-x^4/(20*x - x^2*log(5) + 4*x^2 - x^3 + 12))*exp(
(7*x^2)/(20*x - x^2*log(5) + 4*x^2 - x^3 + 12))*exp((7*x^3)/(20*x - x^2*log(5) + 4*x^2 - x^3 + 12))*exp(-36/(2
0*x - x^2*log(5) + 4*x^2 - x^3 + 12))*exp(-(48*x)/(20*x - x^2*log(5) + 4*x^2 - x^3 + 12))

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sympy [B]  time = 2.32, size = 51, normalized size = 2.12 \begin {gather*} e^{\frac {x^{4} - 7 x^{3} - 7 x^{2} + 48 x + \left (x^{3} - 3 x^{2}\right ) \log {\relax (5 )} + 36}{x^{3} - 4 x^{2} + x^{2} \log {\relax (5 )} - 20 x - 12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**4*ln(5)**2+(2*x**5-8*x**4-40*x**3-24*x**2)*ln(5)+x**6-8*x**5-25*x**4+136*x**3+476*x**2+456*x+144
)*exp(((x**3-3*x**2)*ln(5)+x**4-7*x**3-7*x**2+48*x+36)/(x**2*ln(5)+x**3-4*x**2-20*x-12))/(x**4*ln(5)**2+(2*x**
5-8*x**4-40*x**3-24*x**2)*ln(5)+x**6-8*x**5-24*x**4+136*x**3+496*x**2+480*x+144),x)

[Out]

exp((x**4 - 7*x**3 - 7*x**2 + 48*x + (x**3 - 3*x**2)*log(5) + 36)/(x**3 - 4*x**2 + x**2*log(5) - 20*x - 12))

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