3.88.17 \(\int \frac {5^{\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} (-900 x+20000 x^3+20000 x^4+(-1800 x+900 x^2-40000 x^4-20000 x^5) \log (5 x))}{81-3600 x^2-3600 x^3+40000 x^4+80000 x^5+40000 x^6} \, dx\)

Optimal. Leaf size=45 \[ 5^{\frac {e^{-x}}{2-\frac {9}{100 x^2}+2 x}} x^{\frac {e^{-x}}{2-\frac {9}{100 x^2}+2 x}} \]

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Rubi [F]  time = 25.20, 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 {5^{\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} \left (-900 x+20000 x^3+20000 x^4+\left (-1800 x+900 x^2-40000 x^4-20000 x^5\right ) \log (5 x)\right )}{81-3600 x^2-3600 x^3+40000 x^4+80000 x^5+40000 x^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(5^((100*x^2)/(E^x*(-9 + 200*x^2 + 200*x^3)))*x^((100*x^2)/(E^x*(-9 + 200*x^2 + 200*x^3)))*(-900*x + 20000
*x^3 + 20000*x^4 + (-1800*x + 900*x^2 - 40000*x^4 - 20000*x^5)*Log[5*x]))/(E^x*(81 - 3600*x^2 - 3600*x^3 + 400
00*x^4 + 80000*x^5 + 40000*x^6)),x]

[Out]

-108*Log[5*x]*Defer[Int][(5^(2 + (100*x^2)/(E^x*(-9 + 200*x^2 + 200*x^3)))*x^(1 + (100*x^2)/(E^x*(-9 + 200*x^2
+ 200*x^3))))/(E^x*(-9 + 200*x^2 + 200*x^3)^2), x] + 32*Log[5*x]*Defer[Int][(5^(4 + (100*x^2)/(E^x*(-9 + 200*
x^2 + 200*x^3)))*x^(3 + (100*x^2)/(E^x*(-9 + 200*x^2 + 200*x^3))))/(E^x*(-9 + 200*x^2 + 200*x^3)^2), x] + 4*De
fer[Int][(5^(2 + (100*x^2)/(E^x*(-9 + 200*x^2 + 200*x^3)))*x^(1 + (100*x^2)/(E^x*(-9 + 200*x^2 + 200*x^3))))/(
E^x*(-9 + 200*x^2 + 200*x^3)), x] - 4*Log[5*x]*Defer[Int][(5^(2 + (100*x^2)/(E^x*(-9 + 200*x^2 + 200*x^3)))*x^
(1 + (100*x^2)/(E^x*(-9 + 200*x^2 + 200*x^3))))/(E^x*(-9 + 200*x^2 + 200*x^3)), x] - 4*Log[5*x]*Defer[Int][(5^
(2 + (100*x^2)/(E^x*(-9 + 200*x^2 + 200*x^3)))*x^(2 + (100*x^2)/(E^x*(-9 + 200*x^2 + 200*x^3))))/(E^x*(-9 + 20
0*x^2 + 200*x^3)), x] + 108*Defer[Int][Defer[Int][(5^(2 + (100*x^2)/(E^x*(-9 + 200*x^2 + 200*x^3)))*x^(1 + (10
0*x^2)/(E^x*(-9 + 200*x^2 + 200*x^3))))/(E^x*(-9 + 200*x^2 + 200*x^3)^2), x]/x, x] - 32*Defer[Int][Defer[Int][
(5^(4 + (100*x^2)/(E^x*(-9 + 200*x^2 + 200*x^3)))*x^(3 + (100*x^2)/(E^x*(-9 + 200*x^2 + 200*x^3))))/(E^x*(-9 +
200*x^2 + 200*x^3)^2), x]/x, x] + 4*Defer[Int][Defer[Int][(5^(2 + (100*x^2)/(E^x*(-9 + 200*x^2 + 200*x^3)))*x
^(1 + (100*x^2)/(E^x*(-9 + 200*x^2 + 200*x^3))))/(E^x*(-9 + 200*x^2 + 200*x^3)), x]/x, x] + 4*Defer[Int][Defer
[Int][(5^(2 + (100*x^2)/(E^x*(-9 + 200*x^2 + 200*x^3)))*x^(2 + (100*x^2)/(E^x*(-9 + 200*x^2 + 200*x^3))))/(E^x
*(-9 + 200*x^2 + 200*x^3)), x]/x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {5^{\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} \left (-900 x+20000 x^3+20000 x^4+\left (-1800 x+900 x^2-40000 x^4-20000 x^5\right ) \log (5 x)\right )}{\left (9-200 x^2-200 x^3\right )^2} \, dx\\ &=\int \left (\frac {4\ 5^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{1+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}}}{-9+200 x^2+200 x^3}-\frac {4\ 5^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{1+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} \left (18-9 x+400 x^3+200 x^4\right ) \log (5 x)}{\left (-9+200 x^2+200 x^3\right )^2}\right ) \, dx\\ &=4 \int \frac {5^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{1+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}}}{-9+200 x^2+200 x^3} \, dx-4 \int \frac {5^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{1+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} \left (18-9 x+400 x^3+200 x^4\right ) \log (5 x)}{\left (-9+200 x^2+200 x^3\right )^2} \, dx\\ &=4 \int \frac {5^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{1+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}}}{-9+200 x^2+200 x^3} \, dx+4 \int \frac {27 \int \frac {5^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{1+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}}}{\left (-9+200 x^2+200 x^3\right )^2} \, dx-8 \int \frac {5^{4+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{3+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}}}{\left (-9+200 x^2+200 x^3\right )^2} \, dx+\int \frac {5^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{1+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}}}{-9+200 x^2+200 x^3} \, dx+\int \frac {5^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}}}{-9+200 x^2+200 x^3} \, dx}{x} \, dx-(4 \log (5 x)) \int \frac {5^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{1+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}}}{-9+200 x^2+200 x^3} \, dx-(4 \log (5 x)) \int \frac {5^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}}}{-9+200 x^2+200 x^3} \, dx+(32 \log (5 x)) \int \frac {5^{4+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{3+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}}}{\left (-9+200 x^2+200 x^3\right )^2} \, dx-(108 \log (5 x)) \int \frac {5^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{1+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}}}{\left (-9+200 x^2+200 x^3\right )^2} \, dx\\ &=4 \int \frac {5^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{1+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}}}{-9+200 x^2+200 x^3} \, dx+4 \int \left (\frac {27 \int \frac {5^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{1+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}}}{\left (-9+200 x^2+200 x^3\right )^2} \, dx-8 \int \frac {5^{4+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{3+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}}}{\left (-9+200 x^2+200 x^3\right )^2} \, dx+\int \frac {5^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{1+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}}}{-9+200 x^2+200 x^3} \, dx}{x}+\frac {\int \frac {5^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}}}{-9+200 x^2+200 x^3} \, dx}{x}\right ) \, dx-(4 \log (5 x)) \int \frac {5^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{1+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}}}{-9+200 x^2+200 x^3} \, dx-(4 \log (5 x)) \int \frac {5^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}}}{-9+200 x^2+200 x^3} \, dx+(32 \log (5 x)) \int \frac {5^{4+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{3+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}}}{\left (-9+200 x^2+200 x^3\right )^2} \, dx-(108 \log (5 x)) \int \frac {5^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{1+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}}}{\left (-9+200 x^2+200 x^3\right )^2} \, dx\\ &=4 \int \frac {5^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{1+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}}}{-9+200 x^2+200 x^3} \, dx+4 \int \frac {27 \int \frac {5^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{1+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}}}{\left (-9+200 x^2+200 x^3\right )^2} \, dx-8 \int \frac {5^{4+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{3+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}}}{\left (-9+200 x^2+200 x^3\right )^2} \, dx+\int \frac {5^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{1+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}}}{-9+200 x^2+200 x^3} \, dx}{x} \, dx+4 \int \frac {\int \frac {5^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}}}{-9+200 x^2+200 x^3} \, dx}{x} \, dx-(4 \log (5 x)) \int \frac {5^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{1+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}}}{-9+200 x^2+200 x^3} \, dx-(4 \log (5 x)) \int \frac {5^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}}}{-9+200 x^2+200 x^3} \, dx+(32 \log (5 x)) \int \frac {5^{4+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{3+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}}}{\left (-9+200 x^2+200 x^3\right )^2} \, dx-(108 \log (5 x)) \int \frac {5^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{1+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}}}{\left (-9+200 x^2+200 x^3\right )^2} \, dx\\ &=4 \int \frac {5^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{1+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}}}{-9+200 x^2+200 x^3} \, dx+4 \int \left (\frac {27 \int \frac {5^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{1+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}}}{\left (-9+200 x^2+200 x^3\right )^2} \, dx-8 \int \frac {5^{4+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{3+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}}}{\left (-9+200 x^2+200 x^3\right )^2} \, dx}{x}+\frac {\int \frac {5^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{1+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}}}{-9+200 x^2+200 x^3} \, dx}{x}\right ) \, dx+4 \int \frac {\int \frac {5^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}}}{-9+200 x^2+200 x^3} \, dx}{x} \, dx-(4 \log (5 x)) \int \frac {5^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{1+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}}}{-9+200 x^2+200 x^3} \, dx-(4 \log (5 x)) \int \frac {5^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}}}{-9+200 x^2+200 x^3} \, dx+(32 \log (5 x)) \int \frac {5^{4+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{3+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}}}{\left (-9+200 x^2+200 x^3\right )^2} \, dx-(108 \log (5 x)) \int \frac {5^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{1+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}}}{\left (-9+200 x^2+200 x^3\right )^2} \, dx\\ &=4 \int \frac {5^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{1+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}}}{-9+200 x^2+200 x^3} \, dx+4 \int \frac {27 \int \frac {5^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{1+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}}}{\left (-9+200 x^2+200 x^3\right )^2} \, dx-8 \int \frac {5^{4+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{3+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}}}{\left (-9+200 x^2+200 x^3\right )^2} \, dx}{x} \, dx+4 \int \frac {\int \frac {5^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{1+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}}}{-9+200 x^2+200 x^3} \, dx}{x} \, dx+4 \int \frac {\int \frac {5^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}}}{-9+200 x^2+200 x^3} \, dx}{x} \, dx-(4 \log (5 x)) \int \frac {5^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{1+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}}}{-9+200 x^2+200 x^3} \, dx-(4 \log (5 x)) \int \frac {5^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}}}{-9+200 x^2+200 x^3} \, dx+(32 \log (5 x)) \int \frac {5^{4+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{3+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}}}{\left (-9+200 x^2+200 x^3\right )^2} \, dx-(108 \log (5 x)) \int \frac {5^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{1+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}}}{\left (-9+200 x^2+200 x^3\right )^2} \, dx\\ &=4 \int \frac {5^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{1+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}}}{-9+200 x^2+200 x^3} \, dx+4 \int \left (\frac {27 \int \frac {5^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{1+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}}}{\left (-9+200 x^2+200 x^3\right )^2} \, dx}{x}-\frac {8 \int \frac {5^{4+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{3+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}}}{\left (-9+200 x^2+200 x^3\right )^2} \, dx}{x}\right ) \, dx+4 \int \frac {\int \frac {5^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{1+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}}}{-9+200 x^2+200 x^3} \, dx}{x} \, dx+4 \int \frac {\int \frac {5^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}}}{-9+200 x^2+200 x^3} \, dx}{x} \, dx-(4 \log (5 x)) \int \frac {5^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{1+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}}}{-9+200 x^2+200 x^3} \, dx-(4 \log (5 x)) \int \frac {5^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}}}{-9+200 x^2+200 x^3} \, dx+(32 \log (5 x)) \int \frac {5^{4+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{3+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}}}{\left (-9+200 x^2+200 x^3\right )^2} \, dx-(108 \log (5 x)) \int \frac {5^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{1+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}}}{\left (-9+200 x^2+200 x^3\right )^2} \, dx\\ &=4 \int \frac {5^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{1+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}}}{-9+200 x^2+200 x^3} \, dx+4 \int \frac {\int \frac {5^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{1+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}}}{-9+200 x^2+200 x^3} \, dx}{x} \, dx+4 \int \frac {\int \frac {5^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}}}{-9+200 x^2+200 x^3} \, dx}{x} \, dx-32 \int \frac {\int \frac {5^{4+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{3+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}}}{\left (-9+200 x^2+200 x^3\right )^2} \, dx}{x} \, dx+108 \int \frac {\int \frac {5^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{1+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}}}{\left (-9+200 x^2+200 x^3\right )^2} \, dx}{x} \, dx-(4 \log (5 x)) \int \frac {5^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{1+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}}}{-9+200 x^2+200 x^3} \, dx-(4 \log (5 x)) \int \frac {5^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}}}{-9+200 x^2+200 x^3} \, dx+(32 \log (5 x)) \int \frac {5^{4+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{3+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}}}{\left (-9+200 x^2+200 x^3\right )^2} \, dx-(108 \log (5 x)) \int \frac {5^{2+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{1+\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}}}{\left (-9+200 x^2+200 x^3\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [F]  time = 0.91, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {5^{\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} \left (-900 x+20000 x^3+20000 x^4+\left (-1800 x+900 x^2-40000 x^4-20000 x^5\right ) \log (5 x)\right )}{81-3600 x^2-3600 x^3+40000 x^4+80000 x^5+40000 x^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(5^((100*x^2)/(E^x*(-9 + 200*x^2 + 200*x^3)))*x^((100*x^2)/(E^x*(-9 + 200*x^2 + 200*x^3)))*(-900*x +
20000*x^3 + 20000*x^4 + (-1800*x + 900*x^2 - 40000*x^4 - 20000*x^5)*Log[5*x]))/(E^x*(81 - 3600*x^2 - 3600*x^3
+ 40000*x^4 + 80000*x^5 + 40000*x^6)),x]

[Out]

Integrate[(5^((100*x^2)/(E^x*(-9 + 200*x^2 + 200*x^3)))*x^((100*x^2)/(E^x*(-9 + 200*x^2 + 200*x^3)))*(-900*x +
20000*x^3 + 20000*x^4 + (-1800*x + 900*x^2 - 40000*x^4 - 20000*x^5)*Log[5*x]))/(E^x*(81 - 3600*x^2 - 3600*x^3
+ 40000*x^4 + 80000*x^5 + 40000*x^6)), x]

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fricas [A]  time = 0.77, size = 27, normalized size = 0.60 \begin {gather*} \left (5 \, x\right )^{\frac {100 \, x^{2} e^{\left (-x\right )}}{200 \, x^{3} + 200 \, x^{2} - 9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-20000*x^5-40000*x^4+900*x^2-1800*x)*log(5*x)+20000*x^4+20000*x^3-900*x)*exp(100*x^2*log(5*x)/(200
*x^3+200*x^2-9)/exp(x))/(40000*x^6+80000*x^5+40000*x^4-3600*x^3-3600*x^2+81)/exp(x),x, algorithm="fricas")

[Out]

(5*x)^(100*x^2*e^(-x)/(200*x^3 + 200*x^2 - 9))

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giac [A]  time = 5.19, size = 27, normalized size = 0.60 \begin {gather*} \left (5 \, x\right )^{\frac {100 \, x^{2} e^{\left (-x\right )}}{200 \, x^{3} + 200 \, x^{2} - 9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-20000*x^5-40000*x^4+900*x^2-1800*x)*log(5*x)+20000*x^4+20000*x^3-900*x)*exp(100*x^2*log(5*x)/(200
*x^3+200*x^2-9)/exp(x))/(40000*x^6+80000*x^5+40000*x^4-3600*x^3-3600*x^2+81)/exp(x),x, algorithm="giac")

[Out]

(5*x)^(100*x^2*e^(-x)/(200*x^3 + 200*x^2 - 9))

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maple [A]  time = 0.03, size = 28, normalized size = 0.62

method	result	size
risch	\(\left (5 x \right )^{\frac {100 x^{2} {\mathrm e}^{-x}}{200 x^{3}+200 x^{2}-9}}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-20000*x^5-40000*x^4+900*x^2-1800*x)*ln(5*x)+20000*x^4+20000*x^3-900*x)*exp(100*x^2*ln(5*x)/(200*x^3+200
*x^2-9)/exp(x))/(40000*x^6+80000*x^5+40000*x^4-3600*x^3-3600*x^2+81)/exp(x),x,method=_RETURNVERBOSE)

[Out]

(5*x)^(100*x^2/(200*x^3+200*x^2-9)*exp(-x))

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maxima [B]  time = 0.60, size = 52, normalized size = 1.16 \begin {gather*} e^{\left (\frac {100 \, x^{2} e^{\left (-x\right )} \log \relax (5)}{200 \, x^{3} + 200 \, x^{2} - 9} + \frac {100 \, x^{2} e^{\left (-x\right )} \log \relax (x)}{200 \, x^{3} + 200 \, x^{2} - 9}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-20000*x^5-40000*x^4+900*x^2-1800*x)*log(5*x)+20000*x^4+20000*x^3-900*x)*exp(100*x^2*log(5*x)/(200
*x^3+200*x^2-9)/exp(x))/(40000*x^6+80000*x^5+40000*x^4-3600*x^3-3600*x^2+81)/exp(x),x, algorithm="maxima")

[Out]

e^(100*x^2*e^(-x)*log(5)/(200*x^3 + 200*x^2 - 9) + 100*x^2*e^(-x)*log(x)/(200*x^3 + 200*x^2 - 9))

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mupad [B]  time = 5.72, size = 51, normalized size = 1.13 \begin {gather*} 5^{\frac {100\,x^2\,{\mathrm {e}}^{-x}}{200\,x^3+200\,x^2-9}}\,x^{\frac {100\,x^2\,{\mathrm {e}}^{-x}}{200\,x^3+200\,x^2-9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-x)*exp((100*x^2*log(5*x)*exp(-x))/(200*x^2 + 200*x^3 - 9))*(900*x + log(5*x)*(1800*x - 900*x^2 + 40
000*x^4 + 20000*x^5) - 20000*x^3 - 20000*x^4))/(40000*x^4 - 3600*x^3 - 3600*x^2 + 80000*x^5 + 40000*x^6 + 81),
x)

[Out]

5^((100*x^2*exp(-x))/(200*x^2 + 200*x^3 - 9))*x^((100*x^2*exp(-x))/(200*x^2 + 200*x^3 - 9))

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sympy [A]  time = 0.99, size = 26, normalized size = 0.58 \begin {gather*} e^{\frac {100 x^{2} e^{- x} \log {\left (5 x \right )}}{200 x^{3} + 200 x^{2} - 9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-20000*x**5-40000*x**4+900*x**2-1800*x)*ln(5*x)+20000*x**4+20000*x**3-900*x)*exp(100*x**2*ln(5*x)/
(200*x**3+200*x**2-9)/exp(x))/(40000*x**6+80000*x**5+40000*x**4-3600*x**3-3600*x**2+81)/exp(x),x)

[Out]

exp(100*x**2*exp(-x)*log(5*x)/(200*x**3 + 200*x**2 - 9))

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