Optimal. Leaf size=26 \[ -e^{2 e^{-25/x}}+\frac {2 x}{5 (3+2 x)} \]
________________________________________________________________________________________
Rubi [A] time = 0.64, antiderivative size = 25, normalized size of antiderivative = 0.96, number of steps used = 8, number of rules used = 7, integrand size = 61, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1594, 27, 12, 6742, 6715, 2282, 2194} \begin {gather*} -e^{2 e^{-25/x}}-\frac {3}{5 (2 x+3)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 27
Rule 1594
Rule 2194
Rule 2282
Rule 6715
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-25/x} \left (6 e^{25/x} x^2+e^{2 e^{-25/x}} \left (-2250-3000 x-1000 x^2\right )\right )}{x^2 \left (45+60 x+20 x^2\right )} \, dx\\ &=\int \frac {e^{-25/x} \left (6 e^{25/x} x^2+e^{2 e^{-25/x}} \left (-2250-3000 x-1000 x^2\right )\right )}{5 x^2 (3+2 x)^2} \, dx\\ &=\frac {1}{5} \int \frac {e^{-25/x} \left (6 e^{25/x} x^2+e^{2 e^{-25/x}} \left (-2250-3000 x-1000 x^2\right )\right )}{x^2 (3+2 x)^2} \, dx\\ &=\frac {1}{5} \int \left (-\frac {250 e^{2 e^{-25/x}-\frac {25}{x}}}{x^2}+\frac {6}{(3+2 x)^2}\right ) \, dx\\ &=-\frac {3}{5 (3+2 x)}-50 \int \frac {e^{2 e^{-25/x}-\frac {25}{x}}}{x^2} \, dx\\ &=-\frac {3}{5 (3+2 x)}+50 \operatorname {Subst}\left (\int e^{2 e^{-25 x}-25 x} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {3}{5 (3+2 x)}-2 \operatorname {Subst}\left (\int e^{2 x} \, dx,x,e^{-25/x}\right )\\ &=-e^{2 e^{-25/x}}-\frac {3}{5 (3+2 x)}\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [A] time = 0.14, size = 31, normalized size = 1.19 \begin {gather*} -\frac {2}{5} \left (\frac {5}{2} e^{2 e^{-25/x}}+\frac {3}{2 (3+2 x)}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.52, size = 27, normalized size = 1.04 \begin {gather*} -\frac {5 \, {\left (2 \, x + 3\right )} e^{\left (2 \, e^{\left (-\frac {25}{x}\right )}\right )} + 3}{5 \, {\left (2 \, x + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.98, size = 68, normalized size = 2.62 \begin {gather*} -\frac {10 \, x e^{\left (\frac {2 \, x e^{\left (-\frac {25}{x}\right )} - 25}{x}\right )} + 15 \, e^{\left (\frac {2 \, x e^{\left (-\frac {25}{x}\right )} - 25}{x}\right )} + 3 \, e^{\left (-\frac {25}{x}\right )}}{5 \, {\left (2 \, x e^{\left (-\frac {25}{x}\right )} + 3 \, e^{\left (-\frac {25}{x}\right )}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.10, size = 20, normalized size = 0.77
method | result | size |
risch | \(-\frac {3}{10 \left (x +\frac {3}{2}\right )}-{\mathrm e}^{2 \,{\mathrm e}^{-\frac {25}{x}}}\) | \(20\) |
norman | \(\frac {\left (-\frac {3 x \,{\mathrm e}^{\frac {25}{x}}}{5}-3 \,{\mathrm e}^{\frac {25}{x}} {\mathrm e}^{2 \,{\mathrm e}^{-\frac {25}{x}}} x -2 \,{\mathrm e}^{2 \,{\mathrm e}^{-\frac {25}{x}}} x^{2} {\mathrm e}^{\frac {25}{x}}\right ) {\mathrm e}^{-\frac {25}{x}}}{x \left (2 x +3\right )}\) | \(72\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.45, size = 21, normalized size = 0.81 \begin {gather*} -\frac {3}{5 \, {\left (2 \, x + 3\right )}} - e^{\left (2 \, e^{\left (-\frac {25}{x}\right )}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 3.39, size = 21, normalized size = 0.81 \begin {gather*} -{\mathrm {e}}^{2\,{\mathrm {e}}^{-\frac {25}{x}}}-\frac {3}{5\,\left (2\,x+3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.37, size = 15, normalized size = 0.58 \begin {gather*} - e^{2 e^{- \frac {25}{x}}} - \frac {6}{20 x + 30} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________