Optimal. Leaf size=27 \[ \log \left (e^{-1+x-\frac {1+x}{2 x}}-\frac {x}{5}+3 x^2\right ) \]
________________________________________________________________________________________
Rubi [F] time = 5.80, 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 {-2 x^2+60 x^3+e^{\frac {-1-3 x+2 x^2}{2 x}} \left (5+10 x^2\right )}{10 e^{\frac {-1-3 x+2 x^2}{2 x}} x^2-2 x^3+30 x^4} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {1+2 x^2}{2 x^2}-\frac {e^{\frac {3}{2}+\frac {1}{2 x}} \left (-1+17 x-62 x^2+30 x^3\right )}{2 x \left (5 e^x-e^{\frac {3}{2}+\frac {1}{2 x}} x+15 e^{\frac {3}{2}+\frac {1}{2 x}} x^2\right )}\right ) \, dx\\ &=\frac {1}{2} \int \frac {1+2 x^2}{x^2} \, dx-\frac {1}{2} \int \frac {e^{\frac {3}{2}+\frac {1}{2 x}} \left (-1+17 x-62 x^2+30 x^3\right )}{x \left (5 e^x-e^{\frac {3}{2}+\frac {1}{2 x}} x+15 e^{\frac {3}{2}+\frac {1}{2 x}} x^2\right )} \, dx\\ &=\frac {1}{2} \int \left (2+\frac {1}{x^2}\right ) \, dx-\frac {1}{2} \int \frac {e^{\frac {3}{2}+\frac {1}{2 x}} \left (-1+17 x-62 x^2+30 x^3\right )}{x \left (5 e^x+e^{\frac {1}{2} \left (3+\frac {1}{x}\right )} x (-1+15 x)\right )} \, dx\\ &=-\frac {1}{2 x}+x-\frac {1}{2} \int \left (\frac {17 e^{\frac {3}{2}+\frac {1}{2 x}}}{5 e^x-e^{\frac {3}{2}+\frac {1}{2 x}} x+15 e^{\frac {3}{2}+\frac {1}{2 x}} x^2}-\frac {e^{\frac {3}{2}+\frac {1}{2 x}}}{x \left (5 e^x-e^{\frac {3}{2}+\frac {1}{2 x}} x+15 e^{\frac {3}{2}+\frac {1}{2 x}} x^2\right )}-\frac {62 e^{\frac {3}{2}+\frac {1}{2 x}} x}{5 e^x-e^{\frac {3}{2}+\frac {1}{2 x}} x+15 e^{\frac {3}{2}+\frac {1}{2 x}} x^2}+\frac {30 e^{\frac {3}{2}+\frac {1}{2 x}} x^2}{5 e^x-e^{\frac {3}{2}+\frac {1}{2 x}} x+15 e^{\frac {3}{2}+\frac {1}{2 x}} x^2}\right ) \, dx\\ &=-\frac {1}{2 x}+x+\frac {1}{2} \int \frac {e^{\frac {3}{2}+\frac {1}{2 x}}}{x \left (5 e^x-e^{\frac {3}{2}+\frac {1}{2 x}} x+15 e^{\frac {3}{2}+\frac {1}{2 x}} x^2\right )} \, dx-\frac {17}{2} \int \frac {e^{\frac {3}{2}+\frac {1}{2 x}}}{5 e^x-e^{\frac {3}{2}+\frac {1}{2 x}} x+15 e^{\frac {3}{2}+\frac {1}{2 x}} x^2} \, dx-15 \int \frac {e^{\frac {3}{2}+\frac {1}{2 x}} x^2}{5 e^x-e^{\frac {3}{2}+\frac {1}{2 x}} x+15 e^{\frac {3}{2}+\frac {1}{2 x}} x^2} \, dx+31 \int \frac {e^{\frac {3}{2}+\frac {1}{2 x}} x}{5 e^x-e^{\frac {3}{2}+\frac {1}{2 x}} x+15 e^{\frac {3}{2}+\frac {1}{2 x}} x^2} \, dx\\ &=-\frac {1}{2 x}+x-\frac {17}{2} \int \frac {e^{\frac {3}{2}+\frac {1}{2 x}}}{5 e^x+e^{\frac {1}{2} \left (3+\frac {1}{x}\right )} x (-1+15 x)} \, dx-15 \int \frac {e^{\frac {3}{2}+\frac {1}{2 x}} x^2}{5 e^x+e^{\frac {1}{2} \left (3+\frac {1}{x}\right )} x (-1+15 x)} \, dx+31 \int \frac {e^{\frac {3}{2}+\frac {1}{2 x}} x}{5 e^x+e^{\frac {1}{2} \left (3+\frac {1}{x}\right )} x (-1+15 x)} \, dx-\frac {\operatorname {Subst}\left (\int \frac {\exp \left (\frac {3}{2}+\frac {e^{\frac {3}{2}+\frac {1}{2 x}}}{2 \left (5 e^x+15 e^{\frac {3}{2}+\frac {1}{2 x}} x^2\right )}+\frac {x}{2 \left (5 e^x+15 e^{\frac {3}{2}+\frac {1}{2 x}} x^2\right )}\right )}{x} \, dx,x,\frac {5 e^x-e^{\frac {3}{2}+\frac {1}{2 x}} x+15 e^{\frac {3}{2}+\frac {1}{2 x}} x^2}{x}\right )}{2 \left (5 e^x+15 e^{\frac {3}{2}+\frac {1}{2 x}} x^2\right )}\\ &=-\frac {1}{2 x}+x-\frac {17}{2} \int \frac {e^{\frac {3}{2}+\frac {1}{2 x}}}{5 e^x+e^{\frac {1}{2} \left (3+\frac {1}{x}\right )} x (-1+15 x)} \, dx-15 \int \frac {e^{\frac {3}{2}+\frac {1}{2 x}} x^2}{5 e^x+e^{\frac {1}{2} \left (3+\frac {1}{x}\right )} x (-1+15 x)} \, dx+31 \int \frac {e^{\frac {3}{2}+\frac {1}{2 x}} x}{5 e^x+e^{\frac {1}{2} \left (3+\frac {1}{x}\right )} x (-1+15 x)} \, dx-\frac {\operatorname {Subst}\left (\int \frac {\exp \left (\frac {15 e^x+x+e^{\frac {1}{2} \left (3+\frac {1}{x}\right )} \left (1+45 x^2\right )}{10 \left (e^x+3 e^{\frac {1}{2} \left (3+\frac {1}{x}\right )} x^2\right )}\right )}{x} \, dx,x,\frac {5 e^x-e^{\frac {3}{2}+\frac {1}{2 x}} x+15 e^{\frac {3}{2}+\frac {1}{2 x}} x^2}{x}\right )}{2 \left (5 e^x+15 e^{\frac {3}{2}+\frac {1}{2 x}} x^2\right )}\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [A] time = 1.06, size = 53, normalized size = 1.96 \begin {gather*} \frac {1}{2} \left (-\frac {1}{x}+2 \log \left (5 e^x-e^{\frac {3}{2}+\frac {1}{2 x}} x+15 e^{\frac {3}{2}+\frac {1}{2 x}} x^2\right )\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.60, size = 28, normalized size = 1.04 \begin {gather*} \log \left (15 \, x^{2} - x + 5 \, e^{\left (\frac {2 \, x^{2} - 3 \, x - 1}{2 \, x}\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.17, size = 28, normalized size = 1.04 \begin {gather*} \log \left (15 \, x^{2} - x + 5 \, e^{\left (\frac {2 \, x^{2} - 3 \, x - 1}{2 \, x}\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.11, size = 29, normalized size = 1.07
method | result | size |
norman | \(\ln \left (15 x^{2}+5 \,{\mathrm e}^{\frac {2 x^{2}-3 x -1}{2 x}}-x \right )\) | \(29\) |
risch | \(x -\frac {1}{2 x}-\frac {2 x^{2}-3 x -1}{2 x}+\ln \left (3 x^{2}-\frac {x}{5}+{\mathrm e}^{\frac {2 x^{2}-3 x -1}{2 x}}\right )\) | \(49\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.42, size = 56, normalized size = 2.07 \begin {gather*} -\frac {1}{2 \, x} + \log \left (15 \, x - 1\right ) + \log \relax (x) + \log \left (\frac {{\left (15 \, x^{2} e^{\frac {3}{2}} - x e^{\frac {3}{2}}\right )} e^{\left (\frac {1}{2 \, x}\right )} + 5 \, e^{x}}{15 \, x^{2} e^{\frac {3}{2}} - x e^{\frac {3}{2}}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {60\,x^3-2\,x^2+{\mathrm {e}}^{-\frac {-x^2+\frac {3\,x}{2}+\frac {1}{2}}{x}}\,\left (10\,x^2+5\right )}{10\,x^2\,{\mathrm {e}}^{-\frac {-x^2+\frac {3\,x}{2}+\frac {1}{2}}{x}}-2\,x^3+30\,x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.24, size = 24, normalized size = 0.89 \begin {gather*} \log {\left (3 x^{2} - \frac {x}{5} + e^{\frac {x^{2} - \frac {3 x}{2} - \frac {1}{2}}{x}} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________