Optimal. Leaf size=29 \[ \frac {-25+e^{x+\frac {3+2 x-x^2}{2 x}}}{4+x} \]
________________________________________________________________________________________
Rubi [F] time = 0.96, 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 {50 x^2+e^{\frac {3+2 x+x^2}{2 x}} \left (-12-3 x+2 x^2+x^3\right )}{32 x^2+16 x^3+2 x^4} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {50 x^2+e^{\frac {3+2 x+x^2}{2 x}} \left (-12-3 x+2 x^2+x^3\right )}{x^2 \left (32+16 x+2 x^2\right )} \, dx\\ &=\int \frac {50 x^2+e^{\frac {3+2 x+x^2}{2 x}} \left (-12-3 x+2 x^2+x^3\right )}{2 x^2 (4+x)^2} \, dx\\ &=\frac {1}{2} \int \frac {50 x^2+e^{\frac {3+2 x+x^2}{2 x}} \left (-12-3 x+2 x^2+x^3\right )}{x^2 (4+x)^2} \, dx\\ &=\frac {1}{2} \int \left (\frac {50}{(4+x)^2}+\frac {e^{\frac {3+2 x+x^2}{2 x}} \left (-12-3 x+2 x^2+x^3\right )}{x^2 (4+x)^2}\right ) \, dx\\ &=-\frac {25}{4+x}+\frac {1}{2} \int \frac {e^{\frac {3+2 x+x^2}{2 x}} \left (-12-3 x+2 x^2+x^3\right )}{x^2 (4+x)^2} \, dx\\ &=-\frac {25}{4+x}+\frac {1}{2} \int \left (-\frac {3 e^{\frac {3+2 x+x^2}{2 x}}}{4 x^2}+\frac {3 e^{\frac {3+2 x+x^2}{2 x}}}{16 x}-\frac {2 e^{\frac {3+2 x+x^2}{2 x}}}{(4+x)^2}+\frac {13 e^{\frac {3+2 x+x^2}{2 x}}}{16 (4+x)}\right ) \, dx\\ &=-\frac {25}{4+x}+\frac {3}{32} \int \frac {e^{\frac {3+2 x+x^2}{2 x}}}{x} \, dx-\frac {3}{8} \int \frac {e^{\frac {3+2 x+x^2}{2 x}}}{x^2} \, dx+\frac {13}{32} \int \frac {e^{\frac {3+2 x+x^2}{2 x}}}{4+x} \, dx-\int \frac {e^{\frac {3+2 x+x^2}{2 x}}}{(4+x)^2} \, dx\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [A] time = 0.22, size = 22, normalized size = 0.76 \begin {gather*} \frac {-25+e^{\frac {1}{2} \left (2+\frac {3}{x}+x\right )}}{4+x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.53, size = 22, normalized size = 0.76 \begin {gather*} \frac {e^{\left (\frac {x^{2} + 2 \, x + 3}{2 \, x}\right )} - 25}{x + 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.24, size = 22, normalized size = 0.76 \begin {gather*} \frac {e^{\left (\frac {x^{2} + 2 \, x + 3}{2 \, x}\right )} - 25}{x + 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.10, size = 29, normalized size = 1.00
method | result | size |
risch | \(-\frac {25}{4+x}+\frac {{\mathrm e}^{\frac {x^{2}+2 x +3}{2 x}}}{4+x}\) | \(29\) |
norman | \(\frac {-25 x +{\mathrm e}^{\frac {x^{2}+2 x +3}{2 x}} x}{\left (4+x \right ) x}\) | \(30\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.44, size = 25, normalized size = 0.86 \begin {gather*} \frac {e^{\left (\frac {1}{2} \, x + \frac {3}{2 \, x} + 1\right )}}{x + 4} - \frac {25}{x + 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 3.52, size = 19, normalized size = 0.66 \begin {gather*} \frac {{\mathrm {e}}^{\frac {x}{2}+\frac {3}{2\,x}+1}-25}{x+4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.15, size = 20, normalized size = 0.69 \begin {gather*} \frac {e^{\frac {\frac {x^{2}}{2} + x + \frac {3}{2}}{x}}}{x + 4} - \frac {25}{x + 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________