Optimal. Leaf size=29 \[ \frac {e^2 x}{-x+x^2+\left (3-x-\frac {x}{-2+x}\right )^2} \]
________________________________________________________________________________________
Rubi [F] time = 0.57, 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 {e^2 \left (144-288 x+172 x^2-40 x^4+16 x^5-2 x^6\right )}{1296-3744 x+5296 x^2-4680 x^3+2792 x^4-1144 x^5+313 x^6-52 x^7+4 x^8} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=e^2 \int \frac {144-288 x+172 x^2-40 x^4+16 x^5-2 x^6}{1296-3744 x+5296 x^2-4680 x^3+2792 x^4-1144 x^5+313 x^6-52 x^7+4 x^8} \, dx\\ &=e^2 \int \left (\frac {-540+244 x+28 x^2-21 x^3}{4 \left (36-52 x+36 x^2-13 x^3+2 x^4\right )^2}+\frac {31+6 x-4 x^2}{4 \left (36-52 x+36 x^2-13 x^3+2 x^4\right )}\right ) \, dx\\ &=\frac {1}{4} e^2 \int \frac {-540+244 x+28 x^2-21 x^3}{\left (36-52 x+36 x^2-13 x^3+2 x^4\right )^2} \, dx+\frac {1}{4} e^2 \int \frac {31+6 x-4 x^2}{36-52 x+36 x^2-13 x^3+2 x^4} \, dx\\ &=\frac {21 e^2}{32 \left (36-52 x+36 x^2-13 x^3+2 x^4\right )}+\frac {1}{32} e^2 \int \frac {-5412+3464 x-595 x^2}{\left (36-52 x+36 x^2-13 x^3+2 x^4\right )^2} \, dx+\frac {1}{4} e^2 \int \left (\frac {31}{36-52 x+36 x^2-13 x^3+2 x^4}+\frac {6 x}{36-52 x+36 x^2-13 x^3+2 x^4}-\frac {4 x^2}{36-52 x+36 x^2-13 x^3+2 x^4}\right ) \, dx\\ &=\frac {21 e^2}{32 \left (36-52 x+36 x^2-13 x^3+2 x^4\right )}+\frac {1}{32} e^2 \int \left (-\frac {5412}{\left (36-52 x+36 x^2-13 x^3+2 x^4\right )^2}+\frac {3464 x}{\left (36-52 x+36 x^2-13 x^3+2 x^4\right )^2}-\frac {595 x^2}{\left (36-52 x+36 x^2-13 x^3+2 x^4\right )^2}\right ) \, dx-e^2 \int \frac {x^2}{36-52 x+36 x^2-13 x^3+2 x^4} \, dx+\frac {1}{2} \left (3 e^2\right ) \int \frac {x}{36-52 x+36 x^2-13 x^3+2 x^4} \, dx+\frac {1}{4} \left (31 e^2\right ) \int \frac {1}{36-52 x+36 x^2-13 x^3+2 x^4} \, dx\\ &=\frac {21 e^2}{32 \left (36-52 x+36 x^2-13 x^3+2 x^4\right )}-e^2 \int \frac {x^2}{36-52 x+36 x^2-13 x^3+2 x^4} \, dx+\frac {1}{2} \left (3 e^2\right ) \int \frac {x}{36-52 x+36 x^2-13 x^3+2 x^4} \, dx+\frac {1}{4} \left (31 e^2\right ) \int \frac {1}{36-52 x+36 x^2-13 x^3+2 x^4} \, dx-\frac {1}{32} \left (595 e^2\right ) \int \frac {x^2}{\left (36-52 x+36 x^2-13 x^3+2 x^4\right )^2} \, dx+\frac {1}{4} \left (433 e^2\right ) \int \frac {x}{\left (36-52 x+36 x^2-13 x^3+2 x^4\right )^2} \, dx-\frac {1}{8} \left (1353 e^2\right ) \int \frac {1}{\left (36-52 x+36 x^2-13 x^3+2 x^4\right )^2} \, dx\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 33, normalized size = 1.14 \begin {gather*} \frac {2 e^2 (-2+x)^2 x}{72-104 x+72 x^2-26 x^3+4 x^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.57, size = 37, normalized size = 1.28 \begin {gather*} \frac {{\left (x^{3} - 4 \, x^{2} + 4 \, x\right )} e^{2}}{2 \, x^{4} - 13 \, x^{3} + 36 \, x^{2} - 52 \, x + 36} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.14, size = 37, normalized size = 1.28 \begin {gather*} \frac {{\left (x^{3} - 4 \, x^{2} + 4 \, x\right )} e^{2}}{2 \, x^{4} - 13 \, x^{3} + 36 \, x^{2} - 52 \, x + 36} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.06, size = 32, normalized size = 1.10
method | result | size |
gosper | \(\frac {x \left (x -2\right )^{2} {\mathrm e}^{2}}{2 x^{4}-13 x^{3}+36 x^{2}-52 x +36}\) | \(32\) |
default | \(-\frac {2 \,{\mathrm e}^{2} \left (-\frac {1}{4} x^{3}+x^{2}-x \right )}{x^{4}-\frac {13}{2} x^{3}+18 x^{2}-26 x +18}\) | \(37\) |
risch | \(\frac {{\mathrm e}^{2} \left (\frac {1}{2} x^{3}-2 x^{2}+2 x \right )}{x^{4}-\frac {13}{2} x^{3}+18 x^{2}-26 x +18}\) | \(38\) |
norman | \(\frac {4 \,{\mathrm e}^{2} x -4 x^{2} {\mathrm e}^{2}+x^{3} {\mathrm e}^{2}}{2 x^{4}-13 x^{3}+36 x^{2}-52 x +36}\) | \(43\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.36, size = 37, normalized size = 1.28 \begin {gather*} \frac {{\left (x^{3} - 4 \, x^{2} + 4 \, x\right )} e^{2}}{2 \, x^{4} - 13 \, x^{3} + 36 \, x^{2} - 52 \, x + 36} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 3.13, size = 31, normalized size = 1.07 \begin {gather*} \frac {x\,{\mathrm {e}}^2\,{\left (x-2\right )}^2}{2\,x^4-13\,x^3+36\,x^2-52\,x+36} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.43, size = 42, normalized size = 1.45 \begin {gather*} - \frac {- x^{3} e^{2} + 4 x^{2} e^{2} - 4 x e^{2}}{2 x^{4} - 13 x^{3} + 36 x^{2} - 52 x + 36} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________