Optimal. Leaf size=25 \[ -4+\frac {1}{2 \left (5+x+\frac {x}{e^{5/2}}+\frac {1+x}{2}\right )} \]
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Rubi [A] time = 0.03, antiderivative size = 27, normalized size of antiderivative = 1.08, number of steps used = 5, number of rules used = 5, integrand size = 51, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.098, Rules used = {12, 1981, 27, 6, 32} \begin {gather*} \frac {e^{5/2}}{\left (2+3 e^{5/2}\right ) x+11 e^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 12
Rule 27
Rule 32
Rule 1981
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=-\left (\left (e^{5/2} \left (2+3 e^{5/2}\right )\right ) \int \frac {1}{4 x^2+e^5 \left (121+66 x+9 x^2\right )+e^{5/2} \left (44 x+12 x^2\right )} \, dx\right )\\ &=-\left (\left (e^{5/2} \left (2+3 e^{5/2}\right )\right ) \int \frac {1}{121 e^5+22 e^{5/2} \left (2+3 e^{5/2}\right ) x+\left (2+3 e^{5/2}\right )^2 x^2} \, dx\right )\\ &=-\left (\left (e^{5/2} \left (2+3 e^{5/2}\right )\right ) \int \frac {1}{\left (11 e^{5/2}+2 x+3 e^{5/2} x\right )^2} \, dx\right )\\ &=-\left (\left (e^{5/2} \left (2+3 e^{5/2}\right )\right ) \int \frac {1}{\left (11 e^{5/2}+\left (2+3 e^{5/2}\right ) x\right )^2} \, dx\right )\\ &=\frac {e^{5/2}}{11 e^{5/2}+\left (2+3 e^{5/2}\right ) x}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.01, size = 43, normalized size = 1.72 \begin {gather*} -\frac {-2 e^{5/2}-3 e^5}{\left (2+3 e^{5/2}\right ) \left (2 x+e^{5/2} (11+3 x)\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 17, normalized size = 0.68 \begin {gather*} \frac {e^{\frac {5}{2}}}{{\left (3 \, x + 11\right )} e^{\frac {5}{2}} + 2 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.22, size = 19, normalized size = 0.76
method | result | size |
gosper | \(\frac {{\mathrm e}^{\frac {5}{2}}}{3 \,{\mathrm e}^{\frac {5}{2}} x +11 \,{\mathrm e}^{\frac {5}{2}}+2 x}\) | \(19\) |
norman | \(\frac {\left (-\frac {3 \,{\mathrm e}^{\frac {5}{2}}}{11}-\frac {2}{11}\right ) x}{3 \,{\mathrm e}^{\frac {5}{2}} x +11 \,{\mathrm e}^{\frac {5}{2}}+2 x}\) | \(24\) |
risch | \(\frac {{\mathrm e}^{5}}{\left (3 \,{\mathrm e}^{\frac {5}{2}}+2\right ) \left ({\mathrm e}^{\frac {5}{2}} x +\frac {11 \,{\mathrm e}^{\frac {5}{2}}}{3}+\frac {2 x}{3}\right )}+\frac {2 \,{\mathrm e}^{\frac {5}{2}}}{3 \left (3 \,{\mathrm e}^{\frac {5}{2}}+2\right ) \left ({\mathrm e}^{\frac {5}{2}} x +\frac {11 \,{\mathrm e}^{\frac {5}{2}}}{3}+\frac {2 x}{3}\right )}\) | \(53\) |
meijerg | \(-\frac {3 \left (3 \,{\mathrm e}^{\frac {5}{2}}+2\right )^{2} x}{121 \left (9 \,{\mathrm e}^{5}+12 \,{\mathrm e}^{\frac {5}{2}}+4\right ) \left (1+\frac {x \,{\mathrm e}^{-\frac {5}{2}} \left (3 \,{\mathrm e}^{\frac {5}{2}}+2\right )}{11}\right )}-\frac {2 \,{\mathrm e}^{-\frac {5}{2}} \left (3 \,{\mathrm e}^{\frac {5}{2}}+2\right )^{2} x}{121 \left (9 \,{\mathrm e}^{5}+12 \,{\mathrm e}^{\frac {5}{2}}+4\right ) \left (1+\frac {x \,{\mathrm e}^{-\frac {5}{2}} \left (3 \,{\mathrm e}^{\frac {5}{2}}+2\right )}{11}\right )}\) | \(80\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.36, size = 33, normalized size = 1.32 \begin {gather*} \frac {3 \, e^{5} + 2 \, e^{\frac {5}{2}}}{x {\left (9 \, e^{5} + 12 \, e^{\frac {5}{2}} + 4\right )} + 33 \, e^{5} + 22 \, e^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 18, normalized size = 0.72 \begin {gather*} \frac {{\mathrm {e}}^{5/2}}{11\,{\mathrm {e}}^{5/2}+x\,\left (3\,{\mathrm {e}}^{5/2}+2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.36, size = 114, normalized size = 4.56 \begin {gather*} \frac {\left (- 3 e^{5} - 2 e^{\frac {5}{2}}\right ) \left (- 81 e^{10} - 216 e^{\frac {15}{2}} - 216 e^{5} - 96 e^{\frac {5}{2}} - 16\right )}{x \left (64 + 576 e^{\frac {5}{2}} + 2160 e^{5} + 4320 e^{\frac {15}{2}} + 4860 e^{10} + 2916 e^{\frac {25}{2}} + 729 e^{15}\right ) + 352 e^{\frac {5}{2}} + 2640 e^{5} + 7920 e^{\frac {15}{2}} + 11880 e^{10} + 8910 e^{\frac {25}{2}} + 2673 e^{15}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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