Optimal. Leaf size=30 \[ \frac {\frac {4 e^2}{x^2}+x}{100 (5-2 x)+\frac {1}{2} (-2-x)} \]
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Rubi [A]
time = 0.04, antiderivative size = 39, normalized size of antiderivative = 1.30, number of steps
used = 4, number of rules used = 3, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.088, Rules used = {1608, 27, 1634}
\begin {gather*} \frac {4 e^2}{499 x^2}+\frac {802 e^2}{249001 x}+\frac {2 \left (248502998+64481201 e^2\right )}{99849401 (998-401 x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 1608
Rule 1634
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {1996 x^3+e^2 (-15968+9624 x)}{x^3 \left (996004-800396 x+160801 x^2\right )} \, dx\\ &=\int \frac {1996 x^3+e^2 (-15968+9624 x)}{x^3 (-998+401 x)^2} \, dx\\ &=\int \left (-\frac {8 e^2}{499 x^3}-\frac {802 e^2}{249001 x^2}+\frac {2 \left (248502998+64481201 e^2\right )}{249001 (-998+401 x)^2}\right ) \, dx\\ &=\frac {2 \left (248502998+64481201 e^2\right )}{99849401 (998-401 x)}+\frac {4 e^2}{499 x^2}+\frac {802 e^2}{249001 x}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.01, size = 25, normalized size = 0.83 \begin {gather*} -\frac {4 \left (802 e^2+499 x^2\right )}{401 x^2 (-998+401 x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.23, size = 31, normalized size = 1.03
method | result | size |
norman | \(\frac {-\frac {1996 x^{2}}{401}-8 \,{\mathrm e}^{2}}{x^{2} \left (401 x -998\right )}\) | \(22\) |
gosper | \(-\frac {4 \left (499 x^{2}+802 \,{\mathrm e}^{2}\right )}{401 x^{2} \left (401 x -998\right )}\) | \(23\) |
risch | \(\frac {-\frac {1996 x^{2}}{401}-8 \,{\mathrm e}^{2}}{x^{2} \left (401 x -998\right )}\) | \(23\) |
default | \(\frac {4 \,{\mathrm e}^{2}}{499 x^{2}}+\frac {802 \,{\mathrm e}^{2}}{249001 x}-\frac {4 \left (\frac {499}{401}+\frac {160801 \,{\mathrm e}^{2}}{498002}\right )}{401 x -998}\) | \(31\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 25, normalized size = 0.83 \begin {gather*} -\frac {4 \, {\left (499 \, x^{2} + 802 \, e^{2}\right )}}{401 \, {\left (401 \, x^{3} - 998 \, x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 25, normalized size = 0.83 \begin {gather*} -\frac {4 \, {\left (499 \, x^{2} + 802 \, e^{2}\right )}}{401 \, {\left (401 \, x^{3} - 998 \, x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.20, size = 20, normalized size = 0.67 \begin {gather*} \frac {- 1996 x^{2} - 3208 e^{2}}{160801 x^{3} - 400198 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 31, normalized size = 1.03 \begin {gather*} -\frac {2 \, {\left (64481201 \, e^{2} + 248502998\right )}}{99849401 \, {\left (401 \, x - 998\right )}} + \frac {2 \, {\left (401 \, x e^{2} + 998 \, e^{2}\right )}}{249001 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.09, size = 24, normalized size = 0.80 \begin {gather*} \frac {\frac {1996\,x^2}{401}+8\,{\mathrm {e}}^2}{998\,x^2-401\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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