Optimal. Leaf size=17 \[ \frac {4}{e^{10} \left (1+\frac {625}{x}\right )+x} \]
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Rubi [A]
time = 0.08, antiderivative size = 19, normalized size of antiderivative = 1.12, number of steps
used = 4, number of rules used = 4, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.089, Rules used = {1694, 12, 1828,
8} \begin {gather*} \frac {4 x}{x^2+e^{10} x+625 e^{10}} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 12
Rule 1694
Rule 1828
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\text {Subst}\left (\int \frac {16 \left (e^{10} \left (2500-e^{10}\right )+4 e^{10} x-4 x^2\right )}{\left (2500 e^{10}-e^{20}+4 x^2\right )^2} \, dx,x,\frac {e^{10}}{2}+x\right )\\ &=16 \text {Subst}\left (\int \frac {e^{10} \left (2500-e^{10}\right )+4 e^{10} x-4 x^2}{\left (2500 e^{10}-e^{20}+4 x^2\right )^2} \, dx,x,\frac {e^{10}}{2}+x\right )\\ &=\frac {4 x}{625 e^{10}+e^{10} x+x^2}-\frac {8 \text {Subst}\left (\int 0 \, dx,x,\frac {e^{10}}{2}+x\right )}{e^{10} \left (2500-e^{10}\right )}\\ &=\frac {4 x}{625 e^{10}+e^{10} x+x^2}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.02, size = 16, normalized size = 0.94 \begin {gather*} \frac {4 x}{x^2+e^{10} (625+x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 1.65, size = 83, normalized size = 4.88
method | result | size |
risch | \(\frac {4 x}{x \,{\mathrm e}^{10}+625 \,{\mathrm e}^{10}+x^{2}}\) | \(18\) |
gosper | \(\frac {4 x}{x \,{\mathrm e}^{10}+625 \,{\mathrm e}^{10}+x^{2}}\) | \(22\) |
norman | \(\frac {4 x}{x \,{\mathrm e}^{10}+625 \,{\mathrm e}^{10}+x^{2}}\) | \(22\) |
default | \(2 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}+2 \textit {\_Z}^{3} {\mathrm e}^{10}+\left ({\mathrm e}^{20}+1250 \,{\mathrm e}^{10}\right ) \textit {\_Z}^{2}+1250 \textit {\_Z} \,{\mathrm e}^{20}+390625 \,{\mathrm e}^{20}\right )}{\sum }\frac {\left (625 \,{\mathrm e}^{10}-\textit {\_R}^{2}\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R} \,{\mathrm e}^{20}+625 \,{\mathrm e}^{20}+3 \,{\mathrm e}^{10} \textit {\_R}^{2}+1250 \textit {\_R} \,{\mathrm e}^{10}+2 \textit {\_R}^{3}}\right )\) | \(83\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 17, normalized size = 1.00 \begin {gather*} \frac {4 \, x}{x^{2} + x e^{10} + 625 \, e^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 15, normalized size = 0.88 \begin {gather*} \frac {4 \, x}{x^{2} + {\left (x + 625\right )} e^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.20, size = 15, normalized size = 0.88 \begin {gather*} \frac {4 x}{x^{2} + x e^{10} + 625 e^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 17, normalized size = 1.00 \begin {gather*} \frac {4 \, x}{x^{2} + x e^{10} + 625 \, e^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.19, size = 17, normalized size = 1.00 \begin {gather*} \frac {4\,x}{x^2+{\mathrm {e}}^{10}\,x+625\,{\mathrm {e}}^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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