Optimal. Leaf size=22 \[ -e^{e^{4+\frac {3}{25 x (2+x)}}}+x \]
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Rubi [F]
time = 12.04, 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 {100 x^2+100 x^3+25 x^4+\exp \left (e^{\frac {3+200 x+100 x^2}{50 x+25 x^2}}+\frac {3+200 x+100 x^2}{50 x+25 x^2}\right ) (6+6 x)}{100 x^2+100 x^3+25 x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {100 x^2+100 x^3+25 x^4+\exp \left (e^{\frac {3+200 x+100 x^2}{50 x+25 x^2}}+\frac {3+200 x+100 x^2}{50 x+25 x^2}\right ) (6+6 x)}{x^2 \left (100+100 x+25 x^2\right )} \, dx\\ &=\int \frac {100 x^2+100 x^3+25 x^4+\exp \left (e^{\frac {3+200 x+100 x^2}{50 x+25 x^2}}+\frac {3+200 x+100 x^2}{50 x+25 x^2}\right ) (6+6 x)}{25 x^2 (2+x)^2} \, dx\\ &=\frac {1}{25} \int \frac {100 x^2+100 x^3+25 x^4+\exp \left (e^{\frac {3+200 x+100 x^2}{50 x+25 x^2}}+\frac {3+200 x+100 x^2}{50 x+25 x^2}\right ) (6+6 x)}{x^2 (2+x)^2} \, dx\\ &=\frac {1}{25} \int \left (25+\frac {6 \exp \left (\frac {3+200 x+50 \exp \left (\frac {3}{50 x+25 x^2}+\frac {200 x}{50 x+25 x^2}+\frac {100 x^2}{50 x+25 x^2}\right ) x+100 x^2+25 \exp \left (\frac {3}{50 x+25 x^2}+\frac {200 x}{50 x+25 x^2}+\frac {100 x^2}{50 x+25 x^2}\right ) x^2}{25 x (2+x)}\right ) (1+x)}{x^2 (2+x)^2}\right ) \, dx\\ &=x+\frac {6}{25} \int \frac {\exp \left (\frac {3+200 x+50 \exp \left (\frac {3}{50 x+25 x^2}+\frac {200 x}{50 x+25 x^2}+\frac {100 x^2}{50 x+25 x^2}\right ) x+100 x^2+25 \exp \left (\frac {3}{50 x+25 x^2}+\frac {200 x}{50 x+25 x^2}+\frac {100 x^2}{50 x+25 x^2}\right ) x^2}{25 x (2+x)}\right ) (1+x)}{x^2 (2+x)^2} \, dx\\ &=x+\frac {6}{25} \int \frac {\exp \left (\frac {3+50 \left (4+e^{\frac {3+200 x+100 x^2}{50 x+25 x^2}}\right ) x+25 \left (4+e^{\frac {3+200 x+100 x^2}{50 x+25 x^2}}\right ) x^2}{25 x (2+x)}\right ) (1+x)}{x^2 (2+x)^2} \, dx\\ &=x+\frac {6}{25} \int \left (\frac {\exp \left (\frac {3+50 \left (4+e^{\frac {3+200 x+100 x^2}{50 x+25 x^2}}\right ) x+25 \left (4+e^{\frac {3+200 x+100 x^2}{50 x+25 x^2}}\right ) x^2}{25 x (2+x)}\right )}{4 x^2}-\frac {\exp \left (\frac {3+50 \left (4+e^{\frac {3+200 x+100 x^2}{50 x+25 x^2}}\right ) x+25 \left (4+e^{\frac {3+200 x+100 x^2}{50 x+25 x^2}}\right ) x^2}{25 x (2+x)}\right )}{4 (2+x)^2}\right ) \, dx\\ &=x+\frac {3}{50} \int \frac {\exp \left (\frac {3+50 \left (4+e^{\frac {3+200 x+100 x^2}{50 x+25 x^2}}\right ) x+25 \left (4+e^{\frac {3+200 x+100 x^2}{50 x+25 x^2}}\right ) x^2}{25 x (2+x)}\right )}{x^2} \, dx-\frac {3}{50} \int \frac {\exp \left (\frac {3+50 \left (4+e^{\frac {3+200 x+100 x^2}{50 x+25 x^2}}\right ) x+25 \left (4+e^{\frac {3+200 x+100 x^2}{50 x+25 x^2}}\right ) x^2}{25 x (2+x)}\right )}{(2+x)^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.66, size = 32, normalized size = 1.45 \begin {gather*} \frac {1}{25} \left (-25 e^{e^{4+\frac {3}{50 x}-\frac {3}{50 (2+x)}}}+25 x\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.79, size = 27, normalized size = 1.23
method | result | size |
risch | \(x -{\mathrm e}^{{\mathrm e}^{\frac {100 x^{2}+200 x +3}{25 x \left (2+x \right )}}}\) | \(27\) |
norman | \(\frac {x^{3}-4 x -2 x \,{\mathrm e}^{{\mathrm e}^{\frac {100 x^{2}+200 x +3}{25 x^{2}+50 x}}}-x^{2} {\mathrm e}^{{\mathrm e}^{\frac {100 x^{2}+200 x +3}{25 x^{2}+50 x}}}}{x \left (2+x \right )}\) | \(73\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 103 vs.
\(2 (18) = 36\).
time = 0.42, size = 103, normalized size = 4.68 \begin {gather*} {\left (x e^{\left (\frac {100 \, x^{2} + 200 \, x + 3}{25 \, {\left (x^{2} + 2 \, x\right )}}\right )} - e^{\left (\frac {100 \, x^{2} + 25 \, {\left (x^{2} + 2 \, x\right )} e^{\left (\frac {100 \, x^{2} + 200 \, x + 3}{25 \, {\left (x^{2} + 2 \, x\right )}}\right )} + 200 \, x + 3}{25 \, {\left (x^{2} + 2 \, x\right )}}\right )}\right )} e^{\left (-\frac {100 \, x^{2} + 200 \, x + 3}{25 \, {\left (x^{2} + 2 \, x\right )}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.14, size = 22, normalized size = 1.00 \begin {gather*} x - e^{e^{\frac {100 x^{2} + 200 x + 3}{25 x^{2} + 50 x}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 124 vs.
\(2 (18) = 36\).
time = 0.74, size = 124, normalized size = 5.64 \begin {gather*} {\left (x e^{\left (\frac {100 \, x^{2} + 200 \, x + 3}{25 \, {\left (x^{2} + 2 \, x\right )}}\right )} - e^{\left (\frac {25 \, x^{2} e^{\left (\frac {100 \, x^{2} + 200 \, x + 3}{25 \, {\left (x^{2} + 2 \, x\right )}}\right )} + 100 \, x^{2} + 50 \, x e^{\left (\frac {100 \, x^{2} + 200 \, x + 3}{25 \, {\left (x^{2} + 2 \, x\right )}}\right )} + 200 \, x + 3}{25 \, {\left (x^{2} + 2 \, x\right )}}\right )}\right )} e^{\left (-\frac {100 \, x^{2} + 200 \, x + 3}{25 \, {\left (x^{2} + 2 \, x\right )}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 6.24, size = 37, normalized size = 1.68 \begin {gather*} x-{\mathrm {e}}^{{\mathrm {e}}^{\frac {4\,x}{x+2}}\,{\mathrm {e}}^{\frac {3}{25\,x^2+50\,x}}\,{\mathrm {e}}^{\frac {8}{x+2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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