Optimal. Leaf size=21 \[ -3+x+\frac {9 e^{3 x}}{5 \left (e^{25+x}+x\right )} \]
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Rubi [F]
time = 0.80, 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 {5 e^{50+2 x}+10 e^{25+x} x+5 x^2+e^{3 x} \left (-9+18 e^{25+x}+27 x\right )}{5 e^{50+2 x}+10 e^{25+x} x+5 x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {5 e^{50+2 x}+10 e^{25+x} x+5 x^2+e^{3 x} \left (-9+18 e^{25+x}+27 x\right )}{5 \left (e^{25+x}+x\right )^2} \, dx\\ &=\frac {1}{5} \int \frac {5 e^{50+2 x}+10 e^{25+x} x+5 x^2+e^{3 x} \left (-9+18 e^{25+x}+27 x\right )}{\left (e^{25+x}+x\right )^2} \, dx\\ &=\frac {1}{5} \int \left (18 e^{-25+2 x}-9 e^{-50+x} (1+x)-\frac {9 (-1+x) x^3}{e^{75} \left (e^{25+x}+x\right )^2}+\frac {9 (-3+x) x^2}{e^{75} \left (e^{25+x}+x\right )}+\frac {5 e^{75}+18 x}{e^{75}}\right ) \, dx\\ &=\frac {\left (5 e^{75}+18 x\right )^2}{180 e^{75}}-\frac {9}{5} \int e^{-50+x} (1+x) \, dx+\frac {18}{5} \int e^{-25+2 x} \, dx-\frac {9 \int \frac {(-1+x) x^3}{\left (e^{25+x}+x\right )^2} \, dx}{5 e^{75}}+\frac {9 \int \frac {(-3+x) x^2}{e^{25+x}+x} \, dx}{5 e^{75}}\\ &=\frac {9}{5} e^{-25+2 x}-\frac {9}{5} e^{-50+x} (1+x)+\frac {\left (5 e^{75}+18 x\right )^2}{180 e^{75}}+\frac {9}{5} \int e^{-50+x} \, dx-\frac {9 \int \left (-\frac {x^3}{\left (e^{25+x}+x\right )^2}+\frac {x^4}{\left (e^{25+x}+x\right )^2}\right ) \, dx}{5 e^{75}}+\frac {9 \int \left (-\frac {3 x^2}{e^{25+x}+x}+\frac {x^3}{e^{25+x}+x}\right ) \, dx}{5 e^{75}}\\ &=\frac {9 e^{-50+x}}{5}+\frac {9}{5} e^{-25+2 x}-\frac {9}{5} e^{-50+x} (1+x)+\frac {\left (5 e^{75}+18 x\right )^2}{180 e^{75}}+\frac {9 \int \frac {x^3}{\left (e^{25+x}+x\right )^2} \, dx}{5 e^{75}}-\frac {9 \int \frac {x^4}{\left (e^{25+x}+x\right )^2} \, dx}{5 e^{75}}+\frac {9 \int \frac {x^3}{e^{25+x}+x} \, dx}{5 e^{75}}-\frac {27 \int \frac {x^2}{e^{25+x}+x} \, dx}{5 e^{75}}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 2.34, size = 34, normalized size = 1.62 \begin {gather*} \frac {9 e^{3 x}+5 e^{25+x} x+5 x^2}{5 \left (e^{25+x}+x\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.43, size = 27, normalized size = 1.29
method | result | size |
norman | \(\frac {x^{2}+x \,{\mathrm e}^{x} {\mathrm e}^{25}+\frac {9 \,{\mathrm e}^{3 x}}{5}}{{\mathrm e}^{x} {\mathrm e}^{25}+x}\) | \(27\) |
risch | \({\mathrm e}^{75} {\mathrm e}^{-75} x +\frac {9 \,{\mathrm e}^{-75} x^{2}}{5}+\frac {9 \,{\mathrm e}^{-25+2 x}}{5}-\frac {9 x \,{\mathrm e}^{-50+x}}{5}-\frac {9 x^{3} {\mathrm e}^{-75}}{5 \left ({\mathrm e}^{x +25}+x \right )}\) | \(45\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 29, normalized size = 1.38 \begin {gather*} \frac {5 \, x^{2} + 5 \, x e^{\left (x + 25\right )} + 9 \, e^{\left (3 \, x\right )}}{5 \, {\left (x + e^{\left (x + 25\right )}\right )}} \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. 36 vs.
\(2 (17) = 34\).
time = 0.36, size = 36, normalized size = 1.71 \begin {gather*} \frac {5 \, x^{2} e^{75} + 5 \, x e^{\left (x + 100\right )} + 9 \, e^{\left (3 \, x + 75\right )}}{5 \, {\left (x e^{75} + e^{\left (x + 100\right )}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 70 vs.
\(2 (19) = 38\).
time = 0.11, size = 70, normalized size = 3.33 \begin {gather*} - \frac {9 x^{3}}{5 x e^{75} + 5 e^{100} \sqrt [3]{e^{3 x}}} + \frac {9 x^{2}}{5 e^{75}} + x + \frac {- 45 x e^{25} \sqrt [3]{e^{3 x}} + 45 e^{50} \left (e^{3 x}\right )^{\frac {2}{3}}}{25 e^{75}} \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. 62 vs.
\(2 (17) = 34\).
time = 0.41, size = 62, normalized size = 2.95 \begin {gather*} \frac {5 \, {\left (x + 25\right )}^{2} e^{75} - 125 \, {\left (x + 25\right )} e^{75} + 5 \, {\left (x + 25\right )} e^{\left (x + 100\right )} - 5625 \, x + 9 \, e^{\left (3 \, x + 75\right )} - 5625 \, e^{\left (x + 25\right )}}{5 \, {\left ({\left (x + 25\right )} e^{75} - 25 \, e^{75} + e^{\left (x + 100\right )}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.87, size = 20, normalized size = 0.95 \begin {gather*} x+\frac {9\,{\mathrm {e}}^{3\,x}}{5\,\left (x+{\mathrm {e}}^{x+25}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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