Integrand size = 28, antiderivative size = 18 \[ \int \frac {1+e^{260+85 x+5 x^2} \left (85 x+10 x^2\right )}{x} \, dx=-\frac {4}{3}+e^{5 (4+x) (13+x)}+\log (2 x) \]
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Time = 0.02 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {14, 2268} \[ \int \frac {1+e^{260+85 x+5 x^2} \left (85 x+10 x^2\right )}{x} \, dx=e^{5 x^2+85 x+260}+\log (x) \]
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Rule 14
Rule 2268
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{x}+5 e^{260+85 x+5 x^2} (17+2 x)\right ) \, dx \\ & = \log (x)+5 \int e^{260+85 x+5 x^2} (17+2 x) \, dx \\ & = e^{260+85 x+5 x^2}+\log (x) \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.72 \[ \int \frac {1+e^{260+85 x+5 x^2} \left (85 x+10 x^2\right )}{x} \, dx=e^{5 (4+x) (13+x)}+\log (x) \]
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Time = 0.05 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.72
method | result | size |
risch | \({\mathrm e}^{5 \left (x +13\right ) \left (4+x \right )}+\ln \left (x \right )\) | \(13\) |
norman | \({\mathrm e}^{5 x^{2}+85 x +260}+\ln \left (x \right )\) | \(15\) |
parallelrisch | \({\mathrm e}^{5 x^{2}+85 x +260}+\ln \left (x \right )\) | \(15\) |
default | \(\ln \left (x \right )-\frac {17 i {\mathrm e}^{260} \sqrt {\pi }\, {\mathrm e}^{-\frac {1445}{4}} \sqrt {5}\, \operatorname {erf}\left (i \sqrt {5}\, x +\frac {17 i \sqrt {5}}{2}\right )}{2}+10 \,{\mathrm e}^{260} \left (\frac {{\mathrm e}^{5 x^{2}+85 x}}{10}+\frac {17 i \sqrt {\pi }\, {\mathrm e}^{-\frac {1445}{4}} \sqrt {5}\, \operatorname {erf}\left (i \sqrt {5}\, x +\frac {17 i \sqrt {5}}{2}\right )}{20}\right )\) | \(75\) |
parts | \(\ln \left (x \right )-\frac {17 i {\mathrm e}^{260} \sqrt {\pi }\, {\mathrm e}^{-\frac {1445}{4}} \sqrt {5}\, \operatorname {erf}\left (i \sqrt {5}\, x +\frac {17 i \sqrt {5}}{2}\right )}{2}+10 \,{\mathrm e}^{260} \left (\frac {{\mathrm e}^{5 x^{2}+85 x}}{10}+\frac {17 i \sqrt {\pi }\, {\mathrm e}^{-\frac {1445}{4}} \sqrt {5}\, \operatorname {erf}\left (i \sqrt {5}\, x +\frac {17 i \sqrt {5}}{2}\right )}{20}\right )\) | \(75\) |
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Time = 0.24 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {1+e^{260+85 x+5 x^2} \left (85 x+10 x^2\right )}{x} \, dx=e^{\left (5 \, x^{2} + 85 \, x + 260\right )} + \log \left (x\right ) \]
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Time = 0.05 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {1+e^{260+85 x+5 x^2} \left (85 x+10 x^2\right )}{x} \, dx=e^{5 x^{2} + 85 x + 260} + \log {\left (x \right )} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.31 (sec) , antiderivative size = 89, normalized size of antiderivative = 4.94 \[ \int \frac {1+e^{260+85 x+5 x^2} \left (85 x+10 x^2\right )}{x} \, dx=-\frac {17}{2} i \, \sqrt {5} \sqrt {\pi } \operatorname {erf}\left (i \, \sqrt {5} x + \frac {17}{2} i \, \sqrt {5}\right ) e^{\left (-\frac {405}{4}\right )} - \frac {1}{10} \, \sqrt {5} {\left (\frac {85 \, \sqrt {\pi } {\left (2 \, x + 17\right )} {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {5} \sqrt {-{\left (2 \, x + 17\right )}^{2}}\right ) - 1\right )}}{\sqrt {-{\left (2 \, x + 17\right )}^{2}}} - 2 \, \sqrt {5} e^{\left (\frac {5}{4} \, {\left (2 \, x + 17\right )}^{2}\right )}\right )} e^{\left (-\frac {405}{4}\right )} + \log \left (x\right ) \]
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Time = 0.29 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {1+e^{260+85 x+5 x^2} \left (85 x+10 x^2\right )}{x} \, dx=e^{\left (5 \, x^{2} + 85 \, x + 260\right )} + \log \left (x\right ) \]
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Time = 0.09 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89 \[ \int \frac {1+e^{260+85 x+5 x^2} \left (85 x+10 x^2\right )}{x} \, dx=\ln \left (x\right )+{\mathrm {e}}^{85\,x}\,{\mathrm {e}}^{260}\,{\mathrm {e}}^{5\,x^2} \]
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