Optimal. Leaf size=31 \[ 5 \left (x-\frac {\log \left (7+e^4+x \left (-x+\left (1+x-x^2\right )^2\right )\right )}{x}\right ) \]
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Rubi [A] time = 3.54, antiderivative size = 32, normalized size of antiderivative = 1.03, number of steps used = 17, number of rules used = 7, integrand size = 131, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {6, 6688, 12, 14, 6742, 2100, 2525} \begin {gather*} 5 x-\frac {5 \log \left (x^5-2 x^4-x^3+x^2+x+e^4+7\right )}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 12
Rule 14
Rule 2100
Rule 2525
Rule 6688
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-5 x+25 x^2+5 e^4 x^2+20 x^3+45 x^4-30 x^5-10 x^6+5 x^7+\left (35+5 e^4+5 x+5 x^2-5 x^3-10 x^4+5 x^5\right ) \log \left (7+e^4+x+x^2-x^3-2 x^4+x^5\right )}{\left (7+e^4\right ) x^2+x^3+x^4-x^5-2 x^6+x^7} \, dx\\ &=\int \frac {-5 x+\left (25+5 e^4\right ) x^2+20 x^3+45 x^4-30 x^5-10 x^6+5 x^7+\left (35+5 e^4+5 x+5 x^2-5 x^3-10 x^4+5 x^5\right ) \log \left (7+e^4+x+x^2-x^3-2 x^4+x^5\right )}{\left (7+e^4\right ) x^2+x^3+x^4-x^5-2 x^6+x^7} \, dx\\ &=\int \frac {5 \left (\frac {x \left (-1+\left (5+e^4\right ) x+4 x^2+9 x^3-6 x^4-2 x^5+x^6\right )}{7+e^4+x+x^2-x^3-2 x^4+x^5}+\log \left (7+e^4+x+x^2-x^3-2 x^4+x^5\right )\right )}{x^2} \, dx\\ &=5 \int \frac {\frac {x \left (-1+\left (5+e^4\right ) x+4 x^2+9 x^3-6 x^4-2 x^5+x^6\right )}{7+e^4+x+x^2-x^3-2 x^4+x^5}+\log \left (7+e^4+x+x^2-x^3-2 x^4+x^5\right )}{x^2} \, dx\\ &=5 \int \left (\frac {-1+5 \left (1+\frac {e^4}{5}\right ) x+4 x^2+9 x^3-6 x^4-2 x^5+x^6}{x \left (7 \left (1+\frac {e^4}{7}\right )+x+x^2-x^3-2 x^4+x^5\right )}+\frac {\log \left (7 \left (1+\frac {e^4}{7}\right )+x+x^2-x^3-2 x^4+x^5\right )}{x^2}\right ) \, dx\\ &=5 \int \frac {-1+5 \left (1+\frac {e^4}{5}\right ) x+4 x^2+9 x^3-6 x^4-2 x^5+x^6}{x \left (7 \left (1+\frac {e^4}{7}\right )+x+x^2-x^3-2 x^4+x^5\right )} \, dx+5 \int \frac {\log \left (7 \left (1+\frac {e^4}{7}\right )+x+x^2-x^3-2 x^4+x^5\right )}{x^2} \, dx\\ &=-\frac {5 \log \left (7+e^4+x+x^2-x^3-2 x^4+x^5\right )}{x}+5 \int \frac {1+2 x-3 x^2-8 x^3+5 x^4}{x \left (7 \left (1+\frac {e^4}{7}\right )+x+x^2-x^3-2 x^4+x^5\right )} \, dx+5 \int \left (1+\frac {1}{\left (-7-e^4\right ) x}+\frac {-13-2 e^4+\left (22+3 e^4\right ) x+\left (55+8 e^4\right ) x^2-\left (37+5 e^4\right ) x^3+x^4}{\left (7+e^4\right ) \left (7 \left (1+\frac {e^4}{7}\right )+x+x^2-x^3-2 x^4+x^5\right )}\right ) \, dx\\ &=5 x-\frac {5 \log (x)}{7+e^4}-\frac {5 \log \left (7+e^4+x+x^2-x^3-2 x^4+x^5\right )}{x}+5 \int \left (\frac {1}{\left (7+e^4\right ) x}+\frac {13+2 e^4-\left (22+3 e^4\right ) x-\left (55+8 e^4\right ) x^2+\left (37+5 e^4\right ) x^3-x^4}{\left (7+e^4\right ) \left (7 \left (1+\frac {e^4}{7}\right )+x+x^2-x^3-2 x^4+x^5\right )}\right ) \, dx+\frac {5 \int \frac {-13-2 e^4+\left (22+3 e^4\right ) x+\left (55+8 e^4\right ) x^2-\left (37+5 e^4\right ) x^3+x^4}{7 \left (1+\frac {e^4}{7}\right )+x+x^2-x^3-2 x^4+x^5} \, dx}{7+e^4}\\ &=5 x+\frac {\log \left (7+e^4+x+x^2-x^3-2 x^4+x^5\right )}{7+e^4}-\frac {5 \log \left (7+e^4+x+x^2-x^3-2 x^4+x^5\right )}{x}+\frac {\int \frac {-2 \left (33+5 e^4\right )+3 \left (36+5 e^4\right ) x+2 \left (139+20 e^4\right ) x^2-\left (177+25 e^4\right ) x^3}{7 \left (1+\frac {e^4}{7}\right )+x+x^2-x^3-2 x^4+x^5} \, dx}{7+e^4}+\frac {5 \int \frac {13+2 e^4-\left (22+3 e^4\right ) x-\left (55+8 e^4\right ) x^2+\left (37+5 e^4\right ) x^3-x^4}{7 \left (1+\frac {e^4}{7}\right )+x+x^2-x^3-2 x^4+x^5} \, dx}{7+e^4}\\ &=5 x-\frac {5 \log \left (7+e^4+x+x^2-x^3-2 x^4+x^5\right )}{x}+\frac {\int \frac {2 \left (33+5 e^4\right )-3 \left (36+5 e^4\right ) x-2 \left (139+20 e^4\right ) x^2+\left (177+25 e^4\right ) x^3}{7 \left (1+\frac {e^4}{7}\right )+x+x^2-x^3-2 x^4+x^5} \, dx}{7+e^4}+\frac {\int \left (\frac {2 \left (-33-5 e^4\right )}{7 \left (1+\frac {e^4}{7}\right )+x+x^2-x^3-2 x^4+x^5}+\frac {3 \left (36+5 e^4\right ) x}{7 \left (1+\frac {e^4}{7}\right )+x+x^2-x^3-2 x^4+x^5}+\frac {2 \left (139+20 e^4\right ) x^2}{7 \left (1+\frac {e^4}{7}\right )+x+x^2-x^3-2 x^4+x^5}+\frac {\left (-177-25 e^4\right ) x^3}{7 \left (1+\frac {e^4}{7}\right )+x+x^2-x^3-2 x^4+x^5}\right ) \, dx}{7+e^4}\\ &=5 x-\frac {5 \log \left (7+e^4+x+x^2-x^3-2 x^4+x^5\right )}{x}+\frac {\int \left (\frac {2 \left (33+5 e^4\right )}{7 \left (1+\frac {e^4}{7}\right )+x+x^2-x^3-2 x^4+x^5}+\frac {3 \left (-36-5 e^4\right ) x}{7 \left (1+\frac {e^4}{7}\right )+x+x^2-x^3-2 x^4+x^5}+\frac {2 \left (-139-20 e^4\right ) x^2}{7 \left (1+\frac {e^4}{7}\right )+x+x^2-x^3-2 x^4+x^5}+\frac {\left (177+25 e^4\right ) x^3}{7 \left (1+\frac {e^4}{7}\right )+x+x^2-x^3-2 x^4+x^5}\right ) \, dx}{7+e^4}-\frac {\left (2 \left (33+5 e^4\right )\right ) \int \frac {1}{7 \left (1+\frac {e^4}{7}\right )+x+x^2-x^3-2 x^4+x^5} \, dx}{7+e^4}+\frac {\left (3 \left (36+5 e^4\right )\right ) \int \frac {x}{7 \left (1+\frac {e^4}{7}\right )+x+x^2-x^3-2 x^4+x^5} \, dx}{7+e^4}+\frac {\left (2 \left (139+20 e^4\right )\right ) \int \frac {x^2}{7 \left (1+\frac {e^4}{7}\right )+x+x^2-x^3-2 x^4+x^5} \, dx}{7+e^4}-\frac {\left (177+25 e^4\right ) \int \frac {x^3}{7 \left (1+\frac {e^4}{7}\right )+x+x^2-x^3-2 x^4+x^5} \, dx}{7+e^4}\\ &=5 x-\frac {5 \log \left (7+e^4+x+x^2-x^3-2 x^4+x^5\right )}{x}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.04, size = 32, normalized size = 1.03 \begin {gather*} 5 \left (x-\frac {\log \left (7+e^4+x+x^2-x^3-2 x^4+x^5\right )}{x}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 33, normalized size = 1.06 \begin {gather*} \frac {5 \, {\left (x^{2} - \log \left (x^{5} - 2 \, x^{4} - x^{3} + x^{2} + x + e^{4} + 7\right )\right )}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 33, normalized size = 1.06 \begin {gather*} \frac {5 \, {\left (x^{2} - \log \left (x^{5} - 2 \, x^{4} - x^{3} + x^{2} + x + e^{4} + 7\right )\right )}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.26, size = 32, normalized size = 1.03
method | result | size |
risch | \(-\frac {5 \ln \left ({\mathrm e}^{4}+x^{5}-2 x^{4}-x^{3}+x^{2}+x +7\right )}{x}+5 x\) | \(32\) |
norman | \(\frac {5 x^{2}-5 \ln \left ({\mathrm e}^{4}+x^{5}-2 x^{4}-x^{3}+x^{2}+x +7\right )}{x}\) | \(35\) |
default | \(-\frac {5 \ln \left ({\mathrm e}^{4}+x^{5}-2 x^{4}-x^{3}+x^{2}+x +7\right )}{x}+\frac {5 \left (\munderset {\textit {\_R} =\RootOf \left ({\mathrm e}^{4}+\textit {\_Z}^{5}-2 \textit {\_Z}^{4}-\textit {\_Z}^{3}+\textit {\_Z}^{2}+\textit {\_Z} +7\right )}{\sum }\frac {\left (-\textit {\_R}^{4}+\left (5 \,{\mathrm e}^{4}+37\right ) \textit {\_R}^{3}+\left (-8 \,{\mathrm e}^{4}-55\right ) \textit {\_R}^{2}+\left (-3 \,{\mathrm e}^{4}-22\right ) \textit {\_R} +2 \,{\mathrm e}^{4}+13\right ) \ln \left (x -\textit {\_R} \right )}{5 \textit {\_R}^{4}-8 \textit {\_R}^{3}-3 \textit {\_R}^{2}+2 \textit {\_R} +1}\right )}{{\mathrm e}^{4}+7}+5 x +\frac {5 \left (\munderset {\textit {\_R} =\RootOf \left ({\mathrm e}^{4}+\textit {\_Z}^{5}-2 \textit {\_Z}^{4}-\textit {\_Z}^{3}+\textit {\_Z}^{2}+\textit {\_Z} +7\right )}{\sum }\frac {\left (\textit {\_R}^{4}+\left (-5 \,{\mathrm e}^{4}-37\right ) \textit {\_R}^{3}+\left (8 \,{\mathrm e}^{4}+55\right ) \textit {\_R}^{2}+\left (3 \,{\mathrm e}^{4}+22\right ) \textit {\_R} -2 \,{\mathrm e}^{4}-13\right ) \ln \left (x -\textit {\_R} \right )}{5 \textit {\_R}^{4}-8 \textit {\_R}^{3}-3 \textit {\_R}^{2}+2 \textit {\_R} +1}\right )}{{\mathrm e}^{4}+7}\) | \(232\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 33, normalized size = 1.06 \begin {gather*} \frac {5 \, {\left (x^{2} - \log \left (x^{5} - 2 \, x^{4} - x^{3} + x^{2} + x + e^{4} + 7\right )\right )}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.15, size = 33, normalized size = 1.06 \begin {gather*} -\frac {5\,\left (\ln \left (x^5-2\,x^4-x^3+x^2+x+{\mathrm {e}}^4+7\right )-x^2\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.26, size = 29, normalized size = 0.94 \begin {gather*} 5 x - \frac {5 \log {\left (x^{5} - 2 x^{4} - x^{3} + x^{2} + x + 7 + e^{4} \right )}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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