Optimal. Leaf size=29 \[ \frac {\log \left (-e^{3+x}+\frac {x}{5+\frac {1}{2} x (5+x)^2}\right )}{x^3} \]
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Rubi [A] time = 8.99, antiderivative size = 31, normalized size of antiderivative = 1.07, number of steps used = 24, number of rules used = 4, integrand size = 214, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.019, Rules used = {6688, 6742, 14, 2551} \begin {gather*} \frac {\log \left (\frac {2 x}{x^3+10 x^2+25 x+10}-e^{x+3}\right )}{x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 2551
Rule 6688
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {20 x-20 x^3-4 x^4-e^{3+x} x \left (10+25 x+10 x^2+x^3\right )^2+3 \left (10+25 x+10 x^2+x^3\right ) \left (-2 x+e^{3+x} \left (10+25 x+10 x^2+x^3\right )\right ) \log \left (-e^{3+x}+\frac {2 x}{10+25 x+10 x^2+x^3}\right )}{x^4 \left (10+25 x+10 x^2+x^3\right ) \left (2 x-e^{3+x} \left (10+25 x+10 x^2+x^3\right )\right )} \, dx\\ &=\int \left (\frac {2 \left (-10+10 x+35 x^2+12 x^3+x^4\right )}{x^3 \left (10+25 x+10 x^2+x^3\right ) \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )}+\frac {x-3 \log \left (-e^{3+x}+\frac {2 x}{10+25 x+10 x^2+x^3}\right )}{x^4}\right ) \, dx\\ &=2 \int \frac {-10+10 x+35 x^2+12 x^3+x^4}{x^3 \left (10+25 x+10 x^2+x^3\right ) \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )} \, dx+\int \frac {x-3 \log \left (-e^{3+x}+\frac {2 x}{10+25 x+10 x^2+x^3}\right )}{x^4} \, dx\\ &=2 \int \left (-\frac {1}{x^3 \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )}+\frac {7}{2 x^2 \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )}-\frac {17}{4 x \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )}+\frac {337+160 x+17 x^2}{4 \left (10+25 x+10 x^2+x^3\right ) \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )}\right ) \, dx+\int \left (\frac {1}{x^3}-\frac {3 \log \left (-e^{3+x}+\frac {2 x}{10+25 x+10 x^2+x^3}\right )}{x^4}\right ) \, dx\\ &=-\frac {1}{2 x^2}+\frac {1}{2} \int \frac {337+160 x+17 x^2}{\left (10+25 x+10 x^2+x^3\right ) \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )} \, dx-2 \int \frac {1}{x^3 \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )} \, dx-3 \int \frac {\log \left (-e^{3+x}+\frac {2 x}{10+25 x+10 x^2+x^3}\right )}{x^4} \, dx+7 \int \frac {1}{x^2 \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )} \, dx-\frac {17}{2} \int \frac {1}{x \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )} \, dx\\ &=-\frac {1}{2 x^2}+\frac {\log \left (-e^{3+x}+\frac {2 x}{10+25 x+10 x^2+x^3}\right )}{x^3}+\frac {1}{2} \int \frac {-337-160 x-17 x^2}{\left (10+25 x+10 x^2+x^3\right ) \left (2 x-e^{3+x} \left (10+25 x+10 x^2+x^3\right )\right )} \, dx-2 \int \frac {1}{x^3 \left (-2 x+e^{3+x} \left (10+25 x+10 x^2+x^3\right )\right )} \, dx+7 \int \frac {1}{x^2 \left (-2 x+e^{3+x} \left (10+25 x+10 x^2+x^3\right )\right )} \, dx-\frac {17}{2} \int \frac {1}{x \left (-2 x+e^{3+x} \left (10+25 x+10 x^2+x^3\right )\right )} \, dx-\int \frac {-4 \left (-5+5 x^2+x^3\right )-e^{3+x} \left (10+25 x+10 x^2+x^3\right )^2}{x^3 \left (10+25 x+10 x^2+x^3\right ) \left (2 x-e^{3+x} \left (10+25 x+10 x^2+x^3\right )\right )} \, dx\\ &=-\frac {1}{2 x^2}+\frac {\log \left (-e^{3+x}+\frac {2 x}{10+25 x+10 x^2+x^3}\right )}{x^3}+\frac {1}{2} \int \left (\frac {337}{\left (10+25 x+10 x^2+x^3\right ) \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )}+\frac {160 x}{\left (10+25 x+10 x^2+x^3\right ) \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )}+\frac {17 x^2}{\left (10+25 x+10 x^2+x^3\right ) \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )}\right ) \, dx-2 \int \frac {1}{x^3 \left (-2 x+e^{3+x} \left (10+25 x+10 x^2+x^3\right )\right )} \, dx+7 \int \frac {1}{x^2 \left (-2 x+e^{3+x} \left (10+25 x+10 x^2+x^3\right )\right )} \, dx-\frac {17}{2} \int \frac {1}{x \left (-2 x+e^{3+x} \left (10+25 x+10 x^2+x^3\right )\right )} \, dx-\int \left (\frac {1}{x^3}+\frac {2 \left (-10+10 x+35 x^2+12 x^3+x^4\right )}{x^3 \left (10+25 x+10 x^2+x^3\right ) \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )}\right ) \, dx\\ &=\frac {\log \left (-e^{3+x}+\frac {2 x}{10+25 x+10 x^2+x^3}\right )}{x^3}-2 \int \frac {-10+10 x+35 x^2+12 x^3+x^4}{x^3 \left (10+25 x+10 x^2+x^3\right ) \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )} \, dx-2 \int \frac {1}{x^3 \left (-2 x+e^{3+x} \left (10+25 x+10 x^2+x^3\right )\right )} \, dx+7 \int \frac {1}{x^2 \left (-2 x+e^{3+x} \left (10+25 x+10 x^2+x^3\right )\right )} \, dx+\frac {17}{2} \int \frac {x^2}{\left (10+25 x+10 x^2+x^3\right ) \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )} \, dx-\frac {17}{2} \int \frac {1}{x \left (-2 x+e^{3+x} \left (10+25 x+10 x^2+x^3\right )\right )} \, dx+80 \int \frac {x}{\left (10+25 x+10 x^2+x^3\right ) \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )} \, dx+\frac {337}{2} \int \frac {1}{\left (10+25 x+10 x^2+x^3\right ) \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )} \, dx\\ &=\frac {\log \left (-e^{3+x}+\frac {2 x}{10+25 x+10 x^2+x^3}\right )}{x^3}-2 \int \frac {1}{x^3 \left (-2 x+e^{3+x} \left (10+25 x+10 x^2+x^3\right )\right )} \, dx-2 \int \left (-\frac {1}{x^3 \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )}+\frac {7}{2 x^2 \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )}-\frac {17}{4 x \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )}+\frac {337+160 x+17 x^2}{4 \left (10+25 x+10 x^2+x^3\right ) \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )}\right ) \, dx+7 \int \frac {1}{x^2 \left (-2 x+e^{3+x} \left (10+25 x+10 x^2+x^3\right )\right )} \, dx+\frac {17}{2} \int \frac {x^2}{\left (10+25 x+10 x^2+x^3\right ) \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )} \, dx-\frac {17}{2} \int \frac {1}{x \left (-2 x+e^{3+x} \left (10+25 x+10 x^2+x^3\right )\right )} \, dx+80 \int \frac {x}{\left (10+25 x+10 x^2+x^3\right ) \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )} \, dx+\frac {337}{2} \int \frac {1}{\left (10+25 x+10 x^2+x^3\right ) \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )} \, dx\\ &=\frac {\log \left (-e^{3+x}+\frac {2 x}{10+25 x+10 x^2+x^3}\right )}{x^3}-\frac {1}{2} \int \frac {337+160 x+17 x^2}{\left (10+25 x+10 x^2+x^3\right ) \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )} \, dx+2 \int \frac {1}{x^3 \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )} \, dx-2 \int \frac {1}{x^3 \left (-2 x+e^{3+x} \left (10+25 x+10 x^2+x^3\right )\right )} \, dx-7 \int \frac {1}{x^2 \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )} \, dx+7 \int \frac {1}{x^2 \left (-2 x+e^{3+x} \left (10+25 x+10 x^2+x^3\right )\right )} \, dx+\frac {17}{2} \int \frac {1}{x \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )} \, dx+\frac {17}{2} \int \frac {x^2}{\left (10+25 x+10 x^2+x^3\right ) \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )} \, dx-\frac {17}{2} \int \frac {1}{x \left (-2 x+e^{3+x} \left (10+25 x+10 x^2+x^3\right )\right )} \, dx+80 \int \frac {x}{\left (10+25 x+10 x^2+x^3\right ) \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )} \, dx+\frac {337}{2} \int \frac {1}{\left (10+25 x+10 x^2+x^3\right ) \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )} \, dx\\ &=\frac {\log \left (-e^{3+x}+\frac {2 x}{10+25 x+10 x^2+x^3}\right )}{x^3}-\frac {1}{2} \int \frac {-337-160 x-17 x^2}{\left (10+25 x+10 x^2+x^3\right ) \left (2 x-e^{3+x} \left (10+25 x+10 x^2+x^3\right )\right )} \, dx+\frac {17}{2} \int \frac {x^2}{\left (10+25 x+10 x^2+x^3\right ) \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )} \, dx+80 \int \frac {x}{\left (10+25 x+10 x^2+x^3\right ) \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )} \, dx+\frac {337}{2} \int \frac {1}{\left (10+25 x+10 x^2+x^3\right ) \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )} \, dx\\ &=\frac {\log \left (-e^{3+x}+\frac {2 x}{10+25 x+10 x^2+x^3}\right )}{x^3}-\frac {1}{2} \int \left (\frac {337}{\left (10+25 x+10 x^2+x^3\right ) \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )}+\frac {160 x}{\left (10+25 x+10 x^2+x^3\right ) \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )}+\frac {17 x^2}{\left (10+25 x+10 x^2+x^3\right ) \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )}\right ) \, dx+\frac {17}{2} \int \frac {x^2}{\left (10+25 x+10 x^2+x^3\right ) \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )} \, dx+80 \int \frac {x}{\left (10+25 x+10 x^2+x^3\right ) \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )} \, dx+\frac {337}{2} \int \frac {1}{\left (10+25 x+10 x^2+x^3\right ) \left (10 e^{3+x}-2 x+25 e^{3+x} x+10 e^{3+x} x^2+e^{3+x} x^3\right )} \, dx\\ &=\frac {\log \left (-e^{3+x}+\frac {2 x}{10+25 x+10 x^2+x^3}\right )}{x^3}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.15, size = 31, normalized size = 1.07 \begin {gather*} \frac {\log \left (-e^{3+x}+\frac {2 x}{10+25 x+10 x^2+x^3}\right )}{x^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 44, normalized size = 1.52 \begin {gather*} \frac {\log \left (-\frac {{\left (x^{3} + 10 \, x^{2} + 25 \, x + 10\right )} e^{\left (x + 3\right )} - 2 \, x}{x^{3} + 10 \, x^{2} + 25 \, x + 10}\right )}{x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.59, size = 56, normalized size = 1.93 \begin {gather*} \frac {\log \left (-\frac {x^{3} e^{\left (x + 3\right )} + 10 \, x^{2} e^{\left (x + 3\right )} + 25 \, x e^{\left (x + 3\right )} - 2 \, x + 10 \, e^{\left (x + 3\right )}}{x^{3} + 10 \, x^{2} + 25 \, x + 10}\right )}{x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.18, size = 474, normalized size = 16.34
method | result | size |
risch | \(\frac {\ln \left (x^{3} {\mathrm e}^{3+x}+10 x^{2} {\mathrm e}^{3+x}+\left (25 \,{\mathrm e}^{3+x}-2\right ) x +10 \,{\mathrm e}^{3+x}\right )}{x^{3}}-\frac {i \pi \,\mathrm {csgn}\left (i \left (x^{3} {\mathrm e}^{3+x}+10 x^{2} {\mathrm e}^{3+x}+\left (25 \,{\mathrm e}^{3+x}-2\right ) x +10 \,{\mathrm e}^{3+x}\right )\right ) \mathrm {csgn}\left (\frac {i}{x^{3}+10 x^{2}+25 x +10}\right ) \mathrm {csgn}\left (\frac {i \left (x^{3} {\mathrm e}^{3+x}+10 x^{2} {\mathrm e}^{3+x}+\left (25 \,{\mathrm e}^{3+x}-2\right ) x +10 \,{\mathrm e}^{3+x}\right )}{x^{3}+10 x^{2}+25 x +10}\right )-i \pi \,\mathrm {csgn}\left (i \left (x^{3} {\mathrm e}^{3+x}+10 x^{2} {\mathrm e}^{3+x}+\left (25 \,{\mathrm e}^{3+x}-2\right ) x +10 \,{\mathrm e}^{3+x}\right )\right ) \mathrm {csgn}\left (\frac {i \left (x^{3} {\mathrm e}^{3+x}+10 x^{2} {\mathrm e}^{3+x}+\left (25 \,{\mathrm e}^{3+x}-2\right ) x +10 \,{\mathrm e}^{3+x}\right )}{x^{3}+10 x^{2}+25 x +10}\right )^{2}-i \pi \,\mathrm {csgn}\left (\frac {i}{x^{3}+10 x^{2}+25 x +10}\right ) \mathrm {csgn}\left (\frac {i \left (x^{3} {\mathrm e}^{3+x}+10 x^{2} {\mathrm e}^{3+x}+\left (25 \,{\mathrm e}^{3+x}-2\right ) x +10 \,{\mathrm e}^{3+x}\right )}{x^{3}+10 x^{2}+25 x +10}\right )^{2}-i \pi \mathrm {csgn}\left (\frac {i \left (x^{3} {\mathrm e}^{3+x}+10 x^{2} {\mathrm e}^{3+x}+\left (25 \,{\mathrm e}^{3+x}-2\right ) x +10 \,{\mathrm e}^{3+x}\right )}{x^{3}+10 x^{2}+25 x +10}\right )^{3}+2 i \pi \mathrm {csgn}\left (\frac {i \left (x^{3} {\mathrm e}^{3+x}+10 x^{2} {\mathrm e}^{3+x}+\left (25 \,{\mathrm e}^{3+x}-2\right ) x +10 \,{\mathrm e}^{3+x}\right )}{x^{3}+10 x^{2}+25 x +10}\right )^{2}-2 i \pi +2 \ln \left (x^{3}+10 x^{2}+25 x +10\right )}{2 x^{3}}\) | \(474\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.52, size = 54, normalized size = 1.86 \begin {gather*} -\frac {\log \left (x^{3} + 10 \, x^{2} + 25 \, x + 10\right ) - \log \left (-{\left (x^{3} e^{3} + 10 \, x^{2} e^{3} + 25 \, x e^{3} + 10 \, e^{3}\right )} e^{x} + 2 \, x\right )}{x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.68, size = 44, normalized size = 1.52 \begin {gather*} \frac {\ln \left (\frac {2\,x-{\mathrm {e}}^3\,{\mathrm {e}}^x\,\left (x^3+10\,x^2+25\,x+10\right )}{x^3+10\,x^2+25\,x+10}\right )}{x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.97, size = 41, normalized size = 1.41 \begin {gather*} \frac {\log {\left (\frac {2 x + \left (- x^{3} - 10 x^{2} - 25 x - 10\right ) e^{x + 3}}{x^{3} + 10 x^{2} + 25 x + 10} \right )}}{x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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