Integrand size = 36, antiderivative size = 25 \[ \int \left (15-2 x+e^{-27+24 x+3 x^2} \left (1+24 x+6 x^2\right )-8 \log (x)+\log ^2(x)\right ) \, dx=x \left (e^{3 \left (-25+(4+x)^2\right )}-x+(5-\log (x))^2\right ) \]
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Time = 0.02 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.80, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2326, 2332, 2333} \[ \int \left (15-2 x+e^{-27+24 x+3 x^2} \left (1+24 x+6 x^2\right )-8 \log (x)+\log ^2(x)\right ) \, dx=-x^2+\frac {e^{3 x^2+24 x-27} \left (x^2+4 x\right )}{x+4}+25 x+x \log ^2(x)-10 x \log (x) \]
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Rule 2326
Rule 2332
Rule 2333
Rubi steps \begin{align*} \text {integral}& = 15 x-x^2-8 \int \log (x) \, dx+\int e^{-27+24 x+3 x^2} \left (1+24 x+6 x^2\right ) \, dx+\int \log ^2(x) \, dx \\ & = 23 x-x^2+\frac {e^{-27+24 x+3 x^2} \left (4 x+x^2\right )}{4+x}-8 x \log (x)+x \log ^2(x)-2 \int \log (x) \, dx \\ & = 25 x-x^2+\frac {e^{-27+24 x+3 x^2} \left (4 x+x^2\right )}{4+x}-10 x \log (x)+x \log ^2(x) \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \left (15-2 x+e^{-27+24 x+3 x^2} \left (1+24 x+6 x^2\right )-8 \log (x)+\log ^2(x)\right ) \, dx=x \left (25+e^{-27+3 x (8+x)}-x-10 \log (x)+\log ^2(x)\right ) \]
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Time = 0.10 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.28
method | result | size |
risch | \(x \ln \left (x \right )^{2}+{\mathrm e}^{3 \left (x +9\right ) \left (-1+x \right )} x +25 x -x^{2}-10 x \ln \left (x \right )\) | \(32\) |
norman | \(x \ln \left (x \right )^{2}+{\mathrm e}^{3 x^{2}+24 x -27} x +25 x -x^{2}-10 x \ln \left (x \right )\) | \(34\) |
parallelrisch | \(x \ln \left (x \right )^{2}+{\mathrm e}^{3 x^{2}+24 x -27} x +25 x -x^{2}-10 x \ln \left (x \right )\) | \(34\) |
default | \(25 x +x \ln \left (x \right )^{2}-10 x \ln \left (x \right )-\frac {i {\mathrm e}^{-27} \sqrt {\pi }\, {\mathrm e}^{-48} \sqrt {3}\, \operatorname {erf}\left (i \sqrt {3}\, x +4 i \sqrt {3}\right )}{6}+24 \,{\mathrm e}^{-27} \left (\frac {{\mathrm e}^{3 x^{2}+24 x}}{6}+\frac {2 i \sqrt {\pi }\, {\mathrm e}^{-48} \sqrt {3}\, \operatorname {erf}\left (i \sqrt {3}\, x +4 i \sqrt {3}\right )}{3}\right )+6 \,{\mathrm e}^{-27} \left (\frac {x \,{\mathrm e}^{3 x^{2}+24 x}}{6}-\frac {2 \,{\mathrm e}^{3 x^{2}+24 x}}{3}-\frac {95 i \sqrt {\pi }\, {\mathrm e}^{-48} \sqrt {3}\, \operatorname {erf}\left (i \sqrt {3}\, x +4 i \sqrt {3}\right )}{36}\right )-x^{2}\) | \(148\) |
parts | \(25 x +x \ln \left (x \right )^{2}-10 x \ln \left (x \right )-\frac {i {\mathrm e}^{-27} \sqrt {\pi }\, {\mathrm e}^{-48} \sqrt {3}\, \operatorname {erf}\left (i \sqrt {3}\, x +4 i \sqrt {3}\right )}{6}+24 \,{\mathrm e}^{-27} \left (\frac {{\mathrm e}^{3 x^{2}+24 x}}{6}+\frac {2 i \sqrt {\pi }\, {\mathrm e}^{-48} \sqrt {3}\, \operatorname {erf}\left (i \sqrt {3}\, x +4 i \sqrt {3}\right )}{3}\right )+6 \,{\mathrm e}^{-27} \left (\frac {x \,{\mathrm e}^{3 x^{2}+24 x}}{6}-\frac {2 \,{\mathrm e}^{3 x^{2}+24 x}}{3}-\frac {95 i \sqrt {\pi }\, {\mathrm e}^{-48} \sqrt {3}\, \operatorname {erf}\left (i \sqrt {3}\, x +4 i \sqrt {3}\right )}{36}\right )-x^{2}\) | \(148\) |
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Time = 0.25 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.32 \[ \int \left (15-2 x+e^{-27+24 x+3 x^2} \left (1+24 x+6 x^2\right )-8 \log (x)+\log ^2(x)\right ) \, dx=x \log \left (x\right )^{2} - x^{2} + x e^{\left (3 \, x^{2} + 24 \, x - 27\right )} - 10 \, x \log \left (x\right ) + 25 \, x \]
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Time = 0.09 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.28 \[ \int \left (15-2 x+e^{-27+24 x+3 x^2} \left (1+24 x+6 x^2\right )-8 \log (x)+\log ^2(x)\right ) \, dx=- x^{2} + x e^{3 x^{2} + 24 x - 27} + x \log {\left (x \right )}^{2} - 10 x \log {\left (x \right )} + 25 x \]
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Time = 0.20 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.56 \[ \int \left (15-2 x+e^{-27+24 x+3 x^2} \left (1+24 x+6 x^2\right )-8 \log (x)+\log ^2(x)\right ) \, dx={\left (\log \left (x\right )^{2} - 2 \, \log \left (x\right ) + 2\right )} x - x^{2} + x e^{\left (3 \, x^{2} + 24 \, x - 27\right )} - 8 \, x \log \left (x\right ) + 23 \, x \]
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Time = 0.25 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.32 \[ \int \left (15-2 x+e^{-27+24 x+3 x^2} \left (1+24 x+6 x^2\right )-8 \log (x)+\log ^2(x)\right ) \, dx=x \log \left (x\right )^{2} - x^{2} + x e^{\left (3 \, x^{2} + 24 \, x - 27\right )} - 10 \, x \log \left (x\right ) + 25 \, x \]
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Time = 9.35 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.48 \[ \int \left (15-2 x+e^{-27+24 x+3 x^2} \left (1+24 x+6 x^2\right )-8 \log (x)+\log ^2(x)\right ) \, dx=x\,{\mathrm {e}}^{-27}\,\left ({\mathrm {e}}^{27}\,{\ln \left (x\right )}^2-10\,{\mathrm {e}}^{27}\,\ln \left (x\right )+25\,{\mathrm {e}}^{27}+{\mathrm {e}}^{3\,x^2+24\,x}-x\,{\mathrm {e}}^{27}\right ) \]
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