Integrand size = 116, antiderivative size = 23 \[ \int \frac {125+2100 x+11835 x^2+22792 x^3+2367 x^4+84 x^5+x^6+e^{\frac {12544+896 x+16 x^2}{25+280 x+794 x^2+56 x^3+x^4}} \left (697984 x+75104 x^2+2688 x^3+32 x^4\right )}{125 x+2100 x^2+11835 x^3+22792 x^4+2367 x^5+84 x^6+x^7} \, dx=\log \left (8 e^{-e^{\frac {16}{\left (x+\frac {5}{28+x}\right )^2}}} x\right ) \]
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\[ \int \frac {125+2100 x+11835 x^2+22792 x^3+2367 x^4+84 x^5+x^6+e^{\frac {12544+896 x+16 x^2}{25+280 x+794 x^2+56 x^3+x^4}} \left (697984 x+75104 x^2+2688 x^3+32 x^4\right )}{125 x+2100 x^2+11835 x^3+22792 x^4+2367 x^5+84 x^6+x^7} \, dx=\int \frac {125+2100 x+11835 x^2+22792 x^3+2367 x^4+84 x^5+x^6+e^{\frac {12544+896 x+16 x^2}{25+280 x+794 x^2+56 x^3+x^4}} \left (697984 x+75104 x^2+2688 x^3+32 x^4\right )}{125 x+2100 x^2+11835 x^3+22792 x^4+2367 x^5+84 x^6+x^7} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {125+2100 x+11835 x^2+22792 x^3+2367 x^4+84 x^5+x^6+32 e^{\frac {16 (28+x)^2}{\left (5+28 x+x^2\right )^2}} x \left (21812+2347 x+84 x^2+x^3\right )}{x \left (5+28 x+x^2\right )^3} \, dx \\ & = \int \left (\frac {2100}{\left (5+28 x+x^2\right )^3}+\frac {125}{x \left (5+28 x+x^2\right )^3}+\frac {11835 x}{\left (5+28 x+x^2\right )^3}+\frac {22792 x^2}{\left (5+28 x+x^2\right )^3}+\frac {2367 x^3}{\left (5+28 x+x^2\right )^3}+\frac {84 x^4}{\left (5+28 x+x^2\right )^3}+\frac {x^5}{\left (5+28 x+x^2\right )^3}+\frac {32 e^{\frac {16 (28+x)^2}{\left (5+28 x+x^2\right )^2}} (28+x) \left (779+56 x+x^2\right )}{\left (5+28 x+x^2\right )^3}\right ) \, dx \\ & = 32 \int \frac {e^{\frac {16 (28+x)^2}{\left (5+28 x+x^2\right )^2}} (28+x) \left (779+56 x+x^2\right )}{\left (5+28 x+x^2\right )^3} \, dx+84 \int \frac {x^4}{\left (5+28 x+x^2\right )^3} \, dx+125 \int \frac {1}{x \left (5+28 x+x^2\right )^3} \, dx+2100 \int \frac {1}{\left (5+28 x+x^2\right )^3} \, dx+2367 \int \frac {x^3}{\left (5+28 x+x^2\right )^3} \, dx+11835 \int \frac {x}{\left (5+28 x+x^2\right )^3} \, dx+22792 \int \frac {x^2}{\left (5+28 x+x^2\right )^3} \, dx+\int \frac {x^5}{\left (5+28 x+x^2\right )^3} \, dx \\ & = -\frac {525 (14+x)}{191 \left (5+28 x+x^2\right )^2}-\frac {2367 x^3 (14+x)}{764 \left (5+28 x+x^2\right )^2}+\frac {11835 (5+14 x)}{764 \left (5+28 x+x^2\right )^2}+\frac {5698 x (5+14 x)}{191 \left (5+28 x+x^2\right )^2}+\frac {21 x^3 (5+14 x)}{191 \left (5+28 x+x^2\right )^2}+\frac {x^4 (5+14 x)}{764 \left (5+28 x+x^2\right )^2}+\frac {25 (387+14 x)}{764 \left (5+28 x+x^2\right )^2}-\frac {\int \frac {x^3 (40+28 x)}{\left (5+28 x+x^2\right )^2} \, dx}{1528}-\frac {25 \int \frac {-1528-84 x}{x \left (5+28 x+x^2\right )^2} \, dx}{1528}-\frac {315}{191} \int \frac {x^2}{\left (5+28 x+x^2\right )^2} \, dx-\frac {1575}{191} \int \frac {1}{\left (5+28 x+x^2\right )^2} \, dx-\frac {2849}{191} \int \frac {10-56 x}{\left (5+28 x+x^2\right )^2} \, dx+32 \int \left (\frac {2 e^{\frac {16 (28+x)^2}{\left (5+28 x+x^2\right )^2}} (10766+387 x)}{\left (5+28 x+x^2\right )^3}+\frac {e^{\frac {16 (28+x)^2}{\left (5+28 x+x^2\right )^2}} (56+x)}{\left (5+28 x+x^2\right )^2}\right ) \, dx+\frac {49707}{382} \int \frac {x^2}{\left (5+28 x+x^2\right )^2} \, dx+\frac {248535}{382} \int \frac {1}{\left (5+28 x+x^2\right )^2} \, dx \\ & = -\frac {525 (14+x)}{191 \left (5+28 x+x^2\right )^2}-\frac {2367 x^3 (14+x)}{764 \left (5+28 x+x^2\right )^2}+\frac {11835 (5+14 x)}{764 \left (5+28 x+x^2\right )^2}+\frac {5698 x (5+14 x)}{191 \left (5+28 x+x^2\right )^2}+\frac {21 x^3 (5+14 x)}{191 \left (5+28 x+x^2\right )^2}+\frac {x^4 (5+14 x)}{764 \left (5+28 x+x^2\right )^2}+\frac {25 (387+14 x)}{764 \left (5+28 x+x^2\right )^2}-\frac {245385 (14+x)}{145924 \left (5+28 x+x^2\right )}+\frac {49077 x (5+14 x)}{145924 \left (5+28 x+x^2\right )}+\frac {2849 (210+397 x)}{36481 \left (5+28 x+x^2\right )}+\frac {x^2 (440+2569 x)}{145924 \left (5+28 x+x^2\right )}+\frac {5 (146364+5243 x)}{145924 \left (5+28 x+x^2\right )}-\frac {\int \frac {x (7040+41944 x)}{5+28 x+x^2} \, dx}{1167392}+\frac {5 \int \frac {1167392+41944 x}{x \left (5+28 x+x^2\right )} \, dx}{1167392}+2 \frac {1575 \int \frac {1}{5+28 x+x^2} \, dx}{72962}-2 \frac {248535 \int \frac {1}{5+28 x+x^2} \, dx}{145924}+\frac {1131053 \int \frac {1}{5+28 x+x^2} \, dx}{36481}+32 \int \frac {e^{\frac {16 (28+x)^2}{\left (5+28 x+x^2\right )^2}} (56+x)}{\left (5+28 x+x^2\right )^2} \, dx+64 \int \frac {e^{\frac {16 (28+x)^2}{\left (5+28 x+x^2\right )^2}} (10766+387 x)}{\left (5+28 x+x^2\right )^3} \, dx \\ & = -\frac {5243 x}{145924}-\frac {525 (14+x)}{191 \left (5+28 x+x^2\right )^2}-\frac {2367 x^3 (14+x)}{764 \left (5+28 x+x^2\right )^2}+\frac {11835 (5+14 x)}{764 \left (5+28 x+x^2\right )^2}+\frac {5698 x (5+14 x)}{191 \left (5+28 x+x^2\right )^2}+\frac {21 x^3 (5+14 x)}{191 \left (5+28 x+x^2\right )^2}+\frac {x^4 (5+14 x)}{764 \left (5+28 x+x^2\right )^2}+\frac {25 (387+14 x)}{764 \left (5+28 x+x^2\right )^2}-\frac {245385 (14+x)}{145924 \left (5+28 x+x^2\right )}+\frac {49077 x (5+14 x)}{145924 \left (5+28 x+x^2\right )}+\frac {2849 (210+397 x)}{36481 \left (5+28 x+x^2\right )}+\frac {x^2 (440+2569 x)}{145924 \left (5+28 x+x^2\right )}+\frac {5 (146364+5243 x)}{145924 \left (5+28 x+x^2\right )}-\frac {\int \frac {-209720-1167392 x}{5+28 x+x^2} \, dx}{1167392}+\frac {5 \int \left (\frac {1167392}{5 x}-\frac {8 (4059657+145924 x)}{5 \left (5+28 x+x^2\right )}\right ) \, dx}{1167392}-2 \frac {1575 \text {Subst}\left (\int \frac {1}{764-x^2} \, dx,x,28+2 x\right )}{36481}+2 \frac {248535 \text {Subst}\left (\int \frac {1}{764-x^2} \, dx,x,28+2 x\right )}{72962}+32 \int \left (\frac {56 e^{\frac {16 (28+x)^2}{\left (5+28 x+x^2\right )^2}}}{\left (5+28 x+x^2\right )^2}+\frac {e^{\frac {16 (28+x)^2}{\left (5+28 x+x^2\right )^2}} x}{\left (5+28 x+x^2\right )^2}\right ) \, dx-\frac {2262106 \text {Subst}\left (\int \frac {1}{764-x^2} \, dx,x,28+2 x\right )}{36481}+64 \int \left (\frac {10766 e^{\frac {16 (28+x)^2}{\left (5+28 x+x^2\right )^2}}}{\left (5+28 x+x^2\right )^3}+\frac {387 e^{\frac {16 (28+x)^2}{\left (5+28 x+x^2\right )^2}} x}{\left (5+28 x+x^2\right )^3}\right ) \, dx \\ & = -\frac {5243 x}{145924}-\frac {525 (14+x)}{191 \left (5+28 x+x^2\right )^2}-\frac {2367 x^3 (14+x)}{764 \left (5+28 x+x^2\right )^2}+\frac {11835 (5+14 x)}{764 \left (5+28 x+x^2\right )^2}+\frac {5698 x (5+14 x)}{191 \left (5+28 x+x^2\right )^2}+\frac {21 x^3 (5+14 x)}{191 \left (5+28 x+x^2\right )^2}+\frac {x^4 (5+14 x)}{764 \left (5+28 x+x^2\right )^2}+\frac {25 (387+14 x)}{764 \left (5+28 x+x^2\right )^2}-\frac {245385 (14+x)}{145924 \left (5+28 x+x^2\right )}+\frac {49077 x (5+14 x)}{145924 \left (5+28 x+x^2\right )}+\frac {2849 (210+397 x)}{36481 \left (5+28 x+x^2\right )}+\frac {x^2 (440+2569 x)}{145924 \left (5+28 x+x^2\right )}+\frac {5 (146364+5243 x)}{145924 \left (5+28 x+x^2\right )}-\frac {2016721 \text {arctanh}\left (\frac {14+x}{\sqrt {191}}\right )}{72962 \sqrt {191}}+\log (x)-\frac {\int \frac {4059657+145924 x}{5+28 x+x^2} \, dx}{145924}+32 \int \frac {e^{\frac {16 (28+x)^2}{\left (5+28 x+x^2\right )^2}} x}{\left (5+28 x+x^2\right )^2} \, dx+1792 \int \frac {e^{\frac {16 (28+x)^2}{\left (5+28 x+x^2\right )^2}}}{\left (5+28 x+x^2\right )^2} \, dx+24768 \int \frac {e^{\frac {16 (28+x)^2}{\left (5+28 x+x^2\right )^2}} x}{\left (5+28 x+x^2\right )^3} \, dx+689024 \int \frac {e^{\frac {16 (28+x)^2}{\left (5+28 x+x^2\right )^2}}}{\left (5+28 x+x^2\right )^3} \, dx-\frac {\left (-27871484+2016721 \sqrt {191}\right ) \int \frac {1}{14-\sqrt {191}+x} \, dx}{55742968}+\frac {\left (27871484+2016721 \sqrt {191}\right ) \int \frac {1}{14+\sqrt {191}+x} \, dx}{55742968} \\ & = \text {Too large to display} \\ \end{align*}
Time = 0.55 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {125+2100 x+11835 x^2+22792 x^3+2367 x^4+84 x^5+x^6+e^{\frac {12544+896 x+16 x^2}{25+280 x+794 x^2+56 x^3+x^4}} \left (697984 x+75104 x^2+2688 x^3+32 x^4\right )}{125 x+2100 x^2+11835 x^3+22792 x^4+2367 x^5+84 x^6+x^7} \, dx=-e^{\frac {16 (28+x)^2}{\left (5+28 x+x^2\right )^2}}+\log (x) \]
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Time = 30.02 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04
method | result | size |
risch | \(\ln \left (x \right )-{\mathrm e}^{\frac {16 \left (x +28\right )^{2}}{\left (x^{2}+28 x +5\right )^{2}}}\) | \(24\) |
parallelrisch | \(\ln \left (x \right )-{\mathrm e}^{\frac {16 x^{2}+896 x +12544}{x^{4}+56 x^{3}+794 x^{2}+280 x +25}}\) | \(37\) |
norman | \(\frac {-280 x \,{\mathrm e}^{\frac {16 x^{2}+896 x +12544}{x^{4}+56 x^{3}+794 x^{2}+280 x +25}}-794 x^{2} {\mathrm e}^{\frac {16 x^{2}+896 x +12544}{x^{4}+56 x^{3}+794 x^{2}+280 x +25}}-56 x^{3} {\mathrm e}^{\frac {16 x^{2}+896 x +12544}{x^{4}+56 x^{3}+794 x^{2}+280 x +25}}-x^{4} {\mathrm e}^{\frac {16 x^{2}+896 x +12544}{x^{4}+56 x^{3}+794 x^{2}+280 x +25}}-25 \,{\mathrm e}^{\frac {16 x^{2}+896 x +12544}{x^{4}+56 x^{3}+794 x^{2}+280 x +25}}}{\left (x^{2}+28 x +5\right )^{2}}+\ln \left (x \right )\) | \(196\) |
parts | \(\frac {-280 x \,{\mathrm e}^{\frac {16 x^{2}+896 x +12544}{x^{4}+56 x^{3}+794 x^{2}+280 x +25}}-794 x^{2} {\mathrm e}^{\frac {16 x^{2}+896 x +12544}{x^{4}+56 x^{3}+794 x^{2}+280 x +25}}-56 x^{3} {\mathrm e}^{\frac {16 x^{2}+896 x +12544}{x^{4}+56 x^{3}+794 x^{2}+280 x +25}}-x^{4} {\mathrm e}^{\frac {16 x^{2}+896 x +12544}{x^{4}+56 x^{3}+794 x^{2}+280 x +25}}-25 \,{\mathrm e}^{\frac {16 x^{2}+896 x +12544}{x^{4}+56 x^{3}+794 x^{2}+280 x +25}}}{\left (x^{2}+28 x +5\right )^{2}}+\ln \left (x \right )\) | \(196\) |
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Time = 0.25 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.57 \[ \int \frac {125+2100 x+11835 x^2+22792 x^3+2367 x^4+84 x^5+x^6+e^{\frac {12544+896 x+16 x^2}{25+280 x+794 x^2+56 x^3+x^4}} \left (697984 x+75104 x^2+2688 x^3+32 x^4\right )}{125 x+2100 x^2+11835 x^3+22792 x^4+2367 x^5+84 x^6+x^7} \, dx=-e^{\left (\frac {16 \, {\left (x^{2} + 56 \, x + 784\right )}}{x^{4} + 56 \, x^{3} + 794 \, x^{2} + 280 \, x + 25}\right )} + \log \left (x\right ) \]
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Time = 0.11 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.39 \[ \int \frac {125+2100 x+11835 x^2+22792 x^3+2367 x^4+84 x^5+x^6+e^{\frac {12544+896 x+16 x^2}{25+280 x+794 x^2+56 x^3+x^4}} \left (697984 x+75104 x^2+2688 x^3+32 x^4\right )}{125 x+2100 x^2+11835 x^3+22792 x^4+2367 x^5+84 x^6+x^7} \, dx=- e^{\frac {16 x^{2} + 896 x + 12544}{x^{4} + 56 x^{3} + 794 x^{2} + 280 x + 25}} + \log {\left (x \right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 323 vs. \(2 (21) = 42\).
Time = 2.65 (sec) , antiderivative size = 323, normalized size of antiderivative = 14.04 \[ \int \frac {125+2100 x+11835 x^2+22792 x^3+2367 x^4+84 x^5+x^6+e^{\frac {12544+896 x+16 x^2}{25+280 x+794 x^2+56 x^3+x^4}} \left (697984 x+75104 x^2+2688 x^3+32 x^4\right )}{125 x+2100 x^2+11835 x^3+22792 x^4+2367 x^5+84 x^6+x^7} \, dx=\frac {8197959 \, x^{3} + 173437274 \, x^{2} + 61547605 \, x + 5507025}{145924 \, {\left (x^{4} + 56 \, x^{3} + 794 \, x^{2} + 280 \, x + 25\right )}} - \frac {21 \, {\left (291773 \, x^{3} + 4082722 \, x^{2} + 1439015 \, x + 128450\right )}}{72962 \, {\left (x^{4} + 56 \, x^{3} + 794 \, x^{2} + 280 \, x + 25\right )}} + \frac {5 \, {\left (5243 \, x^{3} + 293168 \, x^{2} + 4137777 \, x + 1101405\right )}}{145924 \, {\left (x^{4} + 56 \, x^{3} + 794 \, x^{2} + 280 \, x + 25\right )}} + \frac {2849 \, {\left (397 \, x^{3} + 16674 \, x^{2} + 9775 \, x + 1050\right )}}{36481 \, {\left (x^{4} + 56 \, x^{3} + 794 \, x^{2} + 280 \, x + 25\right )}} - \frac {2367 \, {\left (105 \, x^{3} + 77372 \, x^{2} + 28315 \, x + 2575\right )}}{145924 \, {\left (x^{4} + 56 \, x^{3} + 794 \, x^{2} + 280 \, x + 25\right )}} - \frac {11835 \, {\left (21 \, x^{3} + 882 \, x^{2} + 5663 \, x + 515\right )}}{145924 \, {\left (x^{4} + 56 \, x^{3} + 794 \, x^{2} + 280 \, x + 25\right )}} + \frac {525 \, {\left (3 \, x^{3} + 126 \, x^{2} + 809 \, x - 5138\right )}}{72962 \, {\left (x^{4} + 56 \, x^{3} + 794 \, x^{2} + 280 \, x + 25\right )}} - e^{\left (\frac {448 \, x}{x^{4} + 56 \, x^{3} + 794 \, x^{2} + 280 \, x + 25} + \frac {12464}{x^{4} + 56 \, x^{3} + 794 \, x^{2} + 280 \, x + 25} + \frac {16}{x^{2} + 28 \, x + 5}\right )} + \log \left (x\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 77 vs. \(2 (21) = 42\).
Time = 0.40 (sec) , antiderivative size = 77, normalized size of antiderivative = 3.35 \[ \int \frac {125+2100 x+11835 x^2+22792 x^3+2367 x^4+84 x^5+x^6+e^{\frac {12544+896 x+16 x^2}{25+280 x+794 x^2+56 x^3+x^4}} \left (697984 x+75104 x^2+2688 x^3+32 x^4\right )}{125 x+2100 x^2+11835 x^3+22792 x^4+2367 x^5+84 x^6+x^7} \, dx=-e^{\left (\frac {16 \, x^{2}}{x^{4} + 56 \, x^{3} + 794 \, x^{2} + 280 \, x + 25} + \frac {896 \, x}{x^{4} + 56 \, x^{3} + 794 \, x^{2} + 280 \, x + 25} + \frac {12544}{x^{4} + 56 \, x^{3} + 794 \, x^{2} + 280 \, x + 25}\right )} + \log \left (x\right ) \]
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Time = 13.36 (sec) , antiderivative size = 78, normalized size of antiderivative = 3.39 \[ \int \frac {125+2100 x+11835 x^2+22792 x^3+2367 x^4+84 x^5+x^6+e^{\frac {12544+896 x+16 x^2}{25+280 x+794 x^2+56 x^3+x^4}} \left (697984 x+75104 x^2+2688 x^3+32 x^4\right )}{125 x+2100 x^2+11835 x^3+22792 x^4+2367 x^5+84 x^6+x^7} \, dx=\ln \left (x\right )-{\mathrm {e}}^{\frac {896\,x}{x^4+56\,x^3+794\,x^2+280\,x+25}}\,{\mathrm {e}}^{\frac {16\,x^2}{x^4+56\,x^3+794\,x^2+280\,x+25}}\,{\mathrm {e}}^{\frac {12544}{x^4+56\,x^3+794\,x^2+280\,x+25}} \]
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