Integrand size = 22, antiderivative size = 296 \[ \int \frac {1}{(1+2 x)^{5/2} \left (2+3 x+5 x^2\right )^2} \, dx=-\frac {820}{4557 (1+2 x)^{3/2}}-\frac {4680}{10633 \sqrt {1+2 x}}+\frac {37+20 x}{217 (1+2 x)^{3/2} \left (2+3 x+5 x^2\right )}+\frac {5 \sqrt {\frac {2}{217} \left (12504542+2632525 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )}{10633}-\frac {5 \sqrt {\frac {2}{217} \left (12504542+2632525 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}+10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )}{10633}-\frac {5 \sqrt {\frac {1}{434} \left (-12504542+2632525 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{10633}+\frac {5 \sqrt {\frac {1}{434} \left (-12504542+2632525 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{10633} \]
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Time = 0.33 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {754, 842, 840, 1183, 648, 632, 210, 642} \[ \int \frac {1}{(1+2 x)^{5/2} \left (2+3 x+5 x^2\right )^2} \, dx=\frac {5 \sqrt {\frac {2}{217} \left (12504542+2632525 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )}{10633}-\frac {5 \sqrt {\frac {2}{217} \left (12504542+2632525 \sqrt {35}\right )} \arctan \left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )}{10633}+\frac {20 x+37}{217 (2 x+1)^{3/2} \left (5 x^2+3 x+2\right )}-\frac {4680}{10633 \sqrt {2 x+1}}-\frac {820}{4557 (2 x+1)^{3/2}}-\frac {5 \sqrt {\frac {1}{434} \left (2632525 \sqrt {35}-12504542\right )} \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{10633}+\frac {5 \sqrt {\frac {1}{434} \left (2632525 \sqrt {35}-12504542\right )} \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{10633} \]
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Rule 210
Rule 632
Rule 642
Rule 648
Rule 754
Rule 840
Rule 842
Rule 1183
Rubi steps \begin{align*} \text {integral}& = \frac {37+20 x}{217 (1+2 x)^{3/2} \left (2+3 x+5 x^2\right )}+\frac {1}{217} \int \frac {255+100 x}{(1+2 x)^{5/2} \left (2+3 x+5 x^2\right )} \, dx \\ & = -\frac {820}{4557 (1+2 x)^{3/2}}+\frac {37+20 x}{217 (1+2 x)^{3/2} \left (2+3 x+5 x^2\right )}+\frac {\int \frac {145-2050 x}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )} \, dx}{1519} \\ & = -\frac {820}{4557 (1+2 x)^{3/2}}-\frac {4680}{10633 \sqrt {1+2 x}}+\frac {37+20 x}{217 (1+2 x)^{3/2} \left (2+3 x+5 x^2\right )}+\frac {\int \frac {-8345-11700 x}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )} \, dx}{10633} \\ & = -\frac {820}{4557 (1+2 x)^{3/2}}-\frac {4680}{10633 \sqrt {1+2 x}}+\frac {37+20 x}{217 (1+2 x)^{3/2} \left (2+3 x+5 x^2\right )}+\frac {2 \text {Subst}\left (\int \frac {-4990-11700 x^2}{7-4 x^2+5 x^4} \, dx,x,\sqrt {1+2 x}\right )}{10633} \\ & = -\frac {820}{4557 (1+2 x)^{3/2}}-\frac {4680}{10633 \sqrt {1+2 x}}+\frac {37+20 x}{217 (1+2 x)^{3/2} \left (2+3 x+5 x^2\right )}+\frac {\text {Subst}\left (\int \frac {-998 \sqrt {10 \left (2+\sqrt {35}\right )}-\left (-4990+2340 \sqrt {35}\right ) x}{\sqrt {\frac {7}{5}}-\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )} x+x^2} \, dx,x,\sqrt {1+2 x}\right )}{10633 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\text {Subst}\left (\int \frac {-998 \sqrt {10 \left (2+\sqrt {35}\right )}+\left (-4990+2340 \sqrt {35}\right ) x}{\sqrt {\frac {7}{5}}+\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )} x+x^2} \, dx,x,\sqrt {1+2 x}\right )}{10633 \sqrt {14 \left (2+\sqrt {35}\right )}} \\ & = -\frac {820}{4557 (1+2 x)^{3/2}}-\frac {4680}{10633 \sqrt {1+2 x}}+\frac {37+20 x}{217 (1+2 x)^{3/2} \left (2+3 x+5 x^2\right )}+\frac {\left (5 \left (499-234 \sqrt {35}\right )\right ) \text {Subst}\left (\int \frac {-\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+2 x}{\sqrt {\frac {7}{5}}-\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )} x+x^2} \, dx,x,\sqrt {1+2 x}\right )}{10633 \sqrt {14 \left (2+\sqrt {35}\right )}}-\frac {\left (5 \left (499-234 \sqrt {35}\right )\right ) \text {Subst}\left (\int \frac {\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+2 x}{\sqrt {\frac {7}{5}}+\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )} x+x^2} \, dx,x,\sqrt {1+2 x}\right )}{10633 \sqrt {14 \left (2+\sqrt {35}\right )}}-\frac {\left (8190+499 \sqrt {35}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {\frac {7}{5}}-\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )} x+x^2} \, dx,x,\sqrt {1+2 x}\right )}{74431}-\frac {\left (8190+499 \sqrt {35}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {\frac {7}{5}}+\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )} x+x^2} \, dx,x,\sqrt {1+2 x}\right )}{74431} \\ & = -\frac {820}{4557 (1+2 x)^{3/2}}-\frac {4680}{10633 \sqrt {1+2 x}}+\frac {37+20 x}{217 (1+2 x)^{3/2} \left (2+3 x+5 x^2\right )}-\frac {5 \sqrt {-\frac {6252271}{217}+\frac {376075 \sqrt {35}}{62}} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{10633}+\frac {5 \sqrt {-\frac {6252271}{217}+\frac {376075 \sqrt {35}}{62}} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{10633}+\frac {\left (2 \left (8190+499 \sqrt {35}\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {2}{5} \left (2-\sqrt {35}\right )-x^2} \, dx,x,-\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+2 \sqrt {1+2 x}\right )}{74431}+\frac {\left (2 \left (8190+499 \sqrt {35}\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {2}{5} \left (2-\sqrt {35}\right )-x^2} \, dx,x,\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+2 \sqrt {1+2 x}\right )}{74431} \\ & = -\frac {820}{4557 (1+2 x)^{3/2}}-\frac {4680}{10633 \sqrt {1+2 x}}+\frac {37+20 x}{217 (1+2 x)^{3/2} \left (2+3 x+5 x^2\right )}+\frac {5 \sqrt {\frac {2}{7 \left (-2+\sqrt {35}\right )}} \left (499+234 \sqrt {35}\right ) \tan ^{-1}\left (\sqrt {\frac {5}{2 \left (-2+\sqrt {35}\right )}} \left (\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}-2 \sqrt {1+2 x}\right )\right )}{10633}-\frac {5 \sqrt {\frac {2}{7 \left (-2+\sqrt {35}\right )}} \left (499+234 \sqrt {35}\right ) \tan ^{-1}\left (\sqrt {\frac {5}{2 \left (-2+\sqrt {35}\right )}} \left (\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+2 \sqrt {1+2 x}\right )\right )}{10633}-\frac {5 \sqrt {-\frac {6252271}{217}+\frac {376075 \sqrt {35}}{62}} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{10633}+\frac {5 \sqrt {-\frac {6252271}{217}+\frac {376075 \sqrt {35}}{62}} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{10633} \\ \end{align*}
Result contains complex when optimal does not.
Time = 1.35 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.48 \[ \int \frac {1}{(1+2 x)^{5/2} \left (2+3 x+5 x^2\right )^2} \, dx=\frac {2 \left (-\frac {217 \left (34121+112560 x+183140 x^2+140400 x^3\right )}{2 (1+2 x)^{3/2} \left (2+3 x+5 x^2\right )}-15 \sqrt {217 \left (12504542-1667459 i \sqrt {31}\right )} \arctan \left (\sqrt {\frac {1}{7} \left (-2-i \sqrt {31}\right )} \sqrt {1+2 x}\right )-15 \sqrt {217 \left (12504542+1667459 i \sqrt {31}\right )} \arctan \left (\sqrt {\frac {1}{7} i \left (2 i+\sqrt {31}\right )} \sqrt {1+2 x}\right )\right )}{6922083} \]
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Time = 0.80 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.19
method | result | size |
pseudoelliptic | \(\frac {1015350 \left (x +\frac {1}{2}\right ) \sqrt {1+2 x}\, \left (\sqrt {5}-\frac {9188 \sqrt {7}}{6769}\right ) \left (x^{2}+\frac {3}{5} x +\frac {2}{5}\right ) \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \ln \left (\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )-1015350 \left (x +\frac {1}{2}\right ) \sqrt {1+2 x}\, \left (\sqrt {5}-\frac {9188 \sqrt {7}}{6769}\right ) \left (x^{2}+\frac {3}{5} x +\frac {2}{5}\right ) \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )+9281400 \left (\arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )-\arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )\right ) \left (x +\frac {1}{2}\right ) \left (x^{2}+\frac {3}{5} x +\frac {2}{5}\right ) \left (\sqrt {5}\, \sqrt {7}+\frac {8190}{499}\right ) \sqrt {1+2 x}-60933600 \left (x^{3}+\frac {9157}{7020} x^{2}+\frac {469}{585} x +\frac {34121}{140400}\right ) \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \left (1+2 x \right )^{\frac {3}{2}} \left (69220830 x^{2}+41532498 x +27688332\right )}\) | \(352\) |
derivativedivides | \(-\frac {16}{147 \left (1+2 x \right )^{\frac {3}{2}}}-\frac {128}{343 \sqrt {1+2 x}}-\frac {16 \left (\frac {89 \left (1+2 x \right )^{\frac {3}{2}}}{62}+\frac {233 \sqrt {1+2 x}}{620}\right )}{343 \left (\left (1+2 x \right )^{2}+\frac {3}{5}-\frac {8 x}{5}\right )}-\frac {\left (-33845 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+45940 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{4614722}-\frac {10 \left (30938 \sqrt {5}\, \sqrt {7}+\frac {\left (-33845 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+45940 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{2307361 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {\left (33845 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-45940 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{4614722}-\frac {10 \left (30938 \sqrt {5}\, \sqrt {7}-\frac {\left (33845 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-45940 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{2307361 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\) | \(433\) |
default | \(-\frac {16}{147 \left (1+2 x \right )^{\frac {3}{2}}}-\frac {128}{343 \sqrt {1+2 x}}-\frac {16 \left (\frac {89 \left (1+2 x \right )^{\frac {3}{2}}}{62}+\frac {233 \sqrt {1+2 x}}{620}\right )}{343 \left (\left (1+2 x \right )^{2}+\frac {3}{5}-\frac {8 x}{5}\right )}-\frac {\left (-33845 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+45940 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{4614722}-\frac {10 \left (30938 \sqrt {5}\, \sqrt {7}+\frac {\left (-33845 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+45940 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{2307361 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {\left (33845 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-45940 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{4614722}-\frac {10 \left (30938 \sqrt {5}\, \sqrt {7}-\frac {\left (33845 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-45940 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{2307361 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\) | \(433\) |
trager | \(-\frac {140400 x^{3}+183140 x^{2}+112560 x +34121}{31899 \left (1+2 x \right )^{\frac {3}{2}} \left (5 x^{2}+3 x +2\right )}-\frac {10 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+3101126416 \textit {\_Z}^{2}+34650939378125\right ) \ln \left (\frac {-7577938592 x \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+3101126416 \textit {\_Z}^{2}+34650939378125\right )^{5}-308754136562536 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+3101126416 \textit {\_Z}^{2}+34650939378125\right )^{3} x +2286118004070180 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+3101126416 \textit {\_Z}^{2}+34650939378125\right )^{2} \sqrt {1+2 x}-116459925407168 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+3101126416 \textit {\_Z}^{2}+34650939378125\right )^{3}-2720797097744165060 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+3101126416 \textit {\_Z}^{2}+34650939378125\right ) x -170548595197901724125 \sqrt {1+2 x}-1501216389977029664 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+3101126416 \textit {\_Z}^{2}+34650939378125\right )}{868 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+3101126416 \textit {\_Z}^{2}+34650939378125\right )^{2} x +7502165 x -6669836}\right )}{10633}+\frac {10 \operatorname {RootOf}\left (\textit {\_Z}^{2}+47089 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+3101126416 \textit {\_Z}^{2}+34650939378125\right )^{2}+1356742807\right ) \ln \left (-\frac {-1082562656 \operatorname {RootOf}\left (\textit {\_Z}^{2}+47089 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+3101126416 \textit {\_Z}^{2}+34650939378125\right )^{2}+1356742807\right ) \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+3101126416 \textit {\_Z}^{2}+34650939378125\right )^{4} x -18274525189608 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+3101126416 \textit {\_Z}^{2}+34650939378125\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+47089 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+3101126416 \textit {\_Z}^{2}+34650939378125\right )^{2}+1356742807\right ) x +70869658126175580 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+3101126416 \textit {\_Z}^{2}+34650939378125\right )^{2} \sqrt {1+2 x}+16637132201024 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+3101126416 \textit {\_Z}^{2}+34650939378125\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+47089 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+3101126416 \textit {\_Z}^{2}+34650939378125\right )^{2}+1356742807\right )-16528108437548820 \operatorname {RootOf}\left (\textit {\_Z}^{2}+47089 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+3101126416 \textit {\_Z}^{2}+34650939378125\right )^{2}+1356742807\right ) x +7328924922481506073415 \sqrt {1+2 x}+264894705503873760 \operatorname {RootOf}\left (\textit {\_Z}^{2}+47089 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+3101126416 \textit {\_Z}^{2}+34650939378125\right )^{2}+1356742807\right )}{868 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+3101126416 \textit {\_Z}^{2}+34650939378125\right )^{2} x +17506919 x +6669836}\right )}{2307361}\) | \(458\) |
risch | \(-\frac {140400 x^{3}+183140 x^{2}+112560 x +34121}{31899 \left (1+2 x \right )^{\frac {3}{2}} \left (5 x^{2}+3 x +2\right )}+\frac {4835 \ln \left (\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{659246}-\frac {22970 \ln \left (\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right ) \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{2307361}+\frac {24175 \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right ) \left (2 \sqrt {5}\, \sqrt {7}+4\right )}{329623 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {45940 \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right ) \sqrt {5}\, \left (2 \sqrt {5}\, \sqrt {7}+4\right ) \sqrt {7}}{2307361 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {9980 \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right ) \sqrt {5}\, \sqrt {7}}{74431 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {4835 \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{659246}+\frac {22970 \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right ) \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{2307361}+\frac {24175 \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right ) \left (2 \sqrt {5}\, \sqrt {7}+4\right )}{329623 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {45940 \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right ) \sqrt {5}\, \left (2 \sqrt {5}\, \sqrt {7}+4\right ) \sqrt {7}}{2307361 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {9980 \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right ) \sqrt {5}\, \sqrt {7}}{74431 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\) | \(643\) |
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Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.04 \[ \int \frac {1}{(1+2 x)^{5/2} \left (2+3 x+5 x^2\right )^2} \, dx=-\frac {3 \, \sqrt {217} {\left (20 \, x^{4} + 32 \, x^{3} + 25 \, x^{2} + 11 \, x + 2\right )} \sqrt {166745900 i \, \sqrt {31} - 1250454200} \log \left (\sqrt {217} \sqrt {166745900 i \, \sqrt {31} - 1250454200} {\left (9188 i \, \sqrt {31} + 15469\right )} + 28562896250 \, \sqrt {2 \, x + 1}\right ) - 3 \, \sqrt {217} {\left (20 \, x^{4} + 32 \, x^{3} + 25 \, x^{2} + 11 \, x + 2\right )} \sqrt {166745900 i \, \sqrt {31} - 1250454200} \log \left (\sqrt {217} \sqrt {166745900 i \, \sqrt {31} - 1250454200} {\left (-9188 i \, \sqrt {31} - 15469\right )} + 28562896250 \, \sqrt {2 \, x + 1}\right ) - 3 \, \sqrt {217} {\left (20 \, x^{4} + 32 \, x^{3} + 25 \, x^{2} + 11 \, x + 2\right )} \sqrt {-166745900 i \, \sqrt {31} - 1250454200} \log \left (\sqrt {217} {\left (9188 i \, \sqrt {31} - 15469\right )} \sqrt {-166745900 i \, \sqrt {31} - 1250454200} + 28562896250 \, \sqrt {2 \, x + 1}\right ) + 3 \, \sqrt {217} {\left (20 \, x^{4} + 32 \, x^{3} + 25 \, x^{2} + 11 \, x + 2\right )} \sqrt {-166745900 i \, \sqrt {31} - 1250454200} \log \left (\sqrt {217} {\left (-9188 i \, \sqrt {31} + 15469\right )} \sqrt {-166745900 i \, \sqrt {31} - 1250454200} + 28562896250 \, \sqrt {2 \, x + 1}\right ) + 434 \, {\left (140400 \, x^{3} + 183140 \, x^{2} + 112560 \, x + 34121\right )} \sqrt {2 \, x + 1}}{13844166 \, {\left (20 \, x^{4} + 32 \, x^{3} + 25 \, x^{2} + 11 \, x + 2\right )}} \]
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\[ \int \frac {1}{(1+2 x)^{5/2} \left (2+3 x+5 x^2\right )^2} \, dx=\int \frac {1}{\left (2 x + 1\right )^{\frac {5}{2}} \left (5 x^{2} + 3 x + 2\right )^{2}}\, dx \]
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\[ \int \frac {1}{(1+2 x)^{5/2} \left (2+3 x+5 x^2\right )^2} \, dx=\int { \frac {1}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{2} {\left (2 \, x + 1\right )}^{\frac {5}{2}}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 638 vs. \(2 (205) = 410\).
Time = 0.53 (sec) , antiderivative size = 638, normalized size of antiderivative = 2.16 \[ \int \frac {1}{(1+2 x)^{5/2} \left (2+3 x+5 x^2\right )^2} \, dx=\text {Too large to display} \]
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Time = 0.19 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.76 \[ \int \frac {1}{(1+2 x)^{5/2} \left (2+3 x+5 x^2\right )^2} \, dx=-\frac {\frac {128\,x}{147}-\frac {5492\,{\left (2\,x+1\right )}^2}{31899}+\frac {4680\,{\left (2\,x+1\right )}^3}{10633}+\frac {144}{245}}{\frac {7\,{\left (2\,x+1\right )}^{3/2}}{5}-\frac {4\,{\left (2\,x+1\right )}^{5/2}}{5}+{\left (2\,x+1\right )}^{7/2}}+\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {-12504542-\sqrt {31}\,1667459{}\mathrm {i}}\,\sqrt {2\,x+1}\,6884992{}\mathrm {i}}{1900211000023\,\left (-\frac {63259306496}{271458714289}+\frac {\sqrt {31}\,3435611008{}\mathrm {i}}{271458714289}\right )}-\frac {13769984\,\sqrt {31}\,\sqrt {217}\,\sqrt {-12504542-\sqrt {31}\,1667459{}\mathrm {i}}\,\sqrt {2\,x+1}}{58906541000713\,\left (-\frac {63259306496}{271458714289}+\frac {\sqrt {31}\,3435611008{}\mathrm {i}}{271458714289}\right )}\right )\,\sqrt {-12504542-\sqrt {31}\,1667459{}\mathrm {i}}\,10{}\mathrm {i}}{2307361}-\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {-12504542+\sqrt {31}\,1667459{}\mathrm {i}}\,\sqrt {2\,x+1}\,6884992{}\mathrm {i}}{1900211000023\,\left (\frac {63259306496}{271458714289}+\frac {\sqrt {31}\,3435611008{}\mathrm {i}}{271458714289}\right )}+\frac {13769984\,\sqrt {31}\,\sqrt {217}\,\sqrt {-12504542+\sqrt {31}\,1667459{}\mathrm {i}}\,\sqrt {2\,x+1}}{58906541000713\,\left (\frac {63259306496}{271458714289}+\frac {\sqrt {31}\,3435611008{}\mathrm {i}}{271458714289}\right )}\right )\,\sqrt {-12504542+\sqrt {31}\,1667459{}\mathrm {i}}\,10{}\mathrm {i}}{2307361} \]
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