Integrand size = 22, antiderivative size = 279 \[ \int \frac {(1+2 x)^{7/2}}{2+3 x+5 x^2} \, dx=-\frac {76}{125} \sqrt {1+2 x}+\frac {16}{75} (1+2 x)^{3/2}+\frac {4}{25} (1+2 x)^{5/2}+\frac {1}{125} \sqrt {\frac {2}{155} \left (-168698+42875 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )-\frac {1}{125} \sqrt {\frac {2}{155} \left (-168698+42875 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}+10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )-\frac {1}{125} \sqrt {\frac {1}{310} \left (168698+42875 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )+\frac {1}{125} \sqrt {\frac {1}{310} \left (168698+42875 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right ) \]
[Out]
Time = 0.43 (sec) , antiderivative size = 279, 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 = {717, 838, 840, 1183, 648, 632, 210, 642} \[ \int \frac {(1+2 x)^{7/2}}{2+3 x+5 x^2} \, dx=\frac {1}{125} \sqrt {\frac {2}{155} \left (42875 \sqrt {35}-168698\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )-\frac {1}{125} \sqrt {\frac {2}{155} \left (42875 \sqrt {35}-168698\right )} \arctan \left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )+\frac {4}{25} (2 x+1)^{5/2}+\frac {16}{75} (2 x+1)^{3/2}-\frac {76}{125} \sqrt {2 x+1}-\frac {1}{125} \sqrt {\frac {1}{310} \left (168698+42875 \sqrt {35}\right )} \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )+\frac {1}{125} \sqrt {\frac {1}{310} \left (168698+42875 \sqrt {35}\right )} \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right ) \]
[In]
[Out]
Rule 210
Rule 632
Rule 642
Rule 648
Rule 717
Rule 838
Rule 840
Rule 1183
Rubi steps \begin{align*} \text {integral}& = \frac {4}{25} (1+2 x)^{5/2}+\frac {1}{5} \int \frac {(1+2 x)^{3/2} (-3+8 x)}{2+3 x+5 x^2} \, dx \\ & = \frac {16}{75} (1+2 x)^{3/2}+\frac {4}{25} (1+2 x)^{5/2}+\frac {1}{25} \int \frac {(-47-38 x) \sqrt {1+2 x}}{2+3 x+5 x^2} \, dx \\ & = -\frac {76}{125} \sqrt {1+2 x}+\frac {16}{75} (1+2 x)^{3/2}+\frac {4}{25} (1+2 x)^{5/2}+\frac {1}{125} \int \frac {-83-432 x}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )} \, dx \\ & = -\frac {76}{125} \sqrt {1+2 x}+\frac {16}{75} (1+2 x)^{3/2}+\frac {4}{25} (1+2 x)^{5/2}+\frac {2}{125} \text {Subst}\left (\int \frac {266-432 x^2}{7-4 x^2+5 x^4} \, dx,x,\sqrt {1+2 x}\right ) \\ & = -\frac {76}{125} \sqrt {1+2 x}+\frac {16}{75} (1+2 x)^{3/2}+\frac {4}{25} (1+2 x)^{5/2}+\frac {\text {Subst}\left (\int \frac {266 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}-\left (266+432 \sqrt {\frac {7}{5}}\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 )}{125 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\text {Subst}\left (\int \frac {266 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+\left (266+432 \sqrt {\frac {7}{5}}\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 )}{125 \sqrt {14 \left (2+\sqrt {35}\right )}} \\ & = -\frac {76}{125} \sqrt {1+2 x}+\frac {16}{75} (1+2 x)^{3/2}+\frac {4}{25} (1+2 x)^{5/2}+\frac {1}{625} \left (-216+19 \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 )+\frac {1}{625} \left (-216+19 \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 )-\frac {1}{125} \sqrt {\frac {1}{310} \left (168698+42875 \sqrt {35}\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 )+\frac {1}{125} \sqrt {\frac {1}{310} \left (168698+42875 \sqrt {35}\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 ) \\ & = -\frac {76}{125} \sqrt {1+2 x}+\frac {16}{75} (1+2 x)^{3/2}+\frac {4}{25} (1+2 x)^{5/2}-\frac {1}{125} \sqrt {\frac {1}{310} \left (168698+42875 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )+\frac {1}{125} \sqrt {\frac {1}{310} \left (168698+42875 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )+\frac {1}{625} \left (2 \left (216-19 \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 )+\frac {1}{625} \left (2 \left (216-19 \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 ) \\ & = -\frac {76}{125} \sqrt {1+2 x}+\frac {16}{75} (1+2 x)^{3/2}+\frac {4}{25} (1+2 x)^{5/2}+\frac {1}{125} \sqrt {\frac {2}{155} \left (-168698+42875 \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 )-\frac {1}{125} \sqrt {\frac {2}{155} \left (-168698+42875 \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 )-\frac {1}{125} \sqrt {\frac {1}{310} \left (168698+42875 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )+\frac {1}{125} \sqrt {\frac {1}{310} \left (168698+42875 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right ) \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.70 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.44 \[ \int \frac {(1+2 x)^{7/2}}{2+3 x+5 x^2} \, dx=\frac {1240 \sqrt {1+2 x} \left (-11+50 x+30 x^2\right )-6 \sqrt {155 \left (-168698-34021 i \sqrt {31}\right )} \arctan \left (\sqrt {\frac {1}{7} \left (-2-i \sqrt {31}\right )} \sqrt {1+2 x}\right )-6 \sqrt {155 \left (-168698+34021 i \sqrt {31}\right )} \arctan \left (\sqrt {\frac {1}{7} i \left (2 i+\sqrt {31}\right )} \sqrt {1+2 x}\right )}{58125} \]
[In]
[Out]
Time = 1.35 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.01
method | result | size |
pseudoelliptic | \(-\frac {38 \left (\frac {233 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \left (\sqrt {5}+\frac {890 \sqrt {7}}{233}\right ) \ln \left (\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{11780}-\frac {233 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \left (\sqrt {5}+\frac {890 \sqrt {7}}{233}\right ) \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{11780}-\frac {40 \left (x^{2}+\frac {5}{3} x -\frac {11}{30}\right ) \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \sqrt {1+2 x}}{19}+\left (\sqrt {5}\, \sqrt {7}-\frac {216}{19}\right ) \left (\arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )-\arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )\right )\right )}{125 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\) | \(281\) |
derivativedivides | \(\frac {4 \left (1+2 x \right )^{\frac {5}{2}}}{25}+\frac {16 \left (1+2 x \right )^{\frac {3}{2}}}{75}-\frac {76 \sqrt {1+2 x}}{125}+\frac {\left (-890 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-233 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{38750}+\frac {2 \left (1178 \sqrt {5}\, \sqrt {7}+\frac {\left (-890 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-233 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{3875 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {\left (890 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+233 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{38750}+\frac {2 \left (1178 \sqrt {5}\, \sqrt {7}-\frac {\left (890 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+233 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{3875 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\) | \(407\) |
default | \(\frac {4 \left (1+2 x \right )^{\frac {5}{2}}}{25}+\frac {16 \left (1+2 x \right )^{\frac {3}{2}}}{75}-\frac {76 \sqrt {1+2 x}}{125}+\frac {\left (-890 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-233 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{38750}+\frac {2 \left (1178 \sqrt {5}\, \sqrt {7}+\frac {\left (-890 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-233 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{3875 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {\left (890 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+233 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{38750}+\frac {2 \left (1178 \sqrt {5}\, \sqrt {7}-\frac {\left (890 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+233 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{3875 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\) | \(407\) |
trager | \(\left (\frac {16}{25} x^{2}+\frac {16}{15} x -\frac {88}{375}\right ) \sqrt {1+2 x}+\frac {\operatorname {RootOf}\left (4805 \textit {\_Z}^{4}-10459276 \textit {\_Z}^{2}+12867859375\right ) \ln \left (\frac {408425 x \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}-10459276 \textit {\_Z}^{2}+12867859375\right )^{5}-990852845 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}-10459276 \textit {\_Z}^{2}+12867859375\right )^{3} x +717162680 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}-10459276 \textit {\_Z}^{2}+12867859375\right )^{3}+7453543125 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}-10459276 \textit {\_Z}^{2}+12867859375\right )^{2} \sqrt {1+2 x}-525953389168 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}-10459276 \textit {\_Z}^{2}+12867859375\right ) x -2061191406976 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}-10459276 \textit {\_Z}^{2}+12867859375\right )-83912677753625 \sqrt {1+2 x}}{155 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}-10459276 \textit {\_Z}^{2}+12867859375\right )^{2} x -270761 x -136084}\right )}{125}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}-10459276 \textit {\_Z}^{2}+12867859375\right )^{2}-52296380\right ) \ln \left (-\frac {81685 \operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}-10459276 \textit {\_Z}^{2}+12867859375\right )^{2}-52296380\right ) \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}-10459276 \textit {\_Z}^{2}+12867859375\right )^{4} x -157444815 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}-10459276 \textit {\_Z}^{2}+12867859375\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}-10459276 \textit {\_Z}^{2}+12867859375\right )^{2}-52296380\right ) x -143432536 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}-10459276 \textit {\_Z}^{2}+12867859375\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}-10459276 \textit {\_Z}^{2}+12867859375\right )^{2}-52296380\right )+231059836875 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}-10459276 \textit {\_Z}^{2}+12867859375\right )^{2} \sqrt {1+2 x}-149515537500 \operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}-10459276 \textit {\_Z}^{2}+12867859375\right )^{2}-52296380\right ) x -100021740000 \operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}-10459276 \textit {\_Z}^{2}+12867859375\right )^{2}-52296380\right )+2098333883121875 \sqrt {1+2 x}}{155 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}-10459276 \textit {\_Z}^{2}+12867859375\right )^{2} x -66635 x +136084}\right )}{19375}\) | \(440\) |
risch | \(\frac {8 \left (30 x^{2}+50 x -11\right ) \sqrt {1+2 x}}{375}-\frac {89 \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}}{3875}-\frac {233 \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}}{38750}-\frac {178 \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}}{3875 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {233 \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 )}{3875 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {76 \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}}{125 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {89 \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}}{3875}+\frac {233 \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}}{38750}-\frac {178 \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}}{3875 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {233 \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 )}{3875 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {76 \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}}{125 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\) | \(626\) |
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.72 \[ \int \frac {(1+2 x)^{7/2}}{2+3 x+5 x^2} \, dx=-\frac {1}{38750} \, \sqrt {155} \sqrt {136084 i \, \sqrt {31} + 674792} \log \left (\sqrt {155} \sqrt {136084 i \, \sqrt {31} + 674792} {\left (178 i \, \sqrt {31} - 589\right )} + 13291250 \, \sqrt {2 \, x + 1}\right ) + \frac {1}{38750} \, \sqrt {155} \sqrt {136084 i \, \sqrt {31} + 674792} \log \left (\sqrt {155} \sqrt {136084 i \, \sqrt {31} + 674792} {\left (-178 i \, \sqrt {31} + 589\right )} + 13291250 \, \sqrt {2 \, x + 1}\right ) + \frac {1}{38750} \, \sqrt {155} \sqrt {-136084 i \, \sqrt {31} + 674792} \log \left (\sqrt {155} {\left (178 i \, \sqrt {31} + 589\right )} \sqrt {-136084 i \, \sqrt {31} + 674792} + 13291250 \, \sqrt {2 \, x + 1}\right ) - \frac {1}{38750} \, \sqrt {155} \sqrt {-136084 i \, \sqrt {31} + 674792} \log \left (\sqrt {155} {\left (-178 i \, \sqrt {31} - 589\right )} \sqrt {-136084 i \, \sqrt {31} + 674792} + 13291250 \, \sqrt {2 \, x + 1}\right ) + \frac {8}{375} \, {\left (30 \, x^{2} + 50 \, x - 11\right )} \sqrt {2 \, x + 1} \]
[In]
[Out]
\[ \int \frac {(1+2 x)^{7/2}}{2+3 x+5 x^2} \, dx=\int \frac {\left (2 x + 1\right )^{\frac {7}{2}}}{5 x^{2} + 3 x + 2}\, dx \]
[In]
[Out]
\[ \int \frac {(1+2 x)^{7/2}}{2+3 x+5 x^2} \, dx=\int { \frac {{\left (2 \, x + 1\right )}^{\frac {7}{2}}}{5 \, x^{2} + 3 \, x + 2} \,d x } \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 614 vs. \(2 (188) = 376\).
Time = 0.70 (sec) , antiderivative size = 614, normalized size of antiderivative = 2.20 \[ \int \frac {(1+2 x)^{7/2}}{2+3 x+5 x^2} \, dx=\text {Too large to display} \]
[In]
[Out]
Time = 9.90 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.72 \[ \int \frac {(1+2 x)^{7/2}}{2+3 x+5 x^2} \, dx=\frac {16\,{\left (2\,x+1\right )}^{3/2}}{75}-\frac {76\,\sqrt {2\,x+1}}{125}+\frac {4\,{\left (2\,x+1\right )}^{5/2}}{25}+\frac {\sqrt {155}\,\mathrm {atan}\left (\frac {\sqrt {155}\,\sqrt {168698+\sqrt {31}\,34021{}\mathrm {i}}\,\sqrt {2\,x+1}\,4354688{}\mathrm {i}}{6103515625\,\left (-\frac {5425941248}{1220703125}+\frac {\sqrt {31}\,579173504{}\mathrm {i}}{1220703125}\right )}+\frac {8709376\,\sqrt {31}\,\sqrt {155}\,\sqrt {168698+\sqrt {31}\,34021{}\mathrm {i}}\,\sqrt {2\,x+1}}{189208984375\,\left (-\frac {5425941248}{1220703125}+\frac {\sqrt {31}\,579173504{}\mathrm {i}}{1220703125}\right )}\right )\,\sqrt {168698+\sqrt {31}\,34021{}\mathrm {i}}\,2{}\mathrm {i}}{19375}-\frac {\sqrt {155}\,\mathrm {atan}\left (\frac {\sqrt {155}\,\sqrt {168698-\sqrt {31}\,34021{}\mathrm {i}}\,\sqrt {2\,x+1}\,4354688{}\mathrm {i}}{6103515625\,\left (\frac {5425941248}{1220703125}+\frac {\sqrt {31}\,579173504{}\mathrm {i}}{1220703125}\right )}-\frac {8709376\,\sqrt {31}\,\sqrt {155}\,\sqrt {168698-\sqrt {31}\,34021{}\mathrm {i}}\,\sqrt {2\,x+1}}{189208984375\,\left (\frac {5425941248}{1220703125}+\frac {\sqrt {31}\,579173504{}\mathrm {i}}{1220703125}\right )}\right )\,\sqrt {168698-\sqrt {31}\,34021{}\mathrm {i}}\,2{}\mathrm {i}}{19375} \]
[In]
[Out]