Integrand size = 22, antiderivative size = 300 \[ \int \frac {(1+2 x)^{3/2}}{\left (2+3 x+5 x^2\right )^3} \, dx=-\frac {(5-4 x) \sqrt {1+2 x}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (67+120 x)}{1922 \left (2+3 x+5 x^2\right )}-\frac {3}{961} \sqrt {\frac {1}{434} \left (15082+2705 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )+\frac {3}{961} \sqrt {\frac {1}{434} \left (15082+2705 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}+10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )-\frac {3 \sqrt {\frac {1}{434} \left (-15082+2705 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{1922}+\frac {3 \sqrt {\frac {1}{434} \left (-15082+2705 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{1922} \]
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Time = 0.30 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {752, 836, 840, 1183, 648, 632, 210, 642} \[ \int \frac {(1+2 x)^{3/2}}{\left (2+3 x+5 x^2\right )^3} \, dx=-\frac {3}{961} \sqrt {\frac {1}{434} \left (15082+2705 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )+\frac {3}{961} \sqrt {\frac {1}{434} \left (15082+2705 \sqrt {35}\right )} \arctan \left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )-\frac {\sqrt {2 x+1} (5-4 x)}{62 \left (5 x^2+3 x+2\right )^2}+\frac {\sqrt {2 x+1} (120 x+67)}{1922 \left (5 x^2+3 x+2\right )}-\frac {3 \sqrt {\frac {1}{434} \left (2705 \sqrt {35}-15082\right )} \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{1922}+\frac {3 \sqrt {\frac {1}{434} \left (2705 \sqrt {35}-15082\right )} \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{1922} \]
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Rule 210
Rule 632
Rule 642
Rule 648
Rule 752
Rule 836
Rule 840
Rule 1183
Rubi steps \begin{align*} \text {integral}& = -\frac {(5-4 x) \sqrt {1+2 x}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {1}{62} \int \frac {17+20 x}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )^2} \, dx \\ & = -\frac {(5-4 x) \sqrt {1+2 x}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (67+120 x)}{1922 \left (2+3 x+5 x^2\right )}+\frac {\int \frac {1239+840 x}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )} \, dx}{13454} \\ & = -\frac {(5-4 x) \sqrt {1+2 x}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (67+120 x)}{1922 \left (2+3 x+5 x^2\right )}+\frac {\text {Subst}\left (\int \frac {1638+840 x^2}{7-4 x^2+5 x^4} \, dx,x,\sqrt {1+2 x}\right )}{6727} \\ & = -\frac {(5-4 x) \sqrt {1+2 x}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (67+120 x)}{1922 \left (2+3 x+5 x^2\right )}+\frac {\text {Subst}\left (\int \frac {1638 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}-\left (1638-168 \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 )}{13454 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\text {Subst}\left (\int \frac {1638 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+\left (1638-168 \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 )}{13454 \sqrt {14 \left (2+\sqrt {35}\right )}} \\ & = -\frac {(5-4 x) \sqrt {1+2 x}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (67+120 x)}{1922 \left (2+3 x+5 x^2\right )}+\frac {\left (3 \left (140+39 \sqrt {35}\right )\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 )}{67270}+\frac {\left (3 \left (140+39 \sqrt {35}\right )\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 )}{67270}-\frac {\left (3 \sqrt {\frac {1}{434} \left (-15082+2705 \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 )}{1922}+\frac {\left (3 \sqrt {\frac {1}{434} \left (-15082+2705 \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 )}{1922} \\ & = -\frac {(5-4 x) \sqrt {1+2 x}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (67+120 x)}{1922 \left (2+3 x+5 x^2\right )}-\frac {3 \sqrt {\frac {1}{434} \left (-15082+2705 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{1922}+\frac {3 \sqrt {\frac {1}{434} \left (-15082+2705 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{1922}-\frac {\left (3 \left (140+39 \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 )}{33635}-\frac {\left (3 \left (140+39 \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 )}{33635} \\ & = -\frac {(5-4 x) \sqrt {1+2 x}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (67+120 x)}{1922 \left (2+3 x+5 x^2\right )}-\frac {3}{961} \sqrt {\frac {1}{434} \left (15082+2705 \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 {3}{961} \sqrt {\frac {1}{434} \left (15082+2705 \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 {3 \sqrt {\frac {1}{434} \left (-15082+2705 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{1922}+\frac {3 \sqrt {\frac {1}{434} \left (-15082+2705 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{1922} \\ \end{align*}
Result contains complex when optimal does not.
Time = 1.40 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.48 \[ \int \frac {(1+2 x)^{3/2}}{\left (2+3 x+5 x^2\right )^3} \, dx=\frac {\frac {217 \sqrt {1+2 x} \left (-21+565 x+695 x^2+600 x^3\right )}{2 \left (2+3 x+5 x^2\right )^2}+3 \sqrt {217 \left (15082+961 i \sqrt {31}\right )} \arctan \left (\sqrt {\frac {1}{7} \left (-2-i \sqrt {31}\right )} \sqrt {1+2 x}\right )+3 \sqrt {217 \left (15082-961 i \sqrt {31}\right )} \arctan \left (\sqrt {\frac {1}{7} i \left (2 i+\sqrt {31}\right )} \sqrt {1+2 x}\right )}{208537} \]
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Time = 0.69 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.09
method | result | size |
pseudoelliptic | \(-\frac {2925 \left (\frac {329 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \left (x^{2}+\frac {3}{5} x +\frac {2}{5}\right )^{2} \left (\sqrt {5}-\frac {218 \sqrt {7}}{329}\right ) \ln \left (\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{4836}-\frac {329 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \left (x^{2}+\frac {3}{5} x +\frac {2}{5}\right )^{2} \left (\sqrt {5}-\frac {218 \sqrt {7}}{329}\right ) \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{4836}-\frac {28 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \left (x^{3}+\frac {139}{120} x^{2}+\frac {113}{120} x -\frac {7}{200}\right ) \sqrt {1+2 x}}{39}+\left (\sqrt {5}\, \sqrt {7}+\frac {140}{39}\right ) \left (x^{2}+\frac {3}{5} x +\frac {2}{5}\right )^{2} \left (\arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )-\arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )\right )\right )}{6727 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \left (5 x^{2}+3 x +2\right )^{2}}\) | \(328\) |
derivativedivides | \(\frac {\frac {600 \left (1+2 x \right )^{\frac {7}{2}}}{961}-\frac {410 \left (1+2 x \right )^{\frac {5}{2}}}{961}+\frac {1280 \left (1+2 x \right )^{\frac {3}{2}}}{961}-\frac {1638 \sqrt {1+2 x}}{961}}{\left (5 \left (1+2 x \right )^{2}+3-8 x \right )^{2}}+\frac {3 \left (-1645 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+1090 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{4170740}+\frac {3 \left (2418 \sqrt {5}\, \sqrt {7}+\frac {\left (-1645 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+1090 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{208537 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {3 \left (1645 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-1090 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{4170740}+\frac {3 \left (2418 \sqrt {5}\, \sqrt {7}-\frac {\left (1645 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-1090 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{208537 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\) | \(435\) |
default | \(\frac {\frac {600 \left (1+2 x \right )^{\frac {7}{2}}}{961}-\frac {410 \left (1+2 x \right )^{\frac {5}{2}}}{961}+\frac {1280 \left (1+2 x \right )^{\frac {3}{2}}}{961}-\frac {1638 \sqrt {1+2 x}}{961}}{\left (5 \left (1+2 x \right )^{2}+3-8 x \right )^{2}}+\frac {3 \left (-1645 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+1090 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{4170740}+\frac {3 \left (2418 \sqrt {5}\, \sqrt {7}+\frac {\left (-1645 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+1090 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{208537 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {3 \left (1645 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-1090 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{4170740}+\frac {3 \left (2418 \sqrt {5}\, \sqrt {7}-\frac {\left (1645 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-1090 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{208537 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\) | \(435\) |
trager | \(\frac {\left (600 x^{3}+695 x^{2}+565 x -21\right ) \sqrt {1+2 x}}{1922 \left (5 x^{2}+3 x +2\right )^{2}}-\frac {6 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}+14961344 \textit {\_Z}^{2}+36585125\right ) \ln \left (\frac {7377527808 x \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}+14961344 \textit {\_Z}^{2}+36585125\right )^{5}+45663063488 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}+14961344 \textit {\_Z}^{2}+36585125\right )^{3} x -16335954432 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}+14961344 \textit {\_Z}^{2}+36585125\right )^{3}-5415288816 \sqrt {1+2 x}\, \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}+14961344 \textit {\_Z}^{2}+36585125\right )^{2}+67162035120 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}+14961344 \textit {\_Z}^{2}+36585125\right ) x -61800326272 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}+14961344 \textit {\_Z}^{2}+36585125\right )-92971296325 \sqrt {1+2 x}}{3472 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}+14961344 \textit {\_Z}^{2}+36585125\right )^{2} x +17965 x +3844}\right )}{961}+\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+188356 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}+14961344 \textit {\_Z}^{2}+36585125\right )^{2}+1636397\right ) \ln \left (-\frac {526966272 \operatorname {RootOf}\left (\textit {\_Z}^{2}+188356 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}+14961344 \textit {\_Z}^{2}+36585125\right )^{2}+1636397\right ) \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}+14961344 \textit {\_Z}^{2}+36585125\right )^{4} x +5894695136 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}+14961344 \textit {\_Z}^{2}+36585125\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+188356 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}+14961344 \textit {\_Z}^{2}+36585125\right )^{2}+1636397\right ) x +1166853888 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}+14961344 \textit {\_Z}^{2}+36585125\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+188356 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}+14961344 \textit {\_Z}^{2}+36585125\right )^{2}+1636397\right )-167873953296 \sqrt {1+2 x}\, \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}+14961344 \textit {\_Z}^{2}+36585125\right )^{2}+16234968544 \operatorname {RootOf}\left (\textit {\_Z}^{2}+188356 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}+14961344 \textit {\_Z}^{2}+36585125\right )^{2}+1636397\right ) x +5723070208 \operatorname {RootOf}\left (\textit {\_Z}^{2}+188356 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}+14961344 \textit {\_Z}^{2}+36585125\right )^{2}+1636397\right )+1423656866023 \sqrt {1+2 x}}{3472 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}+14961344 \textit {\_Z}^{2}+36585125\right )^{2} x +12199 x -3844}\right )}{208537}\) | \(458\) |
risch | \(\frac {\left (600 x^{3}+695 x^{2}+565 x -21\right ) \sqrt {1+2 x}}{1922 \left (5 x^{2}+3 x +2\right )^{2}}-\frac {141 \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}}{119164}+\frac {327 \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}}{417074}-\frac {705 \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 )}{59582 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {327 \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}}{208537 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {234 \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}}{6727 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {141 \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}}{119164}-\frac {327 \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}}{417074}-\frac {705 \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 )}{59582 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {327 \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}}{208537 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {234 \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}}{6727 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\) | \(643\) |
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Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.02 \[ \int \frac {(1+2 x)^{3/2}}{\left (2+3 x+5 x^2\right )^3} \, dx=-\frac {\sqrt {217} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {8649 i \, \sqrt {31} - 135738} \log \left (\sqrt {217} \sqrt {8649 i \, \sqrt {31} - 135738} {\left (218 i \, \sqrt {31} - 1209\right )} + 8804775 \, \sqrt {2 \, x + 1}\right ) - \sqrt {217} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {8649 i \, \sqrt {31} - 135738} \log \left (\sqrt {217} \sqrt {8649 i \, \sqrt {31} - 135738} {\left (-218 i \, \sqrt {31} + 1209\right )} + 8804775 \, \sqrt {2 \, x + 1}\right ) - \sqrt {217} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {-8649 i \, \sqrt {31} - 135738} \log \left (\sqrt {217} {\left (218 i \, \sqrt {31} + 1209\right )} \sqrt {-8649 i \, \sqrt {31} - 135738} + 8804775 \, \sqrt {2 \, x + 1}\right ) + \sqrt {217} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {-8649 i \, \sqrt {31} - 135738} \log \left (\sqrt {217} {\left (-218 i \, \sqrt {31} - 1209\right )} \sqrt {-8649 i \, \sqrt {31} - 135738} + 8804775 \, \sqrt {2 \, x + 1}\right ) - 217 \, {\left (600 \, x^{3} + 695 \, x^{2} + 565 \, x - 21\right )} \sqrt {2 \, x + 1}}{417074 \, {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )}} \]
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\[ \int \frac {(1+2 x)^{3/2}}{\left (2+3 x+5 x^2\right )^3} \, dx=\int \frac {\left (2 x + 1\right )^{\frac {3}{2}}}{\left (5 x^{2} + 3 x + 2\right )^{3}}\, dx \]
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\[ \int \frac {(1+2 x)^{3/2}}{\left (2+3 x+5 x^2\right )^3} \, dx=\int { \frac {{\left (2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{3}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 640 vs. \(2 (213) = 426\).
Time = 0.70 (sec) , antiderivative size = 640, normalized size of antiderivative = 2.13 \[ \int \frac {(1+2 x)^{3/2}}{\left (2+3 x+5 x^2\right )^3} \, dx=\text {Too large to display} \]
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Time = 10.44 (sec) , antiderivative size = 245, normalized size of antiderivative = 0.82 \[ \int \frac {(1+2 x)^{3/2}}{\left (2+3 x+5 x^2\right )^3} \, dx=\frac {\frac {1638\,\sqrt {2\,x+1}}{24025}-\frac {256\,{\left (2\,x+1\right )}^{3/2}}{4805}+\frac {82\,{\left (2\,x+1\right )}^{5/2}}{4805}-\frac {24\,{\left (2\,x+1\right )}^{7/2}}{961}}{\frac {112\,x}{25}-\frac {86\,{\left (2\,x+1\right )}^2}{25}+\frac {8\,{\left (2\,x+1\right )}^3}{5}-{\left (2\,x+1\right )}^4+\frac {7}{25}}+\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {-15082-\sqrt {31}\,961{}\mathrm {i}}\,\sqrt {2\,x+1}\,432{}\mathrm {i}}{5656566125\,\left (\frac {94176}{808080875}+\frac {\sqrt {31}\,16848{}\mathrm {i}}{808080875}\right )}+\frac {864\,\sqrt {31}\,\sqrt {217}\,\sqrt {-15082-\sqrt {31}\,961{}\mathrm {i}}\,\sqrt {2\,x+1}}{175353549875\,\left (\frac {94176}{808080875}+\frac {\sqrt {31}\,16848{}\mathrm {i}}{808080875}\right )}\right )\,\sqrt {-15082-\sqrt {31}\,961{}\mathrm {i}}\,3{}\mathrm {i}}{208537}-\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {-15082+\sqrt {31}\,961{}\mathrm {i}}\,\sqrt {2\,x+1}\,432{}\mathrm {i}}{5656566125\,\left (-\frac {94176}{808080875}+\frac {\sqrt {31}\,16848{}\mathrm {i}}{808080875}\right )}-\frac {864\,\sqrt {31}\,\sqrt {217}\,\sqrt {-15082+\sqrt {31}\,961{}\mathrm {i}}\,\sqrt {2\,x+1}}{175353549875\,\left (-\frac {94176}{808080875}+\frac {\sqrt {31}\,16848{}\mathrm {i}}{808080875}\right )}\right )\,\sqrt {-15082+\sqrt {31}\,961{}\mathrm {i}}\,3{}\mathrm {i}}{208537} \]
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