Integrand size = 22, antiderivative size = 313 \[ \int \frac {1}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )^3} \, dx=-\frac {81090}{329623 \sqrt {1+2 x}}+\frac {37+20 x}{434 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )^2}+\frac {5 (2329+2080 x)}{94178 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )}-\frac {15 \sqrt {\frac {1}{434} \left (-2257111762+387427075 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )}{329623}+\frac {15 \sqrt {\frac {1}{434} \left (-2257111762+387427075 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}+10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )}{329623}-\frac {15 \sqrt {\frac {1}{434} \left (2257111762+387427075 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{659246}+\frac {15 \sqrt {\frac {1}{434} \left (2257111762+387427075 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{659246} \]
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Time = 0.31 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {754, 836, 842, 840, 1183, 648, 632, 210, 642} \[ \int \frac {1}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )^3} \, dx=-\frac {15 \sqrt {\frac {1}{434} \left (387427075 \sqrt {35}-2257111762\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )}{329623}+\frac {15 \sqrt {\frac {1}{434} \left (387427075 \sqrt {35}-2257111762\right )} \arctan \left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )}{329623}+\frac {20 x+37}{434 \sqrt {2 x+1} \left (5 x^2+3 x+2\right )^2}+\frac {5 (2080 x+2329)}{94178 \sqrt {2 x+1} \left (5 x^2+3 x+2\right )}-\frac {81090}{329623 \sqrt {2 x+1}}-\frac {15 \sqrt {\frac {1}{434} \left (2257111762+387427075 \sqrt {35}\right )} \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{659246}+\frac {15 \sqrt {\frac {1}{434} \left (2257111762+387427075 \sqrt {35}\right )} \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{659246} \]
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Rule 210
Rule 632
Rule 642
Rule 648
Rule 754
Rule 836
Rule 840
Rule 842
Rule 1183
Rubi steps \begin{align*} \text {integral}& = \frac {37+20 x}{434 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )^2}+\frac {1}{434} \int \frac {345+140 x}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )^2} \, dx \\ & = \frac {37+20 x}{434 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )^2}+\frac {5 (2329+2080 x)}{94178 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )}+\frac {\int \frac {56145+31200 x}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )} \, dx}{94178} \\ & = -\frac {81090}{329623 \sqrt {1+2 x}}+\frac {37+20 x}{434 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )^2}+\frac {5 (2329+2080 x)}{94178 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )}+\frac {\int \frac {68655-405450 x}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )} \, dx}{659246} \\ & = -\frac {81090}{329623 \sqrt {1+2 x}}+\frac {37+20 x}{434 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )^2}+\frac {5 (2329+2080 x)}{94178 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )}+\frac {\text {Subst}\left (\int \frac {542760-405450 x^2}{7-4 x^2+5 x^4} \, dx,x,\sqrt {1+2 x}\right )}{329623} \\ & = -\frac {81090}{329623 \sqrt {1+2 x}}+\frac {37+20 x}{434 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )^2}+\frac {5 (2329+2080 x)}{94178 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )}+\frac {\text {Subst}\left (\int \frac {108552 \sqrt {10 \left (2+\sqrt {35}\right )}-\left (542760+81090 \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 )}{659246 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\text {Subst}\left (\int \frac {108552 \sqrt {10 \left (2+\sqrt {35}\right )}+\left (542760+81090 \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 )}{659246 \sqrt {14 \left (2+\sqrt {35}\right )}} \\ & = -\frac {81090}{329623 \sqrt {1+2 x}}+\frac {37+20 x}{434 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )^2}+\frac {5 (2329+2080 x)}{94178 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )}-\frac {\left (3 \left (94605-18092 \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 )}{4614722}-\frac {\left (3 \left (94605-18092 \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 )}{4614722}-\frac {\left (15 \sqrt {\frac {1}{434} \left (2257111762+387427075 \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 )}{659246}+\frac {\left (15 \sqrt {\frac {1}{434} \left (2257111762+387427075 \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 )}{659246} \\ & = -\frac {81090}{329623 \sqrt {1+2 x}}+\frac {37+20 x}{434 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )^2}+\frac {5 (2329+2080 x)}{94178 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )}-\frac {15 \sqrt {\frac {1}{434} \left (2257111762+387427075 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{659246}+\frac {15 \sqrt {\frac {1}{434} \left (2257111762+387427075 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{659246}+\frac {\left (3 \left (94605-18092 \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 )}{2307361}+\frac {\left (3 \left (94605-18092 \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 )}{2307361} \\ & = -\frac {81090}{329623 \sqrt {1+2 x}}+\frac {37+20 x}{434 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )^2}+\frac {5 (2329+2080 x)}{94178 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )}-\frac {15 \sqrt {\frac {1}{434} \left (-2257111762+387427075 \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 )}{329623}+\frac {15 \sqrt {\frac {1}{434} \left (-2257111762+387427075 \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 )}{329623}-\frac {15 \sqrt {\frac {1}{434} \left (2257111762+387427075 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{659246}+\frac {15 \sqrt {\frac {1}{434} \left (2257111762+387427075 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{659246} \\ \end{align*}
Result contains complex when optimal does not.
Time = 1.84 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.47 \[ \int \frac {1}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )^3} \, dx=\frac {-\frac {217 \left (429487+1525635 x+4077245 x^2+4501400 x^3+4054500 x^4\right )}{2 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )^2}+15 \sqrt {217 \left (-2257111762+71603149 i \sqrt {31}\right )} \arctan \left (\sqrt {\frac {1}{7} \left (-2-i \sqrt {31}\right )} \sqrt {1+2 x}\right )+15 \sqrt {217 \left (-2257111762-71603149 i \sqrt {31}\right )} \arctan \left (\sqrt {\frac {1}{7} i \left (2 i+\sqrt {31}\right )} \sqrt {1+2 x}\right )}{71528191} \]
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Time = 0.95 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.13
method | result | size |
pseudoelliptic | \(\frac {-\frac {2378625 \sqrt {1+2 x}\, \left (x^{2}+\frac {3}{5} x +\frac {2}{5}\right )^{2} \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \left (\sqrt {5}+\frac {58421 \sqrt {7}}{88802}\right ) \ln \left (\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{20436626}+\frac {2378625 \sqrt {1+2 x}\, \left (x^{2}+\frac {3}{5} x +\frac {2}{5}\right )^{2} \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \left (\sqrt {5}+\frac {58421 \sqrt {7}}{88802}\right ) \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{20436626}+\frac {6784500 \left (\arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )-\arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )\right ) \left (\sqrt {5}\, \sqrt {7}-\frac {94605}{18092}\right ) \left (x^{2}+\frac {3}{5} x +\frac {2}{5}\right )^{2} \sqrt {1+2 x}}{2307361}-\frac {2027250 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \left (x^{4}+\frac {45014}{40545} x^{3}+\frac {815449}{810900} x^{2}+\frac {33903}{90100} x +\frac {429487}{4054500}\right )}{329623}}{\sqrt {1+2 x}\, \left (5 x^{2}+3 x +2\right )^{2} \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\) | \(354\) |
derivativedivides | \(-\frac {64}{343 \sqrt {1+2 x}}-\frac {1600 \left (\frac {9793 \left (1+2 x \right )^{\frac {7}{2}}}{30752}-\frac {14343 \left (1+2 x \right )^{\frac {5}{2}}}{19220}+\frac {762223 \left (1+2 x \right )^{\frac {3}{2}}}{768800}-\frac {170877 \sqrt {1+2 x}}{192200}\right )}{343 \left (5 \left (1+2 x \right )^{2}+3-8 x \right )^{2}}-\frac {3 \left (444010 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+292105 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{286112764}-\frac {15 \left (-1121704 \sqrt {5}\, \sqrt {7}+\frac {\left (444010 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+292105 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{71528191 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {3 \left (-444010 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-292105 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{286112764}-\frac {15 \left (-1121704 \sqrt {5}\, \sqrt {7}-\frac {\left (-444010 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-292105 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{71528191 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\) | \(444\) |
default | \(-\frac {64}{343 \sqrt {1+2 x}}-\frac {1600 \left (\frac {9793 \left (1+2 x \right )^{\frac {7}{2}}}{30752}-\frac {14343 \left (1+2 x \right )^{\frac {5}{2}}}{19220}+\frac {762223 \left (1+2 x \right )^{\frac {3}{2}}}{768800}-\frac {170877 \sqrt {1+2 x}}{192200}\right )}{343 \left (5 \left (1+2 x \right )^{2}+3-8 x \right )^{2}}-\frac {3 \left (444010 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+292105 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{286112764}-\frac {15 \left (-1121704 \sqrt {5}\, \sqrt {7}+\frac {\left (444010 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+292105 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{71528191 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {3 \left (-444010 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-292105 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{286112764}-\frac {15 \left (-1121704 \sqrt {5}\, \sqrt {7}-\frac {\left (-444010 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-292105 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{71528191 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\) | \(444\) |
trager | \(\text {Expression too large to display}\) | \(462\) |
risch | \(-\frac {4054500 x^{4}+4501400 x^{3}+4077245 x^{2}+1525635 x +429487}{659246 \left (5 x^{2}+3 x +2\right )^{2} \sqrt {1+2 x}}-\frac {95145 \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}}{20436626}-\frac {876315 \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}}{286112764}-\frac {475725 \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 )}{10218313 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {876315 \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}}{143056382 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {542760 \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}}{2307361 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {95145 \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}}{20436626}+\frac {876315 \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}}{286112764}-\frac {475725 \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 )}{10218313 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {876315 \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}}{143056382 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {542760 \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}}{2307361 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\) | \(648\) |
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Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.08 \[ \int \frac {1}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )^3} \, dx=\frac {\sqrt {217} {\left (50 \, x^{5} + 85 \, x^{4} + 88 \, x^{3} + 53 \, x^{2} + 20 \, x + 4\right )} \sqrt {16110708525 i \, \sqrt {31} + 507850146450} \log \left (\sqrt {217} \sqrt {16110708525 i \, \sqrt {31} + 507850146450} {\left (58421 i \, \sqrt {31} + 560852\right )} + 6305375645625 \, \sqrt {2 \, x + 1}\right ) - \sqrt {217} {\left (50 \, x^{5} + 85 \, x^{4} + 88 \, x^{3} + 53 \, x^{2} + 20 \, x + 4\right )} \sqrt {16110708525 i \, \sqrt {31} + 507850146450} \log \left (\sqrt {217} \sqrt {16110708525 i \, \sqrt {31} + 507850146450} {\left (-58421 i \, \sqrt {31} - 560852\right )} + 6305375645625 \, \sqrt {2 \, x + 1}\right ) - \sqrt {217} {\left (50 \, x^{5} + 85 \, x^{4} + 88 \, x^{3} + 53 \, x^{2} + 20 \, x + 4\right )} \sqrt {-16110708525 i \, \sqrt {31} + 507850146450} \log \left (\sqrt {217} {\left (58421 i \, \sqrt {31} - 560852\right )} \sqrt {-16110708525 i \, \sqrt {31} + 507850146450} + 6305375645625 \, \sqrt {2 \, x + 1}\right ) + \sqrt {217} {\left (50 \, x^{5} + 85 \, x^{4} + 88 \, x^{3} + 53 \, x^{2} + 20 \, x + 4\right )} \sqrt {-16110708525 i \, \sqrt {31} + 507850146450} \log \left (\sqrt {217} {\left (-58421 i \, \sqrt {31} + 560852\right )} \sqrt {-16110708525 i \, \sqrt {31} + 507850146450} + 6305375645625 \, \sqrt {2 \, x + 1}\right ) - 217 \, {\left (4054500 \, x^{4} + 4501400 \, x^{3} + 4077245 \, x^{2} + 1525635 \, x + 429487\right )} \sqrt {2 \, x + 1}}{143056382 \, {\left (50 \, x^{5} + 85 \, x^{4} + 88 \, x^{3} + 53 \, x^{2} + 20 \, x + 4\right )}} \]
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\[ \int \frac {1}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )^3} \, dx=\int \frac {1}{\left (2 x + 1\right )^{\frac {3}{2}} \left (5 x^{2} + 3 x + 2\right )^{3}}\, dx \]
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\[ \int \frac {1}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )^3} \, dx=\int { \frac {1}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{3} {\left (2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 651 vs. \(2 (222) = 444\).
Time = 0.55 (sec) , antiderivative size = 651, normalized size of antiderivative = 2.08 \[ \int \frac {1}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )^3} \, dx=\text {Too large to display} \]
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Time = 0.20 (sec) , antiderivative size = 252, normalized size of antiderivative = 0.81 \[ \int \frac {1}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )^3} \, dx=\frac {\frac {274928\,x}{235445}-\frac {1362758\,{\left (2\,x+1\right )}^2}{1648115}+\frac {144304\,{\left (2\,x+1\right )}^3}{329623}-\frac {81090\,{\left (2\,x+1\right )}^4}{329623}+\frac {256792}{1177225}}{\frac {49\,\sqrt {2\,x+1}}{25}-\frac {56\,{\left (2\,x+1\right )}^{3/2}}{25}+\frac {86\,{\left (2\,x+1\right )}^{5/2}}{25}-\frac {8\,{\left (2\,x+1\right )}^{7/2}}{5}+{\left (2\,x+1\right )}^{9/2}}+\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {2\,x+1}\,\sqrt {2257111762-\sqrt {31}\,71603149{}\mathrm {i}}\,32187888{}\mathrm {i}}{1826102771022103\,\left (-\frac {1880448604848}{260871824431729}+\frac {\sqrt {31}\,582343269696{}\mathrm {i}}{260871824431729}\right )}+\frac {64375776\,\sqrt {31}\,\sqrt {217}\,\sqrt {2\,x+1}\,\sqrt {2257111762-\sqrt {31}\,71603149{}\mathrm {i}}}{56609185901685193\,\left (-\frac {1880448604848}{260871824431729}+\frac {\sqrt {31}\,582343269696{}\mathrm {i}}{260871824431729}\right )}\right )\,\sqrt {2257111762-\sqrt {31}\,71603149{}\mathrm {i}}\,15{}\mathrm {i}}{71528191}-\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {2\,x+1}\,\sqrt {2257111762+\sqrt {31}\,71603149{}\mathrm {i}}\,32187888{}\mathrm {i}}{1826102771022103\,\left (\frac {1880448604848}{260871824431729}+\frac {\sqrt {31}\,582343269696{}\mathrm {i}}{260871824431729}\right )}-\frac {64375776\,\sqrt {31}\,\sqrt {217}\,\sqrt {2\,x+1}\,\sqrt {2257111762+\sqrt {31}\,71603149{}\mathrm {i}}}{56609185901685193\,\left (\frac {1880448604848}{260871824431729}+\frac {\sqrt {31}\,582343269696{}\mathrm {i}}{260871824431729}\right )}\right )\,\sqrt {2257111762+\sqrt {31}\,71603149{}\mathrm {i}}\,15{}\mathrm {i}}{71528191} \]
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