Integrand size = 20, antiderivative size = 89 \[ \int \frac {1}{(1+2 x) \left (2+3 x+5 x^2\right )^3} \, dx=\frac {37+20 x}{434 \left (2+3 x+5 x^2\right )^2}+\frac {2 (2609+2290 x)}{47089 \left (2+3 x+5 x^2\right )}+\frac {125624 \arctan \left (\frac {3+10 x}{\sqrt {31}}\right )}{329623 \sqrt {31}}+\frac {32}{343} \log (1+2 x)-\frac {16}{343} \log \left (2+3 x+5 x^2\right ) \]
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Time = 0.06 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {754, 836, 814, 648, 632, 210, 642} \[ \int \frac {1}{(1+2 x) \left (2+3 x+5 x^2\right )^3} \, dx=\frac {125624 \arctan \left (\frac {10 x+3}{\sqrt {31}}\right )}{329623 \sqrt {31}}+\frac {20 x+37}{434 \left (5 x^2+3 x+2\right )^2}+\frac {2 (2290 x+2609)}{47089 \left (5 x^2+3 x+2\right )}-\frac {16}{343} \log \left (5 x^2+3 x+2\right )+\frac {32}{343} \log (2 x+1) \]
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Rule 210
Rule 632
Rule 642
Rule 648
Rule 754
Rule 814
Rule 836
Rubi steps \begin{align*} \text {integral}& = \frac {37+20 x}{434 \left (2+3 x+5 x^2\right )^2}+\frac {1}{434} \int \frac {308+120 x}{(1+2 x) \left (2+3 x+5 x^2\right )^2} \, dx \\ & = \frac {37+20 x}{434 \left (2+3 x+5 x^2\right )^2}+\frac {2 (2609+2290 x)}{47089 \left (2+3 x+5 x^2\right )}+\frac {\int \frac {39912+18320 x}{(1+2 x) \left (2+3 x+5 x^2\right )} \, dx}{94178} \\ & = \frac {37+20 x}{434 \left (2+3 x+5 x^2\right )^2}+\frac {2 (2609+2290 x)}{47089 \left (2+3 x+5 x^2\right )}+\frac {\int \left (\frac {123008}{7 (1+2 x)}-\frac {8 (-4171+38440 x)}{7 \left (2+3 x+5 x^2\right )}\right ) \, dx}{94178} \\ & = \frac {37+20 x}{434 \left (2+3 x+5 x^2\right )^2}+\frac {2 (2609+2290 x)}{47089 \left (2+3 x+5 x^2\right )}+\frac {32}{343} \log (1+2 x)-\frac {4 \int \frac {-4171+38440 x}{2+3 x+5 x^2} \, dx}{329623} \\ & = \frac {37+20 x}{434 \left (2+3 x+5 x^2\right )^2}+\frac {2 (2609+2290 x)}{47089 \left (2+3 x+5 x^2\right )}+\frac {32}{343} \log (1+2 x)-\frac {16}{343} \int \frac {3+10 x}{2+3 x+5 x^2} \, dx+\frac {62812 \int \frac {1}{2+3 x+5 x^2} \, dx}{329623} \\ & = \frac {37+20 x}{434 \left (2+3 x+5 x^2\right )^2}+\frac {2 (2609+2290 x)}{47089 \left (2+3 x+5 x^2\right )}+\frac {32}{343} \log (1+2 x)-\frac {16}{343} \log \left (2+3 x+5 x^2\right )-\frac {125624 \text {Subst}\left (\int \frac {1}{-31-x^2} \, dx,x,3+10 x\right )}{329623} \\ & = \frac {37+20 x}{434 \left (2+3 x+5 x^2\right )^2}+\frac {2 (2609+2290 x)}{47089 \left (2+3 x+5 x^2\right )}+\frac {125624 \tan ^{-1}\left (\frac {3+10 x}{\sqrt {31}}\right )}{329623 \sqrt {31}}+\frac {32}{343} \log (1+2 x)-\frac {16}{343} \log \left (2+3 x+5 x^2\right ) \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.88 \[ \int \frac {1}{(1+2 x) \left (2+3 x+5 x^2\right )^3} \, dx=\frac {8 \left (\frac {217 \left (28901+53968 x+79660 x^2+45800 x^3\right )}{16 \left (2+3 x+5 x^2\right )^2}+15703 \sqrt {31} \arctan \left (\frac {3+10 x}{\sqrt {31}}\right )+119164 \log (1+2 x)-59582 \log \left (4 \left (2+3 x+5 x^2\right )\right )\right )}{10218313} \]
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Time = 23.85 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.76
method | result | size |
default | \(\frac {32 \ln \left (1+2 x \right )}{343}-\frac {25 \left (-\frac {6412}{961} x^{3}-\frac {55762}{4805} x^{2}-\frac {188888}{24025} x -\frac {202307}{48050}\right )}{343 \left (5 x^{2}+3 x +2\right )^{2}}-\frac {16 \ln \left (5 x^{2}+3 x +2\right )}{343}+\frac {125624 \arctan \left (\frac {\left (3+10 x \right ) \sqrt {31}}{31}\right ) \sqrt {31}}{10218313}\) | \(68\) |
risch | \(\frac {\frac {22900}{47089} x^{3}+\frac {5690}{6727} x^{2}+\frac {26984}{47089} x +\frac {28901}{94178}}{\left (5 x^{2}+3 x +2\right )^{2}}+\frac {32 \ln \left (1+2 x \right )}{343}-\frac {16 \ln \left (24658420900 x^{2}+14795052540 x +9863368360\right )}{343}+\frac {125624 \sqrt {31}\, \arctan \left (\frac {\left (157030 x +47109\right ) \sqrt {31}}{486793}\right )}{10218313}\) | \(68\) |
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Time = 0.35 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.53 \[ \int \frac {1}{(1+2 x) \left (2+3 x+5 x^2\right )^3} \, dx=\frac {9938600 \, x^{3} + 251248 \, \sqrt {31} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + 17286220 \, x^{2} - 953312 \, {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \log \left (5 \, x^{2} + 3 \, x + 2\right ) + 1906624 \, {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \log \left (2 \, x + 1\right ) + 11711056 \, x + 6271517}{20436626 \, {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )}} \]
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Time = 0.12 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.01 \[ \int \frac {1}{(1+2 x) \left (2+3 x+5 x^2\right )^3} \, dx=\frac {45800 x^{3} + 79660 x^{2} + 53968 x + 28901}{2354450 x^{4} + 2825340 x^{3} + 2731162 x^{2} + 1130136 x + 376712} + \frac {32 \log {\left (x + \frac {1}{2} \right )}}{343} - \frac {16 \log {\left (x^{2} + \frac {3 x}{5} + \frac {2}{5} \right )}}{343} + \frac {125624 \sqrt {31} \operatorname {atan}{\left (\frac {10 \sqrt {31} x}{31} + \frac {3 \sqrt {31}}{31} \right )}}{10218313} \]
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Time = 0.30 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.87 \[ \int \frac {1}{(1+2 x) \left (2+3 x+5 x^2\right )^3} \, dx=\frac {125624}{10218313} \, \sqrt {31} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + \frac {45800 \, x^{3} + 79660 \, x^{2} + 53968 \, x + 28901}{94178 \, {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )}} - \frac {16}{343} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) + \frac {32}{343} \, \log \left (2 \, x + 1\right ) \]
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Time = 0.26 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.76 \[ \int \frac {1}{(1+2 x) \left (2+3 x+5 x^2\right )^3} \, dx=\frac {125624}{10218313} \, \sqrt {31} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + \frac {45800 \, x^{3} + 79660 \, x^{2} + 53968 \, x + 28901}{94178 \, {\left (5 \, x^{2} + 3 \, x + 2\right )}^{2}} - \frac {16}{343} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) + \frac {32}{343} \, \log \left ({\left | 2 \, x + 1 \right |}\right ) \]
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Time = 0.13 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.92 \[ \int \frac {1}{(1+2 x) \left (2+3 x+5 x^2\right )^3} \, dx=\frac {32\,\ln \left (x+\frac {1}{2}\right )}{343}-\ln \left (x+\frac {3}{10}-\frac {\sqrt {31}\,1{}\mathrm {i}}{10}\right )\,\left (\frac {16}{343}+\frac {\sqrt {31}\,62812{}\mathrm {i}}{10218313}\right )+\ln \left (x+\frac {3}{10}+\frac {\sqrt {31}\,1{}\mathrm {i}}{10}\right )\,\left (-\frac {16}{343}+\frac {\sqrt {31}\,62812{}\mathrm {i}}{10218313}\right )+\frac {\frac {916\,x^3}{47089}+\frac {1138\,x^2}{33635}+\frac {26984\,x}{1177225}+\frac {28901}{2354450}}{x^4+\frac {6\,x^3}{5}+\frac {29\,x^2}{25}+\frac {12\,x}{25}+\frac {4}{25}} \]
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