Integrand size = 20, antiderivative size = 102 \[ \int \frac {(1+2 x)^2}{\left (3+5 x+2 x^2\right )^5} \, dx=\frac {(1+2 x) (7+6 x)}{4 \left (3+5 x+2 x^2\right )^4}+\frac {73+62 x}{3 \left (3+5 x+2 x^2\right )^3}-\frac {155 (5+4 x)}{3 \left (3+5 x+2 x^2\right )^2}+\frac {620 (5+4 x)}{3+5 x+2 x^2}+2480 \log (1+x)-2480 \log (3+2 x) \]
[Out]
Time = 0.04 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {752, 652, 628, 630, 31} \[ \int \frac {(1+2 x)^2}{\left (3+5 x+2 x^2\right )^5} \, dx=\frac {620 (4 x+5)}{2 x^2+5 x+3}-\frac {155 (4 x+5)}{3 \left (2 x^2+5 x+3\right )^2}+\frac {62 x+73}{3 \left (2 x^2+5 x+3\right )^3}+\frac {(2 x+1) (6 x+7)}{4 \left (2 x^2+5 x+3\right )^4}+2480 \log (x+1)-2480 \log (2 x+3) \]
[In]
[Out]
Rule 31
Rule 628
Rule 630
Rule 652
Rule 752
Rubi steps \begin{align*} \text {integral}& = \frac {(1+2 x) (7+6 x)}{4 \left (3+5 x+2 x^2\right )^4}-\frac {1}{4} \int \frac {-28-72 x}{\left (3+5 x+2 x^2\right )^4} \, dx \\ & = \frac {(1+2 x) (7+6 x)}{4 \left (3+5 x+2 x^2\right )^4}+\frac {73+62 x}{3 \left (3+5 x+2 x^2\right )^3}+\frac {310}{3} \int \frac {1}{\left (3+5 x+2 x^2\right )^3} \, dx \\ & = \frac {(1+2 x) (7+6 x)}{4 \left (3+5 x+2 x^2\right )^4}+\frac {73+62 x}{3 \left (3+5 x+2 x^2\right )^3}-\frac {155 (5+4 x)}{3 \left (3+5 x+2 x^2\right )^2}-620 \int \frac {1}{\left (3+5 x+2 x^2\right )^2} \, dx \\ & = \frac {(1+2 x) (7+6 x)}{4 \left (3+5 x+2 x^2\right )^4}+\frac {73+62 x}{3 \left (3+5 x+2 x^2\right )^3}-\frac {155 (5+4 x)}{3 \left (3+5 x+2 x^2\right )^2}+\frac {620 (5+4 x)}{3+5 x+2 x^2}+2480 \int \frac {1}{3+5 x+2 x^2} \, dx \\ & = \frac {(1+2 x) (7+6 x)}{4 \left (3+5 x+2 x^2\right )^4}+\frac {73+62 x}{3 \left (3+5 x+2 x^2\right )^3}-\frac {155 (5+4 x)}{3 \left (3+5 x+2 x^2\right )^2}+\frac {620 (5+4 x)}{3+5 x+2 x^2}+4960 \int \frac {1}{2+2 x} \, dx-4960 \int \frac {1}{3+2 x} \, dx \\ & = \frac {(1+2 x) (7+6 x)}{4 \left (3+5 x+2 x^2\right )^4}+\frac {73+62 x}{3 \left (3+5 x+2 x^2\right )^3}-\frac {155 (5+4 x)}{3 \left (3+5 x+2 x^2\right )^2}+\frac {620 (5+4 x)}{3+5 x+2 x^2}+2480 \log (1+x)-2480 \log (3+2 x) \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.97 \[ \int \frac {(1+2 x)^2}{\left (3+5 x+2 x^2\right )^5} \, dx=-\frac {11+10 x}{4 \left (3+5 x+2 x^2\right )^4}+\frac {31 (5+4 x)}{6 \left (3+5 x+2 x^2\right )^3}-\frac {155 (5+4 x)}{3 \left (3+5 x+2 x^2\right )^2}+\frac {620 (5+4 x)}{3+5 x+2 x^2}+2480 \log (2 (1+x))-2480 \log (3+2 x) \]
[In]
[Out]
Time = 0.28 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.63
method | result | size |
norman | \(\frac {19840 x^{7}+173600 x^{6}+1624648 x^{3}+\frac {1428116}{3} x +\frac {1939360}{3} x^{5}+\frac {3552290}{3} x^{2}+\frac {3983500}{3} x^{4}+\frac {325799}{4}}{\left (2 x^{2}+5 x +3\right )^{4}}+2480 \ln \left (1+x \right )-2480 \ln \left (3+2 x \right )\) | \(64\) |
risch | \(\frac {19840 x^{7}+173600 x^{6}+1624648 x^{3}+\frac {1428116}{3} x +\frac {1939360}{3} x^{5}+\frac {3552290}{3} x^{2}+\frac {3983500}{3} x^{4}+\frac {325799}{4}}{\left (2 x^{2}+5 x +3\right )^{4}}+2480 \ln \left (1+x \right )-2480 \ln \left (3+2 x \right )\) | \(65\) |
default | \(\frac {16}{\left (3+2 x \right )^{4}}+\frac {256}{3 \left (3+2 x \right )^{3}}+\frac {328}{\left (3+2 x \right )^{2}}+\frac {1360}{3+2 x}-2480 \ln \left (3+2 x \right )-\frac {1}{4 \left (1+x \right )^{4}}+\frac {14}{3 \left (1+x \right )^{3}}-\frac {52}{\left (1+x \right )^{2}}+\frac {560}{1+x}+2480 \ln \left (1+x \right )\) | \(80\) |
parallelrisch | \(\frac {3909588+8332800 x^{6}+22849856 x +31029760 x^{5}+9642240 \ln \left (1+x \right )+63736000 x^{4}+77983104 x^{3}+56836640 x^{2}+952320 x^{7}+82851840 \ln \left (1+x \right ) x^{6}+204748800 \ln \left (1+x \right ) x^{5}+1904640 \ln \left (1+x \right ) x^{8}+19046400 \ln \left (1+x \right ) x^{7}-1904640 \ln \left (\frac {3}{2}+x \right ) x^{8}-19046400 \ln \left (\frac {3}{2}+x \right ) x^{7}-82851840 \ln \left (\frac {3}{2}+x \right ) x^{6}-204748800 \ln \left (\frac {3}{2}+x \right ) x^{5}-314384640 \ln \left (\frac {3}{2}+x \right ) x^{4}-307123200 \ln \left (\frac {3}{2}+x \right ) x^{3}-186416640 \ln \left (\frac {3}{2}+x \right ) x^{2}-64281600 \ln \left (\frac {3}{2}+x \right ) x -9642240 \ln \left (\frac {3}{2}+x \right )+314384640 \ln \left (1+x \right ) x^{4}+307123200 \ln \left (1+x \right ) x^{3}+186416640 \ln \left (1+x \right ) x^{2}+64281600 \ln \left (1+x \right ) x}{48 \left (2 x^{2}+5 x +3\right )^{4}}\) | \(202\) |
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.70 \[ \int \frac {(1+2 x)^2}{\left (3+5 x+2 x^2\right )^5} \, dx=\frac {238080 \, x^{7} + 2083200 \, x^{6} + 7757440 \, x^{5} + 15934000 \, x^{4} + 19495776 \, x^{3} + 14209160 \, x^{2} - 29760 \, {\left (16 \, x^{8} + 160 \, x^{7} + 696 \, x^{6} + 1720 \, x^{5} + 2641 \, x^{4} + 2580 \, x^{3} + 1566 \, x^{2} + 540 \, x + 81\right )} \log \left (2 \, x + 3\right ) + 29760 \, {\left (16 \, x^{8} + 160 \, x^{7} + 696 \, x^{6} + 1720 \, x^{5} + 2641 \, x^{4} + 2580 \, x^{3} + 1566 \, x^{2} + 540 \, x + 81\right )} \log \left (x + 1\right ) + 5712464 \, x + 977397}{12 \, {\left (16 \, x^{8} + 160 \, x^{7} + 696 \, x^{6} + 1720 \, x^{5} + 2641 \, x^{4} + 2580 \, x^{3} + 1566 \, x^{2} + 540 \, x + 81\right )}} \]
[In]
[Out]
Time = 0.09 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.88 \[ \int \frac {(1+2 x)^2}{\left (3+5 x+2 x^2\right )^5} \, dx=\frac {238080 x^{7} + 2083200 x^{6} + 7757440 x^{5} + 15934000 x^{4} + 19495776 x^{3} + 14209160 x^{2} + 5712464 x + 977397}{192 x^{8} + 1920 x^{7} + 8352 x^{6} + 20640 x^{5} + 31692 x^{4} + 30960 x^{3} + 18792 x^{2} + 6480 x + 972} + 2480 \log {\left (x + 1 \right )} - 2480 \log {\left (x + \frac {3}{2} \right )} \]
[In]
[Out]
none
Time = 0.18 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.92 \[ \int \frac {(1+2 x)^2}{\left (3+5 x+2 x^2\right )^5} \, dx=\frac {238080 \, x^{7} + 2083200 \, x^{6} + 7757440 \, x^{5} + 15934000 \, x^{4} + 19495776 \, x^{3} + 14209160 \, x^{2} + 5712464 \, x + 977397}{12 \, {\left (16 \, x^{8} + 160 \, x^{7} + 696 \, x^{6} + 1720 \, x^{5} + 2641 \, x^{4} + 2580 \, x^{3} + 1566 \, x^{2} + 540 \, x + 81\right )}} - 2480 \, \log \left (2 \, x + 3\right ) + 2480 \, \log \left (x + 1\right ) \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.65 \[ \int \frac {(1+2 x)^2}{\left (3+5 x+2 x^2\right )^5} \, dx=\frac {238080 \, x^{7} + 2083200 \, x^{6} + 7757440 \, x^{5} + 15934000 \, x^{4} + 19495776 \, x^{3} + 14209160 \, x^{2} + 5712464 \, x + 977397}{12 \, {\left (2 \, x^{2} + 5 \, x + 3\right )}^{4}} - 2480 \, \log \left ({\left | 2 \, x + 3 \right |}\right ) + 2480 \, \log \left ({\left | x + 1 \right |}\right ) \]
[In]
[Out]
Time = 0.25 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.83 \[ \int \frac {(1+2 x)^2}{\left (3+5 x+2 x^2\right )^5} \, dx=\frac {1240\,x^7+10850\,x^6+\frac {121210\,x^5}{3}+\frac {995875\,x^4}{12}+\frac {203081\,x^3}{2}+\frac {1776145\,x^2}{24}+\frac {357029\,x}{12}+\frac {325799}{64}}{x^8+10\,x^7+\frac {87\,x^6}{2}+\frac {215\,x^5}{2}+\frac {2641\,x^4}{16}+\frac {645\,x^3}{4}+\frac {783\,x^2}{8}+\frac {135\,x}{4}+\frac {81}{16}}-4960\,\mathrm {atanh}\left (4\,x+5\right ) \]
[In]
[Out]