Integrand size = 20, antiderivative size = 45 \[ \int \frac {(1-2 x) (3+5 x)^2}{(2+3 x)^6} \, dx=-\frac {7}{405 (2+3 x)^5}+\frac {2}{9 (2+3 x)^4}-\frac {65}{81 (2+3 x)^3}+\frac {25}{81 (2+3 x)^2} \]
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Time = 0.01 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {78} \[ \int \frac {(1-2 x) (3+5 x)^2}{(2+3 x)^6} \, dx=\frac {25}{81 (3 x+2)^2}-\frac {65}{81 (3 x+2)^3}+\frac {2}{9 (3 x+2)^4}-\frac {7}{405 (3 x+2)^5} \]
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Rule 78
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {7}{27 (2+3 x)^6}-\frac {8}{3 (2+3 x)^5}+\frac {65}{9 (2+3 x)^4}-\frac {50}{27 (2+3 x)^3}\right ) \, dx \\ & = -\frac {7}{405 (2+3 x)^5}+\frac {2}{9 (2+3 x)^4}-\frac {65}{81 (2+3 x)^3}+\frac {25}{81 (2+3 x)^2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.58 \[ \int \frac {(1-2 x) (3+5 x)^2}{(2+3 x)^6} \, dx=\frac {-127+870 x+3825 x^2+3375 x^3}{405 (2+3 x)^5} \]
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Time = 2.16 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.53
method | result | size |
norman | \(\frac {\frac {25}{3} x^{3}+\frac {58}{27} x +\frac {85}{9} x^{2}-\frac {127}{405}}{\left (2+3 x \right )^{5}}\) | \(24\) |
gosper | \(\frac {3375 x^{3}+3825 x^{2}+870 x -127}{405 \left (2+3 x \right )^{5}}\) | \(25\) |
risch | \(\frac {\frac {25}{3} x^{3}+\frac {58}{27} x +\frac {85}{9} x^{2}-\frac {127}{405}}{\left (2+3 x \right )^{5}}\) | \(25\) |
parallelrisch | \(\frac {1143 x^{5}+3810 x^{4}+9080 x^{3}+7920 x^{2}+2160 x}{480 \left (2+3 x \right )^{5}}\) | \(34\) |
default | \(-\frac {7}{405 \left (2+3 x \right )^{5}}+\frac {2}{9 \left (2+3 x \right )^{4}}-\frac {65}{81 \left (2+3 x \right )^{3}}+\frac {25}{81 \left (2+3 x \right )^{2}}\) | \(38\) |
meijerg | \(\frac {9 x \left (\frac {81}{16} x^{4}+\frac {135}{8} x^{3}+\frac {45}{2} x^{2}+15 x +5\right )}{320 \left (1+\frac {3 x}{2}\right )^{5}}+\frac {3 x^{2} \left (\frac {27}{8} x^{3}+\frac {45}{4} x^{2}+15 x +10\right )}{320 \left (1+\frac {3 x}{2}\right )^{5}}-\frac {7 x^{3} \left (\frac {9}{4} x^{2}+\frac {15}{2} x +10\right )}{384 \left (1+\frac {3 x}{2}\right )^{5}}-\frac {5 x^{4} \left (\frac {3 x}{2}+5\right )}{128 \left (1+\frac {3 x}{2}\right )^{5}}\) | \(98\) |
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Time = 0.22 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.98 \[ \int \frac {(1-2 x) (3+5 x)^2}{(2+3 x)^6} \, dx=\frac {3375 \, x^{3} + 3825 \, x^{2} + 870 \, x - 127}{405 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]
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Time = 0.06 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.91 \[ \int \frac {(1-2 x) (3+5 x)^2}{(2+3 x)^6} \, dx=- \frac {- 3375 x^{3} - 3825 x^{2} - 870 x + 127}{98415 x^{5} + 328050 x^{4} + 437400 x^{3} + 291600 x^{2} + 97200 x + 12960} \]
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Time = 0.21 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.98 \[ \int \frac {(1-2 x) (3+5 x)^2}{(2+3 x)^6} \, dx=\frac {3375 \, x^{3} + 3825 \, x^{2} + 870 \, x - 127}{405 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]
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Time = 0.29 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.53 \[ \int \frac {(1-2 x) (3+5 x)^2}{(2+3 x)^6} \, dx=\frac {3375 \, x^{3} + 3825 \, x^{2} + 870 \, x - 127}{405 \, {\left (3 \, x + 2\right )}^{5}} \]
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Time = 1.32 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.82 \[ \int \frac {(1-2 x) (3+5 x)^2}{(2+3 x)^6} \, dx=\frac {25}{81\,{\left (3\,x+2\right )}^2}-\frac {65}{81\,{\left (3\,x+2\right )}^3}+\frac {2}{9\,{\left (3\,x+2\right )}^4}-\frac {7}{405\,{\left (3\,x+2\right )}^5} \]
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