Integrand size = 25, antiderivative size = 64 \[ \int \frac {(5-x) (3+2 x)^5}{\left (2+5 x+3 x^2\right )^3} \, dx=-\frac {32 x}{27}-\frac {56041+57499 x}{486 \left (2+5 x+3 x^2\right )^2}+\frac {398585+502254 x}{486 \left (2+5 x+3 x^2\right )}-1085 \log (1+x)+\frac {29375}{27} \log (2+3 x) \]
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Time = 0.05 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {830, 1674, 1671, 646, 31} \[ \int \frac {(5-x) (3+2 x)^5}{\left (2+5 x+3 x^2\right )^3} \, dx=\frac {502254 x+398585}{486 \left (3 x^2+5 x+2\right )}-\frac {57499 x+56041}{486 \left (3 x^2+5 x+2\right )^2}-\frac {32 x}{27}-1085 \log (x+1)+\frac {29375}{27} \log (3 x+2) \]
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Rule 31
Rule 646
Rule 830
Rule 1671
Rule 1674
Rubi steps \begin{align*} \text {integral}& = \int \frac {\frac {13}{2} (3+2 x)^5-\frac {1}{2} (3+2 x)^6}{\left (2+5 x+3 x^2\right )^3} \, dx \\ & = -\frac {56041+57499 x}{486 \left (2+5 x+3 x^2\right )^2}-\frac {1}{2} \int \frac {-\frac {72539}{243}-\frac {87920 x}{81}-\frac {9824 x^2}{27}+\frac {160 x^3}{9}+\frac {64 x^4}{3}}{\left (2+5 x+3 x^2\right )^2} \, dx \\ & = -\frac {56041+57499 x}{486 \left (2+5 x+3 x^2\right )^2}+\frac {398585+502254 x}{486 \left (2+5 x+3 x^2\right )}+\frac {1}{2} \int \frac {\frac {58942}{27}+\frac {160 x}{27}-\frac {64 x^2}{9}}{2+5 x+3 x^2} \, dx \\ & = -\frac {56041+57499 x}{486 \left (2+5 x+3 x^2\right )^2}+\frac {398585+502254 x}{486 \left (2+5 x+3 x^2\right )}+\frac {1}{2} \int \left (-\frac {64}{27}+\frac {10 (1969+16 x)}{9 \left (2+5 x+3 x^2\right )}\right ) \, dx \\ & = -\frac {32 x}{27}-\frac {56041+57499 x}{486 \left (2+5 x+3 x^2\right )^2}+\frac {398585+502254 x}{486 \left (2+5 x+3 x^2\right )}+\frac {5}{9} \int \frac {1969+16 x}{2+5 x+3 x^2} \, dx \\ & = -\frac {32 x}{27}-\frac {56041+57499 x}{486 \left (2+5 x+3 x^2\right )^2}+\frac {398585+502254 x}{486 \left (2+5 x+3 x^2\right )}-3255 \int \frac {1}{3+3 x} \, dx+\frac {29375}{9} \int \frac {1}{2+3 x} \, dx \\ & = -\frac {32 x}{27}-\frac {56041+57499 x}{486 \left (2+5 x+3 x^2\right )^2}+\frac {398585+502254 x}{486 \left (2+5 x+3 x^2\right )}-1085 \log (1+x)+\frac {29375}{27} \log (2+3 x) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.27 \[ \int \frac {(5-x) (3+2 x)^5}{\left (2+5 x+3 x^2\right )^3} \, dx=\frac {245891+973450 x+1221179 x^2+486510 x^3-8352 x^4-1728 x^5+176250 \left (2+5 x+3 x^2\right )^2 \log (-4-6 x)-175770 \left (2+5 x+3 x^2\right )^2 \log (-2 (1+x))}{162 \left (2+5 x+3 x^2\right )^2} \]
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Time = 0.32 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.75
method | result | size |
risch | \(-\frac {32 x}{27}+\frac {\frac {9301}{3} x^{3}+\frac {1235675}{162} x^{2}+\frac {489989}{81} x +\frac {247043}{162}}{\left (3 x^{2}+5 x +2\right )^{2}}-1085 \ln \left (1+x \right )+\frac {29375 \ln \left (2+3 x \right )}{27}\) | \(48\) |
norman | \(\frac {3175 x^{3}+\frac {165335}{27} x +\frac {418505}{54} x^{2}-\frac {32}{3} x^{5}+\frac {83201}{54}}{\left (3 x^{2}+5 x +2\right )^{2}}-1085 \ln \left (1+x \right )+\frac {29375 \ln \left (2+3 x \right )}{27}\) | \(49\) |
default | \(-\frac {32 x}{27}-\frac {53125}{162 \left (2+3 x \right )^{2}}+\frac {6250}{9 \left (2+3 x \right )}+\frac {29375 \ln \left (2+3 x \right )}{27}+\frac {3}{\left (1+x \right )^{2}}+\frac {113}{1+x}-1085 \ln \left (1+x \right )\) | \(51\) |
parallelrisch | \(-\frac {2109240 \ln \left (1+x \right ) x^{4}-2115000 \ln \left (x +\frac {2}{3}\right ) x^{4}+2304 x^{5}+7030800 \ln \left (1+x \right ) x^{3}-7050000 \ln \left (x +\frac {2}{3}\right ) x^{3}+748809 x^{4}+8671320 \ln \left (1+x \right ) x^{2}-8695000 \ln \left (x +\frac {2}{3}\right ) x^{2}+1810230 x^{3}+4687200 \ln \left (1+x \right ) x -4700000 \ln \left (x +\frac {2}{3}\right ) x +1404417 x^{2}+937440 \ln \left (1+x \right )-940000 \ln \left (x +\frac {2}{3}\right )+341340 x}{216 \left (3 x^{2}+5 x +2\right )^{2}}\) | \(119\) |
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Time = 0.29 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.61 \[ \int \frac {(5-x) (3+2 x)^5}{\left (2+5 x+3 x^2\right )^3} \, dx=-\frac {1728 \, x^{5} + 5760 \, x^{4} - 495150 \, x^{3} - 1231835 \, x^{2} - 176250 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (3 \, x + 2\right ) + 175770 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (x + 1\right ) - 979210 \, x - 247043}{162 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} \]
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Time = 0.09 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.91 \[ \int \frac {(5-x) (3+2 x)^5}{\left (2+5 x+3 x^2\right )^3} \, dx=- \frac {32 x}{27} - \frac {- 502254 x^{3} - 1235675 x^{2} - 979978 x - 247043}{1458 x^{4} + 4860 x^{3} + 5994 x^{2} + 3240 x + 648} + \frac {29375 \log {\left (x + \frac {2}{3} \right )}}{27} - 1085 \log {\left (x + 1 \right )} \]
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Time = 0.19 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.89 \[ \int \frac {(5-x) (3+2 x)^5}{\left (2+5 x+3 x^2\right )^3} \, dx=-\frac {32}{27} \, x + \frac {502254 \, x^{3} + 1235675 \, x^{2} + 979978 \, x + 247043}{162 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} + \frac {29375}{27} \, \log \left (3 \, x + 2\right ) - 1085 \, \log \left (x + 1\right ) \]
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Time = 0.27 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.77 \[ \int \frac {(5-x) (3+2 x)^5}{\left (2+5 x+3 x^2\right )^3} \, dx=-\frac {32}{27} \, x + \frac {502254 \, x^{3} + 1235675 \, x^{2} + 979978 \, x + 247043}{162 \, {\left (3 \, x + 2\right )}^{2} {\left (x + 1\right )}^{2}} + \frac {29375}{27} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - 1085 \, \log \left ({\left | x + 1 \right |}\right ) \]
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Time = 11.72 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.81 \[ \int \frac {(5-x) (3+2 x)^5}{\left (2+5 x+3 x^2\right )^3} \, dx=\frac {29375\,\ln \left (x+\frac {2}{3}\right )}{27}-1085\,\ln \left (x+1\right )-\frac {32\,x}{27}+\frac {\frac {9301\,x^3}{27}+\frac {1235675\,x^2}{1458}+\frac {489989\,x}{729}+\frac {247043}{1458}}{x^4+\frac {10\,x^3}{3}+\frac {37\,x^2}{9}+\frac {20\,x}{9}+\frac {4}{9}} \]
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