Integrand size = 21, antiderivative size = 116 \[ \int (e x)^m (2-2 a x)^2 (1+a x)^3 \, dx=\frac {4 (e x)^{1+m}}{e (1+m)}+\frac {4 a (e x)^{2+m}}{e^2 (2+m)}-\frac {8 a^2 (e x)^{3+m}}{e^3 (3+m)}-\frac {8 a^3 (e x)^{4+m}}{e^4 (4+m)}+\frac {4 a^4 (e x)^{5+m}}{e^5 (5+m)}+\frac {4 a^5 (e x)^{6+m}}{e^6 (6+m)} \]
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Time = 0.03 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {90} \[ \int (e x)^m (2-2 a x)^2 (1+a x)^3 \, dx=\frac {4 a^5 (e x)^{m+6}}{e^6 (m+6)}+\frac {4 a^4 (e x)^{m+5}}{e^5 (m+5)}-\frac {8 a^3 (e x)^{m+4}}{e^4 (m+4)}-\frac {8 a^2 (e x)^{m+3}}{e^3 (m+3)}+\frac {4 a (e x)^{m+2}}{e^2 (m+2)}+\frac {4 (e x)^{m+1}}{e (m+1)} \]
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Rule 90
Rubi steps \begin{align*} \text {integral}& = \int \left (4 (e x)^m+\frac {4 a (e x)^{1+m}}{e}-\frac {8 a^2 (e x)^{2+m}}{e^2}-\frac {8 a^3 (e x)^{3+m}}{e^3}+\frac {4 a^4 (e x)^{4+m}}{e^4}+\frac {4 a^5 (e x)^{5+m}}{e^5}\right ) \, dx \\ & = \frac {4 (e x)^{1+m}}{e (1+m)}+\frac {4 a (e x)^{2+m}}{e^2 (2+m)}-\frac {8 a^2 (e x)^{3+m}}{e^3 (3+m)}-\frac {8 a^3 (e x)^{4+m}}{e^4 (4+m)}+\frac {4 a^4 (e x)^{5+m}}{e^5 (5+m)}+\frac {4 a^5 (e x)^{6+m}}{e^6 (6+m)} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.62 \[ \int (e x)^m (2-2 a x)^2 (1+a x)^3 \, dx=4 x (e x)^m \left (\frac {1}{1+m}+\frac {a x}{2+m}-\frac {2 a^2 x^2}{3+m}-\frac {2 a^3 x^3}{4+m}+\frac {a^4 x^4}{5+m}+\frac {a^5 x^5}{6+m}\right ) \]
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Time = 0.41 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.99
method | result | size |
norman | \(\frac {4 x \,{\mathrm e}^{m \ln \left (e x \right )}}{1+m}+\frac {4 a \,x^{2} {\mathrm e}^{m \ln \left (e x \right )}}{2+m}-\frac {8 a^{2} x^{3} {\mathrm e}^{m \ln \left (e x \right )}}{3+m}-\frac {8 a^{3} x^{4} {\mathrm e}^{m \ln \left (e x \right )}}{4+m}+\frac {4 a^{4} x^{5} {\mathrm e}^{m \ln \left (e x \right )}}{5+m}+\frac {4 a^{5} x^{6} {\mathrm e}^{m \ln \left (e x \right )}}{6+m}\) | \(115\) |
gosper | \(\frac {4 \left (e x \right )^{m} \left (a^{5} m^{5} x^{5}+15 a^{5} m^{4} x^{5}+85 a^{5} m^{3} x^{5}+a^{4} m^{5} x^{4}+225 a^{5} m^{2} x^{5}+16 a^{4} m^{4} x^{4}+274 a^{5} x^{5} m +95 a^{4} m^{3} x^{4}-2 a^{3} m^{5} x^{3}+120 a^{5} x^{5}+260 a^{4} m^{2} x^{4}-34 a^{3} m^{4} x^{3}+324 a^{4} x^{4} m -214 a^{3} m^{3} x^{3}-2 a^{2} m^{5} x^{2}+144 a^{4} x^{4}-614 a^{3} m^{2} x^{3}-36 a^{2} m^{4} x^{2}-792 a^{3} x^{3} m -242 a^{2} m^{3} x^{2}+a \,m^{5} x -360 a^{3} x^{3}-744 a^{2} m^{2} x^{2}+19 a \,m^{4} x -1016 a^{2} m \,x^{2}+137 a \,m^{3} x +m^{5}-480 a^{2} x^{2}+461 a \,m^{2} x +20 m^{4}+702 a x m +155 m^{3}+360 a x +580 m^{2}+1044 m +720\right ) x}{\left (6+m \right ) \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\) | \(340\) |
risch | \(\frac {4 \left (e x \right )^{m} \left (a^{5} m^{5} x^{5}+15 a^{5} m^{4} x^{5}+85 a^{5} m^{3} x^{5}+a^{4} m^{5} x^{4}+225 a^{5} m^{2} x^{5}+16 a^{4} m^{4} x^{4}+274 a^{5} x^{5} m +95 a^{4} m^{3} x^{4}-2 a^{3} m^{5} x^{3}+120 a^{5} x^{5}+260 a^{4} m^{2} x^{4}-34 a^{3} m^{4} x^{3}+324 a^{4} x^{4} m -214 a^{3} m^{3} x^{3}-2 a^{2} m^{5} x^{2}+144 a^{4} x^{4}-614 a^{3} m^{2} x^{3}-36 a^{2} m^{4} x^{2}-792 a^{3} x^{3} m -242 a^{2} m^{3} x^{2}+a \,m^{5} x -360 a^{3} x^{3}-744 a^{2} m^{2} x^{2}+19 a \,m^{4} x -1016 a^{2} m \,x^{2}+137 a \,m^{3} x +m^{5}-480 a^{2} x^{2}+461 a \,m^{2} x +20 m^{4}+702 a x m +155 m^{3}+360 a x +580 m^{2}+1044 m +720\right ) x}{\left (6+m \right ) \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\) | \(340\) |
parallelrisch | \(\frac {-144 x^{3} \left (e x \right )^{m} a^{2} m^{4}-3168 x^{4} \left (e x \right )^{m} a^{3} m -968 x^{3} \left (e x \right )^{m} a^{2} m^{3}+4 x^{2} \left (e x \right )^{m} a \,m^{5}-2976 x^{3} \left (e x \right )^{m} a^{2} m^{2}+76 x^{2} \left (e x \right )^{m} a \,m^{4}-4064 x^{3} \left (e x \right )^{m} a^{2} m +548 x^{2} \left (e x \right )^{m} a \,m^{3}+1844 x^{2} \left (e x \right )^{m} a \,m^{2}+2808 x^{2} \left (e x \right )^{m} a m +340 x^{6} \left (e x \right )^{m} a^{5} m^{3}+4 x^{5} \left (e x \right )^{m} a^{4} m^{5}+900 x^{6} \left (e x \right )^{m} a^{5} m^{2}+64 x^{5} \left (e x \right )^{m} a^{4} m^{4}+1096 x^{6} \left (e x \right )^{m} a^{5} m +380 x^{5} \left (e x \right )^{m} a^{4} m^{3}-8 x^{4} \left (e x \right )^{m} a^{3} m^{5}+1040 x^{5} \left (e x \right )^{m} a^{4} m^{2}-136 x^{4} \left (e x \right )^{m} a^{3} m^{4}+1296 x^{5} \left (e x \right )^{m} a^{4} m -856 x^{4} \left (e x \right )^{m} a^{3} m^{3}-8 x^{3} \left (e x \right )^{m} a^{2} m^{5}-2456 x^{4} \left (e x \right )^{m} a^{3} m^{2}+4 x^{6} \left (e x \right )^{m} a^{5} m^{5}+60 x^{6} \left (e x \right )^{m} a^{5} m^{4}+480 x^{6} \left (e x \right )^{m} a^{5}+576 x^{5} \left (e x \right )^{m} a^{4}-1440 x^{4} \left (e x \right )^{m} a^{3}+4 x \left (e x \right )^{m} m^{5}-1920 x^{3} \left (e x \right )^{m} a^{2}+80 x \left (e x \right )^{m} m^{4}+620 x \left (e x \right )^{m} m^{3}+1440 x^{2} \left (e x \right )^{m} a +2320 x \left (e x \right )^{m} m^{2}+4176 x \left (e x \right )^{m} m +2880 \left (e x \right )^{m} x}{\left (6+m \right ) \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\) | \(537\) |
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Leaf count of result is larger than twice the leaf count of optimal. 288 vs. \(2 (116) = 232\).
Time = 0.23 (sec) , antiderivative size = 288, normalized size of antiderivative = 2.48 \[ \int (e x)^m (2-2 a x)^2 (1+a x)^3 \, dx=\frac {4 \, {\left ({\left (a^{5} m^{5} + 15 \, a^{5} m^{4} + 85 \, a^{5} m^{3} + 225 \, a^{5} m^{2} + 274 \, a^{5} m + 120 \, a^{5}\right )} x^{6} + {\left (a^{4} m^{5} + 16 \, a^{4} m^{4} + 95 \, a^{4} m^{3} + 260 \, a^{4} m^{2} + 324 \, a^{4} m + 144 \, a^{4}\right )} x^{5} - 2 \, {\left (a^{3} m^{5} + 17 \, a^{3} m^{4} + 107 \, a^{3} m^{3} + 307 \, a^{3} m^{2} + 396 \, a^{3} m + 180 \, a^{3}\right )} x^{4} - 2 \, {\left (a^{2} m^{5} + 18 \, a^{2} m^{4} + 121 \, a^{2} m^{3} + 372 \, a^{2} m^{2} + 508 \, a^{2} m + 240 \, a^{2}\right )} x^{3} + {\left (a m^{5} + 19 \, a m^{4} + 137 \, a m^{3} + 461 \, a m^{2} + 702 \, a m + 360 \, a\right )} x^{2} + {\left (m^{5} + 20 \, m^{4} + 155 \, m^{3} + 580 \, m^{2} + 1044 \, m + 720\right )} x\right )} \left (e x\right )^{m}}{m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1867 vs. \(2 (104) = 208\).
Time = 0.45 (sec) , antiderivative size = 1867, normalized size of antiderivative = 16.09 \[ \int (e x)^m (2-2 a x)^2 (1+a x)^3 \, dx=\text {Too large to display} \]
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Time = 0.21 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.96 \[ \int (e x)^m (2-2 a x)^2 (1+a x)^3 \, dx=\frac {4 \, a^{5} e^{m} x^{6} x^{m}}{m + 6} + \frac {4 \, a^{4} e^{m} x^{5} x^{m}}{m + 5} - \frac {8 \, a^{3} e^{m} x^{4} x^{m}}{m + 4} - \frac {8 \, a^{2} e^{m} x^{3} x^{m}}{m + 3} + \frac {4 \, a e^{m} x^{2} x^{m}}{m + 2} + \frac {4 \, \left (e x\right )^{m + 1}}{e {\left (m + 1\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 533 vs. \(2 (116) = 232\).
Time = 0.27 (sec) , antiderivative size = 533, normalized size of antiderivative = 4.59 \[ \int (e x)^m (2-2 a x)^2 (1+a x)^3 \, dx=\frac {4 \, {\left (\left (e x\right )^{m} a^{5} m^{5} x^{6} + 15 \, \left (e x\right )^{m} a^{5} m^{4} x^{6} + \left (e x\right )^{m} a^{4} m^{5} x^{5} + 85 \, \left (e x\right )^{m} a^{5} m^{3} x^{6} + 16 \, \left (e x\right )^{m} a^{4} m^{4} x^{5} + 225 \, \left (e x\right )^{m} a^{5} m^{2} x^{6} - 2 \, \left (e x\right )^{m} a^{3} m^{5} x^{4} + 95 \, \left (e x\right )^{m} a^{4} m^{3} x^{5} + 274 \, \left (e x\right )^{m} a^{5} m x^{6} - 34 \, \left (e x\right )^{m} a^{3} m^{4} x^{4} + 260 \, \left (e x\right )^{m} a^{4} m^{2} x^{5} + 120 \, \left (e x\right )^{m} a^{5} x^{6} - 2 \, \left (e x\right )^{m} a^{2} m^{5} x^{3} - 214 \, \left (e x\right )^{m} a^{3} m^{3} x^{4} + 324 \, \left (e x\right )^{m} a^{4} m x^{5} - 36 \, \left (e x\right )^{m} a^{2} m^{4} x^{3} - 614 \, \left (e x\right )^{m} a^{3} m^{2} x^{4} + 144 \, \left (e x\right )^{m} a^{4} x^{5} + \left (e x\right )^{m} a m^{5} x^{2} - 242 \, \left (e x\right )^{m} a^{2} m^{3} x^{3} - 792 \, \left (e x\right )^{m} a^{3} m x^{4} + 19 \, \left (e x\right )^{m} a m^{4} x^{2} - 744 \, \left (e x\right )^{m} a^{2} m^{2} x^{3} - 360 \, \left (e x\right )^{m} a^{3} x^{4} + \left (e x\right )^{m} m^{5} x + 137 \, \left (e x\right )^{m} a m^{3} x^{2} - 1016 \, \left (e x\right )^{m} a^{2} m x^{3} + 20 \, \left (e x\right )^{m} m^{4} x + 461 \, \left (e x\right )^{m} a m^{2} x^{2} - 480 \, \left (e x\right )^{m} a^{2} x^{3} + 155 \, \left (e x\right )^{m} m^{3} x + 702 \, \left (e x\right )^{m} a m x^{2} + 580 \, \left (e x\right )^{m} m^{2} x + 360 \, \left (e x\right )^{m} a x^{2} + 1044 \, \left (e x\right )^{m} m x + 720 \, \left (e x\right )^{m} x\right )}}{m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720} \]
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Time = 0.70 (sec) , antiderivative size = 367, normalized size of antiderivative = 3.16 \[ \int (e x)^m (2-2 a x)^2 (1+a x)^3 \, dx={\left (e\,x\right )}^m\,\left (\frac {x\,\left (4\,m^5+80\,m^4+620\,m^3+2320\,m^2+4176\,m+2880\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}+\frac {4\,a\,x^2\,\left (m^5+19\,m^4+137\,m^3+461\,m^2+702\,m+360\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}+\frac {4\,a^5\,x^6\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}+\frac {4\,a^4\,x^5\,\left (m^5+16\,m^4+95\,m^3+260\,m^2+324\,m+144\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}-\frac {8\,a^3\,x^4\,\left (m^5+17\,m^4+107\,m^3+307\,m^2+396\,m+180\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}-\frac {8\,a^2\,x^3\,\left (m^5+18\,m^4+121\,m^3+372\,m^2+508\,m+240\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}\right ) \]
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