Integrand size = 21, antiderivative size = 156 \[ \int (e x)^m (2-2 a x)^3 (1+a x)^4 \, dx=\frac {8 (e x)^{1+m}}{e (1+m)}+\frac {8 a (e x)^{2+m}}{e^2 (2+m)}-\frac {24 a^2 (e x)^{3+m}}{e^3 (3+m)}-\frac {24 a^3 (e x)^{4+m}}{e^4 (4+m)}+\frac {24 a^4 (e x)^{5+m}}{e^5 (5+m)}+\frac {24 a^5 (e x)^{6+m}}{e^6 (6+m)}-\frac {8 a^6 (e x)^{7+m}}{e^7 (7+m)}-\frac {8 a^7 (e x)^{8+m}}{e^8 (8+m)} \]
8*(e*x)^(1+m)/e/(1+m)+8*a*(e*x)^(2+m)/e^2/(2+m)-24*a^2*(e*x)^(3+m)/e^3/(3+ m)-24*a^3*(e*x)^(4+m)/e^4/(4+m)+24*a^4*(e*x)^(5+m)/e^5/(5+m)+24*a^5*(e*x)^ (6+m)/e^6/(6+m)-8*a^6*(e*x)^(7+m)/e^7/(7+m)-8*a^7*(e*x)^(8+m)/e^8/(8+m)
Time = 0.10 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.64 \[ \int (e x)^m (2-2 a x)^3 (1+a x)^4 \, dx=8 x (e x)^m \left (\frac {1}{1+m}+\frac {a x}{2+m}-\frac {3 a^2 x^2}{3+m}-\frac {3 a^3 x^3}{4+m}+\frac {3 a^4 x^4}{5+m}+\frac {3 a^5 x^5}{6+m}-\frac {a^6 x^6}{7+m}-\frac {a^7 x^7}{8+m}\right ) \]
8*x*(e*x)^m*((1 + m)^(-1) + (a*x)/(2 + m) - (3*a^2*x^2)/(3 + m) - (3*a^3*x ^3)/(4 + m) + (3*a^4*x^4)/(5 + m) + (3*a^5*x^5)/(6 + m) - (a^6*x^6)/(7 + m ) - (a^7*x^7)/(8 + m))
Time = 0.29 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {99, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int (2-2 a x)^3 (a x+1)^4 (e x)^m \, dx\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \int \left (-\frac {8 a^7 (e x)^{m+7}}{e^7}-\frac {8 a^6 (e x)^{m+6}}{e^6}+\frac {24 a^5 (e x)^{m+5}}{e^5}+\frac {24 a^4 (e x)^{m+4}}{e^4}-\frac {24 a^3 (e x)^{m+3}}{e^3}-\frac {24 a^2 (e x)^{m+2}}{e^2}+\frac {8 a (e x)^{m+1}}{e}+8 (e x)^m\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {8 a^7 (e x)^{m+8}}{e^8 (m+8)}-\frac {8 a^6 (e x)^{m+7}}{e^7 (m+7)}+\frac {24 a^5 (e x)^{m+6}}{e^6 (m+6)}+\frac {24 a^4 (e x)^{m+5}}{e^5 (m+5)}-\frac {24 a^3 (e x)^{m+4}}{e^4 (m+4)}-\frac {24 a^2 (e x)^{m+3}}{e^3 (m+3)}+\frac {8 a (e x)^{m+2}}{e^2 (m+2)}+\frac {8 (e x)^{m+1}}{e (m+1)}\) |
(8*(e*x)^(1 + m))/(e*(1 + m)) + (8*a*(e*x)^(2 + m))/(e^2*(2 + m)) - (24*a^ 2*(e*x)^(3 + m))/(e^3*(3 + m)) - (24*a^3*(e*x)^(4 + m))/(e^4*(4 + m)) + (2 4*a^4*(e*x)^(5 + m))/(e^5*(5 + m)) + (24*a^5*(e*x)^(6 + m))/(e^6*(6 + m)) - (8*a^6*(e*x)^(7 + m))/(e^7*(7 + m)) - (8*a^7*(e*x)^(8 + m))/(e^8*(8 + m) )
3.1.56.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Time = 1.11 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.99
| method | result | size |
| norman | \(\frac {8 x \,{\mathrm e}^{m \ln \left (e x \right )}}{1+m}+\frac {8 a \,x^{2} {\mathrm e}^{m \ln \left (e x \right )}}{2+m}-\frac {24 a^{2} x^{3} {\mathrm e}^{m \ln \left (e x \right )}}{3+m}-\frac {24 a^{3} x^{4} {\mathrm e}^{m \ln \left (e x \right )}}{4+m}+\frac {24 a^{4} x^{5} {\mathrm e}^{m \ln \left (e x \right )}}{5+m}+\frac {24 a^{5} x^{6} {\mathrm e}^{m \ln \left (e x \right )}}{6+m}-\frac {8 a^{6} x^{7} {\mathrm e}^{m \ln \left (e x \right )}}{7+m}-\frac {8 a^{7} x^{8} {\mathrm e}^{m \ln \left (e x \right )}}{8+m}\) | \(155\) |
| gosper | \(-\frac {8 \left (e x \right )^{m} \left (a^{7} m^{7} x^{7}+28 a^{7} m^{6} x^{7}+322 a^{7} m^{5} x^{7}+a^{6} m^{7} x^{6}+1960 a^{7} m^{4} x^{7}+29 a^{6} m^{6} x^{6}+6769 a^{7} m^{3} x^{7}+343 a^{6} m^{5} x^{6}-3 a^{5} m^{7} x^{5}+13132 a^{7} m^{2} x^{7}+2135 a^{6} m^{4} x^{6}-90 a^{5} m^{6} x^{5}+13068 m \,x^{7} a^{7}+7504 a^{6} m^{3} x^{6}-1098 a^{5} m^{5} x^{5}-3 a^{4} m^{7} x^{4}+5040 a^{7} x^{7}+14756 a^{6} m^{2} x^{6}-7020 a^{5} m^{4} x^{5}-93 a^{4} m^{6} x^{4}+14832 m \,x^{6} a^{6}-25227 a^{5} m^{3} x^{5}-1173 a^{4} m^{5} x^{4}+3 a^{3} m^{7} x^{3}+5760 a^{6} x^{6}-50490 a^{5} m^{2} x^{5}-7743 a^{4} m^{4} x^{4}+96 a^{3} m^{6} x^{3}-51432 a^{5} x^{5} m -28632 a^{4} m^{3} x^{4}+1254 a^{3} m^{5} x^{3}+3 a^{2} m^{7} x^{2}-20160 a^{5} x^{5}-58692 a^{4} m^{2} x^{4}+8592 a^{3} m^{4} x^{3}+99 a^{2} m^{6} x^{2}-60912 a^{4} x^{4} m +32979 a^{3} m^{3} x^{3}+1341 a^{2} m^{5} x^{2}-a \,m^{7} x -24192 a^{4} x^{4}+69936 a^{3} m^{2} x^{3}+9585 a^{2} m^{4} x^{2}-34 a \,m^{6} x +74628 a^{3} x^{3} m +38592 a^{2} m^{3} x^{2}-478 a \,m^{5} x -m^{7}+30240 a^{3} x^{3}+86076 a^{2} m^{2} x^{2}-3580 a \,m^{4} x -35 m^{6}+96144 a^{2} m \,x^{2}-15289 a \,m^{3} x -511 m^{5}+40320 a^{2} x^{2}-36706 a \,m^{2} x -4025 m^{4}-44712 a x m -18424 m^{3}-20160 a x -48860 m^{2}-69264 m -40320\right ) x}{\left (8+m \right ) \left (7+m \right ) \left (6+m \right ) \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\) | \(631\) |
| risch | \(-\frac {8 \left (e x \right )^{m} \left (a^{7} m^{7} x^{7}+28 a^{7} m^{6} x^{7}+322 a^{7} m^{5} x^{7}+a^{6} m^{7} x^{6}+1960 a^{7} m^{4} x^{7}+29 a^{6} m^{6} x^{6}+6769 a^{7} m^{3} x^{7}+343 a^{6} m^{5} x^{6}-3 a^{5} m^{7} x^{5}+13132 a^{7} m^{2} x^{7}+2135 a^{6} m^{4} x^{6}-90 a^{5} m^{6} x^{5}+13068 m \,x^{7} a^{7}+7504 a^{6} m^{3} x^{6}-1098 a^{5} m^{5} x^{5}-3 a^{4} m^{7} x^{4}+5040 a^{7} x^{7}+14756 a^{6} m^{2} x^{6}-7020 a^{5} m^{4} x^{5}-93 a^{4} m^{6} x^{4}+14832 m \,x^{6} a^{6}-25227 a^{5} m^{3} x^{5}-1173 a^{4} m^{5} x^{4}+3 a^{3} m^{7} x^{3}+5760 a^{6} x^{6}-50490 a^{5} m^{2} x^{5}-7743 a^{4} m^{4} x^{4}+96 a^{3} m^{6} x^{3}-51432 a^{5} x^{5} m -28632 a^{4} m^{3} x^{4}+1254 a^{3} m^{5} x^{3}+3 a^{2} m^{7} x^{2}-20160 a^{5} x^{5}-58692 a^{4} m^{2} x^{4}+8592 a^{3} m^{4} x^{3}+99 a^{2} m^{6} x^{2}-60912 a^{4} x^{4} m +32979 a^{3} m^{3} x^{3}+1341 a^{2} m^{5} x^{2}-a \,m^{7} x -24192 a^{4} x^{4}+69936 a^{3} m^{2} x^{3}+9585 a^{2} m^{4} x^{2}-34 a \,m^{6} x +74628 a^{3} x^{3} m +38592 a^{2} m^{3} x^{2}-478 a \,m^{5} x -m^{7}+30240 a^{3} x^{3}+86076 a^{2} m^{2} x^{2}-3580 a \,m^{4} x -35 m^{6}+96144 a^{2} m \,x^{2}-15289 a \,m^{3} x -511 m^{5}+40320 a^{2} x^{2}-36706 a \,m^{2} x -4025 m^{4}-44712 a x m -18424 m^{3}-20160 a x -48860 m^{2}-69264 m -40320\right ) x}{\left (8+m \right ) \left (7+m \right ) \left (6+m \right ) \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\) | \(631\) |
| parallelrisch | \(-\frac {76680 x^{3} \left (e x \right )^{m} a^{2} m^{4}-272 x^{2} \left (e x \right )^{m} a \,m^{6}+597024 x^{4} \left (e x \right )^{m} a^{3} m +308736 x^{3} \left (e x \right )^{m} a^{2} m^{3}-3824 x^{2} \left (e x \right )^{m} a \,m^{5}+688608 x^{3} \left (e x \right )^{m} a^{2} m^{2}-28640 x^{2} \left (e x \right )^{m} a \,m^{4}+769152 x^{3} \left (e x \right )^{m} a^{2} m -122312 x^{2} \left (e x \right )^{m} a \,m^{3}-293648 x^{2} \left (e x \right )^{m} a \,m^{2}-357696 x^{2} \left (e x \right )^{m} a m +118656 x^{7} \left (e x \right )^{m} a^{6} m -201816 x^{6} \left (e x \right )^{m} a^{5} m^{3}-9384 x^{5} \left (e x \right )^{m} a^{4} m^{5}+24 x^{4} \left (e x \right )^{m} a^{3} m^{7}-403920 x^{6} \left (e x \right )^{m} a^{5} m^{2}-61944 x^{5} \left (e x \right )^{m} a^{4} m^{4}+768 x^{4} \left (e x \right )^{m} a^{3} m^{6}-411456 x^{6} \left (e x \right )^{m} a^{5} m -229056 x^{5} \left (e x \right )^{m} a^{4} m^{3}+10032 x^{4} \left (e x \right )^{m} a^{3} m^{5}+24 x^{3} \left (e x \right )^{m} a^{2} m^{7}-469536 x^{5} \left (e x \right )^{m} a^{4} m^{2}+68736 x^{4} \left (e x \right )^{m} a^{3} m^{4}+792 x^{3} \left (e x \right )^{m} a^{2} m^{6}-487296 x^{5} \left (e x \right )^{m} a^{4} m +263832 x^{4} \left (e x \right )^{m} a^{3} m^{3}+10728 x^{3} \left (e x \right )^{m} a^{2} m^{5}-8 x^{2} \left (e x \right )^{m} a \,m^{7}+559488 x^{4} \left (e x \right )^{m} a^{3} m^{2}+8 x^{8} \left (e x \right )^{m} a^{7} m^{7}+224 x^{8} \left (e x \right )^{m} a^{7} m^{6}+2576 x^{8} \left (e x \right )^{m} a^{7} m^{5}+8 x^{7} \left (e x \right )^{m} a^{6} m^{7}+15680 x^{8} \left (e x \right )^{m} a^{7} m^{4}+232 x^{7} \left (e x \right )^{m} a^{6} m^{6}+54152 x^{8} \left (e x \right )^{m} a^{7} m^{3}+2744 x^{7} \left (e x \right )^{m} a^{6} m^{5}-24 x^{6} \left (e x \right )^{m} a^{5} m^{7}+105056 x^{8} \left (e x \right )^{m} a^{7} m^{2}+17080 x^{7} \left (e x \right )^{m} a^{6} m^{4}-720 x^{6} \left (e x \right )^{m} a^{5} m^{6}+104544 x^{8} \left (e x \right )^{m} a^{7} m +60032 x^{7} \left (e x \right )^{m} a^{6} m^{3}-8784 x^{6} \left (e x \right )^{m} a^{5} m^{5}-24 x^{5} \left (e x \right )^{m} a^{4} m^{7}+118048 x^{7} \left (e x \right )^{m} a^{6} m^{2}-56160 x^{6} \left (e x \right )^{m} a^{5} m^{4}-744 x^{5} \left (e x \right )^{m} a^{4} m^{6}+46080 x^{7} \left (e x \right )^{m} a^{6}-161280 x^{6} \left (e x \right )^{m} a^{5}-193536 x^{5} \left (e x \right )^{m} a^{4}-8 x \left (e x \right )^{m} m^{7}+241920 x^{4} \left (e x \right )^{m} a^{3}-280 x \left (e x \right )^{m} m^{6}-4088 x \left (e x \right )^{m} m^{5}+322560 x^{3} \left (e x \right )^{m} a^{2}-32200 x \left (e x \right )^{m} m^{4}-147392 x \left (e x \right )^{m} m^{3}-161280 x^{2} \left (e x \right )^{m} a -390880 x \left (e x \right )^{m} m^{2}-554112 x \left (e x \right )^{m} m -322560 \left (e x \right )^{m} x +40320 x^{8} \left (e x \right )^{m} a^{7}}{\left (8+m \right ) \left (7+m \right ) \left (6+m \right ) \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\) | \(972\) |
8/(1+m)*x*exp(m*ln(e*x))+8*a/(2+m)*x^2*exp(m*ln(e*x))-24*a^2/(3+m)*x^3*exp (m*ln(e*x))-24*a^3/(4+m)*x^4*exp(m*ln(e*x))+24*a^4/(5+m)*x^5*exp(m*ln(e*x) )+24*a^5/(6+m)*x^6*exp(m*ln(e*x))-8*a^6/(7+m)*x^7*exp(m*ln(e*x))-8*a^7/(8+ m)*x^8*exp(m*ln(e*x))
Leaf count of result is larger than twice the leaf count of optimal. 514 vs. \(2 (156) = 312\).
Time = 0.23 (sec) , antiderivative size = 514, normalized size of antiderivative = 3.29 \[ \int (e x)^m (2-2 a x)^3 (1+a x)^4 \, dx=-\frac {8 \, {\left ({\left (a^{7} m^{7} + 28 \, a^{7} m^{6} + 322 \, a^{7} m^{5} + 1960 \, a^{7} m^{4} + 6769 \, a^{7} m^{3} + 13132 \, a^{7} m^{2} + 13068 \, a^{7} m + 5040 \, a^{7}\right )} x^{8} + {\left (a^{6} m^{7} + 29 \, a^{6} m^{6} + 343 \, a^{6} m^{5} + 2135 \, a^{6} m^{4} + 7504 \, a^{6} m^{3} + 14756 \, a^{6} m^{2} + 14832 \, a^{6} m + 5760 \, a^{6}\right )} x^{7} - 3 \, {\left (a^{5} m^{7} + 30 \, a^{5} m^{6} + 366 \, a^{5} m^{5} + 2340 \, a^{5} m^{4} + 8409 \, a^{5} m^{3} + 16830 \, a^{5} m^{2} + 17144 \, a^{5} m + 6720 \, a^{5}\right )} x^{6} - 3 \, {\left (a^{4} m^{7} + 31 \, a^{4} m^{6} + 391 \, a^{4} m^{5} + 2581 \, a^{4} m^{4} + 9544 \, a^{4} m^{3} + 19564 \, a^{4} m^{2} + 20304 \, a^{4} m + 8064 \, a^{4}\right )} x^{5} + 3 \, {\left (a^{3} m^{7} + 32 \, a^{3} m^{6} + 418 \, a^{3} m^{5} + 2864 \, a^{3} m^{4} + 10993 \, a^{3} m^{3} + 23312 \, a^{3} m^{2} + 24876 \, a^{3} m + 10080 \, a^{3}\right )} x^{4} + 3 \, {\left (a^{2} m^{7} + 33 \, a^{2} m^{6} + 447 \, a^{2} m^{5} + 3195 \, a^{2} m^{4} + 12864 \, a^{2} m^{3} + 28692 \, a^{2} m^{2} + 32048 \, a^{2} m + 13440 \, a^{2}\right )} x^{3} - {\left (a m^{7} + 34 \, a m^{6} + 478 \, a m^{5} + 3580 \, a m^{4} + 15289 \, a m^{3} + 36706 \, a m^{2} + 44712 \, a m + 20160 \, a\right )} x^{2} - {\left (m^{7} + 35 \, m^{6} + 511 \, m^{5} + 4025 \, m^{4} + 18424 \, m^{3} + 48860 \, m^{2} + 69264 \, m + 40320\right )} x\right )} \left (e x\right )^{m}}{m^{8} + 36 \, m^{7} + 546 \, m^{6} + 4536 \, m^{5} + 22449 \, m^{4} + 67284 \, m^{3} + 118124 \, m^{2} + 109584 \, m + 40320} \]
-8*((a^7*m^7 + 28*a^7*m^6 + 322*a^7*m^5 + 1960*a^7*m^4 + 6769*a^7*m^3 + 13 132*a^7*m^2 + 13068*a^7*m + 5040*a^7)*x^8 + (a^6*m^7 + 29*a^6*m^6 + 343*a^ 6*m^5 + 2135*a^6*m^4 + 7504*a^6*m^3 + 14756*a^6*m^2 + 14832*a^6*m + 5760*a ^6)*x^7 - 3*(a^5*m^7 + 30*a^5*m^6 + 366*a^5*m^5 + 2340*a^5*m^4 + 8409*a^5* m^3 + 16830*a^5*m^2 + 17144*a^5*m + 6720*a^5)*x^6 - 3*(a^4*m^7 + 31*a^4*m^ 6 + 391*a^4*m^5 + 2581*a^4*m^4 + 9544*a^4*m^3 + 19564*a^4*m^2 + 20304*a^4* m + 8064*a^4)*x^5 + 3*(a^3*m^7 + 32*a^3*m^6 + 418*a^3*m^5 + 2864*a^3*m^4 + 10993*a^3*m^3 + 23312*a^3*m^2 + 24876*a^3*m + 10080*a^3)*x^4 + 3*(a^2*m^7 + 33*a^2*m^6 + 447*a^2*m^5 + 3195*a^2*m^4 + 12864*a^2*m^3 + 28692*a^2*m^2 + 32048*a^2*m + 13440*a^2)*x^3 - (a*m^7 + 34*a*m^6 + 478*a*m^5 + 3580*a*m ^4 + 15289*a*m^3 + 36706*a*m^2 + 44712*a*m + 20160*a)*x^2 - (m^7 + 35*m^6 + 511*m^5 + 4025*m^4 + 18424*m^3 + 48860*m^2 + 69264*m + 40320)*x)*(e*x)^m /(m^8 + 36*m^7 + 546*m^6 + 4536*m^5 + 22449*m^4 + 67284*m^3 + 118124*m^2 + 109584*m + 40320)
Leaf count of result is larger than twice the leaf count of optimal. 4027 vs. \(2 (141) = 282\).
Time = 0.68 (sec) , antiderivative size = 4027, normalized size of antiderivative = 25.81 \[ \int (e x)^m (2-2 a x)^3 (1+a x)^4 \, dx=\text {Too large to display} \]
Piecewise(((-8*a**7*log(x) + 8*a**6/x - 12*a**5/x**2 - 8*a**4/x**3 + 6*a** 3/x**4 + 24*a**2/(5*x**5) - 4*a/(3*x**6) - 8/(7*x**7))/e**8, Eq(m, -8)), ( (-8*a**7*x - 8*a**6*log(x) - 24*a**5/x - 12*a**4/x**2 + 8*a**3/x**3 + 6*a* *2/x**4 - 8*a/(5*x**5) - 4/(3*x**6))/e**7, Eq(m, -7)), ((-4*a**7*x**2 - 8* a**6*x + 24*a**5*log(x) - 24*a**4/x + 12*a**3/x**2 + 8*a**2/x**3 - 2*a/x** 4 - 8/(5*x**5))/e**6, Eq(m, -6)), ((-8*a**7*x**3/3 - 4*a**6*x**2 + 24*a**5 *x + 24*a**4*log(x) + 24*a**3/x + 12*a**2/x**2 - 8*a/(3*x**3) - 2/x**4)/e* *5, Eq(m, -5)), ((-2*a**7*x**4 - 8*a**6*x**3/3 + 12*a**5*x**2 + 24*a**4*x - 24*a**3*log(x) + 24*a**2/x - 4*a/x**2 - 8/(3*x**3))/e**4, Eq(m, -4)), (( -8*a**7*x**5/5 - 2*a**6*x**4 + 8*a**5*x**3 + 12*a**4*x**2 - 24*a**3*x - 24 *a**2*log(x) - 8*a/x - 4/x**2)/e**3, Eq(m, -3)), ((-4*a**7*x**6/3 - 8*a**6 *x**5/5 + 6*a**5*x**4 + 8*a**4*x**3 - 12*a**3*x**2 - 24*a**2*x + 8*a*log(x ) - 8/x)/e**2, Eq(m, -2)), ((-8*a**7*x**7/7 - 4*a**6*x**6/3 + 24*a**5*x**5 /5 + 6*a**4*x**4 - 8*a**3*x**3 - 12*a**2*x**2 + 8*a*x + 8*log(x))/e, Eq(m, -1)), (-8*a**7*m**7*x**8*(e*x)**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) - 224*a**7*m** 6*x**8*(e*x)**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 6728 4*m**3 + 118124*m**2 + 109584*m + 40320) - 2576*a**7*m**5*x**8*(e*x)**m/(m **8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m* *2 + 109584*m + 40320) - 15680*a**7*m**4*x**8*(e*x)**m/(m**8 + 36*m**7 ...
Time = 0.21 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.96 \[ \int (e x)^m (2-2 a x)^3 (1+a x)^4 \, dx=-\frac {8 \, a^{7} e^{m} x^{8} x^{m}}{m + 8} - \frac {8 \, a^{6} e^{m} x^{7} x^{m}}{m + 7} + \frac {24 \, a^{5} e^{m} x^{6} x^{m}}{m + 6} + \frac {24 \, a^{4} e^{m} x^{5} x^{m}}{m + 5} - \frac {24 \, a^{3} e^{m} x^{4} x^{m}}{m + 4} - \frac {24 \, a^{2} e^{m} x^{3} x^{m}}{m + 3} + \frac {8 \, a e^{m} x^{2} x^{m}}{m + 2} + \frac {8 \, \left (e x\right )^{m + 1}}{e {\left (m + 1\right )}} \]
-8*a^7*e^m*x^8*x^m/(m + 8) - 8*a^6*e^m*x^7*x^m/(m + 7) + 24*a^5*e^m*x^6*x^ m/(m + 6) + 24*a^4*e^m*x^5*x^m/(m + 5) - 24*a^3*e^m*x^4*x^m/(m + 4) - 24*a ^2*e^m*x^3*x^m/(m + 3) + 8*a*e^m*x^2*x^m/(m + 2) + 8*(e*x)^(m + 1)/(e*(m + 1))
Leaf count of result is larger than twice the leaf count of optimal. 969 vs. \(2 (156) = 312\).
Time = 0.29 (sec) , antiderivative size = 969, normalized size of antiderivative = 6.21 \[ \int (e x)^m (2-2 a x)^3 (1+a x)^4 \, dx=\text {Too large to display} \]
-8*((e*x)^m*a^7*m^7*x^8 + 28*(e*x)^m*a^7*m^6*x^8 + (e*x)^m*a^6*m^7*x^7 + 3 22*(e*x)^m*a^7*m^5*x^8 + 29*(e*x)^m*a^6*m^6*x^7 + 1960*(e*x)^m*a^7*m^4*x^8 - 3*(e*x)^m*a^5*m^7*x^6 + 343*(e*x)^m*a^6*m^5*x^7 + 6769*(e*x)^m*a^7*m^3* x^8 - 90*(e*x)^m*a^5*m^6*x^6 + 2135*(e*x)^m*a^6*m^4*x^7 + 13132*(e*x)^m*a^ 7*m^2*x^8 - 3*(e*x)^m*a^4*m^7*x^5 - 1098*(e*x)^m*a^5*m^5*x^6 + 7504*(e*x)^ m*a^6*m^3*x^7 + 13068*(e*x)^m*a^7*m*x^8 - 93*(e*x)^m*a^4*m^6*x^5 - 7020*(e *x)^m*a^5*m^4*x^6 + 14756*(e*x)^m*a^6*m^2*x^7 + 5040*(e*x)^m*a^7*x^8 + 3*( e*x)^m*a^3*m^7*x^4 - 1173*(e*x)^m*a^4*m^5*x^5 - 25227*(e*x)^m*a^5*m^3*x^6 + 14832*(e*x)^m*a^6*m*x^7 + 96*(e*x)^m*a^3*m^6*x^4 - 7743*(e*x)^m*a^4*m^4* x^5 - 50490*(e*x)^m*a^5*m^2*x^6 + 5760*(e*x)^m*a^6*x^7 + 3*(e*x)^m*a^2*m^7 *x^3 + 1254*(e*x)^m*a^3*m^5*x^4 - 28632*(e*x)^m*a^4*m^3*x^5 - 51432*(e*x)^ m*a^5*m*x^6 + 99*(e*x)^m*a^2*m^6*x^3 + 8592*(e*x)^m*a^3*m^4*x^4 - 58692*(e *x)^m*a^4*m^2*x^5 - 20160*(e*x)^m*a^5*x^6 - (e*x)^m*a*m^7*x^2 + 1341*(e*x) ^m*a^2*m^5*x^3 + 32979*(e*x)^m*a^3*m^3*x^4 - 60912*(e*x)^m*a^4*m*x^5 - 34* (e*x)^m*a*m^6*x^2 + 9585*(e*x)^m*a^2*m^4*x^3 + 69936*(e*x)^m*a^3*m^2*x^4 - 24192*(e*x)^m*a^4*x^5 - (e*x)^m*m^7*x - 478*(e*x)^m*a*m^5*x^2 + 38592*(e* x)^m*a^2*m^3*x^3 + 74628*(e*x)^m*a^3*m*x^4 - 35*(e*x)^m*m^6*x - 3580*(e*x) ^m*a*m^4*x^2 + 86076*(e*x)^m*a^2*m^2*x^3 + 30240*(e*x)^m*a^3*x^4 - 511*(e* x)^m*m^5*x - 15289*(e*x)^m*a*m^3*x^2 + 96144*(e*x)^m*a^2*m*x^3 - 4025*(e*x )^m*m^4*x - 36706*(e*x)^m*a*m^2*x^2 + 40320*(e*x)^m*a^2*x^3 - 18424*(e*...
Time = 0.84 (sec) , antiderivative size = 683, normalized size of antiderivative = 4.38 \[ \int (e x)^m (2-2 a x)^3 (1+a x)^4 \, dx=\frac {x\,{\left (e\,x\right )}^m\,\left (8\,m^7+280\,m^6+4088\,m^5+32200\,m^4+147392\,m^3+390880\,m^2+554112\,m+322560\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}-\frac {8\,a^7\,x^8\,{\left (e\,x\right )}^m\,\left (m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}-\frac {8\,a^6\,x^7\,{\left (e\,x\right )}^m\,\left (m^7+29\,m^6+343\,m^5+2135\,m^4+7504\,m^3+14756\,m^2+14832\,m+5760\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {24\,a^5\,x^6\,{\left (e\,x\right )}^m\,\left (m^7+30\,m^6+366\,m^5+2340\,m^4+8409\,m^3+16830\,m^2+17144\,m+6720\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {24\,a^4\,x^5\,{\left (e\,x\right )}^m\,\left (m^7+31\,m^6+391\,m^5+2581\,m^4+9544\,m^3+19564\,m^2+20304\,m+8064\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}-\frac {24\,a^3\,x^4\,{\left (e\,x\right )}^m\,\left (m^7+32\,m^6+418\,m^5+2864\,m^4+10993\,m^3+23312\,m^2+24876\,m+10080\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}-\frac {24\,a^2\,x^3\,{\left (e\,x\right )}^m\,\left (m^7+33\,m^6+447\,m^5+3195\,m^4+12864\,m^3+28692\,m^2+32048\,m+13440\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {8\,a\,x^2\,{\left (e\,x\right )}^m\,\left (m^7+34\,m^6+478\,m^5+3580\,m^4+15289\,m^3+36706\,m^2+44712\,m+20160\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320} \]
(x*(e*x)^m*(554112*m + 390880*m^2 + 147392*m^3 + 32200*m^4 + 4088*m^5 + 28 0*m^6 + 8*m^7 + 322560))/(109584*m + 118124*m^2 + 67284*m^3 + 22449*m^4 + 4536*m^5 + 546*m^6 + 36*m^7 + m^8 + 40320) - (8*a^7*x^8*(e*x)^m*(13068*m + 13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040))/(109584 *m + 118124*m^2 + 67284*m^3 + 22449*m^4 + 4536*m^5 + 546*m^6 + 36*m^7 + m^ 8 + 40320) - (8*a^6*x^7*(e*x)^m*(14832*m + 14756*m^2 + 7504*m^3 + 2135*m^4 + 343*m^5 + 29*m^6 + m^7 + 5760))/(109584*m + 118124*m^2 + 67284*m^3 + 22 449*m^4 + 4536*m^5 + 546*m^6 + 36*m^7 + m^8 + 40320) + (24*a^5*x^6*(e*x)^m *(17144*m + 16830*m^2 + 8409*m^3 + 2340*m^4 + 366*m^5 + 30*m^6 + m^7 + 672 0))/(109584*m + 118124*m^2 + 67284*m^3 + 22449*m^4 + 4536*m^5 + 546*m^6 + 36*m^7 + m^8 + 40320) + (24*a^4*x^5*(e*x)^m*(20304*m + 19564*m^2 + 9544*m^ 3 + 2581*m^4 + 391*m^5 + 31*m^6 + m^7 + 8064))/(109584*m + 118124*m^2 + 67 284*m^3 + 22449*m^4 + 4536*m^5 + 546*m^6 + 36*m^7 + m^8 + 40320) - (24*a^3 *x^4*(e*x)^m*(24876*m + 23312*m^2 + 10993*m^3 + 2864*m^4 + 418*m^5 + 32*m^ 6 + m^7 + 10080))/(109584*m + 118124*m^2 + 67284*m^3 + 22449*m^4 + 4536*m^ 5 + 546*m^6 + 36*m^7 + m^8 + 40320) - (24*a^2*x^3*(e*x)^m*(32048*m + 28692 *m^2 + 12864*m^3 + 3195*m^4 + 447*m^5 + 33*m^6 + m^7 + 13440))/(109584*m + 118124*m^2 + 67284*m^3 + 22449*m^4 + 4536*m^5 + 546*m^6 + 36*m^7 + m^8 + 40320) + (8*a*x^2*(e*x)^m*(44712*m + 36706*m^2 + 15289*m^3 + 3580*m^4 + 47 8*m^5 + 34*m^6 + m^7 + 20160))/(109584*m + 118124*m^2 + 67284*m^3 + 224...