Integrand size = 33, antiderivative size = 214 \[ \int (a+b x) (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=-\frac {2 (b d-a e)^7 (d+e x)^{5/2}}{5 e^8}+\frac {2 b (b d-a e)^6 (d+e x)^{7/2}}{e^8}-\frac {14 b^2 (b d-a e)^5 (d+e x)^{9/2}}{3 e^8}+\frac {70 b^3 (b d-a e)^4 (d+e x)^{11/2}}{11 e^8}-\frac {70 b^4 (b d-a e)^3 (d+e x)^{13/2}}{13 e^8}+\frac {14 b^5 (b d-a e)^2 (d+e x)^{15/2}}{5 e^8}-\frac {14 b^6 (b d-a e) (d+e x)^{17/2}}{17 e^8}+\frac {2 b^7 (d+e x)^{19/2}}{19 e^8} \]
-2/5*(-a*e+b*d)^7*(e*x+d)^(5/2)/e^8+2*b*(-a*e+b*d)^6*(e*x+d)^(7/2)/e^8-14/ 3*b^2*(-a*e+b*d)^5*(e*x+d)^(9/2)/e^8+70/11*b^3*(-a*e+b*d)^4*(e*x+d)^(11/2) /e^8-70/13*b^4*(-a*e+b*d)^3*(e*x+d)^(13/2)/e^8+14/5*b^5*(-a*e+b*d)^2*(e*x+ d)^(15/2)/e^8-14/17*b^6*(-a*e+b*d)*(e*x+d)^(17/2)/e^8+2/19*b^7*(e*x+d)^(19 /2)/e^8
Time = 0.19 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.76 \[ \int (a+b x) (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {2 (d+e x)^{5/2} \left (138567 a^7 e^7+138567 a^6 b e^6 (-2 d+5 e x)+46189 a^5 b^2 e^5 \left (8 d^2-20 d e x+35 e^2 x^2\right )+20995 a^4 b^3 e^4 \left (-16 d^3+40 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right )+1615 a^3 b^4 e^3 \left (128 d^4-320 d^3 e x+560 d^2 e^2 x^2-840 d e^3 x^3+1155 e^4 x^4\right )+323 a^2 b^5 e^2 \left (-256 d^5+640 d^4 e x-1120 d^3 e^2 x^2+1680 d^2 e^3 x^3-2310 d e^4 x^4+3003 e^5 x^5\right )+19 a b^6 e \left (1024 d^6-2560 d^5 e x+4480 d^4 e^2 x^2-6720 d^3 e^3 x^3+9240 d^2 e^4 x^4-12012 d e^5 x^5+15015 e^6 x^6\right )+b^7 \left (-2048 d^7+5120 d^6 e x-8960 d^5 e^2 x^2+13440 d^4 e^3 x^3-18480 d^3 e^4 x^4+24024 d^2 e^5 x^5-30030 d e^6 x^6+36465 e^7 x^7\right )\right )}{692835 e^8} \]
(2*(d + e*x)^(5/2)*(138567*a^7*e^7 + 138567*a^6*b*e^6*(-2*d + 5*e*x) + 461 89*a^5*b^2*e^5*(8*d^2 - 20*d*e*x + 35*e^2*x^2) + 20995*a^4*b^3*e^4*(-16*d^ 3 + 40*d^2*e*x - 70*d*e^2*x^2 + 105*e^3*x^3) + 1615*a^3*b^4*e^3*(128*d^4 - 320*d^3*e*x + 560*d^2*e^2*x^2 - 840*d*e^3*x^3 + 1155*e^4*x^4) + 323*a^2*b ^5*e^2*(-256*d^5 + 640*d^4*e*x - 1120*d^3*e^2*x^2 + 1680*d^2*e^3*x^3 - 231 0*d*e^4*x^4 + 3003*e^5*x^5) + 19*a*b^6*e*(1024*d^6 - 2560*d^5*e*x + 4480*d ^4*e^2*x^2 - 6720*d^3*e^3*x^3 + 9240*d^2*e^4*x^4 - 12012*d*e^5*x^5 + 15015 *e^6*x^6) + b^7*(-2048*d^7 + 5120*d^6*e*x - 8960*d^5*e^2*x^2 + 13440*d^4*e ^3*x^3 - 18480*d^3*e^4*x^4 + 24024*d^2*e^5*x^5 - 30030*d*e^6*x^6 + 36465*e ^7*x^7)))/(692835*e^8)
Time = 0.35 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {1184, 27, 53, 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.
| \(\displaystyle \int (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3 (d+e x)^{3/2} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle \frac {\int b^6 (a+b x)^7 (d+e x)^{3/2}dx}{b^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int (a+b x)^7 (d+e x)^{3/2}dx\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \int \left (-\frac {7 b^6 (d+e x)^{15/2} (b d-a e)}{e^7}+\frac {21 b^5 (d+e x)^{13/2} (b d-a e)^2}{e^7}-\frac {35 b^4 (d+e x)^{11/2} (b d-a e)^3}{e^7}+\frac {35 b^3 (d+e x)^{9/2} (b d-a e)^4}{e^7}-\frac {21 b^2 (d+e x)^{7/2} (b d-a e)^5}{e^7}+\frac {7 b (d+e x)^{5/2} (b d-a e)^6}{e^7}+\frac {(d+e x)^{3/2} (a e-b d)^7}{e^7}+\frac {b^7 (d+e x)^{17/2}}{e^7}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {14 b^6 (d+e x)^{17/2} (b d-a e)}{17 e^8}+\frac {14 b^5 (d+e x)^{15/2} (b d-a e)^2}{5 e^8}-\frac {70 b^4 (d+e x)^{13/2} (b d-a e)^3}{13 e^8}+\frac {70 b^3 (d+e x)^{11/2} (b d-a e)^4}{11 e^8}-\frac {14 b^2 (d+e x)^{9/2} (b d-a e)^5}{3 e^8}+\frac {2 b (d+e x)^{7/2} (b d-a e)^6}{e^8}-\frac {2 (d+e x)^{5/2} (b d-a e)^7}{5 e^8}+\frac {2 b^7 (d+e x)^{19/2}}{19 e^8}\) |
(-2*(b*d - a*e)^7*(d + e*x)^(5/2))/(5*e^8) + (2*b*(b*d - a*e)^6*(d + e*x)^ (7/2))/e^8 - (14*b^2*(b*d - a*e)^5*(d + e*x)^(9/2))/(3*e^8) + (70*b^3*(b*d - a*e)^4*(d + e*x)^(11/2))/(11*e^8) - (70*b^4*(b*d - a*e)^3*(d + e*x)^(13 /2))/(13*e^8) + (14*b^5*(b*d - a*e)^2*(d + e*x)^(15/2))/(5*e^8) - (14*b^6* (b*d - a*e)*(d + e*x)^(17/2))/(17*e^8) + (2*b^7*(d + e*x)^(19/2))/(19*e^8)
3.21.60.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/c^p Int[(d + e*x)^m*(f + g*x )^n*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x] && E qQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Time = 0.36 (sec) , antiderivative size = 358, normalized size of antiderivative = 1.67
| method | result | size |
| pseudoelliptic | \(\frac {2 \left (\left (\frac {5}{19} b^{7} x^{7}+a^{7}+\frac {35}{17} a \,b^{6} x^{6}+7 a^{2} b^{5} x^{5}+\frac {175}{13} a^{3} b^{4} x^{4}+\frac {175}{11} a^{4} b^{3} x^{3}+\frac {35}{3} a^{5} b^{2} x^{2}+5 a^{6} b x \right ) e^{7}-2 \left (\frac {35}{323} b^{6} x^{6}+\frac {14}{17} a \,b^{5} x^{5}+\frac {35}{13} a^{2} b^{4} x^{4}+\frac {700}{143} a^{3} b^{3} x^{3}+\frac {175}{33} a^{4} b^{2} x^{2}+\frac {10}{3} a^{5} b x +a^{6}\right ) b d \,e^{6}+\frac {8 b^{2} \left (\frac {21}{323} b^{5} x^{5}+\frac {105}{221} a \,b^{4} x^{4}+\frac {210}{143} a^{2} b^{3} x^{3}+\frac {350}{143} a^{3} b^{2} x^{2}+\frac {25}{11} a^{4} b x +a^{5}\right ) d^{2} e^{5}}{3}-\frac {80 b^{3} \left (\frac {231}{4199} x^{4} b^{4}+\frac {84}{221} a \,b^{3} x^{3}+\frac {14}{13} x^{2} b^{2} a^{2}+\frac {20}{13} b \,a^{3} x +a^{4}\right ) d^{3} e^{4}}{33}+\frac {640 \left (\frac {21}{323} x^{3} b^{3}+\frac {7}{17} a \,b^{2} x^{2}+b \,a^{2} x +a^{3}\right ) b^{4} d^{4} e^{3}}{429}-\frac {256 b^{5} \left (\frac {35}{323} b^{2} x^{2}+\frac {10}{17} a b x +a^{2}\right ) d^{5} e^{2}}{429}+\frac {1024 b^{6} \left (\frac {5 b x}{19}+a \right ) d^{6} e}{7293}-\frac {2048 b^{7} d^{7}}{138567}\right ) \left (e x +d \right )^{\frac {5}{2}}}{5 e^{8}}\) | \(358\) |
| gosper | \(\frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (36465 x^{7} b^{7} e^{7}+285285 x^{6} a \,b^{6} e^{7}-30030 x^{6} b^{7} d \,e^{6}+969969 x^{5} a^{2} b^{5} e^{7}-228228 x^{5} a \,b^{6} d \,e^{6}+24024 x^{5} b^{7} d^{2} e^{5}+1865325 x^{4} a^{3} b^{4} e^{7}-746130 x^{4} a^{2} b^{5} d \,e^{6}+175560 x^{4} a \,b^{6} d^{2} e^{5}-18480 x^{4} b^{7} d^{3} e^{4}+2204475 x^{3} a^{4} b^{3} e^{7}-1356600 x^{3} a^{3} b^{4} d \,e^{6}+542640 x^{3} a^{2} b^{5} d^{2} e^{5}-127680 x^{3} a \,b^{6} d^{3} e^{4}+13440 x^{3} b^{7} d^{4} e^{3}+1616615 x^{2} a^{5} b^{2} e^{7}-1469650 x^{2} a^{4} b^{3} d \,e^{6}+904400 x^{2} a^{3} b^{4} d^{2} e^{5}-361760 x^{2} a^{2} b^{5} d^{3} e^{4}+85120 x^{2} a \,b^{6} d^{4} e^{3}-8960 x^{2} b^{7} d^{5} e^{2}+692835 x \,a^{6} b \,e^{7}-923780 x \,a^{5} b^{2} d \,e^{6}+839800 x \,a^{4} b^{3} d^{2} e^{5}-516800 x \,a^{3} b^{4} d^{3} e^{4}+206720 x \,a^{2} b^{5} d^{4} e^{3}-48640 x a \,b^{6} d^{5} e^{2}+5120 x \,b^{7} d^{6} e +138567 e^{7} a^{7}-277134 b d \,e^{6} a^{6}+369512 b^{2} d^{2} e^{5} a^{5}-335920 b^{3} d^{3} e^{4} a^{4}+206720 b^{4} d^{4} e^{3} a^{3}-82688 b^{5} d^{5} e^{2} a^{2}+19456 b^{6} d^{6} e a -2048 b^{7} d^{7}\right )}{692835 e^{8}}\) | \(498\) |
| trager | \(\frac {2 \left (36465 b^{7} e^{9} x^{9}+285285 a \,b^{6} e^{9} x^{8}+42900 b^{7} d \,e^{8} x^{8}+969969 a^{2} b^{5} e^{9} x^{7}+342342 a \,b^{6} d \,e^{8} x^{7}+429 b^{7} d^{2} e^{7} x^{7}+1865325 a^{3} b^{4} e^{9} x^{6}+1193808 a^{2} b^{5} d \,e^{8} x^{6}+4389 a \,b^{6} d^{2} e^{7} x^{6}-462 b^{7} d^{3} e^{6} x^{6}+2204475 a^{4} b^{3} e^{9} x^{5}+2374050 a^{3} b^{4} d \,e^{8} x^{5}+20349 a^{2} b^{5} d^{2} e^{7} x^{5}-4788 a \,b^{6} d^{3} e^{6} x^{5}+504 b^{7} d^{4} e^{5} x^{5}+1616615 a^{5} b^{2} e^{9} x^{4}+2939300 a^{4} b^{3} d \,e^{8} x^{4}+56525 a^{3} b^{4} d^{2} e^{7} x^{4}-22610 a^{2} b^{5} d^{3} e^{6} x^{4}+5320 a \,b^{6} d^{4} e^{5} x^{4}-560 b^{7} d^{5} e^{4} x^{4}+692835 a^{6} b \,e^{9} x^{3}+2309450 a^{5} b^{2} d \,e^{8} x^{3}+104975 a^{4} b^{3} d^{2} e^{7} x^{3}-64600 a^{3} b^{4} d^{3} e^{6} x^{3}+25840 a^{2} b^{5} d^{4} e^{5} x^{3}-6080 a \,b^{6} d^{5} e^{4} x^{3}+640 b^{7} d^{6} e^{3} x^{3}+138567 a^{7} e^{9} x^{2}+1108536 a^{6} b d \,e^{8} x^{2}+138567 a^{5} b^{2} d^{2} e^{7} x^{2}-125970 a^{4} b^{3} d^{3} e^{6} x^{2}+77520 a^{3} b^{4} d^{4} e^{5} x^{2}-31008 a^{2} b^{5} d^{5} e^{4} x^{2}+7296 a \,b^{6} d^{6} e^{3} x^{2}-768 b^{7} d^{7} e^{2} x^{2}+277134 a^{7} d \,e^{8} x +138567 a^{6} b \,d^{2} e^{7} x -184756 a^{5} b^{2} d^{3} e^{6} x +167960 a^{4} b^{3} d^{4} e^{5} x -103360 a^{3} b^{4} d^{5} e^{4} x +41344 a^{2} b^{5} d^{6} e^{3} x -9728 a \,b^{6} d^{7} e^{2} x +1024 b^{7} d^{8} e x +138567 a^{7} d^{2} e^{7}-277134 a^{6} b \,d^{3} e^{6}+369512 a^{5} b^{2} d^{4} e^{5}-335920 a^{4} b^{3} d^{5} e^{4}+206720 a^{3} b^{4} d^{6} e^{3}-82688 a^{2} b^{5} d^{7} e^{2}+19456 a \,b^{6} d^{8} e -2048 b^{7} d^{9}\right ) \sqrt {e x +d}}{692835 e^{8}}\) | \(746\) |
| risch | \(\frac {2 \left (36465 b^{7} e^{9} x^{9}+285285 a \,b^{6} e^{9} x^{8}+42900 b^{7} d \,e^{8} x^{8}+969969 a^{2} b^{5} e^{9} x^{7}+342342 a \,b^{6} d \,e^{8} x^{7}+429 b^{7} d^{2} e^{7} x^{7}+1865325 a^{3} b^{4} e^{9} x^{6}+1193808 a^{2} b^{5} d \,e^{8} x^{6}+4389 a \,b^{6} d^{2} e^{7} x^{6}-462 b^{7} d^{3} e^{6} x^{6}+2204475 a^{4} b^{3} e^{9} x^{5}+2374050 a^{3} b^{4} d \,e^{8} x^{5}+20349 a^{2} b^{5} d^{2} e^{7} x^{5}-4788 a \,b^{6} d^{3} e^{6} x^{5}+504 b^{7} d^{4} e^{5} x^{5}+1616615 a^{5} b^{2} e^{9} x^{4}+2939300 a^{4} b^{3} d \,e^{8} x^{4}+56525 a^{3} b^{4} d^{2} e^{7} x^{4}-22610 a^{2} b^{5} d^{3} e^{6} x^{4}+5320 a \,b^{6} d^{4} e^{5} x^{4}-560 b^{7} d^{5} e^{4} x^{4}+692835 a^{6} b \,e^{9} x^{3}+2309450 a^{5} b^{2} d \,e^{8} x^{3}+104975 a^{4} b^{3} d^{2} e^{7} x^{3}-64600 a^{3} b^{4} d^{3} e^{6} x^{3}+25840 a^{2} b^{5} d^{4} e^{5} x^{3}-6080 a \,b^{6} d^{5} e^{4} x^{3}+640 b^{7} d^{6} e^{3} x^{3}+138567 a^{7} e^{9} x^{2}+1108536 a^{6} b d \,e^{8} x^{2}+138567 a^{5} b^{2} d^{2} e^{7} x^{2}-125970 a^{4} b^{3} d^{3} e^{6} x^{2}+77520 a^{3} b^{4} d^{4} e^{5} x^{2}-31008 a^{2} b^{5} d^{5} e^{4} x^{2}+7296 a \,b^{6} d^{6} e^{3} x^{2}-768 b^{7} d^{7} e^{2} x^{2}+277134 a^{7} d \,e^{8} x +138567 a^{6} b \,d^{2} e^{7} x -184756 a^{5} b^{2} d^{3} e^{6} x +167960 a^{4} b^{3} d^{4} e^{5} x -103360 a^{3} b^{4} d^{5} e^{4} x +41344 a^{2} b^{5} d^{6} e^{3} x -9728 a \,b^{6} d^{7} e^{2} x +1024 b^{7} d^{8} e x +138567 a^{7} d^{2} e^{7}-277134 a^{6} b \,d^{3} e^{6}+369512 a^{5} b^{2} d^{4} e^{5}-335920 a^{4} b^{3} d^{5} e^{4}+206720 a^{3} b^{4} d^{6} e^{3}-82688 a^{2} b^{5} d^{7} e^{2}+19456 a \,b^{6} d^{8} e -2048 b^{7} d^{9}\right ) \sqrt {e x +d}}{692835 e^{8}}\) | \(746\) |
| derivativedivides | \(\frac {\frac {2 b^{7} \left (e x +d \right )^{\frac {19}{2}}}{19}+\frac {2 \left (\left (a e -b d \right ) b^{6}+3 b^{5} \left (2 a b e -2 b^{2} d \right )\right ) \left (e x +d \right )^{\frac {17}{2}}}{17}+\frac {2 \left (3 \left (a e -b d \right ) \left (2 a b e -2 b^{2} d \right ) b^{4}+b \left (\left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{4}+2 \left (2 a b e -2 b^{2} d \right )^{2} b^{2}+b^{2} \left (2 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right )\right )\right ) \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {2 \left (\left (a e -b d \right ) \left (\left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{4}+2 \left (2 a b e -2 b^{2} d \right )^{2} b^{2}+b^{2} \left (2 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right )\right )+b \left (4 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a b e -2 b^{2} d \right ) b^{2}+\left (2 a b e -2 b^{2} d \right ) \left (2 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right )\right )\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (\left (a e -b d \right ) \left (4 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a b e -2 b^{2} d \right ) b^{2}+\left (2 a b e -2 b^{2} d \right ) \left (2 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right )\right )+b \left (\left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right )+2 \left (2 a b e -2 b^{2} d \right )^{2} \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )+b^{2} \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{2}\right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (a e -b d \right ) \left (\left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right )+2 \left (2 a b e -2 b^{2} d \right )^{2} \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )+b^{2} \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{2}\right )+3 b \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{2} \left (2 a b e -2 b^{2} d \right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (3 \left (a e -b d \right ) \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{2} \left (2 a b e -2 b^{2} d \right )+b \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{3}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (a e -b d \right ) \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{3} \left (e x +d \right )^{\frac {5}{2}}}{5}}{e^{8}}\) | \(936\) |
| default | \(\frac {\frac {2 b^{7} \left (e x +d \right )^{\frac {19}{2}}}{19}+\frac {2 \left (\left (a e -b d \right ) b^{6}+3 b^{5} \left (2 a b e -2 b^{2} d \right )\right ) \left (e x +d \right )^{\frac {17}{2}}}{17}+\frac {2 \left (3 \left (a e -b d \right ) \left (2 a b e -2 b^{2} d \right ) b^{4}+b \left (\left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{4}+2 \left (2 a b e -2 b^{2} d \right )^{2} b^{2}+b^{2} \left (2 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right )\right )\right ) \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {2 \left (\left (a e -b d \right ) \left (\left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{4}+2 \left (2 a b e -2 b^{2} d \right )^{2} b^{2}+b^{2} \left (2 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right )\right )+b \left (4 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a b e -2 b^{2} d \right ) b^{2}+\left (2 a b e -2 b^{2} d \right ) \left (2 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right )\right )\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (\left (a e -b d \right ) \left (4 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a b e -2 b^{2} d \right ) b^{2}+\left (2 a b e -2 b^{2} d \right ) \left (2 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right )\right )+b \left (\left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right )+2 \left (2 a b e -2 b^{2} d \right )^{2} \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )+b^{2} \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{2}\right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (a e -b d \right ) \left (\left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right )+2 \left (2 a b e -2 b^{2} d \right )^{2} \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )+b^{2} \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{2}\right )+3 b \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{2} \left (2 a b e -2 b^{2} d \right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (3 \left (a e -b d \right ) \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{2} \left (2 a b e -2 b^{2} d \right )+b \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{3}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (a e -b d \right ) \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{3} \left (e x +d \right )^{\frac {5}{2}}}{5}}{e^{8}}\) | \(936\) |
2/5*((5/19*b^7*x^7+a^7+35/17*a*b^6*x^6+7*a^2*b^5*x^5+175/13*a^3*b^4*x^4+17 5/11*a^4*b^3*x^3+35/3*a^5*b^2*x^2+5*a^6*b*x)*e^7-2*(35/323*b^6*x^6+14/17*a *b^5*x^5+35/13*a^2*b^4*x^4+700/143*a^3*b^3*x^3+175/33*a^4*b^2*x^2+10/3*a^5 *b*x+a^6)*b*d*e^6+8/3*b^2*(21/323*b^5*x^5+105/221*a*b^4*x^4+210/143*a^2*b^ 3*x^3+350/143*a^3*b^2*x^2+25/11*a^4*b*x+a^5)*d^2*e^5-80/33*b^3*(231/4199*x ^4*b^4+84/221*a*b^3*x^3+14/13*x^2*b^2*a^2+20/13*b*a^3*x+a^4)*d^3*e^4+640/4 29*(21/323*x^3*b^3+7/17*a*b^2*x^2+b*a^2*x+a^3)*b^4*d^4*e^3-256/429*b^5*(35 /323*b^2*x^2+10/17*a*b*x+a^2)*d^5*e^2+1024/7293*b^6*(5/19*b*x+a)*d^6*e-204 8/138567*b^7*d^7)*(e*x+d)^(5/2)/e^8
Leaf count of result is larger than twice the leaf count of optimal. 676 vs. \(2 (184) = 368\).
Time = 0.29 (sec) , antiderivative size = 676, normalized size of antiderivative = 3.16 \[ \int (a+b x) (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {2 \, {\left (36465 \, b^{7} e^{9} x^{9} - 2048 \, b^{7} d^{9} + 19456 \, a b^{6} d^{8} e - 82688 \, a^{2} b^{5} d^{7} e^{2} + 206720 \, a^{3} b^{4} d^{6} e^{3} - 335920 \, a^{4} b^{3} d^{5} e^{4} + 369512 \, a^{5} b^{2} d^{4} e^{5} - 277134 \, a^{6} b d^{3} e^{6} + 138567 \, a^{7} d^{2} e^{7} + 2145 \, {\left (20 \, b^{7} d e^{8} + 133 \, a b^{6} e^{9}\right )} x^{8} + 429 \, {\left (b^{7} d^{2} e^{7} + 798 \, a b^{6} d e^{8} + 2261 \, a^{2} b^{5} e^{9}\right )} x^{7} - 231 \, {\left (2 \, b^{7} d^{3} e^{6} - 19 \, a b^{6} d^{2} e^{7} - 5168 \, a^{2} b^{5} d e^{8} - 8075 \, a^{3} b^{4} e^{9}\right )} x^{6} + 21 \, {\left (24 \, b^{7} d^{4} e^{5} - 228 \, a b^{6} d^{3} e^{6} + 969 \, a^{2} b^{5} d^{2} e^{7} + 113050 \, a^{3} b^{4} d e^{8} + 104975 \, a^{4} b^{3} e^{9}\right )} x^{5} - 35 \, {\left (16 \, b^{7} d^{5} e^{4} - 152 \, a b^{6} d^{4} e^{5} + 646 \, a^{2} b^{5} d^{3} e^{6} - 1615 \, a^{3} b^{4} d^{2} e^{7} - 83980 \, a^{4} b^{3} d e^{8} - 46189 \, a^{5} b^{2} e^{9}\right )} x^{4} + 5 \, {\left (128 \, b^{7} d^{6} e^{3} - 1216 \, a b^{6} d^{5} e^{4} + 5168 \, a^{2} b^{5} d^{4} e^{5} - 12920 \, a^{3} b^{4} d^{3} e^{6} + 20995 \, a^{4} b^{3} d^{2} e^{7} + 461890 \, a^{5} b^{2} d e^{8} + 138567 \, a^{6} b e^{9}\right )} x^{3} - 3 \, {\left (256 \, b^{7} d^{7} e^{2} - 2432 \, a b^{6} d^{6} e^{3} + 10336 \, a^{2} b^{5} d^{5} e^{4} - 25840 \, a^{3} b^{4} d^{4} e^{5} + 41990 \, a^{4} b^{3} d^{3} e^{6} - 46189 \, a^{5} b^{2} d^{2} e^{7} - 369512 \, a^{6} b d e^{8} - 46189 \, a^{7} e^{9}\right )} x^{2} + {\left (1024 \, b^{7} d^{8} e - 9728 \, a b^{6} d^{7} e^{2} + 41344 \, a^{2} b^{5} d^{6} e^{3} - 103360 \, a^{3} b^{4} d^{5} e^{4} + 167960 \, a^{4} b^{3} d^{4} e^{5} - 184756 \, a^{5} b^{2} d^{3} e^{6} + 138567 \, a^{6} b d^{2} e^{7} + 277134 \, a^{7} d e^{8}\right )} x\right )} \sqrt {e x + d}}{692835 \, e^{8}} \]
2/692835*(36465*b^7*e^9*x^9 - 2048*b^7*d^9 + 19456*a*b^6*d^8*e - 82688*a^2 *b^5*d^7*e^2 + 206720*a^3*b^4*d^6*e^3 - 335920*a^4*b^3*d^5*e^4 + 369512*a^ 5*b^2*d^4*e^5 - 277134*a^6*b*d^3*e^6 + 138567*a^7*d^2*e^7 + 2145*(20*b^7*d *e^8 + 133*a*b^6*e^9)*x^8 + 429*(b^7*d^2*e^7 + 798*a*b^6*d*e^8 + 2261*a^2* b^5*e^9)*x^7 - 231*(2*b^7*d^3*e^6 - 19*a*b^6*d^2*e^7 - 5168*a^2*b^5*d*e^8 - 8075*a^3*b^4*e^9)*x^6 + 21*(24*b^7*d^4*e^5 - 228*a*b^6*d^3*e^6 + 969*a^2 *b^5*d^2*e^7 + 113050*a^3*b^4*d*e^8 + 104975*a^4*b^3*e^9)*x^5 - 35*(16*b^7 *d^5*e^4 - 152*a*b^6*d^4*e^5 + 646*a^2*b^5*d^3*e^6 - 1615*a^3*b^4*d^2*e^7 - 83980*a^4*b^3*d*e^8 - 46189*a^5*b^2*e^9)*x^4 + 5*(128*b^7*d^6*e^3 - 1216 *a*b^6*d^5*e^4 + 5168*a^2*b^5*d^4*e^5 - 12920*a^3*b^4*d^3*e^6 + 20995*a^4* b^3*d^2*e^7 + 461890*a^5*b^2*d*e^8 + 138567*a^6*b*e^9)*x^3 - 3*(256*b^7*d^ 7*e^2 - 2432*a*b^6*d^6*e^3 + 10336*a^2*b^5*d^5*e^4 - 25840*a^3*b^4*d^4*e^5 + 41990*a^4*b^3*d^3*e^6 - 46189*a^5*b^2*d^2*e^7 - 369512*a^6*b*d*e^8 - 46 189*a^7*e^9)*x^2 + (1024*b^7*d^8*e - 9728*a*b^6*d^7*e^2 + 41344*a^2*b^5*d^ 6*e^3 - 103360*a^3*b^4*d^5*e^4 + 167960*a^4*b^3*d^4*e^5 - 184756*a^5*b^2*d ^3*e^6 + 138567*a^6*b*d^2*e^7 + 277134*a^7*d*e^8)*x)*sqrt(e*x + d)/e^8
Leaf count of result is larger than twice the leaf count of optimal. 576 vs. \(2 (199) = 398\).
Time = 1.91 (sec) , antiderivative size = 576, normalized size of antiderivative = 2.69 \[ \int (a+b x) (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\begin {cases} \frac {2 \left (\frac {b^{7} \left (d + e x\right )^{\frac {19}{2}}}{19 e^{7}} + \frac {\left (d + e x\right )^{\frac {17}{2}} \cdot \left (7 a b^{6} e - 7 b^{7} d\right )}{17 e^{7}} + \frac {\left (d + e x\right )^{\frac {15}{2}} \cdot \left (21 a^{2} b^{5} e^{2} - 42 a b^{6} d e + 21 b^{7} d^{2}\right )}{15 e^{7}} + \frac {\left (d + e x\right )^{\frac {13}{2}} \cdot \left (35 a^{3} b^{4} e^{3} - 105 a^{2} b^{5} d e^{2} + 105 a b^{6} d^{2} e - 35 b^{7} d^{3}\right )}{13 e^{7}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \cdot \left (35 a^{4} b^{3} e^{4} - 140 a^{3} b^{4} d e^{3} + 210 a^{2} b^{5} d^{2} e^{2} - 140 a b^{6} d^{3} e + 35 b^{7} d^{4}\right )}{11 e^{7}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \cdot \left (21 a^{5} b^{2} e^{5} - 105 a^{4} b^{3} d e^{4} + 210 a^{3} b^{4} d^{2} e^{3} - 210 a^{2} b^{5} d^{3} e^{2} + 105 a b^{6} d^{4} e - 21 b^{7} d^{5}\right )}{9 e^{7}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \cdot \left (7 a^{6} b e^{6} - 42 a^{5} b^{2} d e^{5} + 105 a^{4} b^{3} d^{2} e^{4} - 140 a^{3} b^{4} d^{3} e^{3} + 105 a^{2} b^{5} d^{4} e^{2} - 42 a b^{6} d^{5} e + 7 b^{7} d^{6}\right )}{7 e^{7}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (a^{7} e^{7} - 7 a^{6} b d e^{6} + 21 a^{5} b^{2} d^{2} e^{5} - 35 a^{4} b^{3} d^{3} e^{4} + 35 a^{3} b^{4} d^{4} e^{3} - 21 a^{2} b^{5} d^{5} e^{2} + 7 a b^{6} d^{6} e - b^{7} d^{7}\right )}{5 e^{7}}\right )}{e} & \text {for}\: e \neq 0 \\d^{\frac {3}{2}} \left (\begin {cases} a^{7} x & \text {for}\: b = 0 \\\frac {\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{4}}{8 b} & \text {otherwise} \end {cases}\right ) & \text {otherwise} \end {cases} \]
Piecewise((2*(b**7*(d + e*x)**(19/2)/(19*e**7) + (d + e*x)**(17/2)*(7*a*b* *6*e - 7*b**7*d)/(17*e**7) + (d + e*x)**(15/2)*(21*a**2*b**5*e**2 - 42*a*b **6*d*e + 21*b**7*d**2)/(15*e**7) + (d + e*x)**(13/2)*(35*a**3*b**4*e**3 - 105*a**2*b**5*d*e**2 + 105*a*b**6*d**2*e - 35*b**7*d**3)/(13*e**7) + (d + e*x)**(11/2)*(35*a**4*b**3*e**4 - 140*a**3*b**4*d*e**3 + 210*a**2*b**5*d* *2*e**2 - 140*a*b**6*d**3*e + 35*b**7*d**4)/(11*e**7) + (d + e*x)**(9/2)*( 21*a**5*b**2*e**5 - 105*a**4*b**3*d*e**4 + 210*a**3*b**4*d**2*e**3 - 210*a **2*b**5*d**3*e**2 + 105*a*b**6*d**4*e - 21*b**7*d**5)/(9*e**7) + (d + e*x )**(7/2)*(7*a**6*b*e**6 - 42*a**5*b**2*d*e**5 + 105*a**4*b**3*d**2*e**4 - 140*a**3*b**4*d**3*e**3 + 105*a**2*b**5*d**4*e**2 - 42*a*b**6*d**5*e + 7*b **7*d**6)/(7*e**7) + (d + e*x)**(5/2)*(a**7*e**7 - 7*a**6*b*d*e**6 + 21*a* *5*b**2*d**2*e**5 - 35*a**4*b**3*d**3*e**4 + 35*a**3*b**4*d**4*e**3 - 21*a **2*b**5*d**5*e**2 + 7*a*b**6*d**6*e - b**7*d**7)/(5*e**7))/e, Ne(e, 0)), (d**(3/2)*Piecewise((a**7*x, Eq(b, 0)), ((a**2 + 2*a*b*x + b**2*x**2)**4/( 8*b), True)), True))
Leaf count of result is larger than twice the leaf count of optimal. 456 vs. \(2 (184) = 368\).
Time = 0.20 (sec) , antiderivative size = 456, normalized size of antiderivative = 2.13 \[ \int (a+b x) (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {2 \, {\left (36465 \, {\left (e x + d\right )}^{\frac {19}{2}} b^{7} - 285285 \, {\left (b^{7} d - a b^{6} e\right )} {\left (e x + d\right )}^{\frac {17}{2}} + 969969 \, {\left (b^{7} d^{2} - 2 \, a b^{6} d e + a^{2} b^{5} e^{2}\right )} {\left (e x + d\right )}^{\frac {15}{2}} - 1865325 \, {\left (b^{7} d^{3} - 3 \, a b^{6} d^{2} e + 3 \, a^{2} b^{5} d e^{2} - a^{3} b^{4} e^{3}\right )} {\left (e x + d\right )}^{\frac {13}{2}} + 2204475 \, {\left (b^{7} d^{4} - 4 \, a b^{6} d^{3} e + 6 \, a^{2} b^{5} d^{2} e^{2} - 4 \, a^{3} b^{4} d e^{3} + a^{4} b^{3} e^{4}\right )} {\left (e x + d\right )}^{\frac {11}{2}} - 1616615 \, {\left (b^{7} d^{5} - 5 \, a b^{6} d^{4} e + 10 \, a^{2} b^{5} d^{3} e^{2} - 10 \, a^{3} b^{4} d^{2} e^{3} + 5 \, a^{4} b^{3} d e^{4} - a^{5} b^{2} e^{5}\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 692835 \, {\left (b^{7} d^{6} - 6 \, a b^{6} d^{5} e + 15 \, a^{2} b^{5} d^{4} e^{2} - 20 \, a^{3} b^{4} d^{3} e^{3} + 15 \, a^{4} b^{3} d^{2} e^{4} - 6 \, a^{5} b^{2} d e^{5} + a^{6} b e^{6}\right )} {\left (e x + d\right )}^{\frac {7}{2}} - 138567 \, {\left (b^{7} d^{7} - 7 \, a b^{6} d^{6} e + 21 \, a^{2} b^{5} d^{5} e^{2} - 35 \, a^{3} b^{4} d^{4} e^{3} + 35 \, a^{4} b^{3} d^{3} e^{4} - 21 \, a^{5} b^{2} d^{2} e^{5} + 7 \, a^{6} b d e^{6} - a^{7} e^{7}\right )} {\left (e x + d\right )}^{\frac {5}{2}}\right )}}{692835 \, e^{8}} \]
2/692835*(36465*(e*x + d)^(19/2)*b^7 - 285285*(b^7*d - a*b^6*e)*(e*x + d)^ (17/2) + 969969*(b^7*d^2 - 2*a*b^6*d*e + a^2*b^5*e^2)*(e*x + d)^(15/2) - 1 865325*(b^7*d^3 - 3*a*b^6*d^2*e + 3*a^2*b^5*d*e^2 - a^3*b^4*e^3)*(e*x + d) ^(13/2) + 2204475*(b^7*d^4 - 4*a*b^6*d^3*e + 6*a^2*b^5*d^2*e^2 - 4*a^3*b^4 *d*e^3 + a^4*b^3*e^4)*(e*x + d)^(11/2) - 1616615*(b^7*d^5 - 5*a*b^6*d^4*e + 10*a^2*b^5*d^3*e^2 - 10*a^3*b^4*d^2*e^3 + 5*a^4*b^3*d*e^4 - a^5*b^2*e^5) *(e*x + d)^(9/2) + 692835*(b^7*d^6 - 6*a*b^6*d^5*e + 15*a^2*b^5*d^4*e^2 - 20*a^3*b^4*d^3*e^3 + 15*a^4*b^3*d^2*e^4 - 6*a^5*b^2*d*e^5 + a^6*b*e^6)*(e* x + d)^(7/2) - 138567*(b^7*d^7 - 7*a*b^6*d^6*e + 21*a^2*b^5*d^5*e^2 - 35*a ^3*b^4*d^4*e^3 + 35*a^4*b^3*d^3*e^4 - 21*a^5*b^2*d^2*e^5 + 7*a^6*b*d*e^6 - a^7*e^7)*(e*x + d)^(5/2))/e^8
Leaf count of result is larger than twice the leaf count of optimal. 1746 vs. \(2 (184) = 368\).
Time = 0.32 (sec) , antiderivative size = 1746, normalized size of antiderivative = 8.16 \[ \int (a+b x) (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\text {Too large to display} \]
2/2078505*(2078505*sqrt(e*x + d)*a^7*d^2 + 1385670*((e*x + d)^(3/2) - 3*sq rt(e*x + d)*d)*a^7*d + 4849845*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*a^6*b *d^2/e + 138567*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a^7 + 2909907*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt( e*x + d)*d^2)*a^5*b^2*d^2/e^2 + 1939938*(3*(e*x + d)^(5/2) - 10*(e*x + d)^ (3/2)*d + 15*sqrt(e*x + d)*d^2)*a^6*b*d/e + 2078505*(5*(e*x + d)^(7/2) - 2 1*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*a^4*b ^3*d^2/e^3 + 2494206*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*a^5*b^2*d/e^2 + 415701*(5*(e*x + d)^ (7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d ^3)*a^6*b/e + 230945*(35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 378*(e* x + d)^(5/2)*d^2 - 420*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*a^3*b^ 4*d^2/e^4 + 461890*(35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 378*(e*x + d)^(5/2)*d^2 - 420*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*a^4*b^3* d/e^3 + 138567*(35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 378*(e*x + d) ^(5/2)*d^2 - 420*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*a^5*b^2/e^2 + 62985*(63*(e*x + d)^(11/2) - 385*(e*x + d)^(9/2)*d + 990*(e*x + d)^(7/2) *d^2 - 1386*(e*x + d)^(5/2)*d^3 + 1155*(e*x + d)^(3/2)*d^4 - 693*sqrt(e*x + d)*d^5)*a^2*b^5*d^2/e^5 + 209950*(63*(e*x + d)^(11/2) - 385*(e*x + d)^(9 /2)*d + 990*(e*x + d)^(7/2)*d^2 - 1386*(e*x + d)^(5/2)*d^3 + 1155*(e*x ...
Time = 11.07 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.87 \[ \int (a+b x) (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {2\,b^7\,{\left (d+e\,x\right )}^{19/2}}{19\,e^8}-\frac {\left (14\,b^7\,d-14\,a\,b^6\,e\right )\,{\left (d+e\,x\right )}^{17/2}}{17\,e^8}+\frac {2\,{\left (a\,e-b\,d\right )}^7\,{\left (d+e\,x\right )}^{5/2}}{5\,e^8}+\frac {14\,b^2\,{\left (a\,e-b\,d\right )}^5\,{\left (d+e\,x\right )}^{9/2}}{3\,e^8}+\frac {70\,b^3\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{11/2}}{11\,e^8}+\frac {70\,b^4\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{13/2}}{13\,e^8}+\frac {14\,b^5\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{15/2}}{5\,e^8}+\frac {2\,b\,{\left (a\,e-b\,d\right )}^6\,{\left (d+e\,x\right )}^{7/2}}{e^8} \]
(2*b^7*(d + e*x)^(19/2))/(19*e^8) - ((14*b^7*d - 14*a*b^6*e)*(d + e*x)^(17 /2))/(17*e^8) + (2*(a*e - b*d)^7*(d + e*x)^(5/2))/(5*e^8) + (14*b^2*(a*e - b*d)^5*(d + e*x)^(9/2))/(3*e^8) + (70*b^3*(a*e - b*d)^4*(d + e*x)^(11/2)) /(11*e^8) + (70*b^4*(a*e - b*d)^3*(d + e*x)^(13/2))/(13*e^8) + (14*b^5*(a* e - b*d)^2*(d + e*x)^(15/2))/(5*e^8) + (2*b*(a*e - b*d)^6*(d + e*x)^(7/2)) /e^8