Integrand size = 33, antiderivative size = 308 \[ \int (A+B x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=-\frac {2 (b d-a e)^6 (B d-A e) (d+e x)^{7/2}}{7 e^8}+\frac {2 (b d-a e)^5 (7 b B d-6 A b e-a B e) (d+e x)^{9/2}}{9 e^8}-\frac {6 b (b d-a e)^4 (7 b B d-5 A b e-2 a B e) (d+e x)^{11/2}}{11 e^8}+\frac {10 b^2 (b d-a e)^3 (7 b B d-4 A b e-3 a B e) (d+e x)^{13/2}}{13 e^8}-\frac {2 b^3 (b d-a e)^2 (7 b B d-3 A b e-4 a B e) (d+e x)^{15/2}}{3 e^8}+\frac {6 b^4 (b d-a e) (7 b B d-2 A b e-5 a B e) (d+e x)^{17/2}}{17 e^8}-\frac {2 b^5 (7 b B d-A b e-6 a B e) (d+e x)^{19/2}}{19 e^8}+\frac {2 b^6 B (d+e x)^{21/2}}{21 e^8} \]
-2/7*(-a*e+b*d)^6*(-A*e+B*d)*(e*x+d)^(7/2)/e^8+2/9*(-a*e+b*d)^5*(-6*A*b*e- B*a*e+7*B*b*d)*(e*x+d)^(9/2)/e^8-6/11*b*(-a*e+b*d)^4*(-5*A*b*e-2*B*a*e+7*B *b*d)*(e*x+d)^(11/2)/e^8+10/13*b^2*(-a*e+b*d)^3*(-4*A*b*e-3*B*a*e+7*B*b*d) *(e*x+d)^(13/2)/e^8-2/3*b^3*(-a*e+b*d)^2*(-3*A*b*e-4*B*a*e+7*B*b*d)*(e*x+d )^(15/2)/e^8+6/17*b^4*(-a*e+b*d)*(-2*A*b*e-5*B*a*e+7*B*b*d)*(e*x+d)^(17/2) /e^8-2/19*b^5*(-A*b*e-6*B*a*e+7*B*b*d)*(e*x+d)^(19/2)/e^8+2/21*b^6*B*(e*x+ d)^(21/2)/e^8
Leaf count is larger than twice the leaf count of optimal. \(629\) vs. \(2(308)=616\).
Time = 0.50 (sec) , antiderivative size = 629, normalized size of antiderivative = 2.04 \[ \int (A+B x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {2 (d+e x)^{7/2} \left (46189 a^6 e^6 (-2 B d+9 A e+7 B e x)+25194 a^5 b e^5 \left (11 A e (-2 d+7 e x)+B \left (8 d^2-28 d e x+63 e^2 x^2\right )\right )-4845 a^4 b^2 e^4 \left (-13 A e \left (8 d^2-28 d e x+63 e^2 x^2\right )+3 B \left (16 d^3-56 d^2 e x+126 d e^2 x^2-231 e^3 x^3\right )\right )+1292 a^3 b^3 e^3 \left (15 A e \left (-16 d^3+56 d^2 e x-126 d e^2 x^2+231 e^3 x^3\right )+B \left (128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right )\right )-57 a^2 b^4 e^2 \left (-17 A e \left (128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right )+5 B \left (256 d^5-896 d^4 e x+2016 d^3 e^2 x^2-3696 d^2 e^3 x^3+6006 d e^4 x^4-9009 e^5 x^5\right )\right )+6 a b^5 e \left (19 A e \left (-256 d^5+896 d^4 e x-2016 d^3 e^2 x^2+3696 d^2 e^3 x^3-6006 d e^4 x^4+9009 e^5 x^5\right )+3 B \left (1024 d^6-3584 d^5 e x+8064 d^4 e^2 x^2-14784 d^3 e^3 x^3+24024 d^2 e^4 x^4-36036 d e^5 x^5+51051 e^6 x^6\right )\right )+b^6 \left (3 A e \left (1024 d^6-3584 d^5 e x+8064 d^4 e^2 x^2-14784 d^3 e^3 x^3+24024 d^2 e^4 x^4-36036 d e^5 x^5+51051 e^6 x^6\right )+B \left (-2048 d^7+7168 d^6 e x-16128 d^5 e^2 x^2+29568 d^4 e^3 x^3-48048 d^3 e^4 x^4+72072 d^2 e^5 x^5-102102 d e^6 x^6+138567 e^7 x^7\right )\right )\right )}{2909907 e^8} \]
(2*(d + e*x)^(7/2)*(46189*a^6*e^6*(-2*B*d + 9*A*e + 7*B*e*x) + 25194*a^5*b *e^5*(11*A*e*(-2*d + 7*e*x) + B*(8*d^2 - 28*d*e*x + 63*e^2*x^2)) - 4845*a^ 4*b^2*e^4*(-13*A*e*(8*d^2 - 28*d*e*x + 63*e^2*x^2) + 3*B*(16*d^3 - 56*d^2* e*x + 126*d*e^2*x^2 - 231*e^3*x^3)) + 1292*a^3*b^3*e^3*(15*A*e*(-16*d^3 + 56*d^2*e*x - 126*d*e^2*x^2 + 231*e^3*x^3) + B*(128*d^4 - 448*d^3*e*x + 100 8*d^2*e^2*x^2 - 1848*d*e^3*x^3 + 3003*e^4*x^4)) - 57*a^2*b^4*e^2*(-17*A*e* (128*d^4 - 448*d^3*e*x + 1008*d^2*e^2*x^2 - 1848*d*e^3*x^3 + 3003*e^4*x^4) + 5*B*(256*d^5 - 896*d^4*e*x + 2016*d^3*e^2*x^2 - 3696*d^2*e^3*x^3 + 6006 *d*e^4*x^4 - 9009*e^5*x^5)) + 6*a*b^5*e*(19*A*e*(-256*d^5 + 896*d^4*e*x - 2016*d^3*e^2*x^2 + 3696*d^2*e^3*x^3 - 6006*d*e^4*x^4 + 9009*e^5*x^5) + 3*B *(1024*d^6 - 3584*d^5*e*x + 8064*d^4*e^2*x^2 - 14784*d^3*e^3*x^3 + 24024*d ^2*e^4*x^4 - 36036*d*e^5*x^5 + 51051*e^6*x^6)) + b^6*(3*A*e*(1024*d^6 - 35 84*d^5*e*x + 8064*d^4*e^2*x^2 - 14784*d^3*e^3*x^3 + 24024*d^2*e^4*x^4 - 36 036*d*e^5*x^5 + 51051*e^6*x^6) + B*(-2048*d^7 + 7168*d^6*e*x - 16128*d^5*e ^2*x^2 + 29568*d^4*e^3*x^3 - 48048*d^3*e^4*x^4 + 72072*d^2*e^5*x^5 - 10210 2*d*e^6*x^6 + 138567*e^7*x^7))))/(2909907*e^8)
Time = 0.48 (sec) , antiderivative size = 308, 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, 86, 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.
| \(\displaystyle \int \left (a^2+2 a b x+b^2 x^2\right )^3 (A+B x) (d+e x)^{5/2} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle \frac {\int b^6 (a+b x)^6 (A+B x) (d+e x)^{5/2}dx}{b^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int (a+b x)^6 (A+B x) (d+e x)^{5/2}dx\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \int \left (\frac {b^5 (d+e x)^{17/2} (6 a B e+A b e-7 b B d)}{e^7}-\frac {3 b^4 (d+e x)^{15/2} (b d-a e) (5 a B e+2 A b e-7 b B d)}{e^7}+\frac {5 b^3 (d+e x)^{13/2} (b d-a e)^2 (4 a B e+3 A b e-7 b B d)}{e^7}-\frac {5 b^2 (d+e x)^{11/2} (b d-a e)^3 (3 a B e+4 A b e-7 b B d)}{e^7}+\frac {3 b (d+e x)^{9/2} (b d-a e)^4 (2 a B e+5 A b e-7 b B d)}{e^7}+\frac {(d+e x)^{7/2} (a e-b d)^5 (a B e+6 A b e-7 b B d)}{e^7}+\frac {(d+e x)^{5/2} (a e-b d)^6 (A e-B d)}{e^7}+\frac {b^6 B (d+e x)^{19/2}}{e^7}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 b^5 (d+e x)^{19/2} (-6 a B e-A b e+7 b B d)}{19 e^8}+\frac {6 b^4 (d+e x)^{17/2} (b d-a e) (-5 a B e-2 A b e+7 b B d)}{17 e^8}-\frac {2 b^3 (d+e x)^{15/2} (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{3 e^8}+\frac {10 b^2 (d+e x)^{13/2} (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{13 e^8}-\frac {6 b (d+e x)^{11/2} (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{11 e^8}+\frac {2 (d+e x)^{9/2} (b d-a e)^5 (-a B e-6 A b e+7 b B d)}{9 e^8}-\frac {2 (d+e x)^{7/2} (b d-a e)^6 (B d-A e)}{7 e^8}+\frac {2 b^6 B (d+e x)^{21/2}}{21 e^8}\) |
(-2*(b*d - a*e)^6*(B*d - A*e)*(d + e*x)^(7/2))/(7*e^8) + (2*(b*d - a*e)^5* (7*b*B*d - 6*A*b*e - a*B*e)*(d + e*x)^(9/2))/(9*e^8) - (6*b*(b*d - a*e)^4* (7*b*B*d - 5*A*b*e - 2*a*B*e)*(d + e*x)^(11/2))/(11*e^8) + (10*b^2*(b*d - a*e)^3*(7*b*B*d - 4*A*b*e - 3*a*B*e)*(d + e*x)^(13/2))/(13*e^8) - (2*b^3*( b*d - a*e)^2*(7*b*B*d - 3*A*b*e - 4*a*B*e)*(d + e*x)^(15/2))/(3*e^8) + (6* b^4*(b*d - a*e)*(7*b*B*d - 2*A*b*e - 5*a*B*e)*(d + e*x)^(17/2))/(17*e^8) - (2*b^5*(7*b*B*d - A*b*e - 6*a*B*e)*(d + e*x)^(19/2))/(19*e^8) + (2*b^6*B* (d + e*x)^(21/2))/(21*e^8)
3.19.1.3.1 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_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
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 = 0.46 (sec) , antiderivative size = 516, normalized size of antiderivative = 1.68
| method | result | size |
| pseudoelliptic | \(\frac {2 \left (\left (\left (\frac {1}{3} B \,x^{7}+\frac {7}{19} A \,x^{6}\right ) b^{6}+\frac {42 \left (\frac {17 B x}{19}+A \right ) x^{5} a \,b^{5}}{17}+7 \left (\frac {15 B x}{17}+A \right ) x^{4} a^{2} b^{4}+\frac {140 \left (\frac {13 B x}{15}+A \right ) x^{3} a^{3} b^{3}}{13}+\frac {105 \left (\frac {11 B x}{13}+A \right ) x^{2} a^{4} b^{2}}{11}+\frac {14 \left (\frac {9 B x}{11}+A \right ) x \,a^{5} b}{3}+a^{6} \left (\frac {7 B x}{9}+A \right )\right ) e^{7}-\frac {4 d \left (\left (\frac {7}{38} B \,x^{6}+\frac {63}{323} A \,x^{5}\right ) b^{6}+\frac {21 \left (\frac {18 B x}{19}+A \right ) x^{4} a \,b^{5}}{17}+\frac {42 \left (\frac {65 B x}{68}+A \right ) x^{3} a^{2} b^{4}}{13}+\frac {630 x^{2} \left (\frac {44 B x}{45}+A \right ) a^{3} b^{3}}{143}+\frac {35 x \left (\frac {27 B x}{26}+A \right ) a^{4} b^{2}}{11}+a^{5} \left (\frac {14 B x}{11}+A \right ) b +\frac {B \,a^{6}}{6}\right ) e^{6}}{3}+\frac {40 b \,d^{2} \left (\frac {231 x^{4} \left (B x +A \right ) b^{5}}{1615}+\frac {924 x^{3} \left (\frac {39 B x}{38}+A \right ) a \,b^{4}}{1105}+\frac {126 \left (\frac {55 B x}{51}+A \right ) x^{2} a^{2} b^{3}}{65}+\frac {28 x \left (\frac {6 B x}{5}+A \right ) a^{3} b^{2}}{13}+a^{4} \left (\frac {21 B x}{13}+A \right ) b +\frac {2 a^{5} B}{5}\right ) e^{5}}{33}-\frac {320 b^{2} \left (\frac {231 x^{3} \left (\frac {13 B x}{12}+A \right ) b^{4}}{1615}+\frac {63 \left (\frac {22 B x}{19}+A \right ) x^{2} a \,b^{3}}{85}+\frac {7 x \left (\frac {45 B x}{34}+A \right ) a^{2} b^{2}}{5}+a^{3} \left (\frac {28 B x}{15}+A \right ) b +\frac {3 B \,a^{4}}{4}\right ) d^{3} e^{4}}{429}+\frac {128 b^{3} \left (\frac {63 x^{2} \left (\frac {11 B x}{9}+A \right ) b^{3}}{323}+\frac {14 \left (\frac {27 B x}{19}+A \right ) x a \,b^{2}}{17}+a^{2} \left (\frac {35 B x}{17}+A \right ) b +\frac {4 B \,a^{3}}{3}\right ) d^{4} e^{3}}{429}-\frac {512 b^{4} d^{5} \left (\frac {7 \left (\frac {3 B x}{2}+A \right ) x \,b^{2}}{19}+a \left (\frac {42 B x}{19}+A \right ) b +\frac {5 B \,a^{2}}{2}\right ) e^{2}}{7293}+\frac {1024 b^{5} \left (\left (\frac {7 B x}{3}+A \right ) b +6 B a \right ) d^{6} e}{138567}-\frac {2048 B \,b^{6} d^{7}}{415701}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7 e^{8}}\) | \(516\) |
| gosper | \(\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (138567 b^{6} B \,x^{7} e^{7}+153153 A \,b^{6} e^{7} x^{6}+918918 B a \,b^{5} e^{7} x^{6}-102102 B \,b^{6} d \,e^{6} x^{6}+1027026 A a \,b^{5} e^{7} x^{5}-108108 A \,b^{6} d \,e^{6} x^{5}+2567565 B \,a^{2} b^{4} e^{7} x^{5}-648648 B a \,b^{5} d \,e^{6} x^{5}+72072 B \,b^{6} d^{2} e^{5} x^{5}+2909907 A \,a^{2} b^{4} e^{7} x^{4}-684684 A a \,b^{5} d \,e^{6} x^{4}+72072 A \,b^{6} d^{2} e^{5} x^{4}+3879876 B \,a^{3} b^{3} e^{7} x^{4}-1711710 B \,a^{2} b^{4} d \,e^{6} x^{4}+432432 B a \,b^{5} d^{2} e^{5} x^{4}-48048 B \,b^{6} d^{3} e^{4} x^{4}+4476780 A \,a^{3} b^{3} e^{7} x^{3}-1790712 A \,a^{2} b^{4} d \,e^{6} x^{3}+421344 A a \,b^{5} d^{2} e^{5} x^{3}-44352 A \,b^{6} d^{3} e^{4} x^{3}+3357585 B \,a^{4} b^{2} e^{7} x^{3}-2387616 B \,a^{3} b^{3} d \,e^{6} x^{3}+1053360 B \,a^{2} b^{4} d^{2} e^{5} x^{3}-266112 B a \,b^{5} d^{3} e^{4} x^{3}+29568 B \,b^{6} d^{4} e^{3} x^{3}+3968055 A \,a^{4} b^{2} e^{7} x^{2}-2441880 A \,a^{3} b^{3} d \,e^{6} x^{2}+976752 A \,a^{2} b^{4} d^{2} e^{5} x^{2}-229824 A a \,b^{5} d^{3} e^{4} x^{2}+24192 A \,b^{6} d^{4} e^{3} x^{2}+1587222 B \,a^{5} b \,e^{7} x^{2}-1831410 B \,a^{4} b^{2} d \,e^{6} x^{2}+1302336 B \,a^{3} b^{3} d^{2} e^{5} x^{2}-574560 B \,a^{2} b^{4} d^{3} e^{4} x^{2}+145152 B a \,b^{5} d^{4} e^{3} x^{2}-16128 B \,b^{6} d^{5} e^{2} x^{2}+1939938 A \,a^{5} b \,e^{7} x -1763580 A \,a^{4} b^{2} d \,e^{6} x +1085280 A \,a^{3} b^{3} d^{2} e^{5} x -434112 A \,a^{2} b^{4} d^{3} e^{4} x +102144 A a \,b^{5} d^{4} e^{3} x -10752 A \,b^{6} d^{5} e^{2} x +323323 B \,a^{6} e^{7} x -705432 B \,a^{5} b d \,e^{6} x +813960 B \,a^{4} b^{2} d^{2} e^{5} x -578816 B \,a^{3} b^{3} d^{3} e^{4} x +255360 B \,a^{2} b^{4} d^{4} e^{3} x -64512 B a \,b^{5} d^{5} e^{2} x +7168 B \,b^{6} d^{6} e x +415701 A \,a^{6} e^{7}-554268 A \,a^{5} b d \,e^{6}+503880 A \,a^{4} b^{2} d^{2} e^{5}-310080 A \,a^{3} b^{3} d^{3} e^{4}+124032 A \,a^{2} b^{4} d^{4} e^{3}-29184 A a \,b^{5} d^{5} e^{2}+3072 A \,b^{6} d^{6} e -92378 B \,a^{6} d \,e^{6}+201552 B \,a^{5} b \,d^{2} e^{5}-232560 B \,a^{4} b^{2} d^{3} e^{4}+165376 B \,a^{3} b^{3} d^{4} e^{3}-72960 B \,a^{2} b^{4} d^{5} e^{2}+18432 B a \,b^{5} d^{6} e -2048 B \,b^{6} d^{7}\right )}{2909907 e^{8}}\) | \(913\) |
| derivativedivides | \(\frac {\frac {2 B \,b^{6} \left (e x +d \right )^{\frac {21}{2}}}{21}+\frac {2 \left (\left (A e -B d \right ) b^{6}+3 B \left (2 a b e -2 b^{2} d \right ) b^{4}\right ) \left (e x +d \right )^{\frac {19}{2}}}{19}+\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 {17}{2}}}{17}+\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 {15}{2}}}{15}+\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 {13}{2}}}{13}+\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 {11}{2}}}{11}+\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 {9}{2}}}{9}+\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 {7}{2}}}{7}}{e^{8}}\) | \(938\) |
| default | \(\frac {\frac {2 B \,b^{6} \left (e x +d \right )^{\frac {21}{2}}}{21}+\frac {2 \left (\left (A e -B d \right ) b^{6}+3 B \left (2 a b e -2 b^{2} d \right ) b^{4}\right ) \left (e x +d \right )^{\frac {19}{2}}}{19}+\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 {17}{2}}}{17}+\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 {15}{2}}}{15}+\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 {13}{2}}}{13}+\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 {11}{2}}}{11}+\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 {9}{2}}}{9}+\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 {7}{2}}}{7}}{e^{8}}\) | \(938\) |
| trager | \(\text {Expression too large to display}\) | \(1605\) |
| risch | \(\text {Expression too large to display}\) | \(1605\) |
2/7*(((1/3*B*x^7+7/19*A*x^6)*b^6+42/17*(17/19*B*x+A)*x^5*a*b^5+7*(15/17*B* x+A)*x^4*a^2*b^4+140/13*(13/15*B*x+A)*x^3*a^3*b^3+105/11*(11/13*B*x+A)*x^2 *a^4*b^2+14/3*(9/11*B*x+A)*x*a^5*b+a^6*(7/9*B*x+A))*e^7-4/3*d*((7/38*B*x^6 +63/323*A*x^5)*b^6+21/17*(18/19*B*x+A)*x^4*a*b^5+42/13*(65/68*B*x+A)*x^3*a ^2*b^4+630/143*x^2*(44/45*B*x+A)*a^3*b^3+35/11*x*(27/26*B*x+A)*a^4*b^2+a^5 *(14/11*B*x+A)*b+1/6*B*a^6)*e^6+40/33*b*d^2*(231/1615*x^4*(B*x+A)*b^5+924/ 1105*x^3*(39/38*B*x+A)*a*b^4+126/65*(55/51*B*x+A)*x^2*a^2*b^3+28/13*x*(6/5 *B*x+A)*a^3*b^2+a^4*(21/13*B*x+A)*b+2/5*a^5*B)*e^5-320/429*b^2*(231/1615*x ^3*(13/12*B*x+A)*b^4+63/85*(22/19*B*x+A)*x^2*a*b^3+7/5*x*(45/34*B*x+A)*a^2 *b^2+a^3*(28/15*B*x+A)*b+3/4*B*a^4)*d^3*e^4+128/429*b^3*(63/323*x^2*(11/9* B*x+A)*b^3+14/17*(27/19*B*x+A)*x*a*b^2+a^2*(35/17*B*x+A)*b+4/3*B*a^3)*d^4* e^3-512/7293*b^4*d^5*(7/19*(3/2*B*x+A)*x*b^2+a*(42/19*B*x+A)*b+5/2*B*a^2)* e^2+1024/138567*b^5*((7/3*B*x+A)*b+6*B*a)*d^6*e-2048/415701*B*b^6*d^7)*(e* x+d)^(7/2)/e^8
Leaf count of result is larger than twice the leaf count of optimal. 1293 vs. \(2 (276) = 552\).
Time = 0.33 (sec) , antiderivative size = 1293, normalized size of antiderivative = 4.20 \[ \int (A+B x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\text {Too large to display} \]
2/2909907*(138567*B*b^6*e^10*x^10 - 2048*B*b^6*d^10 + 415701*A*a^6*d^3*e^7 + 3072*(6*B*a*b^5 + A*b^6)*d^9*e - 14592*(5*B*a^2*b^4 + 2*A*a*b^5)*d^8*e^ 2 + 41344*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^7*e^3 - 77520*(3*B*a^4*b^2 + 4*A*a ^3*b^3)*d^6*e^4 + 100776*(2*B*a^5*b + 5*A*a^4*b^2)*d^5*e^5 - 92378*(B*a^6 + 6*A*a^5*b)*d^4*e^6 + 7293*(43*B*b^6*d*e^9 + 21*(6*B*a*b^5 + A*b^6)*e^10) *x^9 + 3861*(47*B*b^6*d^2*e^8 + 91*(6*B*a*b^5 + A*b^6)*d*e^9 + 133*(5*B*a^ 2*b^4 + 2*A*a*b^5)*e^10)*x^8 + 429*(B*b^6*d^3*e^7 + 483*(6*B*a*b^5 + A*b^6 )*d^2*e^8 + 2793*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^9 + 2261*(4*B*a^3*b^3 + 3*A *a^2*b^4)*e^10)*x^7 - 231*(2*B*b^6*d^4*e^6 - 3*(6*B*a*b^5 + A*b^6)*d^3*e^7 - 3135*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^8 - 10013*(4*B*a^3*b^3 + 3*A*a^2*b ^4)*d*e^9 - 4845*(3*B*a^4*b^2 + 4*A*a^3*b^3)*e^10)*x^6 + 63*(8*B*b^6*d^5*e ^5 - 12*(6*B*a*b^5 + A*b^6)*d^4*e^6 + 57*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^7 + 22933*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^8 + 43605*(3*B*a^4*b^2 + 4*A*a^ 3*b^3)*d*e^9 + 12597*(2*B*a^5*b + 5*A*a^4*b^2)*e^10)*x^5 - 7*(80*B*b^6*d^6 *e^4 - 120*(6*B*a*b^5 + A*b^6)*d^5*e^5 + 570*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4 *e^6 - 1615*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^7 - 256785*(3*B*a^4*b^2 + 4* A*a^3*b^3)*d^2*e^8 - 289731*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^9 - 46189*(B*a^6 + 6*A*a^5*b)*e^10)*x^4 + (640*B*b^6*d^7*e^3 + 415701*A*a^6*e^10 - 960*(6* B*a*b^5 + A*b^6)*d^6*e^4 + 4560*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^5 - 12920* (4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^6 + 24225*(3*B*a^4*b^2 + 4*A*a^3*b^3)...
Leaf count of result is larger than twice the leaf count of optimal. 1127 vs. \(2 (316) = 632\).
Time = 2.35 (sec) , antiderivative size = 1127, normalized size of antiderivative = 3.66 \[ \int (A+B x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\begin {cases} \frac {2 \left (\frac {B b^{6} \left (d + e x\right )^{\frac {21}{2}}}{21 e^{7}} + \frac {\left (d + e x\right )^{\frac {19}{2}} \left (A b^{6} e + 6 B a b^{5} e - 7 B b^{6} d\right )}{19 e^{7}} + \frac {\left (d + e x\right )^{\frac {17}{2}} \cdot \left (6 A a b^{5} e^{2} - 6 A b^{6} d e + 15 B a^{2} b^{4} e^{2} - 36 B a b^{5} d e + 21 B b^{6} d^{2}\right )}{17 e^{7}} + \frac {\left (d + e x\right )^{\frac {15}{2}} \cdot \left (15 A a^{2} b^{4} e^{3} - 30 A a b^{5} d e^{2} + 15 A b^{6} d^{2} e + 20 B a^{3} b^{3} e^{3} - 75 B a^{2} b^{4} d e^{2} + 90 B a b^{5} d^{2} e - 35 B b^{6} d^{3}\right )}{15 e^{7}} + \frac {\left (d + e x\right )^{\frac {13}{2}} \cdot \left (20 A a^{3} b^{3} e^{4} - 60 A a^{2} b^{4} d e^{3} + 60 A a b^{5} d^{2} e^{2} - 20 A b^{6} d^{3} e + 15 B a^{4} b^{2} e^{4} - 80 B a^{3} b^{3} d e^{3} + 150 B a^{2} b^{4} d^{2} e^{2} - 120 B a b^{5} d^{3} e + 35 B b^{6} d^{4}\right )}{13 e^{7}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \cdot \left (15 A a^{4} b^{2} e^{5} - 60 A a^{3} b^{3} d e^{4} + 90 A a^{2} b^{4} d^{2} e^{3} - 60 A a b^{5} d^{3} e^{2} + 15 A b^{6} d^{4} e + 6 B a^{5} b e^{5} - 45 B a^{4} b^{2} d e^{4} + 120 B a^{3} b^{3} d^{2} e^{3} - 150 B a^{2} b^{4} d^{3} e^{2} + 90 B a b^{5} d^{4} e - 21 B b^{6} d^{5}\right )}{11 e^{7}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \cdot \left (6 A a^{5} b e^{6} - 30 A a^{4} b^{2} d e^{5} + 60 A a^{3} b^{3} d^{2} e^{4} - 60 A a^{2} b^{4} d^{3} e^{3} + 30 A a b^{5} d^{4} e^{2} - 6 A b^{6} d^{5} e + B a^{6} e^{6} - 12 B a^{5} b d e^{5} + 45 B a^{4} b^{2} d^{2} e^{4} - 80 B a^{3} b^{3} d^{3} e^{3} + 75 B a^{2} b^{4} d^{4} e^{2} - 36 B a b^{5} d^{5} e + 7 B b^{6} d^{6}\right )}{9 e^{7}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (A a^{6} e^{7} - 6 A a^{5} b d e^{6} + 15 A a^{4} b^{2} d^{2} e^{5} - 20 A a^{3} b^{3} d^{3} e^{4} + 15 A a^{2} b^{4} d^{4} e^{3} - 6 A a b^{5} d^{5} e^{2} + A b^{6} d^{6} e - B a^{6} d e^{6} + 6 B a^{5} b d^{2} e^{5} - 15 B a^{4} b^{2} d^{3} e^{4} + 20 B a^{3} b^{3} d^{4} e^{3} - 15 B a^{2} b^{4} d^{5} e^{2} + 6 B a b^{5} d^{6} e - B b^{6} d^{7}\right )}{7 e^{7}}\right )}{e} & \text {for}\: e \neq 0 \\d^{\frac {5}{2}} \left (A a^{6} x + \frac {B b^{6} x^{8}}{8} + \frac {x^{7} \left (A b^{6} + 6 B a b^{5}\right )}{7} + \frac {x^{6} \cdot \left (6 A a b^{5} + 15 B a^{2} b^{4}\right )}{6} + \frac {x^{5} \cdot \left (15 A a^{2} b^{4} + 20 B a^{3} b^{3}\right )}{5} + \frac {x^{4} \cdot \left (20 A a^{3} b^{3} + 15 B a^{4} b^{2}\right )}{4} + \frac {x^{3} \cdot \left (15 A a^{4} b^{2} + 6 B a^{5} b\right )}{3} + \frac {x^{2} \cdot \left (6 A a^{5} b + B a^{6}\right )}{2}\right ) & \text {otherwise} \end {cases} \]
Piecewise((2*(B*b**6*(d + e*x)**(21/2)/(21*e**7) + (d + e*x)**(19/2)*(A*b* *6*e + 6*B*a*b**5*e - 7*B*b**6*d)/(19*e**7) + (d + e*x)**(17/2)*(6*A*a*b** 5*e**2 - 6*A*b**6*d*e + 15*B*a**2*b**4*e**2 - 36*B*a*b**5*d*e + 21*B*b**6* d**2)/(17*e**7) + (d + e*x)**(15/2)*(15*A*a**2*b**4*e**3 - 30*A*a*b**5*d*e **2 + 15*A*b**6*d**2*e + 20*B*a**3*b**3*e**3 - 75*B*a**2*b**4*d*e**2 + 90* B*a*b**5*d**2*e - 35*B*b**6*d**3)/(15*e**7) + (d + e*x)**(13/2)*(20*A*a**3 *b**3*e**4 - 60*A*a**2*b**4*d*e**3 + 60*A*a*b**5*d**2*e**2 - 20*A*b**6*d** 3*e + 15*B*a**4*b**2*e**4 - 80*B*a**3*b**3*d*e**3 + 150*B*a**2*b**4*d**2*e **2 - 120*B*a*b**5*d**3*e + 35*B*b**6*d**4)/(13*e**7) + (d + e*x)**(11/2)* (15*A*a**4*b**2*e**5 - 60*A*a**3*b**3*d*e**4 + 90*A*a**2*b**4*d**2*e**3 - 60*A*a*b**5*d**3*e**2 + 15*A*b**6*d**4*e + 6*B*a**5*b*e**5 - 45*B*a**4*b** 2*d*e**4 + 120*B*a**3*b**3*d**2*e**3 - 150*B*a**2*b**4*d**3*e**2 + 90*B*a* b**5*d**4*e - 21*B*b**6*d**5)/(11*e**7) + (d + e*x)**(9/2)*(6*A*a**5*b*e** 6 - 30*A*a**4*b**2*d*e**5 + 60*A*a**3*b**3*d**2*e**4 - 60*A*a**2*b**4*d**3 *e**3 + 30*A*a*b**5*d**4*e**2 - 6*A*b**6*d**5*e + B*a**6*e**6 - 12*B*a**5* b*d*e**5 + 45*B*a**4*b**2*d**2*e**4 - 80*B*a**3*b**3*d**3*e**3 + 75*B*a**2 *b**4*d**4*e**2 - 36*B*a*b**5*d**5*e + 7*B*b**6*d**6)/(9*e**7) + (d + e*x) **(7/2)*(A*a**6*e**7 - 6*A*a**5*b*d*e**6 + 15*A*a**4*b**2*d**2*e**5 - 20*A *a**3*b**3*d**3*e**4 + 15*A*a**2*b**4*d**4*e**3 - 6*A*a*b**5*d**5*e**2 + A *b**6*d**6*e - B*a**6*d*e**6 + 6*B*a**5*b*d**2*e**5 - 15*B*a**4*b**2*d*...
Leaf count of result is larger than twice the leaf count of optimal. 767 vs. \(2 (276) = 552\).
Time = 0.20 (sec) , antiderivative size = 767, normalized size of antiderivative = 2.49 \[ \int (A+B x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {2 \, {\left (138567 \, {\left (e x + d\right )}^{\frac {21}{2}} B b^{6} - 153153 \, {\left (7 \, B b^{6} d - {\left (6 \, B a b^{5} + A b^{6}\right )} e\right )} {\left (e x + d\right )}^{\frac {19}{2}} + 513513 \, {\left (7 \, B b^{6} d^{2} - 2 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d e + {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {17}{2}} - 969969 \, {\left (7 \, B b^{6} d^{3} - 3 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} e + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d e^{2} - {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} e^{3}\right )} {\left (e x + d\right )}^{\frac {15}{2}} + 1119195 \, {\left (7 \, B b^{6} d^{4} - 4 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} e + 6 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} e^{2} - 4 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d e^{3} + {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} e^{4}\right )} {\left (e x + d\right )}^{\frac {13}{2}} - 793611 \, {\left (7 \, B b^{6} d^{5} - 5 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{4} e + 10 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{3} e^{2} - 10 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} e^{3} + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d e^{4} - {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} e^{5}\right )} {\left (e x + d\right )}^{\frac {11}{2}} + 323323 \, {\left (7 \, B b^{6} d^{6} - 6 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{5} e + 15 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{4} e^{2} - 20 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{3} e^{3} + 15 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} e^{4} - 6 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d e^{5} + {\left (B a^{6} + 6 \, A a^{5} b\right )} e^{6}\right )} {\left (e x + d\right )}^{\frac {9}{2}} - 415701 \, {\left (B b^{6} d^{7} - A a^{6} e^{7} - {\left (6 \, B a b^{5} + A b^{6}\right )} d^{6} e + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{5} e^{2} - 5 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{4} e^{3} + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{3} e^{4} - 3 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2} e^{5} + {\left (B a^{6} + 6 \, A a^{5} b\right )} d e^{6}\right )} {\left (e x + d\right )}^{\frac {7}{2}}\right )}}{2909907 \, e^{8}} \]
2/2909907*(138567*(e*x + d)^(21/2)*B*b^6 - 153153*(7*B*b^6*d - (6*B*a*b^5 + A*b^6)*e)*(e*x + d)^(19/2) + 513513*(7*B*b^6*d^2 - 2*(6*B*a*b^5 + A*b^6) *d*e + (5*B*a^2*b^4 + 2*A*a*b^5)*e^2)*(e*x + d)^(17/2) - 969969*(7*B*b^6*d ^3 - 3*(6*B*a*b^5 + A*b^6)*d^2*e + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^2 - (4* B*a^3*b^3 + 3*A*a^2*b^4)*e^3)*(e*x + d)^(15/2) + 1119195*(7*B*b^6*d^4 - 4* (6*B*a*b^5 + A*b^6)*d^3*e + 6*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^2 - 4*(4*B*a ^3*b^3 + 3*A*a^2*b^4)*d*e^3 + (3*B*a^4*b^2 + 4*A*a^3*b^3)*e^4)*(e*x + d)^( 13/2) - 793611*(7*B*b^6*d^5 - 5*(6*B*a*b^5 + A*b^6)*d^4*e + 10*(5*B*a^2*b^ 4 + 2*A*a*b^5)*d^3*e^2 - 10*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^3 + 5*(3*B*a ^4*b^2 + 4*A*a^3*b^3)*d*e^4 - (2*B*a^5*b + 5*A*a^4*b^2)*e^5)*(e*x + d)^(11 /2) + 323323*(7*B*b^6*d^6 - 6*(6*B*a*b^5 + A*b^6)*d^5*e + 15*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^2 - 20*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^3 + 15*(3*B*a^ 4*b^2 + 4*A*a^3*b^3)*d^2*e^4 - 6*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^5 + (B*a^6 + 6*A*a^5*b)*e^6)*(e*x + d)^(9/2) - 415701*(B*b^6*d^7 - A*a^6*e^7 - (6*B*a *b^5 + A*b^6)*d^6*e + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^2 - 5*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^3 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 - 3*(2*B*a ^5*b + 5*A*a^4*b^2)*d^2*e^5 + (B*a^6 + 6*A*a^5*b)*d*e^6)*(e*x + d)^(7/2))/ e^8
Leaf count of result is larger than twice the leaf count of optimal. 4485 vs. \(2 (276) = 552\).
Time = 0.35 (sec) , antiderivative size = 4485, normalized size of antiderivative = 14.56 \[ \int (A+B x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\text {Too large to display} \]
2/14549535*(14549535*sqrt(e*x + d)*A*a^6*d^3 + 14549535*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*A*a^6*d^2 + 4849845*((e*x + d)^(3/2) - 3*sqrt(e*x + d) *d)*B*a^6*d^3/e + 29099070*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*A*a^5*b*d ^3/e + 2909907*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d )*d^2)*A*a^6*d + 5819814*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sq rt(e*x + d)*d^2)*B*a^5*b*d^3/e^2 + 14549535*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*A*a^4*b^2*d^3/e^2 + 2909907*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*B*a^6*d^2/e + 1745 9442*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*A*a ^5*b*d^2/e + 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*a^6 + 6235515*(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)*B*a^ 4*b^2*d^3/e^3 + 8314020*(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*a^3*b^3*d^3/e^3 + 7482618*(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)*B*a^5*b*d^2/e^2 + 18706545*(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*a^4*b^2*d^2/e^2 + 1247103*(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)*B*a^6*d/e + 7482618*(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*a^5*b...
Time = 10.72 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.91 \[ \int (A+B x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {{\left (d+e\,x\right )}^{19/2}\,\left (2\,A\,b^6\,e-14\,B\,b^6\,d+12\,B\,a\,b^5\,e\right )}{19\,e^8}+\frac {2\,{\left (a\,e-b\,d\right )}^5\,{\left (d+e\,x\right )}^{9/2}\,\left (6\,A\,b\,e+B\,a\,e-7\,B\,b\,d\right )}{9\,e^8}+\frac {2\,B\,b^6\,{\left (d+e\,x\right )}^{21/2}}{21\,e^8}+\frac {2\,\left (A\,e-B\,d\right )\,{\left (a\,e-b\,d\right )}^6\,{\left (d+e\,x\right )}^{7/2}}{7\,e^8}+\frac {6\,b\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{11/2}\,\left (5\,A\,b\,e+2\,B\,a\,e-7\,B\,b\,d\right )}{11\,e^8}+\frac {6\,b^4\,\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{17/2}\,\left (2\,A\,b\,e+5\,B\,a\,e-7\,B\,b\,d\right )}{17\,e^8}+\frac {10\,b^2\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{13/2}\,\left (4\,A\,b\,e+3\,B\,a\,e-7\,B\,b\,d\right )}{13\,e^8}+\frac {2\,b^3\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{15/2}\,\left (3\,A\,b\,e+4\,B\,a\,e-7\,B\,b\,d\right )}{3\,e^8} \]
((d + e*x)^(19/2)*(2*A*b^6*e - 14*B*b^6*d + 12*B*a*b^5*e))/(19*e^8) + (2*( a*e - b*d)^5*(d + e*x)^(9/2)*(6*A*b*e + B*a*e - 7*B*b*d))/(9*e^8) + (2*B*b ^6*(d + e*x)^(21/2))/(21*e^8) + (2*(A*e - B*d)*(a*e - b*d)^6*(d + e*x)^(7/ 2))/(7*e^8) + (6*b*(a*e - b*d)^4*(d + e*x)^(11/2)*(5*A*b*e + 2*B*a*e - 7*B *b*d))/(11*e^8) + (6*b^4*(a*e - b*d)*(d + e*x)^(17/2)*(2*A*b*e + 5*B*a*e - 7*B*b*d))/(17*e^8) + (10*b^2*(a*e - b*d)^3*(d + e*x)^(13/2)*(4*A*b*e + 3* B*a*e - 7*B*b*d))/(13*e^8) + (2*b^3*(a*e - b*d)^2*(d + e*x)^(15/2)*(3*A*b* e + 4*B*a*e - 7*B*b*d))/(3*e^8)