Integrand size = 33, antiderivative size = 300 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{3/2}} \, dx=\frac {2 (b d-a e)^6 (B d-A e)}{e^8 \sqrt {d+e x}}+\frac {2 (b d-a e)^5 (7 b B d-6 A b e-a B e) \sqrt {d+e x}}{e^8}-\frac {2 b (b d-a e)^4 (7 b B d-5 A b e-2 a B e) (d+e x)^{3/2}}{e^8}+\frac {2 b^2 (b d-a e)^3 (7 b B d-4 A b e-3 a B e) (d+e x)^{5/2}}{e^8}-\frac {10 b^3 (b d-a e)^2 (7 b B d-3 A b e-4 a B e) (d+e x)^{7/2}}{7 e^8}+\frac {2 b^4 (b d-a e) (7 b B d-2 A b e-5 a B e) (d+e x)^{9/2}}{3 e^8}-\frac {2 b^5 (7 b B d-A b e-6 a B e) (d+e x)^{11/2}}{11 e^8}+\frac {2 b^6 B (d+e x)^{13/2}}{13 e^8} \]
-2*b*(-a*e+b*d)^4*(-5*A*b*e-2*B*a*e+7*B*b*d)*(e*x+d)^(3/2)/e^8+2*b^2*(-a*e +b*d)^3*(-4*A*b*e-3*B*a*e+7*B*b*d)*(e*x+d)^(5/2)/e^8-10/7*b^3*(-a*e+b*d)^2 *(-3*A*b*e-4*B*a*e+7*B*b*d)*(e*x+d)^(7/2)/e^8+2/3*b^4*(-a*e+b*d)*(-2*A*b*e -5*B*a*e+7*B*b*d)*(e*x+d)^(9/2)/e^8-2/11*b^5*(-A*b*e-6*B*a*e+7*B*b*d)*(e*x +d)^(11/2)/e^8+2/13*b^6*B*(e*x+d)^(13/2)/e^8+2*(-a*e+b*d)^6*(-A*e+B*d)/e^8 /(e*x+d)^(1/2)+2*(-a*e+b*d)^5*(-6*A*b*e-B*a*e+7*B*b*d)*(e*x+d)^(1/2)/e^8
Leaf count is larger than twice the leaf count of optimal. \(624\) vs. \(2(300)=600\).
Time = 0.45 (sec) , antiderivative size = 624, normalized size of antiderivative = 2.08 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{3/2}} \, dx=\frac {2 \left (3003 a^6 e^6 (2 B d-A e+B e x)+6006 a^5 b e^5 \left (3 A e (2 d+e x)+B \left (-8 d^2-4 d e x+e^2 x^2\right )\right )+3003 a^4 b^2 e^4 \left (5 A e \left (-8 d^2-4 d e x+e^2 x^2\right )+3 B \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )\right )-1716 a^3 b^3 e^3 \left (-7 A e \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )+B \left (128 d^4+64 d^3 e x-16 d^2 e^2 x^2+8 d e^3 x^3-5 e^4 x^4\right )\right )+143 a^2 b^4 e^2 \left (9 A e \left (-128 d^4-64 d^3 e x+16 d^2 e^2 x^2-8 d e^3 x^3+5 e^4 x^4\right )+5 B \left (256 d^5+128 d^4 e x-32 d^3 e^2 x^2+16 d^2 e^3 x^3-10 d e^4 x^4+7 e^5 x^5\right )\right )-26 a b^5 e \left (-11 A e \left (256 d^5+128 d^4 e x-32 d^3 e^2 x^2+16 d^2 e^3 x^3-10 d e^4 x^4+7 e^5 x^5\right )+3 B \left (1024 d^6+512 d^5 e x-128 d^4 e^2 x^2+64 d^3 e^3 x^3-40 d^2 e^4 x^4+28 d e^5 x^5-21 e^6 x^6\right )\right )+b^6 \left (13 A e \left (-1024 d^6-512 d^5 e x+128 d^4 e^2 x^2-64 d^3 e^3 x^3+40 d^2 e^4 x^4-28 d e^5 x^5+21 e^6 x^6\right )+7 B \left (2048 d^7+1024 d^6 e x-256 d^5 e^2 x^2+128 d^4 e^3 x^3-80 d^3 e^4 x^4+56 d^2 e^5 x^5-42 d e^6 x^6+33 e^7 x^7\right )\right )\right )}{3003 e^8 \sqrt {d+e x}} \]
(2*(3003*a^6*e^6*(2*B*d - A*e + B*e*x) + 6006*a^5*b*e^5*(3*A*e*(2*d + e*x) + B*(-8*d^2 - 4*d*e*x + e^2*x^2)) + 3003*a^4*b^2*e^4*(5*A*e*(-8*d^2 - 4*d *e*x + e^2*x^2) + 3*B*(16*d^3 + 8*d^2*e*x - 2*d*e^2*x^2 + e^3*x^3)) - 1716 *a^3*b^3*e^3*(-7*A*e*(16*d^3 + 8*d^2*e*x - 2*d*e^2*x^2 + e^3*x^3) + B*(128 *d^4 + 64*d^3*e*x - 16*d^2*e^2*x^2 + 8*d*e^3*x^3 - 5*e^4*x^4)) + 143*a^2*b ^4*e^2*(9*A*e*(-128*d^4 - 64*d^3*e*x + 16*d^2*e^2*x^2 - 8*d*e^3*x^3 + 5*e^ 4*x^4) + 5*B*(256*d^5 + 128*d^4*e*x - 32*d^3*e^2*x^2 + 16*d^2*e^3*x^3 - 10 *d*e^4*x^4 + 7*e^5*x^5)) - 26*a*b^5*e*(-11*A*e*(256*d^5 + 128*d^4*e*x - 32 *d^3*e^2*x^2 + 16*d^2*e^3*x^3 - 10*d*e^4*x^4 + 7*e^5*x^5) + 3*B*(1024*d^6 + 512*d^5*e*x - 128*d^4*e^2*x^2 + 64*d^3*e^3*x^3 - 40*d^2*e^4*x^4 + 28*d*e ^5*x^5 - 21*e^6*x^6)) + b^6*(13*A*e*(-1024*d^6 - 512*d^5*e*x + 128*d^4*e^2 *x^2 - 64*d^3*e^3*x^3 + 40*d^2*e^4*x^4 - 28*d*e^5*x^5 + 21*e^6*x^6) + 7*B* (2048*d^7 + 1024*d^6*e*x - 256*d^5*e^2*x^2 + 128*d^4*e^3*x^3 - 80*d^3*e^4* x^4 + 56*d^2*e^5*x^5 - 42*d*e^6*x^6 + 33*e^7*x^7))))/(3003*e^8*Sqrt[d + e* x])
Time = 0.48 (sec) , antiderivative size = 300, 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.
| \(\displaystyle \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3 (A+B x)}{(d+e x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle \frac {\int \frac {b^6 (a+b x)^6 (A+B x)}{(d+e x)^{3/2}}dx}{b^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{3/2}}dx\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \int \left (\frac {b^5 (d+e x)^{9/2} (6 a B e+A b e-7 b B d)}{e^7}-\frac {3 b^4 (d+e x)^{7/2} (b d-a e) (5 a B e+2 A b e-7 b B d)}{e^7}+\frac {5 b^3 (d+e x)^{5/2} (b d-a e)^2 (4 a B e+3 A b e-7 b B d)}{e^7}-\frac {5 b^2 (d+e x)^{3/2} (b d-a e)^3 (3 a B e+4 A b e-7 b B d)}{e^7}+\frac {3 b \sqrt {d+e x} (b d-a e)^4 (2 a B e+5 A b e-7 b B d)}{e^7}+\frac {(a e-b d)^5 (a B e+6 A b e-7 b B d)}{e^7 \sqrt {d+e x}}+\frac {(a e-b d)^6 (A e-B d)}{e^7 (d+e x)^{3/2}}+\frac {b^6 B (d+e x)^{11/2}}{e^7}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 b^5 (d+e x)^{11/2} (-6 a B e-A b e+7 b B d)}{11 e^8}+\frac {2 b^4 (d+e x)^{9/2} (b d-a e) (-5 a B e-2 A b e+7 b B d)}{3 e^8}-\frac {10 b^3 (d+e x)^{7/2} (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{7 e^8}+\frac {2 b^2 (d+e x)^{5/2} (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{e^8}-\frac {2 b (d+e x)^{3/2} (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{e^8}+\frac {2 \sqrt {d+e x} (b d-a e)^5 (-a B e-6 A b e+7 b B d)}{e^8}+\frac {2 (b d-a e)^6 (B d-A e)}{e^8 \sqrt {d+e x}}+\frac {2 b^6 B (d+e x)^{13/2}}{13 e^8}\) |
(2*(b*d - a*e)^6*(B*d - A*e))/(e^8*Sqrt[d + e*x]) + (2*(b*d - a*e)^5*(7*b* B*d - 6*A*b*e - a*B*e)*Sqrt[d + e*x])/e^8 - (2*b*(b*d - a*e)^4*(7*b*B*d - 5*A*b*e - 2*a*B*e)*(d + e*x)^(3/2))/e^8 + (2*b^2*(b*d - a*e)^3*(7*b*B*d - 4*A*b*e - 3*a*B*e)*(d + e*x)^(5/2))/e^8 - (10*b^3*(b*d - a*e)^2*(7*b*B*d - 3*A*b*e - 4*a*B*e)*(d + e*x)^(7/2))/(7*e^8) + (2*b^4*(b*d - a*e)*(7*b*B*d - 2*A*b*e - 5*a*B*e)*(d + e*x)^(9/2))/(3*e^8) - (2*b^5*(7*b*B*d - A*b*e - 6*a*B*e)*(d + e*x)^(11/2))/(11*e^8) + (2*b^6*B*(d + e*x)^(13/2))/(13*e^8)
3.19.5.3.1 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 = 0.39 (sec) , antiderivative size = 514, normalized size of antiderivative = 1.71
| method | result | size |
| pseudoelliptic | \(-\frac {2 \left (\left (\left (-\frac {1}{13} B \,x^{7}-\frac {1}{11} A \,x^{6}\right ) b^{6}-\frac {2 \left (\frac {9 B x}{11}+A \right ) x^{5} a \,b^{5}}{3}-\frac {15 x^{4} \left (\frac {7 B x}{9}+A \right ) a^{2} b^{4}}{7}-4 x^{3} \left (\frac {5 B x}{7}+A \right ) a^{3} b^{3}-5 \left (\frac {3 B x}{5}+A \right ) x^{2} a^{4} b^{2}-6 \left (\frac {B x}{3}+A \right ) x \,a^{5} b +a^{6} \left (-B x +A \right )\right ) e^{7}-12 \left (-\frac {\left (\frac {21 B x}{26}+A \right ) x^{5} b^{6}}{99}-\frac {5 x^{4} \left (\frac {42 B x}{55}+A \right ) a \,b^{5}}{63}-\frac {2 \left (\frac {25 B x}{36}+A \right ) x^{3} a^{2} b^{4}}{7}-\frac {2 \left (\frac {4 B x}{7}+A \right ) x^{2} a^{3} b^{3}}{3}-\frac {5 x \,a^{4} \left (\frac {3 B x}{10}+A \right ) b^{2}}{3}+a^{5} \left (A -\frac {2 B x}{3}\right ) b +\frac {B \,a^{6}}{6}\right ) d \,e^{6}+40 b \left (-\frac {x^{4} \left (\frac {49 B x}{65}+A \right ) b^{5}}{231}-\frac {4 \left (\frac {15 B x}{22}+A \right ) x^{3} a \,b^{4}}{105}-\frac {6 \left (\frac {5 B x}{9}+A \right ) x^{2} a^{2} b^{3}}{35}-\frac {4 \left (\frac {2 B x}{7}+A \right ) x \,a^{3} b^{2}}{5}+a^{4} \left (-\frac {3 B x}{5}+A \right ) b +\frac {2 a^{5} B}{5}\right ) d^{2} e^{5}-64 b^{2} \left (-\frac {x^{3} \left (\frac {35 B x}{52}+A \right ) b^{4}}{231}-\frac {\left (\frac {6 B x}{11}+A \right ) x^{2} a \,b^{3}}{21}-\frac {3 x \left (\frac {5 B x}{18}+A \right ) a^{2} b^{2}}{7}+a^{3} \left (-\frac {4 B x}{7}+A \right ) b +\frac {3 B \,a^{4}}{4}\right ) d^{3} e^{4}+\frac {384 b^{3} \left (-\frac {\left (\frac {7 B x}{13}+A \right ) x^{2} b^{3}}{99}-\frac {2 x \left (\frac {3 B x}{11}+A \right ) a \,b^{2}}{9}+a^{2} \left (-\frac {5 B x}{9}+A \right ) b +\frac {4 B \,a^{3}}{3}\right ) d^{4} e^{3}}{7}-\frac {512 b^{4} \left (-\frac {\left (\frac {7 B x}{26}+A \right ) x \,b^{2}}{11}+a \left (-\frac {6 B x}{11}+A \right ) b +\frac {5 B \,a^{2}}{2}\right ) d^{5} e^{2}}{21}+\frac {1024 b^{5} \left (\left (-\frac {7 B x}{13}+A \right ) b +6 B a \right ) d^{6} e}{231}-\frac {2048 B \,b^{6} d^{7}}{429}\right )}{\sqrt {e x +d}\, e^{8}}\) | \(514\) |
| risch | \(\frac {2 \left (231 b^{6} B \,x^{6} e^{6}+273 A \,b^{6} e^{6} x^{5}+1638 B a \,b^{5} e^{6} x^{5}-525 B \,b^{6} d \,e^{5} x^{5}+2002 A a \,b^{5} e^{6} x^{4}-637 A \,b^{6} d \,e^{5} x^{4}+5005 B \,a^{2} b^{4} e^{6} x^{4}-3822 B a \,b^{5} d \,e^{5} x^{4}+917 B \,b^{6} d^{2} e^{4} x^{4}+6435 A \,a^{2} b^{4} e^{6} x^{3}-4862 A a \,b^{5} d \,e^{5} x^{3}+1157 A \,b^{6} d^{2} e^{4} x^{3}+8580 B \,a^{3} b^{3} e^{6} x^{3}-12155 B \,a^{2} b^{4} d \,e^{5} x^{3}+6942 B a \,b^{5} d^{2} e^{4} x^{3}-1477 B \,b^{6} d^{3} e^{3} x^{3}+12012 A \,a^{3} b^{3} e^{6} x^{2}-16731 A \,a^{2} b^{4} d \,e^{5} x^{2}+9438 A a \,b^{5} d^{2} e^{4} x^{2}-1989 A \,b^{6} d^{3} e^{3} x^{2}+9009 B \,a^{4} b^{2} e^{6} x^{2}-22308 B \,a^{3} b^{3} d \,e^{5} x^{2}+23595 B \,a^{2} b^{4} d^{2} e^{4} x^{2}-11934 B a \,b^{5} d^{3} e^{3} x^{2}+2373 B \,b^{6} d^{4} e^{2} x^{2}+15015 A \,a^{4} b^{2} e^{6} x -36036 A \,a^{3} b^{3} d \,e^{5} x +37323 A \,a^{2} b^{4} d^{2} e^{4} x -18590 A a \,b^{5} d^{3} e^{3} x +3653 A \,b^{6} d^{4} e^{2} x +6006 B \,a^{5} b \,e^{6} x -27027 B \,a^{4} b^{2} d \,e^{5} x +49764 B \,a^{3} b^{3} d^{2} e^{4} x -46475 B \,a^{2} b^{4} d^{3} e^{3} x +21918 B a \,b^{5} d^{4} e^{2} x -4165 B \,b^{6} d^{5} e x +18018 A \,a^{5} b \,e^{6}-75075 A \,a^{4} b^{2} d \,e^{5}+132132 A \,a^{3} b^{3} d^{2} e^{4}-119691 A \,a^{2} b^{4} d^{3} e^{3}+55198 A a \,b^{5} d^{4} e^{2}-10309 A \,b^{6} d^{5} e +3003 B \,a^{6} e^{6}-30030 B \,a^{5} b d \,e^{5}+99099 B \,a^{4} b^{2} d^{2} e^{4}-159588 B \,a^{3} b^{3} d^{3} e^{3}+137995 B \,a^{2} b^{4} d^{4} e^{2}-61854 B a \,b^{5} d^{5} e +11333 B \,b^{6} d^{6}\right ) \sqrt {e x +d}}{3003 e^{8}}-\frac {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 )}{e^{8} \sqrt {e x +d}}\) | \(883\) |
| gosper | \(-\frac {2 \left (-231 b^{6} B \,x^{7} e^{7}-273 A \,b^{6} e^{7} x^{6}-1638 B a \,b^{5} e^{7} x^{6}+294 B \,b^{6} d \,e^{6} x^{6}-2002 A a \,b^{5} e^{7} x^{5}+364 A \,b^{6} d \,e^{6} x^{5}-5005 B \,a^{2} b^{4} e^{7} x^{5}+2184 B a \,b^{5} d \,e^{6} x^{5}-392 B \,b^{6} d^{2} e^{5} x^{5}-6435 A \,a^{2} b^{4} e^{7} x^{4}+2860 A a \,b^{5} d \,e^{6} x^{4}-520 A \,b^{6} d^{2} e^{5} x^{4}-8580 B \,a^{3} b^{3} e^{7} x^{4}+7150 B \,a^{2} b^{4} d \,e^{6} x^{4}-3120 B a \,b^{5} d^{2} e^{5} x^{4}+560 B \,b^{6} d^{3} e^{4} x^{4}-12012 A \,a^{3} b^{3} e^{7} x^{3}+10296 A \,a^{2} b^{4} d \,e^{6} x^{3}-4576 A a \,b^{5} d^{2} e^{5} x^{3}+832 A \,b^{6} d^{3} e^{4} x^{3}-9009 B \,a^{4} b^{2} e^{7} x^{3}+13728 B \,a^{3} b^{3} d \,e^{6} x^{3}-11440 B \,a^{2} b^{4} d^{2} e^{5} x^{3}+4992 B a \,b^{5} d^{3} e^{4} x^{3}-896 B \,b^{6} d^{4} e^{3} x^{3}-15015 A \,a^{4} b^{2} e^{7} x^{2}+24024 A \,a^{3} b^{3} d \,e^{6} x^{2}-20592 A \,a^{2} b^{4} d^{2} e^{5} x^{2}+9152 A a \,b^{5} d^{3} e^{4} x^{2}-1664 A \,b^{6} d^{4} e^{3} x^{2}-6006 B \,a^{5} b \,e^{7} x^{2}+18018 B \,a^{4} b^{2} d \,e^{6} x^{2}-27456 B \,a^{3} b^{3} d^{2} e^{5} x^{2}+22880 B \,a^{2} b^{4} d^{3} e^{4} x^{2}-9984 B a \,b^{5} d^{4} e^{3} x^{2}+1792 B \,b^{6} d^{5} e^{2} x^{2}-18018 A \,a^{5} b \,e^{7} x +60060 A \,a^{4} b^{2} d \,e^{6} x -96096 A \,a^{3} b^{3} d^{2} e^{5} x +82368 A \,a^{2} b^{4} d^{3} e^{4} x -36608 A a \,b^{5} d^{4} e^{3} x +6656 A \,b^{6} d^{5} e^{2} x -3003 B \,a^{6} e^{7} x +24024 B \,a^{5} b d \,e^{6} x -72072 B \,a^{4} b^{2} d^{2} e^{5} x +109824 B \,a^{3} b^{3} d^{3} e^{4} x -91520 B \,a^{2} b^{4} d^{4} e^{3} x +39936 B a \,b^{5} d^{5} e^{2} x -7168 B \,b^{6} d^{6} e x +3003 A \,a^{6} e^{7}-36036 A \,a^{5} b d \,e^{6}+120120 A \,a^{4} b^{2} d^{2} e^{5}-192192 A \,a^{3} b^{3} d^{3} e^{4}+164736 A \,a^{2} b^{4} d^{4} e^{3}-73216 A a \,b^{5} d^{5} e^{2}+13312 A \,b^{6} d^{6} e -6006 B \,a^{6} d \,e^{6}+48048 B \,a^{5} b \,d^{2} e^{5}-144144 B \,a^{4} b^{2} d^{3} e^{4}+219648 B \,a^{3} b^{3} d^{4} e^{3}-183040 B \,a^{2} b^{4} d^{5} e^{2}+79872 B a \,b^{5} d^{6} e -14336 B \,b^{6} d^{7}\right )}{3003 \sqrt {e x +d}\, e^{8}}\) | \(913\) |
| trager | \(-\frac {2 \left (-231 b^{6} B \,x^{7} e^{7}-273 A \,b^{6} e^{7} x^{6}-1638 B a \,b^{5} e^{7} x^{6}+294 B \,b^{6} d \,e^{6} x^{6}-2002 A a \,b^{5} e^{7} x^{5}+364 A \,b^{6} d \,e^{6} x^{5}-5005 B \,a^{2} b^{4} e^{7} x^{5}+2184 B a \,b^{5} d \,e^{6} x^{5}-392 B \,b^{6} d^{2} e^{5} x^{5}-6435 A \,a^{2} b^{4} e^{7} x^{4}+2860 A a \,b^{5} d \,e^{6} x^{4}-520 A \,b^{6} d^{2} e^{5} x^{4}-8580 B \,a^{3} b^{3} e^{7} x^{4}+7150 B \,a^{2} b^{4} d \,e^{6} x^{4}-3120 B a \,b^{5} d^{2} e^{5} x^{4}+560 B \,b^{6} d^{3} e^{4} x^{4}-12012 A \,a^{3} b^{3} e^{7} x^{3}+10296 A \,a^{2} b^{4} d \,e^{6} x^{3}-4576 A a \,b^{5} d^{2} e^{5} x^{3}+832 A \,b^{6} d^{3} e^{4} x^{3}-9009 B \,a^{4} b^{2} e^{7} x^{3}+13728 B \,a^{3} b^{3} d \,e^{6} x^{3}-11440 B \,a^{2} b^{4} d^{2} e^{5} x^{3}+4992 B a \,b^{5} d^{3} e^{4} x^{3}-896 B \,b^{6} d^{4} e^{3} x^{3}-15015 A \,a^{4} b^{2} e^{7} x^{2}+24024 A \,a^{3} b^{3} d \,e^{6} x^{2}-20592 A \,a^{2} b^{4} d^{2} e^{5} x^{2}+9152 A a \,b^{5} d^{3} e^{4} x^{2}-1664 A \,b^{6} d^{4} e^{3} x^{2}-6006 B \,a^{5} b \,e^{7} x^{2}+18018 B \,a^{4} b^{2} d \,e^{6} x^{2}-27456 B \,a^{3} b^{3} d^{2} e^{5} x^{2}+22880 B \,a^{2} b^{4} d^{3} e^{4} x^{2}-9984 B a \,b^{5} d^{4} e^{3} x^{2}+1792 B \,b^{6} d^{5} e^{2} x^{2}-18018 A \,a^{5} b \,e^{7} x +60060 A \,a^{4} b^{2} d \,e^{6} x -96096 A \,a^{3} b^{3} d^{2} e^{5} x +82368 A \,a^{2} b^{4} d^{3} e^{4} x -36608 A a \,b^{5} d^{4} e^{3} x +6656 A \,b^{6} d^{5} e^{2} x -3003 B \,a^{6} e^{7} x +24024 B \,a^{5} b d \,e^{6} x -72072 B \,a^{4} b^{2} d^{2} e^{5} x +109824 B \,a^{3} b^{3} d^{3} e^{4} x -91520 B \,a^{2} b^{4} d^{4} e^{3} x +39936 B a \,b^{5} d^{5} e^{2} x -7168 B \,b^{6} d^{6} e x +3003 A \,a^{6} e^{7}-36036 A \,a^{5} b d \,e^{6}+120120 A \,a^{4} b^{2} d^{2} e^{5}-192192 A \,a^{3} b^{3} d^{3} e^{4}+164736 A \,a^{2} b^{4} d^{4} e^{3}-73216 A a \,b^{5} d^{5} e^{2}+13312 A \,b^{6} d^{6} e -6006 B \,a^{6} d \,e^{6}+48048 B \,a^{5} b \,d^{2} e^{5}-144144 B \,a^{4} b^{2} d^{3} e^{4}+219648 B \,a^{3} b^{3} d^{4} e^{3}-183040 B \,a^{2} b^{4} d^{5} e^{2}+79872 B a \,b^{5} d^{6} e -14336 B \,b^{6} d^{7}\right )}{3003 \sqrt {e x +d}\, e^{8}}\) | \(913\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(1092\) |
| default | \(\text {Expression too large to display}\) | \(1092\) |
-2*(((-1/13*B*x^7-1/11*A*x^6)*b^6-2/3*(9/11*B*x+A)*x^5*a*b^5-15/7*x^4*(7/9 *B*x+A)*a^2*b^4-4*x^3*(5/7*B*x+A)*a^3*b^3-5*(3/5*B*x+A)*x^2*a^4*b^2-6*(1/3 *B*x+A)*x*a^5*b+a^6*(-B*x+A))*e^7-12*(-1/99*(21/26*B*x+A)*x^5*b^6-5/63*x^4 *(42/55*B*x+A)*a*b^5-2/7*(25/36*B*x+A)*x^3*a^2*b^4-2/3*(4/7*B*x+A)*x^2*a^3 *b^3-5/3*x*a^4*(3/10*B*x+A)*b^2+a^5*(A-2/3*B*x)*b+1/6*B*a^6)*d*e^6+40*b*(- 1/231*x^4*(49/65*B*x+A)*b^5-4/105*(15/22*B*x+A)*x^3*a*b^4-6/35*(5/9*B*x+A) *x^2*a^2*b^3-4/5*(2/7*B*x+A)*x*a^3*b^2+a^4*(-3/5*B*x+A)*b+2/5*a^5*B)*d^2*e ^5-64*b^2*(-1/231*x^3*(35/52*B*x+A)*b^4-1/21*(6/11*B*x+A)*x^2*a*b^3-3/7*x* (5/18*B*x+A)*a^2*b^2+a^3*(-4/7*B*x+A)*b+3/4*B*a^4)*d^3*e^4+384/7*b^3*(-1/9 9*(7/13*B*x+A)*x^2*b^3-2/9*x*(3/11*B*x+A)*a*b^2+a^2*(-5/9*B*x+A)*b+4/3*B*a ^3)*d^4*e^3-512/21*b^4*(-1/11*(7/26*B*x+A)*x*b^2+a*(-6/11*B*x+A)*b+5/2*B*a ^2)*d^5*e^2+1024/231*b^5*((-7/13*B*x+A)*b+6*B*a)*d^6*e-2048/429*B*b^6*d^7) /(e*x+d)^(1/2)/e^8
Leaf count of result is larger than twice the leaf count of optimal. 778 vs. \(2 (276) = 552\).
Time = 0.36 (sec) , antiderivative size = 778, normalized size of antiderivative = 2.59 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (231 \, B b^{6} e^{7} x^{7} + 14336 \, B b^{6} d^{7} - 3003 \, A a^{6} e^{7} - 13312 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{6} e + 36608 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{5} e^{2} - 54912 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{4} e^{3} + 48048 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{3} e^{4} - 24024 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2} e^{5} + 6006 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d e^{6} - 21 \, {\left (14 \, B b^{6} d e^{6} - 13 \, {\left (6 \, B a b^{5} + A b^{6}\right )} e^{7}\right )} x^{6} + 7 \, {\left (56 \, B b^{6} d^{2} e^{5} - 52 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d e^{6} + 143 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} e^{7}\right )} x^{5} - 5 \, {\left (112 \, B b^{6} d^{3} e^{4} - 104 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} e^{5} + 286 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d e^{6} - 429 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} e^{7}\right )} x^{4} + {\left (896 \, B b^{6} d^{4} e^{3} - 832 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} e^{4} + 2288 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} e^{5} - 3432 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d e^{6} + 3003 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} e^{7}\right )} x^{3} - {\left (1792 \, B b^{6} d^{5} e^{2} - 1664 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{4} e^{3} + 4576 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{3} e^{4} - 6864 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} e^{5} + 6006 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d e^{6} - 3003 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} e^{7}\right )} x^{2} + {\left (7168 \, B b^{6} d^{6} e - 6656 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{5} e^{2} + 18304 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{4} e^{3} - 27456 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{3} e^{4} + 24024 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} e^{5} - 12012 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d e^{6} + 3003 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} e^{7}\right )} x\right )} \sqrt {e x + d}}{3003 \, {\left (e^{9} x + d e^{8}\right )}} \]
2/3003*(231*B*b^6*e^7*x^7 + 14336*B*b^6*d^7 - 3003*A*a^6*e^7 - 13312*(6*B* a*b^5 + A*b^6)*d^6*e + 36608*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^2 - 54912*(4* B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^3 + 48048*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e ^4 - 24024*(2*B*a^5*b + 5*A*a^4*b^2)*d^2*e^5 + 6006*(B*a^6 + 6*A*a^5*b)*d* e^6 - 21*(14*B*b^6*d*e^6 - 13*(6*B*a*b^5 + A*b^6)*e^7)*x^6 + 7*(56*B*b^6*d ^2*e^5 - 52*(6*B*a*b^5 + A*b^6)*d*e^6 + 143*(5*B*a^2*b^4 + 2*A*a*b^5)*e^7) *x^5 - 5*(112*B*b^6*d^3*e^4 - 104*(6*B*a*b^5 + A*b^6)*d^2*e^5 + 286*(5*B*a ^2*b^4 + 2*A*a*b^5)*d*e^6 - 429*(4*B*a^3*b^3 + 3*A*a^2*b^4)*e^7)*x^4 + (89 6*B*b^6*d^4*e^3 - 832*(6*B*a*b^5 + A*b^6)*d^3*e^4 + 2288*(5*B*a^2*b^4 + 2* A*a*b^5)*d^2*e^5 - 3432*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d*e^6 + 3003*(3*B*a^4* b^2 + 4*A*a^3*b^3)*e^7)*x^3 - (1792*B*b^6*d^5*e^2 - 1664*(6*B*a*b^5 + A*b^ 6)*d^4*e^3 + 4576*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^4 - 6864*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^5 + 6006*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d*e^6 - 3003*(2*B* a^5*b + 5*A*a^4*b^2)*e^7)*x^2 + (7168*B*b^6*d^6*e - 6656*(6*B*a*b^5 + A*b^ 6)*d^5*e^2 + 18304*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^3 - 27456*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^4 + 24024*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*e^5 - 12012 *(2*B*a^5*b + 5*A*a^4*b^2)*d*e^6 + 3003*(B*a^6 + 6*A*a^5*b)*e^7)*x)*sqrt(e *x + d)/(e^9*x + d*e^8)
Leaf count of result is larger than twice the leaf count of optimal. 942 vs. \(2 (309) = 618\).
Time = 72.05 (sec) , antiderivative size = 942, normalized size of antiderivative = 3.14 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{3/2}} \, dx=\begin {cases} \frac {2 \left (\frac {B b^{6} \left (d + e x\right )^{\frac {13}{2}}}{13 e^{7}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \left (A b^{6} e + 6 B a b^{5} e - 7 B b^{6} d\right )}{11 e^{7}} + \frac {\left (d + e x\right )^{\frac {9}{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 )}{9 e^{7}} + \frac {\left (d + e x\right )^{\frac {7}{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 )}{7 e^{7}} + \frac {\left (d + e x\right )^{\frac {5}{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 )}{5 e^{7}} + \frac {\left (d + e x\right )^{\frac {3}{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 )}{3 e^{7}} + \frac {\sqrt {d + e x} \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 )}{e^{7}} + \frac {\left (- A e + B d\right ) \left (a e - b d\right )^{6}}{e^{7} \sqrt {d + e x}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {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}}{d^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]
Piecewise((2*(B*b**6*(d + e*x)**(13/2)/(13*e**7) + (d + e*x)**(11/2)*(A*b* *6*e + 6*B*a*b**5*e - 7*B*b**6*d)/(11*e**7) + (d + e*x)**(9/2)*(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)/(9*e**7) + (d + e*x)**(7/2)*(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)/(7*e**7) + (d + e*x)**(5/2)*(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)/(5*e**7) + (d + e*x)**(3/2)*(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)/(3*e**7) + sqrt(d + e*x)*(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)/e**7 + (-A*e + B*d)*(a*e - b*d) **6/(e**7*sqrt(d + e*x)))/e, Ne(e, 0)), ((A*a**6*x + B*b**6*x**8/8 + x**7* (A*b**6 + 6*B*a*b**5)/7 + x**6*(6*A*a*b**5 + 15*B*a**2*b**4)/6 + x**5*(15* A*a**2*b**4 + 20*B*a**3*b**3)/5 + x**4*(20*A*a**3*b**3 + 15*B*a**4*b**2...
Leaf count of result is larger than twice the leaf count of optimal. 775 vs. \(2 (276) = 552\).
Time = 0.20 (sec) , antiderivative size = 775, normalized size of antiderivative = 2.58 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (\frac {231 \, {\left (e x + d\right )}^{\frac {13}{2}} B b^{6} - 273 \, {\left (7 \, B b^{6} d - {\left (6 \, B a b^{5} + A b^{6}\right )} e\right )} {\left (e x + d\right )}^{\frac {11}{2}} + 1001 \, {\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 {9}{2}} - 2145 \, {\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 {7}{2}} + 3003 \, {\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 {5}{2}} - 3003 \, {\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 {3}{2}} + 3003 \, {\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 )} \sqrt {e x + d}}{e^{7}} + \frac {3003 \, {\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 )}}{\sqrt {e x + d} e^{7}}\right )}}{3003 \, e} \]
2/3003*((231*(e*x + d)^(13/2)*B*b^6 - 273*(7*B*b^6*d - (6*B*a*b^5 + A*b^6) *e)*(e*x + d)^(11/2) + 1001*(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)^(9/2) - 2145*(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)^(7/2) + 3003*(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)^(5/2) - 3003*(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)^(3/2) + 3003*(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)*sqrt( e*x + d))/e^7 + 3003*(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)/(sqrt(e*x + d)*e^7))/e
Leaf count of result is larger than twice the leaf count of optimal. 1141 vs. \(2 (276) = 552\).
Time = 0.29 (sec) , antiderivative size = 1141, normalized size of antiderivative = 3.80 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{3/2}} \, dx=\text {Too large to display} \]
2*(B*b^6*d^7 - 6*B*a*b^5*d^6*e - A*b^6*d^6*e + 15*B*a^2*b^4*d^5*e^2 + 6*A* a*b^5*d^5*e^2 - 20*B*a^3*b^3*d^4*e^3 - 15*A*a^2*b^4*d^4*e^3 + 15*B*a^4*b^2 *d^3*e^4 + 20*A*a^3*b^3*d^3*e^4 - 6*B*a^5*b*d^2*e^5 - 15*A*a^4*b^2*d^2*e^5 + B*a^6*d*e^6 + 6*A*a^5*b*d*e^6 - A*a^6*e^7)/(sqrt(e*x + d)*e^8) + 2/3003 *(231*(e*x + d)^(13/2)*B*b^6*e^96 - 1911*(e*x + d)^(11/2)*B*b^6*d*e^96 + 7 007*(e*x + d)^(9/2)*B*b^6*d^2*e^96 - 15015*(e*x + d)^(7/2)*B*b^6*d^3*e^96 + 21021*(e*x + d)^(5/2)*B*b^6*d^4*e^96 - 21021*(e*x + d)^(3/2)*B*b^6*d^5*e ^96 + 21021*sqrt(e*x + d)*B*b^6*d^6*e^96 + 1638*(e*x + d)^(11/2)*B*a*b^5*e ^97 + 273*(e*x + d)^(11/2)*A*b^6*e^97 - 12012*(e*x + d)^(9/2)*B*a*b^5*d*e^ 97 - 2002*(e*x + d)^(9/2)*A*b^6*d*e^97 + 38610*(e*x + d)^(7/2)*B*a*b^5*d^2 *e^97 + 6435*(e*x + d)^(7/2)*A*b^6*d^2*e^97 - 72072*(e*x + d)^(5/2)*B*a*b^ 5*d^3*e^97 - 12012*(e*x + d)^(5/2)*A*b^6*d^3*e^97 + 90090*(e*x + d)^(3/2)* B*a*b^5*d^4*e^97 + 15015*(e*x + d)^(3/2)*A*b^6*d^4*e^97 - 108108*sqrt(e*x + d)*B*a*b^5*d^5*e^97 - 18018*sqrt(e*x + d)*A*b^6*d^5*e^97 + 5005*(e*x + d )^(9/2)*B*a^2*b^4*e^98 + 2002*(e*x + d)^(9/2)*A*a*b^5*e^98 - 32175*(e*x + d)^(7/2)*B*a^2*b^4*d*e^98 - 12870*(e*x + d)^(7/2)*A*a*b^5*d*e^98 + 90090*( e*x + d)^(5/2)*B*a^2*b^4*d^2*e^98 + 36036*(e*x + d)^(5/2)*A*a*b^5*d^2*e^98 - 150150*(e*x + d)^(3/2)*B*a^2*b^4*d^3*e^98 - 60060*(e*x + d)^(3/2)*A*a*b ^5*d^3*e^98 + 225225*sqrt(e*x + d)*B*a^2*b^4*d^4*e^98 + 90090*sqrt(e*x + d )*A*a*b^5*d^4*e^98 + 8580*(e*x + d)^(7/2)*B*a^3*b^3*e^99 + 6435*(e*x + ...
Time = 10.73 (sec) , antiderivative size = 438, normalized size of antiderivative = 1.46 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{3/2}} \, dx=\frac {{\left (d+e\,x\right )}^{11/2}\,\left (2\,A\,b^6\,e-14\,B\,b^6\,d+12\,B\,a\,b^5\,e\right )}{11\,e^8}-\frac {-2\,B\,a^6\,d\,e^6+2\,A\,a^6\,e^7+12\,B\,a^5\,b\,d^2\,e^5-12\,A\,a^5\,b\,d\,e^6-30\,B\,a^4\,b^2\,d^3\,e^4+30\,A\,a^4\,b^2\,d^2\,e^5+40\,B\,a^3\,b^3\,d^4\,e^3-40\,A\,a^3\,b^3\,d^3\,e^4-30\,B\,a^2\,b^4\,d^5\,e^2+30\,A\,a^2\,b^4\,d^4\,e^3+12\,B\,a\,b^5\,d^6\,e-12\,A\,a\,b^5\,d^5\,e^2-2\,B\,b^6\,d^7+2\,A\,b^6\,d^6\,e}{e^8\,\sqrt {d+e\,x}}+\frac {2\,{\left (a\,e-b\,d\right )}^5\,\sqrt {d+e\,x}\,\left (6\,A\,b\,e+B\,a\,e-7\,B\,b\,d\right )}{e^8}+\frac {2\,B\,b^6\,{\left (d+e\,x\right )}^{13/2}}{13\,e^8}+\frac {2\,b\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{3/2}\,\left (5\,A\,b\,e+2\,B\,a\,e-7\,B\,b\,d\right )}{e^8}+\frac {2\,b^4\,\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{9/2}\,\left (2\,A\,b\,e+5\,B\,a\,e-7\,B\,b\,d\right )}{3\,e^8}+\frac {2\,b^2\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{5/2}\,\left (4\,A\,b\,e+3\,B\,a\,e-7\,B\,b\,d\right )}{e^8}+\frac {10\,b^3\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{7/2}\,\left (3\,A\,b\,e+4\,B\,a\,e-7\,B\,b\,d\right )}{7\,e^8} \]
((d + e*x)^(11/2)*(2*A*b^6*e - 14*B*b^6*d + 12*B*a*b^5*e))/(11*e^8) - (2*A *a^6*e^7 - 2*B*b^6*d^7 + 2*A*b^6*d^6*e - 2*B*a^6*d*e^6 - 12*A*a*b^5*d^5*e^ 2 + 12*B*a^5*b*d^2*e^5 + 30*A*a^2*b^4*d^4*e^3 - 40*A*a^3*b^3*d^3*e^4 + 30* A*a^4*b^2*d^2*e^5 - 30*B*a^2*b^4*d^5*e^2 + 40*B*a^3*b^3*d^4*e^3 - 30*B*a^4 *b^2*d^3*e^4 - 12*A*a^5*b*d*e^6 + 12*B*a*b^5*d^6*e)/(e^8*(d + e*x)^(1/2)) + (2*(a*e - b*d)^5*(d + e*x)^(1/2)*(6*A*b*e + B*a*e - 7*B*b*d))/e^8 + (2*B *b^6*(d + e*x)^(13/2))/(13*e^8) + (2*b*(a*e - b*d)^4*(d + e*x)^(3/2)*(5*A* b*e + 2*B*a*e - 7*B*b*d))/e^8 + (2*b^4*(a*e - b*d)*(d + e*x)^(9/2)*(2*A*b* e + 5*B*a*e - 7*B*b*d))/(3*e^8) + (2*b^2*(a*e - b*d)^3*(d + e*x)^(5/2)*(4* A*b*e + 3*B*a*e - 7*B*b*d))/e^8 + (10*b^3*(a*e - b*d)^2*(d + e*x)^(7/2)*(3 *A*b*e + 4*B*a*e - 7*B*b*d))/(7*e^8)