Integrand size = 33, antiderivative size = 308 \[ \int (A+B x) \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=-\frac {2 (b d-a e)^6 (B d-A e) (d+e x)^{3/2}}{3 e^8}+\frac {2 (b d-a e)^5 (7 b B d-6 A b e-a B e) (d+e x)^{5/2}}{5 e^8}-\frac {6 b (b d-a e)^4 (7 b B d-5 A b e-2 a B e) (d+e x)^{7/2}}{7 e^8}+\frac {10 b^2 (b d-a e)^3 (7 b B d-4 A b e-3 a B e) (d+e x)^{9/2}}{9 e^8}-\frac {10 b^3 (b d-a e)^2 (7 b B d-3 A b e-4 a B e) (d+e x)^{11/2}}{11 e^8}+\frac {6 b^4 (b d-a e) (7 b B d-2 A b e-5 a B e) (d+e x)^{13/2}}{13 e^8}-\frac {2 b^5 (7 b B d-A b e-6 a B e) (d+e x)^{15/2}}{15 e^8}+\frac {2 b^6 B (d+e x)^{17/2}}{17 e^8} \]
-2/3*(-a*e+b*d)^6*(-A*e+B*d)*(e*x+d)^(3/2)/e^8+2/5*(-a*e+b*d)^5*(-6*A*b*e- B*a*e+7*B*b*d)*(e*x+d)^(5/2)/e^8-6/7*b*(-a*e+b*d)^4*(-5*A*b*e-2*B*a*e+7*B* b*d)*(e*x+d)^(7/2)/e^8+10/9*b^2*(-a*e+b*d)^3*(-4*A*b*e-3*B*a*e+7*B*b*d)*(e *x+d)^(9/2)/e^8-10/11*b^3*(-a*e+b*d)^2*(-3*A*b*e-4*B*a*e+7*B*b*d)*(e*x+d)^ (11/2)/e^8+6/13*b^4*(-a*e+b*d)*(-2*A*b*e-5*B*a*e+7*B*b*d)*(e*x+d)^(13/2)/e ^8-2/15*b^5*(-A*b*e-6*B*a*e+7*B*b*d)*(e*x+d)^(15/2)/e^8+2/17*b^6*B*(e*x+d) ^(17/2)/e^8
Leaf count is larger than twice the leaf count of optimal. \(628\) vs. \(2(308)=616\).
Time = 0.44 (sec) , antiderivative size = 628, normalized size of antiderivative = 2.04 \[ \int (A+B x) \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {2 (d+e x)^{3/2} \left (51051 a^6 e^6 (-2 B d+5 A e+3 B e x)+43758 a^5 b e^5 \left (7 A e (-2 d+3 e x)+B \left (8 d^2-12 d e x+15 e^2 x^2\right )\right )-36465 a^4 b^2 e^4 \left (-3 A e \left (8 d^2-12 d e x+15 e^2 x^2\right )+B \left (16 d^3-24 d^2 e x+30 d e^2 x^2-35 e^3 x^3\right )\right )+4420 a^3 b^3 e^3 \left (11 A e \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )+B \left (128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right )\right )-255 a^2 b^4 e^2 \left (-13 A e \left (128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right )+5 B \left (256 d^5-384 d^4 e x+480 d^3 e^2 x^2-560 d^2 e^3 x^3+630 d e^4 x^4-693 e^5 x^5\right )\right )+102 a b^5 e \left (5 A e \left (-256 d^5+384 d^4 e x-480 d^3 e^2 x^2+560 d^2 e^3 x^3-630 d e^4 x^4+693 e^5 x^5\right )+B \left (1024 d^6-1536 d^5 e x+1920 d^4 e^2 x^2-2240 d^3 e^3 x^3+2520 d^2 e^4 x^4-2772 d e^5 x^5+3003 e^6 x^6\right )\right )+b^6 \left (17 A e \left (1024 d^6-1536 d^5 e x+1920 d^4 e^2 x^2-2240 d^3 e^3 x^3+2520 d^2 e^4 x^4-2772 d e^5 x^5+3003 e^6 x^6\right )-7 B \left (2048 d^7-3072 d^6 e x+3840 d^5 e^2 x^2-4480 d^4 e^3 x^3+5040 d^3 e^4 x^4-5544 d^2 e^5 x^5+6006 d e^6 x^6-6435 e^7 x^7\right )\right )\right )}{765765 e^8} \]
(2*(d + e*x)^(3/2)*(51051*a^6*e^6*(-2*B*d + 5*A*e + 3*B*e*x) + 43758*a^5*b *e^5*(7*A*e*(-2*d + 3*e*x) + B*(8*d^2 - 12*d*e*x + 15*e^2*x^2)) - 36465*a^ 4*b^2*e^4*(-3*A*e*(8*d^2 - 12*d*e*x + 15*e^2*x^2) + B*(16*d^3 - 24*d^2*e*x + 30*d*e^2*x^2 - 35*e^3*x^3)) + 4420*a^3*b^3*e^3*(11*A*e*(-16*d^3 + 24*d^ 2*e*x - 30*d*e^2*x^2 + 35*e^3*x^3) + B*(128*d^4 - 192*d^3*e*x + 240*d^2*e^ 2*x^2 - 280*d*e^3*x^3 + 315*e^4*x^4)) - 255*a^2*b^4*e^2*(-13*A*e*(128*d^4 - 192*d^3*e*x + 240*d^2*e^2*x^2 - 280*d*e^3*x^3 + 315*e^4*x^4) + 5*B*(256* d^5 - 384*d^4*e*x + 480*d^3*e^2*x^2 - 560*d^2*e^3*x^3 + 630*d*e^4*x^4 - 69 3*e^5*x^5)) + 102*a*b^5*e*(5*A*e*(-256*d^5 + 384*d^4*e*x - 480*d^3*e^2*x^2 + 560*d^2*e^3*x^3 - 630*d*e^4*x^4 + 693*e^5*x^5) + B*(1024*d^6 - 1536*d^5 *e*x + 1920*d^4*e^2*x^2 - 2240*d^3*e^3*x^3 + 2520*d^2*e^4*x^4 - 2772*d*e^5 *x^5 + 3003*e^6*x^6)) + b^6*(17*A*e*(1024*d^6 - 1536*d^5*e*x + 1920*d^4*e^ 2*x^2 - 2240*d^3*e^3*x^3 + 2520*d^2*e^4*x^4 - 2772*d*e^5*x^5 + 3003*e^6*x^ 6) - 7*B*(2048*d^7 - 3072*d^6*e*x + 3840*d^5*e^2*x^2 - 4480*d^4*e^3*x^3 + 5040*d^3*e^4*x^4 - 5544*d^2*e^5*x^5 + 6006*d*e^6*x^6 - 6435*e^7*x^7))))/(7 65765*e^8)
Time = 0.47 (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.
| \(\displaystyle \int \left (a^2+2 a b x+b^2 x^2\right )^3 (A+B x) \sqrt {d+e x} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle \frac {\int b^6 (a+b x)^6 (A+B x) \sqrt {d+e x}dx}{b^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int (a+b x)^6 (A+B x) \sqrt {d+e x}dx\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \int \left (\frac {b^5 (d+e x)^{13/2} (6 a B e+A b e-7 b B d)}{e^7}-\frac {3 b^4 (d+e x)^{11/2} (b d-a e) (5 a B e+2 A b e-7 b B d)}{e^7}+\frac {5 b^3 (d+e x)^{9/2} (b d-a e)^2 (4 a B e+3 A b e-7 b B d)}{e^7}-\frac {5 b^2 (d+e x)^{7/2} (b d-a e)^3 (3 a B e+4 A b e-7 b B d)}{e^7}+\frac {3 b (d+e x)^{5/2} (b d-a e)^4 (2 a B e+5 A b e-7 b B d)}{e^7}+\frac {(d+e x)^{3/2} (a e-b d)^5 (a B e+6 A b e-7 b B d)}{e^7}+\frac {\sqrt {d+e x} (a e-b d)^6 (A e-B d)}{e^7}+\frac {b^6 B (d+e x)^{15/2}}{e^7}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 b^5 (d+e x)^{15/2} (-6 a B e-A b e+7 b B d)}{15 e^8}+\frac {6 b^4 (d+e x)^{13/2} (b d-a e) (-5 a B e-2 A b e+7 b B d)}{13 e^8}-\frac {10 b^3 (d+e x)^{11/2} (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{11 e^8}+\frac {10 b^2 (d+e x)^{9/2} (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{9 e^8}-\frac {6 b (d+e x)^{7/2} (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{7 e^8}+\frac {2 (d+e x)^{5/2} (b d-a e)^5 (-a B e-6 A b e+7 b B d)}{5 e^8}-\frac {2 (d+e x)^{3/2} (b d-a e)^6 (B d-A e)}{3 e^8}+\frac {2 b^6 B (d+e x)^{17/2}}{17 e^8}\) |
(-2*(b*d - a*e)^6*(B*d - A*e)*(d + e*x)^(3/2))/(3*e^8) + (2*(b*d - a*e)^5* (7*b*B*d - 6*A*b*e - a*B*e)*(d + e*x)^(5/2))/(5*e^8) - (6*b*(b*d - a*e)^4* (7*b*B*d - 5*A*b*e - 2*a*B*e)*(d + e*x)^(7/2))/(7*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)^(9/2))/(9*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)^(11/2))/(11*e^8) + (6*b^ 4*(b*d - a*e)*(7*b*B*d - 2*A*b*e - 5*a*B*e)*(d + e*x)^(13/2))/(13*e^8) - ( 2*b^5*(7*b*B*d - A*b*e - 6*a*B*e)*(d + e*x)^(15/2))/(15*e^8) + (2*b^6*B*(d + e*x)^(17/2))/(17*e^8)
3.19.3.3.1 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 = 0.35 (sec) , antiderivative size = 518, normalized size of antiderivative = 1.68
| method | result | size |
| pseudoelliptic | \(\frac {2 \left (\left (\left (\frac {3}{17} B \,x^{7}+\frac {1}{5} A \,x^{6}\right ) b^{6}+\frac {18 \left (\frac {13 B x}{15}+A \right ) x^{5} a \,b^{5}}{13}+\frac {45 \left (\frac {11 B x}{13}+A \right ) x^{4} a^{2} b^{4}}{11}+\frac {20 \left (\frac {9 B x}{11}+A \right ) x^{3} a^{3} b^{3}}{3}+\frac {45 x^{2} \left (\frac {7 B x}{9}+A \right ) a^{4} b^{2}}{7}+\frac {18 x \left (\frac {5 B x}{7}+A \right ) a^{5} b}{5}+a^{6} \left (\frac {3 B x}{5}+A \right )\right ) e^{7}-\frac {12 d \left (\left (\frac {7}{102} B \,x^{6}+\frac {1}{13} A \,x^{5}\right ) b^{6}+\frac {75 \left (\frac {22 B x}{25}+A \right ) x^{4} a \,b^{5}}{143}+\frac {50 \left (\frac {45 B x}{52}+A \right ) x^{3} a^{2} b^{4}}{33}+\frac {50 x^{2} \left (\frac {28 B x}{33}+A \right ) a^{3} b^{3}}{21}+\frac {15 x \left (\frac {5 B x}{6}+A \right ) a^{4} b^{2}}{7}+a^{5} \left (\frac {6 B x}{7}+A \right ) b +\frac {B \,a^{6}}{6}\right ) e^{6}}{5}+\frac {24 b \left (\frac {7 x^{4} \left (\frac {77 B x}{85}+A \right ) b^{5}}{143}+\frac {140 \left (\frac {9 B x}{10}+A \right ) x^{3} a \,b^{4}}{429}+\frac {10 x^{2} \left (\frac {35 B x}{39}+A \right ) a^{2} b^{3}}{11}+\frac {4 \left (\frac {10 B x}{11}+A \right ) x \,a^{3} b^{2}}{3}+a^{4} \left (B x +A \right ) b +\frac {2 a^{5} B}{5}\right ) d^{2} e^{5}}{7}-\frac {64 b^{2} d^{3} \left (\frac {7 \left (\frac {63 B x}{68}+A \right ) x^{3} b^{4}}{143}+\frac {45 \left (\frac {14 B x}{15}+A \right ) x^{2} a \,b^{3}}{143}+\frac {9 x \left (\frac {25 B x}{26}+A \right ) a^{2} b^{2}}{11}+a^{3} \left (\frac {12 B x}{11}+A \right ) b +\frac {3 B \,a^{4}}{4}\right ) e^{4}}{21}+\frac {128 b^{3} d^{4} \left (\frac {x^{2} \left (\frac {49 B x}{51}+A \right ) b^{3}}{13}+\frac {6 a x \left (B x +A \right ) b^{2}}{13}+a^{2} \left (\frac {15 B x}{13}+A \right ) b +\frac {4 B \,a^{3}}{3}\right ) e^{3}}{77}-\frac {512 b^{4} \left (\left (\frac {7}{34} B \,x^{2}+\frac {1}{5} A x \right ) b^{2}+a \left (\frac {6 B x}{5}+A \right ) b +\frac {5 B \,a^{2}}{2}\right ) d^{5} e^{2}}{1001}+\frac {1024 b^{5} \left (\left (\frac {21 B x}{17}+A \right ) b +6 B a \right ) d^{6} e}{15015}-\frac {2048 B \,b^{6} d^{7}}{36465}\right ) \left (e x +d \right )^{\frac {3}{2}}}{3 e^{8}}\) | \(518\) |
| gosper | \(\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (45045 b^{6} B \,x^{7} e^{7}+51051 A \,b^{6} e^{7} x^{6}+306306 B a \,b^{5} e^{7} x^{6}-42042 B \,b^{6} d \,e^{6} x^{6}+353430 A a \,b^{5} e^{7} x^{5}-47124 A \,b^{6} d \,e^{6} x^{5}+883575 B \,a^{2} b^{4} e^{7} x^{5}-282744 B a \,b^{5} d \,e^{6} x^{5}+38808 B \,b^{6} d^{2} e^{5} x^{5}+1044225 A \,a^{2} b^{4} e^{7} x^{4}-321300 A a \,b^{5} d \,e^{6} x^{4}+42840 A \,b^{6} d^{2} e^{5} x^{4}+1392300 B \,a^{3} b^{3} e^{7} x^{4}-803250 B \,a^{2} b^{4} d \,e^{6} x^{4}+257040 B a \,b^{5} d^{2} e^{5} x^{4}-35280 B \,b^{6} d^{3} e^{4} x^{4}+1701700 A \,a^{3} b^{3} e^{7} x^{3}-928200 A \,a^{2} b^{4} d \,e^{6} x^{3}+285600 A a \,b^{5} d^{2} e^{5} x^{3}-38080 A \,b^{6} d^{3} e^{4} x^{3}+1276275 B \,a^{4} b^{2} e^{7} x^{3}-1237600 B \,a^{3} b^{3} d \,e^{6} x^{3}+714000 B \,a^{2} b^{4} d^{2} e^{5} x^{3}-228480 B a \,b^{5} d^{3} e^{4} x^{3}+31360 B \,b^{6} d^{4} e^{3} x^{3}+1640925 A \,a^{4} b^{2} e^{7} x^{2}-1458600 A \,a^{3} b^{3} d \,e^{6} x^{2}+795600 A \,a^{2} b^{4} d^{2} e^{5} x^{2}-244800 A a \,b^{5} d^{3} e^{4} x^{2}+32640 A \,b^{6} d^{4} e^{3} x^{2}+656370 B \,a^{5} b \,e^{7} x^{2}-1093950 B \,a^{4} b^{2} d \,e^{6} x^{2}+1060800 B \,a^{3} b^{3} d^{2} e^{5} x^{2}-612000 B \,a^{2} b^{4} d^{3} e^{4} x^{2}+195840 B a \,b^{5} d^{4} e^{3} x^{2}-26880 B \,b^{6} d^{5} e^{2} x^{2}+918918 A \,a^{5} b \,e^{7} x -1312740 A \,a^{4} b^{2} d \,e^{6} x +1166880 A \,a^{3} b^{3} d^{2} e^{5} x -636480 A \,a^{2} b^{4} d^{3} e^{4} x +195840 A a \,b^{5} d^{4} e^{3} x -26112 A \,b^{6} d^{5} e^{2} x +153153 B \,a^{6} e^{7} x -525096 B \,a^{5} b d \,e^{6} x +875160 B \,a^{4} b^{2} d^{2} e^{5} x -848640 B \,a^{3} b^{3} d^{3} e^{4} x +489600 B \,a^{2} b^{4} d^{4} e^{3} x -156672 B a \,b^{5} d^{5} e^{2} x +21504 B \,b^{6} d^{6} e x +255255 A \,a^{6} e^{7}-612612 A \,a^{5} b d \,e^{6}+875160 A \,a^{4} b^{2} d^{2} e^{5}-777920 A \,a^{3} b^{3} d^{3} e^{4}+424320 A \,a^{2} b^{4} d^{4} e^{3}-130560 A a \,b^{5} d^{5} e^{2}+17408 A \,b^{6} d^{6} e -102102 B \,a^{6} d \,e^{6}+350064 B \,a^{5} b \,d^{2} e^{5}-583440 B \,a^{4} b^{2} d^{3} e^{4}+565760 B \,a^{3} b^{3} d^{4} e^{3}-326400 B \,a^{2} b^{4} d^{5} e^{2}+104448 B a \,b^{5} d^{6} e -14336 B \,b^{6} d^{7}\right )}{765765 e^{8}}\) | \(913\) |
| derivativedivides | \(\frac {\frac {2 B \,b^{6} \left (e x +d \right )^{\frac {17}{2}}}{17}+\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 {15}{2}}}{15}+\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 {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 ) 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 {11}{2}}}{11}+\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 {9}{2}}}{9}+\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 {7}{2}}}{7}+\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 {5}{2}}}{5}+\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 {3}{2}}}{3}}{e^{8}}\) | \(938\) |
| default | \(\frac {\frac {2 B \,b^{6} \left (e x +d \right )^{\frac {17}{2}}}{17}+\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 {15}{2}}}{15}+\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 {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 ) 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 {11}{2}}}{11}+\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 {9}{2}}}{9}+\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 {7}{2}}}{7}+\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 {5}{2}}}{5}+\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 {3}{2}}}{3}}{e^{8}}\) | \(938\) |
| trager | \(\text {Expression too large to display}\) | \(1141\) |
| risch | \(\text {Expression too large to display}\) | \(1141\) |
2/3*(((3/17*B*x^7+1/5*A*x^6)*b^6+18/13*(13/15*B*x+A)*x^5*a*b^5+45/11*(11/1 3*B*x+A)*x^4*a^2*b^4+20/3*(9/11*B*x+A)*x^3*a^3*b^3+45/7*x^2*(7/9*B*x+A)*a^ 4*b^2+18/5*x*(5/7*B*x+A)*a^5*b+a^6*(3/5*B*x+A))*e^7-12/5*d*((7/102*B*x^6+1 /13*A*x^5)*b^6+75/143*(22/25*B*x+A)*x^4*a*b^5+50/33*(45/52*B*x+A)*x^3*a^2* b^4+50/21*x^2*(28/33*B*x+A)*a^3*b^3+15/7*x*(5/6*B*x+A)*a^4*b^2+a^5*(6/7*B* x+A)*b+1/6*B*a^6)*e^6+24/7*b*(7/143*x^4*(77/85*B*x+A)*b^5+140/429*(9/10*B* x+A)*x^3*a*b^4+10/11*x^2*(35/39*B*x+A)*a^2*b^3+4/3*(10/11*B*x+A)*x*a^3*b^2 +a^4*(B*x+A)*b+2/5*a^5*B)*d^2*e^5-64/21*b^2*d^3*(7/143*(63/68*B*x+A)*x^3*b ^4+45/143*(14/15*B*x+A)*x^2*a*b^3+9/11*x*(25/26*B*x+A)*a^2*b^2+a^3*(12/11* B*x+A)*b+3/4*B*a^4)*e^4+128/77*b^3*d^4*(1/13*x^2*(49/51*B*x+A)*b^3+6/13*a* x*(B*x+A)*b^2+a^2*(15/13*B*x+A)*b+4/3*B*a^3)*e^3-512/1001*b^4*((7/34*B*x^2 +1/5*A*x)*b^2+a*(6/5*B*x+A)*b+5/2*B*a^2)*d^5*e^2+1024/15015*b^5*((21/17*B* x+A)*b+6*B*a)*d^6*e-2048/36465*B*b^6*d^7)*(e*x+d)^(3/2)/e^8
Leaf count of result is larger than twice the leaf count of optimal. 942 vs. \(2 (276) = 552\).
Time = 0.37 (sec) , antiderivative size = 942, normalized size of antiderivative = 3.06 \[ \int (A+B x) \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {2 \, {\left (45045 \, B b^{6} e^{8} x^{8} - 14336 \, B b^{6} d^{8} + 255255 \, A a^{6} d e^{7} + 17408 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{7} e - 65280 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{6} e^{2} + 141440 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{5} e^{3} - 194480 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{4} e^{4} + 175032 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{3} e^{5} - 102102 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d^{2} e^{6} + 3003 \, {\left (B b^{6} d e^{7} + 17 \, {\left (6 \, B a b^{5} + A b^{6}\right )} e^{8}\right )} x^{7} - 231 \, {\left (14 \, B b^{6} d^{2} e^{6} - 17 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d e^{7} - 765 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} e^{8}\right )} x^{6} + 63 \, {\left (56 \, B b^{6} d^{3} e^{5} - 68 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} e^{6} + 255 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d e^{7} + 5525 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} e^{8}\right )} x^{5} - 35 \, {\left (112 \, B b^{6} d^{4} e^{4} - 136 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} e^{5} + 510 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} e^{6} - 1105 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d e^{7} - 12155 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} e^{8}\right )} x^{4} + 5 \, {\left (896 \, B b^{6} d^{5} e^{3} - 1088 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{4} e^{4} + 4080 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{3} e^{5} - 8840 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} e^{6} + 12155 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d e^{7} + 65637 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} e^{8}\right )} x^{3} - 3 \, {\left (1792 \, B b^{6} d^{6} e^{2} - 2176 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{5} e^{3} + 8160 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{4} e^{4} - 17680 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{3} e^{5} + 24310 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} e^{6} - 21879 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d e^{7} - 51051 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} e^{8}\right )} x^{2} + {\left (7168 \, B b^{6} d^{7} e + 255255 \, A a^{6} e^{8} - 8704 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{6} e^{2} + 32640 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{5} e^{3} - 70720 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{4} e^{4} + 97240 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{3} e^{5} - 87516 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2} e^{6} + 51051 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d e^{7}\right )} x\right )} \sqrt {e x + d}}{765765 \, e^{8}} \]
2/765765*(45045*B*b^6*e^8*x^8 - 14336*B*b^6*d^8 + 255255*A*a^6*d*e^7 + 174 08*(6*B*a*b^5 + A*b^6)*d^7*e - 65280*(5*B*a^2*b^4 + 2*A*a*b^5)*d^6*e^2 + 1 41440*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^5*e^3 - 194480*(3*B*a^4*b^2 + 4*A*a^3* b^3)*d^4*e^4 + 175032*(2*B*a^5*b + 5*A*a^4*b^2)*d^3*e^5 - 102102*(B*a^6 + 6*A*a^5*b)*d^2*e^6 + 3003*(B*b^6*d*e^7 + 17*(6*B*a*b^5 + A*b^6)*e^8)*x^7 - 231*(14*B*b^6*d^2*e^6 - 17*(6*B*a*b^5 + A*b^6)*d*e^7 - 765*(5*B*a^2*b^4 + 2*A*a*b^5)*e^8)*x^6 + 63*(56*B*b^6*d^3*e^5 - 68*(6*B*a*b^5 + A*b^6)*d^2*e ^6 + 255*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^7 + 5525*(4*B*a^3*b^3 + 3*A*a^2*b^4 )*e^8)*x^5 - 35*(112*B*b^6*d^4*e^4 - 136*(6*B*a*b^5 + A*b^6)*d^3*e^5 + 510 *(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^6 - 1105*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d*e^ 7 - 12155*(3*B*a^4*b^2 + 4*A*a^3*b^3)*e^8)*x^4 + 5*(896*B*b^6*d^5*e^3 - 10 88*(6*B*a*b^5 + A*b^6)*d^4*e^4 + 4080*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^5 - 8840*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^6 + 12155*(3*B*a^4*b^2 + 4*A*a^3*b^ 3)*d*e^7 + 65637*(2*B*a^5*b + 5*A*a^4*b^2)*e^8)*x^3 - 3*(1792*B*b^6*d^6*e^ 2 - 2176*(6*B*a*b^5 + A*b^6)*d^5*e^3 + 8160*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4* e^4 - 17680*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^5 + 24310*(3*B*a^4*b^2 + 4*A *a^3*b^3)*d^2*e^6 - 21879*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^7 - 51051*(B*a^6 + 6*A*a^5*b)*e^8)*x^2 + (7168*B*b^6*d^7*e + 255255*A*a^6*e^8 - 8704*(6*B*a* b^5 + A*b^6)*d^6*e^2 + 32640*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^3 - 70720*(4* B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^4 + 97240*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^...
Leaf count of result is larger than twice the leaf count of optimal. 1127 vs. \(2 (316) = 632\).
Time = 2.02 (sec) , antiderivative size = 1127, normalized size of antiderivative = 3.66 \[ \int (A+B x) \sqrt {d+e x} \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 {17}{2}}}{17 e^{7}} + \frac {\left (d + e x\right )^{\frac {15}{2}} \left (A b^{6} e + 6 B a b^{5} e - 7 B b^{6} d\right )}{15 e^{7}} + \frac {\left (d + e x\right )^{\frac {13}{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 )}{13 e^{7}} + \frac {\left (d + e x\right )^{\frac {11}{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 )}{11 e^{7}} + \frac {\left (d + e x\right )^{\frac {9}{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 )}{9 e^{7}} + \frac {\left (d + e x\right )^{\frac {7}{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 )}{7 e^{7}} + \frac {\left (d + e x\right )^{\frac {5}{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 )}{5 e^{7}} + \frac {\left (d + e x\right )^{\frac {3}{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 )}{3 e^{7}}\right )}{e} & \text {for}\: e \neq 0 \\\sqrt {d} \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)**(17/2)/(17*e**7) + (d + e*x)**(15/2)*(A*b* *6*e + 6*B*a*b**5*e - 7*B*b**6*d)/(15*e**7) + (d + e*x)**(13/2)*(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)/(13*e**7) + (d + e*x)**(11/2)*(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)/(11*e**7) + (d + e*x)**(9/2)*(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)/(9*e**7) + (d + e*x)**(7/2)*(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)/(7*e**7) + (d + e*x)**(5/2)*(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)/(5*e**7) + (d + e*x)**(3 /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**3*e...
Leaf count of result is larger than twice the leaf count of optimal. 767 vs. \(2 (276) = 552\).
Time = 0.21 (sec) , antiderivative size = 767, normalized size of antiderivative = 2.49 \[ \int (A+B x) \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {2 \, {\left (45045 \, {\left (e x + d\right )}^{\frac {17}{2}} B b^{6} - 51051 \, {\left (7 \, B b^{6} d - {\left (6 \, B a b^{5} + A b^{6}\right )} e\right )} {\left (e x + d\right )}^{\frac {15}{2}} + 176715 \, {\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 {13}{2}} - 348075 \, {\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 {11}{2}} + 425425 \, {\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 {9}{2}} - 328185 \, {\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 {7}{2}} + 153153 \, {\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 {5}{2}} - 255255 \, {\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 {3}{2}}\right )}}{765765 \, e^{8}} \]
2/765765*(45045*(e*x + d)^(17/2)*B*b^6 - 51051*(7*B*b^6*d - (6*B*a*b^5 + A *b^6)*e)*(e*x + d)^(15/2) + 176715*(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)^(13/2) - 348075*(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)^(11/2) + 425425*(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)^(9/2) - 328185*(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)^(7/2) + 153153*(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)^(5/2) - 255255*(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)^(3/2))/e^8
Leaf count of result is larger than twice the leaf count of optimal. 1871 vs. \(2 (276) = 552\).
Time = 0.30 (sec) , antiderivative size = 1871, normalized size of antiderivative = 6.07 \[ \int (A+B x) \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\text {Too large to display} \]
2/765765*(765765*sqrt(e*x + d)*A*a^6*d + 255255*((e*x + d)^(3/2) - 3*sqrt( e*x + d)*d)*A*a^6 + 255255*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*B*a^6*d/e + 1531530*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*A*a^5*b*d/e + 306306*(3*( e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*B*a^5*b*d/e^ 2 + 765765*(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/e^2 + 51051*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15* sqrt(e*x + d)*d^2)*B*a^6/e + 306306*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2 )*d + 15*sqrt(e*x + d)*d^2)*A*a^5*b/e + 328185*(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/e^3 + 437580*(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/e^3 + 131274*(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/e^2 + 328185*(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/e^2 + 48620*(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)*B*a^3*b^3*d/e^4 + 36465*(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*a^2*b^4*d/e^4 + 36465*(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)*B*a^4*b^2/e^3 + 48620*(35*(e*x + d)^(9/2) - ...
Time = 10.65 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.91 \[ \int (A+B x) \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {{\left (d+e\,x\right )}^{15/2}\,\left (2\,A\,b^6\,e-14\,B\,b^6\,d+12\,B\,a\,b^5\,e\right )}{15\,e^8}+\frac {2\,{\left (a\,e-b\,d\right )}^5\,{\left (d+e\,x\right )}^{5/2}\,\left (6\,A\,b\,e+B\,a\,e-7\,B\,b\,d\right )}{5\,e^8}+\frac {2\,B\,b^6\,{\left (d+e\,x\right )}^{17/2}}{17\,e^8}+\frac {2\,\left (A\,e-B\,d\right )\,{\left (a\,e-b\,d\right )}^6\,{\left (d+e\,x\right )}^{3/2}}{3\,e^8}+\frac {6\,b\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{7/2}\,\left (5\,A\,b\,e+2\,B\,a\,e-7\,B\,b\,d\right )}{7\,e^8}+\frac {6\,b^4\,\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{13/2}\,\left (2\,A\,b\,e+5\,B\,a\,e-7\,B\,b\,d\right )}{13\,e^8}+\frac {10\,b^2\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{9/2}\,\left (4\,A\,b\,e+3\,B\,a\,e-7\,B\,b\,d\right )}{9\,e^8}+\frac {10\,b^3\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{11/2}\,\left (3\,A\,b\,e+4\,B\,a\,e-7\,B\,b\,d\right )}{11\,e^8} \]
((d + e*x)^(15/2)*(2*A*b^6*e - 14*B*b^6*d + 12*B*a*b^5*e))/(15*e^8) + (2*( a*e - b*d)^5*(d + e*x)^(5/2)*(6*A*b*e + B*a*e - 7*B*b*d))/(5*e^8) + (2*B*b ^6*(d + e*x)^(17/2))/(17*e^8) + (2*(A*e - B*d)*(a*e - b*d)^6*(d + e*x)^(3/ 2))/(3*e^8) + (6*b*(a*e - b*d)^4*(d + e*x)^(7/2)*(5*A*b*e + 2*B*a*e - 7*B* b*d))/(7*e^8) + (6*b^4*(a*e - b*d)*(d + e*x)^(13/2)*(2*A*b*e + 5*B*a*e - 7 *B*b*d))/(13*e^8) + (10*b^2*(a*e - b*d)^3*(d + e*x)^(9/2)*(4*A*b*e + 3*B*a *e - 7*B*b*d))/(9*e^8) + (10*b^3*(a*e - b*d)^2*(d + e*x)^(11/2)*(3*A*b*e + 4*B*a*e - 7*B*b*d))/(11*e^8)