Integrand size = 33, antiderivative size = 156 \[ \int (a+b x) \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=-\frac {2 (b d-a e)^5 (d+e x)^{3/2}}{3 e^6}+\frac {2 b (b d-a e)^4 (d+e x)^{5/2}}{e^6}-\frac {20 b^2 (b d-a e)^3 (d+e x)^{7/2}}{7 e^6}+\frac {20 b^3 (b d-a e)^2 (d+e x)^{9/2}}{9 e^6}-\frac {10 b^4 (b d-a e) (d+e x)^{11/2}}{11 e^6}+\frac {2 b^5 (d+e x)^{13/2}}{13 e^6} \]
-2/3*(-a*e+b*d)^5*(e*x+d)^(3/2)/e^6+2*b*(-a*e+b*d)^4*(e*x+d)^(5/2)/e^6-20/ 7*b^2*(-a*e+b*d)^3*(e*x+d)^(7/2)/e^6+20/9*b^3*(-a*e+b*d)^2*(e*x+d)^(9/2)/e ^6-10/11*b^4*(-a*e+b*d)*(e*x+d)^(11/2)/e^6+2/13*b^5*(e*x+d)^(13/2)/e^6
Time = 0.04 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.39 \[ \int (a+b x) \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {2 (d+e x)^{3/2} \left (3003 a^5 e^5+3003 a^4 b e^4 (-2 d+3 e x)+858 a^3 b^2 e^3 \left (8 d^2-12 d e x+15 e^2 x^2\right )+286 a^2 b^3 e^2 \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )+13 a b^4 e \left (128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right )+b^5 \left (-256 d^5+384 d^4 e x-480 d^3 e^2 x^2+560 d^2 e^3 x^3-630 d e^4 x^4+693 e^5 x^5\right )\right )}{9009 e^6} \]
(2*(d + e*x)^(3/2)*(3003*a^5*e^5 + 3003*a^4*b*e^4*(-2*d + 3*e*x) + 858*a^3 *b^2*e^3*(8*d^2 - 12*d*e*x + 15*e^2*x^2) + 286*a^2*b^3*e^2*(-16*d^3 + 24*d ^2*e*x - 30*d*e^2*x^2 + 35*e^3*x^3) + 13*a*b^4*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) + b^5*(-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)))/(9 009*e^6)
Time = 0.30 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {1184, 27, 53, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2 \sqrt {d+e x} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle \frac {\int b^4 (a+b x)^5 \sqrt {d+e x}dx}{b^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int (a+b x)^5 \sqrt {d+e x}dx\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \int \left (-\frac {5 b^4 (d+e x)^{9/2} (b d-a e)}{e^5}+\frac {10 b^3 (d+e x)^{7/2} (b d-a e)^2}{e^5}-\frac {10 b^2 (d+e x)^{5/2} (b d-a e)^3}{e^5}+\frac {5 b (d+e x)^{3/2} (b d-a e)^4}{e^5}+\frac {\sqrt {d+e x} (a e-b d)^5}{e^5}+\frac {b^5 (d+e x)^{11/2}}{e^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {10 b^4 (d+e x)^{11/2} (b d-a e)}{11 e^6}+\frac {20 b^3 (d+e x)^{9/2} (b d-a e)^2}{9 e^6}-\frac {20 b^2 (d+e x)^{7/2} (b d-a e)^3}{7 e^6}+\frac {2 b (d+e x)^{5/2} (b d-a e)^4}{e^6}-\frac {2 (d+e x)^{3/2} (b d-a e)^5}{3 e^6}+\frac {2 b^5 (d+e x)^{13/2}}{13 e^6}\) |
(-2*(b*d - a*e)^5*(d + e*x)^(3/2))/(3*e^6) + (2*b*(b*d - a*e)^4*(d + e*x)^ (5/2))/e^6 - (20*b^2*(b*d - a*e)^3*(d + e*x)^(7/2))/(7*e^6) + (20*b^3*(b*d - a*e)^2*(d + e*x)^(9/2))/(9*e^6) - (10*b^4*(b*d - a*e)*(d + e*x)^(11/2)) /(11*e^6) + (2*b^5*(d + e*x)^(13/2))/(13*e^6)
3.21.53.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/c^p Int[(d + e*x)^m*(f + g*x )^n*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x] && E qQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Time = 0.32 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.30
| method | result | size |
| pseudoelliptic | \(\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (\left (\frac {3}{13} b^{5} x^{5}+\frac {15}{11} a \,b^{4} x^{4}+\frac {10}{3} a^{2} b^{3} x^{3}+\frac {30}{7} a^{3} b^{2} x^{2}+3 a^{4} b x +a^{5}\right ) e^{5}-2 b \left (\frac {15}{143} x^{4} b^{4}+\frac {20}{33} a \,b^{3} x^{3}+\frac {10}{7} x^{2} b^{2} a^{2}+\frac {12}{7} b \,a^{3} x +a^{4}\right ) d \,e^{4}+\frac {16 \left (\frac {35}{429} x^{3} b^{3}+\frac {5}{11} a \,b^{2} x^{2}+b \,a^{2} x +a^{3}\right ) b^{2} d^{2} e^{3}}{7}-\frac {32 b^{3} \left (\frac {15}{143} b^{2} x^{2}+\frac {6}{11} a b x +a^{2}\right ) d^{3} e^{2}}{21}+\frac {128 b^{4} \left (\frac {3 b x}{13}+a \right ) d^{4} e}{231}-\frac {256 b^{5} d^{5}}{3003}\right )}{3 e^{6}}\) | \(203\) |
| gosper | \(\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (693 x^{5} b^{5} e^{5}+4095 x^{4} a \,b^{4} e^{5}-630 x^{4} b^{5} d \,e^{4}+10010 x^{3} a^{2} b^{3} e^{5}-3640 x^{3} a \,b^{4} d \,e^{4}+560 x^{3} b^{5} d^{2} e^{3}+12870 x^{2} a^{3} b^{2} e^{5}-8580 x^{2} a^{2} b^{3} d \,e^{4}+3120 x^{2} a \,b^{4} d^{2} e^{3}-480 x^{2} b^{5} d^{3} e^{2}+9009 x \,a^{4} b \,e^{5}-10296 x \,a^{3} b^{2} d \,e^{4}+6864 x \,a^{2} b^{3} d^{2} e^{3}-2496 x a \,b^{4} d^{3} e^{2}+384 x \,b^{5} d^{4} e +3003 e^{5} a^{5}-6006 b d \,e^{4} a^{4}+6864 b^{2} d^{2} e^{3} a^{3}-4576 b^{3} d^{3} e^{2} a^{2}+1664 b^{4} d^{4} e a -256 b^{5} d^{5}\right )}{9009 e^{6}}\) | \(273\) |
| derivativedivides | \(\frac {\frac {2 b^{5} \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (\left (a e -b d \right ) b^{4}+2 b^{3} \left (2 a b e -2 b^{2} d \right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (2 \left (a e -b d \right ) \left (2 a b e -2 b^{2} d \right ) b^{2}+b \left (2 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (\left (a e -b d \right ) \left (2 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right )+2 b \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a b e -2 b^{2} d \right )\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (2 \left (a e -b d \right ) \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a b e -2 b^{2} d \right )+b \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (a e -b d \right ) \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}}{e^{6}}\) | \(350\) |
| default | \(\frac {\frac {2 b^{5} \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (\left (a e -b d \right ) b^{4}+2 b^{3} \left (2 a b e -2 b^{2} d \right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (2 \left (a e -b d \right ) \left (2 a b e -2 b^{2} d \right ) b^{2}+b \left (2 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (\left (a e -b d \right ) \left (2 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right )+2 b \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a b e -2 b^{2} d \right )\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (2 \left (a e -b d \right ) \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a b e -2 b^{2} d \right )+b \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (a e -b d \right ) \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}}{e^{6}}\) | \(350\) |
| trager | \(\frac {2 \left (693 b^{5} e^{6} x^{6}+4095 b^{4} a \,e^{6} x^{5}+63 b^{5} d \,e^{5} x^{5}+10010 a^{2} b^{3} e^{6} x^{4}+455 b^{4} a d \,e^{5} x^{4}-70 b^{5} d^{2} e^{4} x^{4}+12870 a^{3} b^{2} e^{6} x^{3}+1430 a^{2} b^{3} d \,e^{5} x^{3}-520 a \,b^{4} d^{2} e^{4} x^{3}+80 b^{5} d^{3} e^{3} x^{3}+9009 a^{4} b \,e^{6} x^{2}+2574 a^{3} b^{2} d \,e^{5} x^{2}-1716 a^{2} b^{3} d^{2} e^{4} x^{2}+624 b^{4} a \,d^{3} e^{3} x^{2}-96 b^{5} d^{4} e^{2} x^{2}+3003 a^{5} e^{6} x +3003 a^{4} b d \,e^{5} x -3432 a^{3} b^{2} d^{2} e^{4} x +2288 a^{2} b^{3} d^{3} e^{3} x -832 b^{4} a \,d^{4} e^{2} x +128 b^{5} d^{5} e x +3003 a^{5} d \,e^{5}-6006 a^{4} b \,d^{2} e^{4}+6864 a^{3} b^{2} d^{3} e^{3}-4576 a^{2} b^{3} d^{4} e^{2}+1664 b^{4} a \,d^{5} e -256 b^{5} d^{6}\right ) \sqrt {e x +d}}{9009 e^{6}}\) | \(361\) |
| risch | \(\frac {2 \left (693 b^{5} e^{6} x^{6}+4095 b^{4} a \,e^{6} x^{5}+63 b^{5} d \,e^{5} x^{5}+10010 a^{2} b^{3} e^{6} x^{4}+455 b^{4} a d \,e^{5} x^{4}-70 b^{5} d^{2} e^{4} x^{4}+12870 a^{3} b^{2} e^{6} x^{3}+1430 a^{2} b^{3} d \,e^{5} x^{3}-520 a \,b^{4} d^{2} e^{4} x^{3}+80 b^{5} d^{3} e^{3} x^{3}+9009 a^{4} b \,e^{6} x^{2}+2574 a^{3} b^{2} d \,e^{5} x^{2}-1716 a^{2} b^{3} d^{2} e^{4} x^{2}+624 b^{4} a \,d^{3} e^{3} x^{2}-96 b^{5} d^{4} e^{2} x^{2}+3003 a^{5} e^{6} x +3003 a^{4} b d \,e^{5} x -3432 a^{3} b^{2} d^{2} e^{4} x +2288 a^{2} b^{3} d^{3} e^{3} x -832 b^{4} a \,d^{4} e^{2} x +128 b^{5} d^{5} e x +3003 a^{5} d \,e^{5}-6006 a^{4} b \,d^{2} e^{4}+6864 a^{3} b^{2} d^{3} e^{3}-4576 a^{2} b^{3} d^{4} e^{2}+1664 b^{4} a \,d^{5} e -256 b^{5} d^{6}\right ) \sqrt {e x +d}}{9009 e^{6}}\) | \(361\) |
2/3*(e*x+d)^(3/2)*((3/13*b^5*x^5+15/11*a*b^4*x^4+10/3*a^2*b^3*x^3+30/7*a^3 *b^2*x^2+3*a^4*b*x+a^5)*e^5-2*b*(15/143*x^4*b^4+20/33*a*b^3*x^3+10/7*x^2*b ^2*a^2+12/7*b*a^3*x+a^4)*d*e^4+16/7*(35/429*x^3*b^3+5/11*a*b^2*x^2+b*a^2*x +a^3)*b^2*d^2*e^3-32/21*b^3*(15/143*b^2*x^2+6/11*a*b*x+a^2)*d^3*e^2+128/23 1*b^4*(3/13*b*x+a)*d^4*e-256/3003*b^5*d^5)/e^6
Leaf count of result is larger than twice the leaf count of optimal. 338 vs. \(2 (134) = 268\).
Time = 0.35 (sec) , antiderivative size = 338, normalized size of antiderivative = 2.17 \[ \int (a+b x) \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {2 \, {\left (693 \, b^{5} e^{6} x^{6} - 256 \, b^{5} d^{6} + 1664 \, a b^{4} d^{5} e - 4576 \, a^{2} b^{3} d^{4} e^{2} + 6864 \, a^{3} b^{2} d^{3} e^{3} - 6006 \, a^{4} b d^{2} e^{4} + 3003 \, a^{5} d e^{5} + 63 \, {\left (b^{5} d e^{5} + 65 \, a b^{4} e^{6}\right )} x^{5} - 35 \, {\left (2 \, b^{5} d^{2} e^{4} - 13 \, a b^{4} d e^{5} - 286 \, a^{2} b^{3} e^{6}\right )} x^{4} + 10 \, {\left (8 \, b^{5} d^{3} e^{3} - 52 \, a b^{4} d^{2} e^{4} + 143 \, a^{2} b^{3} d e^{5} + 1287 \, a^{3} b^{2} e^{6}\right )} x^{3} - 3 \, {\left (32 \, b^{5} d^{4} e^{2} - 208 \, a b^{4} d^{3} e^{3} + 572 \, a^{2} b^{3} d^{2} e^{4} - 858 \, a^{3} b^{2} d e^{5} - 3003 \, a^{4} b e^{6}\right )} x^{2} + {\left (128 \, b^{5} d^{5} e - 832 \, a b^{4} d^{4} e^{2} + 2288 \, a^{2} b^{3} d^{3} e^{3} - 3432 \, a^{3} b^{2} d^{2} e^{4} + 3003 \, a^{4} b d e^{5} + 3003 \, a^{5} e^{6}\right )} x\right )} \sqrt {e x + d}}{9009 \, e^{6}} \]
2/9009*(693*b^5*e^6*x^6 - 256*b^5*d^6 + 1664*a*b^4*d^5*e - 4576*a^2*b^3*d^ 4*e^2 + 6864*a^3*b^2*d^3*e^3 - 6006*a^4*b*d^2*e^4 + 3003*a^5*d*e^5 + 63*(b ^5*d*e^5 + 65*a*b^4*e^6)*x^5 - 35*(2*b^5*d^2*e^4 - 13*a*b^4*d*e^5 - 286*a^ 2*b^3*e^6)*x^4 + 10*(8*b^5*d^3*e^3 - 52*a*b^4*d^2*e^4 + 143*a^2*b^3*d*e^5 + 1287*a^3*b^2*e^6)*x^3 - 3*(32*b^5*d^4*e^2 - 208*a*b^4*d^3*e^3 + 572*a^2* b^3*d^2*e^4 - 858*a^3*b^2*d*e^5 - 3003*a^4*b*e^6)*x^2 + (128*b^5*d^5*e - 8 32*a*b^4*d^4*e^2 + 2288*a^2*b^3*d^3*e^3 - 3432*a^3*b^2*d^2*e^4 + 3003*a^4* b*d*e^5 + 3003*a^5*e^6)*x)*sqrt(e*x + d)/e^6
Leaf count of result is larger than twice the leaf count of optimal. 347 vs. \(2 (144) = 288\).
Time = 1.50 (sec) , antiderivative size = 347, normalized size of antiderivative = 2.22 \[ \int (a+b x) \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\begin {cases} \frac {2 \left (\frac {b^{5} \left (d + e x\right )^{\frac {13}{2}}}{13 e^{5}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \cdot \left (5 a b^{4} e - 5 b^{5} d\right )}{11 e^{5}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \cdot \left (10 a^{2} b^{3} e^{2} - 20 a b^{4} d e + 10 b^{5} d^{2}\right )}{9 e^{5}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \cdot \left (10 a^{3} b^{2} e^{3} - 30 a^{2} b^{3} d e^{2} + 30 a b^{4} d^{2} e - 10 b^{5} d^{3}\right )}{7 e^{5}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \left (5 a^{4} b e^{4} - 20 a^{3} b^{2} d e^{3} + 30 a^{2} b^{3} d^{2} e^{2} - 20 a b^{4} d^{3} e + 5 b^{5} d^{4}\right )}{5 e^{5}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (a^{5} e^{5} - 5 a^{4} b d e^{4} + 10 a^{3} b^{2} d^{2} e^{3} - 10 a^{2} b^{3} d^{3} e^{2} + 5 a b^{4} d^{4} e - b^{5} d^{5}\right )}{3 e^{5}}\right )}{e} & \text {for}\: e \neq 0 \\\sqrt {d} \left (\begin {cases} a^{5} x & \text {for}\: b = 0 \\\frac {\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{3}}{6 b} & \text {otherwise} \end {cases}\right ) & \text {otherwise} \end {cases} \]
Piecewise((2*(b**5*(d + e*x)**(13/2)/(13*e**5) + (d + e*x)**(11/2)*(5*a*b* *4*e - 5*b**5*d)/(11*e**5) + (d + e*x)**(9/2)*(10*a**2*b**3*e**2 - 20*a*b* *4*d*e + 10*b**5*d**2)/(9*e**5) + (d + e*x)**(7/2)*(10*a**3*b**2*e**3 - 30 *a**2*b**3*d*e**2 + 30*a*b**4*d**2*e - 10*b**5*d**3)/(7*e**5) + (d + e*x)* *(5/2)*(5*a**4*b*e**4 - 20*a**3*b**2*d*e**3 + 30*a**2*b**3*d**2*e**2 - 20* a*b**4*d**3*e + 5*b**5*d**4)/(5*e**5) + (d + e*x)**(3/2)*(a**5*e**5 - 5*a* *4*b*d*e**4 + 10*a**3*b**2*d**2*e**3 - 10*a**2*b**3*d**3*e**2 + 5*a*b**4*d **4*e - b**5*d**5)/(3*e**5))/e, Ne(e, 0)), (sqrt(d)*Piecewise((a**5*x, Eq( b, 0)), ((a**2 + 2*a*b*x + b**2*x**2)**3/(6*b), True)), True))
Time = 0.19 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.66 \[ \int (a+b x) \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {2 \, {\left (693 \, {\left (e x + d\right )}^{\frac {13}{2}} b^{5} - 4095 \, {\left (b^{5} d - a b^{4} e\right )} {\left (e x + d\right )}^{\frac {11}{2}} + 10010 \, {\left (b^{5} d^{2} - 2 \, a b^{4} d e + a^{2} b^{3} e^{2}\right )} {\left (e x + d\right )}^{\frac {9}{2}} - 12870 \, {\left (b^{5} d^{3} - 3 \, a b^{4} d^{2} e + 3 \, a^{2} b^{3} d e^{2} - a^{3} b^{2} e^{3}\right )} {\left (e x + d\right )}^{\frac {7}{2}} + 9009 \, {\left (b^{5} d^{4} - 4 \, a b^{4} d^{3} e + 6 \, a^{2} b^{3} d^{2} e^{2} - 4 \, a^{3} b^{2} d e^{3} + a^{4} b e^{4}\right )} {\left (e x + d\right )}^{\frac {5}{2}} - 3003 \, {\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\right )} {\left (e x + d\right )}^{\frac {3}{2}}\right )}}{9009 \, e^{6}} \]
2/9009*(693*(e*x + d)^(13/2)*b^5 - 4095*(b^5*d - a*b^4*e)*(e*x + d)^(11/2) + 10010*(b^5*d^2 - 2*a*b^4*d*e + a^2*b^3*e^2)*(e*x + d)^(9/2) - 12870*(b^ 5*d^3 - 3*a*b^4*d^2*e + 3*a^2*b^3*d*e^2 - a^3*b^2*e^3)*(e*x + d)^(7/2) + 9 009*(b^5*d^4 - 4*a*b^4*d^3*e + 6*a^2*b^3*d^2*e^2 - 4*a^3*b^2*d*e^3 + a^4*b *e^4)*(e*x + d)^(5/2) - 3003*(b^5*d^5 - 5*a*b^4*d^4*e + 10*a^2*b^3*d^3*e^2 - 10*a^3*b^2*d^2*e^3 + 5*a^4*b*d*e^4 - a^5*e^5)*(e*x + d)^(3/2))/e^6
Leaf count of result is larger than twice the leaf count of optimal. 641 vs. \(2 (134) = 268\).
Time = 0.27 (sec) , antiderivative size = 641, normalized size of antiderivative = 4.11 \[ \int (a+b x) \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {2 \, {\left (9009 \, \sqrt {e x + d} a^{5} d + 3003 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} a^{5} + \frac {15015 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} a^{4} b d}{e} + \frac {6006 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} a^{3} b^{2} d}{e^{2}} + \frac {3003 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} a^{4} b}{e} + \frac {2574 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} a^{2} b^{3} d}{e^{3}} + \frac {2574 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} a^{3} b^{2}}{e^{2}} + \frac {143 \, {\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} - 180 \, {\left (e x + d\right )}^{\frac {7}{2}} d + 378 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {e x + d} d^{4}\right )} a b^{4} d}{e^{4}} + \frac {286 \, {\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} - 180 \, {\left (e x + d\right )}^{\frac {7}{2}} d + 378 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {e x + d} d^{4}\right )} a^{2} b^{3}}{e^{3}} + \frac {13 \, {\left (63 \, {\left (e x + d\right )}^{\frac {11}{2}} - 385 \, {\left (e x + d\right )}^{\frac {9}{2}} d + 990 \, {\left (e x + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {e x + d} d^{5}\right )} b^{5} d}{e^{5}} + \frac {65 \, {\left (63 \, {\left (e x + d\right )}^{\frac {11}{2}} - 385 \, {\left (e x + d\right )}^{\frac {9}{2}} d + 990 \, {\left (e x + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {e x + d} d^{5}\right )} a b^{4}}{e^{4}} + \frac {3 \, {\left (231 \, {\left (e x + d\right )}^{\frac {13}{2}} - 1638 \, {\left (e x + d\right )}^{\frac {11}{2}} d + 5005 \, {\left (e x + d\right )}^{\frac {9}{2}} d^{2} - 8580 \, {\left (e x + d\right )}^{\frac {7}{2}} d^{3} + 9009 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{4} - 6006 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{5} + 3003 \, \sqrt {e x + d} d^{6}\right )} b^{5}}{e^{5}}\right )}}{9009 \, e} \]
2/9009*(9009*sqrt(e*x + d)*a^5*d + 3003*((e*x + d)^(3/2) - 3*sqrt(e*x + d) *d)*a^5 + 15015*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*a^4*b*d/e + 6006*(3* (e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a^3*b^2*d/e ^2 + 3003*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2 )*a^4*b/e + 2574*(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^2*b^3*d/e^3 + 2574*(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^ 3*b^2/e^2 + 143*(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*b^4*d/e^4 + 286*(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^2*b^3/e^3 + 13*(63 *(e*x + d)^(11/2) - 385*(e*x + d)^(9/2)*d + 990*(e*x + d)^(7/2)*d^2 - 1386 *(e*x + d)^(5/2)*d^3 + 1155*(e*x + d)^(3/2)*d^4 - 693*sqrt(e*x + d)*d^5)*b ^5*d/e^5 + 65*(63*(e*x + d)^(11/2) - 385*(e*x + d)^(9/2)*d + 990*(e*x + d) ^(7/2)*d^2 - 1386*(e*x + d)^(5/2)*d^3 + 1155*(e*x + d)^(3/2)*d^4 - 693*sqr t(e*x + d)*d^5)*a*b^4/e^4 + 3*(231*(e*x + d)^(13/2) - 1638*(e*x + d)^(11/2 )*d + 5005*(e*x + d)^(9/2)*d^2 - 8580*(e*x + d)^(7/2)*d^3 + 9009*(e*x + d) ^(5/2)*d^4 - 6006*(e*x + d)^(3/2)*d^5 + 3003*sqrt(e*x + d)*d^6)*b^5/e^5)/e
Time = 0.05 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.88 \[ \int (a+b x) \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {2\,b^5\,{\left (d+e\,x\right )}^{13/2}}{13\,e^6}-\frac {\left (10\,b^5\,d-10\,a\,b^4\,e\right )\,{\left (d+e\,x\right )}^{11/2}}{11\,e^6}+\frac {2\,{\left (a\,e-b\,d\right )}^5\,{\left (d+e\,x\right )}^{3/2}}{3\,e^6}+\frac {20\,b^2\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{7/2}}{7\,e^6}+\frac {20\,b^3\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{9/2}}{9\,e^6}+\frac {2\,b\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{5/2}}{e^6} \]