Integrand size = 35, antiderivative size = 452 \[ \int (A+B x) \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {2 (b d-a e)^5 (B d-A e) (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x)}-\frac {2 (b d-a e)^4 (6 b B d-5 A b e-a B e) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x)}+\frac {10 b (b d-a e)^3 (3 b B d-2 A b e-a B e) (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x)}-\frac {20 b^2 (b d-a e)^2 (2 b B d-A b e-a B e) (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^7 (a+b x)}+\frac {10 b^3 (b d-a e) (3 b B d-A b e-2 a B e) (d+e x)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x)}-\frac {2 b^4 (6 b B d-A b e-5 a B e) (d+e x)^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^7 (a+b x)}+\frac {2 b^5 B (d+e x)^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}}{15 e^7 (a+b x)} \]
2/3*(-a*e+b*d)^5*(-A*e+B*d)*(e*x+d)^(3/2)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)-2/ 5*(-a*e+b*d)^4*(-5*A*b*e-B*a*e+6*B*b*d)*(e*x+d)^(5/2)*((b*x+a)^2)^(1/2)/e^ 7/(b*x+a)+10/7*b*(-a*e+b*d)^3*(-2*A*b*e-B*a*e+3*B*b*d)*(e*x+d)^(7/2)*((b*x +a)^2)^(1/2)/e^7/(b*x+a)-20/9*b^2*(-a*e+b*d)^2*(-A*b*e-B*a*e+2*B*b*d)*(e*x +d)^(9/2)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)+10/11*b^3*(-a*e+b*d)*(-A*b*e-2*B*a *e+3*B*b*d)*(e*x+d)^(11/2)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)-2/13*b^4*(-A*b*e- 5*B*a*e+6*B*b*d)*(e*x+d)^(13/2)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)+2/15*b^5*B*( e*x+d)^(15/2)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)
Time = 0.36 (sec) , antiderivative size = 490, normalized size of antiderivative = 1.08 \[ \int (A+B x) \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {2 \sqrt {(a+b x)^2} (d+e x)^{3/2} \left (3003 a^5 e^5 (-2 B d+5 A e+3 B e x)+2145 a^4 b e^4 \left (7 A e (-2 d+3 e x)+B \left (8 d^2-12 d e x+15 e^2 x^2\right )\right )-1430 a^3 b^2 e^3 \left (-3 A e \left (8 d^2-12 d e x+15 e^2 x^2\right )+B \left (16 d^3-24 d^2 e x+30 d e^2 x^2-35 e^3 x^3\right )\right )+130 a^2 b^3 e^2 \left (11 A e \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )+B \left (128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right )\right )-5 a b^4 e \left (-13 A e \left (128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right )+5 B \left (256 d^5-384 d^4 e x+480 d^3 e^2 x^2-560 d^2 e^3 x^3+630 d e^4 x^4-693 e^5 x^5\right )\right )+b^5 \left (5 A e \left (-256 d^5+384 d^4 e x-480 d^3 e^2 x^2+560 d^2 e^3 x^3-630 d e^4 x^4+693 e^5 x^5\right )+B \left (1024 d^6-1536 d^5 e x+1920 d^4 e^2 x^2-2240 d^3 e^3 x^3+2520 d^2 e^4 x^4-2772 d e^5 x^5+3003 e^6 x^6\right )\right )\right )}{45045 e^7 (a+b x)} \]
(2*Sqrt[(a + b*x)^2]*(d + e*x)^(3/2)*(3003*a^5*e^5*(-2*B*d + 5*A*e + 3*B*e *x) + 2145*a^4*b*e^4*(7*A*e*(-2*d + 3*e*x) + B*(8*d^2 - 12*d*e*x + 15*e^2* x^2)) - 1430*a^3*b^2*e^3*(-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)) + 130*a^2*b^3*e^2*(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)) - 5*a*b^4*e*(-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 - 693*e^5*x^5)) + b^5*(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))))/(45045*e^7*(a + b*x))
Time = 0.45 (sec) , antiderivative size = 291, normalized size of antiderivative = 0.64, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {1187, 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.
| \(\displaystyle \int \left (a^2+2 a b x+b^2 x^2\right )^{5/2} (A+B x) \sqrt {d+e x} \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int b^5 (a+b x)^5 (A+B x) \sqrt {d+e x}dx}{b^5 (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int (a+b x)^5 (A+B x) \sqrt {d+e x}dx}{a+b x}\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {b^5 B (d+e x)^{13/2}}{e^6}+\frac {b^4 (-6 b B d+A b e+5 a B e) (d+e x)^{11/2}}{e^6}-\frac {5 b^3 (b d-a e) (-3 b B d+A b e+2 a B e) (d+e x)^{9/2}}{e^6}+\frac {10 b^2 (b d-a e)^2 (-2 b B d+A b e+a B e) (d+e x)^{7/2}}{e^6}-\frac {5 b (b d-a e)^3 (-3 b B d+2 A b e+a B e) (d+e x)^{5/2}}{e^6}+\frac {(a e-b d)^4 (-6 b B d+5 A b e+a B e) (d+e x)^{3/2}}{e^6}+\frac {(a e-b d)^5 (A e-B d) \sqrt {d+e x}}{e^6}\right )dx}{a+b x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (-\frac {2 b^4 (d+e x)^{13/2} (-5 a B e-A b e+6 b B d)}{13 e^7}+\frac {10 b^3 (d+e x)^{11/2} (b d-a e) (-2 a B e-A b e+3 b B d)}{11 e^7}-\frac {20 b^2 (d+e x)^{9/2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{9 e^7}+\frac {10 b (d+e x)^{7/2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{7 e^7}-\frac {2 (d+e x)^{5/2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{5 e^7}+\frac {2 (d+e x)^{3/2} (b d-a e)^5 (B d-A e)}{3 e^7}+\frac {2 b^5 B (d+e x)^{15/2}}{15 e^7}\right )}{a+b x}\) |
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*((2*(b*d - a*e)^5*(B*d - A*e)*(d + e*x)^(3/ 2))/(3*e^7) - (2*(b*d - a*e)^4*(6*b*B*d - 5*A*b*e - a*B*e)*(d + e*x)^(5/2) )/(5*e^7) + (10*b*(b*d - a*e)^3*(3*b*B*d - 2*A*b*e - a*B*e)*(d + e*x)^(7/2 ))/(7*e^7) - (20*b^2*(b*d - a*e)^2*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^(9/ 2))/(9*e^7) + (10*b^3*(b*d - a*e)*(3*b*B*d - A*b*e - 2*a*B*e)*(d + e*x)^(1 1/2))/(11*e^7) - (2*b^4*(6*b*B*d - A*b*e - 5*a*B*e)*(d + e*x)^(13/2))/(13* e^7) + (2*b^5*B*(d + e*x)^(15/2))/(15*e^7)))/(a + b*x)
3.19.54.3.1 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[(a + b*x + c*x^2)^FracPart[p]/(c^ IntPart[p]*(b/2 + c*x)^(2*FracPart[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, p}, x] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p]
Time = 0.26 (sec) , antiderivative size = 689, normalized size of antiderivative = 1.52
| method | result | size |
| gosper | \(\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (3003 B \,b^{5} e^{6} x^{6}+3465 A \,b^{5} e^{6} x^{5}+17325 B a \,b^{4} e^{6} x^{5}-2772 B \,b^{5} d \,e^{5} x^{5}+20475 A a \,b^{4} e^{6} x^{4}-3150 A \,b^{5} d \,e^{5} x^{4}+40950 B \,a^{2} b^{3} e^{6} x^{4}-15750 B a \,b^{4} d \,e^{5} x^{4}+2520 B \,b^{5} d^{2} e^{4} x^{4}+50050 A \,a^{2} b^{3} e^{6} x^{3}-18200 A a \,b^{4} d \,e^{5} x^{3}+2800 A \,b^{5} d^{2} e^{4} x^{3}+50050 B \,a^{3} b^{2} e^{6} x^{3}-36400 B \,a^{2} b^{3} d \,e^{5} x^{3}+14000 B a \,b^{4} d^{2} e^{4} x^{3}-2240 B \,b^{5} d^{3} e^{3} x^{3}+64350 A \,a^{3} b^{2} e^{6} x^{2}-42900 A \,a^{2} b^{3} d \,e^{5} x^{2}+15600 A a \,b^{4} d^{2} e^{4} x^{2}-2400 A \,b^{5} d^{3} e^{3} x^{2}+32175 B \,a^{4} b \,e^{6} x^{2}-42900 B \,a^{3} b^{2} d \,e^{5} x^{2}+31200 B \,a^{2} b^{3} d^{2} e^{4} x^{2}-12000 B a \,b^{4} d^{3} e^{3} x^{2}+1920 B \,b^{5} d^{4} e^{2} x^{2}+45045 A \,a^{4} b \,e^{6} x -51480 A \,a^{3} b^{2} d \,e^{5} x +34320 A \,a^{2} b^{3} d^{2} e^{4} x -12480 A a \,b^{4} d^{3} e^{3} x +1920 A \,b^{5} d^{4} e^{2} x +9009 B \,a^{5} e^{6} x -25740 B \,a^{4} b d \,e^{5} x +34320 B \,a^{3} b^{2} d^{2} e^{4} x -24960 B \,a^{2} b^{3} d^{3} e^{3} x +9600 B a \,b^{4} d^{4} e^{2} x -1536 B \,b^{5} d^{5} e x +15015 A \,a^{5} e^{6}-30030 A \,a^{4} b d \,e^{5}+34320 A \,a^{3} b^{2} d^{2} e^{4}-22880 A \,a^{2} b^{3} d^{3} e^{3}+8320 A a \,b^{4} d^{4} e^{2}-1280 A \,b^{5} d^{5} e -6006 B \,a^{5} d \,e^{5}+17160 B \,a^{4} b \,d^{2} e^{4}-22880 B \,a^{3} b^{2} d^{3} e^{3}+16640 B \,a^{2} b^{3} d^{4} e^{2}-6400 B a \,b^{4} d^{5} e +1024 B \,b^{5} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{45045 e^{7} \left (b x +a \right )^{5}}\) | \(689\) |
| default | \(\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (3003 B \,b^{5} e^{6} x^{6}+3465 A \,b^{5} e^{6} x^{5}+17325 B a \,b^{4} e^{6} x^{5}-2772 B \,b^{5} d \,e^{5} x^{5}+20475 A a \,b^{4} e^{6} x^{4}-3150 A \,b^{5} d \,e^{5} x^{4}+40950 B \,a^{2} b^{3} e^{6} x^{4}-15750 B a \,b^{4} d \,e^{5} x^{4}+2520 B \,b^{5} d^{2} e^{4} x^{4}+50050 A \,a^{2} b^{3} e^{6} x^{3}-18200 A a \,b^{4} d \,e^{5} x^{3}+2800 A \,b^{5} d^{2} e^{4} x^{3}+50050 B \,a^{3} b^{2} e^{6} x^{3}-36400 B \,a^{2} b^{3} d \,e^{5} x^{3}+14000 B a \,b^{4} d^{2} e^{4} x^{3}-2240 B \,b^{5} d^{3} e^{3} x^{3}+64350 A \,a^{3} b^{2} e^{6} x^{2}-42900 A \,a^{2} b^{3} d \,e^{5} x^{2}+15600 A a \,b^{4} d^{2} e^{4} x^{2}-2400 A \,b^{5} d^{3} e^{3} x^{2}+32175 B \,a^{4} b \,e^{6} x^{2}-42900 B \,a^{3} b^{2} d \,e^{5} x^{2}+31200 B \,a^{2} b^{3} d^{2} e^{4} x^{2}-12000 B a \,b^{4} d^{3} e^{3} x^{2}+1920 B \,b^{5} d^{4} e^{2} x^{2}+45045 A \,a^{4} b \,e^{6} x -51480 A \,a^{3} b^{2} d \,e^{5} x +34320 A \,a^{2} b^{3} d^{2} e^{4} x -12480 A a \,b^{4} d^{3} e^{3} x +1920 A \,b^{5} d^{4} e^{2} x +9009 B \,a^{5} e^{6} x -25740 B \,a^{4} b d \,e^{5} x +34320 B \,a^{3} b^{2} d^{2} e^{4} x -24960 B \,a^{2} b^{3} d^{3} e^{3} x +9600 B a \,b^{4} d^{4} e^{2} x -1536 B \,b^{5} d^{5} e x +15015 A \,a^{5} e^{6}-30030 A \,a^{4} b d \,e^{5}+34320 A \,a^{3} b^{2} d^{2} e^{4}-22880 A \,a^{2} b^{3} d^{3} e^{3}+8320 A a \,b^{4} d^{4} e^{2}-1280 A \,b^{5} d^{5} e -6006 B \,a^{5} d \,e^{5}+17160 B \,a^{4} b \,d^{2} e^{4}-22880 B \,a^{3} b^{2} d^{3} e^{3}+16640 B \,a^{2} b^{3} d^{4} e^{2}-6400 B a \,b^{4} d^{5} e +1024 B \,b^{5} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{45045 e^{7} \left (b x +a \right )^{5}}\) | \(689\) |
| risch | \(\frac {2 \sqrt {\left (b x +a \right )^{2}}\, \left (3003 B \,b^{5} x^{7} e^{7}+3465 A \,b^{5} e^{7} x^{6}+17325 B a \,b^{4} e^{7} x^{6}+231 B \,b^{5} d \,e^{6} x^{6}+20475 A a \,b^{4} e^{7} x^{5}+315 A \,b^{5} d \,e^{6} x^{5}+40950 B \,a^{2} b^{3} e^{7} x^{5}+1575 B a \,b^{4} d \,e^{6} x^{5}-252 B \,b^{5} d^{2} e^{5} x^{5}+50050 A \,a^{2} b^{3} e^{7} x^{4}+2275 A a \,b^{4} d \,e^{6} x^{4}-350 A \,b^{5} d^{2} e^{5} x^{4}+50050 B \,a^{3} b^{2} e^{7} x^{4}+4550 B \,a^{2} b^{3} d \,e^{6} x^{4}-1750 B a \,b^{4} d^{2} e^{5} x^{4}+280 B \,b^{5} d^{3} e^{4} x^{4}+64350 A \,a^{3} b^{2} e^{7} x^{3}+7150 A \,a^{2} b^{3} d \,e^{6} x^{3}-2600 A a \,b^{4} d^{2} e^{5} x^{3}+400 A \,b^{5} d^{3} e^{4} x^{3}+32175 B \,a^{4} b \,e^{7} x^{3}+7150 B \,a^{3} b^{2} d \,e^{6} x^{3}-5200 B \,a^{2} b^{3} d^{2} e^{5} x^{3}+2000 B a \,b^{4} d^{3} e^{4} x^{3}-320 B \,b^{5} d^{4} e^{3} x^{3}+45045 A \,a^{4} b \,e^{7} x^{2}+12870 A \,a^{3} b^{2} d \,e^{6} x^{2}-8580 A \,a^{2} b^{3} d^{2} e^{5} x^{2}+3120 A a \,b^{4} d^{3} e^{4} x^{2}-480 A \,b^{5} d^{4} e^{3} x^{2}+9009 B \,a^{5} e^{7} x^{2}+6435 B \,a^{4} b d \,e^{6} x^{2}-8580 B \,a^{3} b^{2} d^{2} e^{5} x^{2}+6240 B \,a^{2} b^{3} d^{3} e^{4} x^{2}-2400 B a \,b^{4} d^{4} e^{3} x^{2}+384 B \,b^{5} d^{5} e^{2} x^{2}+15015 A \,a^{5} e^{7} x +15015 A \,a^{4} b d \,e^{6} x -17160 A \,a^{3} b^{2} d^{2} e^{5} x +11440 A \,a^{2} b^{3} d^{3} e^{4} x -4160 A a \,b^{4} d^{4} e^{3} x +640 A \,b^{5} d^{5} e^{2} x +3003 B \,a^{5} d \,e^{6} x -8580 B \,a^{4} b \,d^{2} e^{5} x +11440 B \,a^{3} b^{2} d^{3} e^{4} x -8320 B \,a^{2} b^{3} d^{4} e^{3} x +3200 B a \,b^{4} d^{5} e^{2} x -512 B \,b^{5} d^{6} e x +15015 d A \,a^{5} e^{6}-30030 A \,a^{4} b \,d^{2} e^{5}+34320 A \,a^{3} b^{2} d^{3} e^{4}-22880 A \,a^{2} b^{3} d^{4} e^{3}+8320 A a \,b^{4} d^{5} e^{2}-1280 A \,b^{5} d^{6} e -6006 B \,a^{5} d^{2} e^{5}+17160 B \,a^{4} b \,d^{3} e^{4}-22880 B \,a^{3} b^{2} d^{4} e^{3}+16640 B \,a^{2} b^{3} d^{5} e^{2}-6400 B a \,b^{4} d^{6} e +1024 B \,b^{5} d^{7}\right ) \sqrt {e x +d}}{45045 \left (b x +a \right ) e^{7}}\) | \(881\) |
2/45045*(e*x+d)^(3/2)*(3003*B*b^5*e^6*x^6+3465*A*b^5*e^6*x^5+17325*B*a*b^4 *e^6*x^5-2772*B*b^5*d*e^5*x^5+20475*A*a*b^4*e^6*x^4-3150*A*b^5*d*e^5*x^4+4 0950*B*a^2*b^3*e^6*x^4-15750*B*a*b^4*d*e^5*x^4+2520*B*b^5*d^2*e^4*x^4+5005 0*A*a^2*b^3*e^6*x^3-18200*A*a*b^4*d*e^5*x^3+2800*A*b^5*d^2*e^4*x^3+50050*B *a^3*b^2*e^6*x^3-36400*B*a^2*b^3*d*e^5*x^3+14000*B*a*b^4*d^2*e^4*x^3-2240* B*b^5*d^3*e^3*x^3+64350*A*a^3*b^2*e^6*x^2-42900*A*a^2*b^3*d*e^5*x^2+15600* A*a*b^4*d^2*e^4*x^2-2400*A*b^5*d^3*e^3*x^2+32175*B*a^4*b*e^6*x^2-42900*B*a ^3*b^2*d*e^5*x^2+31200*B*a^2*b^3*d^2*e^4*x^2-12000*B*a*b^4*d^3*e^3*x^2+192 0*B*b^5*d^4*e^2*x^2+45045*A*a^4*b*e^6*x-51480*A*a^3*b^2*d*e^5*x+34320*A*a^ 2*b^3*d^2*e^4*x-12480*A*a*b^4*d^3*e^3*x+1920*A*b^5*d^4*e^2*x+9009*B*a^5*e^ 6*x-25740*B*a^4*b*d*e^5*x+34320*B*a^3*b^2*d^2*e^4*x-24960*B*a^2*b^3*d^3*e^ 3*x+9600*B*a*b^4*d^4*e^2*x-1536*B*b^5*d^5*e*x+15015*A*a^5*e^6-30030*A*a^4* b*d*e^5+34320*A*a^3*b^2*d^2*e^4-22880*A*a^2*b^3*d^3*e^3+8320*A*a*b^4*d^4*e ^2-1280*A*b^5*d^5*e-6006*B*a^5*d*e^5+17160*B*a^4*b*d^2*e^4-22880*B*a^3*b^2 *d^3*e^3+16640*B*a^2*b^3*d^4*e^2-6400*B*a*b^4*d^5*e+1024*B*b^5*d^6)*((b*x+ a)^2)^(5/2)/e^7/(b*x+a)^5
Leaf count of result is larger than twice the leaf count of optimal. 702 vs. \(2 (347) = 694\).
Time = 0.30 (sec) , antiderivative size = 702, normalized size of antiderivative = 1.55 \[ \int (A+B x) \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {2 \, {\left (3003 \, B b^{5} e^{7} x^{7} + 1024 \, B b^{5} d^{7} + 15015 \, A a^{5} d e^{6} - 1280 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{6} e + 8320 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{5} e^{2} - 22880 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{4} e^{3} + 17160 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{3} e^{4} - 6006 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d^{2} e^{5} + 231 \, {\left (B b^{5} d e^{6} + 15 \, {\left (5 \, B a b^{4} + A b^{5}\right )} e^{7}\right )} x^{6} - 63 \, {\left (4 \, B b^{5} d^{2} e^{5} - 5 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d e^{6} - 325 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e^{7}\right )} x^{5} + 35 \, {\left (8 \, B b^{5} d^{3} e^{4} - 10 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e^{5} + 65 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{6} + 1430 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{7}\right )} x^{4} - 5 \, {\left (64 \, B b^{5} d^{4} e^{3} - 80 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} e^{4} + 520 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{5} - 1430 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{6} - 6435 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e^{7}\right )} x^{3} + 3 \, {\left (128 \, B b^{5} d^{5} e^{2} - 160 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} e^{3} + 1040 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e^{4} - 2860 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e^{5} + 2145 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d e^{6} + 3003 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} e^{7}\right )} x^{2} - {\left (512 \, B b^{5} d^{6} e - 15015 \, A a^{5} e^{7} - 640 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} e^{2} + 4160 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} e^{3} - 11440 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} e^{4} + 8580 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} e^{5} - 3003 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d e^{6}\right )} x\right )} \sqrt {e x + d}}{45045 \, e^{7}} \]
2/45045*(3003*B*b^5*e^7*x^7 + 1024*B*b^5*d^7 + 15015*A*a^5*d*e^6 - 1280*(5 *B*a*b^4 + A*b^5)*d^6*e + 8320*(2*B*a^2*b^3 + A*a*b^4)*d^5*e^2 - 22880*(B* a^3*b^2 + A*a^2*b^3)*d^4*e^3 + 17160*(B*a^4*b + 2*A*a^3*b^2)*d^3*e^4 - 600 6*(B*a^5 + 5*A*a^4*b)*d^2*e^5 + 231*(B*b^5*d*e^6 + 15*(5*B*a*b^4 + A*b^5)* e^7)*x^6 - 63*(4*B*b^5*d^2*e^5 - 5*(5*B*a*b^4 + A*b^5)*d*e^6 - 325*(2*B*a^ 2*b^3 + A*a*b^4)*e^7)*x^5 + 35*(8*B*b^5*d^3*e^4 - 10*(5*B*a*b^4 + A*b^5)*d ^2*e^5 + 65*(2*B*a^2*b^3 + A*a*b^4)*d*e^6 + 1430*(B*a^3*b^2 + A*a^2*b^3)*e ^7)*x^4 - 5*(64*B*b^5*d^4*e^3 - 80*(5*B*a*b^4 + A*b^5)*d^3*e^4 + 520*(2*B* a^2*b^3 + A*a*b^4)*d^2*e^5 - 1430*(B*a^3*b^2 + A*a^2*b^3)*d*e^6 - 6435*(B* a^4*b + 2*A*a^3*b^2)*e^7)*x^3 + 3*(128*B*b^5*d^5*e^2 - 160*(5*B*a*b^4 + A* b^5)*d^4*e^3 + 1040*(2*B*a^2*b^3 + A*a*b^4)*d^3*e^4 - 2860*(B*a^3*b^2 + A* a^2*b^3)*d^2*e^5 + 2145*(B*a^4*b + 2*A*a^3*b^2)*d*e^6 + 3003*(B*a^5 + 5*A* a^4*b)*e^7)*x^2 - (512*B*b^5*d^6*e - 15015*A*a^5*e^7 - 640*(5*B*a*b^4 + A* b^5)*d^5*e^2 + 4160*(2*B*a^2*b^3 + A*a*b^4)*d^4*e^3 - 11440*(B*a^3*b^2 + A *a^2*b^3)*d^3*e^4 + 8580*(B*a^4*b + 2*A*a^3*b^2)*d^2*e^5 - 3003*(B*a^5 + 5 *A*a^4*b)*d*e^6)*x)*sqrt(e*x + d)/e^7
Timed out. \[ \int (A+B x) \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 760 vs. \(2 (347) = 694\).
Time = 0.22 (sec) , antiderivative size = 760, normalized size of antiderivative = 1.68 \[ \int (A+B x) \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, 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} A}{9009 \, e^{6}} + \frac {2 \, {\left (3003 \, b^{5} e^{7} x^{7} + 1024 \, b^{5} d^{7} - 6400 \, a b^{4} d^{6} e + 16640 \, a^{2} b^{3} d^{5} e^{2} - 22880 \, a^{3} b^{2} d^{4} e^{3} + 17160 \, a^{4} b d^{3} e^{4} - 6006 \, a^{5} d^{2} e^{5} + 231 \, {\left (b^{5} d e^{6} + 75 \, a b^{4} e^{7}\right )} x^{6} - 63 \, {\left (4 \, b^{5} d^{2} e^{5} - 25 \, a b^{4} d e^{6} - 650 \, a^{2} b^{3} e^{7}\right )} x^{5} + 70 \, {\left (4 \, b^{5} d^{3} e^{4} - 25 \, a b^{4} d^{2} e^{5} + 65 \, a^{2} b^{3} d e^{6} + 715 \, a^{3} b^{2} e^{7}\right )} x^{4} - 5 \, {\left (64 \, b^{5} d^{4} e^{3} - 400 \, a b^{4} d^{3} e^{4} + 1040 \, a^{2} b^{3} d^{2} e^{5} - 1430 \, a^{3} b^{2} d e^{6} - 6435 \, a^{4} b e^{7}\right )} x^{3} + 3 \, {\left (128 \, b^{5} d^{5} e^{2} - 800 \, a b^{4} d^{4} e^{3} + 2080 \, a^{2} b^{3} d^{3} e^{4} - 2860 \, a^{3} b^{2} d^{2} e^{5} + 2145 \, a^{4} b d e^{6} + 3003 \, a^{5} e^{7}\right )} x^{2} - {\left (512 \, b^{5} d^{6} e - 3200 \, a b^{4} d^{5} e^{2} + 8320 \, a^{2} b^{3} d^{4} e^{3} - 11440 \, a^{3} b^{2} d^{3} e^{4} + 8580 \, a^{4} b d^{2} e^{5} - 3003 \, a^{5} d e^{6}\right )} x\right )} \sqrt {e x + d} B}{45045 \, e^{7}} \]
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)*A/e^6 + 2/45045*(3003*b^5*e^7*x^7 + 1024*b^5*d^7 - 6400*a*b^4*d^6*e + 16640*a^2*b^3*d^5*e^2 - 22880*a^3*b^2 *d^4*e^3 + 17160*a^4*b*d^3*e^4 - 6006*a^5*d^2*e^5 + 231*(b^5*d*e^6 + 75*a* b^4*e^7)*x^6 - 63*(4*b^5*d^2*e^5 - 25*a*b^4*d*e^6 - 650*a^2*b^3*e^7)*x^5 + 70*(4*b^5*d^3*e^4 - 25*a*b^4*d^2*e^5 + 65*a^2*b^3*d*e^6 + 715*a^3*b^2*e^7 )*x^4 - 5*(64*b^5*d^4*e^3 - 400*a*b^4*d^3*e^4 + 1040*a^2*b^3*d^2*e^5 - 143 0*a^3*b^2*d*e^6 - 6435*a^4*b*e^7)*x^3 + 3*(128*b^5*d^5*e^2 - 800*a*b^4*d^4 *e^3 + 2080*a^2*b^3*d^3*e^4 - 2860*a^3*b^2*d^2*e^5 + 2145*a^4*b*d*e^6 + 30 03*a^5*e^7)*x^2 - (512*b^5*d^6*e - 3200*a*b^4*d^5*e^2 + 8320*a^2*b^3*d^4*e ^3 - 11440*a^3*b^2*d^3*e^4 + 8580*a^4*b*d^2*e^5 - 3003*a^5*d*e^6)*x)*sqrt( e*x + d)*B/e^7
Leaf count of result is larger than twice the leaf count of optimal. 1597 vs. \(2 (347) = 694\).
Time = 0.32 (sec) , antiderivative size = 1597, normalized size of antiderivative = 3.53 \[ \int (A+B x) \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\text {Too large to display} \]
2/45045*(45045*sqrt(e*x + d)*A*a^5*d*sgn(b*x + a) + 15015*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*A*a^5*sgn(b*x + a) + 15015*((e*x + d)^(3/2) - 3*sqrt (e*x + d)*d)*B*a^5*d*sgn(b*x + a)/e + 75075*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*A*a^4*b*d*sgn(b*x + a)/e + 15015*(3*(e*x + d)^(5/2) - 10*(e*x + d) ^(3/2)*d + 15*sqrt(e*x + d)*d^2)*B*a^4*b*d*sgn(b*x + a)/e^2 + 30030*(3*(e* x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*A*a^3*b^2*d*sg n(b*x + a)/e^2 + 3003*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt( e*x + d)*d^2)*B*a^5*sgn(b*x + a)/e + 15015*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*A*a^4*b*sgn(b*x + a)/e + 12870*(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^3*b^2*d*sgn(b*x + a)/e^3 + 12870*(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^2*b^3*d *sgn(b*x + a)/e^3 + 6435*(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*sgn(b*x + a)/e^2 + 12870 *(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*s qrt(e*x + d)*d^3)*A*a^3*b^2*sgn(b*x + a)/e^2 + 1430*(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^2*b^3*d*sgn(b*x + a)/e^4 + 715*(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*b^4*d*sgn(b*x + a)/e^4 + 1430*(35*...
Timed out. \[ \int (A+B x) \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\int \left (A+B\,x\right )\,\sqrt {d+e\,x}\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2} \,d x \]