Integrand size = 33, antiderivative size = 362 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{16}} \, dx=-\frac {(b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{15 e^7 (a+b x) (d+e x)^{15}}+\frac {3 b (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x) (d+e x)^{14}}-\frac {15 b^2 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^7 (a+b x) (d+e x)^{13}}+\frac {5 b^3 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x) (d+e x)^{12}}-\frac {15 b^4 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x) (d+e x)^{11}}+\frac {3 b^5 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x) (d+e x)^{10}}-\frac {b^6 \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^7 (a+b x) (d+e x)^9} \]
-1/15*(-a*e+b*d)^6*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^15+3/7*b*(-a*e+b* d)^5*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^14-15/13*b^2*(-a*e+b*d)^4*((b*x +a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^13+5/3*b^3*(-a*e+b*d)^3*((b*x+a)^2)^(1/2) /e^7/(b*x+a)/(e*x+d)^12-15/11*b^4*(-a*e+b*d)^2*((b*x+a)^2)^(1/2)/e^7/(b*x+ a)/(e*x+d)^11+3/5*b^5*(-a*e+b*d)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^10- 1/9*b^6*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^9
Time = 1.08 (sec) , antiderivative size = 295, normalized size of antiderivative = 0.81 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{16}} \, dx=-\frac {\sqrt {(a+b x)^2} \left (3003 a^6 e^6+1287 a^5 b e^5 (d+15 e x)+495 a^4 b^2 e^4 \left (d^2+15 d e x+105 e^2 x^2\right )+165 a^3 b^3 e^3 \left (d^3+15 d^2 e x+105 d e^2 x^2+455 e^3 x^3\right )+45 a^2 b^4 e^2 \left (d^4+15 d^3 e x+105 d^2 e^2 x^2+455 d e^3 x^3+1365 e^4 x^4\right )+9 a b^5 e \left (d^5+15 d^4 e x+105 d^3 e^2 x^2+455 d^2 e^3 x^3+1365 d e^4 x^4+3003 e^5 x^5\right )+b^6 \left (d^6+15 d^5 e x+105 d^4 e^2 x^2+455 d^3 e^3 x^3+1365 d^2 e^4 x^4+3003 d e^5 x^5+5005 e^6 x^6\right )\right )}{45045 e^7 (a+b x) (d+e x)^{15}} \]
-1/45045*(Sqrt[(a + b*x)^2]*(3003*a^6*e^6 + 1287*a^5*b*e^5*(d + 15*e*x) + 495*a^4*b^2*e^4*(d^2 + 15*d*e*x + 105*e^2*x^2) + 165*a^3*b^3*e^3*(d^3 + 15 *d^2*e*x + 105*d*e^2*x^2 + 455*e^3*x^3) + 45*a^2*b^4*e^2*(d^4 + 15*d^3*e*x + 105*d^2*e^2*x^2 + 455*d*e^3*x^3 + 1365*e^4*x^4) + 9*a*b^5*e*(d^5 + 15*d ^4*e*x + 105*d^3*e^2*x^2 + 455*d^2*e^3*x^3 + 1365*d*e^4*x^4 + 3003*e^5*x^5 ) + b^6*(d^6 + 15*d^5*e*x + 105*d^4*e^2*x^2 + 455*d^3*e^3*x^3 + 1365*d^2*e ^4*x^4 + 3003*d*e^5*x^5 + 5005*e^6*x^6)))/(e^7*(a + b*x)*(d + e*x)^15)
Time = 0.38 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.56, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {1187, 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.
| \(\displaystyle \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{16}} \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {b^5 (a+b x)^6}{(d+e x)^{16}}dx}{b^5 (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x)^6}{(d+e x)^{16}}dx}{a+b x}\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {b^6}{e^6 (d+e x)^{10}}-\frac {6 (b d-a e) b^5}{e^6 (d+e x)^{11}}+\frac {15 (b d-a e)^2 b^4}{e^6 (d+e x)^{12}}-\frac {20 (b d-a e)^3 b^3}{e^6 (d+e x)^{13}}+\frac {15 (b d-a e)^4 b^2}{e^6 (d+e x)^{14}}-\frac {6 (b d-a e)^5 b}{e^6 (d+e x)^{15}}+\frac {(a e-b d)^6}{e^6 (d+e x)^{16}}\right )dx}{a+b x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {3 b^5 (b d-a e)}{5 e^7 (d+e x)^{10}}-\frac {15 b^4 (b d-a e)^2}{11 e^7 (d+e x)^{11}}+\frac {5 b^3 (b d-a e)^3}{3 e^7 (d+e x)^{12}}-\frac {15 b^2 (b d-a e)^4}{13 e^7 (d+e x)^{13}}+\frac {3 b (b d-a e)^5}{7 e^7 (d+e x)^{14}}-\frac {(b d-a e)^6}{15 e^7 (d+e x)^{15}}-\frac {b^6}{9 e^7 (d+e x)^9}\right )}{a+b x}\) |
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-1/15*(b*d - a*e)^6/(e^7*(d + e*x)^15) + ( 3*b*(b*d - a*e)^5)/(7*e^7*(d + e*x)^14) - (15*b^2*(b*d - a*e)^4)/(13*e^7*( d + e*x)^13) + (5*b^3*(b*d - a*e)^3)/(3*e^7*(d + e*x)^12) - (15*b^4*(b*d - a*e)^2)/(11*e^7*(d + e*x)^11) + (3*b^5*(b*d - a*e))/(5*e^7*(d + e*x)^10) - b^6/(9*e^7*(d + e*x)^9)))/(a + b*x)
3.21.13.3.1 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[(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 = 35.40 (sec) , antiderivative size = 351, normalized size of antiderivative = 0.97
| method | result | size |
| risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {3003 e^{6} a^{6}+1287 b d \,e^{5} a^{5}+495 b^{2} d^{2} e^{4} a^{4}+165 b^{3} d^{3} e^{3} a^{3}+45 b^{4} d^{4} e^{2} a^{2}+9 b^{5} d^{5} e a +b^{6} d^{6}}{45045 e^{7}}-\frac {b \left (1287 e^{5} a^{5}+495 b d \,e^{4} a^{4}+165 b^{2} d^{2} e^{3} a^{3}+45 b^{3} d^{3} e^{2} a^{2}+9 b^{4} d^{4} e a +b^{5} d^{5}\right ) x}{3003 e^{6}}-\frac {b^{2} \left (495 e^{4} a^{4}+165 b d \,e^{3} a^{3}+45 b^{2} d^{2} e^{2} a^{2}+9 b^{3} d^{3} e a +b^{4} d^{4}\right ) x^{2}}{429 e^{5}}-\frac {b^{3} \left (165 a^{3} e^{3}+45 a^{2} b d \,e^{2}+9 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) x^{3}}{99 e^{4}}-\frac {b^{4} \left (45 e^{2} a^{2}+9 a b d e +b^{2} d^{2}\right ) x^{4}}{33 e^{3}}-\frac {b^{5} \left (9 a e +b d \right ) x^{5}}{15 e^{2}}-\frac {b^{6} x^{6}}{9 e}\right )}{\left (b x +a \right ) \left (e x +d \right )^{15}}\) | \(351\) |
| gosper | \(-\frac {\left (5005 b^{6} e^{6} x^{6}+27027 a \,b^{5} e^{6} x^{5}+3003 b^{6} d \,e^{5} x^{5}+61425 a^{2} b^{4} e^{6} x^{4}+12285 a \,b^{5} d \,e^{5} x^{4}+1365 b^{6} d^{2} e^{4} x^{4}+75075 a^{3} b^{3} e^{6} x^{3}+20475 a^{2} b^{4} d \,e^{5} x^{3}+4095 a \,b^{5} d^{2} e^{4} x^{3}+455 b^{6} d^{3} e^{3} x^{3}+51975 a^{4} b^{2} e^{6} x^{2}+17325 a^{3} b^{3} d \,e^{5} x^{2}+4725 a^{2} b^{4} d^{2} e^{4} x^{2}+945 a \,b^{5} d^{3} e^{3} x^{2}+105 b^{6} d^{4} e^{2} x^{2}+19305 a^{5} b \,e^{6} x +7425 a^{4} b^{2} d \,e^{5} x +2475 a^{3} b^{3} d^{2} e^{4} x +675 a^{2} b^{4} d^{3} e^{3} x +135 a \,b^{5} d^{4} e^{2} x +15 b^{6} d^{5} e x +3003 e^{6} a^{6}+1287 b d \,e^{5} a^{5}+495 b^{2} d^{2} e^{4} a^{4}+165 b^{3} d^{3} e^{3} a^{3}+45 b^{4} d^{4} e^{2} a^{2}+9 b^{5} d^{5} e a +b^{6} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{45045 e^{7} \left (e x +d \right )^{15} \left (b x +a \right )^{5}}\) | \(392\) |
| default | \(-\frac {\left (5005 b^{6} e^{6} x^{6}+27027 a \,b^{5} e^{6} x^{5}+3003 b^{6} d \,e^{5} x^{5}+61425 a^{2} b^{4} e^{6} x^{4}+12285 a \,b^{5} d \,e^{5} x^{4}+1365 b^{6} d^{2} e^{4} x^{4}+75075 a^{3} b^{3} e^{6} x^{3}+20475 a^{2} b^{4} d \,e^{5} x^{3}+4095 a \,b^{5} d^{2} e^{4} x^{3}+455 b^{6} d^{3} e^{3} x^{3}+51975 a^{4} b^{2} e^{6} x^{2}+17325 a^{3} b^{3} d \,e^{5} x^{2}+4725 a^{2} b^{4} d^{2} e^{4} x^{2}+945 a \,b^{5} d^{3} e^{3} x^{2}+105 b^{6} d^{4} e^{2} x^{2}+19305 a^{5} b \,e^{6} x +7425 a^{4} b^{2} d \,e^{5} x +2475 a^{3} b^{3} d^{2} e^{4} x +675 a^{2} b^{4} d^{3} e^{3} x +135 a \,b^{5} d^{4} e^{2} x +15 b^{6} d^{5} e x +3003 e^{6} a^{6}+1287 b d \,e^{5} a^{5}+495 b^{2} d^{2} e^{4} a^{4}+165 b^{3} d^{3} e^{3} a^{3}+45 b^{4} d^{4} e^{2} a^{2}+9 b^{5} d^{5} e a +b^{6} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{45045 e^{7} \left (e x +d \right )^{15} \left (b x +a \right )^{5}}\) | \(392\) |
((b*x+a)^2)^(1/2)/(b*x+a)*(-1/45045/e^7*(3003*a^6*e^6+1287*a^5*b*d*e^5+495 *a^4*b^2*d^2*e^4+165*a^3*b^3*d^3*e^3+45*a^2*b^4*d^4*e^2+9*a*b^5*d^5*e+b^6* d^6)-1/3003*b/e^6*(1287*a^5*e^5+495*a^4*b*d*e^4+165*a^3*b^2*d^2*e^3+45*a^2 *b^3*d^3*e^2+9*a*b^4*d^4*e+b^5*d^5)*x-1/429*b^2/e^5*(495*a^4*e^4+165*a^3*b *d*e^3+45*a^2*b^2*d^2*e^2+9*a*b^3*d^3*e+b^4*d^4)*x^2-1/99*b^3/e^4*(165*a^3 *e^3+45*a^2*b*d*e^2+9*a*b^2*d^2*e+b^3*d^3)*x^3-1/33*b^4/e^3*(45*a^2*e^2+9* a*b*d*e+b^2*d^2)*x^4-1/15*b^5/e^2*(9*a*e+b*d)*x^5-1/9*b^6/e*x^6)/(e*x+d)^1 5
Time = 0.28 (sec) , antiderivative size = 507, normalized size of antiderivative = 1.40 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{16}} \, dx=-\frac {5005 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 9 \, a b^{5} d^{5} e + 45 \, a^{2} b^{4} d^{4} e^{2} + 165 \, a^{3} b^{3} d^{3} e^{3} + 495 \, a^{4} b^{2} d^{2} e^{4} + 1287 \, a^{5} b d e^{5} + 3003 \, a^{6} e^{6} + 3003 \, {\left (b^{6} d e^{5} + 9 \, a b^{5} e^{6}\right )} x^{5} + 1365 \, {\left (b^{6} d^{2} e^{4} + 9 \, a b^{5} d e^{5} + 45 \, a^{2} b^{4} e^{6}\right )} x^{4} + 455 \, {\left (b^{6} d^{3} e^{3} + 9 \, a b^{5} d^{2} e^{4} + 45 \, a^{2} b^{4} d e^{5} + 165 \, a^{3} b^{3} e^{6}\right )} x^{3} + 105 \, {\left (b^{6} d^{4} e^{2} + 9 \, a b^{5} d^{3} e^{3} + 45 \, a^{2} b^{4} d^{2} e^{4} + 165 \, a^{3} b^{3} d e^{5} + 495 \, a^{4} b^{2} e^{6}\right )} x^{2} + 15 \, {\left (b^{6} d^{5} e + 9 \, a b^{5} d^{4} e^{2} + 45 \, a^{2} b^{4} d^{3} e^{3} + 165 \, a^{3} b^{3} d^{2} e^{4} + 495 \, a^{4} b^{2} d e^{5} + 1287 \, a^{5} b e^{6}\right )} x}{45045 \, {\left (e^{22} x^{15} + 15 \, d e^{21} x^{14} + 105 \, d^{2} e^{20} x^{13} + 455 \, d^{3} e^{19} x^{12} + 1365 \, d^{4} e^{18} x^{11} + 3003 \, d^{5} e^{17} x^{10} + 5005 \, d^{6} e^{16} x^{9} + 6435 \, d^{7} e^{15} x^{8} + 6435 \, d^{8} e^{14} x^{7} + 5005 \, d^{9} e^{13} x^{6} + 3003 \, d^{10} e^{12} x^{5} + 1365 \, d^{11} e^{11} x^{4} + 455 \, d^{12} e^{10} x^{3} + 105 \, d^{13} e^{9} x^{2} + 15 \, d^{14} e^{8} x + d^{15} e^{7}\right )}} \]
-1/45045*(5005*b^6*e^6*x^6 + b^6*d^6 + 9*a*b^5*d^5*e + 45*a^2*b^4*d^4*e^2 + 165*a^3*b^3*d^3*e^3 + 495*a^4*b^2*d^2*e^4 + 1287*a^5*b*d*e^5 + 3003*a^6* e^6 + 3003*(b^6*d*e^5 + 9*a*b^5*e^6)*x^5 + 1365*(b^6*d^2*e^4 + 9*a*b^5*d*e ^5 + 45*a^2*b^4*e^6)*x^4 + 455*(b^6*d^3*e^3 + 9*a*b^5*d^2*e^4 + 45*a^2*b^4 *d*e^5 + 165*a^3*b^3*e^6)*x^3 + 105*(b^6*d^4*e^2 + 9*a*b^5*d^3*e^3 + 45*a^ 2*b^4*d^2*e^4 + 165*a^3*b^3*d*e^5 + 495*a^4*b^2*e^6)*x^2 + 15*(b^6*d^5*e + 9*a*b^5*d^4*e^2 + 45*a^2*b^4*d^3*e^3 + 165*a^3*b^3*d^2*e^4 + 495*a^4*b^2* d*e^5 + 1287*a^5*b*e^6)*x)/(e^22*x^15 + 15*d*e^21*x^14 + 105*d^2*e^20*x^13 + 455*d^3*e^19*x^12 + 1365*d^4*e^18*x^11 + 3003*d^5*e^17*x^10 + 5005*d^6* e^16*x^9 + 6435*d^7*e^15*x^8 + 6435*d^8*e^14*x^7 + 5005*d^9*e^13*x^6 + 300 3*d^10*e^12*x^5 + 1365*d^11*e^11*x^4 + 455*d^12*e^10*x^3 + 105*d^13*e^9*x^ 2 + 15*d^14*e^8*x + d^15*e^7)
Timed out. \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{16}} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{16}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a*e-b*d>0)', see `assume?` for m ore detail
Leaf count of result is larger than twice the leaf count of optimal. 682 vs. \(2 (271) = 542\).
Time = 0.28 (sec) , antiderivative size = 682, normalized size of antiderivative = 1.88 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{16}} \, dx=\frac {b^{15} \mathrm {sgn}\left (b x + a\right )}{45045 \, {\left (b^{9} d^{9} e^{7} - 9 \, a b^{8} d^{8} e^{8} + 36 \, a^{2} b^{7} d^{7} e^{9} - 84 \, a^{3} b^{6} d^{6} e^{10} + 126 \, a^{4} b^{5} d^{5} e^{11} - 126 \, a^{5} b^{4} d^{4} e^{12} + 84 \, a^{6} b^{3} d^{3} e^{13} - 36 \, a^{7} b^{2} d^{2} e^{14} + 9 \, a^{8} b d e^{15} - a^{9} e^{16}\right )}} - \frac {5005 \, b^{6} e^{6} x^{6} \mathrm {sgn}\left (b x + a\right ) + 3003 \, b^{6} d e^{5} x^{5} \mathrm {sgn}\left (b x + a\right ) + 27027 \, a b^{5} e^{6} x^{5} \mathrm {sgn}\left (b x + a\right ) + 1365 \, b^{6} d^{2} e^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + 12285 \, a b^{5} d e^{5} x^{4} \mathrm {sgn}\left (b x + a\right ) + 61425 \, a^{2} b^{4} e^{6} x^{4} \mathrm {sgn}\left (b x + a\right ) + 455 \, b^{6} d^{3} e^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 4095 \, a b^{5} d^{2} e^{4} x^{3} \mathrm {sgn}\left (b x + a\right ) + 20475 \, a^{2} b^{4} d e^{5} x^{3} \mathrm {sgn}\left (b x + a\right ) + 75075 \, a^{3} b^{3} e^{6} x^{3} \mathrm {sgn}\left (b x + a\right ) + 105 \, b^{6} d^{4} e^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 945 \, a b^{5} d^{3} e^{3} x^{2} \mathrm {sgn}\left (b x + a\right ) + 4725 \, a^{2} b^{4} d^{2} e^{4} x^{2} \mathrm {sgn}\left (b x + a\right ) + 17325 \, a^{3} b^{3} d e^{5} x^{2} \mathrm {sgn}\left (b x + a\right ) + 51975 \, a^{4} b^{2} e^{6} x^{2} \mathrm {sgn}\left (b x + a\right ) + 15 \, b^{6} d^{5} e x \mathrm {sgn}\left (b x + a\right ) + 135 \, a b^{5} d^{4} e^{2} x \mathrm {sgn}\left (b x + a\right ) + 675 \, a^{2} b^{4} d^{3} e^{3} x \mathrm {sgn}\left (b x + a\right ) + 2475 \, a^{3} b^{3} d^{2} e^{4} x \mathrm {sgn}\left (b x + a\right ) + 7425 \, a^{4} b^{2} d e^{5} x \mathrm {sgn}\left (b x + a\right ) + 19305 \, a^{5} b e^{6} x \mathrm {sgn}\left (b x + a\right ) + b^{6} d^{6} \mathrm {sgn}\left (b x + a\right ) + 9 \, a b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) + 45 \, a^{2} b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 165 \, a^{3} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 495 \, a^{4} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 1287 \, a^{5} b d e^{5} \mathrm {sgn}\left (b x + a\right ) + 3003 \, a^{6} e^{6} \mathrm {sgn}\left (b x + a\right )}{45045 \, {\left (e x + d\right )}^{15} e^{7}} \]
1/45045*b^15*sgn(b*x + a)/(b^9*d^9*e^7 - 9*a*b^8*d^8*e^8 + 36*a^2*b^7*d^7* e^9 - 84*a^3*b^6*d^6*e^10 + 126*a^4*b^5*d^5*e^11 - 126*a^5*b^4*d^4*e^12 + 84*a^6*b^3*d^3*e^13 - 36*a^7*b^2*d^2*e^14 + 9*a^8*b*d*e^15 - a^9*e^16) - 1 /45045*(5005*b^6*e^6*x^6*sgn(b*x + a) + 3003*b^6*d*e^5*x^5*sgn(b*x + a) + 27027*a*b^5*e^6*x^5*sgn(b*x + a) + 1365*b^6*d^2*e^4*x^4*sgn(b*x + a) + 122 85*a*b^5*d*e^5*x^4*sgn(b*x + a) + 61425*a^2*b^4*e^6*x^4*sgn(b*x + a) + 455 *b^6*d^3*e^3*x^3*sgn(b*x + a) + 4095*a*b^5*d^2*e^4*x^3*sgn(b*x + a) + 2047 5*a^2*b^4*d*e^5*x^3*sgn(b*x + a) + 75075*a^3*b^3*e^6*x^3*sgn(b*x + a) + 10 5*b^6*d^4*e^2*x^2*sgn(b*x + a) + 945*a*b^5*d^3*e^3*x^2*sgn(b*x + a) + 4725 *a^2*b^4*d^2*e^4*x^2*sgn(b*x + a) + 17325*a^3*b^3*d*e^5*x^2*sgn(b*x + a) + 51975*a^4*b^2*e^6*x^2*sgn(b*x + a) + 15*b^6*d^5*e*x*sgn(b*x + a) + 135*a* b^5*d^4*e^2*x*sgn(b*x + a) + 675*a^2*b^4*d^3*e^3*x*sgn(b*x + a) + 2475*a^3 *b^3*d^2*e^4*x*sgn(b*x + a) + 7425*a^4*b^2*d*e^5*x*sgn(b*x + a) + 19305*a^ 5*b*e^6*x*sgn(b*x + a) + b^6*d^6*sgn(b*x + a) + 9*a*b^5*d^5*e*sgn(b*x + a) + 45*a^2*b^4*d^4*e^2*sgn(b*x + a) + 165*a^3*b^3*d^3*e^3*sgn(b*x + a) + 49 5*a^4*b^2*d^2*e^4*sgn(b*x + a) + 1287*a^5*b*d*e^5*sgn(b*x + a) + 3003*a^6* e^6*sgn(b*x + a))/((e*x + d)^15*e^7)
Time = 11.21 (sec) , antiderivative size = 1010, normalized size of antiderivative = 2.79 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{16}} \, dx=\frac {\left (\frac {-6\,a^5\,b\,e^5+15\,a^4\,b^2\,d\,e^4-20\,a^3\,b^3\,d^2\,e^3+15\,a^2\,b^4\,d^3\,e^2-6\,a\,b^5\,d^4\,e+b^6\,d^5}{14\,e^7}+\frac {d\,\left (\frac {15\,a^4\,b^2\,e^5-20\,a^3\,b^3\,d\,e^4+15\,a^2\,b^4\,d^2\,e^3-6\,a\,b^5\,d^3\,e^2+b^6\,d^4\,e}{14\,e^7}-\frac {d\,\left (\frac {20\,a^3\,b^3\,e^5-15\,a^2\,b^4\,d\,e^4+6\,a\,b^5\,d^2\,e^3-b^6\,d^3\,e^2}{14\,e^7}-\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{14\,e^3}-\frac {b^5\,\left (6\,a\,e-b\,d\right )}{14\,e^3}\right )}{e}+\frac {b^4\,\left (15\,a^2\,e^2-6\,a\,b\,d\,e+b^2\,d^2\right )}{14\,e^4}\right )}{e}\right )}{e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^{14}}-\frac {\left (\frac {15\,a^2\,b^4\,e^2-24\,a\,b^5\,d\,e+10\,b^6\,d^2}{11\,e^7}+\frac {d\,\left (\frac {b^6\,d}{11\,e^6}-\frac {2\,b^5\,\left (3\,a\,e-2\,b\,d\right )}{11\,e^6}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^{11}}-\frac {\left (\frac {a^6}{15\,e}-\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {2\,a\,b^5}{5\,e}-\frac {b^6\,d}{15\,e^2}\right )}{e}-\frac {a^2\,b^4}{e}\right )}{e}+\frac {4\,a^3\,b^3}{3\,e}\right )}{e}-\frac {a^4\,b^2}{e}\right )}{e}+\frac {2\,a^5\,b}{5\,e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^{15}}-\frac {\left (\frac {15\,a^4\,b^2\,e^4-40\,a^3\,b^3\,d\,e^3+45\,a^2\,b^4\,d^2\,e^2-24\,a\,b^5\,d^3\,e+5\,b^6\,d^4}{13\,e^7}+\frac {d\,\left (\frac {-20\,a^3\,b^3\,e^4+30\,a^2\,b^4\,d\,e^3-18\,a\,b^5\,d^2\,e^2+4\,b^6\,d^3\,e}{13\,e^7}+\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{13\,e^4}-\frac {2\,b^5\,\left (3\,a\,e-b\,d\right )}{13\,e^4}\right )}{e}+\frac {3\,b^4\,\left (5\,a^2\,e^2-4\,a\,b\,d\,e+b^2\,d^2\right )}{13\,e^5}\right )}{e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^{13}}+\frac {\left (\frac {5\,b^6\,d-6\,a\,b^5\,e}{10\,e^7}+\frac {b^6\,d}{10\,e^7}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^{10}}+\frac {\left (\frac {-20\,a^3\,b^3\,e^3+45\,a^2\,b^4\,d\,e^2-36\,a\,b^5\,d^2\,e+10\,b^6\,d^3}{12\,e^7}+\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{12\,e^5}-\frac {b^5\,\left (2\,a\,e-b\,d\right )}{4\,e^5}\right )}{e}+\frac {b^4\,\left (5\,a^2\,e^2-6\,a\,b\,d\,e+2\,b^2\,d^2\right )}{4\,e^6}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^{12}}-\frac {b^6\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{9\,e^7\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^9} \]
(((b^6*d^5 - 6*a^5*b*e^5 + 15*a^4*b^2*d*e^4 + 15*a^2*b^4*d^3*e^2 - 20*a^3* b^3*d^2*e^3 - 6*a*b^5*d^4*e)/(14*e^7) + (d*((b^6*d^4*e + 15*a^4*b^2*e^5 - 6*a*b^5*d^3*e^2 - 20*a^3*b^3*d*e^4 + 15*a^2*b^4*d^2*e^3)/(14*e^7) - (d*((2 0*a^3*b^3*e^5 - b^6*d^3*e^2 + 6*a*b^5*d^2*e^3 - 15*a^2*b^4*d*e^4)/(14*e^7) - (d*((d*((b^6*d)/(14*e^3) - (b^5*(6*a*e - b*d))/(14*e^3)))/e + (b^4*(15* a^2*e^2 + b^2*d^2 - 6*a*b*d*e))/(14*e^4)))/e))/e))/e)*(a^2 + b^2*x^2 + 2*a *b*x)^(1/2))/((a + b*x)*(d + e*x)^14) - (((10*b^6*d^2 + 15*a^2*b^4*e^2 - 2 4*a*b^5*d*e)/(11*e^7) + (d*((b^6*d)/(11*e^6) - (2*b^5*(3*a*e - 2*b*d))/(11 *e^6)))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^11) - ((a ^6/(15*e) - (d*((d*((d*((d*((d*((2*a*b^5)/(5*e) - (b^6*d)/(15*e^2)))/e - ( a^2*b^4)/e))/e + (4*a^3*b^3)/(3*e)))/e - (a^4*b^2)/e))/e + (2*a^5*b)/(5*e) ))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^15) - (((5*b^6 *d^4 + 15*a^4*b^2*e^4 - 40*a^3*b^3*d*e^3 + 45*a^2*b^4*d^2*e^2 - 24*a*b^5*d ^3*e)/(13*e^7) + (d*((4*b^6*d^3*e - 20*a^3*b^3*e^4 - 18*a*b^5*d^2*e^2 + 30 *a^2*b^4*d*e^3)/(13*e^7) + (d*((d*((b^6*d)/(13*e^4) - (2*b^5*(3*a*e - b*d) )/(13*e^4)))/e + (3*b^4*(5*a^2*e^2 + b^2*d^2 - 4*a*b*d*e))/(13*e^5)))/e))/ e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^13) + (((5*b^6*d - 6*a*b^5*e)/(10*e^7) + (b^6*d)/(10*e^7))*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2)) /((a + b*x)*(d + e*x)^10) + (((10*b^6*d^3 - 20*a^3*b^3*e^3 + 45*a^2*b^4*d* e^2 - 36*a*b^5*d^2*e)/(12*e^7) + (d*((d*((b^6*d)/(12*e^5) - (b^5*(2*a*e...