Integrand size = 33, antiderivative size = 359 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{12}} \, dx=-\frac {(b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x) (d+e x)^{11}}+\frac {3 b (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x) (d+e x)^{10}}-\frac {5 b^2 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x) (d+e x)^9}+\frac {5 b^3 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^7 (a+b x) (d+e x)^8}-\frac {15 b^4 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x) (d+e x)^7}+\frac {b^5 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)^6}-\frac {b^6 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x) (d+e x)^5} \]
-1/11*(-a*e+b*d)^6*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^11+3/5*b*(-a*e+b* d)^5*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^10-5/3*b^2*(-a*e+b*d)^4*((b*x+a )^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^9+5/2*b^3*(-a*e+b*d)^3*((b*x+a)^2)^(1/2)/e^ 7/(b*x+a)/(e*x+d)^8-15/7*b^4*(-a*e+b*d)^2*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e *x+d)^7+b^5*(-a*e+b*d)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^6-1/5*b^6*((b *x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^5
Time = 1.08 (sec) , antiderivative size = 295, normalized size of antiderivative = 0.82 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{12}} \, dx=-\frac {\sqrt {(a+b x)^2} \left (210 a^6 e^6+126 a^5 b e^5 (d+11 e x)+70 a^4 b^2 e^4 \left (d^2+11 d e x+55 e^2 x^2\right )+35 a^3 b^3 e^3 \left (d^3+11 d^2 e x+55 d e^2 x^2+165 e^3 x^3\right )+15 a^2 b^4 e^2 \left (d^4+11 d^3 e x+55 d^2 e^2 x^2+165 d e^3 x^3+330 e^4 x^4\right )+5 a b^5 e \left (d^5+11 d^4 e x+55 d^3 e^2 x^2+165 d^2 e^3 x^3+330 d e^4 x^4+462 e^5 x^5\right )+b^6 \left (d^6+11 d^5 e x+55 d^4 e^2 x^2+165 d^3 e^3 x^3+330 d^2 e^4 x^4+462 d e^5 x^5+462 e^6 x^6\right )\right )}{2310 e^7 (a+b x) (d+e x)^{11}} \]
-1/2310*(Sqrt[(a + b*x)^2]*(210*a^6*e^6 + 126*a^5*b*e^5*(d + 11*e*x) + 70* a^4*b^2*e^4*(d^2 + 11*d*e*x + 55*e^2*x^2) + 35*a^3*b^3*e^3*(d^3 + 11*d^2*e *x + 55*d*e^2*x^2 + 165*e^3*x^3) + 15*a^2*b^4*e^2*(d^4 + 11*d^3*e*x + 55*d ^2*e^2*x^2 + 165*d*e^3*x^3 + 330*e^4*x^4) + 5*a*b^5*e*(d^5 + 11*d^4*e*x + 55*d^3*e^2*x^2 + 165*d^2*e^3*x^3 + 330*d*e^4*x^4 + 462*e^5*x^5) + b^6*(d^6 + 11*d^5*e*x + 55*d^4*e^2*x^2 + 165*d^3*e^3*x^3 + 330*d^2*e^4*x^4 + 462*d *e^5*x^5 + 462*e^6*x^6)))/(e^7*(a + b*x)*(d + e*x)^11)
Time = 0.40 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.55, 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.
| \(\displaystyle \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{12}} \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {b^5 (a+b x)^6}{(d+e x)^{12}}dx}{b^5 (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x)^6}{(d+e x)^{12}}dx}{a+b x}\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {b^6}{e^6 (d+e x)^6}-\frac {6 (b d-a e) b^5}{e^6 (d+e x)^7}+\frac {15 (b d-a e)^2 b^4}{e^6 (d+e x)^8}-\frac {20 (b d-a e)^3 b^3}{e^6 (d+e x)^9}+\frac {15 (b d-a e)^4 b^2}{e^6 (d+e x)^{10}}-\frac {6 (b d-a e)^5 b}{e^6 (d+e x)^{11}}+\frac {(a e-b d)^6}{e^6 (d+e x)^{12}}\right )dx}{a+b x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {b^5 (b d-a e)}{e^7 (d+e x)^6}-\frac {15 b^4 (b d-a e)^2}{7 e^7 (d+e x)^7}+\frac {5 b^3 (b d-a e)^3}{2 e^7 (d+e x)^8}-\frac {5 b^2 (b d-a e)^4}{3 e^7 (d+e x)^9}+\frac {3 b (b d-a e)^5}{5 e^7 (d+e x)^{10}}-\frac {(b d-a e)^6}{11 e^7 (d+e x)^{11}}-\frac {b^6}{5 e^7 (d+e x)^5}\right )}{a+b x}\) |
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-1/11*(b*d - a*e)^6/(e^7*(d + e*x)^11) + ( 3*b*(b*d - a*e)^5)/(5*e^7*(d + e*x)^10) - (5*b^2*(b*d - a*e)^4)/(3*e^7*(d + e*x)^9) + (5*b^3*(b*d - a*e)^3)/(2*e^7*(d + e*x)^8) - (15*b^4*(b*d - a*e )^2)/(7*e^7*(d + e*x)^7) + (b^5*(b*d - a*e))/(e^7*(d + e*x)^6) - b^6/(5*e^ 7*(d + e*x)^5)))/(a + b*x)
3.21.9.3.1 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 = 9.73 (sec) , antiderivative size = 351, normalized size of antiderivative = 0.98
| method | result | size |
| risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {b^{6} x^{6}}{5 e}-\frac {b^{5} \left (5 a e +b d \right ) x^{5}}{5 e^{2}}-\frac {b^{4} \left (15 e^{2} a^{2}+5 a b d e +b^{2} d^{2}\right ) x^{4}}{7 e^{3}}-\frac {b^{3} \left (35 a^{3} e^{3}+15 a^{2} b d \,e^{2}+5 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) x^{3}}{14 e^{4}}-\frac {b^{2} \left (70 e^{4} a^{4}+35 b d \,e^{3} a^{3}+15 b^{2} d^{2} e^{2} a^{2}+5 b^{3} d^{3} e a +b^{4} d^{4}\right ) x^{2}}{42 e^{5}}-\frac {b \left (126 e^{5} a^{5}+70 b d \,e^{4} a^{4}+35 b^{2} d^{2} e^{3} a^{3}+15 b^{3} d^{3} e^{2} a^{2}+5 b^{4} d^{4} e a +b^{5} d^{5}\right ) x}{210 e^{6}}-\frac {210 e^{6} a^{6}+126 b d \,e^{5} a^{5}+70 b^{2} d^{2} e^{4} a^{4}+35 b^{3} d^{3} e^{3} a^{3}+15 b^{4} d^{4} e^{2} a^{2}+5 b^{5} d^{5} e a +b^{6} d^{6}}{2310 e^{7}}\right )}{\left (b x +a \right ) \left (e x +d \right )^{11}}\) | \(351\) |
| gosper | \(-\frac {\left (462 b^{6} e^{6} x^{6}+2310 a \,b^{5} e^{6} x^{5}+462 b^{6} d \,e^{5} x^{5}+4950 a^{2} b^{4} e^{6} x^{4}+1650 a \,b^{5} d \,e^{5} x^{4}+330 b^{6} d^{2} e^{4} x^{4}+5775 a^{3} b^{3} e^{6} x^{3}+2475 a^{2} b^{4} d \,e^{5} x^{3}+825 a \,b^{5} d^{2} e^{4} x^{3}+165 b^{6} d^{3} e^{3} x^{3}+3850 a^{4} b^{2} e^{6} x^{2}+1925 a^{3} b^{3} d \,e^{5} x^{2}+825 a^{2} b^{4} d^{2} e^{4} x^{2}+275 a \,b^{5} d^{3} e^{3} x^{2}+55 b^{6} d^{4} e^{2} x^{2}+1386 a^{5} b \,e^{6} x +770 a^{4} b^{2} d \,e^{5} x +385 a^{3} b^{3} d^{2} e^{4} x +165 a^{2} b^{4} d^{3} e^{3} x +55 a \,b^{5} d^{4} e^{2} x +11 b^{6} d^{5} e x +210 e^{6} a^{6}+126 b d \,e^{5} a^{5}+70 b^{2} d^{2} e^{4} a^{4}+35 b^{3} d^{3} e^{3} a^{3}+15 b^{4} d^{4} e^{2} a^{2}+5 b^{5} d^{5} e a +b^{6} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{2310 e^{7} \left (e x +d \right )^{11} \left (b x +a \right )^{5}}\) | \(392\) |
| default | \(-\frac {\left (462 b^{6} e^{6} x^{6}+2310 a \,b^{5} e^{6} x^{5}+462 b^{6} d \,e^{5} x^{5}+4950 a^{2} b^{4} e^{6} x^{4}+1650 a \,b^{5} d \,e^{5} x^{4}+330 b^{6} d^{2} e^{4} x^{4}+5775 a^{3} b^{3} e^{6} x^{3}+2475 a^{2} b^{4} d \,e^{5} x^{3}+825 a \,b^{5} d^{2} e^{4} x^{3}+165 b^{6} d^{3} e^{3} x^{3}+3850 a^{4} b^{2} e^{6} x^{2}+1925 a^{3} b^{3} d \,e^{5} x^{2}+825 a^{2} b^{4} d^{2} e^{4} x^{2}+275 a \,b^{5} d^{3} e^{3} x^{2}+55 b^{6} d^{4} e^{2} x^{2}+1386 a^{5} b \,e^{6} x +770 a^{4} b^{2} d \,e^{5} x +385 a^{3} b^{3} d^{2} e^{4} x +165 a^{2} b^{4} d^{3} e^{3} x +55 a \,b^{5} d^{4} e^{2} x +11 b^{6} d^{5} e x +210 e^{6} a^{6}+126 b d \,e^{5} a^{5}+70 b^{2} d^{2} e^{4} a^{4}+35 b^{3} d^{3} e^{3} a^{3}+15 b^{4} d^{4} e^{2} a^{2}+5 b^{5} d^{5} e a +b^{6} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{2310 e^{7} \left (e x +d \right )^{11} \left (b x +a \right )^{5}}\) | \(392\) |
((b*x+a)^2)^(1/2)/(b*x+a)*(-1/5*b^6/e*x^6-1/5*b^5/e^2*(5*a*e+b*d)*x^5-1/7* b^4/e^3*(15*a^2*e^2+5*a*b*d*e+b^2*d^2)*x^4-1/14*b^3/e^4*(35*a^3*e^3+15*a^2 *b*d*e^2+5*a*b^2*d^2*e+b^3*d^3)*x^3-1/42*b^2/e^5*(70*a^4*e^4+35*a^3*b*d*e^ 3+15*a^2*b^2*d^2*e^2+5*a*b^3*d^3*e+b^4*d^4)*x^2-1/210*b/e^6*(126*a^5*e^5+7 0*a^4*b*d*e^4+35*a^3*b^2*d^2*e^3+15*a^2*b^3*d^3*e^2+5*a*b^4*d^4*e+b^5*d^5) *x-1/2310/e^7*(210*a^6*e^6+126*a^5*b*d*e^5+70*a^4*b^2*d^2*e^4+35*a^3*b^3*d ^3*e^3+15*a^2*b^4*d^4*e^2+5*a*b^5*d^5*e+b^6*d^6))/(e*x+d)^11
Time = 0.33 (sec) , antiderivative size = 463, normalized size of antiderivative = 1.29 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{12}} \, dx=-\frac {462 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 5 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} + 35 \, a^{3} b^{3} d^{3} e^{3} + 70 \, a^{4} b^{2} d^{2} e^{4} + 126 \, a^{5} b d e^{5} + 210 \, a^{6} e^{6} + 462 \, {\left (b^{6} d e^{5} + 5 \, a b^{5} e^{6}\right )} x^{5} + 330 \, {\left (b^{6} d^{2} e^{4} + 5 \, a b^{5} d e^{5} + 15 \, a^{2} b^{4} e^{6}\right )} x^{4} + 165 \, {\left (b^{6} d^{3} e^{3} + 5 \, a b^{5} d^{2} e^{4} + 15 \, a^{2} b^{4} d e^{5} + 35 \, a^{3} b^{3} e^{6}\right )} x^{3} + 55 \, {\left (b^{6} d^{4} e^{2} + 5 \, a b^{5} d^{3} e^{3} + 15 \, a^{2} b^{4} d^{2} e^{4} + 35 \, a^{3} b^{3} d e^{5} + 70 \, a^{4} b^{2} e^{6}\right )} x^{2} + 11 \, {\left (b^{6} d^{5} e + 5 \, a b^{5} d^{4} e^{2} + 15 \, a^{2} b^{4} d^{3} e^{3} + 35 \, a^{3} b^{3} d^{2} e^{4} + 70 \, a^{4} b^{2} d e^{5} + 126 \, a^{5} b e^{6}\right )} x}{2310 \, {\left (e^{18} x^{11} + 11 \, d e^{17} x^{10} + 55 \, d^{2} e^{16} x^{9} + 165 \, d^{3} e^{15} x^{8} + 330 \, d^{4} e^{14} x^{7} + 462 \, d^{5} e^{13} x^{6} + 462 \, d^{6} e^{12} x^{5} + 330 \, d^{7} e^{11} x^{4} + 165 \, d^{8} e^{10} x^{3} + 55 \, d^{9} e^{9} x^{2} + 11 \, d^{10} e^{8} x + d^{11} e^{7}\right )}} \]
-1/2310*(462*b^6*e^6*x^6 + b^6*d^6 + 5*a*b^5*d^5*e + 15*a^2*b^4*d^4*e^2 + 35*a^3*b^3*d^3*e^3 + 70*a^4*b^2*d^2*e^4 + 126*a^5*b*d*e^5 + 210*a^6*e^6 + 462*(b^6*d*e^5 + 5*a*b^5*e^6)*x^5 + 330*(b^6*d^2*e^4 + 5*a*b^5*d*e^5 + 15* a^2*b^4*e^6)*x^4 + 165*(b^6*d^3*e^3 + 5*a*b^5*d^2*e^4 + 15*a^2*b^4*d*e^5 + 35*a^3*b^3*e^6)*x^3 + 55*(b^6*d^4*e^2 + 5*a*b^5*d^3*e^3 + 15*a^2*b^4*d^2* e^4 + 35*a^3*b^3*d*e^5 + 70*a^4*b^2*e^6)*x^2 + 11*(b^6*d^5*e + 5*a*b^5*d^4 *e^2 + 15*a^2*b^4*d^3*e^3 + 35*a^3*b^3*d^2*e^4 + 70*a^4*b^2*d*e^5 + 126*a^ 5*b*e^6)*x)/(e^18*x^11 + 11*d*e^17*x^10 + 55*d^2*e^16*x^9 + 165*d^3*e^15*x ^8 + 330*d^4*e^14*x^7 + 462*d^5*e^13*x^6 + 462*d^6*e^12*x^5 + 330*d^7*e^11 *x^4 + 165*d^8*e^10*x^3 + 55*d^9*e^9*x^2 + 11*d^10*e^8*x + d^11*e^7)
Exception generated. \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{12}} \, dx=\text {Exception raised: HeuristicGCDFailed} \]
Exception generated. \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{12}} \, dx=\text {Exception raised: ValueError} \]
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. 626 vs. \(2 (270) = 540\).
Time = 0.28 (sec) , antiderivative size = 626, normalized size of antiderivative = 1.74 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{12}} \, dx=\frac {b^{11} \mathrm {sgn}\left (b x + a\right )}{2310 \, {\left (b^{5} d^{5} e^{7} - 5 \, a b^{4} d^{4} e^{8} + 10 \, a^{2} b^{3} d^{3} e^{9} - 10 \, a^{3} b^{2} d^{2} e^{10} + 5 \, a^{4} b d e^{11} - a^{5} e^{12}\right )}} - \frac {462 \, b^{6} e^{6} x^{6} \mathrm {sgn}\left (b x + a\right ) + 462 \, b^{6} d e^{5} x^{5} \mathrm {sgn}\left (b x + a\right ) + 2310 \, a b^{5} e^{6} x^{5} \mathrm {sgn}\left (b x + a\right ) + 330 \, b^{6} d^{2} e^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + 1650 \, a b^{5} d e^{5} x^{4} \mathrm {sgn}\left (b x + a\right ) + 4950 \, a^{2} b^{4} e^{6} x^{4} \mathrm {sgn}\left (b x + a\right ) + 165 \, b^{6} d^{3} e^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 825 \, a b^{5} d^{2} e^{4} x^{3} \mathrm {sgn}\left (b x + a\right ) + 2475 \, a^{2} b^{4} d e^{5} x^{3} \mathrm {sgn}\left (b x + a\right ) + 5775 \, a^{3} b^{3} e^{6} x^{3} \mathrm {sgn}\left (b x + a\right ) + 55 \, b^{6} d^{4} e^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 275 \, a b^{5} d^{3} e^{3} x^{2} \mathrm {sgn}\left (b x + a\right ) + 825 \, a^{2} b^{4} d^{2} e^{4} x^{2} \mathrm {sgn}\left (b x + a\right ) + 1925 \, a^{3} b^{3} d e^{5} x^{2} \mathrm {sgn}\left (b x + a\right ) + 3850 \, a^{4} b^{2} e^{6} x^{2} \mathrm {sgn}\left (b x + a\right ) + 11 \, b^{6} d^{5} e x \mathrm {sgn}\left (b x + a\right ) + 55 \, a b^{5} d^{4} e^{2} x \mathrm {sgn}\left (b x + a\right ) + 165 \, a^{2} b^{4} d^{3} e^{3} x \mathrm {sgn}\left (b x + a\right ) + 385 \, a^{3} b^{3} d^{2} e^{4} x \mathrm {sgn}\left (b x + a\right ) + 770 \, a^{4} b^{2} d e^{5} x \mathrm {sgn}\left (b x + a\right ) + 1386 \, a^{5} b e^{6} x \mathrm {sgn}\left (b x + a\right ) + b^{6} d^{6} \mathrm {sgn}\left (b x + a\right ) + 5 \, a b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{2} b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 35 \, a^{3} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 70 \, a^{4} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 126 \, a^{5} b d e^{5} \mathrm {sgn}\left (b x + a\right ) + 210 \, a^{6} e^{6} \mathrm {sgn}\left (b x + a\right )}{2310 \, {\left (e x + d\right )}^{11} e^{7}} \]
1/2310*b^11*sgn(b*x + a)/(b^5*d^5*e^7 - 5*a*b^4*d^4*e^8 + 10*a^2*b^3*d^3*e ^9 - 10*a^3*b^2*d^2*e^10 + 5*a^4*b*d*e^11 - a^5*e^12) - 1/2310*(462*b^6*e^ 6*x^6*sgn(b*x + a) + 462*b^6*d*e^5*x^5*sgn(b*x + a) + 2310*a*b^5*e^6*x^5*s gn(b*x + a) + 330*b^6*d^2*e^4*x^4*sgn(b*x + a) + 1650*a*b^5*d*e^5*x^4*sgn( b*x + a) + 4950*a^2*b^4*e^6*x^4*sgn(b*x + a) + 165*b^6*d^3*e^3*x^3*sgn(b*x + a) + 825*a*b^5*d^2*e^4*x^3*sgn(b*x + a) + 2475*a^2*b^4*d*e^5*x^3*sgn(b* x + a) + 5775*a^3*b^3*e^6*x^3*sgn(b*x + a) + 55*b^6*d^4*e^2*x^2*sgn(b*x + a) + 275*a*b^5*d^3*e^3*x^2*sgn(b*x + a) + 825*a^2*b^4*d^2*e^4*x^2*sgn(b*x + a) + 1925*a^3*b^3*d*e^5*x^2*sgn(b*x + a) + 3850*a^4*b^2*e^6*x^2*sgn(b*x + a) + 11*b^6*d^5*e*x*sgn(b*x + a) + 55*a*b^5*d^4*e^2*x*sgn(b*x + a) + 165 *a^2*b^4*d^3*e^3*x*sgn(b*x + a) + 385*a^3*b^3*d^2*e^4*x*sgn(b*x + a) + 770 *a^4*b^2*d*e^5*x*sgn(b*x + a) + 1386*a^5*b*e^6*x*sgn(b*x + a) + b^6*d^6*sg n(b*x + a) + 5*a*b^5*d^5*e*sgn(b*x + a) + 15*a^2*b^4*d^4*e^2*sgn(b*x + a) + 35*a^3*b^3*d^3*e^3*sgn(b*x + a) + 70*a^4*b^2*d^2*e^4*sgn(b*x + a) + 126* a^5*b*d*e^5*sgn(b*x + a) + 210*a^6*e^6*sgn(b*x + a))/((e*x + d)^11*e^7)
Time = 11.16 (sec) , antiderivative size = 1010, normalized size of antiderivative = 2.81 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{12}} \, 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}{10\,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}{10\,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}{10\,e^7}-\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{10\,e^3}-\frac {b^5\,\left (6\,a\,e-b\,d\right )}{10\,e^3}\right )}{e}+\frac {b^4\,\left (15\,a^2\,e^2-6\,a\,b\,d\,e+b^2\,d^2\right )}{10\,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 )}^{10}}-\frac {\left (\frac {15\,a^2\,b^4\,e^2-24\,a\,b^5\,d\,e+10\,b^6\,d^2}{7\,e^7}+\frac {d\,\left (\frac {b^6\,d}{7\,e^6}-\frac {2\,b^5\,\left (3\,a\,e-2\,b\,d\right )}{7\,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 )}^7}-\frac {\left (\frac {a^6}{11\,e}-\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {6\,a\,b^5}{11\,e}-\frac {b^6\,d}{11\,e^2}\right )}{e}-\frac {15\,a^2\,b^4}{11\,e}\right )}{e}+\frac {20\,a^3\,b^3}{11\,e}\right )}{e}-\frac {15\,a^4\,b^2}{11\,e}\right )}{e}+\frac {6\,a^5\,b}{11\,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 )}^{11}}-\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}{9\,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}{9\,e^7}+\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{9\,e^4}-\frac {2\,b^5\,\left (3\,a\,e-b\,d\right )}{9\,e^4}\right )}{e}+\frac {b^4\,\left (5\,a^2\,e^2-4\,a\,b\,d\,e+b^2\,d^2\right )}{3\,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 )}^9}+\frac {\left (\frac {5\,b^6\,d-6\,a\,b^5\,e}{6\,e^7}+\frac {b^6\,d}{6\,e^7}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^6}+\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}{8\,e^7}+\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{8\,e^5}-\frac {3\,b^5\,\left (2\,a\,e-b\,d\right )}{8\,e^5}\right )}{e}+\frac {3\,b^4\,\left (5\,a^2\,e^2-6\,a\,b\,d\,e+2\,b^2\,d^2\right )}{8\,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 )}^8}-\frac {b^6\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{5\,e^7\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^5} \]
(((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)/(10*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)/(10*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)/(10*e^7) - (d*((d*((b^6*d)/(10*e^3) - (b^5*(6*a*e - b*d))/(10*e^3)))/e + (b^4*(15* a^2*e^2 + b^2*d^2 - 6*a*b*d*e))/(10*e^4)))/e))/e))/e)*(a^2 + b^2*x^2 + 2*a *b*x)^(1/2))/((a + b*x)*(d + e*x)^10) - (((10*b^6*d^2 + 15*a^2*b^4*e^2 - 2 4*a*b^5*d*e)/(7*e^7) + (d*((b^6*d)/(7*e^6) - (2*b^5*(3*a*e - 2*b*d))/(7*e^ 6)))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^7) - ((a^6/( 11*e) - (d*((d*((d*((d*((d*((6*a*b^5)/(11*e) - (b^6*d)/(11*e^2)))/e - (15* a^2*b^4)/(11*e)))/e + (20*a^3*b^3)/(11*e)))/e - (15*a^4*b^2)/(11*e)))/e + (6*a^5*b)/(11*e)))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x )^11) - (((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)/(9*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)/(9*e^7) + (d*((d*((b^6*d)/(9*e^4) - (2*b^5* (3*a*e - b*d))/(9*e^4)))/e + (b^4*(5*a^2*e^2 + b^2*d^2 - 4*a*b*d*e))/(3*e^ 5)))/e))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^9) + ((( 5*b^6*d - 6*a*b^5*e)/(6*e^7) + (b^6*d)/(6*e^7))*(a^2 + b^2*x^2 + 2*a*b*x)^ (1/2))/((a + b*x)*(d + e*x)^6) + (((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)/(8*e^7) + (d*((d*((b^6*d)/(8*e^5) - (3*b^5*(...