Integrand size = 33, antiderivative size = 362 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{13}} \, dx=-\frac {(b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{12 e^7 (a+b x) (d+e x)^{12}}+\frac {6 b (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x) (d+e x)^{11}}-\frac {3 b^2 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^7 (a+b x) (d+e x)^{10}}+\frac {20 b^3 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^7 (a+b x) (d+e x)^9}-\frac {15 b^4 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{8 e^7 (a+b x) (d+e x)^8}+\frac {6 b^5 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x) (d+e x)^7}-\frac {b^6 \sqrt {a^2+2 a b x+b^2 x^2}}{6 e^7 (a+b x) (d+e x)^6} \]
-1/12*(-a*e+b*d)^6*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^12+6/11*b*(-a*e+b *d)^5*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^11-3/2*b^2*(-a*e+b*d)^4*((b*x+ a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^10+20/9*b^3*(-a*e+b*d)^3*((b*x+a)^2)^(1/2) /e^7/(b*x+a)/(e*x+d)^9-15/8*b^4*(-a*e+b*d)^2*((b*x+a)^2)^(1/2)/e^7/(b*x+a) /(e*x+d)^8+6/7*b^5*(-a*e+b*d)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^7-1/6* b^6*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^6
Time = 1.07 (sec) , antiderivative size = 295, normalized size of antiderivative = 0.81 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{13}} \, dx=-\frac {\sqrt {(a+b x)^2} \left (462 a^6 e^6+252 a^5 b e^5 (d+12 e x)+126 a^4 b^2 e^4 \left (d^2+12 d e x+66 e^2 x^2\right )+56 a^3 b^3 e^3 \left (d^3+12 d^2 e x+66 d e^2 x^2+220 e^3 x^3\right )+21 a^2 b^4 e^2 \left (d^4+12 d^3 e x+66 d^2 e^2 x^2+220 d e^3 x^3+495 e^4 x^4\right )+6 a b^5 e \left (d^5+12 d^4 e x+66 d^3 e^2 x^2+220 d^2 e^3 x^3+495 d e^4 x^4+792 e^5 x^5\right )+b^6 \left (d^6+12 d^5 e x+66 d^4 e^2 x^2+220 d^3 e^3 x^3+495 d^2 e^4 x^4+792 d e^5 x^5+924 e^6 x^6\right )\right )}{5544 e^7 (a+b x) (d+e x)^{12}} \]
-1/5544*(Sqrt[(a + b*x)^2]*(462*a^6*e^6 + 252*a^5*b*e^5*(d + 12*e*x) + 126 *a^4*b^2*e^4*(d^2 + 12*d*e*x + 66*e^2*x^2) + 56*a^3*b^3*e^3*(d^3 + 12*d^2* e*x + 66*d*e^2*x^2 + 220*e^3*x^3) + 21*a^2*b^4*e^2*(d^4 + 12*d^3*e*x + 66* d^2*e^2*x^2 + 220*d*e^3*x^3 + 495*e^4*x^4) + 6*a*b^5*e*(d^5 + 12*d^4*e*x + 66*d^3*e^2*x^2 + 220*d^2*e^3*x^3 + 495*d*e^4*x^4 + 792*e^5*x^5) + b^6*(d^ 6 + 12*d^5*e*x + 66*d^4*e^2*x^2 + 220*d^3*e^3*x^3 + 495*d^2*e^4*x^4 + 792* d*e^5*x^5 + 924*e^6*x^6)))/(e^7*(a + b*x)*(d + e*x)^12)
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.
| \(\displaystyle \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{13}} \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {b^5 (a+b x)^6}{(d+e x)^{13}}dx}{b^5 (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x)^6}{(d+e x)^{13}}dx}{a+b x}\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {b^6}{e^6 (d+e x)^7}-\frac {6 (b d-a e) b^5}{e^6 (d+e x)^8}+\frac {15 (b d-a e)^2 b^4}{e^6 (d+e x)^9}-\frac {20 (b d-a e)^3 b^3}{e^6 (d+e x)^{10}}+\frac {15 (b d-a e)^4 b^2}{e^6 (d+e x)^{11}}-\frac {6 (b d-a e)^5 b}{e^6 (d+e x)^{12}}+\frac {(a e-b d)^6}{e^6 (d+e x)^{13}}\right )dx}{a+b x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {6 b^5 (b d-a e)}{7 e^7 (d+e x)^7}-\frac {15 b^4 (b d-a e)^2}{8 e^7 (d+e x)^8}+\frac {20 b^3 (b d-a e)^3}{9 e^7 (d+e x)^9}-\frac {3 b^2 (b d-a e)^4}{2 e^7 (d+e x)^{10}}+\frac {6 b (b d-a e)^5}{11 e^7 (d+e x)^{11}}-\frac {(b d-a e)^6}{12 e^7 (d+e x)^{12}}-\frac {b^6}{6 e^7 (d+e x)^6}\right )}{a+b x}\) |
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-1/12*(b*d - a*e)^6/(e^7*(d + e*x)^12) + ( 6*b*(b*d - a*e)^5)/(11*e^7*(d + e*x)^11) - (3*b^2*(b*d - a*e)^4)/(2*e^7*(d + e*x)^10) + (20*b^3*(b*d - a*e)^3)/(9*e^7*(d + e*x)^9) - (15*b^4*(b*d - a*e)^2)/(8*e^7*(d + e*x)^8) + (6*b^5*(b*d - a*e))/(7*e^7*(d + e*x)^7) - b^ 6/(6*e^7*(d + e*x)^6)))/(a + b*x)
3.21.10.3.1 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 = 14.52 (sec) , antiderivative size = 351, normalized size of antiderivative = 0.97
| method | result | size |
| risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {b^{6} x^{6}}{6 e}-\frac {b^{5} \left (6 a e +b d \right ) x^{5}}{7 e^{2}}-\frac {5 b^{4} \left (21 e^{2} a^{2}+6 a b d e +b^{2} d^{2}\right ) x^{4}}{56 e^{3}}-\frac {5 b^{3} \left (56 a^{3} e^{3}+21 a^{2} b d \,e^{2}+6 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) x^{3}}{126 e^{4}}-\frac {b^{2} \left (126 e^{4} a^{4}+56 b d \,e^{3} a^{3}+21 b^{2} d^{2} e^{2} a^{2}+6 b^{3} d^{3} e a +b^{4} d^{4}\right ) x^{2}}{84 e^{5}}-\frac {b \left (252 e^{5} a^{5}+126 b d \,e^{4} a^{4}+56 b^{2} d^{2} e^{3} a^{3}+21 b^{3} d^{3} e^{2} a^{2}+6 b^{4} d^{4} e a +b^{5} d^{5}\right ) x}{462 e^{6}}-\frac {462 e^{6} a^{6}+252 b d \,e^{5} a^{5}+126 b^{2} d^{2} e^{4} a^{4}+56 b^{3} d^{3} e^{3} a^{3}+21 b^{4} d^{4} e^{2} a^{2}+6 b^{5} d^{5} e a +b^{6} d^{6}}{5544 e^{7}}\right )}{\left (b x +a \right ) \left (e x +d \right )^{12}}\) | \(351\) |
| gosper | \(-\frac {\left (924 b^{6} e^{6} x^{6}+4752 a \,b^{5} e^{6} x^{5}+792 b^{6} d \,e^{5} x^{5}+10395 a^{2} b^{4} e^{6} x^{4}+2970 a \,b^{5} d \,e^{5} x^{4}+495 b^{6} d^{2} e^{4} x^{4}+12320 a^{3} b^{3} e^{6} x^{3}+4620 a^{2} b^{4} d \,e^{5} x^{3}+1320 a \,b^{5} d^{2} e^{4} x^{3}+220 b^{6} d^{3} e^{3} x^{3}+8316 a^{4} b^{2} e^{6} x^{2}+3696 a^{3} b^{3} d \,e^{5} x^{2}+1386 a^{2} b^{4} d^{2} e^{4} x^{2}+396 a \,b^{5} d^{3} e^{3} x^{2}+66 b^{6} d^{4} e^{2} x^{2}+3024 a^{5} b \,e^{6} x +1512 a^{4} b^{2} d \,e^{5} x +672 a^{3} b^{3} d^{2} e^{4} x +252 a^{2} b^{4} d^{3} e^{3} x +72 a \,b^{5} d^{4} e^{2} x +12 b^{6} d^{5} e x +462 e^{6} a^{6}+252 b d \,e^{5} a^{5}+126 b^{2} d^{2} e^{4} a^{4}+56 b^{3} d^{3} e^{3} a^{3}+21 b^{4} d^{4} e^{2} a^{2}+6 b^{5} d^{5} e a +b^{6} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{5544 e^{7} \left (e x +d \right )^{12} \left (b x +a \right )^{5}}\) | \(392\) |
| default | \(-\frac {\left (924 b^{6} e^{6} x^{6}+4752 a \,b^{5} e^{6} x^{5}+792 b^{6} d \,e^{5} x^{5}+10395 a^{2} b^{4} e^{6} x^{4}+2970 a \,b^{5} d \,e^{5} x^{4}+495 b^{6} d^{2} e^{4} x^{4}+12320 a^{3} b^{3} e^{6} x^{3}+4620 a^{2} b^{4} d \,e^{5} x^{3}+1320 a \,b^{5} d^{2} e^{4} x^{3}+220 b^{6} d^{3} e^{3} x^{3}+8316 a^{4} b^{2} e^{6} x^{2}+3696 a^{3} b^{3} d \,e^{5} x^{2}+1386 a^{2} b^{4} d^{2} e^{4} x^{2}+396 a \,b^{5} d^{3} e^{3} x^{2}+66 b^{6} d^{4} e^{2} x^{2}+3024 a^{5} b \,e^{6} x +1512 a^{4} b^{2} d \,e^{5} x +672 a^{3} b^{3} d^{2} e^{4} x +252 a^{2} b^{4} d^{3} e^{3} x +72 a \,b^{5} d^{4} e^{2} x +12 b^{6} d^{5} e x +462 e^{6} a^{6}+252 b d \,e^{5} a^{5}+126 b^{2} d^{2} e^{4} a^{4}+56 b^{3} d^{3} e^{3} a^{3}+21 b^{4} d^{4} e^{2} a^{2}+6 b^{5} d^{5} e a +b^{6} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{5544 e^{7} \left (e x +d \right )^{12} \left (b x +a \right )^{5}}\) | \(392\) |
((b*x+a)^2)^(1/2)/(b*x+a)*(-1/6*b^6/e*x^6-1/7*b^5/e^2*(6*a*e+b*d)*x^5-5/56 *b^4/e^3*(21*a^2*e^2+6*a*b*d*e+b^2*d^2)*x^4-5/126*b^3/e^4*(56*a^3*e^3+21*a ^2*b*d*e^2+6*a*b^2*d^2*e+b^3*d^3)*x^3-1/84*b^2/e^5*(126*a^4*e^4+56*a^3*b*d *e^3+21*a^2*b^2*d^2*e^2+6*a*b^3*d^3*e+b^4*d^4)*x^2-1/462*b/e^6*(252*a^5*e^ 5+126*a^4*b*d*e^4+56*a^3*b^2*d^2*e^3+21*a^2*b^3*d^3*e^2+6*a*b^4*d^4*e+b^5* d^5)*x-1/5544/e^7*(462*a^6*e^6+252*a^5*b*d*e^5+126*a^4*b^2*d^2*e^4+56*a^3* b^3*d^3*e^3+21*a^2*b^4*d^4*e^2+6*a*b^5*d^5*e+b^6*d^6))/(e*x+d)^12
Time = 0.30 (sec) , antiderivative size = 474, normalized size of antiderivative = 1.31 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{13}} \, dx=-\frac {924 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 6 \, a b^{5} d^{5} e + 21 \, a^{2} b^{4} d^{4} e^{2} + 56 \, a^{3} b^{3} d^{3} e^{3} + 126 \, a^{4} b^{2} d^{2} e^{4} + 252 \, a^{5} b d e^{5} + 462 \, a^{6} e^{6} + 792 \, {\left (b^{6} d e^{5} + 6 \, a b^{5} e^{6}\right )} x^{5} + 495 \, {\left (b^{6} d^{2} e^{4} + 6 \, a b^{5} d e^{5} + 21 \, a^{2} b^{4} e^{6}\right )} x^{4} + 220 \, {\left (b^{6} d^{3} e^{3} + 6 \, a b^{5} d^{2} e^{4} + 21 \, a^{2} b^{4} d e^{5} + 56 \, a^{3} b^{3} e^{6}\right )} x^{3} + 66 \, {\left (b^{6} d^{4} e^{2} + 6 \, a b^{5} d^{3} e^{3} + 21 \, a^{2} b^{4} d^{2} e^{4} + 56 \, a^{3} b^{3} d e^{5} + 126 \, a^{4} b^{2} e^{6}\right )} x^{2} + 12 \, {\left (b^{6} d^{5} e + 6 \, a b^{5} d^{4} e^{2} + 21 \, a^{2} b^{4} d^{3} e^{3} + 56 \, a^{3} b^{3} d^{2} e^{4} + 126 \, a^{4} b^{2} d e^{5} + 252 \, a^{5} b e^{6}\right )} x}{5544 \, {\left (e^{19} x^{12} + 12 \, d e^{18} x^{11} + 66 \, d^{2} e^{17} x^{10} + 220 \, d^{3} e^{16} x^{9} + 495 \, d^{4} e^{15} x^{8} + 792 \, d^{5} e^{14} x^{7} + 924 \, d^{6} e^{13} x^{6} + 792 \, d^{7} e^{12} x^{5} + 495 \, d^{8} e^{11} x^{4} + 220 \, d^{9} e^{10} x^{3} + 66 \, d^{10} e^{9} x^{2} + 12 \, d^{11} e^{8} x + d^{12} e^{7}\right )}} \]
-1/5544*(924*b^6*e^6*x^6 + b^6*d^6 + 6*a*b^5*d^5*e + 21*a^2*b^4*d^4*e^2 + 56*a^3*b^3*d^3*e^3 + 126*a^4*b^2*d^2*e^4 + 252*a^5*b*d*e^5 + 462*a^6*e^6 + 792*(b^6*d*e^5 + 6*a*b^5*e^6)*x^5 + 495*(b^6*d^2*e^4 + 6*a*b^5*d*e^5 + 21 *a^2*b^4*e^6)*x^4 + 220*(b^6*d^3*e^3 + 6*a*b^5*d^2*e^4 + 21*a^2*b^4*d*e^5 + 56*a^3*b^3*e^6)*x^3 + 66*(b^6*d^4*e^2 + 6*a*b^5*d^3*e^3 + 21*a^2*b^4*d^2 *e^4 + 56*a^3*b^3*d*e^5 + 126*a^4*b^2*e^6)*x^2 + 12*(b^6*d^5*e + 6*a*b^5*d ^4*e^2 + 21*a^2*b^4*d^3*e^3 + 56*a^3*b^3*d^2*e^4 + 126*a^4*b^2*d*e^5 + 252 *a^5*b*e^6)*x)/(e^19*x^12 + 12*d*e^18*x^11 + 66*d^2*e^17*x^10 + 220*d^3*e^ 16*x^9 + 495*d^4*e^15*x^8 + 792*d^5*e^14*x^7 + 924*d^6*e^13*x^6 + 792*d^7* e^12*x^5 + 495*d^8*e^11*x^4 + 220*d^9*e^10*x^3 + 66*d^10*e^9*x^2 + 12*d^11 *e^8*x + d^12*e^7)
Exception generated. \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{13}} \, dx=\text {Exception raised: HeuristicGCDFailed} \]
Exception generated. \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{13}} \, dx=\text {Exception raised: ValueError} \]
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. 639 vs. \(2 (271) = 542\).
Time = 0.28 (sec) , antiderivative size = 639, normalized size of antiderivative = 1.77 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{13}} \, dx=\frac {b^{12} \mathrm {sgn}\left (b x + a\right )}{5544 \, {\left (b^{6} d^{6} e^{7} - 6 \, a b^{5} d^{5} e^{8} + 15 \, a^{2} b^{4} d^{4} e^{9} - 20 \, a^{3} b^{3} d^{3} e^{10} + 15 \, a^{4} b^{2} d^{2} e^{11} - 6 \, a^{5} b d e^{12} + a^{6} e^{13}\right )}} - \frac {924 \, b^{6} e^{6} x^{6} \mathrm {sgn}\left (b x + a\right ) + 792 \, b^{6} d e^{5} x^{5} \mathrm {sgn}\left (b x + a\right ) + 4752 \, a b^{5} e^{6} x^{5} \mathrm {sgn}\left (b x + a\right ) + 495 \, b^{6} d^{2} e^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + 2970 \, a b^{5} d e^{5} x^{4} \mathrm {sgn}\left (b x + a\right ) + 10395 \, a^{2} b^{4} e^{6} x^{4} \mathrm {sgn}\left (b x + a\right ) + 220 \, b^{6} d^{3} e^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 1320 \, a b^{5} d^{2} e^{4} x^{3} \mathrm {sgn}\left (b x + a\right ) + 4620 \, a^{2} b^{4} d e^{5} x^{3} \mathrm {sgn}\left (b x + a\right ) + 12320 \, a^{3} b^{3} e^{6} x^{3} \mathrm {sgn}\left (b x + a\right ) + 66 \, b^{6} d^{4} e^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 396 \, a b^{5} d^{3} e^{3} x^{2} \mathrm {sgn}\left (b x + a\right ) + 1386 \, a^{2} b^{4} d^{2} e^{4} x^{2} \mathrm {sgn}\left (b x + a\right ) + 3696 \, a^{3} b^{3} d e^{5} x^{2} \mathrm {sgn}\left (b x + a\right ) + 8316 \, a^{4} b^{2} e^{6} x^{2} \mathrm {sgn}\left (b x + a\right ) + 12 \, b^{6} d^{5} e x \mathrm {sgn}\left (b x + a\right ) + 72 \, a b^{5} d^{4} e^{2} x \mathrm {sgn}\left (b x + a\right ) + 252 \, a^{2} b^{4} d^{3} e^{3} x \mathrm {sgn}\left (b x + a\right ) + 672 \, a^{3} b^{3} d^{2} e^{4} x \mathrm {sgn}\left (b x + a\right ) + 1512 \, a^{4} b^{2} d e^{5} x \mathrm {sgn}\left (b x + a\right ) + 3024 \, a^{5} b e^{6} x \mathrm {sgn}\left (b x + a\right ) + b^{6} d^{6} \mathrm {sgn}\left (b x + a\right ) + 6 \, a b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) + 21 \, a^{2} b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 56 \, a^{3} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 126 \, a^{4} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 252 \, a^{5} b d e^{5} \mathrm {sgn}\left (b x + a\right ) + 462 \, a^{6} e^{6} \mathrm {sgn}\left (b x + a\right )}{5544 \, {\left (e x + d\right )}^{12} e^{7}} \]
1/5544*b^12*sgn(b*x + a)/(b^6*d^6*e^7 - 6*a*b^5*d^5*e^8 + 15*a^2*b^4*d^4*e ^9 - 20*a^3*b^3*d^3*e^10 + 15*a^4*b^2*d^2*e^11 - 6*a^5*b*d*e^12 + a^6*e^13 ) - 1/5544*(924*b^6*e^6*x^6*sgn(b*x + a) + 792*b^6*d*e^5*x^5*sgn(b*x + a) + 4752*a*b^5*e^6*x^5*sgn(b*x + a) + 495*b^6*d^2*e^4*x^4*sgn(b*x + a) + 297 0*a*b^5*d*e^5*x^4*sgn(b*x + a) + 10395*a^2*b^4*e^6*x^4*sgn(b*x + a) + 220* b^6*d^3*e^3*x^3*sgn(b*x + a) + 1320*a*b^5*d^2*e^4*x^3*sgn(b*x + a) + 4620* a^2*b^4*d*e^5*x^3*sgn(b*x + a) + 12320*a^3*b^3*e^6*x^3*sgn(b*x + a) + 66*b ^6*d^4*e^2*x^2*sgn(b*x + a) + 396*a*b^5*d^3*e^3*x^2*sgn(b*x + a) + 1386*a^ 2*b^4*d^2*e^4*x^2*sgn(b*x + a) + 3696*a^3*b^3*d*e^5*x^2*sgn(b*x + a) + 831 6*a^4*b^2*e^6*x^2*sgn(b*x + a) + 12*b^6*d^5*e*x*sgn(b*x + a) + 72*a*b^5*d^ 4*e^2*x*sgn(b*x + a) + 252*a^2*b^4*d^3*e^3*x*sgn(b*x + a) + 672*a^3*b^3*d^ 2*e^4*x*sgn(b*x + a) + 1512*a^4*b^2*d*e^5*x*sgn(b*x + a) + 3024*a^5*b*e^6* x*sgn(b*x + a) + b^6*d^6*sgn(b*x + a) + 6*a*b^5*d^5*e*sgn(b*x + a) + 21*a^ 2*b^4*d^4*e^2*sgn(b*x + a) + 56*a^3*b^3*d^3*e^3*sgn(b*x + a) + 126*a^4*b^2 *d^2*e^4*sgn(b*x + a) + 252*a^5*b*d*e^5*sgn(b*x + a) + 462*a^6*e^6*sgn(b*x + a))/((e*x + d)^12*e^7)
Time = 11.13 (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)^{13}} \, 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}{11\,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}{11\,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}{11\,e^7}-\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{11\,e^3}-\frac {b^5\,\left (6\,a\,e-b\,d\right )}{11\,e^3}\right )}{e}+\frac {b^4\,\left (15\,a^2\,e^2-6\,a\,b\,d\,e+b^2\,d^2\right )}{11\,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 )}^{11}}-\frac {\left (\frac {15\,a^2\,b^4\,e^2-24\,a\,b^5\,d\,e+10\,b^6\,d^2}{8\,e^7}+\frac {d\,\left (\frac {b^6\,d}{8\,e^6}-\frac {b^5\,\left (3\,a\,e-2\,b\,d\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 )}^8}-\frac {\left (\frac {a^6}{12\,e}-\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {a\,b^5}{2\,e}-\frac {b^6\,d}{12\,e^2}\right )}{e}-\frac {5\,a^2\,b^4}{4\,e}\right )}{e}+\frac {5\,a^3\,b^3}{3\,e}\right )}{e}-\frac {5\,a^4\,b^2}{4\,e}\right )}{e}+\frac {a^5\,b}{2\,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 )}^{12}}-\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}{10\,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}{10\,e^7}+\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{10\,e^4}-\frac {b^5\,\left (3\,a\,e-b\,d\right )}{5\,e^4}\right )}{e}+\frac {3\,b^4\,\left (5\,a^2\,e^2-4\,a\,b\,d\,e+b^2\,d^2\right )}{10\,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 )}^{10}}+\frac {\left (\frac {5\,b^6\,d-6\,a\,b^5\,e}{7\,e^7}+\frac {b^6\,d}{7\,e^7}\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 {-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}{9\,e^7}+\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{9\,e^5}-\frac {b^5\,\left (2\,a\,e-b\,d\right )}{3\,e^5}\right )}{e}+\frac {b^4\,\left (5\,a^2\,e^2-6\,a\,b\,d\,e+2\,b^2\,d^2\right )}{3\,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 )}^9}-\frac {b^6\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{6\,e^7\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^6} \]
(((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)/(11*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)/(11*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)/(11*e^7) - (d*((d*((b^6*d)/(11*e^3) - (b^5*(6*a*e - b*d))/(11*e^3)))/e + (b^4*(15* a^2*e^2 + b^2*d^2 - 6*a*b*d*e))/(11*e^4)))/e))/e))/e)*(a^2 + b^2*x^2 + 2*a *b*x)^(1/2))/((a + b*x)*(d + e*x)^11) - (((10*b^6*d^2 + 15*a^2*b^4*e^2 - 2 4*a*b^5*d*e)/(8*e^7) + (d*((b^6*d)/(8*e^6) - (b^5*(3*a*e - 2*b*d))/(4*e^6) ))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^8) - ((a^6/(12 *e) - (d*((d*((d*((d*((d*((a*b^5)/(2*e) - (b^6*d)/(12*e^2)))/e - (5*a^2*b^ 4)/(4*e)))/e + (5*a^3*b^3)/(3*e)))/e - (5*a^4*b^2)/(4*e)))/e + (a^5*b)/(2* e)))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^12) - (((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)/(10*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)/(10*e^7) + (d*((d*((b^6*d)/(10*e^4) - (b^5*(3*a*e - b*d) )/(5*e^4)))/e + (3*b^4*(5*a^2*e^2 + b^2*d^2 - 4*a*b*d*e))/(10*e^5)))/e))/e )*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^10) + (((5*b^6*d - 6*a*b^5*e)/(7*e^7) + (b^6*d)/(7*e^7))*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(( a + b*x)*(d + e*x)^7) + (((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)/(9*e^7) + (d*((d*((b^6*d)/(9*e^5) - (b^5*(2*a*e - b*d...