Integrand size = 28, antiderivative size = 308 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{12}} \, dx=\frac {(b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^6 (a+b x) (d+e x)^{11}}-\frac {b (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^6 (a+b x) (d+e x)^{10}}+\frac {10 b^2 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^6 (a+b x) (d+e x)^9}-\frac {5 b^3 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^6 (a+b x) (d+e x)^8}+\frac {5 b^4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^6 (a+b x) (d+e x)^7}-\frac {b^5 \sqrt {a^2+2 a b x+b^2 x^2}}{6 e^6 (a+b x) (d+e x)^6} \]
1/11*(-a*e+b*d)^5*((b*x+a)^2)^(1/2)/e^6/(b*x+a)/(e*x+d)^11-1/2*b*(-a*e+b*d )^4*((b*x+a)^2)^(1/2)/e^6/(b*x+a)/(e*x+d)^10+10/9*b^2*(-a*e+b*d)^3*((b*x+a )^2)^(1/2)/e^6/(b*x+a)/(e*x+d)^9-5/4*b^3*(-a*e+b*d)^2*((b*x+a)^2)^(1/2)/e^ 6/(b*x+a)/(e*x+d)^8+5/7*b^4*(-a*e+b*d)*((b*x+a)^2)^(1/2)/e^6/(b*x+a)/(e*x+ d)^7-1/6*b^5*((b*x+a)^2)^(1/2)/e^6/(b*x+a)/(e*x+d)^6
Time = 1.05 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.72 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{12}} \, dx=-\frac {\sqrt {(a+b x)^2} \left (252 a^5 e^5+126 a^4 b e^4 (d+11 e x)+56 a^3 b^2 e^3 \left (d^2+11 d e x+55 e^2 x^2\right )+21 a^2 b^3 e^2 \left (d^3+11 d^2 e x+55 d e^2 x^2+165 e^3 x^3\right )+6 a b^4 e \left (d^4+11 d^3 e x+55 d^2 e^2 x^2+165 d e^3 x^3+330 e^4 x^4\right )+b^5 \left (d^5+11 d^4 e x+55 d^3 e^2 x^2+165 d^2 e^3 x^3+330 d e^4 x^4+462 e^5 x^5\right )\right )}{2772 e^6 (a+b x) (d+e x)^{11}} \]
-1/2772*(Sqrt[(a + b*x)^2]*(252*a^5*e^5 + 126*a^4*b*e^4*(d + 11*e*x) + 56* a^3*b^2*e^3*(d^2 + 11*d*e*x + 55*e^2*x^2) + 21*a^2*b^3*e^2*(d^3 + 11*d^2*e *x + 55*d*e^2*x^2 + 165*e^3*x^3) + 6*a*b^4*e*(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) + b^5*(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)))/(e^6*(a + b*x)*(d + e*x)^11)
Time = 0.34 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.56, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1102, 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.
| \(\displaystyle \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{12}} \, dx\) |
| \(\Big \downarrow \) 1102 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {b^5 (a+b x)^5}{(d+e x)^{12}}dx}{b^5 (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x)^5}{(d+e x)^{12}}dx}{a+b x}\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {b^5}{e^5 (d+e x)^7}-\frac {5 (b d-a e) b^4}{e^5 (d+e x)^8}+\frac {10 (b d-a e)^2 b^3}{e^5 (d+e x)^9}-\frac {10 (b d-a e)^3 b^2}{e^5 (d+e x)^{10}}+\frac {5 (b d-a e)^4 b}{e^5 (d+e x)^{11}}+\frac {(a e-b d)^5}{e^5 (d+e x)^{12}}\right )dx}{a+b x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {5 b^4 (b d-a e)}{7 e^6 (d+e x)^7}-\frac {5 b^3 (b d-a e)^2}{4 e^6 (d+e x)^8}+\frac {10 b^2 (b d-a e)^3}{9 e^6 (d+e x)^9}-\frac {b (b d-a e)^4}{2 e^6 (d+e x)^{10}}+\frac {(b d-a e)^5}{11 e^6 (d+e x)^{11}}-\frac {b^5}{6 e^6 (d+e x)^6}\right )}{a+b x}\) |
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*((b*d - a*e)^5/(11*e^6*(d + e*x)^11) - (b*( b*d - a*e)^4)/(2*e^6*(d + e*x)^10) + (10*b^2*(b*d - a*e)^3)/(9*e^6*(d + e* x)^9) - (5*b^3*(b*d - a*e)^2)/(4*e^6*(d + e*x)^8) + (5*b^4*(b*d - a*e))/(7 *e^6*(d + e*x)^7) - b^5/(6*e^6*(d + e*x)^6)))/(a + b*x)
3.16.86.3.1 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_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*F racPart[p])) Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0]
Time = 10.12 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.85
| method | result | size |
| risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {b^{5} x^{5}}{6 e}-\frac {5 b^{4} \left (6 a e +b d \right ) x^{4}}{42 e^{2}}-\frac {5 b^{3} \left (21 a^{2} e^{2}+6 a b d e +b^{2} d^{2}\right ) x^{3}}{84 e^{3}}-\frac {5 b^{2} \left (56 a^{3} e^{3}+21 a^{2} b d \,e^{2}+6 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) x^{2}}{252 e^{4}}-\frac {b \left (126 e^{4} a^{4}+56 b \,e^{3} d \,a^{3}+21 b^{2} e^{2} d^{2} a^{2}+6 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) x}{252 e^{5}}-\frac {252 a^{5} e^{5}+126 a^{4} b d \,e^{4}+56 a^{3} b^{2} d^{2} e^{3}+21 a^{2} b^{3} d^{3} e^{2}+6 a \,b^{4} d^{4} e +b^{5} d^{5}}{2772 e^{6}}\right )}{\left (b x +a \right ) \left (e x +d \right )^{11}}\) | \(262\) |
| gosper | \(-\frac {\left (462 x^{5} e^{5} b^{5}+1980 x^{4} a \,b^{4} e^{5}+330 x^{4} b^{5} d \,e^{4}+3465 x^{3} a^{2} b^{3} e^{5}+990 x^{3} a \,b^{4} d \,e^{4}+165 x^{3} b^{5} d^{2} e^{3}+3080 x^{2} a^{3} b^{2} e^{5}+1155 x^{2} a^{2} b^{3} d \,e^{4}+330 x^{2} a \,b^{4} d^{2} e^{3}+55 x^{2} b^{5} d^{3} e^{2}+1386 a^{4} b \,e^{5} x +616 a^{3} b^{2} d \,e^{4} x +231 x \,a^{2} b^{3} d^{2} e^{3}+66 x a \,b^{4} d^{3} e^{2}+11 b^{5} d^{4} e x +252 a^{5} e^{5}+126 a^{4} b d \,e^{4}+56 a^{3} b^{2} d^{2} e^{3}+21 a^{2} b^{3} d^{3} e^{2}+6 a \,b^{4} d^{4} e +b^{5} d^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{2772 e^{6} \left (e x +d \right )^{11} \left (b x +a \right )^{5}}\) | \(288\) |
| default | \(-\frac {\left (462 x^{5} e^{5} b^{5}+1980 x^{4} a \,b^{4} e^{5}+330 x^{4} b^{5} d \,e^{4}+3465 x^{3} a^{2} b^{3} e^{5}+990 x^{3} a \,b^{4} d \,e^{4}+165 x^{3} b^{5} d^{2} e^{3}+3080 x^{2} a^{3} b^{2} e^{5}+1155 x^{2} a^{2} b^{3} d \,e^{4}+330 x^{2} a \,b^{4} d^{2} e^{3}+55 x^{2} b^{5} d^{3} e^{2}+1386 a^{4} b \,e^{5} x +616 a^{3} b^{2} d \,e^{4} x +231 x \,a^{2} b^{3} d^{2} e^{3}+66 x a \,b^{4} d^{3} e^{2}+11 b^{5} d^{4} e x +252 a^{5} e^{5}+126 a^{4} b d \,e^{4}+56 a^{3} b^{2} d^{2} e^{3}+21 a^{2} b^{3} d^{3} e^{2}+6 a \,b^{4} d^{4} e +b^{5} d^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{2772 e^{6} \left (e x +d \right )^{11} \left (b x +a \right )^{5}}\) | \(288\) |
((b*x+a)^2)^(1/2)/(b*x+a)*(-1/6*b^5/e*x^5-5/42*b^4/e^2*(6*a*e+b*d)*x^4-5/8 4*b^3/e^3*(21*a^2*e^2+6*a*b*d*e+b^2*d^2)*x^3-5/252*b^2/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^2-1/252*b/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-1/2772/e^6*(252*a^5*e^5+1 26*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 ))/(e*x+d)^11
Time = 0.28 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.20 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{12}} \, dx=-\frac {462 \, b^{5} e^{5} x^{5} + b^{5} d^{5} + 6 \, a b^{4} d^{4} e + 21 \, a^{2} b^{3} d^{3} e^{2} + 56 \, a^{3} b^{2} d^{2} e^{3} + 126 \, a^{4} b d e^{4} + 252 \, a^{5} e^{5} + 330 \, {\left (b^{5} d e^{4} + 6 \, a b^{4} e^{5}\right )} x^{4} + 165 \, {\left (b^{5} d^{2} e^{3} + 6 \, a b^{4} d e^{4} + 21 \, a^{2} b^{3} e^{5}\right )} x^{3} + 55 \, {\left (b^{5} d^{3} e^{2} + 6 \, a b^{4} d^{2} e^{3} + 21 \, a^{2} b^{3} d e^{4} + 56 \, a^{3} b^{2} e^{5}\right )} x^{2} + 11 \, {\left (b^{5} d^{4} e + 6 \, a b^{4} d^{3} e^{2} + 21 \, a^{2} b^{3} d^{2} e^{3} + 56 \, a^{3} b^{2} d e^{4} + 126 \, a^{4} b e^{5}\right )} x}{2772 \, {\left (e^{17} x^{11} + 11 \, d e^{16} x^{10} + 55 \, d^{2} e^{15} x^{9} + 165 \, d^{3} e^{14} x^{8} + 330 \, d^{4} e^{13} x^{7} + 462 \, d^{5} e^{12} x^{6} + 462 \, d^{6} e^{11} x^{5} + 330 \, d^{7} e^{10} x^{4} + 165 \, d^{8} e^{9} x^{3} + 55 \, d^{9} e^{8} x^{2} + 11 \, d^{10} e^{7} x + d^{11} e^{6}\right )}} \]
-1/2772*(462*b^5*e^5*x^5 + b^5*d^5 + 6*a*b^4*d^4*e + 21*a^2*b^3*d^3*e^2 + 56*a^3*b^2*d^2*e^3 + 126*a^4*b*d*e^4 + 252*a^5*e^5 + 330*(b^5*d*e^4 + 6*a* b^4*e^5)*x^4 + 165*(b^5*d^2*e^3 + 6*a*b^4*d*e^4 + 21*a^2*b^3*e^5)*x^3 + 55 *(b^5*d^3*e^2 + 6*a*b^4*d^2*e^3 + 21*a^2*b^3*d*e^4 + 56*a^3*b^2*e^5)*x^2 + 11*(b^5*d^4*e + 6*a*b^4*d^3*e^2 + 21*a^2*b^3*d^2*e^3 + 56*a^3*b^2*d*e^4 + 126*a^4*b*e^5)*x)/(e^17*x^11 + 11*d*e^16*x^10 + 55*d^2*e^15*x^9 + 165*d^3 *e^14*x^8 + 330*d^4*e^13*x^7 + 462*d^5*e^12*x^6 + 462*d^6*e^11*x^5 + 330*d ^7*e^10*x^4 + 165*d^8*e^9*x^3 + 55*d^9*e^8*x^2 + 11*d^10*e^7*x + d^11*e^6)
Exception generated. \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{12}} \, dx=\text {Exception raised: HeuristicGCDFailed} \]
Exception generated. \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{12}} \, dx=\text {Exception raised: ValueError} \]
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. 493 vs. \(2 (230) = 460\).
Time = 0.28 (sec) , antiderivative size = 493, normalized size of antiderivative = 1.60 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{12}} \, dx=\frac {b^{11} \mathrm {sgn}\left (b x + a\right )}{2772 \, {\left (b^{6} d^{6} e^{6} - 6 \, a b^{5} d^{5} e^{7} + 15 \, a^{2} b^{4} d^{4} e^{8} - 20 \, a^{3} b^{3} d^{3} e^{9} + 15 \, a^{4} b^{2} d^{2} e^{10} - 6 \, a^{5} b d e^{11} + a^{6} e^{12}\right )}} - \frac {462 \, b^{5} e^{5} x^{5} \mathrm {sgn}\left (b x + a\right ) + 330 \, b^{5} d e^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + 1980 \, a b^{4} e^{5} x^{4} \mathrm {sgn}\left (b x + a\right ) + 165 \, b^{5} d^{2} e^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 990 \, a b^{4} d e^{4} x^{3} \mathrm {sgn}\left (b x + a\right ) + 3465 \, a^{2} b^{3} e^{5} x^{3} \mathrm {sgn}\left (b x + a\right ) + 55 \, b^{5} d^{3} e^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 330 \, a b^{4} d^{2} e^{3} x^{2} \mathrm {sgn}\left (b x + a\right ) + 1155 \, a^{2} b^{3} d e^{4} x^{2} \mathrm {sgn}\left (b x + a\right ) + 3080 \, a^{3} b^{2} e^{5} x^{2} \mathrm {sgn}\left (b x + a\right ) + 11 \, b^{5} d^{4} e x \mathrm {sgn}\left (b x + a\right ) + 66 \, a b^{4} d^{3} e^{2} x \mathrm {sgn}\left (b x + a\right ) + 231 \, a^{2} b^{3} d^{2} e^{3} x \mathrm {sgn}\left (b x + a\right ) + 616 \, a^{3} b^{2} d e^{4} x \mathrm {sgn}\left (b x + a\right ) + 1386 \, a^{4} b e^{5} x \mathrm {sgn}\left (b x + a\right ) + b^{5} d^{5} \mathrm {sgn}\left (b x + a\right ) + 6 \, a b^{4} d^{4} e \mathrm {sgn}\left (b x + a\right ) + 21 \, a^{2} b^{3} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + 56 \, a^{3} b^{2} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 126 \, a^{4} b d e^{4} \mathrm {sgn}\left (b x + a\right ) + 252 \, a^{5} e^{5} \mathrm {sgn}\left (b x + a\right )}{2772 \, {\left (e x + d\right )}^{11} e^{6}} \]
1/2772*b^11*sgn(b*x + a)/(b^6*d^6*e^6 - 6*a*b^5*d^5*e^7 + 15*a^2*b^4*d^4*e ^8 - 20*a^3*b^3*d^3*e^9 + 15*a^4*b^2*d^2*e^10 - 6*a^5*b*d*e^11 + a^6*e^12) - 1/2772*(462*b^5*e^5*x^5*sgn(b*x + a) + 330*b^5*d*e^4*x^4*sgn(b*x + a) + 1980*a*b^4*e^5*x^4*sgn(b*x + a) + 165*b^5*d^2*e^3*x^3*sgn(b*x + a) + 990* a*b^4*d*e^4*x^3*sgn(b*x + a) + 3465*a^2*b^3*e^5*x^3*sgn(b*x + a) + 55*b^5* d^3*e^2*x^2*sgn(b*x + a) + 330*a*b^4*d^2*e^3*x^2*sgn(b*x + a) + 1155*a^2*b ^3*d*e^4*x^2*sgn(b*x + a) + 3080*a^3*b^2*e^5*x^2*sgn(b*x + a) + 11*b^5*d^4 *e*x*sgn(b*x + a) + 66*a*b^4*d^3*e^2*x*sgn(b*x + a) + 231*a^2*b^3*d^2*e^3* x*sgn(b*x + a) + 616*a^3*b^2*d*e^4*x*sgn(b*x + a) + 1386*a^4*b*e^5*x*sgn(b *x + a) + b^5*d^5*sgn(b*x + a) + 6*a*b^4*d^4*e*sgn(b*x + a) + 21*a^2*b^3*d ^3*e^2*sgn(b*x + a) + 56*a^3*b^2*d^2*e^3*sgn(b*x + a) + 126*a^4*b*d*e^4*sg n(b*x + a) + 252*a^5*e^5*sgn(b*x + a))/((e*x + d)^11*e^6)
Time = 9.63 (sec) , antiderivative size = 687, normalized size of antiderivative = 2.23 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{12}} \, dx=\frac {\left (\frac {4\,b^5\,d-5\,a\,b^4\,e}{7\,e^6}+\frac {b^5\,d}{7\,e^6}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^7}-\frac {\left (\frac {5\,a^4\,b\,e^4-10\,a^3\,b^2\,d\,e^3+10\,a^2\,b^3\,d^2\,e^2-5\,a\,b^4\,d^3\,e+b^5\,d^4}{10\,e^6}+\frac {d\,\left (\frac {-10\,a^3\,b^2\,e^4+10\,a^2\,b^3\,d\,e^3-5\,a\,b^4\,d^2\,e^2+b^5\,d^3\,e}{10\,e^6}+\frac {d\,\left (\frac {d\,\left (\frac {b^5\,d}{10\,e^3}-\frac {b^4\,\left (5\,a\,e-b\,d\right )}{10\,e^3}\right )}{e}+\frac {b^3\,\left (10\,a^2\,e^2-5\,a\,b\,d\,e+b^2\,d^2\right )}{10\,e^4}\right )}{e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^{10}}-\frac {\left (\frac {10\,a^2\,b^3\,e^2-15\,a\,b^4\,d\,e+6\,b^5\,d^2}{8\,e^6}+\frac {d\,\left (\frac {b^5\,d}{8\,e^5}-\frac {b^4\,\left (5\,a\,e-3\,b\,d\right )}{8\,e^5}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^8}-\frac {\left (\frac {a^5}{11\,e}-\frac {d\,\left (\frac {5\,a^4\,b}{11\,e}-\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {5\,a\,b^4}{11\,e}-\frac {b^5\,d}{11\,e^2}\right )}{e}-\frac {10\,a^2\,b^3}{11\,e}\right )}{e}+\frac {10\,a^3\,b^2}{11\,e}\right )}{e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^{11}}+\frac {\left (\frac {-10\,a^3\,b^2\,e^3+20\,a^2\,b^3\,d\,e^2-15\,a\,b^4\,d^2\,e+4\,b^5\,d^3}{9\,e^6}+\frac {d\,\left (\frac {d\,\left (\frac {b^5\,d}{9\,e^4}-\frac {b^4\,\left (5\,a\,e-2\,b\,d\right )}{9\,e^4}\right )}{e}+\frac {b^3\,\left (10\,a^2\,e^2-10\,a\,b\,d\,e+3\,b^2\,d^2\right )}{9\,e^5}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^9}-\frac {b^5\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{6\,e^6\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^6} \]
(((4*b^5*d - 5*a*b^4*e)/(7*e^6) + (b^5*d)/(7*e^6))*(a^2 + b^2*x^2 + 2*a*b* x)^(1/2))/((a + b*x)*(d + e*x)^7) - (((b^5*d^4 + 5*a^4*b*e^4 - 10*a^3*b^2* d*e^3 + 10*a^2*b^3*d^2*e^2 - 5*a*b^4*d^3*e)/(10*e^6) + (d*((b^5*d^3*e - 10 *a^3*b^2*e^4 - 5*a*b^4*d^2*e^2 + 10*a^2*b^3*d*e^3)/(10*e^6) + (d*((d*((b^5 *d)/(10*e^3) - (b^4*(5*a*e - b*d))/(10*e^3)))/e + (b^3*(10*a^2*e^2 + b^2*d ^2 - 5*a*b*d*e))/(10*e^4)))/e))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^10) - (((6*b^5*d^2 + 10*a^2*b^3*e^2 - 15*a*b^4*d*e)/(8*e^6) + (d*((b^5*d)/(8*e^5) - (b^4*(5*a*e - 3*b*d))/(8*e^5)))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^8) - ((a^5/(11*e) - (d*((5*a^4*b)/ (11*e) - (d*((d*((d*((5*a*b^4)/(11*e) - (b^5*d)/(11*e^2)))/e - (10*a^2*b^3 )/(11*e)))/e + (10*a^3*b^2)/(11*e)))/e))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2 ))/((a + b*x)*(d + e*x)^11) + (((4*b^5*d^3 - 10*a^3*b^2*e^3 + 20*a^2*b^3*d *e^2 - 15*a*b^4*d^2*e)/(9*e^6) + (d*((d*((b^5*d)/(9*e^4) - (b^4*(5*a*e - 2 *b*d))/(9*e^4)))/e + (b^3*(10*a^2*e^2 + 3*b^2*d^2 - 10*a*b*d*e))/(9*e^5))) /e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^9) - (b^5*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(6*e^6*(a + b*x)*(d + e*x)^6)