Integrand size = 33, antiderivative size = 41 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^8} \, dx=\frac {\left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{7 (b d-a e) (d+e x)^7} \]
Leaf count is larger than twice the leaf count of optimal. \(289\) vs. \(2(41)=82\).
Time = 1.07 (sec) , antiderivative size = 289, normalized size of antiderivative = 7.05 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^8} \, dx=-\frac {\sqrt {(a+b x)^2} \left (a^6 e^6+a^5 b e^5 (d+7 e x)+a^4 b^2 e^4 \left (d^2+7 d e x+21 e^2 x^2\right )+a^3 b^3 e^3 \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )+a^2 b^4 e^2 \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )+a b^5 e \left (d^5+7 d^4 e x+21 d^3 e^2 x^2+35 d^2 e^3 x^3+35 d e^4 x^4+21 e^5 x^5\right )+b^6 \left (d^6+7 d^5 e x+21 d^4 e^2 x^2+35 d^3 e^3 x^3+35 d^2 e^4 x^4+21 d e^5 x^5+7 e^6 x^6\right )\right )}{7 e^7 (a+b x) (d+e x)^7} \]
-1/7*(Sqrt[(a + b*x)^2]*(a^6*e^6 + a^5*b*e^5*(d + 7*e*x) + a^4*b^2*e^4*(d^ 2 + 7*d*e*x + 21*e^2*x^2) + a^3*b^3*e^3*(d^3 + 7*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3) + a^2*b^4*e^2*(d^4 + 7*d^3*e*x + 21*d^2*e^2*x^2 + 35*d*e^3*x^3 + 35*e^4*x^4) + a*b^5*e*(d^5 + 7*d^4*e*x + 21*d^3*e^2*x^2 + 35*d^2*e^3*x^ 3 + 35*d*e^4*x^4 + 21*e^5*x^5) + b^6*(d^6 + 7*d^5*e*x + 21*d^4*e^2*x^2 + 3 5*d^3*e^3*x^3 + 35*d^2*e^4*x^4 + 21*d*e^5*x^5 + 7*e^6*x^6)))/(e^7*(a + b*x )*(d + e*x)^7)
Time = 0.18 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {1185}
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.
| \(\displaystyle \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^8} \, dx\) |
| \(\Big \downarrow \) 1185 |
| \(\displaystyle \frac {\left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{7 (d+e x)^7 (b d-a e)}\) |
3.21.5.3.1 Defintions of rubi rules used
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_) + (g_.)*(x_))*((a_) + (b_.)*(x_) + (c_ .)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-f)*g*(d + e*x)^(m + 1)*((a + b*x + c* x^2)^(p + 1)/(b*(p + 1)*(e*f - d*g))), x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[b^2 - 4*a*c, 0] && EqQ[m + 2*p + 3, 0] && EqQ[2*c*f - b*g, 0 ]
Leaf count of result is larger than twice the leaf count of optimal. \(329\) vs. \(2(37)=74\).
Time = 2.09 (sec) , antiderivative size = 330, normalized size of antiderivative = 8.05
| method | result | size |
| risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {b^{6} x^{6}}{e}-\frac {3 b^{5} \left (a e +b d \right ) x^{5}}{e^{2}}-\frac {5 b^{4} \left (e^{2} a^{2}+a b d e +b^{2} d^{2}\right ) x^{4}}{e^{3}}-\frac {5 b^{3} \left (a^{3} e^{3}+a^{2} b d \,e^{2}+a \,b^{2} d^{2} e +b^{3} d^{3}\right ) x^{3}}{e^{4}}-\frac {3 b^{2} \left (e^{4} a^{4}+b d \,e^{3} a^{3}+b^{2} d^{2} e^{2} a^{2}+b^{3} d^{3} e a +b^{4} d^{4}\right ) x^{2}}{e^{5}}-\frac {b \left (e^{5} a^{5}+b d \,e^{4} a^{4}+b^{2} d^{2} e^{3} a^{3}+b^{3} d^{3} e^{2} a^{2}+b^{4} d^{4} e a +b^{5} d^{5}\right ) x}{e^{6}}-\frac {e^{6} a^{6}+b d \,e^{5} a^{5}+b^{2} d^{2} e^{4} a^{4}+b^{3} d^{3} e^{3} a^{3}+b^{4} d^{4} e^{2} a^{2}+b^{5} d^{5} e a +b^{6} d^{6}}{7 e^{7}}\right )}{\left (b x +a \right ) \left (e x +d \right )^{7}}\) | \(330\) |
| gosper | \(-\frac {\left (7 b^{6} e^{6} x^{6}+21 a \,b^{5} e^{6} x^{5}+21 b^{6} d \,e^{5} x^{5}+35 a^{2} b^{4} e^{6} x^{4}+35 a \,b^{5} d \,e^{5} x^{4}+35 b^{6} d^{2} e^{4} x^{4}+35 a^{3} b^{3} e^{6} x^{3}+35 a^{2} b^{4} d \,e^{5} x^{3}+35 a \,b^{5} d^{2} e^{4} x^{3}+35 b^{6} d^{3} e^{3} x^{3}+21 a^{4} b^{2} e^{6} x^{2}+21 a^{3} b^{3} d \,e^{5} x^{2}+21 a^{2} b^{4} d^{2} e^{4} x^{2}+21 a \,b^{5} d^{3} e^{3} x^{2}+21 b^{6} d^{4} e^{2} x^{2}+7 a^{5} b \,e^{6} x +7 a^{4} b^{2} d \,e^{5} x +7 a^{3} b^{3} d^{2} e^{4} x +7 a^{2} b^{4} d^{3} e^{3} x +7 a \,b^{5} d^{4} e^{2} x +7 b^{6} d^{5} e x +e^{6} a^{6}+b d \,e^{5} a^{5}+b^{2} d^{2} e^{4} a^{4}+b^{3} d^{3} e^{3} a^{3}+b^{4} d^{4} e^{2} a^{2}+b^{5} d^{5} e a +b^{6} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{7 \left (e x +d \right )^{7} e^{7} \left (b x +a \right )^{5}}\) | \(386\) |
| default | \(-\frac {\left (7 b^{6} e^{6} x^{6}+21 a \,b^{5} e^{6} x^{5}+21 b^{6} d \,e^{5} x^{5}+35 a^{2} b^{4} e^{6} x^{4}+35 a \,b^{5} d \,e^{5} x^{4}+35 b^{6} d^{2} e^{4} x^{4}+35 a^{3} b^{3} e^{6} x^{3}+35 a^{2} b^{4} d \,e^{5} x^{3}+35 a \,b^{5} d^{2} e^{4} x^{3}+35 b^{6} d^{3} e^{3} x^{3}+21 a^{4} b^{2} e^{6} x^{2}+21 a^{3} b^{3} d \,e^{5} x^{2}+21 a^{2} b^{4} d^{2} e^{4} x^{2}+21 a \,b^{5} d^{3} e^{3} x^{2}+21 b^{6} d^{4} e^{2} x^{2}+7 a^{5} b \,e^{6} x +7 a^{4} b^{2} d \,e^{5} x +7 a^{3} b^{3} d^{2} e^{4} x +7 a^{2} b^{4} d^{3} e^{3} x +7 a \,b^{5} d^{4} e^{2} x +7 b^{6} d^{5} e x +e^{6} a^{6}+b d \,e^{5} a^{5}+b^{2} d^{2} e^{4} a^{4}+b^{3} d^{3} e^{3} a^{3}+b^{4} d^{4} e^{2} a^{2}+b^{5} d^{5} e a +b^{6} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{7 \left (e x +d \right )^{7} e^{7} \left (b x +a \right )^{5}}\) | \(386\) |
((b*x+a)^2)^(1/2)/(b*x+a)*(-b^6/e*x^6-3*b^5*(a*e+b*d)/e^2*x^5-5*b^4*(a^2*e ^2+a*b*d*e+b^2*d^2)/e^3*x^4-5*b^3*(a^3*e^3+a^2*b*d*e^2+a*b^2*d^2*e+b^3*d^3 )/e^4*x^3-3*b^2*(a^4*e^4+a^3*b*d*e^3+a^2*b^2*d^2*e^2+a*b^3*d^3*e+b^4*d^4)/ e^5*x^2-b*(a^5*e^5+a^4*b*d*e^4+a^3*b^2*d^2*e^3+a^2*b^3*d^3*e^2+a*b^4*d^4*e +b^5*d^5)/e^6*x-1/7*(a^6*e^6+a^5*b*d*e^5+a^4*b^2*d^2*e^4+a^3*b^3*d^3*e^3+a ^2*b^4*d^4*e^2+a*b^5*d^5*e+b^6*d^6)/e^7)/(e*x+d)^7
Leaf count of result is larger than twice the leaf count of optimal. 398 vs. \(2 (37) = 74\).
Time = 0.33 (sec) , antiderivative size = 398, normalized size of antiderivative = 9.71 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^8} \, dx=-\frac {7 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + a b^{5} d^{5} e + a^{2} b^{4} d^{4} e^{2} + a^{3} b^{3} d^{3} e^{3} + a^{4} b^{2} d^{2} e^{4} + a^{5} b d e^{5} + a^{6} e^{6} + 21 \, {\left (b^{6} d e^{5} + a b^{5} e^{6}\right )} x^{5} + 35 \, {\left (b^{6} d^{2} e^{4} + a b^{5} d e^{5} + a^{2} b^{4} e^{6}\right )} x^{4} + 35 \, {\left (b^{6} d^{3} e^{3} + a b^{5} d^{2} e^{4} + a^{2} b^{4} d e^{5} + a^{3} b^{3} e^{6}\right )} x^{3} + 21 \, {\left (b^{6} d^{4} e^{2} + a b^{5} d^{3} e^{3} + a^{2} b^{4} d^{2} e^{4} + a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{2} + 7 \, {\left (b^{6} d^{5} e + a b^{5} d^{4} e^{2} + a^{2} b^{4} d^{3} e^{3} + a^{3} b^{3} d^{2} e^{4} + a^{4} b^{2} d e^{5} + a^{5} b e^{6}\right )} x}{7 \, {\left (e^{14} x^{7} + 7 \, d e^{13} x^{6} + 21 \, d^{2} e^{12} x^{5} + 35 \, d^{3} e^{11} x^{4} + 35 \, d^{4} e^{10} x^{3} + 21 \, d^{5} e^{9} x^{2} + 7 \, d^{6} e^{8} x + d^{7} e^{7}\right )}} \]
-1/7*(7*b^6*e^6*x^6 + b^6*d^6 + a*b^5*d^5*e + a^2*b^4*d^4*e^2 + a^3*b^3*d^ 3*e^3 + a^4*b^2*d^2*e^4 + a^5*b*d*e^5 + a^6*e^6 + 21*(b^6*d*e^5 + a*b^5*e^ 6)*x^5 + 35*(b^6*d^2*e^4 + a*b^5*d*e^5 + a^2*b^4*e^6)*x^4 + 35*(b^6*d^3*e^ 3 + a*b^5*d^2*e^4 + a^2*b^4*d*e^5 + a^3*b^3*e^6)*x^3 + 21*(b^6*d^4*e^2 + a *b^5*d^3*e^3 + a^2*b^4*d^2*e^4 + a^3*b^3*d*e^5 + a^4*b^2*e^6)*x^2 + 7*(b^6 *d^5*e + a*b^5*d^4*e^2 + a^2*b^4*d^3*e^3 + a^3*b^3*d^2*e^4 + a^4*b^2*d*e^5 + a^5*b*e^6)*x)/(e^14*x^7 + 7*d*e^13*x^6 + 21*d^2*e^12*x^5 + 35*d^3*e^11* x^4 + 35*d^4*e^10*x^3 + 21*d^5*e^9*x^2 + 7*d^6*e^8*x + d^7*e^7)
Timed out. \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^8} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^8} \, dx=\text {Exception raised: ValueError} \]
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. 564 vs. \(2 (37) = 74\).
Time = 0.28 (sec) , antiderivative size = 564, normalized size of antiderivative = 13.76 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^8} \, dx=\frac {b^{7} \mathrm {sgn}\left (b x + a\right )}{7 \, {\left (b d e^{7} - a e^{8}\right )}} - \frac {7 \, b^{6} e^{6} x^{6} \mathrm {sgn}\left (b x + a\right ) + 21 \, b^{6} d e^{5} x^{5} \mathrm {sgn}\left (b x + a\right ) + 21 \, a b^{5} e^{6} x^{5} \mathrm {sgn}\left (b x + a\right ) + 35 \, b^{6} d^{2} e^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + 35 \, a b^{5} d e^{5} x^{4} \mathrm {sgn}\left (b x + a\right ) + 35 \, a^{2} b^{4} e^{6} x^{4} \mathrm {sgn}\left (b x + a\right ) + 35 \, b^{6} d^{3} e^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 35 \, a b^{5} d^{2} e^{4} x^{3} \mathrm {sgn}\left (b x + a\right ) + 35 \, a^{2} b^{4} d e^{5} x^{3} \mathrm {sgn}\left (b x + a\right ) + 35 \, a^{3} b^{3} e^{6} x^{3} \mathrm {sgn}\left (b x + a\right ) + 21 \, b^{6} d^{4} e^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 21 \, a b^{5} d^{3} e^{3} x^{2} \mathrm {sgn}\left (b x + a\right ) + 21 \, a^{2} b^{4} d^{2} e^{4} x^{2} \mathrm {sgn}\left (b x + a\right ) + 21 \, a^{3} b^{3} d e^{5} x^{2} \mathrm {sgn}\left (b x + a\right ) + 21 \, a^{4} b^{2} e^{6} x^{2} \mathrm {sgn}\left (b x + a\right ) + 7 \, b^{6} d^{5} e x \mathrm {sgn}\left (b x + a\right ) + 7 \, a b^{5} d^{4} e^{2} x \mathrm {sgn}\left (b x + a\right ) + 7 \, a^{2} b^{4} d^{3} e^{3} x \mathrm {sgn}\left (b x + a\right ) + 7 \, a^{3} b^{3} d^{2} e^{4} x \mathrm {sgn}\left (b x + a\right ) + 7 \, a^{4} b^{2} d e^{5} x \mathrm {sgn}\left (b x + a\right ) + 7 \, a^{5} b e^{6} x \mathrm {sgn}\left (b x + a\right ) + b^{6} d^{6} \mathrm {sgn}\left (b x + a\right ) + a b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) + a^{2} b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + a^{3} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + a^{4} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + a^{5} b d e^{5} \mathrm {sgn}\left (b x + a\right ) + a^{6} e^{6} \mathrm {sgn}\left (b x + a\right )}{7 \, {\left (e x + d\right )}^{7} e^{7}} \]
1/7*b^7*sgn(b*x + a)/(b*d*e^7 - a*e^8) - 1/7*(7*b^6*e^6*x^6*sgn(b*x + a) + 21*b^6*d*e^5*x^5*sgn(b*x + a) + 21*a*b^5*e^6*x^5*sgn(b*x + a) + 35*b^6*d^ 2*e^4*x^4*sgn(b*x + a) + 35*a*b^5*d*e^5*x^4*sgn(b*x + a) + 35*a^2*b^4*e^6* x^4*sgn(b*x + a) + 35*b^6*d^3*e^3*x^3*sgn(b*x + a) + 35*a*b^5*d^2*e^4*x^3* sgn(b*x + a) + 35*a^2*b^4*d*e^5*x^3*sgn(b*x + a) + 35*a^3*b^3*e^6*x^3*sgn( b*x + a) + 21*b^6*d^4*e^2*x^2*sgn(b*x + a) + 21*a*b^5*d^3*e^3*x^2*sgn(b*x + a) + 21*a^2*b^4*d^2*e^4*x^2*sgn(b*x + a) + 21*a^3*b^3*d*e^5*x^2*sgn(b*x + a) + 21*a^4*b^2*e^6*x^2*sgn(b*x + a) + 7*b^6*d^5*e*x*sgn(b*x + a) + 7*a* b^5*d^4*e^2*x*sgn(b*x + a) + 7*a^2*b^4*d^3*e^3*x*sgn(b*x + a) + 7*a^3*b^3* d^2*e^4*x*sgn(b*x + a) + 7*a^4*b^2*d*e^5*x*sgn(b*x + a) + 7*a^5*b*e^6*x*sg n(b*x + a) + b^6*d^6*sgn(b*x + a) + a*b^5*d^5*e*sgn(b*x + a) + a^2*b^4*d^4 *e^2*sgn(b*x + a) + a^3*b^3*d^3*e^3*sgn(b*x + a) + a^4*b^2*d^2*e^4*sgn(b*x + a) + a^5*b*d*e^5*sgn(b*x + a) + a^6*e^6*sgn(b*x + a))/((e*x + d)^7*e^7)
Time = 11.16 (sec) , antiderivative size = 1010, normalized size of antiderivative = 24.63 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^8} \, 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}{6\,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}{6\,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}{6\,e^7}-\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{6\,e^3}-\frac {b^5\,\left (6\,a\,e-b\,d\right )}{6\,e^3}\right )}{e}+\frac {b^4\,\left (15\,a^2\,e^2-6\,a\,b\,d\,e+b^2\,d^2\right )}{6\,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 )}^6}-\frac {\left (\frac {15\,a^2\,b^4\,e^2-24\,a\,b^5\,d\,e+10\,b^6\,d^2}{3\,e^7}+\frac {d\,\left (\frac {b^6\,d}{3\,e^6}-\frac {2\,b^5\,\left (3\,a\,e-2\,b\,d\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 )}^3}-\frac {\left (\frac {a^6}{7\,e}-\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {6\,a\,b^5}{7\,e}-\frac {b^6\,d}{7\,e^2}\right )}{e}-\frac {15\,a^2\,b^4}{7\,e}\right )}{e}+\frac {20\,a^3\,b^3}{7\,e}\right )}{e}-\frac {15\,a^4\,b^2}{7\,e}\right )}{e}+\frac {6\,a^5\,b}{7\,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 )}^7}-\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}{5\,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}{5\,e^7}+\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{5\,e^4}-\frac {2\,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 )}{5\,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 )}^5}+\frac {\left (\frac {5\,b^6\,d-6\,a\,b^5\,e}{2\,e^7}+\frac {b^6\,d}{2\,e^7}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^2}+\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}{4\,e^7}+\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{4\,e^5}-\frac {3\,b^5\,\left (2\,a\,e-b\,d\right )}{4\,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 )}{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 )}^4}-\frac {b^6\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{e^7\,\left (a+b\,x\right )\,\left (d+e\,x\right )} \]
(((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)/(6*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)/(6*e^7) - (d*((20* 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)/(6*e^7) - (d*((d*((b^6*d)/(6*e^3) - (b^5*(6*a*e - b*d))/(6*e^3)))/e + (b^4*(15*a^2*e ^2 + b^2*d^2 - 6*a*b*d*e))/(6*e^4)))/e))/e))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^ (1/2))/((a + b*x)*(d + e*x)^6) - (((10*b^6*d^2 + 15*a^2*b^4*e^2 - 24*a*b^5 *d*e)/(3*e^7) + (d*((b^6*d)/(3*e^6) - (2*b^5*(3*a*e - 2*b*d))/(3*e^6)))/e) *(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^3) - ((a^6/(7*e) - (d*((d*((d*((d*((d*((6*a*b^5)/(7*e) - (b^6*d)/(7*e^2)))/e - (15*a^2*b^4)/( 7*e)))/e + (20*a^3*b^3)/(7*e)))/e - (15*a^4*b^2)/(7*e)))/e + (6*a^5*b)/(7* e)))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^7) - (((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)/(5*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)/(5*e^7) + (d*((d*((b^6*d)/(5*e^4) - (2*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))/(5*e^5)))/e))/e)*( a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^5) + (((5*b^6*d - 6*a *b^5*e)/(2*e^7) + (b^6*d)/(2*e^7))*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^2) + (((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)/(4*e^7) + (d*((d*((b^6*d)/(4*e^5) - (3*b^5*(2*a*e - b*d))...