Integrand size = 33, antiderivative size = 98 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^9} \, dx=\frac {(a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{8 (b d-a e) (d+e x)^8}+\frac {b (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{56 (b d-a e)^2 (d+e x)^7} \]
1/8*(b*x+a)^6*((b*x+a)^2)^(1/2)/(-a*e+b*d)/(e*x+d)^8+1/56*b*(b*x+a)^6*((b* x+a)^2)^(1/2)/(-a*e+b*d)^2/(e*x+d)^7
Leaf count is larger than twice the leaf count of optimal. \(295\) vs. \(2(98)=196\).
Time = 1.08 (sec) , antiderivative size = 295, normalized size of antiderivative = 3.01 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^9} \, dx=-\frac {\sqrt {(a+b x)^2} \left (7 a^6 e^6+6 a^5 b e^5 (d+8 e x)+5 a^4 b^2 e^4 \left (d^2+8 d e x+28 e^2 x^2\right )+4 a^3 b^3 e^3 \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )+3 a^2 b^4 e^2 \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )+2 a b^5 e \left (d^5+8 d^4 e x+28 d^3 e^2 x^2+56 d^2 e^3 x^3+70 d e^4 x^4+56 e^5 x^5\right )+b^6 \left (d^6+8 d^5 e x+28 d^4 e^2 x^2+56 d^3 e^3 x^3+70 d^2 e^4 x^4+56 d e^5 x^5+28 e^6 x^6\right )\right )}{56 e^7 (a+b x) (d+e x)^8} \]
-1/56*(Sqrt[(a + b*x)^2]*(7*a^6*e^6 + 6*a^5*b*e^5*(d + 8*e*x) + 5*a^4*b^2* e^4*(d^2 + 8*d*e*x + 28*e^2*x^2) + 4*a^3*b^3*e^3*(d^3 + 8*d^2*e*x + 28*d*e ^2*x^2 + 56*e^3*x^3) + 3*a^2*b^4*e^2*(d^4 + 8*d^3*e*x + 28*d^2*e^2*x^2 + 5 6*d*e^3*x^3 + 70*e^4*x^4) + 2*a*b^5*e*(d^5 + 8*d^4*e*x + 28*d^3*e^2*x^2 + 56*d^2*e^3*x^3 + 70*d*e^4*x^4 + 56*e^5*x^5) + b^6*(d^6 + 8*d^5*e*x + 28*d^ 4*e^2*x^2 + 56*d^3*e^3*x^3 + 70*d^2*e^4*x^4 + 56*d*e^5*x^5 + 28*e^6*x^6))) /(e^7*(a + b*x)*(d + e*x)^8)
Time = 0.22 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.88, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {1187, 27, 55, 48}
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.
| \(\displaystyle \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^9} \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {b^5 (a+b x)^6}{(d+e x)^9}dx}{b^5 (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x)^6}{(d+e x)^9}dx}{a+b x}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {b \int \frac {(a+b x)^6}{(d+e x)^8}dx}{8 (b d-a e)}+\frac {(a+b x)^7}{8 (d+e x)^8 (b d-a e)}\right )}{a+b x}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {b (a+b x)^7}{56 (d+e x)^7 (b d-a e)^2}+\frac {(a+b x)^7}{8 (d+e x)^8 (b d-a e)}\right )}{a+b x}\) |
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*((a + b*x)^7/(8*(b*d - a*e)*(d + e*x)^8) + (b*(a + b*x)^7)/(56*(b*d - a*e)^2*(d + e*x)^7)))/(a + b*x)
3.21.6.3.1 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] :> Simp [(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(S implify[m + n + 2]/((b*c - a*d)*(m + 1))) Int[(a + b*x)^Simplify[m + 1]*( c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 2], 0] && NeQ[m, -1] && !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] || !SumSimp lerQ[n, 1])
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]
Leaf count of result is larger than twice the leaf count of optimal. \(350\) vs. \(2(72)=144\).
Time = 2.92 (sec) , antiderivative size = 351, normalized size of antiderivative = 3.58
| method | result | size |
| risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {b^{6} x^{6}}{2 e}-\frac {b^{5} \left (2 a e +b d \right ) x^{5}}{e^{2}}-\frac {5 b^{4} \left (3 e^{2} a^{2}+2 a b d e +b^{2} d^{2}\right ) x^{4}}{4 e^{3}}-\frac {b^{3} \left (4 a^{3} e^{3}+3 a^{2} b d \,e^{2}+2 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) x^{3}}{e^{4}}-\frac {b^{2} \left (5 e^{4} a^{4}+4 b d \,e^{3} a^{3}+3 b^{2} d^{2} e^{2} a^{2}+2 b^{3} d^{3} e a +b^{4} d^{4}\right ) x^{2}}{2 e^{5}}-\frac {b \left (6 e^{5} a^{5}+5 b d \,e^{4} a^{4}+4 b^{2} d^{2} e^{3} a^{3}+3 b^{3} d^{3} e^{2} a^{2}+2 b^{4} d^{4} e a +b^{5} d^{5}\right ) x}{7 e^{6}}-\frac {7 e^{6} a^{6}+6 b d \,e^{5} a^{5}+5 b^{2} d^{2} e^{4} a^{4}+4 b^{3} d^{3} e^{3} a^{3}+3 b^{4} d^{4} e^{2} a^{2}+2 b^{5} d^{5} e a +b^{6} d^{6}}{56 e^{7}}\right )}{\left (b x +a \right ) \left (e x +d \right )^{8}}\) | \(351\) |
| gosper | \(-\frac {\left (28 b^{6} e^{6} x^{6}+112 a \,b^{5} e^{6} x^{5}+56 b^{6} d \,e^{5} x^{5}+210 a^{2} b^{4} e^{6} x^{4}+140 a \,b^{5} d \,e^{5} x^{4}+70 b^{6} d^{2} e^{4} x^{4}+224 a^{3} b^{3} e^{6} x^{3}+168 a^{2} b^{4} d \,e^{5} x^{3}+112 a \,b^{5} d^{2} e^{4} x^{3}+56 b^{6} d^{3} e^{3} x^{3}+140 a^{4} b^{2} e^{6} x^{2}+112 a^{3} b^{3} d \,e^{5} x^{2}+84 a^{2} b^{4} d^{2} e^{4} x^{2}+56 a \,b^{5} d^{3} e^{3} x^{2}+28 b^{6} d^{4} e^{2} x^{2}+48 a^{5} b \,e^{6} x +40 a^{4} b^{2} d \,e^{5} x +32 a^{3} b^{3} d^{2} e^{4} x +24 a^{2} b^{4} d^{3} e^{3} x +16 a \,b^{5} d^{4} e^{2} x +8 b^{6} d^{5} e x +7 e^{6} a^{6}+6 b d \,e^{5} a^{5}+5 b^{2} d^{2} e^{4} a^{4}+4 b^{3} d^{3} e^{3} a^{3}+3 b^{4} d^{4} e^{2} a^{2}+2 b^{5} d^{5} e a +b^{6} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{56 e^{7} \left (e x +d \right )^{8} \left (b x +a \right )^{5}}\) | \(392\) |
| default | \(-\frac {\left (28 b^{6} e^{6} x^{6}+112 a \,b^{5} e^{6} x^{5}+56 b^{6} d \,e^{5} x^{5}+210 a^{2} b^{4} e^{6} x^{4}+140 a \,b^{5} d \,e^{5} x^{4}+70 b^{6} d^{2} e^{4} x^{4}+224 a^{3} b^{3} e^{6} x^{3}+168 a^{2} b^{4} d \,e^{5} x^{3}+112 a \,b^{5} d^{2} e^{4} x^{3}+56 b^{6} d^{3} e^{3} x^{3}+140 a^{4} b^{2} e^{6} x^{2}+112 a^{3} b^{3} d \,e^{5} x^{2}+84 a^{2} b^{4} d^{2} e^{4} x^{2}+56 a \,b^{5} d^{3} e^{3} x^{2}+28 b^{6} d^{4} e^{2} x^{2}+48 a^{5} b \,e^{6} x +40 a^{4} b^{2} d \,e^{5} x +32 a^{3} b^{3} d^{2} e^{4} x +24 a^{2} b^{4} d^{3} e^{3} x +16 a \,b^{5} d^{4} e^{2} x +8 b^{6} d^{5} e x +7 e^{6} a^{6}+6 b d \,e^{5} a^{5}+5 b^{2} d^{2} e^{4} a^{4}+4 b^{3} d^{3} e^{3} a^{3}+3 b^{4} d^{4} e^{2} a^{2}+2 b^{5} d^{5} e a +b^{6} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{56 e^{7} \left (e x +d \right )^{8} \left (b x +a \right )^{5}}\) | \(392\) |
((b*x+a)^2)^(1/2)/(b*x+a)*(-1/2*b^6/e*x^6-b^5/e^2*(2*a*e+b*d)*x^5-5/4*b^4/ e^3*(3*a^2*e^2+2*a*b*d*e+b^2*d^2)*x^4-b^3/e^4*(4*a^3*e^3+3*a^2*b*d*e^2+2*a *b^2*d^2*e+b^3*d^3)*x^3-1/2*b^2/e^5*(5*a^4*e^4+4*a^3*b*d*e^3+3*a^2*b^2*d^2 *e^2+2*a*b^3*d^3*e+b^4*d^4)*x^2-1/7*b/e^6*(6*a^5*e^5+5*a^4*b*d*e^4+4*a^3*b ^2*d^2*e^3+3*a^2*b^3*d^3*e^2+2*a*b^4*d^4*e+b^5*d^5)*x-1/56/e^7*(7*a^6*e^6+ 6*a^5*b*d*e^5+5*a^4*b^2*d^2*e^4+4*a^3*b^3*d^3*e^3+3*a^2*b^4*d^4*e^2+2*a*b^ 5*d^5*e+b^6*d^6))/(e*x+d)^8
Leaf count of result is larger than twice the leaf count of optimal. 430 vs. \(2 (72) = 144\).
Time = 0.28 (sec) , antiderivative size = 430, normalized size of antiderivative = 4.39 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^9} \, dx=-\frac {28 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 2 \, a b^{5} d^{5} e + 3 \, a^{2} b^{4} d^{4} e^{2} + 4 \, a^{3} b^{3} d^{3} e^{3} + 5 \, a^{4} b^{2} d^{2} e^{4} + 6 \, a^{5} b d e^{5} + 7 \, a^{6} e^{6} + 56 \, {\left (b^{6} d e^{5} + 2 \, a b^{5} e^{6}\right )} x^{5} + 70 \, {\left (b^{6} d^{2} e^{4} + 2 \, a b^{5} d e^{5} + 3 \, a^{2} b^{4} e^{6}\right )} x^{4} + 56 \, {\left (b^{6} d^{3} e^{3} + 2 \, a b^{5} d^{2} e^{4} + 3 \, a^{2} b^{4} d e^{5} + 4 \, a^{3} b^{3} e^{6}\right )} x^{3} + 28 \, {\left (b^{6} d^{4} e^{2} + 2 \, a b^{5} d^{3} e^{3} + 3 \, a^{2} b^{4} d^{2} e^{4} + 4 \, a^{3} b^{3} d e^{5} + 5 \, a^{4} b^{2} e^{6}\right )} x^{2} + 8 \, {\left (b^{6} d^{5} e + 2 \, a b^{5} d^{4} e^{2} + 3 \, a^{2} b^{4} d^{3} e^{3} + 4 \, a^{3} b^{3} d^{2} e^{4} + 5 \, a^{4} b^{2} d e^{5} + 6 \, a^{5} b e^{6}\right )} x}{56 \, {\left (e^{15} x^{8} + 8 \, d e^{14} x^{7} + 28 \, d^{2} e^{13} x^{6} + 56 \, d^{3} e^{12} x^{5} + 70 \, d^{4} e^{11} x^{4} + 56 \, d^{5} e^{10} x^{3} + 28 \, d^{6} e^{9} x^{2} + 8 \, d^{7} e^{8} x + d^{8} e^{7}\right )}} \]
-1/56*(28*b^6*e^6*x^6 + b^6*d^6 + 2*a*b^5*d^5*e + 3*a^2*b^4*d^4*e^2 + 4*a^ 3*b^3*d^3*e^3 + 5*a^4*b^2*d^2*e^4 + 6*a^5*b*d*e^5 + 7*a^6*e^6 + 56*(b^6*d* e^5 + 2*a*b^5*e^6)*x^5 + 70*(b^6*d^2*e^4 + 2*a*b^5*d*e^5 + 3*a^2*b^4*e^6)* x^4 + 56*(b^6*d^3*e^3 + 2*a*b^5*d^2*e^4 + 3*a^2*b^4*d*e^5 + 4*a^3*b^3*e^6) *x^3 + 28*(b^6*d^4*e^2 + 2*a*b^5*d^3*e^3 + 3*a^2*b^4*d^2*e^4 + 4*a^3*b^3*d *e^5 + 5*a^4*b^2*e^6)*x^2 + 8*(b^6*d^5*e + 2*a*b^5*d^4*e^2 + 3*a^2*b^4*d^3 *e^3 + 4*a^3*b^3*d^2*e^4 + 5*a^4*b^2*d*e^5 + 6*a^5*b*e^6)*x)/(e^15*x^8 + 8 *d*e^14*x^7 + 28*d^2*e^13*x^6 + 56*d^3*e^12*x^5 + 70*d^4*e^11*x^4 + 56*d^5 *e^10*x^3 + 28*d^6*e^9*x^2 + 8*d^7*e^8*x + d^8*e^7)
\[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^9} \, dx=\int \frac {\left (a + b x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}{\left (d + e x\right )^{9}}\, dx \]
Exception generated. \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^9} \, dx=\text {Exception raised: ValueError} \]
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. 583 vs. \(2 (72) = 144\).
Time = 0.27 (sec) , antiderivative size = 583, normalized size of antiderivative = 5.95 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^9} \, dx=\frac {b^{8} \mathrm {sgn}\left (b x + a\right )}{56 \, {\left (b^{2} d^{2} e^{7} - 2 \, a b d e^{8} + a^{2} e^{9}\right )}} - \frac {28 \, b^{6} e^{6} x^{6} \mathrm {sgn}\left (b x + a\right ) + 56 \, b^{6} d e^{5} x^{5} \mathrm {sgn}\left (b x + a\right ) + 112 \, a b^{5} e^{6} x^{5} \mathrm {sgn}\left (b x + a\right ) + 70 \, b^{6} d^{2} e^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + 140 \, a b^{5} d e^{5} x^{4} \mathrm {sgn}\left (b x + a\right ) + 210 \, a^{2} b^{4} e^{6} x^{4} \mathrm {sgn}\left (b x + a\right ) + 56 \, b^{6} d^{3} e^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 112 \, a b^{5} d^{2} e^{4} x^{3} \mathrm {sgn}\left (b x + a\right ) + 168 \, a^{2} b^{4} d e^{5} x^{3} \mathrm {sgn}\left (b x + a\right ) + 224 \, a^{3} b^{3} e^{6} x^{3} \mathrm {sgn}\left (b x + a\right ) + 28 \, b^{6} d^{4} e^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 56 \, a b^{5} d^{3} e^{3} x^{2} \mathrm {sgn}\left (b x + a\right ) + 84 \, a^{2} b^{4} d^{2} e^{4} x^{2} \mathrm {sgn}\left (b x + a\right ) + 112 \, a^{3} b^{3} d e^{5} x^{2} \mathrm {sgn}\left (b x + a\right ) + 140 \, a^{4} b^{2} e^{6} x^{2} \mathrm {sgn}\left (b x + a\right ) + 8 \, b^{6} d^{5} e x \mathrm {sgn}\left (b x + a\right ) + 16 \, a b^{5} d^{4} e^{2} x \mathrm {sgn}\left (b x + a\right ) + 24 \, a^{2} b^{4} d^{3} e^{3} x \mathrm {sgn}\left (b x + a\right ) + 32 \, a^{3} b^{3} d^{2} e^{4} x \mathrm {sgn}\left (b x + a\right ) + 40 \, a^{4} b^{2} d e^{5} x \mathrm {sgn}\left (b x + a\right ) + 48 \, a^{5} b e^{6} x \mathrm {sgn}\left (b x + a\right ) + b^{6} d^{6} \mathrm {sgn}\left (b x + a\right ) + 2 \, a b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{2} b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 4 \, a^{3} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{5} b d e^{5} \mathrm {sgn}\left (b x + a\right ) + 7 \, a^{6} e^{6} \mathrm {sgn}\left (b x + a\right )}{56 \, {\left (e x + d\right )}^{8} e^{7}} \]
1/56*b^8*sgn(b*x + a)/(b^2*d^2*e^7 - 2*a*b*d*e^8 + a^2*e^9) - 1/56*(28*b^6 *e^6*x^6*sgn(b*x + a) + 56*b^6*d*e^5*x^5*sgn(b*x + a) + 112*a*b^5*e^6*x^5* sgn(b*x + a) + 70*b^6*d^2*e^4*x^4*sgn(b*x + a) + 140*a*b^5*d*e^5*x^4*sgn(b *x + a) + 210*a^2*b^4*e^6*x^4*sgn(b*x + a) + 56*b^6*d^3*e^3*x^3*sgn(b*x + a) + 112*a*b^5*d^2*e^4*x^3*sgn(b*x + a) + 168*a^2*b^4*d*e^5*x^3*sgn(b*x + a) + 224*a^3*b^3*e^6*x^3*sgn(b*x + a) + 28*b^6*d^4*e^2*x^2*sgn(b*x + a) + 56*a*b^5*d^3*e^3*x^2*sgn(b*x + a) + 84*a^2*b^4*d^2*e^4*x^2*sgn(b*x + a) + 112*a^3*b^3*d*e^5*x^2*sgn(b*x + a) + 140*a^4*b^2*e^6*x^2*sgn(b*x + a) + 8* b^6*d^5*e*x*sgn(b*x + a) + 16*a*b^5*d^4*e^2*x*sgn(b*x + a) + 24*a^2*b^4*d^ 3*e^3*x*sgn(b*x + a) + 32*a^3*b^3*d^2*e^4*x*sgn(b*x + a) + 40*a^4*b^2*d*e^ 5*x*sgn(b*x + a) + 48*a^5*b*e^6*x*sgn(b*x + a) + b^6*d^6*sgn(b*x + a) + 2* a*b^5*d^5*e*sgn(b*x + a) + 3*a^2*b^4*d^4*e^2*sgn(b*x + a) + 4*a^3*b^3*d^3* e^3*sgn(b*x + a) + 5*a^4*b^2*d^2*e^4*sgn(b*x + a) + 6*a^5*b*d*e^5*sgn(b*x + a) + 7*a^6*e^6*sgn(b*x + a))/((e*x + d)^8*e^7)
Time = 11.09 (sec) , antiderivative size = 1010, normalized size of antiderivative = 10.31 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^9} \, 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}{7\,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}{7\,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}{7\,e^7}-\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{7\,e^3}-\frac {b^5\,\left (6\,a\,e-b\,d\right )}{7\,e^3}\right )}{e}+\frac {b^4\,\left (15\,a^2\,e^2-6\,a\,b\,d\,e+b^2\,d^2\right )}{7\,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 )}^7}-\frac {\left (\frac {15\,a^2\,b^4\,e^2-24\,a\,b^5\,d\,e+10\,b^6\,d^2}{4\,e^7}+\frac {d\,\left (\frac {b^6\,d}{4\,e^6}-\frac {b^5\,\left (3\,a\,e-2\,b\,d\right )}{2\,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 {\left (\frac {a^6}{8\,e}-\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {3\,a\,b^5}{4\,e}-\frac {b^6\,d}{8\,e^2}\right )}{e}-\frac {15\,a^2\,b^4}{8\,e}\right )}{e}+\frac {5\,a^3\,b^3}{2\,e}\right )}{e}-\frac {15\,a^4\,b^2}{8\,e}\right )}{e}+\frac {3\,a^5\,b}{4\,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 )}^8}-\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}{6\,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}{6\,e^7}+\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{6\,e^4}-\frac {b^5\,\left (3\,a\,e-b\,d\right )}{3\,e^4}\right )}{e}+\frac {b^4\,\left (5\,a^2\,e^2-4\,a\,b\,d\,e+b^2\,d^2\right )}{2\,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 )}^6}+\frac {\left (\frac {5\,b^6\,d-6\,a\,b^5\,e}{3\,e^7}+\frac {b^6\,d}{3\,e^7}\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 {-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}{5\,e^7}+\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{5\,e^5}-\frac {3\,b^5\,\left (2\,a\,e-b\,d\right )}{5\,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 )}{5\,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 )}^5}-\frac {b^6\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{2\,e^7\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^2} \]
(((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)/(7*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)/(7*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)/(7*e^7) - (d*((d*((b^6*d)/(7*e^3) - (b^5*(6*a*e - b*d))/(7*e^3)))/e + (b^4*(15*a^2*e ^2 + b^2*d^2 - 6*a*b*d*e))/(7*e^4)))/e))/e))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^ (1/2))/((a + b*x)*(d + e*x)^7) - (((10*b^6*d^2 + 15*a^2*b^4*e^2 - 24*a*b^5 *d*e)/(4*e^7) + (d*((b^6*d)/(4*e^6) - (b^5*(3*a*e - 2*b*d))/(2*e^6)))/e)*( a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^4) - ((a^6/(8*e) - (d *((d*((d*((d*((d*((3*a*b^5)/(4*e) - (b^6*d)/(8*e^2)))/e - (15*a^2*b^4)/(8* e)))/e + (5*a^3*b^3)/(2*e)))/e - (15*a^4*b^2)/(8*e)))/e + (3*a^5*b)/(4*e)) )/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^8) - (((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)/(6*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)/(6*e^7) + (d*((d*((b^6*d)/(6*e^4) - (b^5*(3*a*e - b*d))/(3*e^ 4)))/e + (b^4*(5*a^2*e^2 + b^2*d^2 - 4*a*b*d*e))/(2*e^5)))/e))/e)*(a^2 + b ^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^6) + (((5*b^6*d - 6*a*b^5*e) /(3*e^7) + (b^6*d)/(3*e^7))*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^3) + (((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)/(5*e^7) + (d*((d*((b^6*d)/(5*e^5) - (3*b^5*(2*a*e - b*d))/(5*e^5...