\(\int \frac {(a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^8} \, dx\) [1582]

Optimal result

Integrand size = 28, antiderivative size = 98 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^8} \, dx=\frac {(a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{7 (b d-a e) (d+e x)^7}+\frac {b (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{42 (b d-a e)^2 (d+e x)^6} \]

[Out]

1/7*(b*x+a)^5*((b*x+a)^2)^(1/2)/(-a*e+b*d)/(e*x+d)^7+1/42*b*(b*x+a)^5*((b*x+a)^2)^(1/2)/(-a*e+b*d)^2/(e*x+d)^6

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {660, 47, 37} \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^8} \, dx=\frac {b \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5}{42 (d+e x)^6 (b d-a e)^2}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5}{7 (d+e x)^7 (b d-a e)} \]

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^8,x]

[Out]

((a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*(b*d - a*e)*(d + e*x)^7) + (b*(a + b*x)^5*Sqrt[a^2 + 2*a*b*x +
b^2*x^2])/(42*(b*d - a*e)^2*(d + e*x)^6)

Rule 37

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] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 47

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] - Dist[d*(Simplify[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] && NeQ[b*c - a*d, 0] && 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] ||  !SumSimplerQ[n, 1])

Rule 660

Int[((d_.) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(a + b*x + c*x^2)^Fra
cPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[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] &&  !IntegerQ[p] && NeQ[2*c*d - b*e, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^5}{(d+e x)^8} \, dx}{b^4 \left (a b+b^2 x\right )} \\ & = \frac {(a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{7 (b d-a e) (d+e x)^7}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^5}{(d+e x)^7} \, dx}{7 b^3 (b d-a e) \left (a b+b^2 x\right )} \\ & = \frac {(a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{7 (b d-a e) (d+e x)^7}+\frac {b (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{42 (b d-a e)^2 (d+e x)^6} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(223\) vs. \(2(98)=196\).

Time = 1.06 (sec) , antiderivative size = 223, normalized size of antiderivative = 2.28 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^8} \, dx=-\frac {\sqrt {(a+b x)^2} \left (6 a^5 e^5+5 a^4 b e^4 (d+7 e x)+4 a^3 b^2 e^3 \left (d^2+7 d e x+21 e^2 x^2\right )+3 a^2 b^3 e^2 \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )+2 a b^4 e \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )+b^5 \left (d^5+7 d^4 e x+21 d^3 e^2 x^2+35 d^2 e^3 x^3+35 d e^4 x^4+21 e^5 x^5\right )\right )}{42 e^6 (a+b x) (d+e x)^7} \]

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^8,x]

[Out]

-1/42*(Sqrt[(a + b*x)^2]*(6*a^5*e^5 + 5*a^4*b*e^4*(d + 7*e*x) + 4*a^3*b^2*e^3*(d^2 + 7*d*e*x + 21*e^2*x^2) + 3
*a^2*b^3*e^2*(d^3 + 7*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3) + 2*a*b^4*e*(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) + b^5*(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))
)/(e^6*(a + b*x)*(d + e*x)^7)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(261\) vs. \(2(72)=144\).

Time = 4.15 (sec) , antiderivative size = 262, normalized size of antiderivative = 2.67

method	result	size
risch	\(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {b^{5} x^{5}}{2 e}-\frac {5 b^{4} \left (2 a e +b d \right ) x^{4}}{6 e^{2}}-\frac {5 b^{3} \left (3 a^{2} e^{2}+2 a b d e +b^{2} d^{2}\right ) x^{3}}{6 e^{3}}-\frac {b^{2} \left (4 a^{3} e^{3}+3 a^{2} b d \,e^{2}+2 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) x^{2}}{2 e^{4}}-\frac {b \left (5 e^{4} a^{4}+4 b \,e^{3} d \,a^{3}+3 b^{2} e^{2} d^{2} a^{2}+2 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) x}{6 e^{5}}-\frac {6 a^{5} e^{5}+5 a^{4} b d \,e^{4}+4 a^{3} b^{2} d^{2} e^{3}+3 a^{2} b^{3} d^{3} e^{2}+2 a \,b^{4} d^{4} e +b^{5} d^{5}}{42 e^{6}}\right )}{\left (b x +a \right ) \left (e x +d \right )^{7}}\)	\(262\)
gosper	\(-\frac {\left (21 x^{5} e^{5} b^{5}+70 x^{4} a \,b^{4} e^{5}+35 x^{4} b^{5} d \,e^{4}+105 x^{3} a^{2} b^{3} e^{5}+70 x^{3} a \,b^{4} d \,e^{4}+35 x^{3} b^{5} d^{2} e^{3}+84 x^{2} a^{3} b^{2} e^{5}+63 x^{2} a^{2} b^{3} d \,e^{4}+42 x^{2} a \,b^{4} d^{2} e^{3}+21 x^{2} b^{5} d^{3} e^{2}+35 a^{4} b \,e^{5} x +28 a^{3} b^{2} d \,e^{4} x +21 x \,a^{2} b^{3} d^{2} e^{3}+14 x a \,b^{4} d^{3} e^{2}+7 b^{5} d^{4} e x +6 a^{5} e^{5}+5 a^{4} b d \,e^{4}+4 a^{3} b^{2} d^{2} e^{3}+3 a^{2} b^{3} d^{3} e^{2}+2 a \,b^{4} d^{4} e +b^{5} d^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{42 e^{6} \left (e x +d \right )^{7} \left (b x +a \right )^{5}}\)	\(288\)
default	\(-\frac {\left (21 x^{5} e^{5} b^{5}+70 x^{4} a \,b^{4} e^{5}+35 x^{4} b^{5} d \,e^{4}+105 x^{3} a^{2} b^{3} e^{5}+70 x^{3} a \,b^{4} d \,e^{4}+35 x^{3} b^{5} d^{2} e^{3}+84 x^{2} a^{3} b^{2} e^{5}+63 x^{2} a^{2} b^{3} d \,e^{4}+42 x^{2} a \,b^{4} d^{2} e^{3}+21 x^{2} b^{5} d^{3} e^{2}+35 a^{4} b \,e^{5} x +28 a^{3} b^{2} d \,e^{4} x +21 x \,a^{2} b^{3} d^{2} e^{3}+14 x a \,b^{4} d^{3} e^{2}+7 b^{5} d^{4} e x +6 a^{5} e^{5}+5 a^{4} b d \,e^{4}+4 a^{3} b^{2} d^{2} e^{3}+3 a^{2} b^{3} d^{3} e^{2}+2 a \,b^{4} d^{4} e +b^{5} d^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{42 e^{6} \left (e x +d \right )^{7} \left (b x +a \right )^{5}}\)	\(288\)

[In]

int((b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^8,x,method=_RETURNVERBOSE)

[Out]

((b*x+a)^2)^(1/2)/(b*x+a)*(-1/2*b^5/e*x^5-5/6/e^2*b^4*(2*a*e+b*d)*x^4-5/6*b^3/e^3*(3*a^2*e^2+2*a*b*d*e+b^2*d^2
)*x^3-1/2*b^2/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^2-1/6*b/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-1/42/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))/(e*x+d)^7

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 326 vs. \(2 (72) = 144\).

Time = 0.46 (sec) , antiderivative size = 326, normalized size of antiderivative = 3.33 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^8} \, dx=-\frac {21 \, b^{5} e^{5} x^{5} + b^{5} d^{5} + 2 \, a b^{4} d^{4} e + 3 \, a^{2} b^{3} d^{3} e^{2} + 4 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} + 6 \, a^{5} e^{5} + 35 \, {\left (b^{5} d e^{4} + 2 \, a b^{4} e^{5}\right )} x^{4} + 35 \, {\left (b^{5} d^{2} e^{3} + 2 \, a b^{4} d e^{4} + 3 \, a^{2} b^{3} e^{5}\right )} x^{3} + 21 \, {\left (b^{5} d^{3} e^{2} + 2 \, a b^{4} d^{2} e^{3} + 3 \, a^{2} b^{3} d e^{4} + 4 \, a^{3} b^{2} e^{5}\right )} x^{2} + 7 \, {\left (b^{5} d^{4} e + 2 \, a b^{4} d^{3} e^{2} + 3 \, a^{2} b^{3} d^{2} e^{3} + 4 \, a^{3} b^{2} d e^{4} + 5 \, a^{4} b e^{5}\right )} x}{42 \, {\left (e^{13} x^{7} + 7 \, d e^{12} x^{6} + 21 \, d^{2} e^{11} x^{5} + 35 \, d^{3} e^{10} x^{4} + 35 \, d^{4} e^{9} x^{3} + 21 \, d^{5} e^{8} x^{2} + 7 \, d^{6} e^{7} x + d^{7} e^{6}\right )}} \]

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^8,x, algorithm="fricas")

[Out]

-1/42*(21*b^5*e^5*x^5 + b^5*d^5 + 2*a*b^4*d^4*e + 3*a^2*b^3*d^3*e^2 + 4*a^3*b^2*d^2*e^3 + 5*a^4*b*d*e^4 + 6*a^
5*e^5 + 35*(b^5*d*e^4 + 2*a*b^4*e^5)*x^4 + 35*(b^5*d^2*e^3 + 2*a*b^4*d*e^4 + 3*a^2*b^3*e^5)*x^3 + 21*(b^5*d^3*
e^2 + 2*a*b^4*d^2*e^3 + 3*a^2*b^3*d*e^4 + 4*a^3*b^2*e^5)*x^2 + 7*(b^5*d^4*e + 2*a*b^4*d^3*e^2 + 3*a^2*b^3*d^2*
e^3 + 4*a^3*b^2*d*e^4 + 5*a^4*b*e^5)*x)/(e^13*x^7 + 7*d*e^12*x^6 + 21*d^2*e^11*x^5 + 35*d^3*e^10*x^4 + 35*d^4*
e^9*x^3 + 21*d^5*e^8*x^2 + 7*d^6*e^7*x + d^7*e^6)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^8} \, dx=\text {Timed out} \]

[In]

integrate((b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**8,x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^8} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*e-b*d>0)', see `assume?` for
more detail

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 437 vs. \(2 (72) = 144\).

Time = 0.29 (sec) , antiderivative size = 437, normalized size of antiderivative = 4.46 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^8} \, dx=\frac {b^{7} \mathrm {sgn}\left (b x + a\right )}{42 \, {\left (b^{2} d^{2} e^{6} - 2 \, a b d e^{7} + a^{2} e^{8}\right )}} - \frac {21 \, b^{5} e^{5} x^{5} \mathrm {sgn}\left (b x + a\right ) + 35 \, b^{5} d e^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + 70 \, a b^{4} e^{5} x^{4} \mathrm {sgn}\left (b x + a\right ) + 35 \, b^{5} d^{2} e^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 70 \, a b^{4} d e^{4} x^{3} \mathrm {sgn}\left (b x + a\right ) + 105 \, a^{2} b^{3} e^{5} x^{3} \mathrm {sgn}\left (b x + a\right ) + 21 \, b^{5} d^{3} e^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 42 \, a b^{4} d^{2} e^{3} x^{2} \mathrm {sgn}\left (b x + a\right ) + 63 \, a^{2} b^{3} d e^{4} x^{2} \mathrm {sgn}\left (b x + a\right ) + 84 \, a^{3} b^{2} e^{5} x^{2} \mathrm {sgn}\left (b x + a\right ) + 7 \, b^{5} d^{4} e x \mathrm {sgn}\left (b x + a\right ) + 14 \, a b^{4} d^{3} e^{2} x \mathrm {sgn}\left (b x + a\right ) + 21 \, a^{2} b^{3} d^{2} e^{3} x \mathrm {sgn}\left (b x + a\right ) + 28 \, a^{3} b^{2} d e^{4} x \mathrm {sgn}\left (b x + a\right ) + 35 \, a^{4} b e^{5} x \mathrm {sgn}\left (b x + a\right ) + b^{5} d^{5} \mathrm {sgn}\left (b x + a\right ) + 2 \, a b^{4} d^{4} e \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{2} b^{3} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + 4 \, a^{3} b^{2} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} b d e^{4} \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{5} e^{5} \mathrm {sgn}\left (b x + a\right )}{42 \, {\left (e x + d\right )}^{7} e^{6}} \]

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^8,x, algorithm="giac")

[Out]

1/42*b^7*sgn(b*x + a)/(b^2*d^2*e^6 - 2*a*b*d*e^7 + a^2*e^8) - 1/42*(21*b^5*e^5*x^5*sgn(b*x + a) + 35*b^5*d*e^4
*x^4*sgn(b*x + a) + 70*a*b^4*e^5*x^4*sgn(b*x + a) + 35*b^5*d^2*e^3*x^3*sgn(b*x + a) + 70*a*b^4*d*e^4*x^3*sgn(b
*x + a) + 105*a^2*b^3*e^5*x^3*sgn(b*x + a) + 21*b^5*d^3*e^2*x^2*sgn(b*x + a) + 42*a*b^4*d^2*e^3*x^2*sgn(b*x +
a) + 63*a^2*b^3*d*e^4*x^2*sgn(b*x + a) + 84*a^3*b^2*e^5*x^2*sgn(b*x + a) + 7*b^5*d^4*e*x*sgn(b*x + a) + 14*a*b
^4*d^3*e^2*x*sgn(b*x + a) + 21*a^2*b^3*d^2*e^3*x*sgn(b*x + a) + 28*a^3*b^2*d*e^4*x*sgn(b*x + a) + 35*a^4*b*e^5
*x*sgn(b*x + a) + b^5*d^5*sgn(b*x + a) + 2*a*b^4*d^4*e*sgn(b*x + a) + 3*a^2*b^3*d^3*e^2*sgn(b*x + a) + 4*a^3*b
^2*d^2*e^3*sgn(b*x + a) + 5*a^4*b*d*e^4*sgn(b*x + a) + 6*a^5*e^5*sgn(b*x + a))/((e*x + d)^7*e^6)

Mupad [B] (verification not implemented)

Time = 9.60 (sec) , antiderivative size = 687, normalized size of antiderivative = 7.01 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^8} \, dx=\frac {\left (\frac {4\,b^5\,d-5\,a\,b^4\,e}{3\,e^6}+\frac {b^5\,d}{3\,e^6}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^3}-\frac {\left (\frac {5\,a^4\,b\,e^4-10\,a^3\,b^2\,d\,e^3+10\,a^2\,b^3\,d^2\,e^2-5\,a\,b^4\,d^3\,e+b^5\,d^4}{6\,e^6}+\frac {d\,\left (\frac {-10\,a^3\,b^2\,e^4+10\,a^2\,b^3\,d\,e^3-5\,a\,b^4\,d^2\,e^2+b^5\,d^3\,e}{6\,e^6}+\frac {d\,\left (\frac {d\,\left (\frac {b^5\,d}{6\,e^3}-\frac {b^4\,\left (5\,a\,e-b\,d\right )}{6\,e^3}\right )}{e}+\frac {b^3\,\left (10\,a^2\,e^2-5\,a\,b\,d\,e+b^2\,d^2\right )}{6\,e^4}\right )}{e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^6}-\frac {\left (\frac {10\,a^2\,b^3\,e^2-15\,a\,b^4\,d\,e+6\,b^5\,d^2}{4\,e^6}+\frac {d\,\left (\frac {b^5\,d}{4\,e^5}-\frac {b^4\,\left (5\,a\,e-3\,b\,d\right )}{4\,e^5}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^4}-\frac {\left (\frac {a^5}{7\,e}-\frac {d\,\left (\frac {5\,a^4\,b}{7\,e}-\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {5\,a\,b^4}{7\,e}-\frac {b^5\,d}{7\,e^2}\right )}{e}-\frac {10\,a^2\,b^3}{7\,e}\right )}{e}+\frac {10\,a^3\,b^2}{7\,e}\right )}{e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^7}+\frac {\left (\frac {-10\,a^3\,b^2\,e^3+20\,a^2\,b^3\,d\,e^2-15\,a\,b^4\,d^2\,e+4\,b^5\,d^3}{5\,e^6}+\frac {d\,\left (\frac {d\,\left (\frac {b^5\,d}{5\,e^4}-\frac {b^4\,\left (5\,a\,e-2\,b\,d\right )}{5\,e^4}\right )}{e}+\frac {b^3\,\left (10\,a^2\,e^2-10\,a\,b\,d\,e+3\,b^2\,d^2\right )}{5\,e^5}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^5}-\frac {b^5\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{2\,e^6\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^2} \]

[In]

int((a^2 + b^2*x^2 + 2*a*b*x)^(5/2)/(d + e*x)^8,x)

[Out]

(((4*b^5*d - 5*a*b^4*e)/(3*e^6) + (b^5*d)/(3*e^6))*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^3) -
(((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)/(6*e^6) + (d*((b^5*d^3*e - 1
0*a^3*b^2*e^4 - 5*a*b^4*d^2*e^2 + 10*a^2*b^3*d*e^3)/(6*e^6) + (d*((d*((b^5*d)/(6*e^3) - (b^4*(5*a*e - b*d))/(6
*e^3)))/e + (b^3*(10*a^2*e^2 + b^2*d^2 - 5*a*b*d*e))/(6*e^4)))/e))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b
*x)*(d + e*x)^6) - (((6*b^5*d^2 + 10*a^2*b^3*e^2 - 15*a*b^4*d*e)/(4*e^6) + (d*((b^5*d)/(4*e^5) - (b^4*(5*a*e -
3*b*d))/(4*e^5)))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^4) - ((a^5/(7*e) - (d*((5*a^4*b)/(
7*e) - (d*((d*((d*((5*a*b^4)/(7*e) - (b^5*d)/(7*e^2)))/e - (10*a^2*b^3)/(7*e)))/e + (10*a^3*b^2)/(7*e)))/e))/e
)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^7) + (((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)/(5*e^6) + (d*((d*((b^5*d)/(5*e^4) - (b^4*(5*a*e - 2*b*d))/(5*e^4)))/e + (b^3*(10*a^2*e^2 + 3
*b^2*d^2 - 10*a*b*d*e))/(5*e^5)))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^5) - (b^5*(a^2 + b^
2*x^2 + 2*a*b*x)^(1/2))/(2*e^6*(a + b*x)*(d + e*x)^2)