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

Optimal. Leaf size=98 \[ \frac {(a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{7 (b d-a e) (d+e x)^7}+\frac {b (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{42 (b d-a e)^2 (d+e x)^6} \]

[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

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Rubi [A]
time = 0.03, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {660, 47, 37} \begin {gather*} \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)} \end {gather*}

Antiderivative was successfully verified.

[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*} \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^8} \, dx &=\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*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(223\) vs. \(2(98)=196\).
time = 0.05, size = 223, normalized size = 2.28 \begin {gather*} -\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} \end {gather*}

Antiderivative was successfully verified.

[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)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(287\) vs. \(2(72)=144\).
time = 0.62, size = 288, normalized size = 2.94

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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(b*d-%e*a>0)', see `assume?` fo
r more detai

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 296 vs. \(2 (76) = 152\).
time = 2.65, size = 296, normalized size = 3.02 \begin {gather*} -\frac {b^{5} d^{5} + {\left (21 \, b^{5} x^{5} + 70 \, a b^{4} x^{4} + 105 \, a^{2} b^{3} x^{3} + 84 \, a^{3} b^{2} x^{2} + 35 \, a^{4} b x + 6 \, a^{5}\right )} e^{5} + {\left (35 \, b^{5} d x^{4} + 70 \, a b^{4} d x^{3} + 63 \, a^{2} b^{3} d x^{2} + 28 \, a^{3} b^{2} d x + 5 \, a^{4} b d\right )} e^{4} + {\left (35 \, b^{5} d^{2} x^{3} + 42 \, a b^{4} d^{2} x^{2} + 21 \, a^{2} b^{3} d^{2} x + 4 \, a^{3} b^{2} d^{2}\right )} e^{3} + {\left (21 \, b^{5} d^{3} x^{2} + 14 \, a b^{4} d^{3} x + 3 \, a^{2} b^{3} d^{3}\right )} e^{2} + {\left (7 \, b^{5} d^{4} x + 2 \, a b^{4} d^{4}\right )} e}{42 \, {\left (x^{7} e^{13} + 7 \, d x^{6} e^{12} + 21 \, d^{2} x^{5} e^{11} + 35 \, d^{3} x^{4} e^{10} + 35 \, d^{4} x^{3} e^{9} + 21 \, d^{5} x^{2} e^{8} + 7 \, d^{6} x e^{7} + d^{7} e^{6}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/42*(b^5*d^5 + (21*b^5*x^5 + 70*a*b^4*x^4 + 105*a^2*b^3*x^3 + 84*a^3*b^2*x^2 + 35*a^4*b*x + 6*a^5)*e^5 + (35
*b^5*d*x^4 + 70*a*b^4*d*x^3 + 63*a^2*b^3*d*x^2 + 28*a^3*b^2*d*x + 5*a^4*b*d)*e^4 + (35*b^5*d^2*x^3 + 42*a*b^4*
d^2*x^2 + 21*a^2*b^3*d^2*x + 4*a^3*b^2*d^2)*e^3 + (21*b^5*d^3*x^2 + 14*a*b^4*d^3*x + 3*a^2*b^3*d^3)*e^2 + (7*b
^5*d^4*x + 2*a*b^4*d^4)*e)/(x^7*e^13 + 7*d*x^6*e^12 + 21*d^2*x^5*e^11 + 35*d^3*x^4*e^10 + 35*d^4*x^3*e^9 + 21*
d^5*x^2*e^8 + 7*d^6*x*e^7 + d^7*e^6)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 381 vs. \(2 (76) = 152\).
time = 0.98, size = 381, normalized size = 3.89 \begin {gather*} -\frac {{\left (21 \, b^{5} x^{5} e^{5} \mathrm {sgn}\left (b x + a\right ) + 35 \, b^{5} d x^{4} e^{4} \mathrm {sgn}\left (b x + a\right ) + 35 \, b^{5} d^{2} x^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 21 \, b^{5} d^{3} x^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 7 \, b^{5} d^{4} x e \mathrm {sgn}\left (b x + a\right ) + b^{5} d^{5} \mathrm {sgn}\left (b x + a\right ) + 70 \, a b^{4} x^{4} e^{5} \mathrm {sgn}\left (b x + a\right ) + 70 \, a b^{4} d x^{3} e^{4} \mathrm {sgn}\left (b x + a\right ) + 42 \, a b^{4} d^{2} x^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 14 \, a b^{4} d^{3} x e^{2} \mathrm {sgn}\left (b x + a\right ) + 2 \, a b^{4} d^{4} e \mathrm {sgn}\left (b x + a\right ) + 105 \, a^{2} b^{3} x^{3} e^{5} \mathrm {sgn}\left (b x + a\right ) + 63 \, a^{2} b^{3} d x^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 21 \, a^{2} b^{3} d^{2} x e^{3} \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{2} b^{3} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + 84 \, a^{3} b^{2} x^{2} e^{5} \mathrm {sgn}\left (b x + a\right ) + 28 \, a^{3} b^{2} d x e^{4} \mathrm {sgn}\left (b x + a\right ) + 4 \, a^{3} b^{2} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 35 \, a^{4} b x e^{5} \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 )\right )} e^{\left (-6\right )}}{42 \, {\left (x e + d\right )}^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.74, size = 687, normalized size = 7.01 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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