3.9.58 \(\int \frac {1}{(d+e x)^4 (d^2-e^2 x^2)^{7/2}} \, dx\) [858]

Optimal. Leaf size=205 \[ \frac {48 x}{715 d^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {1}{13 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {9}{143 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8}{143 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8}{143 d^4 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {64 x}{715 d^8 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {128 x}{715 d^{10} \sqrt {d^2-e^2 x^2}} \]

[Out]

48/715*x/d^6/(-e^2*x^2+d^2)^(5/2)-1/13/d/e/(e*x+d)^4/(-e^2*x^2+d^2)^(5/2)-9/143/d^2/e/(e*x+d)^3/(-e^2*x^2+d^2)
^(5/2)-8/143/d^3/e/(e*x+d)^2/(-e^2*x^2+d^2)^(5/2)-8/143/d^4/e/(e*x+d)/(-e^2*x^2+d^2)^(5/2)+64/715*x/d^8/(-e^2*
x^2+d^2)^(3/2)+128/715*x/d^10/(-e^2*x^2+d^2)^(1/2)

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Rubi [A]
time = 0.06, antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {673, 198, 197} \begin {gather*} -\frac {9}{143 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {1}{13 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {128 x}{715 d^{10} \sqrt {d^2-e^2 x^2}}+\frac {64 x}{715 d^8 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {48 x}{715 d^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8}{143 d^4 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8}{143 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^4*(d^2 - e^2*x^2)^(7/2)),x]

[Out]

(48*x)/(715*d^6*(d^2 - e^2*x^2)^(5/2)) - 1/(13*d*e*(d + e*x)^4*(d^2 - e^2*x^2)^(5/2)) - 9/(143*d^2*e*(d + e*x)
^3*(d^2 - e^2*x^2)^(5/2)) - 8/(143*d^3*e*(d + e*x)^2*(d^2 - e^2*x^2)^(5/2)) - 8/(143*d^4*e*(d + e*x)*(d^2 - e^
2*x^2)^(5/2)) + (64*x)/(715*d^8*(d^2 - e^2*x^2)^(3/2)) + (128*x)/(715*d^10*Sqrt[d^2 - e^2*x^2])

Rule 197

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1)/a), x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 198

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, n, p}, x] && ILtQ[Simplify[1/n + p +
1], 0] && NeQ[p, -1]

Rule 673

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-e)*(d + e*x)^m*((a + c*x^2)^(p +
1)/(2*c*d*(m + p + 1))), x] + Dist[Simplify[m + 2*p + 2]/(2*d*(m + p + 1)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^
p, x], x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && ILtQ[Simplify[m + 2*p +
2], 0]

Rubi steps

\begin {align*} \int \frac {1}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx &=-\frac {1}{13 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {9 \int \frac {1}{(d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{13 d}\\ &=-\frac {1}{13 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {9}{143 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {72 \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{143 d^2}\\ &=-\frac {1}{13 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {9}{143 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8}{143 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {56 \int \frac {1}{(d+e x) \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{143 d^3}\\ &=-\frac {1}{13 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {9}{143 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8}{143 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8}{143 d^4 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {48 \int \frac {1}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx}{143 d^4}\\ &=\frac {48 x}{715 d^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {1}{13 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {9}{143 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8}{143 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8}{143 d^4 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {192 \int \frac {1}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{715 d^6}\\ &=\frac {48 x}{715 d^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {1}{13 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {9}{143 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8}{143 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8}{143 d^4 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {64 x}{715 d^8 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {128 \int \frac {1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{715 d^8}\\ &=\frac {48 x}{715 d^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {1}{13 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {9}{143 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8}{143 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8}{143 d^4 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {64 x}{715 d^8 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {128 x}{715 d^{10} \sqrt {d^2-e^2 x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.57, size = 137, normalized size = 0.67 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (-180 d^9-5 d^8 e x+800 d^7 e^2 x^2+1080 d^6 e^3 x^3-320 d^5 e^4 x^4-1552 d^4 e^5 x^5-768 d^3 e^6 x^6+448 d^2 e^7 x^7+512 d e^8 x^8+128 e^9 x^9\right )}{715 d^{10} e (d-e x)^3 (d+e x)^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^4*(d^2 - e^2*x^2)^(7/2)),x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(-180*d^9 - 5*d^8*e*x + 800*d^7*e^2*x^2 + 1080*d^6*e^3*x^3 - 320*d^5*e^4*x^4 - 1552*d^4*e
^5*x^5 - 768*d^3*e^6*x^6 + 448*d^2*e^7*x^7 + 512*d*e^8*x^8 + 128*e^9*x^9))/(715*d^10*e*(d - e*x)^3*(d + e*x)^7
)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(378\) vs. \(2(177)=354\).
time = 0.49, size = 379, normalized size = 1.85

method	result	size
gosper	\(-\frac {\left (-e x +d \right ) \left (-128 e^{9} x^{9}-512 e^{8} x^{8} d -448 e^{7} x^{7} d^{2}+768 d^{3} e^{6} x^{6}+1552 d^{4} e^{5} x^{5}+320 d^{5} e^{4} x^{4}-1080 e^{3} x^{3} d^{6}-800 d^{7} e^{2} x^{2}+5 x \,d^{8} e +180 d^{9}\right )}{715 \left (e x +d \right )^{3} d^{10} e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}\)	\(132\)
trager	\(-\frac {\left (-128 e^{9} x^{9}-512 e^{8} x^{8} d -448 e^{7} x^{7} d^{2}+768 d^{3} e^{6} x^{6}+1552 d^{4} e^{5} x^{5}+320 d^{5} e^{4} x^{4}-1080 e^{3} x^{3} d^{6}-800 d^{7} e^{2} x^{2}+5 x \,d^{8} e +180 d^{9}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{715 d^{10} \left (e x +d \right )^{7} \left (-e x +d \right )^{3} e}\)	\(134\)
default	\(\frac {-\frac {1}{13 d e \left (x +\frac {d}{e}\right )^{4} \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}+\frac {9 e \left (-\frac {1}{11 d e \left (x +\frac {d}{e}\right )^{3} \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}+\frac {8 e \left (-\frac {1}{9 d e \left (x +\frac {d}{e}\right )^{2} \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}+\frac {7 e \left (-\frac {1}{7 d e \left (x +\frac {d}{e}\right ) \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}+\frac {6 e \left (-\frac {-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e}{10 d^{2} e^{2} \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}+\frac {-\frac {2 \left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right )}{15 d^{2} e^{2} \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}-\frac {4 \left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right )}{15 e^{2} d^{4} \sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}}{d^{2}}\right )}{7 d}\right )}{9 d}\right )}{11 d}\right )}{13 d}}{e^{4}}\)	\(379\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^4/(-e^2*x^2+d^2)^(7/2),x,method=_RETURNVERBOSE)

[Out]

1/e^4*(-1/13/d/e/(x+d/e)^4/(-e^2*(x+d/e)^2+2*d*e*(x+d/e))^(5/2)+9/13*e/d*(-1/11/d/e/(x+d/e)^3/(-e^2*(x+d/e)^2+
2*d*e*(x+d/e))^(5/2)+8/11*e/d*(-1/9/d/e/(x+d/e)^2/(-e^2*(x+d/e)^2+2*d*e*(x+d/e))^(5/2)+7/9*e/d*(-1/7/d/e/(x+d/
e)/(-e^2*(x+d/e)^2+2*d*e*(x+d/e))^(5/2)+6/7*e/d*(-1/10*(-2*e^2*(x+d/e)+2*d*e)/d^2/e^2/(-e^2*(x+d/e)^2+2*d*e*(x
+d/e))^(5/2)+4/5/d^2*(-1/6*(-2*e^2*(x+d/e)+2*d*e)/d^2/e^2/(-e^2*(x+d/e)^2+2*d*e*(x+d/e))^(3/2)-1/3/e^2/d^4*(-2
*e^2*(x+d/e)+2*d*e)/(-e^2*(x+d/e)^2+2*d*e*(x+d/e))^(1/2)))))))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 370 vs. \(2 (170) = 340\).
time = 0.29, size = 370, normalized size = 1.80 \begin {gather*} -\frac {1}{13 \, {\left ({\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d x^{4} e^{5} + 4 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} x^{3} e^{4} + 6 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} x^{2} e^{3} + 4 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4} x e^{2} + {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{5} e\right )}} - \frac {9}{143 \, {\left ({\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} x^{3} e^{4} + 3 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} x^{2} e^{3} + 3 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4} x e^{2} + {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{5} e\right )}} - \frac {8}{143 \, {\left ({\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} x^{2} e^{3} + 2 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4} x e^{2} + {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{5} e\right )}} - \frac {8}{143 \, {\left ({\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4} x e^{2} + {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{5} e\right )}} + \frac {48 \, x}{715 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{6}} + \frac {64 \, x}{715 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{8}} + \frac {128 \, x}{715 \, \sqrt {-x^{2} e^{2} + d^{2}} d^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^4/(-e^2*x^2+d^2)^(7/2),x, algorithm="maxima")

[Out]

-1/13/((-x^2*e^2 + d^2)^(5/2)*d*x^4*e^5 + 4*(-x^2*e^2 + d^2)^(5/2)*d^2*x^3*e^4 + 6*(-x^2*e^2 + d^2)^(5/2)*d^3*
x^2*e^3 + 4*(-x^2*e^2 + d^2)^(5/2)*d^4*x*e^2 + (-x^2*e^2 + d^2)^(5/2)*d^5*e) - 9/143/((-x^2*e^2 + d^2)^(5/2)*d
^2*x^3*e^4 + 3*(-x^2*e^2 + d^2)^(5/2)*d^3*x^2*e^3 + 3*(-x^2*e^2 + d^2)^(5/2)*d^4*x*e^2 + (-x^2*e^2 + d^2)^(5/2
)*d^5*e) - 8/143/((-x^2*e^2 + d^2)^(5/2)*d^3*x^2*e^3 + 2*(-x^2*e^2 + d^2)^(5/2)*d^4*x*e^2 + (-x^2*e^2 + d^2)^(
5/2)*d^5*e) - 8/143/((-x^2*e^2 + d^2)^(5/2)*d^4*x*e^2 + (-x^2*e^2 + d^2)^(5/2)*d^5*e) + 48/715*x/((-x^2*e^2 +
d^2)^(5/2)*d^6) + 64/715*x/((-x^2*e^2 + d^2)^(3/2)*d^8) + 128/715*x/(sqrt(-x^2*e^2 + d^2)*d^10)

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Fricas [A]
time = 4.09, size = 291, normalized size = 1.42 \begin {gather*} -\frac {180 \, x^{10} e^{10} + 720 \, d x^{9} e^{9} + 540 \, d^{2} x^{8} e^{8} - 1440 \, d^{3} x^{7} e^{7} - 2520 \, d^{4} x^{6} e^{6} + 2520 \, d^{6} x^{4} e^{4} + 1440 \, d^{7} x^{3} e^{3} - 540 \, d^{8} x^{2} e^{2} - 720 \, d^{9} x e - 180 \, d^{10} + {\left (128 \, x^{9} e^{9} + 512 \, d x^{8} e^{8} + 448 \, d^{2} x^{7} e^{7} - 768 \, d^{3} x^{6} e^{6} - 1552 \, d^{4} x^{5} e^{5} - 320 \, d^{5} x^{4} e^{4} + 1080 \, d^{6} x^{3} e^{3} + 800 \, d^{7} x^{2} e^{2} - 5 \, d^{8} x e - 180 \, d^{9}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{715 \, {\left (d^{10} x^{10} e^{11} + 4 \, d^{11} x^{9} e^{10} + 3 \, d^{12} x^{8} e^{9} - 8 \, d^{13} x^{7} e^{8} - 14 \, d^{14} x^{6} e^{7} + 14 \, d^{16} x^{4} e^{5} + 8 \, d^{17} x^{3} e^{4} - 3 \, d^{18} x^{2} e^{3} - 4 \, d^{19} x e^{2} - d^{20} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^4/(-e^2*x^2+d^2)^(7/2),x, algorithm="fricas")

[Out]

-1/715*(180*x^10*e^10 + 720*d*x^9*e^9 + 540*d^2*x^8*e^8 - 1440*d^3*x^7*e^7 - 2520*d^4*x^6*e^6 + 2520*d^6*x^4*e
^4 + 1440*d^7*x^3*e^3 - 540*d^8*x^2*e^2 - 720*d^9*x*e - 180*d^10 + (128*x^9*e^9 + 512*d*x^8*e^8 + 448*d^2*x^7*
e^7 - 768*d^3*x^6*e^6 - 1552*d^4*x^5*e^5 - 320*d^5*x^4*e^4 + 1080*d^6*x^3*e^3 + 800*d^7*x^2*e^2 - 5*d^8*x*e -
180*d^9)*sqrt(-x^2*e^2 + d^2))/(d^10*x^10*e^11 + 4*d^11*x^9*e^10 + 3*d^12*x^8*e^9 - 8*d^13*x^7*e^8 - 14*d^14*x
^6*e^7 + 14*d^16*x^4*e^5 + 8*d^17*x^3*e^4 - 3*d^18*x^2*e^3 - 4*d^19*x*e^2 - d^20*e)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}} \left (d + e x\right )^{4}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)**4/(-e**2*x**2+d**2)**(7/2),x)

[Out]

Integral(1/((-(-d + e*x)*(d + e*x))**(7/2)*(d + e*x)**4), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^4/(-e^2*x^2+d^2)^(7/2),x, algorithm="giac")

[Out]

integrate(1/((-x^2*e^2 + d^2)^(7/2)*(x*e + d)^4), x)

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Mupad [B]
time = 0.96, size = 242, normalized size = 1.18 \begin {gather*} \frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {64\,x}{715\,d^8}+\frac {189}{4576\,d^7\,e}\right )}{{\left (d+e\,x\right )}^2\,{\left (d-e\,x\right )}^2}+\frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {1139\,x}{5720\,d^6}-\frac {427}{2288\,d^5\,e}\right )}{{\left (d+e\,x\right )}^3\,{\left (d-e\,x\right )}^3}-\frac {\sqrt {d^2-e^2\,x^2}}{104\,d^4\,e\,{\left (d+e\,x\right )}^7}-\frac {51\,\sqrt {d^2-e^2\,x^2}}{2288\,d^5\,e\,{\left (d+e\,x\right )}^6}-\frac {19\,\sqrt {d^2-e^2\,x^2}}{572\,d^6\,e\,{\left (d+e\,x\right )}^5}-\frac {189\,\sqrt {d^2-e^2\,x^2}}{4576\,d^7\,e\,{\left (d+e\,x\right )}^4}+\frac {128\,x\,\sqrt {d^2-e^2\,x^2}}{715\,d^{10}\,\left (d+e\,x\right )\,\left (d-e\,x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((d^2 - e^2*x^2)^(7/2)*(d + e*x)^4),x)

[Out]

((d^2 - e^2*x^2)^(1/2)*((64*x)/(715*d^8) + 189/(4576*d^7*e)))/((d + e*x)^2*(d - e*x)^2) + ((d^2 - e^2*x^2)^(1/
2)*((1139*x)/(5720*d^6) - 427/(2288*d^5*e)))/((d + e*x)^3*(d - e*x)^3) - (d^2 - e^2*x^2)^(1/2)/(104*d^4*e*(d +
e*x)^7) - (51*(d^2 - e^2*x^2)^(1/2))/(2288*d^5*e*(d + e*x)^6) - (19*(d^2 - e^2*x^2)^(1/2))/(572*d^6*e*(d + e*
x)^5) - (189*(d^2 - e^2*x^2)^(1/2))/(4576*d^7*e*(d + e*x)^4) + (128*x*(d^2 - e^2*x^2)^(1/2))/(715*d^10*(d + e*
x)*(d - e*x))

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