### 3.145 $$\int \frac{a+b x^2+c x^4}{x^{10} \sqrt{d-e x} \sqrt{d+e x}} \, dx$$

Optimal. Leaf size=292 $-\frac{8 e^4 \left (d^2-e^2 x^2\right ) \left (16 a e^4+18 b d^2 e^2+21 c d^4\right )}{315 d^{10} x \sqrt{d-e x} \sqrt{d+e x}}-\frac{4 e^2 \left (d^2-e^2 x^2\right ) \left (16 a e^4+18 b d^2 e^2+21 c d^4\right )}{315 d^8 x^3 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (d^2-e^2 x^2\right ) \left (16 a e^4+18 b d^2 e^2+21 c d^4\right )}{105 d^6 x^5 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (d^2-e^2 x^2\right ) \left (8 a e^2+9 b d^2\right )}{63 d^4 x^7 \sqrt{d-e x} \sqrt{d+e x}}-\frac{a \left (d^2-e^2 x^2\right )}{9 d^2 x^9 \sqrt{d-e x} \sqrt{d+e x}}$

[Out]

-(a*(d^2 - e^2*x^2))/(9*d^2*x^9*Sqrt[d - e*x]*Sqrt[d + e*x]) - ((9*b*d^2 + 8*a*e^2)*(d^2 - e^2*x^2))/(63*d^4*x
^7*Sqrt[d - e*x]*Sqrt[d + e*x]) - ((21*c*d^4 + 18*b*d^2*e^2 + 16*a*e^4)*(d^2 - e^2*x^2))/(105*d^6*x^5*Sqrt[d -
e*x]*Sqrt[d + e*x]) - (4*e^2*(21*c*d^4 + 18*b*d^2*e^2 + 16*a*e^4)*(d^2 - e^2*x^2))/(315*d^8*x^3*Sqrt[d - e*x]
*Sqrt[d + e*x]) - (8*e^4*(21*c*d^4 + 18*b*d^2*e^2 + 16*a*e^4)*(d^2 - e^2*x^2))/(315*d^10*x*Sqrt[d - e*x]*Sqrt[
d + e*x])

________________________________________________________________________________________

Rubi [A]  time = 0.241533, antiderivative size = 292, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 35, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.143, Rules used = {520, 1265, 453, 271, 264} $-\frac{8 e^4 \left (d^2-e^2 x^2\right ) \left (16 a e^4+18 b d^2 e^2+21 c d^4\right )}{315 d^{10} x \sqrt{d-e x} \sqrt{d+e x}}-\frac{4 e^2 \left (d^2-e^2 x^2\right ) \left (16 a e^4+18 b d^2 e^2+21 c d^4\right )}{315 d^8 x^3 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (d^2-e^2 x^2\right ) \left (16 a e^4+18 b d^2 e^2+21 c d^4\right )}{105 d^6 x^5 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (d^2-e^2 x^2\right ) \left (8 a e^2+9 b d^2\right )}{63 d^4 x^7 \sqrt{d-e x} \sqrt{d+e x}}-\frac{a \left (d^2-e^2 x^2\right )}{9 d^2 x^9 \sqrt{d-e x} \sqrt{d+e x}}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + b*x^2 + c*x^4)/(x^10*Sqrt[d - e*x]*Sqrt[d + e*x]),x]

[Out]

-(a*(d^2 - e^2*x^2))/(9*d^2*x^9*Sqrt[d - e*x]*Sqrt[d + e*x]) - ((9*b*d^2 + 8*a*e^2)*(d^2 - e^2*x^2))/(63*d^4*x
^7*Sqrt[d - e*x]*Sqrt[d + e*x]) - ((21*c*d^4 + 18*b*d^2*e^2 + 16*a*e^4)*(d^2 - e^2*x^2))/(105*d^6*x^5*Sqrt[d -
e*x]*Sqrt[d + e*x]) - (4*e^2*(21*c*d^4 + 18*b*d^2*e^2 + 16*a*e^4)*(d^2 - e^2*x^2))/(315*d^8*x^3*Sqrt[d - e*x]
*Sqrt[d + e*x]) - (8*e^4*(21*c*d^4 + 18*b*d^2*e^2 + 16*a*e^4)*(d^2 - e^2*x^2))/(315*d^10*x*Sqrt[d - e*x]*Sqrt[
d + e*x])

Rule 520

Int[(u_.)*((c_) + (d_.)*(x_)^(n_.) + (e_.)*(x_)^(n2_.))^(q_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b
2_.)*(x_)^(non2_.))^(p_.), x_Symbol] :> Dist[((a1 + b1*x^(n/2))^FracPart[p]*(a2 + b2*x^(n/2))^FracPart[p])/(a1
*a2 + b1*b2*x^n)^FracPart[p], Int[u*(a1*a2 + b1*b2*x^n)^p*(c + d*x^n + e*x^(2*n))^q, x], x] /; FreeQ[{a1, b1,
a2, b2, c, d, e, n, p, q}, x] && EqQ[non2, n/2] && EqQ[n2, 2*n] && EqQ[a2*b1 + a1*b2, 0]

Rule 1265

Int[((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Wit
h[{Qx = PolynomialQuotient[(a + b*x^2 + c*x^4)^p, f*x, x], R = PolynomialRemainder[(a + b*x^2 + c*x^4)^p, f*x,
x]}, Simp[(R*(f*x)^(m + 1)*(d + e*x^2)^(q + 1))/(d*f*(m + 1)), x] + Dist[1/(d*f^2*(m + 1)), Int[(f*x)^(m + 2)
*(d + e*x^2)^q*ExpandToSum[(d*f*(m + 1)*Qx)/x - e*R*(m + 2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e, f, q},
x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && LtQ[m, -1]

Rule 453

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(c*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(a*e*(m + 1)), x] + Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)), In
t[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[n] ||
GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) &&  !ILtQ[p, -1]

Rule 271

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

Rule 264

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

Rubi steps

\begin{align*} \int \frac{a+b x^2+c x^4}{x^{10} \sqrt{d-e x} \sqrt{d+e x}} \, dx &=\frac{\sqrt{d^2-e^2 x^2} \int \frac{a+b x^2+c x^4}{x^{10} \sqrt{d^2-e^2 x^2}} \, dx}{\sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{a \left (d^2-e^2 x^2\right )}{9 d^2 x^9 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\sqrt{d^2-e^2 x^2} \int \frac{-9 b d^2-8 a e^2-9 c d^2 x^2}{x^8 \sqrt{d^2-e^2 x^2}} \, dx}{9 d^2 \sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{a \left (d^2-e^2 x^2\right )}{9 d^2 x^9 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (9 b d^2+8 a e^2\right ) \left (d^2-e^2 x^2\right )}{63 d^4 x^7 \sqrt{d-e x} \sqrt{d+e x}}+\frac{\left (\left (63 c d^4-6 e^2 \left (-9 b d^2-8 a e^2\right )\right ) \sqrt{d^2-e^2 x^2}\right ) \int \frac{1}{x^6 \sqrt{d^2-e^2 x^2}} \, dx}{63 d^4 \sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{a \left (d^2-e^2 x^2\right )}{9 d^2 x^9 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (9 b d^2+8 a e^2\right ) \left (d^2-e^2 x^2\right )}{63 d^4 x^7 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (21 c d^4+18 b d^2 e^2+16 a e^4\right ) \left (d^2-e^2 x^2\right )}{105 d^6 x^5 \sqrt{d-e x} \sqrt{d+e x}}+\frac{\left (4 e^2 \left (63 c d^4-6 e^2 \left (-9 b d^2-8 a e^2\right )\right ) \sqrt{d^2-e^2 x^2}\right ) \int \frac{1}{x^4 \sqrt{d^2-e^2 x^2}} \, dx}{315 d^6 \sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{a \left (d^2-e^2 x^2\right )}{9 d^2 x^9 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (9 b d^2+8 a e^2\right ) \left (d^2-e^2 x^2\right )}{63 d^4 x^7 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (21 c d^4+18 b d^2 e^2+16 a e^4\right ) \left (d^2-e^2 x^2\right )}{105 d^6 x^5 \sqrt{d-e x} \sqrt{d+e x}}-\frac{4 e^2 \left (21 c d^4+18 b d^2 e^2+16 a e^4\right ) \left (d^2-e^2 x^2\right )}{315 d^8 x^3 \sqrt{d-e x} \sqrt{d+e x}}+\frac{\left (8 e^4 \left (63 c d^4-6 e^2 \left (-9 b d^2-8 a e^2\right )\right ) \sqrt{d^2-e^2 x^2}\right ) \int \frac{1}{x^2 \sqrt{d^2-e^2 x^2}} \, dx}{945 d^8 \sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{a \left (d^2-e^2 x^2\right )}{9 d^2 x^9 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (9 b d^2+8 a e^2\right ) \left (d^2-e^2 x^2\right )}{63 d^4 x^7 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (21 c d^4+18 b d^2 e^2+16 a e^4\right ) \left (d^2-e^2 x^2\right )}{105 d^6 x^5 \sqrt{d-e x} \sqrt{d+e x}}-\frac{4 e^2 \left (21 c d^4+18 b d^2 e^2+16 a e^4\right ) \left (d^2-e^2 x^2\right )}{315 d^8 x^3 \sqrt{d-e x} \sqrt{d+e x}}-\frac{8 e^4 \left (21 c d^4+18 b d^2 e^2+16 a e^4\right ) \left (d^2-e^2 x^2\right )}{315 d^{10} x \sqrt{d-e x} \sqrt{d+e x}}\\ \end{align*}

Mathematica [A]  time = 0.180761, size = 158, normalized size = 0.54 $-\frac{\sqrt{d-e x} \sqrt{d+e x} \left (a \left (40 d^6 e^2 x^2+48 d^4 e^4 x^4+64 d^2 e^6 x^6+35 d^8+128 e^8 x^8\right )+9 b \left (6 d^6 e^2 x^4+8 d^4 e^4 x^6+16 d^2 e^6 x^8+5 d^8 x^2\right )+21 c d^4 x^4 \left (4 d^2 e^2 x^2+3 d^4+8 e^4 x^4\right )\right )}{315 d^{10} x^9}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a + b*x^2 + c*x^4)/(x^10*Sqrt[d - e*x]*Sqrt[d + e*x]),x]

[Out]

-(Sqrt[d - e*x]*Sqrt[d + e*x]*(21*c*d^4*x^4*(3*d^4 + 4*d^2*e^2*x^2 + 8*e^4*x^4) + 9*b*(5*d^8*x^2 + 6*d^6*e^2*x
^4 + 8*d^4*e^4*x^6 + 16*d^2*e^6*x^8) + a*(35*d^8 + 40*d^6*e^2*x^2 + 48*d^4*e^4*x^4 + 64*d^2*e^6*x^6 + 128*e^8*
x^8)))/(315*d^10*x^9)

________________________________________________________________________________________

Maple [A]  time = 0.007, size = 154, normalized size = 0.5 \begin{align*} -{\frac{128\,a{e}^{8}{x}^{8}+144\,b{d}^{2}{e}^{6}{x}^{8}+168\,c{d}^{4}{e}^{4}{x}^{8}+64\,a{d}^{2}{e}^{6}{x}^{6}+72\,b{d}^{4}{e}^{4}{x}^{6}+84\,c{d}^{6}{e}^{2}{x}^{6}+48\,a{d}^{4}{e}^{4}{x}^{4}+54\,b{d}^{6}{e}^{2}{x}^{4}+63\,c{d}^{8}{x}^{4}+40\,a{d}^{6}{e}^{2}{x}^{2}+45\,b{d}^{8}{x}^{2}+35\,a{d}^{8}}{315\,{x}^{9}{d}^{10}}\sqrt{ex+d}\sqrt{-ex+d}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2+a)/x^10/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x)

[Out]

-1/315*(e*x+d)^(1/2)*(-e*x+d)^(1/2)*(128*a*e^8*x^8+144*b*d^2*e^6*x^8+168*c*d^4*e^4*x^8+64*a*d^2*e^6*x^6+72*b*d
^4*e^4*x^6+84*c*d^6*e^2*x^6+48*a*d^4*e^4*x^4+54*b*d^6*e^2*x^4+63*c*d^8*x^4+40*a*d^6*e^2*x^2+45*b*d^8*x^2+35*a*
d^8)/x^9/d^10

________________________________________________________________________________________

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)/x^10/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.86087, size = 327, normalized size = 1.12 \begin{align*} -\frac{{\left (35 \, a d^{8} + 8 \,{\left (21 \, c d^{4} e^{4} + 18 \, b d^{2} e^{6} + 16 \, a e^{8}\right )} x^{8} + 4 \,{\left (21 \, c d^{6} e^{2} + 18 \, b d^{4} e^{4} + 16 \, a d^{2} e^{6}\right )} x^{6} + 3 \,{\left (21 \, c d^{8} + 18 \, b d^{6} e^{2} + 16 \, a d^{4} e^{4}\right )} x^{4} + 5 \,{\left (9 \, b d^{8} + 8 \, a d^{6} e^{2}\right )} x^{2}\right )} \sqrt{e x + d} \sqrt{-e x + d}}{315 \, d^{10} x^{9}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)/x^10/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

-1/315*(35*a*d^8 + 8*(21*c*d^4*e^4 + 18*b*d^2*e^6 + 16*a*e^8)*x^8 + 4*(21*c*d^6*e^2 + 18*b*d^4*e^4 + 16*a*d^2*
e^6)*x^6 + 3*(21*c*d^8 + 18*b*d^6*e^2 + 16*a*d^4*e^4)*x^4 + 5*(9*b*d^8 + 8*a*d^6*e^2)*x^2)*sqrt(e*x + d)*sqrt(
-e*x + d)/(d^10*x^9)

________________________________________________________________________________________

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2+a)/x**10/(-e*x+d)**(1/2)/(e*x+d)**(1/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 3.4791, size = 2607, normalized size = 8.93 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)/x^10/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

-4/315*(315*c*d^4*((sqrt(2)*sqrt(d) - sqrt(-x*e + d))/sqrt(x*e + d) - sqrt(x*e + d)/(sqrt(2)*sqrt(d) - sqrt(-x
*e + d)))^17*e^6 + 315*b*d^2*((sqrt(2)*sqrt(d) - sqrt(-x*e + d))/sqrt(x*e + d) - sqrt(x*e + d)/(sqrt(2)*sqrt(d
) - sqrt(-x*e + d)))^17*e^8 - 6720*c*d^4*((sqrt(2)*sqrt(d) - sqrt(-x*e + d))/sqrt(x*e + d) - sqrt(x*e + d)/(sq
rt(2)*sqrt(d) - sqrt(-x*e + d)))^15*e^6 + 315*a*((sqrt(2)*sqrt(d) - sqrt(-x*e + d))/sqrt(x*e + d) - sqrt(x*e +
d)/(sqrt(2)*sqrt(d) - sqrt(-x*e + d)))^17*e^10 - 5040*b*d^2*((sqrt(2)*sqrt(d) - sqrt(-x*e + d))/sqrt(x*e + d)
- sqrt(x*e + d)/(sqrt(2)*sqrt(d) - sqrt(-x*e + d)))^15*e^8 + 76608*c*d^4*((sqrt(2)*sqrt(d) - sqrt(-x*e + d))/
sqrt(x*e + d) - sqrt(x*e + d)/(sqrt(2)*sqrt(d) - sqrt(-x*e + d)))^13*e^6 - 3360*a*((sqrt(2)*sqrt(d) - sqrt(-x*
e + d))/sqrt(x*e + d) - sqrt(x*e + d)/(sqrt(2)*sqrt(d) - sqrt(-x*e + d)))^15*e^10 + 68544*b*d^2*((sqrt(2)*sqrt
(d) - sqrt(-x*e + d))/sqrt(x*e + d) - sqrt(x*e + d)/(sqrt(2)*sqrt(d) - sqrt(-x*e + d)))^13*e^8 - 580608*c*d^4*
((sqrt(2)*sqrt(d) - sqrt(-x*e + d))/sqrt(x*e + d) - sqrt(x*e + d)/(sqrt(2)*sqrt(d) - sqrt(-x*e + d)))^11*e^6 +
76608*a*((sqrt(2)*sqrt(d) - sqrt(-x*e + d))/sqrt(x*e + d) - sqrt(x*e + d)/(sqrt(2)*sqrt(d) - sqrt(-x*e + d)))
^13*e^10 - 509184*b*d^2*((sqrt(2)*sqrt(d) - sqrt(-x*e + d))/sqrt(x*e + d) - sqrt(x*e + d)/(sqrt(2)*sqrt(d) - s
qrt(-x*e + d)))^11*e^8 + 2892288*c*d^4*((sqrt(2)*sqrt(d) - sqrt(-x*e + d))/sqrt(x*e + d) - sqrt(x*e + d)/(sqrt
(2)*sqrt(d) - sqrt(-x*e + d)))^9*e^6 - 327168*a*((sqrt(2)*sqrt(d) - sqrt(-x*e + d))/sqrt(x*e + d) - sqrt(x*e +
d)/(sqrt(2)*sqrt(d) - sqrt(-x*e + d)))^11*e^10 + 2363904*b*d^2*((sqrt(2)*sqrt(d) - sqrt(-x*e + d))/sqrt(x*e +
d) - sqrt(x*e + d)/(sqrt(2)*sqrt(d) - sqrt(-x*e + d)))^9*e^8 - 9289728*c*d^4*((sqrt(2)*sqrt(d) - sqrt(-x*e +
d))/sqrt(x*e + d) - sqrt(x*e + d)/(sqrt(2)*sqrt(d) - sqrt(-x*e + d)))^7*e^6 + 2728448*a*((sqrt(2)*sqrt(d) - sq
rt(-x*e + d))/sqrt(x*e + d) - sqrt(x*e + d)/(sqrt(2)*sqrt(d) - sqrt(-x*e + d)))^9*e^10 - 8146944*b*d^2*((sqrt(
2)*sqrt(d) - sqrt(-x*e + d))/sqrt(x*e + d) - sqrt(x*e + d)/(sqrt(2)*sqrt(d) - sqrt(-x*e + d)))^7*e^8 + 1961164
8*c*d^4*((sqrt(2)*sqrt(d) - sqrt(-x*e + d))/sqrt(x*e + d) - sqrt(x*e + d)/(sqrt(2)*sqrt(d) - sqrt(-x*e + d)))^
5*e^6 - 5234688*a*((sqrt(2)*sqrt(d) - sqrt(-x*e + d))/sqrt(x*e + d) - sqrt(x*e + d)/(sqrt(2)*sqrt(d) - sqrt(-x
*e + d)))^7*e^10 + 17547264*b*d^2*((sqrt(2)*sqrt(d) - sqrt(-x*e + d))/sqrt(x*e + d) - sqrt(x*e + d)/(sqrt(2)*s
qrt(d) - sqrt(-x*e + d)))^5*e^8 - 27525120*c*d^4*((sqrt(2)*sqrt(d) - sqrt(-x*e + d))/sqrt(x*e + d) - sqrt(x*e
+ d)/(sqrt(2)*sqrt(d) - sqrt(-x*e + d)))^3*e^6 + 19611648*a*((sqrt(2)*sqrt(d) - sqrt(-x*e + d))/sqrt(x*e + d)
- sqrt(x*e + d)/(sqrt(2)*sqrt(d) - sqrt(-x*e + d)))^5*e^10 - 20643840*b*d^2*((sqrt(2)*sqrt(d) - sqrt(-x*e + d)
)/sqrt(x*e + d) - sqrt(x*e + d)/(sqrt(2)*sqrt(d) - sqrt(-x*e + d)))^3*e^8 + 20643840*c*d^4*((sqrt(2)*sqrt(d) -
sqrt(-x*e + d))/sqrt(x*e + d) - sqrt(x*e + d)/(sqrt(2)*sqrt(d) - sqrt(-x*e + d)))*e^6 - 13762560*a*((sqrt(2)*
sqrt(d) - sqrt(-x*e + d))/sqrt(x*e + d) - sqrt(x*e + d)/(sqrt(2)*sqrt(d) - sqrt(-x*e + d)))^3*e^10 + 20643840*
b*d^2*((sqrt(2)*sqrt(d) - sqrt(-x*e + d))/sqrt(x*e + d) - sqrt(x*e + d)/(sqrt(2)*sqrt(d) - sqrt(-x*e + d)))*e^
8 + 20643840*a*((sqrt(2)*sqrt(d) - sqrt(-x*e + d))/sqrt(x*e + d) - sqrt(x*e + d)/(sqrt(2)*sqrt(d) - sqrt(-x*e
+ d)))*e^10)*e^(-1)/((((sqrt(2)*sqrt(d) - sqrt(-x*e + d))/sqrt(x*e + d) - sqrt(x*e + d)/(sqrt(2)*sqrt(d) - sqr
t(-x*e + d)))^2 - 4)^9*d^10)