3.168 \(\int \frac{A+B x^2+C x^4+D x^6}{x^{10} (a+b x^2)^{9/2}} \, dx\)

Optimal. Leaf size=392 \[ -\frac{256 b^2 x \left (128 A b^3-3 a \left (5 a^2 D-12 a b C+24 b^2 B\right )\right )}{315 a^9 \sqrt{a+b x^2}}-\frac{128 b^2 x \left (128 A b^3-3 a \left (5 a^2 D-12 a b C+24 b^2 B\right )\right )}{315 a^8 \left (a+b x^2\right )^{3/2}}-\frac{32 b^2 x \left (128 A b^3-3 a \left (5 a^2 D-12 a b C+24 b^2 B\right )\right )}{105 a^7 \left (a+b x^2\right )^{5/2}}-\frac{16 b^2 x \left (128 A b^3-3 a \left (5 a^2 D-12 a b C+24 b^2 B\right )\right )}{63 a^6 \left (a+b x^2\right )^{7/2}}-\frac{2 b \left (128 A b^3-3 a \left (5 a^2 D-12 a b C+24 b^2 B\right )\right )}{9 a^5 x \left (a+b x^2\right )^{7/2}}+\frac{128 A b^3-3 a \left (5 a^2 D-12 a b C+24 b^2 B\right )}{45 a^4 x^3 \left (a+b x^2\right )^{7/2}}-\frac{32 A b^2-9 a (2 b B-a C)}{45 a^3 x^5 \left (a+b x^2\right )^{7/2}}+\frac{16 A b-9 a B}{63 a^2 x^7 \left (a+b x^2\right )^{7/2}}-\frac{A}{9 a x^9 \left (a+b x^2\right )^{7/2}} \]

[Out]

-A/(9*a*x^9*(a + b*x^2)^(7/2)) + (16*A*b - 9*a*B)/(63*a^2*x^7*(a + b*x^2)^(7/2)) - (32*A*b^2 - 9*a*(2*b*B - a*
C))/(45*a^3*x^5*(a + b*x^2)^(7/2)) + (128*A*b^3 - 3*a*(24*b^2*B - 12*a*b*C + 5*a^2*D))/(45*a^4*x^3*(a + b*x^2)
^(7/2)) - (2*b*(128*A*b^3 - 3*a*(24*b^2*B - 12*a*b*C + 5*a^2*D)))/(9*a^5*x*(a + b*x^2)^(7/2)) - (16*b^2*(128*A
*b^3 - 3*a*(24*b^2*B - 12*a*b*C + 5*a^2*D))*x)/(63*a^6*(a + b*x^2)^(7/2)) - (32*b^2*(128*A*b^3 - 3*a*(24*b^2*B
- 12*a*b*C + 5*a^2*D))*x)/(105*a^7*(a + b*x^2)^(5/2)) - (128*b^2*(128*A*b^3 - 3*a*(24*b^2*B - 12*a*b*C + 5*a^
2*D))*x)/(315*a^8*(a + b*x^2)^(3/2)) - (256*b^2*(128*A*b^3 - 3*a*(24*b^2*B - 12*a*b*C + 5*a^2*D))*x)/(315*a^9*
Sqrt[a + b*x^2])

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Rubi [A]  time = 0.546361, antiderivative size = 380, normalized size of antiderivative = 0.97, number of steps used = 10, number of rules used = 5, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.156, Rules used = {1803, 12, 271, 192, 191} \[ -\frac{256 b^2 x \left (-15 a^3 D-36 a b (2 b B-a C)+128 A b^3\right )}{315 a^9 \sqrt{a+b x^2}}-\frac{128 b^2 x \left (-15 a^3 D-36 a b (2 b B-a C)+128 A b^3\right )}{315 a^8 \left (a+b x^2\right )^{3/2}}-\frac{32 b^2 x \left (-15 a^3 D-36 a b (2 b B-a C)+128 A b^3\right )}{105 a^7 \left (a+b x^2\right )^{5/2}}-\frac{16 b^2 x \left (-15 a^3 D-36 a b (2 b B-a C)+128 A b^3\right )}{63 a^6 \left (a+b x^2\right )^{7/2}}-\frac{2 b \left (-15 a^3 D-36 a b (2 b B-a C)+128 A b^3\right )}{9 a^5 x \left (a+b x^2\right )^{7/2}}+\frac{-15 a^3 D-36 a b (2 b B-a C)+128 A b^3}{45 a^4 x^3 \left (a+b x^2\right )^{7/2}}-\frac{32 A b^2-9 a (2 b B-a C)}{45 a^3 x^5 \left (a+b x^2\right )^{7/2}}+\frac{16 A b-9 a B}{63 a^2 x^7 \left (a+b x^2\right )^{7/2}}-\frac{A}{9 a x^9 \left (a+b x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x^2 + C*x^4 + D*x^6)/(x^10*(a + b*x^2)^(9/2)),x]

[Out]

-A/(9*a*x^9*(a + b*x^2)^(7/2)) + (16*A*b - 9*a*B)/(63*a^2*x^7*(a + b*x^2)^(7/2)) - (32*A*b^2 - 9*a*(2*b*B - a*
C))/(45*a^3*x^5*(a + b*x^2)^(7/2)) + (128*A*b^3 - 36*a*b*(2*b*B - a*C) - 15*a^3*D)/(45*a^4*x^3*(a + b*x^2)^(7/
2)) - (2*b*(128*A*b^3 - 36*a*b*(2*b*B - a*C) - 15*a^3*D))/(9*a^5*x*(a + b*x^2)^(7/2)) - (16*b^2*(128*A*b^3 - 3
6*a*b*(2*b*B - a*C) - 15*a^3*D)*x)/(63*a^6*(a + b*x^2)^(7/2)) - (32*b^2*(128*A*b^3 - 36*a*b*(2*b*B - a*C) - 15
*a^3*D)*x)/(105*a^7*(a + b*x^2)^(5/2)) - (128*b^2*(128*A*b^3 - 36*a*b*(2*b*B - a*C) - 15*a^3*D)*x)/(315*a^8*(a
+ b*x^2)^(3/2)) - (256*b^2*(128*A*b^3 - 36*a*b*(2*b*B - a*C) - 15*a^3*D)*x)/(315*a^9*Sqrt[a + b*x^2])

Rule 1803

Int[(Pq_)*(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{A = Coeff[Pq, x, 0], Q = PolynomialQuotient
[Pq - Coeff[Pq, x, 0], x^2, x]}, Simp[(A*x^(m + 1)*(a + b*x^2)^(p + 1))/(a*(m + 1)), x] + Dist[1/(a*(m + 1)),
Int[x^(m + 2)*(a + b*x^2)^p*(a*(m + 1)*Q - A*b*(m + 2*(p + 1) + 1)), x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq,
x^2] && IntegerQ[m/2] && ILtQ[(m + 1)/2 + p, 0] && LtQ[m + Expon[Pq, x] + 2*p + 1, 0]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 192

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 191

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]

Rubi steps

\begin{align*} \int \frac{A+B x^2+C x^4+D x^6}{x^{10} \left (a+b x^2\right )^{9/2}} \, dx &=-\frac{A}{9 a x^9 \left (a+b x^2\right )^{7/2}}-\frac{\int \frac{16 A b-9 a \left (B+C x^2+D x^4\right )}{x^8 \left (a+b x^2\right )^{9/2}} \, dx}{9 a}\\ &=-\frac{A}{9 a x^9 \left (a+b x^2\right )^{7/2}}+\frac{16 A b-9 a B}{63 a^2 x^7 \left (a+b x^2\right )^{7/2}}+\frac{\int \frac{14 b (16 A b-9 a B)-7 a \left (-9 a C-9 a D x^2\right )}{x^6 \left (a+b x^2\right )^{9/2}} \, dx}{63 a^2}\\ &=-\frac{A}{9 a x^9 \left (a+b x^2\right )^{7/2}}+\frac{16 A b-9 a B}{63 a^2 x^7 \left (a+b x^2\right )^{7/2}}-\frac{32 A b^2-9 a (2 b B-a C)}{45 a^3 x^5 \left (a+b x^2\right )^{7/2}}-\frac{\int \frac{12 b \left (224 A b^2-126 a b B+63 a^2 C\right )-315 a^3 D}{x^4 \left (a+b x^2\right )^{9/2}} \, dx}{315 a^3}\\ &=-\frac{A}{9 a x^9 \left (a+b x^2\right )^{7/2}}+\frac{16 A b-9 a B}{63 a^2 x^7 \left (a+b x^2\right )^{7/2}}-\frac{32 A b^2-9 a (2 b B-a C)}{45 a^3 x^5 \left (a+b x^2\right )^{7/2}}-\frac{\left (128 A b^3-36 a b (2 b B-a C)-15 a^3 D\right ) \int \frac{1}{x^4 \left (a+b x^2\right )^{9/2}} \, dx}{15 a^3}\\ &=-\frac{A}{9 a x^9 \left (a+b x^2\right )^{7/2}}+\frac{16 A b-9 a B}{63 a^2 x^7 \left (a+b x^2\right )^{7/2}}-\frac{32 A b^2-9 a (2 b B-a C)}{45 a^3 x^5 \left (a+b x^2\right )^{7/2}}+\frac{128 A b^3-36 a b (2 b B-a C)-15 a^3 D}{45 a^4 x^3 \left (a+b x^2\right )^{7/2}}+\frac{\left (2 b \left (128 A b^3-36 a b (2 b B-a C)-15 a^3 D\right )\right ) \int \frac{1}{x^2 \left (a+b x^2\right )^{9/2}} \, dx}{9 a^4}\\ &=-\frac{A}{9 a x^9 \left (a+b x^2\right )^{7/2}}+\frac{16 A b-9 a B}{63 a^2 x^7 \left (a+b x^2\right )^{7/2}}-\frac{32 A b^2-9 a (2 b B-a C)}{45 a^3 x^5 \left (a+b x^2\right )^{7/2}}+\frac{128 A b^3-36 a b (2 b B-a C)-15 a^3 D}{45 a^4 x^3 \left (a+b x^2\right )^{7/2}}-\frac{2 b \left (128 A b^3-36 a b (2 b B-a C)-15 a^3 D\right )}{9 a^5 x \left (a+b x^2\right )^{7/2}}-\frac{\left (16 b^2 \left (128 A b^3-36 a b (2 b B-a C)-15 a^3 D\right )\right ) \int \frac{1}{\left (a+b x^2\right )^{9/2}} \, dx}{9 a^5}\\ &=-\frac{A}{9 a x^9 \left (a+b x^2\right )^{7/2}}+\frac{16 A b-9 a B}{63 a^2 x^7 \left (a+b x^2\right )^{7/2}}-\frac{32 A b^2-9 a (2 b B-a C)}{45 a^3 x^5 \left (a+b x^2\right )^{7/2}}+\frac{128 A b^3-36 a b (2 b B-a C)-15 a^3 D}{45 a^4 x^3 \left (a+b x^2\right )^{7/2}}-\frac{2 b \left (128 A b^3-36 a b (2 b B-a C)-15 a^3 D\right )}{9 a^5 x \left (a+b x^2\right )^{7/2}}-\frac{16 b^2 \left (128 A b^3-36 a b (2 b B-a C)-15 a^3 D\right ) x}{63 a^6 \left (a+b x^2\right )^{7/2}}-\frac{\left (32 b^2 \left (128 A b^3-36 a b (2 b B-a C)-15 a^3 D\right )\right ) \int \frac{1}{\left (a+b x^2\right )^{7/2}} \, dx}{21 a^6}\\ &=-\frac{A}{9 a x^9 \left (a+b x^2\right )^{7/2}}+\frac{16 A b-9 a B}{63 a^2 x^7 \left (a+b x^2\right )^{7/2}}-\frac{32 A b^2-9 a (2 b B-a C)}{45 a^3 x^5 \left (a+b x^2\right )^{7/2}}+\frac{128 A b^3-36 a b (2 b B-a C)-15 a^3 D}{45 a^4 x^3 \left (a+b x^2\right )^{7/2}}-\frac{2 b \left (128 A b^3-36 a b (2 b B-a C)-15 a^3 D\right )}{9 a^5 x \left (a+b x^2\right )^{7/2}}-\frac{16 b^2 \left (128 A b^3-36 a b (2 b B-a C)-15 a^3 D\right ) x}{63 a^6 \left (a+b x^2\right )^{7/2}}-\frac{32 b^2 \left (128 A b^3-36 a b (2 b B-a C)-15 a^3 D\right ) x}{105 a^7 \left (a+b x^2\right )^{5/2}}-\frac{\left (128 b^2 \left (128 A b^3-36 a b (2 b B-a C)-15 a^3 D\right )\right ) \int \frac{1}{\left (a+b x^2\right )^{5/2}} \, dx}{105 a^7}\\ &=-\frac{A}{9 a x^9 \left (a+b x^2\right )^{7/2}}+\frac{16 A b-9 a B}{63 a^2 x^7 \left (a+b x^2\right )^{7/2}}-\frac{32 A b^2-9 a (2 b B-a C)}{45 a^3 x^5 \left (a+b x^2\right )^{7/2}}+\frac{128 A b^3-36 a b (2 b B-a C)-15 a^3 D}{45 a^4 x^3 \left (a+b x^2\right )^{7/2}}-\frac{2 b \left (128 A b^3-36 a b (2 b B-a C)-15 a^3 D\right )}{9 a^5 x \left (a+b x^2\right )^{7/2}}-\frac{16 b^2 \left (128 A b^3-36 a b (2 b B-a C)-15 a^3 D\right ) x}{63 a^6 \left (a+b x^2\right )^{7/2}}-\frac{32 b^2 \left (128 A b^3-36 a b (2 b B-a C)-15 a^3 D\right ) x}{105 a^7 \left (a+b x^2\right )^{5/2}}-\frac{128 b^2 \left (128 A b^3-36 a b (2 b B-a C)-15 a^3 D\right ) x}{315 a^8 \left (a+b x^2\right )^{3/2}}-\frac{\left (256 b^2 \left (128 A b^3-36 a b (2 b B-a C)-15 a^3 D\right )\right ) \int \frac{1}{\left (a+b x^2\right )^{3/2}} \, dx}{315 a^8}\\ &=-\frac{A}{9 a x^9 \left (a+b x^2\right )^{7/2}}+\frac{16 A b-9 a B}{63 a^2 x^7 \left (a+b x^2\right )^{7/2}}-\frac{32 A b^2-9 a (2 b B-a C)}{45 a^3 x^5 \left (a+b x^2\right )^{7/2}}+\frac{128 A b^3-36 a b (2 b B-a C)-15 a^3 D}{45 a^4 x^3 \left (a+b x^2\right )^{7/2}}-\frac{2 b \left (128 A b^3-36 a b (2 b B-a C)-15 a^3 D\right )}{9 a^5 x \left (a+b x^2\right )^{7/2}}-\frac{16 b^2 \left (128 A b^3-36 a b (2 b B-a C)-15 a^3 D\right ) x}{63 a^6 \left (a+b x^2\right )^{7/2}}-\frac{32 b^2 \left (128 A b^3-36 a b (2 b B-a C)-15 a^3 D\right ) x}{105 a^7 \left (a+b x^2\right )^{5/2}}-\frac{128 b^2 \left (128 A b^3-36 a b (2 b B-a C)-15 a^3 D\right ) x}{315 a^8 \left (a+b x^2\right )^{3/2}}-\frac{256 b^2 \left (128 A b^3-36 a b (2 b B-a C)-15 a^3 D\right ) x}{315 a^9 \sqrt{a+b x^2}}\\ \end{align*}

Mathematica [A]  time = 0.162759, size = 270, normalized size = 0.69 \[ \frac{256 a^3 b^5 x^{10} \left (-280 A+315 B x^2-126 C x^4+15 D x^6\right )+4480 a^4 b^4 x^8 \left (-2 A+9 B x^2-9 C x^4+3 D x^6\right )+112 a^5 b^3 x^6 \left (8 A+45 B x^2-180 C x^4+150 D x^6\right )-56 a^6 b^2 x^4 \left (4 A+9 B x^2+45 C x^4-150 D x^6\right )-1024 a^2 b^6 x^{12} \left (140 A-63 B x^2+9 C x^4\right )+2 a^7 b x^2 \left (40 A+21 \left (3 B x^2+6 C x^4+25 D x^6\right )\right )-a^8 \left (35 A+45 B x^2+63 C x^4+105 D x^6\right )+2048 a b^7 x^{14} \left (9 B x^2-56 A\right )-32768 A b^8 x^{16}}{315 a^9 x^9 \left (a+b x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x^2 + C*x^4 + D*x^6)/(x^10*(a + b*x^2)^(9/2)),x]

[Out]

(-32768*A*b^8*x^16 + 2048*a*b^7*x^14*(-56*A + 9*B*x^2) - 1024*a^2*b^6*x^12*(140*A - 63*B*x^2 + 9*C*x^4) - 56*a
^6*b^2*x^4*(4*A + 9*B*x^2 + 45*C*x^4 - 150*D*x^6) + 4480*a^4*b^4*x^8*(-2*A + 9*B*x^2 - 9*C*x^4 + 3*D*x^6) + 25
6*a^3*b^5*x^10*(-280*A + 315*B*x^2 - 126*C*x^4 + 15*D*x^6) - a^8*(35*A + 45*B*x^2 + 63*C*x^4 + 105*D*x^6) + 11
2*a^5*b^3*x^6*(8*A + 45*B*x^2 - 180*C*x^4 + 150*D*x^6) + 2*a^7*b*x^2*(40*A + 21*(3*B*x^2 + 6*C*x^4 + 25*D*x^6)
))/(315*a^9*x^9*(a + b*x^2)^(7/2))

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Maple [A]  time = 0.008, size = 349, normalized size = 0.9 \begin{align*} -{\frac{32768\,A{b}^{8}{x}^{16}-18432\,Ba{b}^{7}{x}^{16}+9216\,C{a}^{2}{b}^{6}{x}^{16}-3840\,D{a}^{3}{b}^{5}{x}^{16}+114688\,Aa{b}^{7}{x}^{14}-64512\,B{a}^{2}{b}^{6}{x}^{14}+32256\,C{a}^{3}{b}^{5}{x}^{14}-13440\,D{a}^{4}{b}^{4}{x}^{14}+143360\,A{a}^{2}{b}^{6}{x}^{12}-80640\,B{a}^{3}{b}^{5}{x}^{12}+40320\,C{a}^{4}{b}^{4}{x}^{12}-16800\,D{a}^{5}{b}^{3}{x}^{12}+71680\,A{a}^{3}{b}^{5}{x}^{10}-40320\,B{a}^{4}{b}^{4}{x}^{10}+20160\,C{a}^{5}{b}^{3}{x}^{10}-8400\,D{a}^{6}{b}^{2}{x}^{10}+8960\,A{a}^{4}{b}^{4}{x}^{8}-5040\,B{a}^{5}{b}^{3}{x}^{8}+2520\,C{a}^{6}{b}^{2}{x}^{8}-1050\,D{a}^{7}b{x}^{8}-896\,A{a}^{5}{b}^{3}{x}^{6}+504\,B{a}^{6}{b}^{2}{x}^{6}-252\,C{a}^{7}b{x}^{6}+105\,D{a}^{8}{x}^{6}+224\,A{a}^{6}{b}^{2}{x}^{4}-126\,B{a}^{7}b{x}^{4}+63\,C{a}^{8}{x}^{4}-80\,A{a}^{7}b{x}^{2}+45\,B{a}^{8}{x}^{2}+35\,A{a}^{8}}{315\,{x}^{9}{a}^{9}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((D*x^6+C*x^4+B*x^2+A)/x^10/(b*x^2+a)^(9/2),x)

[Out]

-1/315*(32768*A*b^8*x^16-18432*B*a*b^7*x^16+9216*C*a^2*b^6*x^16-3840*D*a^3*b^5*x^16+114688*A*a*b^7*x^14-64512*
B*a^2*b^6*x^14+32256*C*a^3*b^5*x^14-13440*D*a^4*b^4*x^14+143360*A*a^2*b^6*x^12-80640*B*a^3*b^5*x^12+40320*C*a^
4*b^4*x^12-16800*D*a^5*b^3*x^12+71680*A*a^3*b^5*x^10-40320*B*a^4*b^4*x^10+20160*C*a^5*b^3*x^10-8400*D*a^6*b^2*
x^10+8960*A*a^4*b^4*x^8-5040*B*a^5*b^3*x^8+2520*C*a^6*b^2*x^8-1050*D*a^7*b*x^8-896*A*a^5*b^3*x^6+504*B*a^6*b^2
*x^6-252*C*a^7*b*x^6+105*D*a^8*x^6+224*A*a^6*b^2*x^4-126*B*a^7*b*x^4+63*C*a^8*x^4-80*A*a^7*b*x^2+45*B*a^8*x^2+
35*A*a^8)/x^9/(b*x^2+a)^(7/2)/a^9

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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((D*x^6+C*x^4+B*x^2+A)/x^10/(b*x^2+a)^(9/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((D*x^6+C*x^4+B*x^2+A)/x^10/(b*x^2+a)^(9/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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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((D*x**6+C*x**4+B*x**2+A)/x**10/(b*x**2+a)**(9/2),x)

[Out]

Timed out

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Giac [B]  time = 1.32324, size = 1569, normalized size = 4. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((D*x^6+C*x^4+B*x^2+A)/x^10/(b*x^2+a)^(9/2),x, algorithm="giac")

[Out]

1/105*((x^2*((790*D*a^24*b^8 - 1686*C*a^23*b^9 + 3072*B*a^22*b^10 - 5053*A*a^21*b^11)*x^2/(a^30*b^3) + 7*(365*
D*a^25*b^7 - 768*C*a^24*b^8 + 1386*B*a^23*b^9 - 2264*A*a^22*b^10)/(a^30*b^3)) + 35*(80*D*a^26*b^6 - 165*C*a^25
*b^7 + 294*B*a^24*b^8 - 476*A*a^23*b^9)/(a^30*b^3))*x^2 + 105*(10*D*a^27*b^5 - 20*C*a^26*b^6 + 35*B*a^25*b^7 -
56*A*a^24*b^8)/(a^30*b^3))*x/(b*x^2 + a)^(7/2) - 2/315*(1260*(sqrt(b)*x - sqrt(b*x^2 + a))^16*D*a^3*b^(3/2) -
3150*(sqrt(b)*x - sqrt(b*x^2 + a))^16*C*a^2*b^(5/2) + 6300*(sqrt(b)*x - sqrt(b*x^2 + a))^16*B*a*b^(7/2) - 110
25*(sqrt(b)*x - sqrt(b*x^2 + a))^16*A*b^(9/2) - 10710*(sqrt(b)*x - sqrt(b*x^2 + a))^14*D*a^4*b^(3/2) + 27720*(
sqrt(b)*x - sqrt(b*x^2 + a))^14*C*a^3*b^(5/2) - 56700*(sqrt(b)*x - sqrt(b*x^2 + a))^14*B*a^2*b^(7/2) + 100800*
(sqrt(b)*x - sqrt(b*x^2 + a))^14*A*a*b^(9/2) + 39270*(sqrt(b)*x - sqrt(b*x^2 + a))^12*D*a^5*b^(3/2) - 105840*(
sqrt(b)*x - sqrt(b*x^2 + a))^12*C*a^4*b^(5/2) + 223020*(sqrt(b)*x - sqrt(b*x^2 + a))^12*B*a^3*b^(7/2) - 405300
*(sqrt(b)*x - sqrt(b*x^2 + a))^12*A*a^2*b^(9/2) - 81270*(sqrt(b)*x - sqrt(b*x^2 + a))^10*D*a^6*b^(3/2) + 22680
0*(sqrt(b)*x - sqrt(b*x^2 + a))^10*C*a^5*b^(5/2) - 495180*(sqrt(b)*x - sqrt(b*x^2 + a))^10*B*a^4*b^(7/2) + 927
360*(sqrt(b)*x - sqrt(b*x^2 + a))^10*A*a^3*b^(9/2) + 103950*(sqrt(b)*x - sqrt(b*x^2 + a))^8*D*a^7*b^(3/2) - 29
7108*(sqrt(b)*x - sqrt(b*x^2 + a))^8*C*a^6*b^(5/2) + 666036*(sqrt(b)*x - sqrt(b*x^2 + a))^8*B*a^5*b^(7/2) - 12
91374*(sqrt(b)*x - sqrt(b*x^2 + a))^8*A*a^4*b^(9/2) - 84210*(sqrt(b)*x - sqrt(b*x^2 + a))^6*D*a^8*b^(3/2) + 24
3432*(sqrt(b)*x - sqrt(b*x^2 + a))^6*C*a^7*b^(5/2) - 551124*(sqrt(b)*x - sqrt(b*x^2 + a))^6*B*a^6*b^(7/2) + 10
73856*(sqrt(b)*x - sqrt(b*x^2 + a))^6*A*a^5*b^(9/2) + 42210*(sqrt(b)*x - sqrt(b*x^2 + a))^4*D*a^9*b^(3/2) - 12
1968*(sqrt(b)*x - sqrt(b*x^2 + a))^4*C*a^8*b^(5/2) + 275076*(sqrt(b)*x - sqrt(b*x^2 + a))^4*B*a^7*b^(7/2) - 53
3124*(sqrt(b)*x - sqrt(b*x^2 + a))^4*A*a^6*b^(9/2) - 11970*(sqrt(b)*x - sqrt(b*x^2 + a))^2*D*a^10*b^(3/2) + 34
272*(sqrt(b)*x - sqrt(b*x^2 + a))^2*C*a^9*b^(5/2) - 76644*(sqrt(b)*x - sqrt(b*x^2 + a))^2*B*a^8*b^(7/2) + 1474
56*(sqrt(b)*x - sqrt(b*x^2 + a))^2*A*a^7*b^(9/2) + 1470*D*a^11*b^(3/2) - 4158*C*a^10*b^(5/2) + 9216*B*a^9*b^(7
/2) - 17609*A*a^8*b^(9/2))/(((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^9*a^8)