3.167 \(\int \frac{A+B x^2+C x^4+D x^6}{x^8 (a+b x^2)^{9/2}} \, dx\)

Optimal. Leaf size=334 \[ \frac{128 b x \left (48 A b^3-a \left (3 a^2 D-10 a b C+24 b^2 B\right )\right )}{105 a^8 \sqrt{a+b x^2}}+\frac{64 b x \left (48 A b^3-a \left (3 a^2 D-10 a b C+24 b^2 B\right )\right )}{105 a^7 \left (a+b x^2\right )^{3/2}}+\frac{16 b x \left (48 A b^3-a \left (3 a^2 D-10 a b C+24 b^2 B\right )\right )}{35 a^6 \left (a+b x^2\right )^{5/2}}+\frac{8 b x \left (48 A b^3-a \left (3 a^2 D-10 a b C+24 b^2 B\right )\right )}{21 a^5 \left (a+b x^2\right )^{7/2}}+\frac{48 A b^3-a \left (3 a^2 D-10 a b C+24 b^2 B\right )}{3 a^4 x \left (a+b x^2\right )^{7/2}}-\frac{24 A b^2-a (12 b B-5 a C)}{15 a^3 x^3 \left (a+b x^2\right )^{7/2}}+\frac{2 A b-a B}{5 a^2 x^5 \left (a+b x^2\right )^{7/2}}-\frac{A}{7 a x^7 \left (a+b x^2\right )^{7/2}} \]

[Out]

-A/(7*a*x^7*(a + b*x^2)^(7/2)) + (2*A*b - a*B)/(5*a^2*x^5*(a + b*x^2)^(7/2)) - (24*A*b^2 - a*(12*b*B - 5*a*C))
/(15*a^3*x^3*(a + b*x^2)^(7/2)) + (48*A*b^3 - a*(24*b^2*B - 10*a*b*C + 3*a^2*D))/(3*a^4*x*(a + b*x^2)^(7/2)) +
(8*b*(48*A*b^3 - a*(24*b^2*B - 10*a*b*C + 3*a^2*D))*x)/(21*a^5*(a + b*x^2)^(7/2)) + (16*b*(48*A*b^3 - a*(24*b
^2*B - 10*a*b*C + 3*a^2*D))*x)/(35*a^6*(a + b*x^2)^(5/2)) + (64*b*(48*A*b^3 - a*(24*b^2*B - 10*a*b*C + 3*a^2*D
))*x)/(105*a^7*(a + b*x^2)^(3/2)) + (128*b*(48*A*b^3 - a*(24*b^2*B - 10*a*b*C + 3*a^2*D))*x)/(105*a^8*Sqrt[a +
b*x^2])

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Rubi [A]  time = 0.480311, antiderivative size = 334, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ \frac{128 b x \left (48 A b^3-a \left (3 a^2 D-10 a b C+24 b^2 B\right )\right )}{105 a^8 \sqrt{a+b x^2}}+\frac{64 b x \left (48 A b^3-a \left (3 a^2 D-10 a b C+24 b^2 B\right )\right )}{105 a^7 \left (a+b x^2\right )^{3/2}}+\frac{16 b x \left (48 A b^3-a \left (3 a^2 D-10 a b C+24 b^2 B\right )\right )}{35 a^6 \left (a+b x^2\right )^{5/2}}+\frac{8 b x \left (48 A b^3-a \left (3 a^2 D-10 a b C+24 b^2 B\right )\right )}{21 a^5 \left (a+b x^2\right )^{7/2}}+\frac{48 A b^3-a \left (3 a^2 D-10 a b C+24 b^2 B\right )}{3 a^4 x \left (a+b x^2\right )^{7/2}}-\frac{24 A b^2-a (12 b B-5 a C)}{15 a^3 x^3 \left (a+b x^2\right )^{7/2}}+\frac{2 A b-a B}{5 a^2 x^5 \left (a+b x^2\right )^{7/2}}-\frac{A}{7 a x^7 \left (a+b x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x^2 + C*x^4 + D*x^6)/(x^8*(a + b*x^2)^(9/2)),x]

[Out]

-A/(7*a*x^7*(a + b*x^2)^(7/2)) + (2*A*b - a*B)/(5*a^2*x^5*(a + b*x^2)^(7/2)) - (24*A*b^2 - a*(12*b*B - 5*a*C))
/(15*a^3*x^3*(a + b*x^2)^(7/2)) + (48*A*b^3 - a*(24*b^2*B - 10*a*b*C + 3*a^2*D))/(3*a^4*x*(a + b*x^2)^(7/2)) +
(8*b*(48*A*b^3 - a*(24*b^2*B - 10*a*b*C + 3*a^2*D))*x)/(21*a^5*(a + b*x^2)^(7/2)) + (16*b*(48*A*b^3 - a*(24*b
^2*B - 10*a*b*C + 3*a^2*D))*x)/(35*a^6*(a + b*x^2)^(5/2)) + (64*b*(48*A*b^3 - a*(24*b^2*B - 10*a*b*C + 3*a^2*D
))*x)/(105*a^7*(a + b*x^2)^(3/2)) + (128*b*(48*A*b^3 - a*(24*b^2*B - 10*a*b*C + 3*a^2*D))*x)/(105*a^8*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^8 \left (a+b x^2\right )^{9/2}} \, dx &=-\frac{A}{7 a x^7 \left (a+b x^2\right )^{7/2}}-\frac{\int \frac{14 A b-7 a \left (B+C x^2+D x^4\right )}{x^6 \left (a+b x^2\right )^{9/2}} \, dx}{7 a}\\ &=-\frac{A}{7 a x^7 \left (a+b x^2\right )^{7/2}}+\frac{2 A b-a B}{5 a^2 x^5 \left (a+b x^2\right )^{7/2}}+\frac{\int \frac{12 b (14 A b-7 a B)-5 a \left (-7 a C-7 a D x^2\right )}{x^4 \left (a+b x^2\right )^{9/2}} \, dx}{35 a^2}\\ &=-\frac{A}{7 a x^7 \left (a+b x^2\right )^{7/2}}+\frac{2 A b-a B}{5 a^2 x^5 \left (a+b x^2\right )^{7/2}}-\frac{24 A b^2-a (12 b B-5 a C)}{15 a^3 x^3 \left (a+b x^2\right )^{7/2}}-\frac{\int \frac{10 b \left (168 A b^2-84 a b B+35 a^2 C\right )-105 a^3 D}{x^2 \left (a+b x^2\right )^{9/2}} \, dx}{105 a^3}\\ &=-\frac{A}{7 a x^7 \left (a+b x^2\right )^{7/2}}+\frac{2 A b-a B}{5 a^2 x^5 \left (a+b x^2\right )^{7/2}}-\frac{24 A b^2-a (12 b B-5 a C)}{15 a^3 x^3 \left (a+b x^2\right )^{7/2}}-\frac{\left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right ) \int \frac{1}{x^2 \left (a+b x^2\right )^{9/2}} \, dx}{3 a^3}\\ &=-\frac{A}{7 a x^7 \left (a+b x^2\right )^{7/2}}+\frac{2 A b-a B}{5 a^2 x^5 \left (a+b x^2\right )^{7/2}}-\frac{24 A b^2-a (12 b B-5 a C)}{15 a^3 x^3 \left (a+b x^2\right )^{7/2}}+\frac{48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )}{3 a^4 x \left (a+b x^2\right )^{7/2}}+\frac{\left (8 b \left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right )\right ) \int \frac{1}{\left (a+b x^2\right )^{9/2}} \, dx}{3 a^4}\\ &=-\frac{A}{7 a x^7 \left (a+b x^2\right )^{7/2}}+\frac{2 A b-a B}{5 a^2 x^5 \left (a+b x^2\right )^{7/2}}-\frac{24 A b^2-a (12 b B-5 a C)}{15 a^3 x^3 \left (a+b x^2\right )^{7/2}}+\frac{48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )}{3 a^4 x \left (a+b x^2\right )^{7/2}}+\frac{8 b \left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right ) x}{21 a^5 \left (a+b x^2\right )^{7/2}}+\frac{\left (16 b \left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right )\right ) \int \frac{1}{\left (a+b x^2\right )^{7/2}} \, dx}{7 a^5}\\ &=-\frac{A}{7 a x^7 \left (a+b x^2\right )^{7/2}}+\frac{2 A b-a B}{5 a^2 x^5 \left (a+b x^2\right )^{7/2}}-\frac{24 A b^2-a (12 b B-5 a C)}{15 a^3 x^3 \left (a+b x^2\right )^{7/2}}+\frac{48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )}{3 a^4 x \left (a+b x^2\right )^{7/2}}+\frac{8 b \left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right ) x}{21 a^5 \left (a+b x^2\right )^{7/2}}+\frac{16 b \left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right ) x}{35 a^6 \left (a+b x^2\right )^{5/2}}+\frac{\left (64 b \left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right )\right ) \int \frac{1}{\left (a+b x^2\right )^{5/2}} \, dx}{35 a^6}\\ &=-\frac{A}{7 a x^7 \left (a+b x^2\right )^{7/2}}+\frac{2 A b-a B}{5 a^2 x^5 \left (a+b x^2\right )^{7/2}}-\frac{24 A b^2-a (12 b B-5 a C)}{15 a^3 x^3 \left (a+b x^2\right )^{7/2}}+\frac{48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )}{3 a^4 x \left (a+b x^2\right )^{7/2}}+\frac{8 b \left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right ) x}{21 a^5 \left (a+b x^2\right )^{7/2}}+\frac{16 b \left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right ) x}{35 a^6 \left (a+b x^2\right )^{5/2}}+\frac{64 b \left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right ) x}{105 a^7 \left (a+b x^2\right )^{3/2}}+\frac{\left (128 b \left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right )\right ) \int \frac{1}{\left (a+b x^2\right )^{3/2}} \, dx}{105 a^7}\\ &=-\frac{A}{7 a x^7 \left (a+b x^2\right )^{7/2}}+\frac{2 A b-a B}{5 a^2 x^5 \left (a+b x^2\right )^{7/2}}-\frac{24 A b^2-a (12 b B-5 a C)}{15 a^3 x^3 \left (a+b x^2\right )^{7/2}}+\frac{48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )}{3 a^4 x \left (a+b x^2\right )^{7/2}}+\frac{8 b \left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right ) x}{21 a^5 \left (a+b x^2\right )^{7/2}}+\frac{16 b \left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right ) x}{35 a^6 \left (a+b x^2\right )^{5/2}}+\frac{64 b \left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right ) x}{105 a^7 \left (a+b x^2\right )^{3/2}}+\frac{128 b \left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right ) x}{105 a^8 \sqrt{a+b x^2}}\\ \end{align*}

Mathematica [A]  time = 0.146353, size = 234, normalized size = 0.7 \[ \frac{128 a^3 b^4 x^8 \left (105 A-105 B x^2+35 C x^4-3 D x^6\right )+112 a^4 b^3 x^6 \left (15 A-60 B x^2+50 C x^4-12 D x^6\right )-56 a^5 b^2 x^4 \left (3 A+15 B x^2-50 C x^4+30 D x^6\right )+256 a^2 b^5 x^{10} \left (105 A-42 B x^2+5 C x^4\right )+14 a^6 b x^2 \left (3 A+6 B x^2+25 C x^4-60 D x^6\right )-a^7 \left (15 A+21 B x^2+35 x^4 \left (C+3 D x^2\right )\right )-3072 a b^6 x^{12} \left (B x^2-7 A\right )+6144 A b^7 x^{14}}{105 a^8 x^7 \left (a+b x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x^2 + C*x^4 + D*x^6)/(x^8*(a + b*x^2)^(9/2)),x]

[Out]

(6144*A*b^7*x^14 - 3072*a*b^6*x^12*(-7*A + B*x^2) + 256*a^2*b^5*x^10*(105*A - 42*B*x^2 + 5*C*x^4) + 14*a^6*b*x
^2*(3*A + 6*B*x^2 + 25*C*x^4 - 60*D*x^6) + 112*a^4*b^3*x^6*(15*A - 60*B*x^2 + 50*C*x^4 - 12*D*x^6) + 128*a^3*b
^4*x^8*(105*A - 105*B*x^2 + 35*C*x^4 - 3*D*x^6) - 56*a^5*b^2*x^4*(3*A + 15*B*x^2 - 50*C*x^4 + 30*D*x^6) - a^7*
(15*A + 21*B*x^2 + 35*x^4*(C + 3*D*x^2)))/(105*a^8*x^7*(a + b*x^2)^(7/2))

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Maple [A]  time = 0.008, size = 301, normalized size = 0.9 \begin{align*} -{\frac{-6144\,A{b}^{7}{x}^{14}+3072\,Ba{b}^{6}{x}^{14}-1280\,C{a}^{2}{b}^{5}{x}^{14}+384\,D{a}^{3}{b}^{4}{x}^{14}-21504\,Aa{b}^{6}{x}^{12}+10752\,B{a}^{2}{b}^{5}{x}^{12}-4480\,C{a}^{3}{b}^{4}{x}^{12}+1344\,D{a}^{4}{b}^{3}{x}^{12}-26880\,A{a}^{2}{b}^{5}{x}^{10}+13440\,B{a}^{3}{b}^{4}{x}^{10}-5600\,C{a}^{4}{b}^{3}{x}^{10}+1680\,D{a}^{5}{b}^{2}{x}^{10}-13440\,A{a}^{3}{b}^{4}{x}^{8}+6720\,B{a}^{4}{b}^{3}{x}^{8}-2800\,C{a}^{5}{b}^{2}{x}^{8}+840\,D{a}^{6}b{x}^{8}-1680\,A{a}^{4}{b}^{3}{x}^{6}+840\,B{a}^{5}{b}^{2}{x}^{6}-350\,C{a}^{6}b{x}^{6}+105\,D{a}^{7}{x}^{6}+168\,A{a}^{5}{b}^{2}{x}^{4}-84\,B{a}^{6}b{x}^{4}+35\,C{a}^{7}{x}^{4}-42\,A{a}^{6}b{x}^{2}+21\,B{a}^{7}{x}^{2}+15\,A{a}^{7}}{105\,{x}^{7}{a}^{8}} \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^8/(b*x^2+a)^(9/2),x)

[Out]

-1/105*(-6144*A*b^7*x^14+3072*B*a*b^6*x^14-1280*C*a^2*b^5*x^14+384*D*a^3*b^4*x^14-21504*A*a*b^6*x^12+10752*B*a
^2*b^5*x^12-4480*C*a^3*b^4*x^12+1344*D*a^4*b^3*x^12-26880*A*a^2*b^5*x^10+13440*B*a^3*b^4*x^10-5600*C*a^4*b^3*x
^10+1680*D*a^5*b^2*x^10-13440*A*a^3*b^4*x^8+6720*B*a^4*b^3*x^8-2800*C*a^5*b^2*x^8+840*D*a^6*b*x^8-1680*A*a^4*b
^3*x^6+840*B*a^5*b^2*x^6-350*C*a^6*b*x^6+105*D*a^7*x^6+168*A*a^5*b^2*x^4-84*B*a^6*b*x^4+35*C*a^7*x^4-42*A*a^6*
b*x^2+21*B*a^7*x^2+15*A*a^7)/x^7/(b*x^2+a)^(7/2)/a^8

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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^8/(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^8/(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**8/(b*x**2+a)**(9/2),x)

[Out]

Timed out

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Giac [B]  time = 1.33665, size = 1266, normalized size = 3.79 \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^8/(b*x^2+a)^(9/2),x, algorithm="giac")

[Out]

-1/105*((x^2*((279*D*a^21*b^7 - 790*C*a^20*b^8 + 1686*B*a^19*b^9 - 3072*A*a^18*b^10)*x^2/(a^26*b^3) + 7*(132*D
*a^22*b^6 - 365*C*a^21*b^7 + 768*B*a^20*b^8 - 1386*A*a^19*b^9)/(a^26*b^3)) + 35*(30*D*a^23*b^5 - 80*C*a^22*b^6
+ 165*B*a^21*b^7 - 294*A*a^20*b^8)/(a^26*b^3))*x^2 + 105*(4*D*a^24*b^4 - 10*C*a^23*b^5 + 20*B*a^22*b^6 - 35*A
*a^21*b^7)/(a^26*b^3))*x/(b*x^2 + a)^(7/2) + 2/105*(105*(sqrt(b)*x - sqrt(b*x^2 + a))^12*D*a^3*sqrt(b) - 420*(
sqrt(b)*x - sqrt(b*x^2 + a))^12*C*a^2*b^(3/2) + 1050*(sqrt(b)*x - sqrt(b*x^2 + a))^12*B*a*b^(5/2) - 2100*(sqrt
(b)*x - sqrt(b*x^2 + a))^12*A*b^(7/2) - 630*(sqrt(b)*x - sqrt(b*x^2 + a))^10*D*a^4*sqrt(b) + 2730*(sqrt(b)*x -
sqrt(b*x^2 + a))^10*C*a^3*b^(3/2) - 7140*(sqrt(b)*x - sqrt(b*x^2 + a))^10*B*a^2*b^(5/2) + 14700*(sqrt(b)*x -
sqrt(b*x^2 + a))^10*A*a*b^(7/2) + 1575*(sqrt(b)*x - sqrt(b*x^2 + a))^8*D*a^5*sqrt(b) - 7210*(sqrt(b)*x - sqrt(
b*x^2 + a))^8*C*a^4*b^(3/2) + 19950*(sqrt(b)*x - sqrt(b*x^2 + a))^8*B*a^3*b^(5/2) - 42840*(sqrt(b)*x - sqrt(b*
x^2 + a))^8*A*a^2*b^(7/2) - 2100*(sqrt(b)*x - sqrt(b*x^2 + a))^6*D*a^6*sqrt(b) + 9940*(sqrt(b)*x - sqrt(b*x^2
+ a))^6*C*a^5*b^(3/2) - 28560*(sqrt(b)*x - sqrt(b*x^2 + a))^6*B*a^4*b^(5/2) + 64680*(sqrt(b)*x - sqrt(b*x^2 +
a))^6*A*a^3*b^(7/2) + 1575*(sqrt(b)*x - sqrt(b*x^2 + a))^4*D*a^7*sqrt(b) - 7560*(sqrt(b)*x - sqrt(b*x^2 + a))^
4*C*a^6*b^(3/2) + 21966*(sqrt(b)*x - sqrt(b*x^2 + a))^4*B*a^5*b^(5/2) - 49812*(sqrt(b)*x - sqrt(b*x^2 + a))^4*
A*a^4*b^(7/2) - 630*(sqrt(b)*x - sqrt(b*x^2 + a))^2*D*a^8*sqrt(b) + 3010*(sqrt(b)*x - sqrt(b*x^2 + a))^2*C*a^7
*b^(3/2) - 8652*(sqrt(b)*x - sqrt(b*x^2 + a))^2*B*a^6*b^(5/2) + 19404*(sqrt(b)*x - sqrt(b*x^2 + a))^2*A*a^5*b^
(7/2) + 105*D*a^9*sqrt(b) - 490*C*a^8*b^(3/2) + 1386*B*a^7*b^(5/2) - 3072*A*a^6*b^(7/2))/(((sqrt(b)*x - sqrt(b
*x^2 + a))^2 - a)^7*a^7)