3.520 \(\int \frac{\sqrt{a+b x^2} (A+B x^2)}{x^{11}} \, dx\)

Optimal. Leaf size=189 \[ \frac{b^3 \sqrt{a+b x^2} (7 A b-10 a B)}{256 a^4 x^2}-\frac{b^2 \sqrt{a+b x^2} (7 A b-10 a B)}{384 a^3 x^4}-\frac{b^4 (7 A b-10 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{256 a^{9/2}}+\frac{b \sqrt{a+b x^2} (7 A b-10 a B)}{480 a^2 x^6}+\frac{\sqrt{a+b x^2} (7 A b-10 a B)}{80 a x^8}-\frac{A \left (a+b x^2\right )^{3/2}}{10 a x^{10}} \]

[Out]

((7*A*b - 10*a*B)*Sqrt[a + b*x^2])/(80*a*x^8) + (b*(7*A*b - 10*a*B)*Sqrt[a + b*x^2])/(480*a^2*x^6) - (b^2*(7*A
*b - 10*a*B)*Sqrt[a + b*x^2])/(384*a^3*x^4) + (b^3*(7*A*b - 10*a*B)*Sqrt[a + b*x^2])/(256*a^4*x^2) - (A*(a + b
*x^2)^(3/2))/(10*a*x^10) - (b^4*(7*A*b - 10*a*B)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(256*a^(9/2))

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Rubi [A]  time = 0.146803, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {446, 78, 47, 51, 63, 208} \[ \frac{b^3 \sqrt{a+b x^2} (7 A b-10 a B)}{256 a^4 x^2}-\frac{b^2 \sqrt{a+b x^2} (7 A b-10 a B)}{384 a^3 x^4}-\frac{b^4 (7 A b-10 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{256 a^{9/2}}+\frac{b \sqrt{a+b x^2} (7 A b-10 a B)}{480 a^2 x^6}+\frac{\sqrt{a+b x^2} (7 A b-10 a B)}{80 a x^8}-\frac{A \left (a+b x^2\right )^{3/2}}{10 a x^{10}} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[a + b*x^2]*(A + B*x^2))/x^11,x]

[Out]

((7*A*b - 10*a*B)*Sqrt[a + b*x^2])/(80*a*x^8) + (b*(7*A*b - 10*a*B)*Sqrt[a + b*x^2])/(480*a^2*x^6) - (b^2*(7*A
*b - 10*a*B)*Sqrt[a + b*x^2])/(384*a^3*x^4) + (b^3*(7*A*b - 10*a*B)*Sqrt[a + b*x^2])/(256*a^4*x^2) - (A*(a + b
*x^2)^(3/2))/(10*a*x^10) - (b^4*(7*A*b - 10*a*B)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(256*a^(9/2))

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 51

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*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \int \frac{\sqrt{a+b x^2} \left (A+B x^2\right )}{x^{11}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{\sqrt{a+b x} (A+B x)}{x^6} \, dx,x,x^2\right )\\ &=-\frac{A \left (a+b x^2\right )^{3/2}}{10 a x^{10}}+\frac{\left (-\frac{7 A b}{2}+5 a B\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x^5} \, dx,x,x^2\right )}{10 a}\\ &=\frac{(7 A b-10 a B) \sqrt{a+b x^2}}{80 a x^8}-\frac{A \left (a+b x^2\right )^{3/2}}{10 a x^{10}}-\frac{(b (7 A b-10 a B)) \operatorname{Subst}\left (\int \frac{1}{x^4 \sqrt{a+b x}} \, dx,x,x^2\right )}{160 a}\\ &=\frac{(7 A b-10 a B) \sqrt{a+b x^2}}{80 a x^8}+\frac{b (7 A b-10 a B) \sqrt{a+b x^2}}{480 a^2 x^6}-\frac{A \left (a+b x^2\right )^{3/2}}{10 a x^{10}}+\frac{\left (b^2 (7 A b-10 a B)\right ) \operatorname{Subst}\left (\int \frac{1}{x^3 \sqrt{a+b x}} \, dx,x,x^2\right )}{192 a^2}\\ &=\frac{(7 A b-10 a B) \sqrt{a+b x^2}}{80 a x^8}+\frac{b (7 A b-10 a B) \sqrt{a+b x^2}}{480 a^2 x^6}-\frac{b^2 (7 A b-10 a B) \sqrt{a+b x^2}}{384 a^3 x^4}-\frac{A \left (a+b x^2\right )^{3/2}}{10 a x^{10}}-\frac{\left (b^3 (7 A b-10 a B)\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b x}} \, dx,x,x^2\right )}{256 a^3}\\ &=\frac{(7 A b-10 a B) \sqrt{a+b x^2}}{80 a x^8}+\frac{b (7 A b-10 a B) \sqrt{a+b x^2}}{480 a^2 x^6}-\frac{b^2 (7 A b-10 a B) \sqrt{a+b x^2}}{384 a^3 x^4}+\frac{b^3 (7 A b-10 a B) \sqrt{a+b x^2}}{256 a^4 x^2}-\frac{A \left (a+b x^2\right )^{3/2}}{10 a x^{10}}+\frac{\left (b^4 (7 A b-10 a B)\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^2\right )}{512 a^4}\\ &=\frac{(7 A b-10 a B) \sqrt{a+b x^2}}{80 a x^8}+\frac{b (7 A b-10 a B) \sqrt{a+b x^2}}{480 a^2 x^6}-\frac{b^2 (7 A b-10 a B) \sqrt{a+b x^2}}{384 a^3 x^4}+\frac{b^3 (7 A b-10 a B) \sqrt{a+b x^2}}{256 a^4 x^2}-\frac{A \left (a+b x^2\right )^{3/2}}{10 a x^{10}}+\frac{\left (b^3 (7 A b-10 a B)\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^2}\right )}{256 a^4}\\ &=\frac{(7 A b-10 a B) \sqrt{a+b x^2}}{80 a x^8}+\frac{b (7 A b-10 a B) \sqrt{a+b x^2}}{480 a^2 x^6}-\frac{b^2 (7 A b-10 a B) \sqrt{a+b x^2}}{384 a^3 x^4}+\frac{b^3 (7 A b-10 a B) \sqrt{a+b x^2}}{256 a^4 x^2}-\frac{A \left (a+b x^2\right )^{3/2}}{10 a x^{10}}-\frac{b^4 (7 A b-10 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{256 a^{9/2}}\\ \end{align*}

Mathematica [C]  time = 0.0220841, size = 62, normalized size = 0.33 \[ -\frac{\left (a+b x^2\right )^{3/2} \left (3 a^5 A+b^4 x^{10} (10 a B-7 A b) \, _2F_1\left (\frac{3}{2},5;\frac{5}{2};\frac{b x^2}{a}+1\right )\right )}{30 a^6 x^{10}} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[a + b*x^2]*(A + B*x^2))/x^11,x]

[Out]

-((a + b*x^2)^(3/2)*(3*a^5*A + b^4*(-7*A*b + 10*a*B)*x^10*Hypergeometric2F1[3/2, 5, 5/2, 1 + (b*x^2)/a]))/(30*
a^6*x^10)

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Maple [A]  time = 0.024, size = 281, normalized size = 1.5 \begin{align*} -{\frac{A}{10\,a{x}^{10}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{7\,Ab}{80\,{a}^{2}{x}^{8}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{7\,A{b}^{2}}{96\,{a}^{3}{x}^{6}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{7\,A{b}^{3}}{128\,{a}^{4}{x}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{7\,A{b}^{4}}{256\,{a}^{5}{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{7\,A{b}^{5}}{256}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{9}{2}}}}+{\frac{7\,A{b}^{5}}{256\,{a}^{5}}\sqrt{b{x}^{2}+a}}-{\frac{B}{8\,a{x}^{8}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,Bb}{48\,{a}^{2}{x}^{6}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{5\,B{b}^{2}}{64\,{a}^{3}{x}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,B{b}^{3}}{128\,{a}^{4}{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,B{b}^{4}}{128}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{7}{2}}}}-{\frac{5\,B{b}^{4}}{128\,{a}^{4}}\sqrt{b{x}^{2}+a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x^2+A)*(b*x^2+a)^(1/2)/x^11,x)

[Out]

-1/10*A*(b*x^2+a)^(3/2)/a/x^10+7/80*A*b/a^2/x^8*(b*x^2+a)^(3/2)-7/96*A*b^2/a^3/x^6*(b*x^2+a)^(3/2)+7/128*A*b^3
/a^4/x^4*(b*x^2+a)^(3/2)-7/256*A*b^4/a^5/x^2*(b*x^2+a)^(3/2)-7/256*A*b^5/a^(9/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(
1/2))/x)+7/256*A*b^5/a^5*(b*x^2+a)^(1/2)-1/8*B/a/x^8*(b*x^2+a)^(3/2)+5/48*B*b/a^2/x^6*(b*x^2+a)^(3/2)-5/64*B*b
^2/a^3/x^4*(b*x^2+a)^(3/2)+5/128*B*b^3/a^4/x^2*(b*x^2+a)^(3/2)+5/128*B*b^4/a^(7/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)
^(1/2))/x)-5/128*B*b^4/a^4*(b*x^2+a)^(1/2)

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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((B*x^2+A)*(b*x^2+a)^(1/2)/x^11,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.33929, size = 745, normalized size = 3.94 \begin{align*} \left [-\frac{15 \,{\left (10 \, B a b^{4} - 7 \, A b^{5}\right )} \sqrt{a} x^{10} \log \left (-\frac{b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) + 2 \,{\left (15 \,{\left (10 \, B a^{2} b^{3} - 7 \, A a b^{4}\right )} x^{8} - 10 \,{\left (10 \, B a^{3} b^{2} - 7 \, A a^{2} b^{3}\right )} x^{6} + 384 \, A a^{5} + 8 \,{\left (10 \, B a^{4} b - 7 \, A a^{3} b^{2}\right )} x^{4} + 48 \,{\left (10 \, B a^{5} + A a^{4} b\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{7680 \, a^{5} x^{10}}, -\frac{15 \,{\left (10 \, B a b^{4} - 7 \, A b^{5}\right )} \sqrt{-a} x^{10} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) +{\left (15 \,{\left (10 \, B a^{2} b^{3} - 7 \, A a b^{4}\right )} x^{8} - 10 \,{\left (10 \, B a^{3} b^{2} - 7 \, A a^{2} b^{3}\right )} x^{6} + 384 \, A a^{5} + 8 \,{\left (10 \, B a^{4} b - 7 \, A a^{3} b^{2}\right )} x^{4} + 48 \,{\left (10 \, B a^{5} + A a^{4} b\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{3840 \, a^{5} x^{10}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)*(b*x^2+a)^(1/2)/x^11,x, algorithm="fricas")

[Out]

[-1/7680*(15*(10*B*a*b^4 - 7*A*b^5)*sqrt(a)*x^10*log(-(b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) + 2*(15*(
10*B*a^2*b^3 - 7*A*a*b^4)*x^8 - 10*(10*B*a^3*b^2 - 7*A*a^2*b^3)*x^6 + 384*A*a^5 + 8*(10*B*a^4*b - 7*A*a^3*b^2)
*x^4 + 48*(10*B*a^5 + A*a^4*b)*x^2)*sqrt(b*x^2 + a))/(a^5*x^10), -1/3840*(15*(10*B*a*b^4 - 7*A*b^5)*sqrt(-a)*x
^10*arctan(sqrt(-a)/sqrt(b*x^2 + a)) + (15*(10*B*a^2*b^3 - 7*A*a*b^4)*x^8 - 10*(10*B*a^3*b^2 - 7*A*a^2*b^3)*x^
6 + 384*A*a^5 + 8*(10*B*a^4*b - 7*A*a^3*b^2)*x^4 + 48*(10*B*a^5 + A*a^4*b)*x^2)*sqrt(b*x^2 + a))/(a^5*x^10)]

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Sympy [A]  time = 114.89, size = 347, normalized size = 1.84 \begin{align*} - \frac{A a}{10 \sqrt{b} x^{11} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{9 A \sqrt{b}}{80 x^{9} \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{A b^{\frac{3}{2}}}{480 a x^{7} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{7 A b^{\frac{5}{2}}}{1920 a^{2} x^{5} \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{7 A b^{\frac{7}{2}}}{768 a^{3} x^{3} \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{7 A b^{\frac{9}{2}}}{256 a^{4} x \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{7 A b^{5} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{256 a^{\frac{9}{2}}} - \frac{B a}{8 \sqrt{b} x^{9} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{7 B \sqrt{b}}{48 x^{7} \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{B b^{\frac{3}{2}}}{192 a x^{5} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{5 B b^{\frac{5}{2}}}{384 a^{2} x^{3} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{5 B b^{\frac{7}{2}}}{128 a^{3} x \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{5 B b^{4} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{128 a^{\frac{7}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x**2+A)*(b*x**2+a)**(1/2)/x**11,x)

[Out]

-A*a/(10*sqrt(b)*x**11*sqrt(a/(b*x**2) + 1)) - 9*A*sqrt(b)/(80*x**9*sqrt(a/(b*x**2) + 1)) + A*b**(3/2)/(480*a*
x**7*sqrt(a/(b*x**2) + 1)) - 7*A*b**(5/2)/(1920*a**2*x**5*sqrt(a/(b*x**2) + 1)) + 7*A*b**(7/2)/(768*a**3*x**3*
sqrt(a/(b*x**2) + 1)) + 7*A*b**(9/2)/(256*a**4*x*sqrt(a/(b*x**2) + 1)) - 7*A*b**5*asinh(sqrt(a)/(sqrt(b)*x))/(
256*a**(9/2)) - B*a/(8*sqrt(b)*x**9*sqrt(a/(b*x**2) + 1)) - 7*B*sqrt(b)/(48*x**7*sqrt(a/(b*x**2) + 1)) + B*b**
(3/2)/(192*a*x**5*sqrt(a/(b*x**2) + 1)) - 5*B*b**(5/2)/(384*a**2*x**3*sqrt(a/(b*x**2) + 1)) - 5*B*b**(7/2)/(12
8*a**3*x*sqrt(a/(b*x**2) + 1)) + 5*B*b**4*asinh(sqrt(a)/(sqrt(b)*x))/(128*a**(7/2))

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Giac [A]  time = 1.15335, size = 311, normalized size = 1.65 \begin{align*} -\frac{\frac{15 \,{\left (10 \, B a b^{5} - 7 \, A b^{6}\right )} \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{4}} + \frac{150 \,{\left (b x^{2} + a\right )}^{\frac{9}{2}} B a b^{5} - 700 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} B a^{2} b^{5} + 1280 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} B a^{3} b^{5} - 580 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} B a^{4} b^{5} - 150 \, \sqrt{b x^{2} + a} B a^{5} b^{5} - 105 \,{\left (b x^{2} + a\right )}^{\frac{9}{2}} A b^{6} + 490 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} A a b^{6} - 896 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} A a^{2} b^{6} + 790 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} A a^{3} b^{6} + 105 \, \sqrt{b x^{2} + a} A a^{4} b^{6}}{a^{4} b^{5} x^{10}}}{3840 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)*(b*x^2+a)^(1/2)/x^11,x, algorithm="giac")

[Out]

-1/3840*(15*(10*B*a*b^5 - 7*A*b^6)*arctan(sqrt(b*x^2 + a)/sqrt(-a))/(sqrt(-a)*a^4) + (150*(b*x^2 + a)^(9/2)*B*
a*b^5 - 700*(b*x^2 + a)^(7/2)*B*a^2*b^5 + 1280*(b*x^2 + a)^(5/2)*B*a^3*b^5 - 580*(b*x^2 + a)^(3/2)*B*a^4*b^5 -
150*sqrt(b*x^2 + a)*B*a^5*b^5 - 105*(b*x^2 + a)^(9/2)*A*b^6 + 490*(b*x^2 + a)^(7/2)*A*a*b^6 - 896*(b*x^2 + a)
^(5/2)*A*a^2*b^6 + 790*(b*x^2 + a)^(3/2)*A*a^3*b^6 + 105*sqrt(b*x^2 + a)*A*a^4*b^6)/(a^4*b^5*x^10))/b