3.530 \(\int \frac {1}{x^3 (a+b x^2)^{9/2}} \, dx\)
Optimal. Leaf size=126 \[ \frac {9 b \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 a^{11/2}}-\frac {9 b}{2 a^5 \sqrt {a+b x^2}}-\frac {3 b}{2 a^4 \left (a+b x^2\right )^{3/2}}-\frac {9 b}{10 a^3 \left (a+b x^2\right )^{5/2}}-\frac {9 b}{14 a^2 \left (a+b x^2\right )^{7/2}}-\frac {1}{2 a x^2 \left (a+b x^2\right )^{7/2}} \]
[Out]
-9/14*b/a^2/(b*x^2+a)^(7/2)-1/2/a/x^2/(b*x^2+a)^(7/2)-9/10*b/a^3/(b*x^2+a)^(5/2)-3/2*b/a^4/(b*x^2+a)^(3/2)+9/2
*b*arctanh((b*x^2+a)^(1/2)/a^(1/2))/a^(11/2)-9/2*b/a^5/(b*x^2+a)^(1/2)
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Rubi [A] time = 0.08, antiderivative size = 132, normalized size of antiderivative = 1.05,
number of steps used = 8, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used =
{266, 51, 63, 208} \[ -\frac {9 \sqrt {a+b x^2}}{2 a^5 x^2}+\frac {3}{a^4 x^2 \sqrt {a+b x^2}}+\frac {3}{5 a^3 x^2 \left (a+b x^2\right )^{3/2}}+\frac {9}{35 a^2 x^2 \left (a+b x^2\right )^{5/2}}+\frac {9 b \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 a^{11/2}}+\frac {1}{7 a x^2 \left (a+b x^2\right )^{7/2}} \]
Antiderivative was successfully verified.
[In]
Int[1/(x^3*(a + b*x^2)^(9/2)),x]
[Out]
1/(7*a*x^2*(a + b*x^2)^(7/2)) + 9/(35*a^2*x^2*(a + b*x^2)^(5/2)) + 3/(5*a^3*x^2*(a + b*x^2)^(3/2)) + 3/(a^4*x^
2*Sqrt[a + b*x^2]) - (9*Sqrt[a + b*x^2])/(2*a^5*x^2) + (9*b*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(2*a^(11/2))
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]
Rule 266
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (a+b x^2\right )^{9/2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x^2 (a+b x)^{9/2}} \, dx,x,x^2\right )\\ &=\frac {1}{7 a x^2 \left (a+b x^2\right )^{7/2}}+\frac {9 \operatorname {Subst}\left (\int \frac {1}{x^2 (a+b x)^{7/2}} \, dx,x,x^2\right )}{14 a}\\ &=\frac {1}{7 a x^2 \left (a+b x^2\right )^{7/2}}+\frac {9}{35 a^2 x^2 \left (a+b x^2\right )^{5/2}}+\frac {9 \operatorname {Subst}\left (\int \frac {1}{x^2 (a+b x)^{5/2}} \, dx,x,x^2\right )}{10 a^2}\\ &=\frac {1}{7 a x^2 \left (a+b x^2\right )^{7/2}}+\frac {9}{35 a^2 x^2 \left (a+b x^2\right )^{5/2}}+\frac {3}{5 a^3 x^2 \left (a+b x^2\right )^{3/2}}+\frac {3 \operatorname {Subst}\left (\int \frac {1}{x^2 (a+b x)^{3/2}} \, dx,x,x^2\right )}{2 a^3}\\ &=\frac {1}{7 a x^2 \left (a+b x^2\right )^{7/2}}+\frac {9}{35 a^2 x^2 \left (a+b x^2\right )^{5/2}}+\frac {3}{5 a^3 x^2 \left (a+b x^2\right )^{3/2}}+\frac {3}{a^4 x^2 \sqrt {a+b x^2}}+\frac {9 \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b x}} \, dx,x,x^2\right )}{2 a^4}\\ &=\frac {1}{7 a x^2 \left (a+b x^2\right )^{7/2}}+\frac {9}{35 a^2 x^2 \left (a+b x^2\right )^{5/2}}+\frac {3}{5 a^3 x^2 \left (a+b x^2\right )^{3/2}}+\frac {3}{a^4 x^2 \sqrt {a+b x^2}}-\frac {9 \sqrt {a+b x^2}}{2 a^5 x^2}-\frac {(9 b) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^2\right )}{4 a^5}\\ &=\frac {1}{7 a x^2 \left (a+b x^2\right )^{7/2}}+\frac {9}{35 a^2 x^2 \left (a+b x^2\right )^{5/2}}+\frac {3}{5 a^3 x^2 \left (a+b x^2\right )^{3/2}}+\frac {3}{a^4 x^2 \sqrt {a+b x^2}}-\frac {9 \sqrt {a+b x^2}}{2 a^5 x^2}-\frac {9 \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^2}\right )}{2 a^5}\\ &=\frac {1}{7 a x^2 \left (a+b x^2\right )^{7/2}}+\frac {9}{35 a^2 x^2 \left (a+b x^2\right )^{5/2}}+\frac {3}{5 a^3 x^2 \left (a+b x^2\right )^{3/2}}+\frac {3}{a^4 x^2 \sqrt {a+b x^2}}-\frac {9 \sqrt {a+b x^2}}{2 a^5 x^2}+\frac {9 b \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 a^{11/2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 37, normalized size = 0.29 \[ -\frac {b \, _2F_1\left (-\frac {7}{2},2;-\frac {5}{2};\frac {b x^2}{a}+1\right )}{7 a^2 \left (a+b x^2\right )^{7/2}} \]
Antiderivative was successfully verified.
[In]
Integrate[1/(x^3*(a + b*x^2)^(9/2)),x]
[Out]
-1/7*(b*Hypergeometric2F1[-7/2, 2, -5/2, 1 + (b*x^2)/a])/(a^2*(a + b*x^2)^(7/2))
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fricas [A] time = 1.11, size = 373, normalized size = 2.96 \[ \left [\frac {315 \, {\left (b^{5} x^{10} + 4 \, a b^{4} x^{8} + 6 \, a^{2} b^{3} x^{6} + 4 \, a^{3} b^{2} x^{4} + a^{4} b x^{2}\right )} \sqrt {a} \log \left (-\frac {b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (315 \, a b^{4} x^{8} + 1050 \, a^{2} b^{3} x^{6} + 1218 \, a^{3} b^{2} x^{4} + 528 \, a^{4} b x^{2} + 35 \, a^{5}\right )} \sqrt {b x^{2} + a}}{140 \, {\left (a^{6} b^{4} x^{10} + 4 \, a^{7} b^{3} x^{8} + 6 \, a^{8} b^{2} x^{6} + 4 \, a^{9} b x^{4} + a^{10} x^{2}\right )}}, -\frac {315 \, {\left (b^{5} x^{10} + 4 \, a b^{4} x^{8} + 6 \, a^{2} b^{3} x^{6} + 4 \, a^{3} b^{2} x^{4} + a^{4} b x^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) + {\left (315 \, a b^{4} x^{8} + 1050 \, a^{2} b^{3} x^{6} + 1218 \, a^{3} b^{2} x^{4} + 528 \, a^{4} b x^{2} + 35 \, a^{5}\right )} \sqrt {b x^{2} + a}}{70 \, {\left (a^{6} b^{4} x^{10} + 4 \, a^{7} b^{3} x^{8} + 6 \, a^{8} b^{2} x^{6} + 4 \, a^{9} b x^{4} + a^{10} x^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^3/(b*x^2+a)^(9/2),x, algorithm="fricas")
[Out]
[1/140*(315*(b^5*x^10 + 4*a*b^4*x^8 + 6*a^2*b^3*x^6 + 4*a^3*b^2*x^4 + a^4*b*x^2)*sqrt(a)*log(-(b*x^2 + 2*sqrt(
b*x^2 + a)*sqrt(a) + 2*a)/x^2) - 2*(315*a*b^4*x^8 + 1050*a^2*b^3*x^6 + 1218*a^3*b^2*x^4 + 528*a^4*b*x^2 + 35*a
^5)*sqrt(b*x^2 + a))/(a^6*b^4*x^10 + 4*a^7*b^3*x^8 + 6*a^8*b^2*x^6 + 4*a^9*b*x^4 + a^10*x^2), -1/70*(315*(b^5*
x^10 + 4*a*b^4*x^8 + 6*a^2*b^3*x^6 + 4*a^3*b^2*x^4 + a^4*b*x^2)*sqrt(-a)*arctan(sqrt(-a)/sqrt(b*x^2 + a)) + (3
15*a*b^4*x^8 + 1050*a^2*b^3*x^6 + 1218*a^3*b^2*x^4 + 528*a^4*b*x^2 + 35*a^5)*sqrt(b*x^2 + a))/(a^6*b^4*x^10 +
4*a^7*b^3*x^8 + 6*a^8*b^2*x^6 + 4*a^9*b*x^4 + a^10*x^2)]
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giac [A] time = 1.13, size = 104, normalized size = 0.83 \[ -\frac {9 \, b \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{2 \, \sqrt {-a} a^{5}} - \frac {\sqrt {b x^{2} + a}}{2 \, a^{5} x^{2}} - \frac {140 \, {\left (b x^{2} + a\right )}^{3} b + 35 \, {\left (b x^{2} + a\right )}^{2} a b + 14 \, {\left (b x^{2} + a\right )} a^{2} b + 5 \, a^{3} b}{35 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^3/(b*x^2+a)^(9/2),x, algorithm="giac")
[Out]
-9/2*b*arctan(sqrt(b*x^2 + a)/sqrt(-a))/(sqrt(-a)*a^5) - 1/2*sqrt(b*x^2 + a)/(a^5*x^2) - 1/35*(140*(b*x^2 + a)
^3*b + 35*(b*x^2 + a)^2*a*b + 14*(b*x^2 + a)*a^2*b + 5*a^3*b)/((b*x^2 + a)^(7/2)*a^5)
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maple [A] time = 0.01, size = 108, normalized size = 0.86 \[ -\frac {9 b}{14 \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{2}}-\frac {9 b}{10 \left (b \,x^{2}+a \right )^{\frac {5}{2}} a^{3}}-\frac {1}{2 \left (b \,x^{2}+a \right )^{\frac {7}{2}} a \,x^{2}}-\frac {3 b}{2 \left (b \,x^{2}+a \right )^{\frac {3}{2}} a^{4}}+\frac {9 b \ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )}{2 a^{\frac {11}{2}}}-\frac {9 b}{2 \sqrt {b \,x^{2}+a}\, a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/x^3/(b*x^2+a)^(9/2),x)
[Out]
-1/2/a/x^2/(b*x^2+a)^(7/2)-9/14*b/a^2/(b*x^2+a)^(7/2)-9/10*b/a^3/(b*x^2+a)^(5/2)-3/2*b/a^4/(b*x^2+a)^(3/2)-9/2
*b/a^5/(b*x^2+a)^(1/2)+9/2/a^(11/2)*b*ln((2*a+2*(b*x^2+a)^(1/2)*a^(1/2))/x)
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maxima [A] time = 1.39, size = 96, normalized size = 0.76 \[ \frac {9 \, b \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{2 \, a^{\frac {11}{2}}} - \frac {9 \, b}{2 \, \sqrt {b x^{2} + a} a^{5}} - \frac {3 \, b}{2 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{4}} - \frac {9 \, b}{10 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{3}} - \frac {9 \, b}{14 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{2}} - \frac {1}{2 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^3/(b*x^2+a)^(9/2),x, algorithm="maxima")
[Out]
9/2*b*arcsinh(a/(sqrt(a*b)*abs(x)))/a^(11/2) - 9/2*b/(sqrt(b*x^2 + a)*a^5) - 3/2*b/((b*x^2 + a)^(3/2)*a^4) - 9
/10*b/((b*x^2 + a)^(5/2)*a^3) - 9/14*b/((b*x^2 + a)^(7/2)*a^2) - 1/2/((b*x^2 + a)^(7/2)*a*x^2)
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mupad [B] time = 4.96, size = 113, normalized size = 0.90 \[ \frac {9\,b\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{2\,a^{11/2}}-\frac {\frac {b}{7\,a}+\frac {3\,b\,{\left (b\,x^2+a\right )}^2}{5\,a^3}+\frac {3\,b\,{\left (b\,x^2+a\right )}^3}{a^4}-\frac {9\,b\,{\left (b\,x^2+a\right )}^4}{2\,a^5}+\frac {9\,b\,\left (b\,x^2+a\right )}{35\,a^2}}{a\,{\left (b\,x^2+a\right )}^{7/2}-{\left (b\,x^2+a\right )}^{9/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(x^3*(a + b*x^2)^(9/2)),x)
[Out]
(9*b*atanh((a + b*x^2)^(1/2)/a^(1/2)))/(2*a^(11/2)) - (b/(7*a) + (3*b*(a + b*x^2)^2)/(5*a^3) + (3*b*(a + b*x^2
)^3)/a^4 - (9*b*(a + b*x^2)^4)/(2*a^5) + (9*b*(a + b*x^2))/(35*a^2))/(a*(a + b*x^2)^(7/2) - (a + b*x^2)^(9/2))
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sympy [B] time = 12.43, size = 5540, normalized size = 43.97 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x**3/(b*x**2+a)**(9/2),x)
[Out]
-70*a**49*sqrt(1 + b*x**2/a)/(140*a**(107/2)*x**2 + 1400*a**(105/2)*b*x**4 + 6300*a**(103/2)*b**2*x**6 + 16800
*a**(101/2)*b**3*x**8 + 29400*a**(99/2)*b**4*x**10 + 35280*a**(97/2)*b**5*x**12 + 29400*a**(95/2)*b**6*x**14 +
16800*a**(93/2)*b**7*x**16 + 6300*a**(91/2)*b**8*x**18 + 1400*a**(89/2)*b**9*x**20 + 140*a**(87/2)*b**10*x**2
2) - 1476*a**48*b*x**2*sqrt(1 + b*x**2/a)/(140*a**(107/2)*x**2 + 1400*a**(105/2)*b*x**4 + 6300*a**(103/2)*b**2
*x**6 + 16800*a**(101/2)*b**3*x**8 + 29400*a**(99/2)*b**4*x**10 + 35280*a**(97/2)*b**5*x**12 + 29400*a**(95/2)
*b**6*x**14 + 16800*a**(93/2)*b**7*x**16 + 6300*a**(91/2)*b**8*x**18 + 1400*a**(89/2)*b**9*x**20 + 140*a**(87/
2)*b**10*x**22) - 315*a**48*b*x**2*log(b*x**2/a)/(140*a**(107/2)*x**2 + 1400*a**(105/2)*b*x**4 + 6300*a**(103/
2)*b**2*x**6 + 16800*a**(101/2)*b**3*x**8 + 29400*a**(99/2)*b**4*x**10 + 35280*a**(97/2)*b**5*x**12 + 29400*a*
*(95/2)*b**6*x**14 + 16800*a**(93/2)*b**7*x**16 + 6300*a**(91/2)*b**8*x**18 + 1400*a**(89/2)*b**9*x**20 + 140*
a**(87/2)*b**10*x**22) + 630*a**48*b*x**2*log(sqrt(1 + b*x**2/a) + 1)/(140*a**(107/2)*x**2 + 1400*a**(105/2)*b
*x**4 + 6300*a**(103/2)*b**2*x**6 + 16800*a**(101/2)*b**3*x**8 + 29400*a**(99/2)*b**4*x**10 + 35280*a**(97/2)*
b**5*x**12 + 29400*a**(95/2)*b**6*x**14 + 16800*a**(93/2)*b**7*x**16 + 6300*a**(91/2)*b**8*x**18 + 1400*a**(89
/2)*b**9*x**20 + 140*a**(87/2)*b**10*x**22) - 9822*a**47*b**2*x**4*sqrt(1 + b*x**2/a)/(140*a**(107/2)*x**2 + 1
400*a**(105/2)*b*x**4 + 6300*a**(103/2)*b**2*x**6 + 16800*a**(101/2)*b**3*x**8 + 29400*a**(99/2)*b**4*x**10 +
35280*a**(97/2)*b**5*x**12 + 29400*a**(95/2)*b**6*x**14 + 16800*a**(93/2)*b**7*x**16 + 6300*a**(91/2)*b**8*x**
18 + 1400*a**(89/2)*b**9*x**20 + 140*a**(87/2)*b**10*x**22) - 3150*a**47*b**2*x**4*log(b*x**2/a)/(140*a**(107/
2)*x**2 + 1400*a**(105/2)*b*x**4 + 6300*a**(103/2)*b**2*x**6 + 16800*a**(101/2)*b**3*x**8 + 29400*a**(99/2)*b*
*4*x**10 + 35280*a**(97/2)*b**5*x**12 + 29400*a**(95/2)*b**6*x**14 + 16800*a**(93/2)*b**7*x**16 + 6300*a**(91/
2)*b**8*x**18 + 1400*a**(89/2)*b**9*x**20 + 140*a**(87/2)*b**10*x**22) + 6300*a**47*b**2*x**4*log(sqrt(1 + b*x
**2/a) + 1)/(140*a**(107/2)*x**2 + 1400*a**(105/2)*b*x**4 + 6300*a**(103/2)*b**2*x**6 + 16800*a**(101/2)*b**3*
x**8 + 29400*a**(99/2)*b**4*x**10 + 35280*a**(97/2)*b**5*x**12 + 29400*a**(95/2)*b**6*x**14 + 16800*a**(93/2)*
b**7*x**16 + 6300*a**(91/2)*b**8*x**18 + 1400*a**(89/2)*b**9*x**20 + 140*a**(87/2)*b**10*x**22) - 33956*a**46*
b**3*x**6*sqrt(1 + b*x**2/a)/(140*a**(107/2)*x**2 + 1400*a**(105/2)*b*x**4 + 6300*a**(103/2)*b**2*x**6 + 16800
*a**(101/2)*b**3*x**8 + 29400*a**(99/2)*b**4*x**10 + 35280*a**(97/2)*b**5*x**12 + 29400*a**(95/2)*b**6*x**14 +
16800*a**(93/2)*b**7*x**16 + 6300*a**(91/2)*b**8*x**18 + 1400*a**(89/2)*b**9*x**20 + 140*a**(87/2)*b**10*x**2
2) - 14175*a**46*b**3*x**6*log(b*x**2/a)/(140*a**(107/2)*x**2 + 1400*a**(105/2)*b*x**4 + 6300*a**(103/2)*b**2*
x**6 + 16800*a**(101/2)*b**3*x**8 + 29400*a**(99/2)*b**4*x**10 + 35280*a**(97/2)*b**5*x**12 + 29400*a**(95/2)*
b**6*x**14 + 16800*a**(93/2)*b**7*x**16 + 6300*a**(91/2)*b**8*x**18 + 1400*a**(89/2)*b**9*x**20 + 140*a**(87/2
)*b**10*x**22) + 28350*a**46*b**3*x**6*log(sqrt(1 + b*x**2/a) + 1)/(140*a**(107/2)*x**2 + 1400*a**(105/2)*b*x*
*4 + 6300*a**(103/2)*b**2*x**6 + 16800*a**(101/2)*b**3*x**8 + 29400*a**(99/2)*b**4*x**10 + 35280*a**(97/2)*b**
5*x**12 + 29400*a**(95/2)*b**6*x**14 + 16800*a**(93/2)*b**7*x**16 + 6300*a**(91/2)*b**8*x**18 + 1400*a**(89/2)
*b**9*x**20 + 140*a**(87/2)*b**10*x**22) - 71940*a**45*b**4*x**8*sqrt(1 + b*x**2/a)/(140*a**(107/2)*x**2 + 140
0*a**(105/2)*b*x**4 + 6300*a**(103/2)*b**2*x**6 + 16800*a**(101/2)*b**3*x**8 + 29400*a**(99/2)*b**4*x**10 + 35
280*a**(97/2)*b**5*x**12 + 29400*a**(95/2)*b**6*x**14 + 16800*a**(93/2)*b**7*x**16 + 6300*a**(91/2)*b**8*x**18
+ 1400*a**(89/2)*b**9*x**20 + 140*a**(87/2)*b**10*x**22) - 37800*a**45*b**4*x**8*log(b*x**2/a)/(140*a**(107/2
)*x**2 + 1400*a**(105/2)*b*x**4 + 6300*a**(103/2)*b**2*x**6 + 16800*a**(101/2)*b**3*x**8 + 29400*a**(99/2)*b**
4*x**10 + 35280*a**(97/2)*b**5*x**12 + 29400*a**(95/2)*b**6*x**14 + 16800*a**(93/2)*b**7*x**16 + 6300*a**(91/2
)*b**8*x**18 + 1400*a**(89/2)*b**9*x**20 + 140*a**(87/2)*b**10*x**22) + 75600*a**45*b**4*x**8*log(sqrt(1 + b*x
**2/a) + 1)/(140*a**(107/2)*x**2 + 1400*a**(105/2)*b*x**4 + 6300*a**(103/2)*b**2*x**6 + 16800*a**(101/2)*b**3*
x**8 + 29400*a**(99/2)*b**4*x**10 + 35280*a**(97/2)*b**5*x**12 + 29400*a**(95/2)*b**6*x**14 + 16800*a**(93/2)*
b**7*x**16 + 6300*a**(91/2)*b**8*x**18 + 1400*a**(89/2)*b**9*x**20 + 140*a**(87/2)*b**10*x**22) - 100260*a**44
*b**5*x**10*sqrt(1 + b*x**2/a)/(140*a**(107/2)*x**2 + 1400*a**(105/2)*b*x**4 + 6300*a**(103/2)*b**2*x**6 + 168
00*a**(101/2)*b**3*x**8 + 29400*a**(99/2)*b**4*x**10 + 35280*a**(97/2)*b**5*x**12 + 29400*a**(95/2)*b**6*x**14
+ 16800*a**(93/2)*b**7*x**16 + 6300*a**(91/2)*b**8*x**18 + 1400*a**(89/2)*b**9*x**20 + 140*a**(87/2)*b**10*x*
*22) - 66150*a**44*b**5*x**10*log(b*x**2/a)/(140*a**(107/2)*x**2 + 1400*a**(105/2)*b*x**4 + 6300*a**(103/2)*b*
*2*x**6 + 16800*a**(101/2)*b**3*x**8 + 29400*a**(99/2)*b**4*x**10 + 35280*a**(97/2)*b**5*x**12 + 29400*a**(95/
2)*b**6*x**14 + 16800*a**(93/2)*b**7*x**16 + 6300*a**(91/2)*b**8*x**18 + 1400*a**(89/2)*b**9*x**20 + 140*a**(8
7/2)*b**10*x**22) + 132300*a**44*b**5*x**10*log(sqrt(1 + b*x**2/a) + 1)/(140*a**(107/2)*x**2 + 1400*a**(105/2)
*b*x**4 + 6300*a**(103/2)*b**2*x**6 + 16800*a**(101/2)*b**3*x**8 + 29400*a**(99/2)*b**4*x**10 + 35280*a**(97/2
)*b**5*x**12 + 29400*a**(95/2)*b**6*x**14 + 16800*a**(93/2)*b**7*x**16 + 6300*a**(91/2)*b**8*x**18 + 1400*a**(
89/2)*b**9*x**20 + 140*a**(87/2)*b**10*x**22) - 94396*a**43*b**6*x**12*sqrt(1 + b*x**2/a)/(140*a**(107/2)*x**2
+ 1400*a**(105/2)*b*x**4 + 6300*a**(103/2)*b**2*x**6 + 16800*a**(101/2)*b**3*x**8 + 29400*a**(99/2)*b**4*x**1
0 + 35280*a**(97/2)*b**5*x**12 + 29400*a**(95/2)*b**6*x**14 + 16800*a**(93/2)*b**7*x**16 + 6300*a**(91/2)*b**8
*x**18 + 1400*a**(89/2)*b**9*x**20 + 140*a**(87/2)*b**10*x**22) - 79380*a**43*b**6*x**12*log(b*x**2/a)/(140*a*
*(107/2)*x**2 + 1400*a**(105/2)*b*x**4 + 6300*a**(103/2)*b**2*x**6 + 16800*a**(101/2)*b**3*x**8 + 29400*a**(99
/2)*b**4*x**10 + 35280*a**(97/2)*b**5*x**12 + 29400*a**(95/2)*b**6*x**14 + 16800*a**(93/2)*b**7*x**16 + 6300*a
**(91/2)*b**8*x**18 + 1400*a**(89/2)*b**9*x**20 + 140*a**(87/2)*b**10*x**22) + 158760*a**43*b**6*x**12*log(sqr
t(1 + b*x**2/a) + 1)/(140*a**(107/2)*x**2 + 1400*a**(105/2)*b*x**4 + 6300*a**(103/2)*b**2*x**6 + 16800*a**(101
/2)*b**3*x**8 + 29400*a**(99/2)*b**4*x**10 + 35280*a**(97/2)*b**5*x**12 + 29400*a**(95/2)*b**6*x**14 + 16800*a
**(93/2)*b**7*x**16 + 6300*a**(91/2)*b**8*x**18 + 1400*a**(89/2)*b**9*x**20 + 140*a**(87/2)*b**10*x**22) - 597
72*a**42*b**7*x**14*sqrt(1 + b*x**2/a)/(140*a**(107/2)*x**2 + 1400*a**(105/2)*b*x**4 + 6300*a**(103/2)*b**2*x*
*6 + 16800*a**(101/2)*b**3*x**8 + 29400*a**(99/2)*b**4*x**10 + 35280*a**(97/2)*b**5*x**12 + 29400*a**(95/2)*b*
*6*x**14 + 16800*a**(93/2)*b**7*x**16 + 6300*a**(91/2)*b**8*x**18 + 1400*a**(89/2)*b**9*x**20 + 140*a**(87/2)*
b**10*x**22) - 66150*a**42*b**7*x**14*log(b*x**2/a)/(140*a**(107/2)*x**2 + 1400*a**(105/2)*b*x**4 + 6300*a**(1
03/2)*b**2*x**6 + 16800*a**(101/2)*b**3*x**8 + 29400*a**(99/2)*b**4*x**10 + 35280*a**(97/2)*b**5*x**12 + 29400
*a**(95/2)*b**6*x**14 + 16800*a**(93/2)*b**7*x**16 + 6300*a**(91/2)*b**8*x**18 + 1400*a**(89/2)*b**9*x**20 + 1
40*a**(87/2)*b**10*x**22) + 132300*a**42*b**7*x**14*log(sqrt(1 + b*x**2/a) + 1)/(140*a**(107/2)*x**2 + 1400*a*
*(105/2)*b*x**4 + 6300*a**(103/2)*b**2*x**6 + 16800*a**(101/2)*b**3*x**8 + 29400*a**(99/2)*b**4*x**10 + 35280*
a**(97/2)*b**5*x**12 + 29400*a**(95/2)*b**6*x**14 + 16800*a**(93/2)*b**7*x**16 + 6300*a**(91/2)*b**8*x**18 + 1
400*a**(89/2)*b**9*x**20 + 140*a**(87/2)*b**10*x**22) - 24486*a**41*b**8*x**16*sqrt(1 + b*x**2/a)/(140*a**(107
/2)*x**2 + 1400*a**(105/2)*b*x**4 + 6300*a**(103/2)*b**2*x**6 + 16800*a**(101/2)*b**3*x**8 + 29400*a**(99/2)*b
**4*x**10 + 35280*a**(97/2)*b**5*x**12 + 29400*a**(95/2)*b**6*x**14 + 16800*a**(93/2)*b**7*x**16 + 6300*a**(91
/2)*b**8*x**18 + 1400*a**(89/2)*b**9*x**20 + 140*a**(87/2)*b**10*x**22) - 37800*a**41*b**8*x**16*log(b*x**2/a)
/(140*a**(107/2)*x**2 + 1400*a**(105/2)*b*x**4 + 6300*a**(103/2)*b**2*x**6 + 16800*a**(101/2)*b**3*x**8 + 2940
0*a**(99/2)*b**4*x**10 + 35280*a**(97/2)*b**5*x**12 + 29400*a**(95/2)*b**6*x**14 + 16800*a**(93/2)*b**7*x**16
+ 6300*a**(91/2)*b**8*x**18 + 1400*a**(89/2)*b**9*x**20 + 140*a**(87/2)*b**10*x**22) + 75600*a**41*b**8*x**16*
log(sqrt(1 + b*x**2/a) + 1)/(140*a**(107/2)*x**2 + 1400*a**(105/2)*b*x**4 + 6300*a**(103/2)*b**2*x**6 + 16800*
a**(101/2)*b**3*x**8 + 29400*a**(99/2)*b**4*x**10 + 35280*a**(97/2)*b**5*x**12 + 29400*a**(95/2)*b**6*x**14 +
16800*a**(93/2)*b**7*x**16 + 6300*a**(91/2)*b**8*x**18 + 1400*a**(89/2)*b**9*x**20 + 140*a**(87/2)*b**10*x**22
) - 5880*a**40*b**9*x**18*sqrt(1 + b*x**2/a)/(140*a**(107/2)*x**2 + 1400*a**(105/2)*b*x**4 + 6300*a**(103/2)*b
**2*x**6 + 16800*a**(101/2)*b**3*x**8 + 29400*a**(99/2)*b**4*x**10 + 35280*a**(97/2)*b**5*x**12 + 29400*a**(95
/2)*b**6*x**14 + 16800*a**(93/2)*b**7*x**16 + 6300*a**(91/2)*b**8*x**18 + 1400*a**(89/2)*b**9*x**20 + 140*a**(
87/2)*b**10*x**22) - 14175*a**40*b**9*x**18*log(b*x**2/a)/(140*a**(107/2)*x**2 + 1400*a**(105/2)*b*x**4 + 6300
*a**(103/2)*b**2*x**6 + 16800*a**(101/2)*b**3*x**8 + 29400*a**(99/2)*b**4*x**10 + 35280*a**(97/2)*b**5*x**12 +
29400*a**(95/2)*b**6*x**14 + 16800*a**(93/2)*b**7*x**16 + 6300*a**(91/2)*b**8*x**18 + 1400*a**(89/2)*b**9*x**
20 + 140*a**(87/2)*b**10*x**22) + 28350*a**40*b**9*x**18*log(sqrt(1 + b*x**2/a) + 1)/(140*a**(107/2)*x**2 + 14
00*a**(105/2)*b*x**4 + 6300*a**(103/2)*b**2*x**6 + 16800*a**(101/2)*b**3*x**8 + 29400*a**(99/2)*b**4*x**10 + 3
5280*a**(97/2)*b**5*x**12 + 29400*a**(95/2)*b**6*x**14 + 16800*a**(93/2)*b**7*x**16 + 6300*a**(91/2)*b**8*x**1
8 + 1400*a**(89/2)*b**9*x**20 + 140*a**(87/2)*b**10*x**22) - 630*a**39*b**10*x**20*sqrt(1 + b*x**2/a)/(140*a**
(107/2)*x**2 + 1400*a**(105/2)*b*x**4 + 6300*a**(103/2)*b**2*x**6 + 16800*a**(101/2)*b**3*x**8 + 29400*a**(99/
2)*b**4*x**10 + 35280*a**(97/2)*b**5*x**12 + 29400*a**(95/2)*b**6*x**14 + 16800*a**(93/2)*b**7*x**16 + 6300*a*
*(91/2)*b**8*x**18 + 1400*a**(89/2)*b**9*x**20 + 140*a**(87/2)*b**10*x**22) - 3150*a**39*b**10*x**20*log(b*x**
2/a)/(140*a**(107/2)*x**2 + 1400*a**(105/2)*b*x**4 + 6300*a**(103/2)*b**2*x**6 + 16800*a**(101/2)*b**3*x**8 +
29400*a**(99/2)*b**4*x**10 + 35280*a**(97/2)*b**5*x**12 + 29400*a**(95/2)*b**6*x**14 + 16800*a**(93/2)*b**7*x*
*16 + 6300*a**(91/2)*b**8*x**18 + 1400*a**(89/2)*b**9*x**20 + 140*a**(87/2)*b**10*x**22) + 6300*a**39*b**10*x*
*20*log(sqrt(1 + b*x**2/a) + 1)/(140*a**(107/2)*x**2 + 1400*a**(105/2)*b*x**4 + 6300*a**(103/2)*b**2*x**6 + 16
800*a**(101/2)*b**3*x**8 + 29400*a**(99/2)*b**4*x**10 + 35280*a**(97/2)*b**5*x**12 + 29400*a**(95/2)*b**6*x**1
4 + 16800*a**(93/2)*b**7*x**16 + 6300*a**(91/2)*b**8*x**18 + 1400*a**(89/2)*b**9*x**20 + 140*a**(87/2)*b**10*x
**22) - 315*a**38*b**11*x**22*log(b*x**2/a)/(140*a**(107/2)*x**2 + 1400*a**(105/2)*b*x**4 + 6300*a**(103/2)*b*
*2*x**6 + 16800*a**(101/2)*b**3*x**8 + 29400*a**(99/2)*b**4*x**10 + 35280*a**(97/2)*b**5*x**12 + 29400*a**(95/
2)*b**6*x**14 + 16800*a**(93/2)*b**7*x**16 + 6300*a**(91/2)*b**8*x**18 + 1400*a**(89/2)*b**9*x**20 + 140*a**(8
7/2)*b**10*x**22) + 630*a**38*b**11*x**22*log(sqrt(1 + b*x**2/a) + 1)/(140*a**(107/2)*x**2 + 1400*a**(105/2)*b
*x**4 + 6300*a**(103/2)*b**2*x**6 + 16800*a**(101/2)*b**3*x**8 + 29400*a**(99/2)*b**4*x**10 + 35280*a**(97/2)*
b**5*x**12 + 29400*a**(95/2)*b**6*x**14 + 16800*a**(93/2)*b**7*x**16 + 6300*a**(91/2)*b**8*x**18 + 1400*a**(89
/2)*b**9*x**20 + 140*a**(87/2)*b**10*x**22)
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