Integrand size = 22, antiderivative size = 153 \[ \int \frac {A+B x^2}{x^7 \left (a+b x^2\right )^{3/2}} \, dx=-\frac {5 b^2 (7 A b-6 a B)}{16 a^4 \sqrt {a+b x^2}}-\frac {A}{6 a x^6 \sqrt {a+b x^2}}+\frac {7 A b-6 a B}{24 a^2 x^4 \sqrt {a+b x^2}}-\frac {5 b (7 A b-6 a B)}{48 a^3 x^2 \sqrt {a+b x^2}}+\frac {5 b^2 (7 A b-6 a B) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{16 a^{9/2}} \]
5/16*b^2*(7*A*b-6*B*a)*arctanh((b*x^2+a)^(1/2)/a^(1/2))/a^(9/2)-5/16*b^2*( 7*A*b-6*B*a)/a^4/(b*x^2+a)^(1/2)-1/6*A/a/x^6/(b*x^2+a)^(1/2)+1/24*(7*A*b-6 *B*a)/a^2/x^4/(b*x^2+a)^(1/2)-5/48*b*(7*A*b-6*B*a)/a^3/x^2/(b*x^2+a)^(1/2)
Time = 0.21 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.82 \[ \int \frac {A+B x^2}{x^7 \left (a+b x^2\right )^{3/2}} \, dx=\frac {-8 a^3 A+14 a^2 A b x^2-12 a^3 B x^2-35 a A b^2 x^4+30 a^2 b B x^4-105 A b^3 x^6+90 a b^2 B x^6}{48 a^4 x^6 \sqrt {a+b x^2}}-\frac {5 b^2 (-7 A b+6 a B) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{16 a^{9/2}} \]
(-8*a^3*A + 14*a^2*A*b*x^2 - 12*a^3*B*x^2 - 35*a*A*b^2*x^4 + 30*a^2*b*B*x^ 4 - 105*A*b^3*x^6 + 90*a*b^2*B*x^6)/(48*a^4*x^6*Sqrt[a + b*x^2]) - (5*b^2* (-7*A*b + 6*a*B)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(16*a^(9/2))
Time = 0.24 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.93, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {354, 87, 52, 52, 61, 73, 221}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {A+B x^2}{x^7 \left (a+b x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 354 |
| \(\displaystyle \frac {1}{2} \int \frac {B x^2+A}{x^8 \left (b x^2+a\right )^{3/2}}dx^2\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {1}{2} \left (-\frac {(7 A b-6 a B) \int \frac {1}{x^6 \left (b x^2+a\right )^{3/2}}dx^2}{6 a}-\frac {A}{3 a x^6 \sqrt {a+b x^2}}\right )\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {1}{2} \left (-\frac {(7 A b-6 a B) \left (-\frac {5 b \int \frac {1}{x^4 \left (b x^2+a\right )^{3/2}}dx^2}{4 a}-\frac {1}{2 a x^4 \sqrt {a+b x^2}}\right )}{6 a}-\frac {A}{3 a x^6 \sqrt {a+b x^2}}\right )\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {1}{2} \left (-\frac {(7 A b-6 a B) \left (-\frac {5 b \left (-\frac {3 b \int \frac {1}{x^2 \left (b x^2+a\right )^{3/2}}dx^2}{2 a}-\frac {1}{a x^2 \sqrt {a+b x^2}}\right )}{4 a}-\frac {1}{2 a x^4 \sqrt {a+b x^2}}\right )}{6 a}-\frac {A}{3 a x^6 \sqrt {a+b x^2}}\right )\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {1}{2} \left (-\frac {(7 A b-6 a B) \left (-\frac {5 b \left (-\frac {3 b \left (\frac {\int \frac {1}{x^2 \sqrt {b x^2+a}}dx^2}{a}+\frac {2}{a \sqrt {a+b x^2}}\right )}{2 a}-\frac {1}{a x^2 \sqrt {a+b x^2}}\right )}{4 a}-\frac {1}{2 a x^4 \sqrt {a+b x^2}}\right )}{6 a}-\frac {A}{3 a x^6 \sqrt {a+b x^2}}\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{2} \left (-\frac {(7 A b-6 a B) \left (-\frac {5 b \left (-\frac {3 b \left (\frac {2 \int \frac {1}{\frac {x^4}{b}-\frac {a}{b}}d\sqrt {b x^2+a}}{a b}+\frac {2}{a \sqrt {a+b x^2}}\right )}{2 a}-\frac {1}{a x^2 \sqrt {a+b x^2}}\right )}{4 a}-\frac {1}{2 a x^4 \sqrt {a+b x^2}}\right )}{6 a}-\frac {A}{3 a x^6 \sqrt {a+b x^2}}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {1}{2} \left (-\frac {(7 A b-6 a B) \left (-\frac {5 b \left (-\frac {3 b \left (\frac {2}{a \sqrt {a+b x^2}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{3/2}}\right )}{2 a}-\frac {1}{a x^2 \sqrt {a+b x^2}}\right )}{4 a}-\frac {1}{2 a x^4 \sqrt {a+b x^2}}\right )}{6 a}-\frac {A}{3 a x^6 \sqrt {a+b x^2}}\right )\) |
(-1/3*A/(a*x^6*Sqrt[a + b*x^2]) - ((7*A*b - 6*a*B)*(-1/2*1/(a*x^4*Sqrt[a + b*x^2]) - (5*b*(-(1/(a*x^2*Sqrt[a + b*x^2])) - (3*b*(2/(a*Sqrt[a + b*x^2] ) - (2*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/a^(3/2)))/(2*a)))/(4*a)))/(6*a))/ 2
3.6.82.3.1 Defintions of rubi rules used
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] - Simp[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] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
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] - Simp[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] && 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]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[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] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(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] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x , x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ [(m - 1)/2]
Time = 2.88 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.78
| method | result | size |
| pseudoelliptic | \(\frac {\frac {35 x^{6} \left (A b -\frac {6 B a}{7}\right ) b^{2} \sqrt {b \,x^{2}+a}\, \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{\sqrt {a}}\right )}{16}-\frac {35 x^{4} \left (-\frac {18 x^{2} B}{7}+A \right ) b^{2} a^{\frac {3}{2}}}{48}+\frac {7 b \,x^{2} \left (\frac {15 x^{2} B}{7}+A \right ) a^{\frac {5}{2}}}{24}+\frac {\left (-3 x^{2} B -2 A \right ) a^{\frac {7}{2}}}{12}-\frac {35 A \sqrt {a}\, b^{3} x^{6}}{16}}{x^{6} a^{\frac {9}{2}} \sqrt {b \,x^{2}+a}}\) | \(120\) |
| risch | \(-\frac {\sqrt {b \,x^{2}+a}\, \left (57 A \,b^{2} x^{4}-42 B a b \,x^{4}-22 a A b \,x^{2}+12 a^{2} B \,x^{2}+8 a^{2} A \right )}{48 a^{4} x^{6}}-\frac {b^{2} \left (-\frac {19 A b -14 B a}{\sqrt {b \,x^{2}+a}}+5 a \left (7 A b -6 B a \right ) \left (\frac {1}{a \sqrt {b \,x^{2}+a}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{a^{\frac {3}{2}}}\right )\right )}{16 a^{4}}\) | \(143\) |
| default | \(A \left (-\frac {1}{6 a \,x^{6} \sqrt {b \,x^{2}+a}}-\frac {7 b \left (-\frac {1}{4 a \,x^{4} \sqrt {b \,x^{2}+a}}-\frac {5 b \left (-\frac {1}{2 a \,x^{2} \sqrt {b \,x^{2}+a}}-\frac {3 b \left (\frac {1}{a \sqrt {b \,x^{2}+a}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{a^{\frac {3}{2}}}\right )}{2 a}\right )}{4 a}\right )}{6 a}\right )+B \left (-\frac {1}{4 a \,x^{4} \sqrt {b \,x^{2}+a}}-\frac {5 b \left (-\frac {1}{2 a \,x^{2} \sqrt {b \,x^{2}+a}}-\frac {3 b \left (\frac {1}{a \sqrt {b \,x^{2}+a}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{a^{\frac {3}{2}}}\right )}{2 a}\right )}{4 a}\right )\) | \(210\) |
7/24*(15/2*x^6*(A*b-6/7*B*a)*b^2*(b*x^2+a)^(1/2)*arctanh((b*x^2+a)^(1/2)/a ^(1/2))-5/2*x^4*(-18/7*x^2*B+A)*b^2*a^(3/2)+b*x^2*(15/7*x^2*B+A)*a^(5/2)+2 /7*(-3*B*x^2-2*A)*a^(7/2)-15/2*A*a^(1/2)*b^3*x^6)/(b*x^2+a)^(1/2)/a^(9/2)/ x^6
Time = 0.28 (sec) , antiderivative size = 341, normalized size of antiderivative = 2.23 \[ \int \frac {A+B x^2}{x^7 \left (a+b x^2\right )^{3/2}} \, dx=\left [-\frac {15 \, {\left ({\left (6 \, B a b^{3} - 7 \, A b^{4}\right )} x^{8} + {\left (6 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{6}\right )} \sqrt {a} \log \left (-\frac {b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (15 \, {\left (6 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{6} - 8 \, A a^{4} + 5 \, {\left (6 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x^{4} - 2 \, {\left (6 \, B a^{4} - 7 \, A a^{3} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{96 \, {\left (a^{5} b x^{8} + a^{6} x^{6}\right )}}, \frac {15 \, {\left ({\left (6 \, B a b^{3} - 7 \, A b^{4}\right )} x^{8} + {\left (6 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{6}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) + {\left (15 \, {\left (6 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{6} - 8 \, A a^{4} + 5 \, {\left (6 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x^{4} - 2 \, {\left (6 \, B a^{4} - 7 \, A a^{3} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{48 \, {\left (a^{5} b x^{8} + a^{6} x^{6}\right )}}\right ] \]
[-1/96*(15*((6*B*a*b^3 - 7*A*b^4)*x^8 + (6*B*a^2*b^2 - 7*A*a*b^3)*x^6)*sqr t(a)*log(-(b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) - 2*(15*(6*B*a^2* b^2 - 7*A*a*b^3)*x^6 - 8*A*a^4 + 5*(6*B*a^3*b - 7*A*a^2*b^2)*x^4 - 2*(6*B* a^4 - 7*A*a^3*b)*x^2)*sqrt(b*x^2 + a))/(a^5*b*x^8 + a^6*x^6), 1/48*(15*((6 *B*a*b^3 - 7*A*b^4)*x^8 + (6*B*a^2*b^2 - 7*A*a*b^3)*x^6)*sqrt(-a)*arctan(s qrt(-a)/sqrt(b*x^2 + a)) + (15*(6*B*a^2*b^2 - 7*A*a*b^3)*x^6 - 8*A*a^4 + 5 *(6*B*a^3*b - 7*A*a^2*b^2)*x^4 - 2*(6*B*a^4 - 7*A*a^3*b)*x^2)*sqrt(b*x^2 + a))/(a^5*b*x^8 + a^6*x^6)]
Time = 54.82 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.54 \[ \int \frac {A+B x^2}{x^7 \left (a+b x^2\right )^{3/2}} \, dx=A \left (- \frac {1}{6 a \sqrt {b} x^{7} \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {7 \sqrt {b}}{24 a^{2} x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {35 b^{\frac {3}{2}}}{48 a^{3} x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {35 b^{\frac {5}{2}}}{16 a^{4} x \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {35 b^{3} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{16 a^{\frac {9}{2}}}\right ) + B \left (- \frac {1}{4 a \sqrt {b} x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {5 \sqrt {b}}{8 a^{2} x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {15 b^{\frac {3}{2}}}{8 a^{3} x \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {15 b^{2} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{8 a^{\frac {7}{2}}}\right ) \]
A*(-1/(6*a*sqrt(b)*x**7*sqrt(a/(b*x**2) + 1)) + 7*sqrt(b)/(24*a**2*x**5*sq rt(a/(b*x**2) + 1)) - 35*b**(3/2)/(48*a**3*x**3*sqrt(a/(b*x**2) + 1)) - 35 *b**(5/2)/(16*a**4*x*sqrt(a/(b*x**2) + 1)) + 35*b**3*asinh(sqrt(a)/(sqrt(b )*x))/(16*a**(9/2))) + B*(-1/(4*a*sqrt(b)*x**5*sqrt(a/(b*x**2) + 1)) + 5*s qrt(b)/(8*a**2*x**3*sqrt(a/(b*x**2) + 1)) + 15*b**(3/2)/(8*a**3*x*sqrt(a/( b*x**2) + 1)) - 15*b**2*asinh(sqrt(a)/(sqrt(b)*x))/(8*a**(7/2)))
Time = 0.19 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.14 \[ \int \frac {A+B x^2}{x^7 \left (a+b x^2\right )^{3/2}} \, dx=-\frac {15 \, B b^{2} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{8 \, a^{\frac {7}{2}}} + \frac {35 \, A b^{3} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{16 \, a^{\frac {9}{2}}} + \frac {15 \, B b^{2}}{8 \, \sqrt {b x^{2} + a} a^{3}} - \frac {35 \, A b^{3}}{16 \, \sqrt {b x^{2} + a} a^{4}} + \frac {5 \, B b}{8 \, \sqrt {b x^{2} + a} a^{2} x^{2}} - \frac {35 \, A b^{2}}{48 \, \sqrt {b x^{2} + a} a^{3} x^{2}} - \frac {B}{4 \, \sqrt {b x^{2} + a} a x^{4}} + \frac {7 \, A b}{24 \, \sqrt {b x^{2} + a} a^{2} x^{4}} - \frac {A}{6 \, \sqrt {b x^{2} + a} a x^{6}} \]
-15/8*B*b^2*arcsinh(a/(sqrt(a*b)*abs(x)))/a^(7/2) + 35/16*A*b^3*arcsinh(a/ (sqrt(a*b)*abs(x)))/a^(9/2) + 15/8*B*b^2/(sqrt(b*x^2 + a)*a^3) - 35/16*A*b ^3/(sqrt(b*x^2 + a)*a^4) + 5/8*B*b/(sqrt(b*x^2 + a)*a^2*x^2) - 35/48*A*b^2 /(sqrt(b*x^2 + a)*a^3*x^2) - 1/4*B/(sqrt(b*x^2 + a)*a*x^4) + 7/24*A*b/(sqr t(b*x^2 + a)*a^2*x^4) - 1/6*A/(sqrt(b*x^2 + a)*a*x^6)
Time = 0.28 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.18 \[ \int \frac {A+B x^2}{x^7 \left (a+b x^2\right )^{3/2}} \, dx=\frac {5 \, {\left (6 \, B a b^{2} - 7 \, A b^{3}\right )} \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{16 \, \sqrt {-a} a^{4}} + \frac {B a b^{2} - A b^{3}}{\sqrt {b x^{2} + a} a^{4}} + \frac {42 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} B a b^{2} - 96 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B a^{2} b^{2} + 54 \, \sqrt {b x^{2} + a} B a^{3} b^{2} - 57 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} A b^{3} + 136 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A a b^{3} - 87 \, \sqrt {b x^{2} + a} A a^{2} b^{3}}{48 \, a^{4} b^{3} x^{6}} \]
5/16*(6*B*a*b^2 - 7*A*b^3)*arctan(sqrt(b*x^2 + a)/sqrt(-a))/(sqrt(-a)*a^4) + (B*a*b^2 - A*b^3)/(sqrt(b*x^2 + a)*a^4) + 1/48*(42*(b*x^2 + a)^(5/2)*B* a*b^2 - 96*(b*x^2 + a)^(3/2)*B*a^2*b^2 + 54*sqrt(b*x^2 + a)*B*a^3*b^2 - 57 *(b*x^2 + a)^(5/2)*A*b^3 + 136*(b*x^2 + a)^(3/2)*A*a*b^3 - 87*sqrt(b*x^2 + a)*A*a^2*b^3)/(a^4*b^3*x^6)
Time = 6.57 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.16 \[ \int \frac {A+B x^2}{x^7 \left (a+b x^2\right )^{3/2}} \, dx=\frac {35\,A\,b^3\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{16\,a^{9/2}}-\frac {15\,B\,b^2\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{8\,a^{7/2}}-\frac {35\,A\,b^3}{16\,a^4\,\sqrt {b\,x^2+a}}+\frac {15\,B\,b^2}{8\,a^3\,\sqrt {b\,x^2+a}}-\frac {A}{6\,a\,x^6\,\sqrt {b\,x^2+a}}-\frac {B}{4\,a\,x^4\,\sqrt {b\,x^2+a}}+\frac {7\,A\,b}{24\,a^2\,x^4\,\sqrt {b\,x^2+a}}+\frac {5\,B\,b}{8\,a^2\,x^2\,\sqrt {b\,x^2+a}}-\frac {35\,A\,b^2}{48\,a^3\,x^2\,\sqrt {b\,x^2+a}} \]
(35*A*b^3*atanh((a + b*x^2)^(1/2)/a^(1/2)))/(16*a^(9/2)) - (15*B*b^2*atanh ((a + b*x^2)^(1/2)/a^(1/2)))/(8*a^(7/2)) - (35*A*b^3)/(16*a^4*(a + b*x^2)^ (1/2)) + (15*B*b^2)/(8*a^3*(a + b*x^2)^(1/2)) - A/(6*a*x^6*(a + b*x^2)^(1/ 2)) - B/(4*a*x^4*(a + b*x^2)^(1/2)) + (7*A*b)/(24*a^2*x^4*(a + b*x^2)^(1/2 )) + (5*B*b)/(8*a^2*x^2*(a + b*x^2)^(1/2)) - (35*A*b^2)/(48*a^3*x^2*(a + b *x^2)^(1/2))