Integrand size = 22, antiderivative size = 149 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^7} \, dx=\frac {5 b^2 (A b+6 a B) \sqrt {a+b x^2}}{16 a}-\frac {5 b (A b+6 a B) \left (a+b x^2\right )^{3/2}}{48 a x^2}-\frac {(A b+6 a B) \left (a+b x^2\right )^{5/2}}{24 a x^4}-\frac {A \left (a+b x^2\right )^{7/2}}{6 a x^6}-\frac {5 b^2 (A b+6 a B) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{16 \sqrt {a}} \]
-5/48*b*(A*b+6*B*a)*(b*x^2+a)^(3/2)/a/x^2-1/24*(A*b+6*B*a)*(b*x^2+a)^(5/2) /a/x^4-1/6*A*(b*x^2+a)^(7/2)/a/x^6-5/16*b^2*(A*b+6*B*a)*arctanh((b*x^2+a)^ (1/2)/a^(1/2))/a^(1/2)+5/16*b^2*(A*b+6*B*a)*(b*x^2+a)^(1/2)/a
Time = 0.20 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.72 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^7} \, dx=\frac {\sqrt {a+b x^2} \left (-8 a^2 A-26 a A b x^2-12 a^2 B x^2-33 A b^2 x^4-54 a b B x^4+48 b^2 B x^6\right )}{48 x^6}-\frac {5 b^2 (A b+6 a B) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{16 \sqrt {a}} \]
(Sqrt[a + b*x^2]*(-8*a^2*A - 26*a*A*b*x^2 - 12*a^2*B*x^2 - 33*A*b^2*x^4 - 54*a*b*B*x^4 + 48*b^2*B*x^6))/(48*x^6) - (5*b^2*(A*b + 6*a*B)*ArcTanh[Sqrt [a + b*x^2]/Sqrt[a]])/(16*Sqrt[a])
Time = 0.22 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.85, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {354, 87, 51, 51, 60, 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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^7} \, dx\) |
| \(\Big \downarrow \) 354 |
| \(\displaystyle \frac {1}{2} \int \frac {\left (b x^2+a\right )^{5/2} \left (B x^2+A\right )}{x^8}dx^2\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {1}{2} \left (\frac {(6 a B+A b) \int \frac {\left (b x^2+a\right )^{5/2}}{x^6}dx^2}{6 a}-\frac {A \left (a+b x^2\right )^{7/2}}{3 a x^6}\right )\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {1}{2} \left (\frac {(6 a B+A b) \left (\frac {5}{4} b \int \frac {\left (b x^2+a\right )^{3/2}}{x^4}dx^2-\frac {\left (a+b x^2\right )^{5/2}}{2 x^4}\right )}{6 a}-\frac {A \left (a+b x^2\right )^{7/2}}{3 a x^6}\right )\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {1}{2} \left (\frac {(6 a B+A b) \left (\frac {5}{4} b \left (\frac {3}{2} b \int \frac {\sqrt {b x^2+a}}{x^2}dx^2-\frac {\left (a+b x^2\right )^{3/2}}{x^2}\right )-\frac {\left (a+b x^2\right )^{5/2}}{2 x^4}\right )}{6 a}-\frac {A \left (a+b x^2\right )^{7/2}}{3 a x^6}\right )\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{2} \left (\frac {(6 a B+A b) \left (\frac {5}{4} b \left (\frac {3}{2} b \left (a \int \frac {1}{x^2 \sqrt {b x^2+a}}dx^2+2 \sqrt {a+b x^2}\right )-\frac {\left (a+b x^2\right )^{3/2}}{x^2}\right )-\frac {\left (a+b x^2\right )^{5/2}}{2 x^4}\right )}{6 a}-\frac {A \left (a+b x^2\right )^{7/2}}{3 a x^6}\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{2} \left (\frac {(6 a B+A b) \left (\frac {5}{4} b \left (\frac {3}{2} b \left (\frac {2 a \int \frac {1}{\frac {x^4}{b}-\frac {a}{b}}d\sqrt {b x^2+a}}{b}+2 \sqrt {a+b x^2}\right )-\frac {\left (a+b x^2\right )^{3/2}}{x^2}\right )-\frac {\left (a+b x^2\right )^{5/2}}{2 x^4}\right )}{6 a}-\frac {A \left (a+b x^2\right )^{7/2}}{3 a x^6}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {1}{2} \left (\frac {(6 a B+A b) \left (\frac {5}{4} b \left (\frac {3}{2} b \left (2 \sqrt {a+b x^2}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )\right )-\frac {\left (a+b x^2\right )^{3/2}}{x^2}\right )-\frac {\left (a+b x^2\right )^{5/2}}{2 x^4}\right )}{6 a}-\frac {A \left (a+b x^2\right )^{7/2}}{3 a x^6}\right )\) |
(-1/3*(A*(a + b*x^2)^(7/2))/(a*x^6) + ((A*b + 6*a*B)*(-1/2*(a + b*x^2)^(5/ 2)/x^4 + (5*b*(-((a + b*x^2)^(3/2)/x^2) + (3*b*(2*Sqrt[a + b*x^2] - 2*Sqrt [a]*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]]))/2))/4))/(6*a))/2
3.6.50.3.1 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/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x ] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[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 = 2.85 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.67
| method | result | size |
| pseudoelliptic | \(-\frac {11 \left (\frac {5 b^{2} x^{6} \left (A b +6 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{\sqrt {a}}\right )}{11}+\left (\frac {26 x^{2} \left (\frac {27 x^{2} B}{13}+A \right ) b \,a^{\frac {3}{2}}}{33}+\frac {4 \left (x^{2} B +\frac {2 A}{3}\right ) a^{\frac {5}{2}}}{11}+b^{2} x^{4} \sqrt {a}\, \left (-\frac {16 x^{2} B}{11}+A \right )\right ) \sqrt {b \,x^{2}+a}\right )}{16 \sqrt {a}\, x^{6}}\) | \(100\) |
| risch | \(-\frac {\sqrt {b \,x^{2}+a}\, \left (33 A \,b^{2} x^{4}+54 B a b \,x^{4}+26 a A b \,x^{2}+12 a^{2} B \,x^{2}+8 a^{2} A \right )}{48 x^{6}}+\frac {b^{2} \left (16 \sqrt {b \,x^{2}+a}\, B -\frac {\left (5 A b +30 B a \right ) \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{\sqrt {a}}\right )}{16}\) | \(112\) |
| default | \(A \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{6 a \,x^{6}}+\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{4 a \,x^{4}}+\frac {3 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{2 a \,x^{2}}+\frac {5 b \left (\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{5}+a \left (\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\right )\right )}{2 a}\right )}{4 a}\right )}{6 a}\right )+B \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{4 a \,x^{4}}+\frac {3 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{2 a \,x^{2}}+\frac {5 b \left (\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{5}+a \left (\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\right )\right )}{2 a}\right )}{4 a}\right )\) | \(258\) |
-11/16/a^(1/2)*(5/11*b^2*x^6*(A*b+6*B*a)*arctanh((b*x^2+a)^(1/2)/a^(1/2))+ (26/33*x^2*(27/13*x^2*B+A)*b*a^(3/2)+4/11*(x^2*B+2/3*A)*a^(5/2)+b^2*x^4*a^ (1/2)*(-16/11*x^2*B+A))*(b*x^2+a)^(1/2))/x^6
Time = 0.28 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.62 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^7} \, dx=\left [\frac {15 \, {\left (6 \, B a b^{2} + A b^{3}\right )} \sqrt {a} x^{6} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (48 \, B a b^{2} x^{6} - 3 \, {\left (18 \, B a^{2} b + 11 \, A a b^{2}\right )} x^{4} - 8 \, A a^{3} - 2 \, {\left (6 \, B a^{3} + 13 \, A a^{2} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{96 \, a x^{6}}, \frac {15 \, {\left (6 \, B a b^{2} + A b^{3}\right )} \sqrt {-a} x^{6} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) + {\left (48 \, B a b^{2} x^{6} - 3 \, {\left (18 \, B a^{2} b + 11 \, A a b^{2}\right )} x^{4} - 8 \, A a^{3} - 2 \, {\left (6 \, B a^{3} + 13 \, A a^{2} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{48 \, a x^{6}}\right ] \]
[1/96*(15*(6*B*a*b^2 + A*b^3)*sqrt(a)*x^6*log(-(b*x^2 - 2*sqrt(b*x^2 + a)* sqrt(a) + 2*a)/x^2) + 2*(48*B*a*b^2*x^6 - 3*(18*B*a^2*b + 11*A*a*b^2)*x^4 - 8*A*a^3 - 2*(6*B*a^3 + 13*A*a^2*b)*x^2)*sqrt(b*x^2 + a))/(a*x^6), 1/48*( 15*(6*B*a*b^2 + A*b^3)*sqrt(-a)*x^6*arctan(sqrt(-a)/sqrt(b*x^2 + a)) + (48 *B*a*b^2*x^6 - 3*(18*B*a^2*b + 11*A*a*b^2)*x^4 - 8*A*a^3 - 2*(6*B*a^3 + 13 *A*a^2*b)*x^2)*sqrt(b*x^2 + a))/(a*x^6)]
Leaf count of result is larger than twice the leaf count of optimal. 306 vs. \(2 (136) = 272\).
Time = 68.52 (sec) , antiderivative size = 306, normalized size of antiderivative = 2.05 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^7} \, dx=- \frac {A a^{3}}{6 \sqrt {b} x^{7} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {17 A a^{2} \sqrt {b}}{24 x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {35 A a b^{\frac {3}{2}}}{48 x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {A b^{\frac {5}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{2 x} - \frac {3 A b^{\frac {5}{2}}}{16 x \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {5 A b^{3} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{16 \sqrt {a}} - \frac {15 B \sqrt {a} b^{2} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{8} - \frac {B a^{3}}{4 \sqrt {b} x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {3 B a^{2} \sqrt {b}}{8 x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {B a b^{\frac {3}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{x} + \frac {7 B a b^{\frac {3}{2}}}{8 x \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {B b^{\frac {5}{2}} x}{\sqrt {\frac {a}{b x^{2}} + 1}} \]
-A*a**3/(6*sqrt(b)*x**7*sqrt(a/(b*x**2) + 1)) - 17*A*a**2*sqrt(b)/(24*x**5 *sqrt(a/(b*x**2) + 1)) - 35*A*a*b**(3/2)/(48*x**3*sqrt(a/(b*x**2) + 1)) - A*b**(5/2)*sqrt(a/(b*x**2) + 1)/(2*x) - 3*A*b**(5/2)/(16*x*sqrt(a/(b*x**2) + 1)) - 5*A*b**3*asinh(sqrt(a)/(sqrt(b)*x))/(16*sqrt(a)) - 15*B*sqrt(a)*b **2*asinh(sqrt(a)/(sqrt(b)*x))/8 - B*a**3/(4*sqrt(b)*x**5*sqrt(a/(b*x**2) + 1)) - 3*B*a**2*sqrt(b)/(8*x**3*sqrt(a/(b*x**2) + 1)) - B*a*b**(3/2)*sqrt (a/(b*x**2) + 1)/x + 7*B*a*b**(3/2)/(8*x*sqrt(a/(b*x**2) + 1)) + B*b**(5/2 )*x/sqrt(a/(b*x**2) + 1)
Time = 0.21 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.63 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^7} \, dx=-\frac {15}{8} \, B \sqrt {a} b^{2} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right ) - \frac {5 \, A b^{3} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{16 \, \sqrt {a}} + \frac {15}{8} \, \sqrt {b x^{2} + a} B b^{2} + \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} B b^{2}}{8 \, a^{2}} + \frac {5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B b^{2}}{8 \, a} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A b^{3}}{16 \, a^{3}} + \frac {5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A b^{3}}{48 \, a^{2}} + \frac {5 \, \sqrt {b x^{2} + a} A b^{3}}{16 \, a} - \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} B b}{8 \, a^{2} x^{2}} - \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} A b^{2}}{16 \, a^{3} x^{2}} - \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} B}{4 \, a x^{4}} - \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} A b}{24 \, a^{2} x^{4}} - \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} A}{6 \, a x^{6}} \]
-15/8*B*sqrt(a)*b^2*arcsinh(a/(sqrt(a*b)*abs(x))) - 5/16*A*b^3*arcsinh(a/( sqrt(a*b)*abs(x)))/sqrt(a) + 15/8*sqrt(b*x^2 + a)*B*b^2 + 3/8*(b*x^2 + a)^ (5/2)*B*b^2/a^2 + 5/8*(b*x^2 + a)^(3/2)*B*b^2/a + 1/16*(b*x^2 + a)^(5/2)*A *b^3/a^3 + 5/48*(b*x^2 + a)^(3/2)*A*b^3/a^2 + 5/16*sqrt(b*x^2 + a)*A*b^3/a - 3/8*(b*x^2 + a)^(7/2)*B*b/(a^2*x^2) - 1/16*(b*x^2 + a)^(7/2)*A*b^2/(a^3 *x^2) - 1/4*(b*x^2 + a)^(7/2)*B/(a*x^4) - 1/24*(b*x^2 + a)^(7/2)*A*b/(a^2* x^4) - 1/6*(b*x^2 + a)^(7/2)*A/(a*x^6)
Time = 0.31 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.12 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^7} \, dx=\frac {48 \, \sqrt {b x^{2} + a} B b^{3} + \frac {15 \, {\left (6 \, B a b^{3} + A b^{4}\right )} \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} - \frac {54 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} B a b^{3} - 96 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B a^{2} b^{3} + 42 \, \sqrt {b x^{2} + a} B a^{3} b^{3} + 33 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} A b^{4} - 40 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A a b^{4} + 15 \, \sqrt {b x^{2} + a} A a^{2} b^{4}}{b^{3} x^{6}}}{48 \, b} \]
1/48*(48*sqrt(b*x^2 + a)*B*b^3 + 15*(6*B*a*b^3 + A*b^4)*arctan(sqrt(b*x^2 + a)/sqrt(-a))/sqrt(-a) - (54*(b*x^2 + a)^(5/2)*B*a*b^3 - 96*(b*x^2 + a)^( 3/2)*B*a^2*b^3 + 42*sqrt(b*x^2 + a)*B*a^3*b^3 + 33*(b*x^2 + a)^(5/2)*A*b^4 - 40*(b*x^2 + a)^(3/2)*A*a*b^4 + 15*sqrt(b*x^2 + a)*A*a^2*b^4)/(b^3*x^6)) /b
Time = 7.38 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.01 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^7} \, dx=B\,b^2\,\sqrt {b\,x^2+a}-\frac {11\,A\,{\left (b\,x^2+a\right )}^{5/2}}{16\,x^6}+\frac {5\,A\,a\,{\left (b\,x^2+a\right )}^{3/2}}{6\,x^6}-\frac {9\,B\,a\,{\left (b\,x^2+a\right )}^{3/2}}{8\,x^4}-\frac {5\,A\,a^2\,\sqrt {b\,x^2+a}}{16\,x^6}+\frac {7\,B\,a^2\,\sqrt {b\,x^2+a}}{8\,x^4}+\frac {A\,b^3\,\mathrm {atan}\left (\frac {\sqrt {b\,x^2+a}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,5{}\mathrm {i}}{16\,\sqrt {a}}+\frac {B\,\sqrt {a}\,b^2\,\mathrm {atan}\left (\frac {\sqrt {b\,x^2+a}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,15{}\mathrm {i}}{8} \]
B*b^2*(a + b*x^2)^(1/2) - (11*A*(a + b*x^2)^(5/2))/(16*x^6) + (A*b^3*atan( ((a + b*x^2)^(1/2)*1i)/a^(1/2))*5i)/(16*a^(1/2)) + (B*a^(1/2)*b^2*atan(((a + b*x^2)^(1/2)*1i)/a^(1/2))*15i)/8 + (5*A*a*(a + b*x^2)^(3/2))/(6*x^6) - (9*B*a*(a + b*x^2)^(3/2))/(8*x^4) - (5*A*a^2*(a + b*x^2)^(1/2))/(16*x^6) + (7*B*a^2*(a + b*x^2)^(1/2))/(8*x^4)