Integrand size = 22, antiderivative size = 146 \[ \int \frac {A+B x^2}{x^6 \left (a+b x^2\right )^{5/2}} \, dx=-\frac {A}{5 a x^5 \left (a+b x^2\right )^{3/2}}+\frac {8 A b-5 a B}{15 a^2 x^3 \left (a+b x^2\right )^{3/2}}-\frac {2 b (8 A b-5 a B)}{5 a^3 x \left (a+b x^2\right )^{3/2}}-\frac {8 b^2 (8 A b-5 a B) x}{15 a^4 \left (a+b x^2\right )^{3/2}}-\frac {16 b^2 (8 A b-5 a B) x}{15 a^5 \sqrt {a+b x^2}} \]
-1/5*A/a/x^5/(b*x^2+a)^(3/2)+1/15*(8*A*b-5*B*a)/a^2/x^3/(b*x^2+a)^(3/2)-2/ 5*b*(8*A*b-5*B*a)/a^3/x/(b*x^2+a)^(3/2)-8/15*b^2*(8*A*b-5*B*a)*x/a^4/(b*x^ 2+a)^(3/2)-16/15*b^2*(8*A*b-5*B*a)*x/a^5/(b*x^2+a)^(1/2)
Time = 0.16 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.72 \[ \int \frac {A+B x^2}{x^6 \left (a+b x^2\right )^{5/2}} \, dx=\frac {-128 A b^4 x^8+16 a b^3 x^6 \left (-12 A+5 B x^2\right )+24 a^2 b^2 x^4 \left (-2 A+5 B x^2\right )-a^4 \left (3 A+5 B x^2\right )+a^3 \left (8 A b x^2+30 b B x^4\right )}{15 a^5 x^5 \left (a+b x^2\right )^{3/2}} \]
(-128*A*b^4*x^8 + 16*a*b^3*x^6*(-12*A + 5*B*x^2) + 24*a^2*b^2*x^4*(-2*A + 5*B*x^2) - a^4*(3*A + 5*B*x^2) + a^3*(8*A*b*x^2 + 30*b*B*x^4))/(15*a^5*x^5 *(a + b*x^2)^(3/2))
Time = 0.22 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.90, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {359, 245, 245, 209, 208}
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.
| \(\displaystyle \int \frac {A+B x^2}{x^6 \left (a+b x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 359 |
| \(\displaystyle -\frac {(8 A b-5 a B) \int \frac {1}{x^4 \left (b x^2+a\right )^{5/2}}dx}{5 a}-\frac {A}{5 a x^5 \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle -\frac {(8 A b-5 a B) \left (-\frac {2 b \int \frac {1}{x^2 \left (b x^2+a\right )^{5/2}}dx}{a}-\frac {1}{3 a x^3 \left (a+b x^2\right )^{3/2}}\right )}{5 a}-\frac {A}{5 a x^5 \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle -\frac {(8 A b-5 a B) \left (-\frac {2 b \left (-\frac {4 b \int \frac {1}{\left (b x^2+a\right )^{5/2}}dx}{a}-\frac {1}{a x \left (a+b x^2\right )^{3/2}}\right )}{a}-\frac {1}{3 a x^3 \left (a+b x^2\right )^{3/2}}\right )}{5 a}-\frac {A}{5 a x^5 \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 209 |
| \(\displaystyle -\frac {(8 A b-5 a B) \left (-\frac {2 b \left (-\frac {4 b \left (\frac {2 \int \frac {1}{\left (b x^2+a\right )^{3/2}}dx}{3 a}+\frac {x}{3 a \left (a+b x^2\right )^{3/2}}\right )}{a}-\frac {1}{a x \left (a+b x^2\right )^{3/2}}\right )}{a}-\frac {1}{3 a x^3 \left (a+b x^2\right )^{3/2}}\right )}{5 a}-\frac {A}{5 a x^5 \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 208 |
| \(\displaystyle -\frac {\left (-\frac {2 b \left (-\frac {4 b \left (\frac {2 x}{3 a^2 \sqrt {a+b x^2}}+\frac {x}{3 a \left (a+b x^2\right )^{3/2}}\right )}{a}-\frac {1}{a x \left (a+b x^2\right )^{3/2}}\right )}{a}-\frac {1}{3 a x^3 \left (a+b x^2\right )^{3/2}}\right ) (8 A b-5 a B)}{5 a}-\frac {A}{5 a x^5 \left (a+b x^2\right )^{3/2}}\) |
-1/5*A/(a*x^5*(a + b*x^2)^(3/2)) - ((8*A*b - 5*a*B)*(-1/3*1/(a*x^3*(a + b* x^2)^(3/2)) - (2*b*(-(1/(a*x*(a + b*x^2)^(3/2))) - (4*b*(x/(3*a*(a + b*x^2 )^(3/2)) + (2*x)/(3*a^2*Sqrt[a + b*x^2])))/a))/a))/(5*a)
3.6.97.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), x] /; FreeQ[{a, b}, x]
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && ILtQ[p + 3/2, 0]
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))), x] - Simp[b*((m + 2*(p + 1) + 1)/(a*(m + 1))) Int[x^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, m, p}, x] && ILtQ[Si mplify[(m + 1)/2 + p + 1], 0] && NeQ[m, -1]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(a*e^2*(m + 1)) Int[(e*x)^(m + 2)* (a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && !ILtQ[p, -1]
Time = 2.90 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.65
| method | result | size |
| pseudoelliptic | \(\frac {\left (-5 x^{2} B -3 A \right ) a^{4}+8 x^{2} b \left (\frac {15 x^{2} B}{4}+A \right ) a^{3}-48 x^{4} \left (-\frac {5 x^{2} B}{2}+A \right ) b^{2} a^{2}-192 x^{6} \left (-\frac {5 x^{2} B}{12}+A \right ) b^{3} a -128 A \,b^{4} x^{8}}{15 \left (b \,x^{2}+a \right )^{\frac {3}{2}} x^{5} a^{5}}\) | \(95\) |
| gosper | \(-\frac {128 A \,b^{4} x^{8}-80 B a \,b^{3} x^{8}+192 A a \,b^{3} x^{6}-120 B \,a^{2} b^{2} x^{6}+48 A \,a^{2} b^{2} x^{4}-30 B \,a^{3} b \,x^{4}-8 A \,a^{3} b \,x^{2}+5 B \,a^{4} x^{2}+3 A \,a^{4}}{15 x^{5} \left (b \,x^{2}+a \right )^{\frac {3}{2}} a^{5}}\) | \(107\) |
| trager | \(-\frac {128 A \,b^{4} x^{8}-80 B a \,b^{3} x^{8}+192 A a \,b^{3} x^{6}-120 B \,a^{2} b^{2} x^{6}+48 A \,a^{2} b^{2} x^{4}-30 B \,a^{3} b \,x^{4}-8 A \,a^{3} b \,x^{2}+5 B \,a^{4} x^{2}+3 A \,a^{4}}{15 x^{5} \left (b \,x^{2}+a \right )^{\frac {3}{2}} a^{5}}\) | \(107\) |
| risch | \(-\frac {\sqrt {b \,x^{2}+a}\, \left (73 A \,b^{2} x^{4}-40 B a b \,x^{4}-14 a A b \,x^{2}+5 a^{2} B \,x^{2}+3 a^{2} A \right )}{15 a^{5} x^{5}}-\frac {\sqrt {b \,x^{2}+a}\, x \left (11 A \,b^{2} x^{2}-8 B a b \,x^{2}+12 a b A -9 a^{2} B \right ) b^{2}}{3 a^{5} \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )}\) | \(127\) |
| default | \(B \left (-\frac {1}{3 a \,x^{3} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {2 b \left (-\frac {1}{a x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {4 b \left (\frac {x}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {b \,x^{2}+a}}\right )}{a}\right )}{a}\right )+A \left (-\frac {1}{5 a \,x^{5} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {8 b \left (-\frac {1}{3 a \,x^{3} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {2 b \left (-\frac {1}{a x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {4 b \left (\frac {x}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {b \,x^{2}+a}}\right )}{a}\right )}{a}\right )}{5 a}\right )\) | \(188\) |
1/15*((-5*B*x^2-3*A)*a^4+8*x^2*b*(15/4*x^2*B+A)*a^3-48*x^4*(-5/2*x^2*B+A)* b^2*a^2-192*x^6*(-5/12*x^2*B+A)*b^3*a-128*A*b^4*x^8)/(b*x^2+a)^(3/2)/x^5/a ^5
Time = 0.29 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.88 \[ \int \frac {A+B x^2}{x^6 \left (a+b x^2\right )^{5/2}} \, dx=\frac {{\left (16 \, {\left (5 \, B a b^{3} - 8 \, A b^{4}\right )} x^{8} + 24 \, {\left (5 \, B a^{2} b^{2} - 8 \, A a b^{3}\right )} x^{6} - 3 \, A a^{4} + 6 \, {\left (5 \, B a^{3} b - 8 \, A a^{2} b^{2}\right )} x^{4} - {\left (5 \, B a^{4} - 8 \, A a^{3} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{15 \, {\left (a^{5} b^{2} x^{9} + 2 \, a^{6} b x^{7} + a^{7} x^{5}\right )}} \]
1/15*(16*(5*B*a*b^3 - 8*A*b^4)*x^8 + 24*(5*B*a^2*b^2 - 8*A*a*b^3)*x^6 - 3* A*a^4 + 6*(5*B*a^3*b - 8*A*a^2*b^2)*x^4 - (5*B*a^4 - 8*A*a^3*b)*x^2)*sqrt( b*x^2 + a)/(a^5*b^2*x^9 + 2*a^6*b*x^7 + a^7*x^5)
Leaf count of result is larger than twice the leaf count of optimal. 944 vs. \(2 (141) = 282\).
Time = 15.02 (sec) , antiderivative size = 944, normalized size of antiderivative = 6.47 \[ \int \frac {A+B x^2}{x^6 \left (a+b x^2\right )^{5/2}} \, dx=A \left (- \frac {3 a^{6} b^{\frac {33}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{15 a^{9} b^{16} x^{4} + 60 a^{8} b^{17} x^{6} + 90 a^{7} b^{18} x^{8} + 60 a^{6} b^{19} x^{10} + 15 a^{5} b^{20} x^{12}} + \frac {2 a^{5} b^{\frac {35}{2}} x^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{15 a^{9} b^{16} x^{4} + 60 a^{8} b^{17} x^{6} + 90 a^{7} b^{18} x^{8} + 60 a^{6} b^{19} x^{10} + 15 a^{5} b^{20} x^{12}} - \frac {35 a^{4} b^{\frac {37}{2}} x^{4} \sqrt {\frac {a}{b x^{2}} + 1}}{15 a^{9} b^{16} x^{4} + 60 a^{8} b^{17} x^{6} + 90 a^{7} b^{18} x^{8} + 60 a^{6} b^{19} x^{10} + 15 a^{5} b^{20} x^{12}} - \frac {280 a^{3} b^{\frac {39}{2}} x^{6} \sqrt {\frac {a}{b x^{2}} + 1}}{15 a^{9} b^{16} x^{4} + 60 a^{8} b^{17} x^{6} + 90 a^{7} b^{18} x^{8} + 60 a^{6} b^{19} x^{10} + 15 a^{5} b^{20} x^{12}} - \frac {560 a^{2} b^{\frac {41}{2}} x^{8} \sqrt {\frac {a}{b x^{2}} + 1}}{15 a^{9} b^{16} x^{4} + 60 a^{8} b^{17} x^{6} + 90 a^{7} b^{18} x^{8} + 60 a^{6} b^{19} x^{10} + 15 a^{5} b^{20} x^{12}} - \frac {448 a b^{\frac {43}{2}} x^{10} \sqrt {\frac {a}{b x^{2}} + 1}}{15 a^{9} b^{16} x^{4} + 60 a^{8} b^{17} x^{6} + 90 a^{7} b^{18} x^{8} + 60 a^{6} b^{19} x^{10} + 15 a^{5} b^{20} x^{12}} - \frac {128 b^{\frac {45}{2}} x^{12} \sqrt {\frac {a}{b x^{2}} + 1}}{15 a^{9} b^{16} x^{4} + 60 a^{8} b^{17} x^{6} + 90 a^{7} b^{18} x^{8} + 60 a^{6} b^{19} x^{10} + 15 a^{5} b^{20} x^{12}}\right ) + B \left (- \frac {a^{4} b^{\frac {19}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{7} b^{9} x^{2} + 9 a^{6} b^{10} x^{4} + 9 a^{5} b^{11} x^{6} + 3 a^{4} b^{12} x^{8}} + \frac {5 a^{3} b^{\frac {21}{2}} x^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{7} b^{9} x^{2} + 9 a^{6} b^{10} x^{4} + 9 a^{5} b^{11} x^{6} + 3 a^{4} b^{12} x^{8}} + \frac {30 a^{2} b^{\frac {23}{2}} x^{4} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{7} b^{9} x^{2} + 9 a^{6} b^{10} x^{4} + 9 a^{5} b^{11} x^{6} + 3 a^{4} b^{12} x^{8}} + \frac {40 a b^{\frac {25}{2}} x^{6} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{7} b^{9} x^{2} + 9 a^{6} b^{10} x^{4} + 9 a^{5} b^{11} x^{6} + 3 a^{4} b^{12} x^{8}} + \frac {16 b^{\frac {27}{2}} x^{8} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{7} b^{9} x^{2} + 9 a^{6} b^{10} x^{4} + 9 a^{5} b^{11} x^{6} + 3 a^{4} b^{12} x^{8}}\right ) \]
A*(-3*a**6*b**(33/2)*sqrt(a/(b*x**2) + 1)/(15*a**9*b**16*x**4 + 60*a**8*b* *17*x**6 + 90*a**7*b**18*x**8 + 60*a**6*b**19*x**10 + 15*a**5*b**20*x**12) + 2*a**5*b**(35/2)*x**2*sqrt(a/(b*x**2) + 1)/(15*a**9*b**16*x**4 + 60*a** 8*b**17*x**6 + 90*a**7*b**18*x**8 + 60*a**6*b**19*x**10 + 15*a**5*b**20*x* *12) - 35*a**4*b**(37/2)*x**4*sqrt(a/(b*x**2) + 1)/(15*a**9*b**16*x**4 + 6 0*a**8*b**17*x**6 + 90*a**7*b**18*x**8 + 60*a**6*b**19*x**10 + 15*a**5*b** 20*x**12) - 280*a**3*b**(39/2)*x**6*sqrt(a/(b*x**2) + 1)/(15*a**9*b**16*x* *4 + 60*a**8*b**17*x**6 + 90*a**7*b**18*x**8 + 60*a**6*b**19*x**10 + 15*a* *5*b**20*x**12) - 560*a**2*b**(41/2)*x**8*sqrt(a/(b*x**2) + 1)/(15*a**9*b* *16*x**4 + 60*a**8*b**17*x**6 + 90*a**7*b**18*x**8 + 60*a**6*b**19*x**10 + 15*a**5*b**20*x**12) - 448*a*b**(43/2)*x**10*sqrt(a/(b*x**2) + 1)/(15*a** 9*b**16*x**4 + 60*a**8*b**17*x**6 + 90*a**7*b**18*x**8 + 60*a**6*b**19*x** 10 + 15*a**5*b**20*x**12) - 128*b**(45/2)*x**12*sqrt(a/(b*x**2) + 1)/(15*a **9*b**16*x**4 + 60*a**8*b**17*x**6 + 90*a**7*b**18*x**8 + 60*a**6*b**19*x **10 + 15*a**5*b**20*x**12)) + B*(-a**4*b**(19/2)*sqrt(a/(b*x**2) + 1)/(3* a**7*b**9*x**2 + 9*a**6*b**10*x**4 + 9*a**5*b**11*x**6 + 3*a**4*b**12*x**8 ) + 5*a**3*b**(21/2)*x**2*sqrt(a/(b*x**2) + 1)/(3*a**7*b**9*x**2 + 9*a**6* b**10*x**4 + 9*a**5*b**11*x**6 + 3*a**4*b**12*x**8) + 30*a**2*b**(23/2)*x* *4*sqrt(a/(b*x**2) + 1)/(3*a**7*b**9*x**2 + 9*a**6*b**10*x**4 + 9*a**5*b** 11*x**6 + 3*a**4*b**12*x**8) + 40*a*b**(25/2)*x**6*sqrt(a/(b*x**2) + 1)...
Time = 0.20 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.18 \[ \int \frac {A+B x^2}{x^6 \left (a+b x^2\right )^{5/2}} \, dx=\frac {16 \, B b^{2} x}{3 \, \sqrt {b x^{2} + a} a^{4}} + \frac {8 \, B b^{2} x}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{3}} - \frac {128 \, A b^{3} x}{15 \, \sqrt {b x^{2} + a} a^{5}} - \frac {64 \, A b^{3} x}{15 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{4}} + \frac {2 \, B b}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2} x} - \frac {16 \, A b^{2}}{5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{3} x} - \frac {B}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a x^{3}} + \frac {8 \, A b}{15 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2} x^{3}} - \frac {A}{5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a x^{5}} \]
16/3*B*b^2*x/(sqrt(b*x^2 + a)*a^4) + 8/3*B*b^2*x/((b*x^2 + a)^(3/2)*a^3) - 128/15*A*b^3*x/(sqrt(b*x^2 + a)*a^5) - 64/15*A*b^3*x/((b*x^2 + a)^(3/2)*a ^4) + 2*B*b/((b*x^2 + a)^(3/2)*a^2*x) - 16/5*A*b^2/((b*x^2 + a)^(3/2)*a^3* x) - 1/3*B/((b*x^2 + a)^(3/2)*a*x^3) + 8/15*A*b/((b*x^2 + a)^(3/2)*a^2*x^3 ) - 1/5*A/((b*x^2 + a)^(3/2)*a*x^5)
Leaf count of result is larger than twice the leaf count of optimal. 336 vs. \(2 (126) = 252\).
Time = 0.32 (sec) , antiderivative size = 336, normalized size of antiderivative = 2.30 \[ \int \frac {A+B x^2}{x^6 \left (a+b x^2\right )^{5/2}} \, dx=\frac {x {\left (\frac {{\left (8 \, B a^{5} b^{4} - 11 \, A a^{4} b^{5}\right )} x^{2}}{a^{9} b} + \frac {3 \, {\left (3 \, B a^{6} b^{3} - 4 \, A a^{5} b^{4}\right )}}{a^{9} b}\right )}}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}}} - \frac {2 \, {\left (30 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} B a b^{\frac {3}{2}} - 45 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} A b^{\frac {5}{2}} - 150 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} B a^{2} b^{\frac {3}{2}} + 240 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} A a b^{\frac {5}{2}} + 250 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} B a^{3} b^{\frac {3}{2}} - 490 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} A a^{2} b^{\frac {5}{2}} - 170 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a^{4} b^{\frac {3}{2}} + 320 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} A a^{3} b^{\frac {5}{2}} + 40 \, B a^{5} b^{\frac {3}{2}} - 73 \, A a^{4} b^{\frac {5}{2}}\right )}}{15 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{5} a^{4}} \]
1/3*x*((8*B*a^5*b^4 - 11*A*a^4*b^5)*x^2/(a^9*b) + 3*(3*B*a^6*b^3 - 4*A*a^5 *b^4)/(a^9*b))/(b*x^2 + a)^(3/2) - 2/15*(30*(sqrt(b)*x - sqrt(b*x^2 + a))^ 8*B*a*b^(3/2) - 45*(sqrt(b)*x - sqrt(b*x^2 + a))^8*A*b^(5/2) - 150*(sqrt(b )*x - sqrt(b*x^2 + a))^6*B*a^2*b^(3/2) + 240*(sqrt(b)*x - sqrt(b*x^2 + a)) ^6*A*a*b^(5/2) + 250*(sqrt(b)*x - sqrt(b*x^2 + a))^4*B*a^3*b^(3/2) - 490*( sqrt(b)*x - sqrt(b*x^2 + a))^4*A*a^2*b^(5/2) - 170*(sqrt(b)*x - sqrt(b*x^2 + a))^2*B*a^4*b^(3/2) + 320*(sqrt(b)*x - sqrt(b*x^2 + a))^2*A*a^3*b^(5/2) + 40*B*a^5*b^(3/2) - 73*A*a^4*b^(5/2))/(((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^5*a^4)
Time = 5.48 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.58 \[ \int \frac {A+B x^2}{x^6 \left (a+b x^2\right )^{5/2}} \, dx=\frac {\frac {a\,\left (\frac {b^2\,\left (73\,A\,b-40\,B\,a\right )}{18\,a^4}+\frac {b^2\,\left (86\,A\,b-35\,B\,a\right )}{30\,a^4}+\frac {a\,\left (\frac {28\,A\,b^4-10\,B\,a\,b^3}{45\,a^5}-\frac {b^3\,\left (86\,A\,b-35\,B\,a\right )}{18\,a^5}\right )}{b}\right )}{b}-\frac {b\,\left (73\,A\,b-40\,B\,a\right )}{30\,a^3}}{x\,{\left (b\,x^2+a\right )}^{3/2}}+\frac {x^2\,\left (\frac {28\,A\,b^3-10\,B\,a\,b^2}{15\,a^5}-\frac {2\,b^2\,\left (26\,A\,b-15\,B\,a\right )}{5\,a^5}\right )-\frac {b\,\left (26\,A\,b-15\,B\,a\right )}{5\,a^4}}{x\,\sqrt {b\,x^2+a}}-\frac {\sqrt {b\,x^2+a}\,\left (5\,B\,a^3-14\,A\,a^2\,b\right )}{15\,a^6\,x^3}-\frac {A\,\sqrt {b\,x^2+a}}{5\,a^3\,x^5} \]
((a*((b^2*(73*A*b - 40*B*a))/(18*a^4) + (b^2*(86*A*b - 35*B*a))/(30*a^4) + (a*((28*A*b^4 - 10*B*a*b^3)/(45*a^5) - (b^3*(86*A*b - 35*B*a))/(18*a^5))) /b))/b - (b*(73*A*b - 40*B*a))/(30*a^3))/(x*(a + b*x^2)^(3/2)) + (x^2*((28 *A*b^3 - 10*B*a*b^2)/(15*a^5) - (2*b^2*(26*A*b - 15*B*a))/(5*a^5)) - (b*(2 6*A*b - 15*B*a))/(5*a^4))/(x*(a + b*x^2)^(1/2)) - ((a + b*x^2)^(1/2)*(5*B* a^3 - 14*A*a^2*b))/(15*a^6*x^3) - (A*(a + b*x^2)^(1/2))/(5*a^3*x^5)