Integrand size = 22, antiderivative size = 148 \[ \int \frac {A+B x^2}{x^8 \left (a+b x^2\right )^{3/2}} \, dx=-\frac {A}{7 a x^7 \sqrt {a+b x^2}}+\frac {8 A b-7 a B}{35 a^2 x^5 \sqrt {a+b x^2}}-\frac {2 b (8 A b-7 a B)}{35 a^3 x^3 \sqrt {a+b x^2}}+\frac {8 b^2 (8 A b-7 a B)}{35 a^4 x \sqrt {a+b x^2}}+\frac {16 b^3 (8 A b-7 a B) x}{35 a^5 \sqrt {a+b x^2}} \]
-1/7*A/a/x^7/(b*x^2+a)^(1/2)+1/35*(8*A*b-7*B*a)/a^2/x^5/(b*x^2+a)^(1/2)-2/ 35*b*(8*A*b-7*B*a)/a^3/x^3/(b*x^2+a)^(1/2)+8/35*b^2*(8*A*b-7*B*a)/a^4/x/(b *x^2+a)^(1/2)+16/35*b^3*(8*A*b-7*B*a)*x/a^5/(b*x^2+a)^(1/2)
Time = 0.15 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.71 \[ \int \frac {A+B x^2}{x^8 \left (a+b x^2\right )^{3/2}} \, dx=\frac {128 A b^4 x^8+16 a b^3 x^6 \left (4 A-7 B x^2\right )-8 a^2 b^2 x^4 \left (2 A+7 B x^2\right )+2 a^3 b x^2 \left (4 A+7 B x^2\right )-a^4 \left (5 A+7 B x^2\right )}{35 a^5 x^7 \sqrt {a+b x^2}} \]
(128*A*b^4*x^8 + 16*a*b^3*x^6*(4*A - 7*B*x^2) - 8*a^2*b^2*x^4*(2*A + 7*B*x ^2) + 2*a^3*b*x^2*(4*A + 7*B*x^2) - a^4*(5*A + 7*B*x^2))/(35*a^5*x^7*Sqrt[ a + b*x^2])
Time = 0.23 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.93, 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, 245, 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.
| \(\displaystyle \int \frac {A+B x^2}{x^8 \left (a+b x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 359 |
| \(\displaystyle -\frac {(8 A b-7 a B) \int \frac {1}{x^6 \left (b x^2+a\right )^{3/2}}dx}{7 a}-\frac {A}{7 a x^7 \sqrt {a+b x^2}}\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle -\frac {(8 A b-7 a B) \left (-\frac {6 b \int \frac {1}{x^4 \left (b x^2+a\right )^{3/2}}dx}{5 a}-\frac {1}{5 a x^5 \sqrt {a+b x^2}}\right )}{7 a}-\frac {A}{7 a x^7 \sqrt {a+b x^2}}\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle -\frac {(8 A b-7 a B) \left (-\frac {6 b \left (-\frac {4 b \int \frac {1}{x^2 \left (b x^2+a\right )^{3/2}}dx}{3 a}-\frac {1}{3 a x^3 \sqrt {a+b x^2}}\right )}{5 a}-\frac {1}{5 a x^5 \sqrt {a+b x^2}}\right )}{7 a}-\frac {A}{7 a x^7 \sqrt {a+b x^2}}\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle -\frac {(8 A b-7 a B) \left (-\frac {6 b \left (-\frac {4 b \left (-\frac {2 b \int \frac {1}{\left (b x^2+a\right )^{3/2}}dx}{a}-\frac {1}{a x \sqrt {a+b x^2}}\right )}{3 a}-\frac {1}{3 a x^3 \sqrt {a+b x^2}}\right )}{5 a}-\frac {1}{5 a x^5 \sqrt {a+b x^2}}\right )}{7 a}-\frac {A}{7 a x^7 \sqrt {a+b x^2}}\) |
| \(\Big \downarrow \) 208 |
| \(\displaystyle -\frac {\left (-\frac {6 b \left (-\frac {4 b \left (-\frac {2 b x}{a^2 \sqrt {a+b x^2}}-\frac {1}{a x \sqrt {a+b x^2}}\right )}{3 a}-\frac {1}{3 a x^3 \sqrt {a+b x^2}}\right )}{5 a}-\frac {1}{5 a x^5 \sqrt {a+b x^2}}\right ) (8 A b-7 a B)}{7 a}-\frac {A}{7 a x^7 \sqrt {a+b x^2}}\) |
-1/7*A/(a*x^7*Sqrt[a + b*x^2]) - ((8*A*b - 7*a*B)*(-1/5*1/(a*x^5*Sqrt[a + b*x^2]) - (6*b*(-1/3*1/(a*x^3*Sqrt[a + b*x^2]) - (4*b*(-(1/(a*x*Sqrt[a + b *x^2])) - (2*b*x)/(a^2*Sqrt[a + b*x^2])))/(3*a)))/(5*a)))/(7*a)
3.6.83.3.1 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[(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 = 2.91 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.63
| method | result | size |
| pseudoelliptic | \(-\frac {\left (\frac {7 x^{2} B}{5}+A \right ) a^{4}-\frac {8 x^{2} b \left (\frac {7 x^{2} B}{4}+A \right ) a^{3}}{5}+\frac {16 x^{4} b^{2} \left (\frac {7 x^{2} B}{2}+A \right ) a^{2}}{5}-\frac {64 x^{6} \left (-\frac {7 x^{2} B}{4}+A \right ) b^{3} a}{5}-\frac {128 A \,b^{4} x^{8}}{5}}{7 \sqrt {b \,x^{2}+a}\, x^{7} a^{5}}\) | \(93\) |
| gosper | \(-\frac {-128 A \,b^{4} x^{8}+112 B a \,b^{3} x^{8}-64 A a \,b^{3} x^{6}+56 B \,a^{2} b^{2} x^{6}+16 A \,a^{2} b^{2} x^{4}-14 B \,a^{3} b \,x^{4}-8 A \,a^{3} b \,x^{2}+7 B \,a^{4} x^{2}+5 A \,a^{4}}{35 x^{7} \sqrt {b \,x^{2}+a}\, a^{5}}\) | \(107\) |
| trager | \(-\frac {-128 A \,b^{4} x^{8}+112 B a \,b^{3} x^{8}-64 A a \,b^{3} x^{6}+56 B \,a^{2} b^{2} x^{6}+16 A \,a^{2} b^{2} x^{4}-14 B \,a^{3} b \,x^{4}-8 A \,a^{3} b \,x^{2}+7 B \,a^{4} x^{2}+5 A \,a^{4}}{35 x^{7} \sqrt {b \,x^{2}+a}\, a^{5}}\) | \(107\) |
| risch | \(-\frac {\sqrt {b \,x^{2}+a}\, \left (-93 x^{6} b^{3} A +77 x^{6} a \,b^{2} B +29 A a \,b^{2} x^{4}-21 B \,a^{2} b \,x^{4}-13 A \,a^{2} b \,x^{2}+7 B \,a^{3} x^{2}+5 a^{3} A \right )}{35 a^{5} x^{7}}+\frac {x \,b^{3} \left (A b -B a \right )}{\sqrt {b \,x^{2}+a}\, a^{5}}\) | \(109\) |
| default | \(A \left (-\frac {1}{7 a \,x^{7} \sqrt {b \,x^{2}+a}}-\frac {8 b \left (-\frac {1}{5 a \,x^{5} \sqrt {b \,x^{2}+a}}-\frac {6 b \left (-\frac {1}{3 a \,x^{3} \sqrt {b \,x^{2}+a}}-\frac {4 b \left (-\frac {1}{a x \sqrt {b \,x^{2}+a}}-\frac {2 b x}{a^{2} \sqrt {b \,x^{2}+a}}\right )}{3 a}\right )}{5 a}\right )}{7 a}\right )+B \left (-\frac {1}{5 a \,x^{5} \sqrt {b \,x^{2}+a}}-\frac {6 b \left (-\frac {1}{3 a \,x^{3} \sqrt {b \,x^{2}+a}}-\frac {4 b \left (-\frac {1}{a x \sqrt {b \,x^{2}+a}}-\frac {2 b x}{a^{2} \sqrt {b \,x^{2}+a}}\right )}{3 a}\right )}{5 a}\right )\) | \(194\) |
-1/7/(b*x^2+a)^(1/2)*((7/5*x^2*B+A)*a^4-8/5*x^2*b*(7/4*x^2*B+A)*a^3+16/5*x ^4*b^2*(7/2*x^2*B+A)*a^2-64/5*x^6*(-7/4*x^2*B+A)*b^3*a-128/5*A*b^4*x^8)/x^ 7/a^5
Time = 0.27 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.79 \[ \int \frac {A+B x^2}{x^8 \left (a+b x^2\right )^{3/2}} \, dx=-\frac {{\left (16 \, {\left (7 \, B a b^{3} - 8 \, A b^{4}\right )} x^{8} + 8 \, {\left (7 \, B a^{2} b^{2} - 8 \, A a b^{3}\right )} x^{6} + 5 \, A a^{4} - 2 \, {\left (7 \, B a^{3} b - 8 \, A a^{2} b^{2}\right )} x^{4} + {\left (7 \, B a^{4} - 8 \, A a^{3} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{35 \, {\left (a^{5} b x^{9} + a^{6} x^{7}\right )}} \]
-1/35*(16*(7*B*a*b^3 - 8*A*b^4)*x^8 + 8*(7*B*a^2*b^2 - 8*A*a*b^3)*x^6 + 5* A*a^4 - 2*(7*B*a^3*b - 8*A*a^2*b^2)*x^4 + (7*B*a^4 - 8*A*a^3*b)*x^2)*sqrt( b*x^2 + a)/(a^5*b*x^9 + a^6*x^7)
Leaf count of result is larger than twice the leaf count of optimal. 1030 vs. \(2 (143) = 286\).
Time = 5.99 (sec) , antiderivative size = 1030, normalized size of antiderivative = 6.96 \[ \int \frac {A+B x^2}{x^8 \left (a+b x^2\right )^{3/2}} \, dx=A \left (- \frac {5 a^{7} b^{\frac {33}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{35 a^{9} b^{16} x^{6} + 140 a^{8} b^{17} x^{8} + 210 a^{7} b^{18} x^{10} + 140 a^{6} b^{19} x^{12} + 35 a^{5} b^{20} x^{14}} - \frac {7 a^{6} b^{\frac {35}{2}} x^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{35 a^{9} b^{16} x^{6} + 140 a^{8} b^{17} x^{8} + 210 a^{7} b^{18} x^{10} + 140 a^{6} b^{19} x^{12} + 35 a^{5} b^{20} x^{14}} - \frac {7 a^{5} b^{\frac {37}{2}} x^{4} \sqrt {\frac {a}{b x^{2}} + 1}}{35 a^{9} b^{16} x^{6} + 140 a^{8} b^{17} x^{8} + 210 a^{7} b^{18} x^{10} + 140 a^{6} b^{19} x^{12} + 35 a^{5} b^{20} x^{14}} + \frac {35 a^{4} b^{\frac {39}{2}} x^{6} \sqrt {\frac {a}{b x^{2}} + 1}}{35 a^{9} b^{16} x^{6} + 140 a^{8} b^{17} x^{8} + 210 a^{7} b^{18} x^{10} + 140 a^{6} b^{19} x^{12} + 35 a^{5} b^{20} x^{14}} + \frac {280 a^{3} b^{\frac {41}{2}} x^{8} \sqrt {\frac {a}{b x^{2}} + 1}}{35 a^{9} b^{16} x^{6} + 140 a^{8} b^{17} x^{8} + 210 a^{7} b^{18} x^{10} + 140 a^{6} b^{19} x^{12} + 35 a^{5} b^{20} x^{14}} + \frac {560 a^{2} b^{\frac {43}{2}} x^{10} \sqrt {\frac {a}{b x^{2}} + 1}}{35 a^{9} b^{16} x^{6} + 140 a^{8} b^{17} x^{8} + 210 a^{7} b^{18} x^{10} + 140 a^{6} b^{19} x^{12} + 35 a^{5} b^{20} x^{14}} + \frac {448 a b^{\frac {45}{2}} x^{12} \sqrt {\frac {a}{b x^{2}} + 1}}{35 a^{9} b^{16} x^{6} + 140 a^{8} b^{17} x^{8} + 210 a^{7} b^{18} x^{10} + 140 a^{6} b^{19} x^{12} + 35 a^{5} b^{20} x^{14}} + \frac {128 b^{\frac {47}{2}} x^{14} \sqrt {\frac {a}{b x^{2}} + 1}}{35 a^{9} b^{16} x^{6} + 140 a^{8} b^{17} x^{8} + 210 a^{7} b^{18} x^{10} + 140 a^{6} b^{19} x^{12} + 35 a^{5} b^{20} x^{14}}\right ) + B \left (- \frac {a^{5} b^{\frac {19}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{5 a^{7} b^{9} x^{4} + 15 a^{6} b^{10} x^{6} + 15 a^{5} b^{11} x^{8} + 5 a^{4} b^{12} x^{10}} - \frac {5 a^{3} b^{\frac {23}{2}} x^{4} \sqrt {\frac {a}{b x^{2}} + 1}}{5 a^{7} b^{9} x^{4} + 15 a^{6} b^{10} x^{6} + 15 a^{5} b^{11} x^{8} + 5 a^{4} b^{12} x^{10}} - \frac {30 a^{2} b^{\frac {25}{2}} x^{6} \sqrt {\frac {a}{b x^{2}} + 1}}{5 a^{7} b^{9} x^{4} + 15 a^{6} b^{10} x^{6} + 15 a^{5} b^{11} x^{8} + 5 a^{4} b^{12} x^{10}} - \frac {40 a b^{\frac {27}{2}} x^{8} \sqrt {\frac {a}{b x^{2}} + 1}}{5 a^{7} b^{9} x^{4} + 15 a^{6} b^{10} x^{6} + 15 a^{5} b^{11} x^{8} + 5 a^{4} b^{12} x^{10}} - \frac {16 b^{\frac {29}{2}} x^{10} \sqrt {\frac {a}{b x^{2}} + 1}}{5 a^{7} b^{9} x^{4} + 15 a^{6} b^{10} x^{6} + 15 a^{5} b^{11} x^{8} + 5 a^{4} b^{12} x^{10}}\right ) \]
A*(-5*a**7*b**(33/2)*sqrt(a/(b*x**2) + 1)/(35*a**9*b**16*x**6 + 140*a**8*b **17*x**8 + 210*a**7*b**18*x**10 + 140*a**6*b**19*x**12 + 35*a**5*b**20*x* *14) - 7*a**6*b**(35/2)*x**2*sqrt(a/(b*x**2) + 1)/(35*a**9*b**16*x**6 + 14 0*a**8*b**17*x**8 + 210*a**7*b**18*x**10 + 140*a**6*b**19*x**12 + 35*a**5* b**20*x**14) - 7*a**5*b**(37/2)*x**4*sqrt(a/(b*x**2) + 1)/(35*a**9*b**16*x **6 + 140*a**8*b**17*x**8 + 210*a**7*b**18*x**10 + 140*a**6*b**19*x**12 + 35*a**5*b**20*x**14) + 35*a**4*b**(39/2)*x**6*sqrt(a/(b*x**2) + 1)/(35*a** 9*b**16*x**6 + 140*a**8*b**17*x**8 + 210*a**7*b**18*x**10 + 140*a**6*b**19 *x**12 + 35*a**5*b**20*x**14) + 280*a**3*b**(41/2)*x**8*sqrt(a/(b*x**2) + 1)/(35*a**9*b**16*x**6 + 140*a**8*b**17*x**8 + 210*a**7*b**18*x**10 + 140* a**6*b**19*x**12 + 35*a**5*b**20*x**14) + 560*a**2*b**(43/2)*x**10*sqrt(a/ (b*x**2) + 1)/(35*a**9*b**16*x**6 + 140*a**8*b**17*x**8 + 210*a**7*b**18*x **10 + 140*a**6*b**19*x**12 + 35*a**5*b**20*x**14) + 448*a*b**(45/2)*x**12 *sqrt(a/(b*x**2) + 1)/(35*a**9*b**16*x**6 + 140*a**8*b**17*x**8 + 210*a**7 *b**18*x**10 + 140*a**6*b**19*x**12 + 35*a**5*b**20*x**14) + 128*b**(47/2) *x**14*sqrt(a/(b*x**2) + 1)/(35*a**9*b**16*x**6 + 140*a**8*b**17*x**8 + 21 0*a**7*b**18*x**10 + 140*a**6*b**19*x**12 + 35*a**5*b**20*x**14)) + B*(-a* *5*b**(19/2)*sqrt(a/(b*x**2) + 1)/(5*a**7*b**9*x**4 + 15*a**6*b**10*x**6 + 15*a**5*b**11*x**8 + 5*a**4*b**12*x**10) - 5*a**3*b**(23/2)*x**4*sqrt(a/( b*x**2) + 1)/(5*a**7*b**9*x**4 + 15*a**6*b**10*x**6 + 15*a**5*b**11*x**...
Time = 0.19 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.19 \[ \int \frac {A+B x^2}{x^8 \left (a+b x^2\right )^{3/2}} \, dx=-\frac {16 \, B b^{3} x}{5 \, \sqrt {b x^{2} + a} a^{4}} + \frac {128 \, A b^{4} x}{35 \, \sqrt {b x^{2} + a} a^{5}} - \frac {8 \, B b^{2}}{5 \, \sqrt {b x^{2} + a} a^{3} x} + \frac {64 \, A b^{3}}{35 \, \sqrt {b x^{2} + a} a^{4} x} + \frac {2 \, B b}{5 \, \sqrt {b x^{2} + a} a^{2} x^{3}} - \frac {16 \, A b^{2}}{35 \, \sqrt {b x^{2} + a} a^{3} x^{3}} - \frac {B}{5 \, \sqrt {b x^{2} + a} a x^{5}} + \frac {8 \, A b}{35 \, \sqrt {b x^{2} + a} a^{2} x^{5}} - \frac {A}{7 \, \sqrt {b x^{2} + a} a x^{7}} \]
-16/5*B*b^3*x/(sqrt(b*x^2 + a)*a^4) + 128/35*A*b^4*x/(sqrt(b*x^2 + a)*a^5) - 8/5*B*b^2/(sqrt(b*x^2 + a)*a^3*x) + 64/35*A*b^3/(sqrt(b*x^2 + a)*a^4*x) + 2/5*B*b/(sqrt(b*x^2 + a)*a^2*x^3) - 16/35*A*b^2/(sqrt(b*x^2 + a)*a^3*x^ 3) - 1/5*B/(sqrt(b*x^2 + a)*a*x^5) + 8/35*A*b/(sqrt(b*x^2 + a)*a^2*x^5) - 1/7*A/(sqrt(b*x^2 + a)*a*x^7)
Leaf count of result is larger than twice the leaf count of optimal. 407 vs. \(2 (128) = 256\).
Time = 0.30 (sec) , antiderivative size = 407, normalized size of antiderivative = 2.75 \[ \int \frac {A+B x^2}{x^8 \left (a+b x^2\right )^{3/2}} \, dx=-\frac {{\left (B a b^{3} - A b^{4}\right )} x}{\sqrt {b x^{2} + a} a^{5}} + \frac {2 \, {\left (35 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} B a b^{\frac {5}{2}} - 35 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} A b^{\frac {7}{2}} - 280 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} B a^{2} b^{\frac {5}{2}} + 280 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} A a b^{\frac {7}{2}} + 1015 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} B a^{3} b^{\frac {5}{2}} - 1015 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} A a^{2} b^{\frac {7}{2}} - 1680 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} B a^{4} b^{\frac {5}{2}} + 2240 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} A a^{3} b^{\frac {7}{2}} + 1337 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} B a^{5} b^{\frac {5}{2}} - 1673 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} A a^{4} b^{\frac {7}{2}} - 504 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a^{6} b^{\frac {5}{2}} + 616 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} A a^{5} b^{\frac {7}{2}} + 77 \, B a^{7} b^{\frac {5}{2}} - 93 \, A a^{6} b^{\frac {7}{2}}\right )}}{35 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{7} a^{4}} \]
-(B*a*b^3 - A*b^4)*x/(sqrt(b*x^2 + a)*a^5) + 2/35*(35*(sqrt(b)*x - sqrt(b* x^2 + a))^12*B*a*b^(5/2) - 35*(sqrt(b)*x - sqrt(b*x^2 + a))^12*A*b^(7/2) - 280*(sqrt(b)*x - sqrt(b*x^2 + a))^10*B*a^2*b^(5/2) + 280*(sqrt(b)*x - sqr t(b*x^2 + a))^10*A*a*b^(7/2) + 1015*(sqrt(b)*x - sqrt(b*x^2 + a))^8*B*a^3* b^(5/2) - 1015*(sqrt(b)*x - sqrt(b*x^2 + a))^8*A*a^2*b^(7/2) - 1680*(sqrt( b)*x - sqrt(b*x^2 + a))^6*B*a^4*b^(5/2) + 2240*(sqrt(b)*x - sqrt(b*x^2 + a ))^6*A*a^3*b^(7/2) + 1337*(sqrt(b)*x - sqrt(b*x^2 + a))^4*B*a^5*b^(5/2) - 1673*(sqrt(b)*x - sqrt(b*x^2 + a))^4*A*a^4*b^(7/2) - 504*(sqrt(b)*x - sqrt (b*x^2 + a))^2*B*a^6*b^(5/2) + 616*(sqrt(b)*x - sqrt(b*x^2 + a))^2*A*a^5*b ^(7/2) + 77*B*a^7*b^(5/2) - 93*A*a^6*b^(7/2))/(((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^7*a^4)
Time = 5.42 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.00 \[ \int \frac {A+B x^2}{x^8 \left (a+b x^2\right )^{3/2}} \, dx=-\frac {x^2\,\left (\frac {58\,A\,b^4-42\,B\,a\,b^3}{35\,a^5}-\frac {2\,b^3\,\left (93\,A\,b-77\,B\,a\right )}{35\,a^5}\right )-\frac {b^2\,\left (93\,A\,b-77\,B\,a\right )}{35\,a^4}}{x\,\sqrt {b\,x^2+a}}-\frac {\left (7\,B\,a^2-13\,A\,a\,b\right )\,\sqrt {b\,x^2+a}}{35\,a^4\,x^5}-\frac {A\,\sqrt {b\,x^2+a}}{7\,a^2\,x^7}-\frac {b\,\sqrt {b\,x^2+a}\,\left (29\,A\,b-21\,B\,a\right )}{35\,a^4\,x^3} \]