Integrand size = 37, antiderivative size = 193 \[ \int \frac {A+B x^2+C x^4+D x^6+F x^8}{x^2 \left (a+b x^2\right )^{9/2}} \, dx=-\frac {A}{a x \left (a+b x^2\right )^{7/2}}-\frac {(8 A b-a B) x}{a^2 \left (a+b x^2\right )^{7/2}}-\frac {\left (48 A b^2-a (6 b B+a C)\right ) x^3}{3 a^3 \left (a+b x^2\right )^{7/2}}-\frac {\left (192 A b^3-a \left (24 b^2 B+4 a b C+3 a^2 D\right )\right ) x^5}{15 a^4 \left (a+b x^2\right )^{7/2}}-\frac {\left (384 A b^4-a \left (48 b^3 B+8 a b^2 C+6 a^2 b D+15 a^3 F\right )\right ) x^7}{105 a^5 \left (a+b x^2\right )^{7/2}} \]
-A/a/x/(b*x^2+a)^(7/2)-(8*A*b-B*a)*x/a^2/(b*x^2+a)^(7/2)-1/3*(48*A*b^2-a*( 6*B*b+C*a))*x^3/a^3/(b*x^2+a)^(7/2)-1/15*(192*A*b^3-a*(24*B*b^2+4*C*a*b+3* D*a^2))*x^5/a^4/(b*x^2+a)^(7/2)-1/105*(384*A*b^4-a*(48*B*b^3+8*C*a*b^2+6*D *a^2*b+15*F*a^3))*x^7/a^5/(b*x^2+a)^(7/2)
Time = 0.50 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.72 \[ \int \frac {A+B x^2+C x^4+D x^6+F x^8}{x^2 \left (a+b x^2\right )^{9/2}} \, dx=\frac {-384 A b^4 x^8+48 a b^3 x^6 \left (-28 A+B x^2\right )+8 a^2 b^2 x^4 \left (-210 A+21 B x^2+C x^4\right )+2 a^3 b x^2 \left (-420 A+105 B x^2+14 C x^4+3 D x^6\right )+a^4 \left (-105 A+105 B x^2+35 C x^4+21 D x^6+15 F x^8\right )}{105 a^5 x \left (a+b x^2\right )^{7/2}} \]
(-384*A*b^4*x^8 + 48*a*b^3*x^6*(-28*A + B*x^2) + 8*a^2*b^2*x^4*(-210*A + 2 1*B*x^2 + C*x^4) + 2*a^3*b*x^2*(-420*A + 105*B*x^2 + 14*C*x^4 + 3*D*x^6) + a^4*(-105*A + 105*B*x^2 + 35*C*x^4 + 21*D*x^6 + 15*F*x^8))/(105*a^5*x*(a + b*x^2)^(7/2))
Time = 0.64 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.27, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.216, Rules used = {2334, 2344, 1586, 9, 25, 27, 362, 242}
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.
| \(\displaystyle \int \frac {A+B x^2+C x^4+D x^6+F x^8}{x^2 \left (a+b x^2\right )^{9/2}} \, dx\) |
| \(\Big \downarrow \) 2334 |
| \(\displaystyle -\frac {\int \frac {8 A b-a \left (F x^6+D x^4+C x^2+B\right )}{\left (b x^2+a\right )^{9/2}}dx}{a}-\frac {A}{a x \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 2344 |
| \(\displaystyle -\frac {\frac {\int \frac {x^2 \left (-a^2 F x^4-a^2 D x^2+6 b (8 A b-a B)-a^2 C\right )}{\left (b x^2+a\right )^{9/2}}dx}{a}+\frac {x (8 A b-a B)}{a \left (a+b x^2\right )^{7/2}}}{a}-\frac {A}{a x \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 1586 |
| \(\displaystyle -\frac {\frac {\frac {x^3 \left (48 A b^2-a \left (\frac {a^3 F}{b^2}-\frac {a^2 D}{b}+a C+6 b B\right )\right )}{7 a \left (a+b x^2\right )^{7/2}}-\frac {\int -\frac {x \left (\left (192 A b^2-a \left (-\frac {3 F a^3}{b^2}+\frac {3 D a^2}{b}+4 C a+24 b B\right )\right ) x-\frac {7 a^3 F x^3}{b}\right )}{\left (b x^2+a\right )^{7/2}}dx}{7 a}}{a}+\frac {x (8 A b-a B)}{a \left (a+b x^2\right )^{7/2}}}{a}-\frac {A}{a x \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 9 |
| \(\displaystyle -\frac {\frac {\frac {x^3 \left (48 A b^2-a \left (\frac {a^3 F}{b^2}-\frac {a^2 D}{b}+a C+6 b B\right )\right )}{7 a \left (a+b x^2\right )^{7/2}}-\frac {\int -\frac {x^2 \left (b \left (192 A b^2-a \left (-\frac {3 F a^3}{b^2}+\frac {3 D a^2}{b}+4 C a+24 b B\right )\right )-7 a^3 F x^2\right )}{b \left (b x^2+a\right )^{7/2}}dx}{7 a}}{a}+\frac {x (8 A b-a B)}{a \left (a+b x^2\right )^{7/2}}}{a}-\frac {A}{a x \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\frac {\int \frac {x^2 \left (-7 F x^2 a^3-\left (-\frac {3 F a^3}{b}+3 D a^2+4 b C a+24 b^2 B\right ) a+192 A b^3\right )}{b \left (b x^2+a\right )^{7/2}}dx}{7 a}+\frac {x^3 \left (48 A b^2-a \left (\frac {a^3 F}{b^2}-\frac {a^2 D}{b}+a C+6 b B\right )\right )}{7 a \left (a+b x^2\right )^{7/2}}}{a}+\frac {x (8 A b-a B)}{a \left (a+b x^2\right )^{7/2}}}{a}-\frac {A}{a x \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {\int \frac {x^2 \left (-7 F x^2 a^3-\left (-\frac {3 F a^3}{b}+3 D a^2+4 b C a+24 b^2 B\right ) a+192 A b^3\right )}{\left (b x^2+a\right )^{7/2}}dx}{7 a b}+\frac {x^3 \left (48 A b^2-a \left (\frac {a^3 F}{b^2}-\frac {a^2 D}{b}+a C+6 b B\right )\right )}{7 a \left (a+b x^2\right )^{7/2}}}{a}+\frac {x (8 A b-a B)}{a \left (a+b x^2\right )^{7/2}}}{a}-\frac {A}{a x \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 362 |
| \(\displaystyle -\frac {\frac {\frac {\frac {\left (384 A b^4-a \left (15 a^3 F+6 a^2 b D+8 a b^2 C+48 b^3 B\right )\right ) \int \frac {x^2}{\left (b x^2+a\right )^{5/2}}dx}{5 a b}+\frac {x^3 \left (192 A b^4-a \left (-10 a^3 F+3 a^2 b D+4 a b^2 C+24 b^3 B\right )\right )}{5 a b \left (a+b x^2\right )^{5/2}}}{7 a b}+\frac {x^3 \left (48 A b^2-a \left (\frac {a^3 F}{b^2}-\frac {a^2 D}{b}+a C+6 b B\right )\right )}{7 a \left (a+b x^2\right )^{7/2}}}{a}+\frac {x (8 A b-a B)}{a \left (a+b x^2\right )^{7/2}}}{a}-\frac {A}{a x \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 242 |
| \(\displaystyle -\frac {\frac {\frac {x^3 \left (48 A b^2-a \left (\frac {a^3 F}{b^2}-\frac {a^2 D}{b}+a C+6 b B\right )\right )}{7 a \left (a+b x^2\right )^{7/2}}+\frac {\frac {x^3 \left (384 A b^4-a \left (15 a^3 F+6 a^2 b D+8 a b^2 C+48 b^3 B\right )\right )}{15 a^2 b \left (a+b x^2\right )^{3/2}}+\frac {x^3 \left (192 A b^4-a \left (-10 a^3 F+3 a^2 b D+4 a b^2 C+24 b^3 B\right )\right )}{5 a b \left (a+b x^2\right )^{5/2}}}{7 a b}}{a}+\frac {x (8 A b-a B)}{a \left (a+b x^2\right )^{7/2}}}{a}-\frac {A}{a x \left (a+b x^2\right )^{7/2}}\) |
-(A/(a*x*(a + b*x^2)^(7/2))) - (((8*A*b - a*B)*x)/(a*(a + b*x^2)^(7/2)) + (((48*A*b^2 - a*(6*b*B + a*C - (a^2*D)/b + (a^3*F)/b^2))*x^3)/(7*a*(a + b* x^2)^(7/2)) + (((192*A*b^4 - a*(24*b^3*B + 4*a*b^2*C + 3*a^2*b*D - 10*a^3* F))*x^3)/(5*a*b*(a + b*x^2)^(5/2)) + ((384*A*b^4 - a*(48*b^3*B + 8*a*b^2*C + 6*a^2*b*D + 15*a^3*F))*x^3)/(15*a^2*b*(a + b*x^2)^(3/2)))/(7*a*b))/a)/a
3.2.74.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[1/e^(p*r) Int[u*(e*x)^(m + p*r)*ExpandToSum[Px/x^r, x]^p, x], x] /; IGtQ[r, 0]] /; FreeQ[{e, m}, x] && PolyQ[Px, x] && IntegerQ[p] && !MonomialQ[Px, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ (m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] /; FreeQ[{a, b, c, m, p}, x ] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[(-(b*c - a*d))*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*b*e *(p + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(2*a*b*(p + 1)) I nt[(e*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && N eQ[b*c - a*d, 0] && LtQ[p, -1] && (( !IntegerQ[p + 1/2] && NeQ[p, -5/4]) || !RationalQ[m] || (ILtQ[p + 1/2, 0] && LeQ[-1, m, -2*(p + 1)]))
Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c _.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQuotient[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^ p, d + e*x^2, x], x, 0]}, Simp[(-R)*(f*x)^(m + 1)*((d + e*x^2)^(q + 1)/(2*d *f*(q + 1))), x] + Simp[f/(2*d*(q + 1)) Int[(f*x)^(m - 1)*(d + e*x^2)^(q + 1)*ExpandToSum[2*d*(q + 1)*x*Qx + R*(m + 2*q + 3)*x, x], x], x]] /; FreeQ [{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && LtQ[q, -1] && GtQ[m, 0]
Int[(Pq_)*(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{A = Coef f[Pq, x, 0], Q = PolynomialQuotient[Pq - Coeff[Pq, x, 0], x^2, x]}, Simp[A* x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))), x] + Simp[1/(a*(m + 1)) Int[ x^(m + 2)*(a + b*x^2)^p*(a*(m + 1)*Q - A*b*(m + 2*(p + 1) + 1)), x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x^2] && IntegerQ[m/2] && ILtQ[(m + 1)/2 + p, 0] && LtQ[m + Expon[Pq, x] + 2*p + 1, 0]
Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{A = Coeff[Pq, x, 0 ], Q = PolynomialQuotient[Pq - Coeff[Pq, x, 0], x^2, x]}, Simp[A*x*((a + b* x^2)^(p + 1)/a), x] + Simp[1/a Int[x^2*(a + b*x^2)^p*(a*Q - A*b*(2*p + 3) ), x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x^2] && ILtQ[p + 1/2, 0] && LtQ [Expon[Pq, x] + 2*p + 1, 0]
Time = 3.61 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.68
| method | result | size |
| pseudoelliptic | \(\frac {\left (15 F \,x^{8}+21 D x^{6}+35 C \,x^{4}+105 x^{2} B -105 A \right ) a^{4}-840 b \left (-\frac {1}{140} D x^{6}-\frac {1}{30} C \,x^{4}-\frac {1}{4} x^{2} B +A \right ) x^{2} a^{3}-1680 \left (-\frac {1}{210} C \,x^{4}-\frac {1}{10} x^{2} B +A \right ) b^{2} x^{4} a^{2}-1344 b^{3} x^{6} \left (-\frac {x^{2} B}{28}+A \right ) a -384 A \,b^{4} x^{8}}{105 \left (b \,x^{2}+a \right )^{\frac {7}{2}} x \,a^{5}}\) | \(131\) |
| gosper | \(-\frac {384 A \,b^{4} x^{8}-48 B a \,b^{3} x^{8}-8 C \,a^{2} b^{2} x^{8}-6 D a^{3} b \,x^{8}-15 F \,a^{4} x^{8}+1344 A a \,b^{3} x^{6}-168 B \,a^{2} b^{2} x^{6}-28 C \,a^{3} b \,x^{6}-21 D a^{4} x^{6}+1680 A \,a^{2} b^{2} x^{4}-210 B \,a^{3} b \,x^{4}-35 C \,a^{4} x^{4}+840 A \,a^{3} b \,x^{2}-105 B \,a^{4} x^{2}+105 A \,a^{4}}{105 x \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{5}}\) | \(166\) |
| trager | \(-\frac {384 A \,b^{4} x^{8}-48 B a \,b^{3} x^{8}-8 C \,a^{2} b^{2} x^{8}-6 D a^{3} b \,x^{8}-15 F \,a^{4} x^{8}+1344 A a \,b^{3} x^{6}-168 B \,a^{2} b^{2} x^{6}-28 C \,a^{3} b \,x^{6}-21 D a^{4} x^{6}+1680 A \,a^{2} b^{2} x^{4}-210 B \,a^{3} b \,x^{4}-35 C \,a^{4} x^{4}+840 A \,a^{3} b \,x^{2}-105 B \,a^{4} x^{2}+105 A \,a^{4}}{105 x \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{5}}\) | \(166\) |
| default | \(B \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )+F \left (-\frac {x^{5}}{2 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {5 a \left (-\frac {x^{3}}{4 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {3 a \left (-\frac {x}{6 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {a \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )}{6 b}\right )}{4 b}\right )}{2 b}\right )+D \left (-\frac {x^{3}}{4 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {3 a \left (-\frac {x}{6 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {a \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )}{6 b}\right )}{4 b}\right )+C \left (-\frac {x}{6 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {a \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )}{6 b}\right )+A \left (-\frac {1}{a x \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {8 b \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )}{a}\right )\) | \(539\) |
1/105*((15*F*x^8+21*D*x^6+35*C*x^4+105*B*x^2-105*A)*a^4-840*b*(-1/140*D*x^ 6-1/30*C*x^4-1/4*x^2*B+A)*x^2*a^3-1680*(-1/210*C*x^4-1/10*x^2*B+A)*b^2*x^4 *a^2-1344*b^3*x^6*(-1/28*x^2*B+A)*a-384*A*b^4*x^8)/(b*x^2+a)^(7/2)/x/a^5
Time = 0.34 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.97 \[ \int \frac {A+B x^2+C x^4+D x^6+F x^8}{x^2 \left (a+b x^2\right )^{9/2}} \, dx=\frac {{\left ({\left (15 \, F a^{4} + 6 \, D a^{3} b + 8 \, C a^{2} b^{2} + 48 \, B a b^{3} - 384 \, A b^{4}\right )} x^{8} + 7 \, {\left (3 \, D a^{4} + 4 \, C a^{3} b + 24 \, B a^{2} b^{2} - 192 \, A a b^{3}\right )} x^{6} - 105 \, A a^{4} + 35 \, {\left (C a^{4} + 6 \, B a^{3} b - 48 \, A a^{2} b^{2}\right )} x^{4} + 105 \, {\left (B a^{4} - 8 \, A a^{3} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{105 \, {\left (a^{5} b^{4} x^{9} + 4 \, a^{6} b^{3} x^{7} + 6 \, a^{7} b^{2} x^{5} + 4 \, a^{8} b x^{3} + a^{9} x\right )}} \]
1/105*((15*F*a^4 + 6*D*a^3*b + 8*C*a^2*b^2 + 48*B*a*b^3 - 384*A*b^4)*x^8 + 7*(3*D*a^4 + 4*C*a^3*b + 24*B*a^2*b^2 - 192*A*a*b^3)*x^6 - 105*A*a^4 + 35 *(C*a^4 + 6*B*a^3*b - 48*A*a^2*b^2)*x^4 + 105*(B*a^4 - 8*A*a^3*b)*x^2)*sqr t(b*x^2 + a)/(a^5*b^4*x^9 + 4*a^6*b^3*x^7 + 6*a^7*b^2*x^5 + 4*a^8*b*x^3 + a^9*x)
Leaf count of result is larger than twice the leaf count of optimal. 2490 vs. \(2 (184) = 368\).
Time = 98.39 (sec) , antiderivative size = 2490, normalized size of antiderivative = 12.90 \[ \int \frac {A+B x^2+C x^4+D x^6+F x^8}{x^2 \left (a+b x^2\right )^{9/2}} \, dx=\text {Too large to display} \]
A*(-35*a**4*b**(33/2)*sqrt(a/(b*x**2) + 1)/(35*a**9*b**16 + 140*a**8*b**17 *x**2 + 210*a**7*b**18*x**4 + 140*a**6*b**19*x**6 + 35*a**5*b**20*x**8) - 280*a**3*b**(35/2)*x**2*sqrt(a/(b*x**2) + 1)/(35*a**9*b**16 + 140*a**8*b** 17*x**2 + 210*a**7*b**18*x**4 + 140*a**6*b**19*x**6 + 35*a**5*b**20*x**8) - 560*a**2*b**(37/2)*x**4*sqrt(a/(b*x**2) + 1)/(35*a**9*b**16 + 140*a**8*b **17*x**2 + 210*a**7*b**18*x**4 + 140*a**6*b**19*x**6 + 35*a**5*b**20*x**8 ) - 448*a*b**(39/2)*x**6*sqrt(a/(b*x**2) + 1)/(35*a**9*b**16 + 140*a**8*b* *17*x**2 + 210*a**7*b**18*x**4 + 140*a**6*b**19*x**6 + 35*a**5*b**20*x**8) - 128*b**(41/2)*x**8*sqrt(a/(b*x**2) + 1)/(35*a**9*b**16 + 140*a**8*b**17 *x**2 + 210*a**7*b**18*x**4 + 140*a**6*b**19*x**6 + 35*a**5*b**20*x**8)) + B*(35*a**14*x/(35*a**(37/2)*sqrt(1 + b*x**2/a) + 210*a**(35/2)*b*x**2*sqr t(1 + b*x**2/a) + 525*a**(33/2)*b**2*x**4*sqrt(1 + b*x**2/a) + 700*a**(31/ 2)*b**3*x**6*sqrt(1 + b*x**2/a) + 525*a**(29/2)*b**4*x**8*sqrt(1 + b*x**2/ a) + 210*a**(27/2)*b**5*x**10*sqrt(1 + b*x**2/a) + 35*a**(25/2)*b**6*x**12 *sqrt(1 + b*x**2/a)) + 175*a**13*b*x**3/(35*a**(37/2)*sqrt(1 + b*x**2/a) + 210*a**(35/2)*b*x**2*sqrt(1 + b*x**2/a) + 525*a**(33/2)*b**2*x**4*sqrt(1 + b*x**2/a) + 700*a**(31/2)*b**3*x**6*sqrt(1 + b*x**2/a) + 525*a**(29/2)*b **4*x**8*sqrt(1 + b*x**2/a) + 210*a**(27/2)*b**5*x**10*sqrt(1 + b*x**2/a) + 35*a**(25/2)*b**6*x**12*sqrt(1 + b*x**2/a)) + 371*a**12*b**2*x**5/(35*a* *(37/2)*sqrt(1 + b*x**2/a) + 210*a**(35/2)*b*x**2*sqrt(1 + b*x**2/a) + ...
Leaf count of result is larger than twice the leaf count of optimal. 421 vs. \(2 (175) = 350\).
Time = 0.21 (sec) , antiderivative size = 421, normalized size of antiderivative = 2.18 \[ \int \frac {A+B x^2+C x^4+D x^6+F x^8}{x^2 \left (a+b x^2\right )^{9/2}} \, dx=-\frac {F x^{5}}{2 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} - \frac {5 \, F a x^{3}}{8 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}} - \frac {D x^{3}}{4 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} + \frac {16 \, B x}{35 \, \sqrt {b x^{2} + a} a^{4}} + \frac {8 \, B x}{35 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{3}} + \frac {6 \, B x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{2}} + \frac {B x}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a} + \frac {F x}{14 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{3}} + \frac {F x}{7 \, \sqrt {b x^{2} + a} a b^{3}} + \frac {3 \, F a x}{56 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{3}} - \frac {15 \, F a^{2} x}{56 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{3}} + \frac {3 \, D x}{140 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{2}} + \frac {2 \, D x}{35 \, \sqrt {b x^{2} + a} a^{2} b^{2}} + \frac {D x}{35 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a b^{2}} - \frac {3 \, D a x}{28 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}} - \frac {C x}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} + \frac {8 \, C x}{105 \, \sqrt {b x^{2} + a} a^{3} b} + \frac {4 \, C x}{105 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2} b} + \frac {C x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a b} - \frac {128 \, A b x}{35 \, \sqrt {b x^{2} + a} a^{5}} - \frac {64 \, A b x}{35 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{4}} - \frac {48 \, A b x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{3}} - \frac {8 \, A b x}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{2}} - \frac {A}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} a x} \]
-1/2*F*x^5/((b*x^2 + a)^(7/2)*b) - 5/8*F*a*x^3/((b*x^2 + a)^(7/2)*b^2) - 1 /4*D*x^3/((b*x^2 + a)^(7/2)*b) + 16/35*B*x/(sqrt(b*x^2 + a)*a^4) + 8/35*B* x/((b*x^2 + a)^(3/2)*a^3) + 6/35*B*x/((b*x^2 + a)^(5/2)*a^2) + 1/7*B*x/((b *x^2 + a)^(7/2)*a) + 1/14*F*x/((b*x^2 + a)^(3/2)*b^3) + 1/7*F*x/(sqrt(b*x^ 2 + a)*a*b^3) + 3/56*F*a*x/((b*x^2 + a)^(5/2)*b^3) - 15/56*F*a^2*x/((b*x^2 + a)^(7/2)*b^3) + 3/140*D*x/((b*x^2 + a)^(5/2)*b^2) + 2/35*D*x/(sqrt(b*x^ 2 + a)*a^2*b^2) + 1/35*D*x/((b*x^2 + a)^(3/2)*a*b^2) - 3/28*D*a*x/((b*x^2 + a)^(7/2)*b^2) - 1/7*C*x/((b*x^2 + a)^(7/2)*b) + 8/105*C*x/(sqrt(b*x^2 + a)*a^3*b) + 4/105*C*x/((b*x^2 + a)^(3/2)*a^2*b) + 1/35*C*x/((b*x^2 + a)^(5 /2)*a*b) - 128/35*A*b*x/(sqrt(b*x^2 + a)*a^5) - 64/35*A*b*x/((b*x^2 + a)^( 3/2)*a^4) - 48/35*A*b*x/((b*x^2 + a)^(5/2)*a^3) - 8/7*A*b*x/((b*x^2 + a)^( 7/2)*a^2) - A/((b*x^2 + a)^(7/2)*a*x)
Time = 0.31 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.14 \[ \int \frac {A+B x^2+C x^4+D x^6+F x^8}{x^2 \left (a+b x^2\right )^{9/2}} \, dx=\frac {{\left ({\left (x^{2} {\left (\frac {{\left (15 \, F a^{13} b^{3} + 6 \, D a^{12} b^{4} + 8 \, C a^{11} b^{5} + 48 \, B a^{10} b^{6} - 279 \, A a^{9} b^{7}\right )} x^{2}}{a^{14} b^{3}} + \frac {7 \, {\left (3 \, D a^{13} b^{3} + 4 \, C a^{12} b^{4} + 24 \, B a^{11} b^{5} - 132 \, A a^{10} b^{6}\right )}}{a^{14} b^{3}}\right )} + \frac {35 \, {\left (C a^{13} b^{3} + 6 \, B a^{12} b^{4} - 30 \, A a^{11} b^{5}\right )}}{a^{14} b^{3}}\right )} x^{2} + \frac {105 \, {\left (B a^{13} b^{3} - 4 \, A a^{12} b^{4}\right )}}{a^{14} b^{3}}\right )} x}{105 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} + \frac {2 \, A \sqrt {b}}{{\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )} a^{4}} \]
1/105*((x^2*((15*F*a^13*b^3 + 6*D*a^12*b^4 + 8*C*a^11*b^5 + 48*B*a^10*b^6 - 279*A*a^9*b^7)*x^2/(a^14*b^3) + 7*(3*D*a^13*b^3 + 4*C*a^12*b^4 + 24*B*a^ 11*b^5 - 132*A*a^10*b^6)/(a^14*b^3)) + 35*(C*a^13*b^3 + 6*B*a^12*b^4 - 30* A*a^11*b^5)/(a^14*b^3))*x^2 + 105*(B*a^13*b^3 - 4*A*a^12*b^4)/(a^14*b^3))* x/(b*x^2 + a)^(7/2) + 2*A*sqrt(b)/(((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)*a ^4)
Timed out. \[ \int \frac {A+B x^2+C x^4+D x^6+F x^8}{x^2 \left (a+b x^2\right )^{9/2}} \, dx=\int \frac {A+B\,x^2+C\,x^4+F\,x^8+x^6\,D}{x^2\,{\left (b\,x^2+a\right )}^{9/2}} \,d x \]