Integrand size = 30, antiderivative size = 287 \[ \int \frac {x^8 \left (c+d x^2+e x^4+f x^6\right )}{\left (a+b x^2\right )^3} \, dx=-\frac {a \left (15 b^3 c-27 a b^2 d+43 a^2 b e-63 a^3 f\right ) x}{4 b^7}+\frac {\left (5 b^3 c-9 a b^2 d+15 a^2 b e-23 a^3 f\right ) x^3}{6 b^6}-\frac {\left (5 b^3 c-9 a b^2 d+17 a^2 b e-29 a^3 f\right ) x^5}{20 a b^5}+\frac {(b e-3 a f) x^7}{7 b^4}+\frac {f x^9}{9 b^3}+\frac {\left (c-\frac {a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^9}{4 a \left (a+b x^2\right )^2}-\frac {a^2 \left (5 b^3 c-9 a b^2 d+13 a^2 b e-17 a^3 f\right ) x}{8 b^7 \left (a+b x^2\right )}+\frac {a^{3/2} \left (35 b^3 c-63 a b^2 d+99 a^2 b e-143 a^3 f\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 b^{15/2}} \]
-1/4*a*(-63*a^3*f+43*a^2*b*e-27*a*b^2*d+15*b^3*c)*x/b^7+1/6*(-23*a^3*f+15* a^2*b*e-9*a*b^2*d+5*b^3*c)*x^3/b^6-1/20*(-29*a^3*f+17*a^2*b*e-9*a*b^2*d+5* b^3*c)*x^5/a/b^5+1/7*(-3*a*f+b*e)*x^7/b^4+1/9*f*x^9/b^3+1/4*(c-a*(a^2*f-a* b*e+b^2*d)/b^3)*x^9/a/(b*x^2+a)^2-1/8*a^2*(-17*a^3*f+13*a^2*b*e-9*a*b^2*d+ 5*b^3*c)*x/b^7/(b*x^2+a)+1/8*a^(3/2)*(-143*a^3*f+99*a^2*b*e-63*a*b^2*d+35* b^3*c)*arctan(x*b^(1/2)/a^(1/2))/b^(15/2)
Time = 0.11 (sec) , antiderivative size = 272, normalized size of antiderivative = 0.95 \[ \int \frac {x^8 \left (c+d x^2+e x^4+f x^6\right )}{\left (a+b x^2\right )^3} \, dx=\frac {a \left (-3 b^3 c+6 a b^2 d-10 a^2 b e+15 a^3 f\right ) x}{b^7}+\frac {\left (b^3 c-3 a b^2 d+6 a^2 b e-10 a^3 f\right ) x^3}{3 b^6}+\frac {\left (b^2 d-3 a b e+6 a^2 f\right ) x^5}{5 b^5}+\frac {(b e-3 a f) x^7}{7 b^4}+\frac {f x^9}{9 b^3}+\frac {a^3 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{4 b^7 \left (a+b x^2\right )^2}+\frac {a^2 \left (-13 b^3 c+17 a b^2 d-21 a^2 b e+25 a^3 f\right ) x}{8 b^7 \left (a+b x^2\right )}-\frac {a^{3/2} \left (-35 b^3 c+63 a b^2 d-99 a^2 b e+143 a^3 f\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 b^{15/2}} \]
(a*(-3*b^3*c + 6*a*b^2*d - 10*a^2*b*e + 15*a^3*f)*x)/b^7 + ((b^3*c - 3*a*b ^2*d + 6*a^2*b*e - 10*a^3*f)*x^3)/(3*b^6) + ((b^2*d - 3*a*b*e + 6*a^2*f)*x ^5)/(5*b^5) + ((b*e - 3*a*f)*x^7)/(7*b^4) + (f*x^9)/(9*b^3) + (a^3*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(4*b^7*(a + b*x^2)^2) + (a^2*(-13*b^3*c + 17*a*b^2*d - 21*a^2*b*e + 25*a^3*f)*x)/(8*b^7*(a + b*x^2)) - (a^(3/2)*(-35 *b^3*c + 63*a*b^2*d - 99*a^2*b*e + 143*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]]) /(8*b^(15/2))
Time = 0.86 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2335, 9, 1580, 2341, 2009}
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.
| \(\displaystyle \int \frac {x^8 \left (c+d x^2+e x^4+f x^6\right )}{\left (a+b x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 2335 |
| \(\displaystyle \frac {x^9 \left (c-\frac {a \left (a^2 f-a b e+b^2 d\right )}{b^3}\right )}{4 a \left (a+b x^2\right )^2}-\frac {\int \frac {x^7 \left (-4 a f x^5-4 a \left (e-\frac {a f}{b}\right ) x^3+\left (-\frac {9 f a^3}{b^2}+\frac {9 e a^2}{b}-9 d a+5 b c\right ) x\right )}{\left (b x^2+a\right )^2}dx}{4 a b}\) |
| \(\Big \downarrow \) 9 |
| \(\displaystyle \frac {x^9 \left (c-\frac {a \left (a^2 f-a b e+b^2 d\right )}{b^3}\right )}{4 a \left (a+b x^2\right )^2}-\frac {\int \frac {x^8 \left (-4 a f x^4-4 a \left (e-\frac {a f}{b}\right ) x^2+5 b c-9 a d+\frac {9 a^2 e}{b}-\frac {9 a^3 f}{b^2}\right )}{\left (b x^2+a\right )^2}dx}{4 a b}\) |
| \(\Big \downarrow \) 1580 |
| \(\displaystyle \frac {x^9 \left (c-\frac {a \left (a^2 f-a b e+b^2 d\right )}{b^3}\right )}{4 a \left (a+b x^2\right )^2}-\frac {\frac {a^3 x \left (-17 a^3 f+13 a^2 b e-9 a b^2 d+5 b^3 c\right )}{2 b^6 \left (a+b x^2\right )}-\frac {\int \frac {8 a b^5 f x^{10}+8 a b^4 (b e-2 a f) x^8-2 b^3 \left (-17 f a^3+13 b e a^2-9 b^2 d a+5 b^3 c\right ) x^6+2 a b^2 \left (-17 f a^3+13 b e a^2-9 b^2 d a+5 b^3 c\right ) x^4-2 a^2 b \left (-17 f a^3+13 b e a^2-9 b^2 d a+5 b^3 c\right ) x^2+a^3 \left (-17 f a^3+13 b e a^2-9 b^2 d a+5 b^3 c\right )}{b x^2+a}dx}{2 b^6}}{4 a b}\) |
| \(\Big \downarrow \) 2341 |
| \(\displaystyle \frac {x^9 \left (c-\frac {a \left (a^2 f-a b e+b^2 d\right )}{b^3}\right )}{4 a \left (a+b x^2\right )^2}-\frac {\frac {a^3 x \left (-17 a^3 f+13 a^2 b e-9 a b^2 d+5 b^3 c\right )}{2 b^6 \left (a+b x^2\right )}-\frac {\int \left (8 a b^4 f x^8+8 a b^3 (b e-3 a f) x^6-2 b^2 \left (-29 f a^3+17 b e a^2-9 b^2 d a+5 b^3 c\right ) x^4+4 a b \left (-23 f a^3+15 b e a^2-9 b^2 d a+5 b^3 c\right ) x^2-2 a^2 \left (-63 f a^3+43 b e a^2-27 b^2 d a+15 b^3 c\right )+\frac {-143 f a^6+99 b e a^5-63 b^2 d a^4+35 b^3 c a^3}{b x^2+a}\right )dx}{2 b^6}}{4 a b}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {x^9 \left (c-\frac {a \left (a^2 f-a b e+b^2 d\right )}{b^3}\right )}{4 a \left (a+b x^2\right )^2}-\frac {\frac {a^3 x \left (-17 a^3 f+13 a^2 b e-9 a b^2 d+5 b^3 c\right )}{2 b^6 \left (a+b x^2\right )}-\frac {-\frac {2}{5} b^2 x^5 \left (-29 a^3 f+17 a^2 b e-9 a b^2 d+5 b^3 c\right )+\frac {4}{3} a b x^3 \left (-23 a^3 f+15 a^2 b e-9 a b^2 d+5 b^3 c\right )-2 a^2 x \left (-63 a^3 f+43 a^2 b e-27 a b^2 d+15 b^3 c\right )+\frac {a^{5/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (-143 a^3 f+99 a^2 b e-63 a b^2 d+35 b^3 c\right )}{\sqrt {b}}+\frac {8}{9} a b^4 f x^9+\frac {8}{7} a b^3 x^7 (b e-3 a f)}{2 b^6}}{4 a b}\) |
((c - (a*(b^2*d - a*b*e + a^2*f))/b^3)*x^9)/(4*a*(a + b*x^2)^2) - ((a^3*(5 *b^3*c - 9*a*b^2*d + 13*a^2*b*e - 17*a^3*f)*x)/(2*b^6*(a + b*x^2)) - (-2*a ^2*(15*b^3*c - 27*a*b^2*d + 43*a^2*b*e - 63*a^3*f)*x + (4*a*b*(5*b^3*c - 9 *a*b^2*d + 15*a^2*b*e - 23*a^3*f)*x^3)/3 - (2*b^2*(5*b^3*c - 9*a*b^2*d + 1 7*a^2*b*e - 29*a^3*f)*x^5)/5 + (8*a*b^3*(b*e - 3*a*f)*x^7)/7 + (8*a*b^4*f* x^9)/9 + (a^(5/2)*(35*b^3*c - 63*a*b^2*d + 99*a^2*b*e - 143*a^3*f)*ArcTan[ (Sqrt[b]*x)/Sqrt[a]])/Sqrt[b])/(2*b^6))/(4*a*b)
3.2.33.3.1 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[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_) ^4)^(p_.), x_Symbol] :> Simp[(-d)^(m/2 - 1)*(c*d^2 - b*d*e + a*e^2)^p*x*((d + e*x^2)^(q + 1)/(2*e^(2*p + m/2)*(q + 1))), x] + Simp[1/(2*e^(2*p + m/2)* (q + 1)) Int[(d + e*x^2)^(q + 1)*ExpandToSum[Together[(1/(d + e*x^2))*(2* e^(2*p + m/2)*(q + 1)*x^m*(a + b*x^2 + c*x^4)^p - (-d)^(m/2 - 1)*(c*d^2 - b *d*e + a*e^2)^p*(d + e*(2*q + 3)*x^2))], x], x], x] /; FreeQ[{a, b, c, d, e }, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && ILtQ[q, -1] && IGtQ[m/2, 0]
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ {Q = PolynomialQuotient[Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[Pq , a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[(c*x)^m*(a + b*x^2)^(p + 1)*((a*g - b*f*x)/(2*a*b*(p + 1))), x] + Simp[c/(2*a*b*(p + 1)) Int[(c*x)^(m - 1)*(a + b*x^2)^(p + 1)*ExpandToSu m[2*a*b*(p + 1)*x*Q - a*g*m + b*f*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && GtQ[m, 0]
Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq* (a + b*x^2)^p, x], x] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
Time = 3.62 (sec) , antiderivative size = 268, normalized size of antiderivative = 0.93
| method | result | size |
| default | \(\frac {\frac {1}{9} f \,x^{9} b^{4}-\frac {3}{7} a \,b^{3} f \,x^{7}+\frac {1}{7} b^{4} e \,x^{7}+\frac {6}{5} a^{2} b^{2} f \,x^{5}-\frac {3}{5} a \,b^{3} e \,x^{5}+\frac {1}{5} b^{4} d \,x^{5}-\frac {10}{3} a^{3} b f \,x^{3}+2 a^{2} b^{2} e \,x^{3}-a \,b^{3} d \,x^{3}+\frac {1}{3} b^{4} c \,x^{3}+15 a^{4} f x -10 a^{3} b e x +6 a^{2} b^{2} d x -3 a \,b^{3} c x}{b^{7}}-\frac {a^{2} \left (\frac {\left (-\frac {25}{8} a^{3} b f +\frac {21}{8} a^{2} e \,b^{2}-\frac {17}{8} a \,b^{3} d +\frac {13}{8} b^{4} c \right ) x^{3}-\frac {a \left (23 f \,a^{3}-19 a^{2} b e +15 a \,b^{2} d -11 b^{3} c \right ) x}{8}}{\left (b \,x^{2}+a \right )^{2}}+\frac {\left (143 f \,a^{3}-99 a^{2} b e +63 a \,b^{2} d -35 b^{3} c \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{b^{7}}\) | \(268\) |
| risch | \(\frac {f \,x^{9}}{9 b^{3}}-\frac {3 a f \,x^{7}}{7 b^{4}}+\frac {e \,x^{7}}{7 b^{3}}+\frac {6 a^{2} f \,x^{5}}{5 b^{5}}-\frac {3 a e \,x^{5}}{5 b^{4}}+\frac {d \,x^{5}}{5 b^{3}}-\frac {10 a^{3} f \,x^{3}}{3 b^{6}}+\frac {2 a^{2} e \,x^{3}}{b^{5}}-\frac {a d \,x^{3}}{b^{4}}+\frac {c \,x^{3}}{3 b^{3}}+\frac {15 a^{4} f x}{b^{7}}-\frac {10 a^{3} e x}{b^{6}}+\frac {6 a^{2} d x}{b^{5}}-\frac {3 a c x}{b^{4}}+\frac {\left (\frac {25}{8} a^{5} b f -\frac {21}{8} a^{4} e \,b^{2}+\frac {17}{8} a^{3} d \,b^{3}-\frac {13}{8} a^{2} c \,b^{4}\right ) x^{3}+\frac {a^{3} \left (23 f \,a^{3}-19 a^{2} b e +15 a \,b^{2} d -11 b^{3} c \right ) x}{8}}{b^{7} \left (b \,x^{2}+a \right )^{2}}+\frac {143 \sqrt {-a b}\, a^{4} \ln \left (-\sqrt {-a b}\, x -a \right ) f}{16 b^{8}}-\frac {99 \sqrt {-a b}\, a^{3} \ln \left (-\sqrt {-a b}\, x -a \right ) e}{16 b^{7}}+\frac {63 \sqrt {-a b}\, a^{2} \ln \left (-\sqrt {-a b}\, x -a \right ) d}{16 b^{6}}-\frac {35 \sqrt {-a b}\, a \ln \left (-\sqrt {-a b}\, x -a \right ) c}{16 b^{5}}-\frac {143 \sqrt {-a b}\, a^{4} \ln \left (\sqrt {-a b}\, x -a \right ) f}{16 b^{8}}+\frac {99 \sqrt {-a b}\, a^{3} \ln \left (\sqrt {-a b}\, x -a \right ) e}{16 b^{7}}-\frac {63 \sqrt {-a b}\, a^{2} \ln \left (\sqrt {-a b}\, x -a \right ) d}{16 b^{6}}+\frac {35 \sqrt {-a b}\, a \ln \left (\sqrt {-a b}\, x -a \right ) c}{16 b^{5}}\) | \(452\) |
1/b^7*(1/9*f*x^9*b^4-3/7*a*b^3*f*x^7+1/7*b^4*e*x^7+6/5*a^2*b^2*f*x^5-3/5*a *b^3*e*x^5+1/5*b^4*d*x^5-10/3*a^3*b*f*x^3+2*a^2*b^2*e*x^3-a*b^3*d*x^3+1/3* b^4*c*x^3+15*a^4*f*x-10*a^3*b*e*x+6*a^2*b^2*d*x-3*a*b^3*c*x)-a^2/b^7*(((-2 5/8*a^3*b*f+21/8*a^2*e*b^2-17/8*a*b^3*d+13/8*b^4*c)*x^3-1/8*a*(23*a^3*f-19 *a^2*b*e+15*a*b^2*d-11*b^3*c)*x)/(b*x^2+a)^2+1/8*(143*a^3*f-99*a^2*b*e+63* a*b^2*d-35*b^3*c)/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2)))
Time = 0.27 (sec) , antiderivative size = 762, normalized size of antiderivative = 2.66 \[ \int \frac {x^8 \left (c+d x^2+e x^4+f x^6\right )}{\left (a+b x^2\right )^3} \, dx=\left [\frac {560 \, b^{6} f x^{13} + 80 \, {\left (9 \, b^{6} e - 13 \, a b^{5} f\right )} x^{11} + 16 \, {\left (63 \, b^{6} d - 99 \, a b^{5} e + 143 \, a^{2} b^{4} f\right )} x^{9} + 48 \, {\left (35 \, b^{6} c - 63 \, a b^{5} d + 99 \, a^{2} b^{4} e - 143 \, a^{3} b^{3} f\right )} x^{7} - 336 \, {\left (35 \, a b^{5} c - 63 \, a^{2} b^{4} d + 99 \, a^{3} b^{3} e - 143 \, a^{4} b^{2} f\right )} x^{5} - 1050 \, {\left (35 \, a^{2} b^{4} c - 63 \, a^{3} b^{3} d + 99 \, a^{4} b^{2} e - 143 \, a^{5} b f\right )} x^{3} - 315 \, {\left (35 \, a^{3} b^{3} c - 63 \, a^{4} b^{2} d + 99 \, a^{5} b e - 143 \, a^{6} f + {\left (35 \, a b^{5} c - 63 \, a^{2} b^{4} d + 99 \, a^{3} b^{3} e - 143 \, a^{4} b^{2} f\right )} x^{4} + 2 \, {\left (35 \, a^{2} b^{4} c - 63 \, a^{3} b^{3} d + 99 \, a^{4} b^{2} e - 143 \, a^{5} b f\right )} x^{2}\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} - 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right ) - 630 \, {\left (35 \, a^{3} b^{3} c - 63 \, a^{4} b^{2} d + 99 \, a^{5} b e - 143 \, a^{6} f\right )} x}{5040 \, {\left (b^{9} x^{4} + 2 \, a b^{8} x^{2} + a^{2} b^{7}\right )}}, \frac {280 \, b^{6} f x^{13} + 40 \, {\left (9 \, b^{6} e - 13 \, a b^{5} f\right )} x^{11} + 8 \, {\left (63 \, b^{6} d - 99 \, a b^{5} e + 143 \, a^{2} b^{4} f\right )} x^{9} + 24 \, {\left (35 \, b^{6} c - 63 \, a b^{5} d + 99 \, a^{2} b^{4} e - 143 \, a^{3} b^{3} f\right )} x^{7} - 168 \, {\left (35 \, a b^{5} c - 63 \, a^{2} b^{4} d + 99 \, a^{3} b^{3} e - 143 \, a^{4} b^{2} f\right )} x^{5} - 525 \, {\left (35 \, a^{2} b^{4} c - 63 \, a^{3} b^{3} d + 99 \, a^{4} b^{2} e - 143 \, a^{5} b f\right )} x^{3} + 315 \, {\left (35 \, a^{3} b^{3} c - 63 \, a^{4} b^{2} d + 99 \, a^{5} b e - 143 \, a^{6} f + {\left (35 \, a b^{5} c - 63 \, a^{2} b^{4} d + 99 \, a^{3} b^{3} e - 143 \, a^{4} b^{2} f\right )} x^{4} + 2 \, {\left (35 \, a^{2} b^{4} c - 63 \, a^{3} b^{3} d + 99 \, a^{4} b^{2} e - 143 \, a^{5} b f\right )} x^{2}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) - 315 \, {\left (35 \, a^{3} b^{3} c - 63 \, a^{4} b^{2} d + 99 \, a^{5} b e - 143 \, a^{6} f\right )} x}{2520 \, {\left (b^{9} x^{4} + 2 \, a b^{8} x^{2} + a^{2} b^{7}\right )}}\right ] \]
[1/5040*(560*b^6*f*x^13 + 80*(9*b^6*e - 13*a*b^5*f)*x^11 + 16*(63*b^6*d - 99*a*b^5*e + 143*a^2*b^4*f)*x^9 + 48*(35*b^6*c - 63*a*b^5*d + 99*a^2*b^4*e - 143*a^3*b^3*f)*x^7 - 336*(35*a*b^5*c - 63*a^2*b^4*d + 99*a^3*b^3*e - 14 3*a^4*b^2*f)*x^5 - 1050*(35*a^2*b^4*c - 63*a^3*b^3*d + 99*a^4*b^2*e - 143* a^5*b*f)*x^3 - 315*(35*a^3*b^3*c - 63*a^4*b^2*d + 99*a^5*b*e - 143*a^6*f + (35*a*b^5*c - 63*a^2*b^4*d + 99*a^3*b^3*e - 143*a^4*b^2*f)*x^4 + 2*(35*a^ 2*b^4*c - 63*a^3*b^3*d + 99*a^4*b^2*e - 143*a^5*b*f)*x^2)*sqrt(-a/b)*log(( b*x^2 - 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)) - 630*(35*a^3*b^3*c - 63*a^4*b^ 2*d + 99*a^5*b*e - 143*a^6*f)*x)/(b^9*x^4 + 2*a*b^8*x^2 + a^2*b^7), 1/2520 *(280*b^6*f*x^13 + 40*(9*b^6*e - 13*a*b^5*f)*x^11 + 8*(63*b^6*d - 99*a*b^5 *e + 143*a^2*b^4*f)*x^9 + 24*(35*b^6*c - 63*a*b^5*d + 99*a^2*b^4*e - 143*a ^3*b^3*f)*x^7 - 168*(35*a*b^5*c - 63*a^2*b^4*d + 99*a^3*b^3*e - 143*a^4*b^ 2*f)*x^5 - 525*(35*a^2*b^4*c - 63*a^3*b^3*d + 99*a^4*b^2*e - 143*a^5*b*f)* x^3 + 315*(35*a^3*b^3*c - 63*a^4*b^2*d + 99*a^5*b*e - 143*a^6*f + (35*a*b^ 5*c - 63*a^2*b^4*d + 99*a^3*b^3*e - 143*a^4*b^2*f)*x^4 + 2*(35*a^2*b^4*c - 63*a^3*b^3*d + 99*a^4*b^2*e - 143*a^5*b*f)*x^2)*sqrt(a/b)*arctan(b*x*sqrt (a/b)/a) - 315*(35*a^3*b^3*c - 63*a^4*b^2*d + 99*a^5*b*e - 143*a^6*f)*x)/( b^9*x^4 + 2*a*b^8*x^2 + a^2*b^7)]
Time = 19.39 (sec) , antiderivative size = 503, normalized size of antiderivative = 1.75 \[ \int \frac {x^8 \left (c+d x^2+e x^4+f x^6\right )}{\left (a+b x^2\right )^3} \, dx=x^{7} \left (- \frac {3 a f}{7 b^{4}} + \frac {e}{7 b^{3}}\right ) + x^{5} \cdot \left (\frac {6 a^{2} f}{5 b^{5}} - \frac {3 a e}{5 b^{4}} + \frac {d}{5 b^{3}}\right ) + x^{3} \left (- \frac {10 a^{3} f}{3 b^{6}} + \frac {2 a^{2} e}{b^{5}} - \frac {a d}{b^{4}} + \frac {c}{3 b^{3}}\right ) + x \left (\frac {15 a^{4} f}{b^{7}} - \frac {10 a^{3} e}{b^{6}} + \frac {6 a^{2} d}{b^{5}} - \frac {3 a c}{b^{4}}\right ) + \frac {\sqrt {- \frac {a^{3}}{b^{15}}} \cdot \left (143 a^{3} f - 99 a^{2} b e + 63 a b^{2} d - 35 b^{3} c\right ) \log {\left (- \frac {b^{7} \sqrt {- \frac {a^{3}}{b^{15}}} \cdot \left (143 a^{3} f - 99 a^{2} b e + 63 a b^{2} d - 35 b^{3} c\right )}{143 a^{4} f - 99 a^{3} b e + 63 a^{2} b^{2} d - 35 a b^{3} c} + x \right )}}{16} - \frac {\sqrt {- \frac {a^{3}}{b^{15}}} \cdot \left (143 a^{3} f - 99 a^{2} b e + 63 a b^{2} d - 35 b^{3} c\right ) \log {\left (\frac {b^{7} \sqrt {- \frac {a^{3}}{b^{15}}} \cdot \left (143 a^{3} f - 99 a^{2} b e + 63 a b^{2} d - 35 b^{3} c\right )}{143 a^{4} f - 99 a^{3} b e + 63 a^{2} b^{2} d - 35 a b^{3} c} + x \right )}}{16} + \frac {x^{3} \cdot \left (25 a^{5} b f - 21 a^{4} b^{2} e + 17 a^{3} b^{3} d - 13 a^{2} b^{4} c\right ) + x \left (23 a^{6} f - 19 a^{5} b e + 15 a^{4} b^{2} d - 11 a^{3} b^{3} c\right )}{8 a^{2} b^{7} + 16 a b^{8} x^{2} + 8 b^{9} x^{4}} + \frac {f x^{9}}{9 b^{3}} \]
x**7*(-3*a*f/(7*b**4) + e/(7*b**3)) + x**5*(6*a**2*f/(5*b**5) - 3*a*e/(5*b **4) + d/(5*b**3)) + x**3*(-10*a**3*f/(3*b**6) + 2*a**2*e/b**5 - a*d/b**4 + c/(3*b**3)) + x*(15*a**4*f/b**7 - 10*a**3*e/b**6 + 6*a**2*d/b**5 - 3*a*c /b**4) + sqrt(-a**3/b**15)*(143*a**3*f - 99*a**2*b*e + 63*a*b**2*d - 35*b* *3*c)*log(-b**7*sqrt(-a**3/b**15)*(143*a**3*f - 99*a**2*b*e + 63*a*b**2*d - 35*b**3*c)/(143*a**4*f - 99*a**3*b*e + 63*a**2*b**2*d - 35*a*b**3*c) + x )/16 - sqrt(-a**3/b**15)*(143*a**3*f - 99*a**2*b*e + 63*a*b**2*d - 35*b**3 *c)*log(b**7*sqrt(-a**3/b**15)*(143*a**3*f - 99*a**2*b*e + 63*a*b**2*d - 3 5*b**3*c)/(143*a**4*f - 99*a**3*b*e + 63*a**2*b**2*d - 35*a*b**3*c) + x)/1 6 + (x**3*(25*a**5*b*f - 21*a**4*b**2*e + 17*a**3*b**3*d - 13*a**2*b**4*c) + x*(23*a**6*f - 19*a**5*b*e + 15*a**4*b**2*d - 11*a**3*b**3*c))/(8*a**2* b**7 + 16*a*b**8*x**2 + 8*b**9*x**4) + f*x**9/(9*b**3)
Time = 0.28 (sec) , antiderivative size = 281, normalized size of antiderivative = 0.98 \[ \int \frac {x^8 \left (c+d x^2+e x^4+f x^6\right )}{\left (a+b x^2\right )^3} \, dx=-\frac {{\left (13 \, a^{2} b^{4} c - 17 \, a^{3} b^{3} d + 21 \, a^{4} b^{2} e - 25 \, a^{5} b f\right )} x^{3} + {\left (11 \, a^{3} b^{3} c - 15 \, a^{4} b^{2} d + 19 \, a^{5} b e - 23 \, a^{6} f\right )} x}{8 \, {\left (b^{9} x^{4} + 2 \, a b^{8} x^{2} + a^{2} b^{7}\right )}} + \frac {{\left (35 \, a^{2} b^{3} c - 63 \, a^{3} b^{2} d + 99 \, a^{4} b e - 143 \, a^{5} f\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} b^{7}} + \frac {35 \, b^{4} f x^{9} + 45 \, {\left (b^{4} e - 3 \, a b^{3} f\right )} x^{7} + 63 \, {\left (b^{4} d - 3 \, a b^{3} e + 6 \, a^{2} b^{2} f\right )} x^{5} + 105 \, {\left (b^{4} c - 3 \, a b^{3} d + 6 \, a^{2} b^{2} e - 10 \, a^{3} b f\right )} x^{3} - 315 \, {\left (3 \, a b^{3} c - 6 \, a^{2} b^{2} d + 10 \, a^{3} b e - 15 \, a^{4} f\right )} x}{315 \, b^{7}} \]
-1/8*((13*a^2*b^4*c - 17*a^3*b^3*d + 21*a^4*b^2*e - 25*a^5*b*f)*x^3 + (11* a^3*b^3*c - 15*a^4*b^2*d + 19*a^5*b*e - 23*a^6*f)*x)/(b^9*x^4 + 2*a*b^8*x^ 2 + a^2*b^7) + 1/8*(35*a^2*b^3*c - 63*a^3*b^2*d + 99*a^4*b*e - 143*a^5*f)* arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^7) + 1/315*(35*b^4*f*x^9 + 45*(b^4*e - 3*a*b^3*f)*x^7 + 63*(b^4*d - 3*a*b^3*e + 6*a^2*b^2*f)*x^5 + 105*(b^4*c - 3 *a*b^3*d + 6*a^2*b^2*e - 10*a^3*b*f)*x^3 - 315*(3*a*b^3*c - 6*a^2*b^2*d + 10*a^3*b*e - 15*a^4*f)*x)/b^7
Time = 0.31 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.02 \[ \int \frac {x^8 \left (c+d x^2+e x^4+f x^6\right )}{\left (a+b x^2\right )^3} \, dx=\frac {{\left (35 \, a^{2} b^{3} c - 63 \, a^{3} b^{2} d + 99 \, a^{4} b e - 143 \, a^{5} f\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} b^{7}} - \frac {13 \, a^{2} b^{4} c x^{3} - 17 \, a^{3} b^{3} d x^{3} + 21 \, a^{4} b^{2} e x^{3} - 25 \, a^{5} b f x^{3} + 11 \, a^{3} b^{3} c x - 15 \, a^{4} b^{2} d x + 19 \, a^{5} b e x - 23 \, a^{6} f x}{8 \, {\left (b x^{2} + a\right )}^{2} b^{7}} + \frac {35 \, b^{24} f x^{9} + 45 \, b^{24} e x^{7} - 135 \, a b^{23} f x^{7} + 63 \, b^{24} d x^{5} - 189 \, a b^{23} e x^{5} + 378 \, a^{2} b^{22} f x^{5} + 105 \, b^{24} c x^{3} - 315 \, a b^{23} d x^{3} + 630 \, a^{2} b^{22} e x^{3} - 1050 \, a^{3} b^{21} f x^{3} - 945 \, a b^{23} c x + 1890 \, a^{2} b^{22} d x - 3150 \, a^{3} b^{21} e x + 4725 \, a^{4} b^{20} f x}{315 \, b^{27}} \]
1/8*(35*a^2*b^3*c - 63*a^3*b^2*d + 99*a^4*b*e - 143*a^5*f)*arctan(b*x/sqrt (a*b))/(sqrt(a*b)*b^7) - 1/8*(13*a^2*b^4*c*x^3 - 17*a^3*b^3*d*x^3 + 21*a^4 *b^2*e*x^3 - 25*a^5*b*f*x^3 + 11*a^3*b^3*c*x - 15*a^4*b^2*d*x + 19*a^5*b*e *x - 23*a^6*f*x)/((b*x^2 + a)^2*b^7) + 1/315*(35*b^24*f*x^9 + 45*b^24*e*x^ 7 - 135*a*b^23*f*x^7 + 63*b^24*d*x^5 - 189*a*b^23*e*x^5 + 378*a^2*b^22*f*x ^5 + 105*b^24*c*x^3 - 315*a*b^23*d*x^3 + 630*a^2*b^22*e*x^3 - 1050*a^3*b^2 1*f*x^3 - 945*a*b^23*c*x + 1890*a^2*b^22*d*x - 3150*a^3*b^21*e*x + 4725*a^ 4*b^20*f*x)/b^27
Time = 5.59 (sec) , antiderivative size = 506, normalized size of antiderivative = 1.76 \[ \int \frac {x^8 \left (c+d x^2+e x^4+f x^6\right )}{\left (a+b x^2\right )^3} \, dx=x^7\,\left (\frac {e}{7\,b^3}-\frac {3\,a\,f}{7\,b^4}\right )+x^3\,\left (\frac {c}{3\,b^3}-\frac {a^3\,f}{3\,b^6}-\frac {a^2\,\left (\frac {e}{b^3}-\frac {3\,a\,f}{b^4}\right )}{b^2}+\frac {a\,\left (\frac {3\,a^2\,f}{b^5}-\frac {d}{b^3}+\frac {3\,a\,\left (\frac {e}{b^3}-\frac {3\,a\,f}{b^4}\right )}{b}\right )}{b}\right )+\frac {x\,\left (\frac {23\,f\,a^6}{8}-\frac {19\,e\,a^5\,b}{8}+\frac {15\,d\,a^4\,b^2}{8}-\frac {11\,c\,a^3\,b^3}{8}\right )-x^3\,\left (-\frac {25\,f\,a^5\,b}{8}+\frac {21\,e\,a^4\,b^2}{8}-\frac {17\,d\,a^3\,b^3}{8}+\frac {13\,c\,a^2\,b^4}{8}\right )}{a^2\,b^7+2\,a\,b^8\,x^2+b^9\,x^4}-x\,\left (\frac {3\,a\,\left (\frac {c}{b^3}-\frac {a^3\,f}{b^6}-\frac {3\,a^2\,\left (\frac {e}{b^3}-\frac {3\,a\,f}{b^4}\right )}{b^2}+\frac {3\,a\,\left (\frac {3\,a^2\,f}{b^5}-\frac {d}{b^3}+\frac {3\,a\,\left (\frac {e}{b^3}-\frac {3\,a\,f}{b^4}\right )}{b}\right )}{b}\right )}{b}-\frac {3\,a^2\,\left (\frac {3\,a^2\,f}{b^5}-\frac {d}{b^3}+\frac {3\,a\,\left (\frac {e}{b^3}-\frac {3\,a\,f}{b^4}\right )}{b}\right )}{b^2}+\frac {a^3\,\left (\frac {e}{b^3}-\frac {3\,a\,f}{b^4}\right )}{b^3}\right )-x^5\,\left (\frac {3\,a^2\,f}{5\,b^5}-\frac {d}{5\,b^3}+\frac {3\,a\,\left (\frac {e}{b^3}-\frac {3\,a\,f}{b^4}\right )}{5\,b}\right )+\frac {f\,x^9}{9\,b^3}-\frac {a^{3/2}\,\mathrm {atan}\left (\frac {a^{3/2}\,\sqrt {b}\,x\,\left (-143\,f\,a^3+99\,e\,a^2\,b-63\,d\,a\,b^2+35\,c\,b^3\right )}{143\,f\,a^5-99\,e\,a^4\,b+63\,d\,a^3\,b^2-35\,c\,a^2\,b^3}\right )\,\left (-143\,f\,a^3+99\,e\,a^2\,b-63\,d\,a\,b^2+35\,c\,b^3\right )}{8\,b^{15/2}} \]
x^7*(e/(7*b^3) - (3*a*f)/(7*b^4)) + x^3*(c/(3*b^3) - (a^3*f)/(3*b^6) - (a^ 2*(e/b^3 - (3*a*f)/b^4))/b^2 + (a*((3*a^2*f)/b^5 - d/b^3 + (3*a*(e/b^3 - ( 3*a*f)/b^4))/b))/b) + (x*((23*a^6*f)/8 - (11*a^3*b^3*c)/8 + (15*a^4*b^2*d) /8 - (19*a^5*b*e)/8) - x^3*((13*a^2*b^4*c)/8 - (17*a^3*b^3*d)/8 + (21*a^4* b^2*e)/8 - (25*a^5*b*f)/8))/(a^2*b^7 + b^9*x^4 + 2*a*b^8*x^2) - x*((3*a*(c /b^3 - (a^3*f)/b^6 - (3*a^2*(e/b^3 - (3*a*f)/b^4))/b^2 + (3*a*((3*a^2*f)/b ^5 - d/b^3 + (3*a*(e/b^3 - (3*a*f)/b^4))/b))/b))/b - (3*a^2*((3*a^2*f)/b^5 - d/b^3 + (3*a*(e/b^3 - (3*a*f)/b^4))/b))/b^2 + (a^3*(e/b^3 - (3*a*f)/b^4 ))/b^3) - x^5*((3*a^2*f)/(5*b^5) - d/(5*b^3) + (3*a*(e/b^3 - (3*a*f)/b^4)) /(5*b)) + (f*x^9)/(9*b^3) - (a^(3/2)*atan((a^(3/2)*b^(1/2)*x*(35*b^3*c - 1 43*a^3*f - 63*a*b^2*d + 99*a^2*b*e))/(143*a^5*f - 35*a^2*b^3*c + 63*a^3*b^ 2*d - 99*a^4*b*e))*(35*b^3*c - 143*a^3*f - 63*a*b^2*d + 99*a^2*b*e))/(8*b^ (15/2))