Integrand size = 30, antiderivative size = 240 \[ \int \frac {x^6 \left (c+d x^2+e x^4+f x^6\right )}{\left (a+b x^2\right )^2} \, dx=-\frac {a \left (5 b^3 c-7 a b^2 d+9 a^2 b e-11 a^3 f\right ) x}{2 b^6}+\frac {\left (5 b^3 c-7 a b^2 d+9 a^2 b e-11 a^3 f\right ) x^3}{6 b^5}-\frac {\left (5 b^3 c-7 a b^2 d+9 a^2 b e-11 a^3 f\right ) x^5}{10 a b^4}+\frac {(b e-2 a f) x^7}{7 b^3}+\frac {f x^9}{9 b^2}+\frac {\left (c-\frac {a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^7}{2 a \left (a+b x^2\right )}+\frac {a^{3/2} \left (5 b^3 c-7 a b^2 d+9 a^2 b e-11 a^3 f\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{13/2}} \]
-1/2*a*(-11*a^3*f+9*a^2*b*e-7*a*b^2*d+5*b^3*c)*x/b^6+1/6*(-11*a^3*f+9*a^2* b*e-7*a*b^2*d+5*b^3*c)*x^3/b^5-1/10*(-11*a^3*f+9*a^2*b*e-7*a*b^2*d+5*b^3*c )*x^5/a/b^4+1/7*(-2*a*f+b*e)*x^7/b^3+1/9*f*x^9/b^2+1/2*(c-a*(a^2*f-a*b*e+b ^2*d)/b^3)*x^7/a/(b*x^2+a)+1/2*a^(3/2)*(-11*a^3*f+9*a^2*b*e-7*a*b^2*d+5*b^ 3*c)*arctan(x*b^(1/2)/a^(1/2))/b^(13/2)
Time = 0.08 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.95 \[ \int \frac {x^6 \left (c+d x^2+e x^4+f x^6\right )}{\left (a+b x^2\right )^2} \, dx=\frac {a \left (-2 b^3 c+3 a b^2 d-4 a^2 b e+5 a^3 f\right ) x}{b^6}+\frac {\left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x^3}{3 b^5}+\frac {\left (b^2 d-2 a b e+3 a^2 f\right ) x^5}{5 b^4}+\frac {(b e-2 a f) x^7}{7 b^3}+\frac {f x^9}{9 b^2}-\frac {\left (a^2 b^3 c-a^3 b^2 d+a^4 b e-a^5 f\right ) x}{2 b^6 \left (a+b x^2\right )}-\frac {a^{3/2} \left (-5 b^3 c+7 a b^2 d-9 a^2 b e+11 a^3 f\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{13/2}} \]
(a*(-2*b^3*c + 3*a*b^2*d - 4*a^2*b*e + 5*a^3*f)*x)/b^6 + ((b^3*c - 2*a*b^2 *d + 3*a^2*b*e - 4*a^3*f)*x^3)/(3*b^5) + ((b^2*d - 2*a*b*e + 3*a^2*f)*x^5) /(5*b^4) + ((b*e - 2*a*f)*x^7)/(7*b^3) + (f*x^9)/(9*b^2) - ((a^2*b^3*c - a ^3*b^2*d + a^4*b*e - a^5*f)*x)/(2*b^6*(a + b*x^2)) - (a^(3/2)*(-5*b^3*c + 7*a*b^2*d - 9*a^2*b*e + 11*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(13/2) )
Time = 0.53 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.03, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2335, 9, 1584, 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.
| \(\displaystyle \int \frac {x^6 \left (c+d x^2+e x^4+f x^6\right )}{\left (a+b x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 2335 |
| \(\displaystyle \frac {x^7 \left (c-\frac {a \left (a^2 f-a b e+b^2 d\right )}{b^3}\right )}{2 a \left (a+b x^2\right )}-\frac {\int \frac {x^5 \left (-2 a f x^5-2 a \left (e-\frac {a f}{b}\right ) x^3+\left (-\frac {7 f a^3}{b^2}+\frac {7 e a^2}{b}-7 d a+5 b c\right ) x\right )}{b x^2+a}dx}{2 a b}\) |
| \(\Big \downarrow \) 9 |
| \(\displaystyle \frac {x^7 \left (c-\frac {a \left (a^2 f-a b e+b^2 d\right )}{b^3}\right )}{2 a \left (a+b x^2\right )}-\frac {\int \frac {x^6 \left (-2 a f x^4-2 a \left (e-\frac {a f}{b}\right ) x^2+5 b c-7 a d+\frac {7 a^2 e}{b}-\frac {7 a^3 f}{b^2}\right )}{b x^2+a}dx}{2 a b}\) |
| \(\Big \downarrow \) 1584 |
| \(\displaystyle \frac {x^7 \left (c-\frac {a \left (a^2 f-a b e+b^2 d\right )}{b^3}\right )}{2 a \left (a+b x^2\right )}-\frac {\int \left (-\frac {2 a f x^8}{b}-\frac {2 a (b e-2 a f) x^6}{b^2}+\frac {\left (-11 f a^3+9 b e a^2-7 b^2 d a+5 b^3 c\right ) x^4}{b^3}-\frac {a \left (-11 f a^3+9 b e a^2-7 b^2 d a+5 b^3 c\right ) x^2}{b^4}+\frac {a^2 \left (-11 f a^3+9 b e a^2-7 b^2 d a+5 b^3 c\right )}{b^5}+\frac {11 f a^6-9 b e a^5+7 b^2 d a^4-5 b^3 c a^3}{b^5 \left (b x^2+a\right )}\right )dx}{2 a b}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {x^7 \left (c-\frac {a \left (a^2 f-a b e+b^2 d\right )}{b^3}\right )}{2 a \left (a+b x^2\right )}-\frac {\frac {x^5 \left (-11 a^3 f+9 a^2 b e-7 a b^2 d+5 b^3 c\right )}{5 b^3}+\frac {a^2 x \left (-11 a^3 f+9 a^2 b e-7 a b^2 d+5 b^3 c\right )}{b^5}-\frac {a x^3 \left (-11 a^3 f+9 a^2 b e-7 a b^2 d+5 b^3 c\right )}{3 b^4}-\frac {a^{5/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (-11 a^3 f+9 a^2 b e-7 a b^2 d+5 b^3 c\right )}{b^{11/2}}-\frac {2 a x^7 (b e-2 a f)}{7 b^2}-\frac {2 a f x^9}{9 b}}{2 a b}\) |
((c - (a*(b^2*d - a*b*e + a^2*f))/b^3)*x^7)/(2*a*(a + b*x^2)) - ((a^2*(5*b ^3*c - 7*a*b^2*d + 9*a^2*b*e - 11*a^3*f)*x)/b^5 - (a*(5*b^3*c - 7*a*b^2*d + 9*a^2*b*e - 11*a^3*f)*x^3)/(3*b^4) + ((5*b^3*c - 7*a*b^2*d + 9*a^2*b*e - 11*a^3*f)*x^5)/(5*b^3) - (2*a*(b*e - 2*a*f)*x^7)/(7*b^2) - (2*a*f*x^9)/(9 *b) - (a^(5/2)*(5*b^3*c - 7*a*b^2*d + 9*a^2*b*e - 11*a^3*f)*ArcTan[(Sqrt[b ]*x)/Sqrt[a]])/b^(11/2))/(2*a*b)
3.2.24.3.1 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[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + ( c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q* (a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[ b^2 - 4*a*c, 0] && IGtQ[p, 0] && IGtQ[q, -2]
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]
Time = 3.59 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.95
| method | result | size |
| default | \(\frac {\frac {1}{9} f \,x^{9} b^{4}-\frac {2}{7} a \,b^{3} f \,x^{7}+\frac {1}{7} b^{4} e \,x^{7}+\frac {3}{5} a^{2} b^{2} f \,x^{5}-\frac {2}{5} a \,b^{3} e \,x^{5}+\frac {1}{5} b^{4} d \,x^{5}-\frac {4}{3} a^{3} b f \,x^{3}+a^{2} b^{2} e \,x^{3}-\frac {2}{3} a \,b^{3} d \,x^{3}+\frac {1}{3} b^{4} c \,x^{3}+5 a^{4} f x -4 a^{3} b e x +3 a^{2} b^{2} d x -2 a \,b^{3} c x}{b^{6}}-\frac {a^{2} \left (\frac {\left (-\frac {1}{2} f \,a^{3}+\frac {1}{2} a^{2} b e -\frac {1}{2} a \,b^{2} d +\frac {1}{2} b^{3} c \right ) x}{b \,x^{2}+a}+\frac {\left (11 f \,a^{3}-9 a^{2} b e +7 a \,b^{2} d -5 b^{3} c \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{b^{6}}\) | \(229\) |
| risch | \(\frac {f \,x^{9}}{9 b^{2}}-\frac {2 a f \,x^{7}}{7 b^{3}}+\frac {e \,x^{7}}{7 b^{2}}+\frac {3 a^{2} f \,x^{5}}{5 b^{4}}-\frac {2 a e \,x^{5}}{5 b^{3}}+\frac {d \,x^{5}}{5 b^{2}}-\frac {4 a^{3} f \,x^{3}}{3 b^{5}}+\frac {a^{2} e \,x^{3}}{b^{4}}-\frac {2 a d \,x^{3}}{3 b^{3}}+\frac {c \,x^{3}}{3 b^{2}}+\frac {5 a^{4} f x}{b^{6}}-\frac {4 a^{3} e x}{b^{5}}+\frac {3 a^{2} d x}{b^{4}}-\frac {2 a c x}{b^{3}}+\frac {\left (\frac {1}{2} f \,a^{5}-\frac {1}{2} a^{4} e b +\frac {1}{2} a^{3} d \,b^{2}-\frac {1}{2} a^{2} c \,b^{3}\right ) x}{b^{6} \left (b \,x^{2}+a \right )}+\frac {11 \sqrt {-a b}\, a^{4} \ln \left (-\sqrt {-a b}\, x -a \right ) f}{4 b^{7}}-\frac {9 \sqrt {-a b}\, a^{3} \ln \left (-\sqrt {-a b}\, x -a \right ) e}{4 b^{6}}+\frac {7 \sqrt {-a b}\, a^{2} \ln \left (-\sqrt {-a b}\, x -a \right ) d}{4 b^{5}}-\frac {5 \sqrt {-a b}\, a \ln \left (-\sqrt {-a b}\, x -a \right ) c}{4 b^{4}}-\frac {11 \sqrt {-a b}\, a^{4} \ln \left (\sqrt {-a b}\, x -a \right ) f}{4 b^{7}}+\frac {9 \sqrt {-a b}\, a^{3} \ln \left (\sqrt {-a b}\, x -a \right ) e}{4 b^{6}}-\frac {7 \sqrt {-a b}\, a^{2} \ln \left (\sqrt {-a b}\, x -a \right ) d}{4 b^{5}}+\frac {5 \sqrt {-a b}\, a \ln \left (\sqrt {-a b}\, x -a \right ) c}{4 b^{4}}\) | \(411\) |
1/b^6*(1/9*f*x^9*b^4-2/7*a*b^3*f*x^7+1/7*b^4*e*x^7+3/5*a^2*b^2*f*x^5-2/5*a *b^3*e*x^5+1/5*b^4*d*x^5-4/3*a^3*b*f*x^3+a^2*b^2*e*x^3-2/3*a*b^3*d*x^3+1/3 *b^4*c*x^3+5*a^4*f*x-4*a^3*b*e*x+3*a^2*b^2*d*x-2*a*b^3*c*x)-a^2/b^6*((-1/2 *f*a^3+1/2*a^2*b*e-1/2*a*b^2*d+1/2*b^3*c)*x/(b*x^2+a)+1/2*(11*a^3*f-9*a^2* b*e+7*a*b^2*d-5*b^3*c)/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2)))
Time = 0.30 (sec) , antiderivative size = 572, normalized size of antiderivative = 2.38 \[ \int \frac {x^6 \left (c+d x^2+e x^4+f x^6\right )}{\left (a+b x^2\right )^2} \, dx=\left [\frac {140 \, b^{5} f x^{11} + 20 \, {\left (9 \, b^{5} e - 11 \, a b^{4} f\right )} x^{9} + 36 \, {\left (7 \, b^{5} d - 9 \, a b^{4} e + 11 \, a^{2} b^{3} f\right )} x^{7} + 84 \, {\left (5 \, b^{5} c - 7 \, a b^{4} d + 9 \, a^{2} b^{3} e - 11 \, a^{3} b^{2} f\right )} x^{5} - 420 \, {\left (5 \, a b^{4} c - 7 \, a^{2} b^{3} d + 9 \, a^{3} b^{2} e - 11 \, a^{4} b f\right )} x^{3} - 315 \, {\left (5 \, a^{2} b^{3} c - 7 \, a^{3} b^{2} d + 9 \, a^{4} b e - 11 \, a^{5} f + {\left (5 \, a b^{4} c - 7 \, a^{2} b^{3} d + 9 \, a^{3} b^{2} e - 11 \, a^{4} b f\right )} x^{2}\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} - 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right ) - 630 \, {\left (5 \, a^{2} b^{3} c - 7 \, a^{3} b^{2} d + 9 \, a^{4} b e - 11 \, a^{5} f\right )} x}{1260 \, {\left (b^{7} x^{2} + a b^{6}\right )}}, \frac {70 \, b^{5} f x^{11} + 10 \, {\left (9 \, b^{5} e - 11 \, a b^{4} f\right )} x^{9} + 18 \, {\left (7 \, b^{5} d - 9 \, a b^{4} e + 11 \, a^{2} b^{3} f\right )} x^{7} + 42 \, {\left (5 \, b^{5} c - 7 \, a b^{4} d + 9 \, a^{2} b^{3} e - 11 \, a^{3} b^{2} f\right )} x^{5} - 210 \, {\left (5 \, a b^{4} c - 7 \, a^{2} b^{3} d + 9 \, a^{3} b^{2} e - 11 \, a^{4} b f\right )} x^{3} + 315 \, {\left (5 \, a^{2} b^{3} c - 7 \, a^{3} b^{2} d + 9 \, a^{4} b e - 11 \, a^{5} f + {\left (5 \, a b^{4} c - 7 \, a^{2} b^{3} d + 9 \, a^{3} b^{2} e - 11 \, a^{4} b f\right )} x^{2}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) - 315 \, {\left (5 \, a^{2} b^{3} c - 7 \, a^{3} b^{2} d + 9 \, a^{4} b e - 11 \, a^{5} f\right )} x}{630 \, {\left (b^{7} x^{2} + a b^{6}\right )}}\right ] \]
[1/1260*(140*b^5*f*x^11 + 20*(9*b^5*e - 11*a*b^4*f)*x^9 + 36*(7*b^5*d - 9* a*b^4*e + 11*a^2*b^3*f)*x^7 + 84*(5*b^5*c - 7*a*b^4*d + 9*a^2*b^3*e - 11*a ^3*b^2*f)*x^5 - 420*(5*a*b^4*c - 7*a^2*b^3*d + 9*a^3*b^2*e - 11*a^4*b*f)*x ^3 - 315*(5*a^2*b^3*c - 7*a^3*b^2*d + 9*a^4*b*e - 11*a^5*f + (5*a*b^4*c - 7*a^2*b^3*d + 9*a^3*b^2*e - 11*a^4*b*f)*x^2)*sqrt(-a/b)*log((b*x^2 - 2*b*x *sqrt(-a/b) - a)/(b*x^2 + a)) - 630*(5*a^2*b^3*c - 7*a^3*b^2*d + 9*a^4*b*e - 11*a^5*f)*x)/(b^7*x^2 + a*b^6), 1/630*(70*b^5*f*x^11 + 10*(9*b^5*e - 11 *a*b^4*f)*x^9 + 18*(7*b^5*d - 9*a*b^4*e + 11*a^2*b^3*f)*x^7 + 42*(5*b^5*c - 7*a*b^4*d + 9*a^2*b^3*e - 11*a^3*b^2*f)*x^5 - 210*(5*a*b^4*c - 7*a^2*b^3 *d + 9*a^3*b^2*e - 11*a^4*b*f)*x^3 + 315*(5*a^2*b^3*c - 7*a^3*b^2*d + 9*a^ 4*b*e - 11*a^5*f + (5*a*b^4*c - 7*a^2*b^3*d + 9*a^3*b^2*e - 11*a^4*b*f)*x^ 2)*sqrt(a/b)*arctan(b*x*sqrt(a/b)/a) - 315*(5*a^2*b^3*c - 7*a^3*b^2*d + 9* a^4*b*e - 11*a^5*f)*x)/(b^7*x^2 + a*b^6)]
Time = 1.16 (sec) , antiderivative size = 444, normalized size of antiderivative = 1.85 \[ \int \frac {x^6 \left (c+d x^2+e x^4+f x^6\right )}{\left (a+b x^2\right )^2} \, dx=x^{7} \left (- \frac {2 a f}{7 b^{3}} + \frac {e}{7 b^{2}}\right ) + x^{5} \cdot \left (\frac {3 a^{2} f}{5 b^{4}} - \frac {2 a e}{5 b^{3}} + \frac {d}{5 b^{2}}\right ) + x^{3} \left (- \frac {4 a^{3} f}{3 b^{5}} + \frac {a^{2} e}{b^{4}} - \frac {2 a d}{3 b^{3}} + \frac {c}{3 b^{2}}\right ) + x \left (\frac {5 a^{4} f}{b^{6}} - \frac {4 a^{3} e}{b^{5}} + \frac {3 a^{2} d}{b^{4}} - \frac {2 a c}{b^{3}}\right ) + \frac {x \left (a^{5} f - a^{4} b e + a^{3} b^{2} d - a^{2} b^{3} c\right )}{2 a b^{6} + 2 b^{7} x^{2}} + \frac {\sqrt {- \frac {a^{3}}{b^{13}}} \cdot \left (11 a^{3} f - 9 a^{2} b e + 7 a b^{2} d - 5 b^{3} c\right ) \log {\left (- \frac {b^{6} \sqrt {- \frac {a^{3}}{b^{13}}} \cdot \left (11 a^{3} f - 9 a^{2} b e + 7 a b^{2} d - 5 b^{3} c\right )}{11 a^{4} f - 9 a^{3} b e + 7 a^{2} b^{2} d - 5 a b^{3} c} + x \right )}}{4} - \frac {\sqrt {- \frac {a^{3}}{b^{13}}} \cdot \left (11 a^{3} f - 9 a^{2} b e + 7 a b^{2} d - 5 b^{3} c\right ) \log {\left (\frac {b^{6} \sqrt {- \frac {a^{3}}{b^{13}}} \cdot \left (11 a^{3} f - 9 a^{2} b e + 7 a b^{2} d - 5 b^{3} c\right )}{11 a^{4} f - 9 a^{3} b e + 7 a^{2} b^{2} d - 5 a b^{3} c} + x \right )}}{4} + \frac {f x^{9}}{9 b^{2}} \]
x**7*(-2*a*f/(7*b**3) + e/(7*b**2)) + x**5*(3*a**2*f/(5*b**4) - 2*a*e/(5*b **3) + d/(5*b**2)) + x**3*(-4*a**3*f/(3*b**5) + a**2*e/b**4 - 2*a*d/(3*b** 3) + c/(3*b**2)) + x*(5*a**4*f/b**6 - 4*a**3*e/b**5 + 3*a**2*d/b**4 - 2*a* c/b**3) + x*(a**5*f - a**4*b*e + a**3*b**2*d - a**2*b**3*c)/(2*a*b**6 + 2* b**7*x**2) + sqrt(-a**3/b**13)*(11*a**3*f - 9*a**2*b*e + 7*a*b**2*d - 5*b* *3*c)*log(-b**6*sqrt(-a**3/b**13)*(11*a**3*f - 9*a**2*b*e + 7*a*b**2*d - 5 *b**3*c)/(11*a**4*f - 9*a**3*b*e + 7*a**2*b**2*d - 5*a*b**3*c) + x)/4 - sq rt(-a**3/b**13)*(11*a**3*f - 9*a**2*b*e + 7*a*b**2*d - 5*b**3*c)*log(b**6* sqrt(-a**3/b**13)*(11*a**3*f - 9*a**2*b*e + 7*a*b**2*d - 5*b**3*c)/(11*a** 4*f - 9*a**3*b*e + 7*a**2*b**2*d - 5*a*b**3*c) + x)/4 + f*x**9/(9*b**2)
Time = 0.27 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.95 \[ \int \frac {x^6 \left (c+d x^2+e x^4+f x^6\right )}{\left (a+b x^2\right )^2} \, dx=-\frac {{\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e - a^{5} f\right )} x}{2 \, {\left (b^{7} x^{2} + a b^{6}\right )}} + \frac {{\left (5 \, a^{2} b^{3} c - 7 \, a^{3} b^{2} d + 9 \, a^{4} b e - 11 \, a^{5} f\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b^{6}} + \frac {35 \, b^{4} f x^{9} + 45 \, {\left (b^{4} e - 2 \, a b^{3} f\right )} x^{7} + 63 \, {\left (b^{4} d - 2 \, a b^{3} e + 3 \, a^{2} b^{2} f\right )} x^{5} + 105 \, {\left (b^{4} c - 2 \, a b^{3} d + 3 \, a^{2} b^{2} e - 4 \, a^{3} b f\right )} x^{3} - 315 \, {\left (2 \, a b^{3} c - 3 \, a^{2} b^{2} d + 4 \, a^{3} b e - 5 \, a^{4} f\right )} x}{315 \, b^{6}} \]
-1/2*(a^2*b^3*c - a^3*b^2*d + a^4*b*e - a^5*f)*x/(b^7*x^2 + a*b^6) + 1/2*( 5*a^2*b^3*c - 7*a^3*b^2*d + 9*a^4*b*e - 11*a^5*f)*arctan(b*x/sqrt(a*b))/(s qrt(a*b)*b^6) + 1/315*(35*b^4*f*x^9 + 45*(b^4*e - 2*a*b^3*f)*x^7 + 63*(b^4 *d - 2*a*b^3*e + 3*a^2*b^2*f)*x^5 + 105*(b^4*c - 2*a*b^3*d + 3*a^2*b^2*e - 4*a^3*b*f)*x^3 - 315*(2*a*b^3*c - 3*a^2*b^2*d + 4*a^3*b*e - 5*a^4*f)*x)/b ^6
Time = 0.31 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.02 \[ \int \frac {x^6 \left (c+d x^2+e x^4+f x^6\right )}{\left (a+b x^2\right )^2} \, dx=\frac {{\left (5 \, a^{2} b^{3} c - 7 \, a^{3} b^{2} d + 9 \, a^{4} b e - 11 \, a^{5} f\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b^{6}} - \frac {a^{2} b^{3} c x - a^{3} b^{2} d x + a^{4} b e x - a^{5} f x}{2 \, {\left (b x^{2} + a\right )} b^{6}} + \frac {35 \, b^{16} f x^{9} + 45 \, b^{16} e x^{7} - 90 \, a b^{15} f x^{7} + 63 \, b^{16} d x^{5} - 126 \, a b^{15} e x^{5} + 189 \, a^{2} b^{14} f x^{5} + 105 \, b^{16} c x^{3} - 210 \, a b^{15} d x^{3} + 315 \, a^{2} b^{14} e x^{3} - 420 \, a^{3} b^{13} f x^{3} - 630 \, a b^{15} c x + 945 \, a^{2} b^{14} d x - 1260 \, a^{3} b^{13} e x + 1575 \, a^{4} b^{12} f x}{315 \, b^{18}} \]
1/2*(5*a^2*b^3*c - 7*a^3*b^2*d + 9*a^4*b*e - 11*a^5*f)*arctan(b*x/sqrt(a*b ))/(sqrt(a*b)*b^6) - 1/2*(a^2*b^3*c*x - a^3*b^2*d*x + a^4*b*e*x - a^5*f*x) /((b*x^2 + a)*b^6) + 1/315*(35*b^16*f*x^9 + 45*b^16*e*x^7 - 90*a*b^15*f*x^ 7 + 63*b^16*d*x^5 - 126*a*b^15*e*x^5 + 189*a^2*b^14*f*x^5 + 105*b^16*c*x^3 - 210*a*b^15*d*x^3 + 315*a^2*b^14*e*x^3 - 420*a^3*b^13*f*x^3 - 630*a*b^15 *c*x + 945*a^2*b^14*d*x - 1260*a^3*b^13*e*x + 1575*a^4*b^12*f*x)/b^18
Time = 0.11 (sec) , antiderivative size = 413, normalized size of antiderivative = 1.72 \[ \int \frac {x^6 \left (c+d x^2+e x^4+f x^6\right )}{\left (a+b x^2\right )^2} \, dx=x^7\,\left (\frac {e}{7\,b^2}-\frac {2\,a\,f}{7\,b^3}\right )-x\,\left (\frac {2\,a\,\left (\frac {c}{b^2}-\frac {a^2\,\left (\frac {e}{b^2}-\frac {2\,a\,f}{b^3}\right )}{b^2}+\frac {2\,a\,\left (\frac {a^2\,f}{b^4}-\frac {d}{b^2}+\frac {2\,a\,\left (\frac {e}{b^2}-\frac {2\,a\,f}{b^3}\right )}{b}\right )}{b}\right )}{b}-\frac {a^2\,\left (\frac {a^2\,f}{b^4}-\frac {d}{b^2}+\frac {2\,a\,\left (\frac {e}{b^2}-\frac {2\,a\,f}{b^3}\right )}{b}\right )}{b^2}\right )-x^5\,\left (\frac {a^2\,f}{5\,b^4}-\frac {d}{5\,b^2}+\frac {2\,a\,\left (\frac {e}{b^2}-\frac {2\,a\,f}{b^3}\right )}{5\,b}\right )+x^3\,\left (\frac {c}{3\,b^2}-\frac {a^2\,\left (\frac {e}{b^2}-\frac {2\,a\,f}{b^3}\right )}{3\,b^2}+\frac {2\,a\,\left (\frac {a^2\,f}{b^4}-\frac {d}{b^2}+\frac {2\,a\,\left (\frac {e}{b^2}-\frac {2\,a\,f}{b^3}\right )}{b}\right )}{3\,b}\right )+\frac {f\,x^9}{9\,b^2}+\frac {x\,\left (\frac {f\,a^5}{2}-\frac {e\,a^4\,b}{2}+\frac {d\,a^3\,b^2}{2}-\frac {c\,a^2\,b^3}{2}\right )}{b^7\,x^2+a\,b^6}-\frac {a^{3/2}\,\mathrm {atan}\left (\frac {a^{3/2}\,\sqrt {b}\,x\,\left (-11\,f\,a^3+9\,e\,a^2\,b-7\,d\,a\,b^2+5\,c\,b^3\right )}{11\,f\,a^5-9\,e\,a^4\,b+7\,d\,a^3\,b^2-5\,c\,a^2\,b^3}\right )\,\left (-11\,f\,a^3+9\,e\,a^2\,b-7\,d\,a\,b^2+5\,c\,b^3\right )}{2\,b^{13/2}} \]
x^7*(e/(7*b^2) - (2*a*f)/(7*b^3)) - x*((2*a*(c/b^2 - (a^2*(e/b^2 - (2*a*f) /b^3))/b^2 + (2*a*((a^2*f)/b^4 - d/b^2 + (2*a*(e/b^2 - (2*a*f)/b^3))/b))/b ))/b - (a^2*((a^2*f)/b^4 - d/b^2 + (2*a*(e/b^2 - (2*a*f)/b^3))/b))/b^2) - x^5*((a^2*f)/(5*b^4) - d/(5*b^2) + (2*a*(e/b^2 - (2*a*f)/b^3))/(5*b)) + x^ 3*(c/(3*b^2) - (a^2*(e/b^2 - (2*a*f)/b^3))/(3*b^2) + (2*a*((a^2*f)/b^4 - d /b^2 + (2*a*(e/b^2 - (2*a*f)/b^3))/b))/(3*b)) + (f*x^9)/(9*b^2) + (x*((a^5 *f)/2 - (a^2*b^3*c)/2 + (a^3*b^2*d)/2 - (a^4*b*e)/2))/(a*b^6 + b^7*x^2) - (a^(3/2)*atan((a^(3/2)*b^(1/2)*x*(5*b^3*c - 11*a^3*f - 7*a*b^2*d + 9*a^2*b *e))/(11*a^5*f - 5*a^2*b^3*c + 7*a^3*b^2*d - 9*a^4*b*e))*(5*b^3*c - 11*a^3 *f - 7*a*b^2*d + 9*a^2*b*e))/(2*b^(13/2))