Integrand size = 30, antiderivative size = 247 \[ \int \frac {x^6 \left (c+d x^2+e x^4+f x^6\right )}{\left (a+b x^2\right )^3} \, dx=\frac {\left (3 b^3 c-7 a b^2 d+13 a^2 b e-21 a^3 f\right ) x}{2 b^6}-\frac {\left (3 b^3 c-7 a b^2 d+15 a^2 b e-27 a^3 f\right ) x^3}{12 a b^5}+\frac {(b e-3 a f) x^5}{5 b^4}+\frac {f x^7}{7 b^3}+\frac {\left (c-\frac {a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^7}{4 a \left (a+b x^2\right )^2}+\frac {a \left (3 b^3 c-7 a b^2 d+11 a^2 b e-15 a^3 f\right ) x}{8 b^6 \left (a+b x^2\right )}-\frac {\sqrt {a} \left (15 b^3 c-35 a b^2 d+63 a^2 b e-99 a^3 f\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 b^{13/2}} \]
1/2*(-21*a^3*f+13*a^2*b*e-7*a*b^2*d+3*b^3*c)*x/b^6-1/12*(-27*a^3*f+15*a^2* b*e-7*a*b^2*d+3*b^3*c)*x^3/a/b^5+1/5*(-3*a*f+b*e)*x^5/b^4+1/7*f*x^7/b^3+1/ 4*(c-a*(a^2*f-a*b*e+b^2*d)/b^3)*x^7/a/(b*x^2+a)^2+1/8*a*(-15*a^3*f+11*a^2* b*e-7*a*b^2*d+3*b^3*c)*x/b^6/(b*x^2+a)-1/8*(-99*a^3*f+63*a^2*b*e-35*a*b^2* d+15*b^3*c)*arctan(x*b^(1/2)/a^(1/2))*a^(1/2)/b^(13/2)
Time = 0.09 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.94 \[ \int \frac {x^6 \left (c+d x^2+e x^4+f x^6\right )}{\left (a+b x^2\right )^3} \, dx=\frac {\left (b^3 c-3 a b^2 d+6 a^2 b e-10 a^3 f\right ) x}{b^6}+\frac {\left (b^2 d-3 a b e+6 a^2 f\right ) x^3}{3 b^5}+\frac {(b e-3 a f) x^5}{5 b^4}+\frac {f x^7}{7 b^3}+\frac {a^2 \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) x}{4 b^6 \left (a+b x^2\right )^2}+\frac {a \left (9 b^3 c-13 a b^2 d+17 a^2 b e-21 a^3 f\right ) x}{8 b^6 \left (a+b x^2\right )}+\frac {\sqrt {a} \left (-15 b^3 c+35 a b^2 d-63 a^2 b e+99 a^3 f\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 b^{13/2}} \]
((b^3*c - 3*a*b^2*d + 6*a^2*b*e - 10*a^3*f)*x)/b^6 + ((b^2*d - 3*a*b*e + 6 *a^2*f)*x^3)/(3*b^5) + ((b*e - 3*a*f)*x^5)/(5*b^4) + (f*x^7)/(7*b^3) + (a^ 2*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x)/(4*b^6*(a + b*x^2)^2) + (a*(9* b^3*c - 13*a*b^2*d + 17*a^2*b*e - 21*a^3*f)*x)/(8*b^6*(a + b*x^2)) + (Sqrt [a]*(-15*b^3*c + 35*a*b^2*d - 63*a^2*b*e + 99*a^3*f)*ArcTan[(Sqrt[b]*x)/Sq rt[a]])/(8*b^(13/2))
Time = 0.78 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2335, 9, 1580, 25, 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.
| \(\displaystyle \int \frac {x^6 \left (c+d x^2+e x^4+f x^6\right )}{\left (a+b x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 2335 |
| \(\displaystyle \frac {x^7 \left (c-\frac {a \left (a^2 f-a b e+b^2 d\right )}{b^3}\right )}{4 a \left (a+b x^2\right )^2}-\frac {\int \frac {x^5 \left (-4 a f x^5-4 a \left (e-\frac {a f}{b}\right ) x^3+\left (-\frac {7 f a^3}{b^2}+\frac {7 e a^2}{b}-7 d a+3 b c\right ) x\right )}{\left (b x^2+a\right )^2}dx}{4 a b}\) |
| \(\Big \downarrow \) 9 |
| \(\displaystyle \frac {x^7 \left (c-\frac {a \left (a^2 f-a b e+b^2 d\right )}{b^3}\right )}{4 a \left (a+b x^2\right )^2}-\frac {\int \frac {x^6 \left (-4 a f x^4-4 a \left (e-\frac {a f}{b}\right ) x^2+3 b c-7 a d+\frac {7 a^2 e}{b}-\frac {7 a^3 f}{b^2}\right )}{\left (b x^2+a\right )^2}dx}{4 a b}\) |
| \(\Big \downarrow \) 1580 |
| \(\displaystyle \frac {x^7 \left (c-\frac {a \left (a^2 f-a b e+b^2 d\right )}{b^3}\right )}{4 a \left (a+b x^2\right )^2}-\frac {-\frac {\int -\frac {-8 a b^4 f x^8-8 a b^3 (b e-2 a f) x^6+2 b^2 \left (-15 f a^3+11 b e a^2-7 b^2 d a+3 b^3 c\right ) x^4-2 a b \left (-15 f a^3+11 b e a^2-7 b^2 d a+3 b^3 c\right ) x^2+a^2 \left (-15 f a^3+11 b e a^2-7 b^2 d a+3 b^3 c\right )}{b x^2+a}dx}{2 b^5}-\frac {a^2 x \left (-15 a^3 f+11 a^2 b e-7 a b^2 d+3 b^3 c\right )}{2 b^5 \left (a+b x^2\right )}}{4 a b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {x^7 \left (c-\frac {a \left (a^2 f-a b e+b^2 d\right )}{b^3}\right )}{4 a \left (a+b x^2\right )^2}-\frac {\frac {\int \frac {-8 a b^4 f x^8-8 a b^3 (b e-2 a f) x^6+2 b^2 \left (-15 f a^3+11 b e a^2-7 b^2 d a+3 b^3 c\right ) x^4-2 a b \left (-15 f a^3+11 b e a^2-7 b^2 d a+3 b^3 c\right ) x^2+a^2 \left (-15 f a^3+11 b e a^2-7 b^2 d a+3 b^3 c\right )}{b x^2+a}dx}{2 b^5}-\frac {a^2 x \left (-15 a^3 f+11 a^2 b e-7 a b^2 d+3 b^3 c\right )}{2 b^5 \left (a+b x^2\right )}}{4 a b}\) |
| \(\Big \downarrow \) 2341 |
| \(\displaystyle \frac {x^7 \left (c-\frac {a \left (a^2 f-a b e+b^2 d\right )}{b^3}\right )}{4 a \left (a+b x^2\right )^2}-\frac {\frac {\int \left (-8 a b^3 f x^6-8 a b^2 (b e-3 a f) x^4+2 b \left (-27 f a^3+15 b e a^2-7 b^2 d a+3 b^3 c\right ) x^2-4 a \left (-21 f a^3+13 b e a^2-7 b^2 d a+3 b^3 c\right )+\frac {-99 f a^5+63 b e a^4-35 b^2 d a^3+15 b^3 c a^2}{b x^2+a}\right )dx}{2 b^5}-\frac {a^2 x \left (-15 a^3 f+11 a^2 b e-7 a b^2 d+3 b^3 c\right )}{2 b^5 \left (a+b x^2\right )}}{4 a b}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {x^7 \left (c-\frac {a \left (a^2 f-a b e+b^2 d\right )}{b^3}\right )}{4 a \left (a+b x^2\right )^2}-\frac {\frac {\frac {2}{3} b x^3 \left (-27 a^3 f+15 a^2 b e-7 a b^2 d+3 b^3 c\right )-4 a x \left (-21 a^3 f+13 a^2 b e-7 a b^2 d+3 b^3 c\right )+\frac {a^{3/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (-99 a^3 f+63 a^2 b e-35 a b^2 d+15 b^3 c\right )}{\sqrt {b}}-\frac {8}{7} a b^3 f x^7-\frac {8}{5} a b^2 x^5 (b e-3 a f)}{2 b^5}-\frac {a^2 x \left (-15 a^3 f+11 a^2 b e-7 a b^2 d+3 b^3 c\right )}{2 b^5 \left (a+b x^2\right )}}{4 a b}\) |
((c - (a*(b^2*d - a*b*e + a^2*f))/b^3)*x^7)/(4*a*(a + b*x^2)^2) - (-1/2*(a ^2*(3*b^3*c - 7*a*b^2*d + 11*a^2*b*e - 15*a^3*f)*x)/(b^5*(a + b*x^2)) + (- 4*a*(3*b^3*c - 7*a*b^2*d + 13*a^2*b*e - 21*a^3*f)*x + (2*b*(3*b^3*c - 7*a* b^2*d + 15*a^2*b*e - 27*a^3*f)*x^3)/3 - (8*a*b^2*(b*e - 3*a*f)*x^5)/5 - (8 *a*b^3*f*x^7)/7 + (a^(3/2)*(15*b^3*c - 35*a*b^2*d + 63*a^2*b*e - 99*a^3*f) *ArcTan[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[b])/(2*b^5))/(4*a*b)
3.2.34.3.1 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 = 3.52 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.89
| method | result | size |
| default | \(-\frac {-\frac {1}{7} f \,x^{7} b^{3}+\frac {3}{5} a \,b^{2} f \,x^{5}-\frac {1}{5} b^{3} e \,x^{5}-2 a^{2} b f \,x^{3}+a \,b^{2} e \,x^{3}-\frac {1}{3} b^{3} d \,x^{3}+10 f \,a^{3} x -6 a^{2} b e x +3 a \,b^{2} d x -b^{3} c x}{b^{6}}+\frac {a \left (\frac {\left (-\frac {21}{8} a^{3} b f +\frac {17}{8} a^{2} e \,b^{2}-\frac {13}{8} a \,b^{3} d +\frac {9}{8} b^{4} c \right ) x^{3}-\frac {a \left (19 f \,a^{3}-15 a^{2} b e +11 a \,b^{2} d -7 b^{3} c \right ) x}{8}}{\left (b \,x^{2}+a \right )^{2}}+\frac {\left (99 f \,a^{3}-63 a^{2} b e +35 a \,b^{2} d -15 b^{3} c \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{b^{6}}\) | \(219\) |
| risch | \(\frac {f \,x^{7}}{7 b^{3}}-\frac {3 a f \,x^{5}}{5 b^{4}}+\frac {e \,x^{5}}{5 b^{3}}+\frac {2 a^{2} f \,x^{3}}{b^{5}}-\frac {a e \,x^{3}}{b^{4}}+\frac {d \,x^{3}}{3 b^{3}}-\frac {10 f \,a^{3} x}{b^{6}}+\frac {6 a^{2} e x}{b^{5}}-\frac {3 a d x}{b^{4}}+\frac {c x}{b^{3}}+\frac {\left (-\frac {21}{8} a^{4} b f +\frac {17}{8} a^{3} b^{2} e -\frac {13}{8} a^{2} b^{3} d +\frac {9}{8} a \,b^{4} c \right ) x^{3}-\frac {a^{2} \left (19 f \,a^{3}-15 a^{2} b e +11 a \,b^{2} d -7 b^{3} c \right ) x}{8}}{b^{6} \left (b \,x^{2}+a \right )^{2}}+\frac {99 \sqrt {-a b}\, \ln \left (-\sqrt {-a b}\, x +a \right ) f \,a^{3}}{16 b^{7}}-\frac {63 \sqrt {-a b}\, \ln \left (-\sqrt {-a b}\, x +a \right ) a^{2} e}{16 b^{6}}+\frac {35 \sqrt {-a b}\, \ln \left (-\sqrt {-a b}\, x +a \right ) a d}{16 b^{5}}-\frac {15 \sqrt {-a b}\, \ln \left (-\sqrt {-a b}\, x +a \right ) c}{16 b^{4}}-\frac {99 \sqrt {-a b}\, \ln \left (\sqrt {-a b}\, x +a \right ) f \,a^{3}}{16 b^{7}}+\frac {63 \sqrt {-a b}\, \ln \left (\sqrt {-a b}\, x +a \right ) a^{2} e}{16 b^{6}}-\frac {35 \sqrt {-a b}\, \ln \left (\sqrt {-a b}\, x +a \right ) a d}{16 b^{5}}+\frac {15 \sqrt {-a b}\, \ln \left (\sqrt {-a b}\, x +a \right ) c}{16 b^{4}}\) | \(381\) |
-1/b^6*(-1/7*f*x^7*b^3+3/5*a*b^2*f*x^5-1/5*b^3*e*x^5-2*a^2*b*f*x^3+a*b^2*e *x^3-1/3*b^3*d*x^3+10*f*a^3*x-6*a^2*b*e*x+3*a*b^2*d*x-b^3*c*x)+a/b^6*(((-2 1/8*a^3*b*f+17/8*a^2*e*b^2-13/8*a*b^3*d+9/8*b^4*c)*x^3-1/8*a*(19*a^3*f-15* a^2*b*e+11*a*b^2*d-7*b^3*c)*x)/(b*x^2+a)^2+1/8*(99*a^3*f-63*a^2*b*e+35*a*b ^2*d-15*b^3*c)/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2)))
Time = 0.29 (sec) , antiderivative size = 668, normalized size of antiderivative = 2.70 \[ \int \frac {x^6 \left (c+d x^2+e x^4+f x^6\right )}{\left (a+b x^2\right )^3} \, dx=\left [\frac {240 \, b^{5} f x^{11} + 48 \, {\left (7 \, b^{5} e - 11 \, a b^{4} f\right )} x^{9} + 16 \, {\left (35 \, b^{5} d - 63 \, a b^{4} e + 99 \, a^{2} b^{3} f\right )} x^{7} + 112 \, {\left (15 \, b^{5} c - 35 \, a b^{4} d + 63 \, a^{2} b^{3} e - 99 \, a^{3} b^{2} f\right )} x^{5} + 350 \, {\left (15 \, a b^{4} c - 35 \, a^{2} b^{3} d + 63 \, a^{3} b^{2} e - 99 \, a^{4} b f\right )} x^{3} - 105 \, {\left (15 \, a^{2} b^{3} c - 35 \, a^{3} b^{2} d + 63 \, a^{4} b e - 99 \, a^{5} f + {\left (15 \, b^{5} c - 35 \, a b^{4} d + 63 \, a^{2} b^{3} e - 99 \, a^{3} b^{2} f\right )} x^{4} + 2 \, {\left (15 \, a b^{4} c - 35 \, a^{2} b^{3} d + 63 \, a^{3} b^{2} e - 99 \, 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 ) + 210 \, {\left (15 \, a^{2} b^{3} c - 35 \, a^{3} b^{2} d + 63 \, a^{4} b e - 99 \, a^{5} f\right )} x}{1680 \, {\left (b^{8} x^{4} + 2 \, a b^{7} x^{2} + a^{2} b^{6}\right )}}, \frac {120 \, b^{5} f x^{11} + 24 \, {\left (7 \, b^{5} e - 11 \, a b^{4} f\right )} x^{9} + 8 \, {\left (35 \, b^{5} d - 63 \, a b^{4} e + 99 \, a^{2} b^{3} f\right )} x^{7} + 56 \, {\left (15 \, b^{5} c - 35 \, a b^{4} d + 63 \, a^{2} b^{3} e - 99 \, a^{3} b^{2} f\right )} x^{5} + 175 \, {\left (15 \, a b^{4} c - 35 \, a^{2} b^{3} d + 63 \, a^{3} b^{2} e - 99 \, a^{4} b f\right )} x^{3} - 105 \, {\left (15 \, a^{2} b^{3} c - 35 \, a^{3} b^{2} d + 63 \, a^{4} b e - 99 \, a^{5} f + {\left (15 \, b^{5} c - 35 \, a b^{4} d + 63 \, a^{2} b^{3} e - 99 \, a^{3} b^{2} f\right )} x^{4} + 2 \, {\left (15 \, a b^{4} c - 35 \, a^{2} b^{3} d + 63 \, a^{3} b^{2} e - 99 \, a^{4} b f\right )} x^{2}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) + 105 \, {\left (15 \, a^{2} b^{3} c - 35 \, a^{3} b^{2} d + 63 \, a^{4} b e - 99 \, a^{5} f\right )} x}{840 \, {\left (b^{8} x^{4} + 2 \, a b^{7} x^{2} + a^{2} b^{6}\right )}}\right ] \]
[1/1680*(240*b^5*f*x^11 + 48*(7*b^5*e - 11*a*b^4*f)*x^9 + 16*(35*b^5*d - 6 3*a*b^4*e + 99*a^2*b^3*f)*x^7 + 112*(15*b^5*c - 35*a*b^4*d + 63*a^2*b^3*e - 99*a^3*b^2*f)*x^5 + 350*(15*a*b^4*c - 35*a^2*b^3*d + 63*a^3*b^2*e - 99*a ^4*b*f)*x^3 - 105*(15*a^2*b^3*c - 35*a^3*b^2*d + 63*a^4*b*e - 99*a^5*f + ( 15*b^5*c - 35*a*b^4*d + 63*a^2*b^3*e - 99*a^3*b^2*f)*x^4 + 2*(15*a*b^4*c - 35*a^2*b^3*d + 63*a^3*b^2*e - 99*a^4*b*f)*x^2)*sqrt(-a/b)*log((b*x^2 + 2* b*x*sqrt(-a/b) - a)/(b*x^2 + a)) + 210*(15*a^2*b^3*c - 35*a^3*b^2*d + 63*a ^4*b*e - 99*a^5*f)*x)/(b^8*x^4 + 2*a*b^7*x^2 + a^2*b^6), 1/840*(120*b^5*f* x^11 + 24*(7*b^5*e - 11*a*b^4*f)*x^9 + 8*(35*b^5*d - 63*a*b^4*e + 99*a^2*b ^3*f)*x^7 + 56*(15*b^5*c - 35*a*b^4*d + 63*a^2*b^3*e - 99*a^3*b^2*f)*x^5 + 175*(15*a*b^4*c - 35*a^2*b^3*d + 63*a^3*b^2*e - 99*a^4*b*f)*x^3 - 105*(15 *a^2*b^3*c - 35*a^3*b^2*d + 63*a^4*b*e - 99*a^5*f + (15*b^5*c - 35*a*b^4*d + 63*a^2*b^3*e - 99*a^3*b^2*f)*x^4 + 2*(15*a*b^4*c - 35*a^2*b^3*d + 63*a^ 3*b^2*e - 99*a^4*b*f)*x^2)*sqrt(a/b)*arctan(b*x*sqrt(a/b)/a) + 105*(15*a^2 *b^3*c - 35*a^3*b^2*d + 63*a^4*b*e - 99*a^5*f)*x)/(b^8*x^4 + 2*a*b^7*x^2 + a^2*b^6)]
Time = 17.92 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.28 \[ \int \frac {x^6 \left (c+d x^2+e x^4+f x^6\right )}{\left (a+b x^2\right )^3} \, dx=x^{5} \left (- \frac {3 a f}{5 b^{4}} + \frac {e}{5 b^{3}}\right ) + x^{3} \cdot \left (\frac {2 a^{2} f}{b^{5}} - \frac {a e}{b^{4}} + \frac {d}{3 b^{3}}\right ) + x \left (- \frac {10 a^{3} f}{b^{6}} + \frac {6 a^{2} e}{b^{5}} - \frac {3 a d}{b^{4}} + \frac {c}{b^{3}}\right ) - \frac {\sqrt {- \frac {a}{b^{13}}} \cdot \left (99 a^{3} f - 63 a^{2} b e + 35 a b^{2} d - 15 b^{3} c\right ) \log {\left (- b^{6} \sqrt {- \frac {a}{b^{13}}} + x \right )}}{16} + \frac {\sqrt {- \frac {a}{b^{13}}} \cdot \left (99 a^{3} f - 63 a^{2} b e + 35 a b^{2} d - 15 b^{3} c\right ) \log {\left (b^{6} \sqrt {- \frac {a}{b^{13}}} + x \right )}}{16} + \frac {x^{3} \left (- 21 a^{4} b f + 17 a^{3} b^{2} e - 13 a^{2} b^{3} d + 9 a b^{4} c\right ) + x \left (- 19 a^{5} f + 15 a^{4} b e - 11 a^{3} b^{2} d + 7 a^{2} b^{3} c\right )}{8 a^{2} b^{6} + 16 a b^{7} x^{2} + 8 b^{8} x^{4}} + \frac {f x^{7}}{7 b^{3}} \]
x**5*(-3*a*f/(5*b**4) + e/(5*b**3)) + x**3*(2*a**2*f/b**5 - a*e/b**4 + d/( 3*b**3)) + x*(-10*a**3*f/b**6 + 6*a**2*e/b**5 - 3*a*d/b**4 + c/b**3) - sqr t(-a/b**13)*(99*a**3*f - 63*a**2*b*e + 35*a*b**2*d - 15*b**3*c)*log(-b**6* sqrt(-a/b**13) + x)/16 + sqrt(-a/b**13)*(99*a**3*f - 63*a**2*b*e + 35*a*b* *2*d - 15*b**3*c)*log(b**6*sqrt(-a/b**13) + x)/16 + (x**3*(-21*a**4*b*f + 17*a**3*b**2*e - 13*a**2*b**3*d + 9*a*b**4*c) + x*(-19*a**5*f + 15*a**4*b* e - 11*a**3*b**2*d + 7*a**2*b**3*c))/(8*a**2*b**6 + 16*a*b**7*x**2 + 8*b** 8*x**4) + f*x**7/(7*b**3)
Time = 0.30 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.96 \[ \int \frac {x^6 \left (c+d x^2+e x^4+f x^6\right )}{\left (a+b x^2\right )^3} \, dx=\frac {{\left (9 \, a b^{4} c - 13 \, a^{2} b^{3} d + 17 \, a^{3} b^{2} e - 21 \, a^{4} b f\right )} x^{3} + {\left (7 \, a^{2} b^{3} c - 11 \, a^{3} b^{2} d + 15 \, a^{4} b e - 19 \, a^{5} f\right )} x}{8 \, {\left (b^{8} x^{4} + 2 \, a b^{7} x^{2} + a^{2} b^{6}\right )}} - \frac {{\left (15 \, a b^{3} c - 35 \, a^{2} b^{2} d + 63 \, a^{3} b e - 99 \, a^{4} f\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} b^{6}} + \frac {15 \, b^{3} f x^{7} + 21 \, {\left (b^{3} e - 3 \, a b^{2} f\right )} x^{5} + 35 \, {\left (b^{3} d - 3 \, a b^{2} e + 6 \, a^{2} b f\right )} x^{3} + 105 \, {\left (b^{3} c - 3 \, a b^{2} d + 6 \, a^{2} b e - 10 \, a^{3} f\right )} x}{105 \, b^{6}} \]
1/8*((9*a*b^4*c - 13*a^2*b^3*d + 17*a^3*b^2*e - 21*a^4*b*f)*x^3 + (7*a^2*b ^3*c - 11*a^3*b^2*d + 15*a^4*b*e - 19*a^5*f)*x)/(b^8*x^4 + 2*a*b^7*x^2 + a ^2*b^6) - 1/8*(15*a*b^3*c - 35*a^2*b^2*d + 63*a^3*b*e - 99*a^4*f)*arctan(b *x/sqrt(a*b))/(sqrt(a*b)*b^6) + 1/105*(15*b^3*f*x^7 + 21*(b^3*e - 3*a*b^2* f)*x^5 + 35*(b^3*d - 3*a*b^2*e + 6*a^2*b*f)*x^3 + 105*(b^3*c - 3*a*b^2*d + 6*a^2*b*e - 10*a^3*f)*x)/b^6
Time = 0.28 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.99 \[ \int \frac {x^6 \left (c+d x^2+e x^4+f x^6\right )}{\left (a+b x^2\right )^3} \, dx=-\frac {{\left (15 \, a b^{3} c - 35 \, a^{2} b^{2} d + 63 \, a^{3} b e - 99 \, a^{4} f\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} b^{6}} + \frac {9 \, a b^{4} c x^{3} - 13 \, a^{2} b^{3} d x^{3} + 17 \, a^{3} b^{2} e x^{3} - 21 \, a^{4} b f x^{3} + 7 \, a^{2} b^{3} c x - 11 \, a^{3} b^{2} d x + 15 \, a^{4} b e x - 19 \, a^{5} f x}{8 \, {\left (b x^{2} + a\right )}^{2} b^{6}} + \frac {15 \, b^{18} f x^{7} + 21 \, b^{18} e x^{5} - 63 \, a b^{17} f x^{5} + 35 \, b^{18} d x^{3} - 105 \, a b^{17} e x^{3} + 210 \, a^{2} b^{16} f x^{3} + 105 \, b^{18} c x - 315 \, a b^{17} d x + 630 \, a^{2} b^{16} e x - 1050 \, a^{3} b^{15} f x}{105 \, b^{21}} \]
-1/8*(15*a*b^3*c - 35*a^2*b^2*d + 63*a^3*b*e - 99*a^4*f)*arctan(b*x/sqrt(a *b))/(sqrt(a*b)*b^6) + 1/8*(9*a*b^4*c*x^3 - 13*a^2*b^3*d*x^3 + 17*a^3*b^2* e*x^3 - 21*a^4*b*f*x^3 + 7*a^2*b^3*c*x - 11*a^3*b^2*d*x + 15*a^4*b*e*x - 1 9*a^5*f*x)/((b*x^2 + a)^2*b^6) + 1/105*(15*b^18*f*x^7 + 21*b^18*e*x^5 - 63 *a*b^17*f*x^5 + 35*b^18*d*x^3 - 105*a*b^17*e*x^3 + 210*a^2*b^16*f*x^3 + 10 5*b^18*c*x - 315*a*b^17*d*x + 630*a^2*b^16*e*x - 1050*a^3*b^15*f*x)/b^21
Time = 0.12 (sec) , antiderivative size = 348, normalized size of antiderivative = 1.41 \[ \int \frac {x^6 \left (c+d x^2+e x^4+f x^6\right )}{\left (a+b x^2\right )^3} \, dx=x^5\,\left (\frac {e}{5\,b^3}-\frac {3\,a\,f}{5\,b^4}\right )+x\,\left (\frac {c}{b^3}-\frac {a^3\,f}{b^6}-\frac {3\,a^2\,\left (\frac {e}{b^3}-\frac {3\,a\,f}{b^4}\right )}{b^2}+\frac {3\,a\,\left (\frac {3\,a^2\,f}{b^5}-\frac {d}{b^3}+\frac {3\,a\,\left (\frac {e}{b^3}-\frac {3\,a\,f}{b^4}\right )}{b}\right )}{b}\right )-x^3\,\left (\frac {a^2\,f}{b^5}-\frac {d}{3\,b^3}+\frac {a\,\left (\frac {e}{b^3}-\frac {3\,a\,f}{b^4}\right )}{b}\right )-\frac {\left (\frac {21\,f\,a^4\,b}{8}-\frac {17\,e\,a^3\,b^2}{8}+\frac {13\,d\,a^2\,b^3}{8}-\frac {9\,c\,a\,b^4}{8}\right )\,x^3+\left (\frac {19\,f\,a^5}{8}-\frac {15\,e\,a^4\,b}{8}+\frac {11\,d\,a^3\,b^2}{8}-\frac {7\,c\,a^2\,b^3}{8}\right )\,x}{a^2\,b^6+2\,a\,b^7\,x^2+b^8\,x^4}+\frac {f\,x^7}{7\,b^3}+\frac {\sqrt {a}\,\mathrm {atan}\left (\frac {\sqrt {a}\,\sqrt {b}\,x\,\left (-99\,f\,a^3+63\,e\,a^2\,b-35\,d\,a\,b^2+15\,c\,b^3\right )}{99\,f\,a^4-63\,e\,a^3\,b+35\,d\,a^2\,b^2-15\,c\,a\,b^3}\right )\,\left (-99\,f\,a^3+63\,e\,a^2\,b-35\,d\,a\,b^2+15\,c\,b^3\right )}{8\,b^{13/2}} \]
x^5*(e/(5*b^3) - (3*a*f)/(5*b^4)) + x*(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) - x^3*((a^2*f)/b^5 - d/(3*b^3) + (a*(e/b^3 - (3*a*f)/b^4))/b ) - (x*((19*a^5*f)/8 - (7*a^2*b^3*c)/8 + (11*a^3*b^2*d)/8 - (15*a^4*b*e)/8 ) + x^3*((13*a^2*b^3*d)/8 - (17*a^3*b^2*e)/8 - (9*a*b^4*c)/8 + (21*a^4*b*f )/8))/(a^2*b^6 + b^8*x^4 + 2*a*b^7*x^2) + (f*x^7)/(7*b^3) + (a^(1/2)*atan( (a^(1/2)*b^(1/2)*x*(15*b^3*c - 99*a^3*f - 35*a*b^2*d + 63*a^2*b*e))/(99*a^ 4*f + 35*a^2*b^2*d - 15*a*b^3*c - 63*a^3*b*e))*(15*b^3*c - 99*a^3*f - 35*a *b^2*d + 63*a^2*b*e))/(8*b^(13/2))