Integrand size = 25, antiderivative size = 461 \[ \int \frac {x^6 \left (A+B x^2\right )}{\left (a+b x^2+c x^4\right )^3} \, dx=-\frac {\left (b^2 B-12 A b c+20 a B c\right ) x}{8 c \left (b^2-4 a c\right )^2}-\frac {x^5 \left (A b-2 a B-(b B-2 A c) x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {x^3 \left (5 A b^2-12 a b B+4 a A c-\left (b^2 B-12 A b c+20 a B c\right ) x^2\right )}{8 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {\left (b^3 B+3 A b^2 c-16 a b B c+12 a A c^2-\frac {b^4 B+3 A b^3 c-18 a b^2 B c+36 a A b c^2-40 a^2 B c^2}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{8 \sqrt {2} c^{3/2} \left (b^2-4 a c\right )^2 \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\left (b^3 B+3 A b^2 c-16 a b B c+12 a A c^2+\frac {b^4 B+3 A b^3 c-18 a b^2 B c+36 a A b c^2-40 a^2 B c^2}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{8 \sqrt {2} c^{3/2} \left (b^2-4 a c\right )^2 \sqrt {b+\sqrt {b^2-4 a c}}} \]
-1/8*(-12*A*b*c+20*B*a*c+B*b^2)*x/c/(-4*a*c+b^2)^2-1/4*x^5*(A*b-2*B*a-(-2* A*c+B*b)*x^2)/(-4*a*c+b^2)/(c*x^4+b*x^2+a)^2-1/8*x^3*(5*A*b^2-12*a*b*B+4*A *a*c-(-12*A*b*c+20*B*a*c+B*b^2)*x^2)/(-4*a*c+b^2)^2/(c*x^4+b*x^2+a)+1/16*a rctan(x*2^(1/2)*c^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2))*(B*b^3+3*A*b^2*c-16* B*a*b*c+12*A*a*c^2+(-36*A*a*b*c^2-3*A*b^3*c+40*B*a^2*c^2+18*B*a*b^2*c-B*b^ 4)/(-4*a*c+b^2)^(1/2))/c^(3/2)/(-4*a*c+b^2)^2*2^(1/2)/(b-(-4*a*c+b^2)^(1/2 ))^(1/2)+1/16*arctan(x*2^(1/2)*c^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2))*(B*b^ 3+3*A*b^2*c-16*B*a*b*c+12*A*a*c^2+(36*A*a*b*c^2+3*A*b^3*c-40*B*a^2*c^2-18* B*a*b^2*c+B*b^4)/(-4*a*c+b^2)^(1/2))/c^(3/2)/(-4*a*c+b^2)^2*2^(1/2)/(b+(-4 *a*c+b^2)^(1/2))^(1/2)
Time = 1.27 (sec) , antiderivative size = 543, normalized size of antiderivative = 1.18 \[ \int \frac {x^6 \left (A+B x^2\right )}{\left (a+b x^2+c x^4\right )^3} \, dx=\frac {\frac {2 x \left (-2 b^4 B+4 a b c^2 \left (A-4 B x^2\right )+b^3 c \left (2 A+B x^2\right )+12 a c^2 \left (-3 a B+A c x^2\right )+b^2 c \left (11 a B+3 A c x^2\right )\right )}{\left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac {4 x \left (2 a^2 B c+b^2 (-b B+A c) x^2+a \left (-b^2 B-2 A c^2 x^2+b c \left (A+3 B x^2\right )\right )\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {\sqrt {2} \sqrt {c} \left (-b^4 B+3 b^2 c \left (6 a B+A \sqrt {b^2-4 a c}\right )+4 a c^2 \left (10 a B+3 A \sqrt {b^2-4 a c}\right )+b^3 \left (-3 A c+B \sqrt {b^2-4 a c}\right )-4 a b c \left (9 A c+4 B \sqrt {b^2-4 a c}\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{5/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {2} \sqrt {c} \left (b^4 B+3 b^2 c \left (-6 a B+A \sqrt {b^2-4 a c}\right )+4 a c^2 \left (-10 a B+3 A \sqrt {b^2-4 a c}\right )+4 a b c \left (9 A c-4 B \sqrt {b^2-4 a c}\right )+b^3 \left (3 A c+B \sqrt {b^2-4 a c}\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{5/2} \sqrt {b+\sqrt {b^2-4 a c}}}}{16 c^2} \]
((2*x*(-2*b^4*B + 4*a*b*c^2*(A - 4*B*x^2) + b^3*c*(2*A + B*x^2) + 12*a*c^2 *(-3*a*B + A*c*x^2) + b^2*c*(11*a*B + 3*A*c*x^2)))/((b^2 - 4*a*c)^2*(a + b *x^2 + c*x^4)) - (4*x*(2*a^2*B*c + b^2*(-(b*B) + A*c)*x^2 + a*(-(b^2*B) - 2*A*c^2*x^2 + b*c*(A + 3*B*x^2))))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (Sqrt[2]*Sqrt[c]*(-(b^4*B) + 3*b^2*c*(6*a*B + A*Sqrt[b^2 - 4*a*c]) + 4*a* c^2*(10*a*B + 3*A*Sqrt[b^2 - 4*a*c]) + b^3*(-3*A*c + B*Sqrt[b^2 - 4*a*c]) - 4*a*b*c*(9*A*c + 4*B*Sqrt[b^2 - 4*a*c]))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt [b - Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]] ) + (Sqrt[2]*Sqrt[c]*(b^4*B + 3*b^2*c*(-6*a*B + A*Sqrt[b^2 - 4*a*c]) + 4*a *c^2*(-10*a*B + 3*A*Sqrt[b^2 - 4*a*c]) + 4*a*b*c*(9*A*c - 4*B*Sqrt[b^2 - 4 *a*c]) + b^3*(3*A*c + B*Sqrt[b^2 - 4*a*c]))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqr t[b + Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(5/2)*Sqrt[b + Sqrt[b^2 - 4*a*c] ]))/(16*c^2)
Time = 0.82 (sec) , antiderivative size = 458, normalized size of antiderivative = 0.99, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {1598, 1598, 1602, 25, 1480, 218}
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.
| \(\displaystyle \int \frac {x^6 \left (A+B x^2\right )}{\left (a+b x^2+c x^4\right )^3} \, dx\) |
| \(\Big \downarrow \) 1598 |
| \(\displaystyle \frac {\int \frac {x^4 \left ((b B-2 A c) x^2+5 (A b-2 a B)\right )}{\left (c x^4+b x^2+a\right )^2}dx}{4 \left (b^2-4 a c\right )}-\frac {x^5 \left (-2 a B-\left (x^2 (b B-2 A c)\right )+A b\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\) |
| \(\Big \downarrow \) 1598 |
| \(\displaystyle \frac {\frac {\int \frac {x^2 \left (3 \left (5 A b^2-12 a B b+4 a A c\right )-\left (B b^2-12 A c b+20 a B c\right ) x^2\right )}{c x^4+b x^2+a}dx}{2 \left (b^2-4 a c\right )}-\frac {x^3 \left (-x^2 \left (20 a B c-12 A b c+b^2 B\right )+4 a A c-12 a b B+5 A b^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{4 \left (b^2-4 a c\right )}-\frac {x^5 \left (-2 a B-\left (x^2 (b B-2 A c)\right )+A b\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\) |
| \(\Big \downarrow \) 1602 |
| \(\displaystyle \frac {\frac {-\frac {\int -\frac {\left (B b^3+3 A c b^2-16 a B c b+12 a A c^2\right ) x^2+a \left (B b^2-12 A c b+20 a B c\right )}{c x^4+b x^2+a}dx}{c}-\frac {x \left (20 a B c-12 A b c+b^2 B\right )}{c}}{2 \left (b^2-4 a c\right )}-\frac {x^3 \left (-x^2 \left (20 a B c-12 A b c+b^2 B\right )+4 a A c-12 a b B+5 A b^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{4 \left (b^2-4 a c\right )}-\frac {x^5 \left (-2 a B-\left (x^2 (b B-2 A c)\right )+A b\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {\left (B b^3+3 A c b^2-16 a B c b+12 a A c^2\right ) x^2+a \left (B b^2-12 A c b+20 a B c\right )}{c x^4+b x^2+a}dx}{c}-\frac {x \left (20 a B c-12 A b c+b^2 B\right )}{c}}{2 \left (b^2-4 a c\right )}-\frac {x^3 \left (-x^2 \left (20 a B c-12 A b c+b^2 B\right )+4 a A c-12 a b B+5 A b^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{4 \left (b^2-4 a c\right )}-\frac {x^5 \left (-2 a B-\left (x^2 (b B-2 A c)\right )+A b\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {\frac {\frac {\frac {1}{2} \left (-\frac {-40 a^2 B c^2+36 a A b c^2-18 a b^2 B c+3 A b^3 c+b^4 B}{\sqrt {b^2-4 a c}}+12 a A c^2-16 a b B c+3 A b^2 c+b^3 B\right ) \int \frac {1}{c x^2+\frac {1}{2} \left (b-\sqrt {b^2-4 a c}\right )}dx+\frac {1}{2} \left (\frac {-40 a^2 B c^2+36 a A b c^2-18 a b^2 B c+3 A b^3 c+b^4 B}{\sqrt {b^2-4 a c}}+12 a A c^2-16 a b B c+3 A b^2 c+b^3 B\right ) \int \frac {1}{c x^2+\frac {1}{2} \left (b+\sqrt {b^2-4 a c}\right )}dx}{c}-\frac {x \left (20 a B c-12 A b c+b^2 B\right )}{c}}{2 \left (b^2-4 a c\right )}-\frac {x^3 \left (-x^2 \left (20 a B c-12 A b c+b^2 B\right )+4 a A c-12 a b B+5 A b^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{4 \left (b^2-4 a c\right )}-\frac {x^5 \left (-2 a B-\left (x^2 (b B-2 A c)\right )+A b\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {\frac {\frac {\left (-\frac {-40 a^2 B c^2+36 a A b c^2-18 a b^2 B c+3 A b^3 c+b^4 B}{\sqrt {b^2-4 a c}}+12 a A c^2-16 a b B c+3 A b^2 c+b^3 B\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} \sqrt {c} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\left (\frac {-40 a^2 B c^2+36 a A b c^2-18 a b^2 B c+3 A b^3 c+b^4 B}{\sqrt {b^2-4 a c}}+12 a A c^2-16 a b B c+3 A b^2 c+b^3 B\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {2} \sqrt {c} \sqrt {\sqrt {b^2-4 a c}+b}}}{c}-\frac {x \left (20 a B c-12 A b c+b^2 B\right )}{c}}{2 \left (b^2-4 a c\right )}-\frac {x^3 \left (-x^2 \left (20 a B c-12 A b c+b^2 B\right )+4 a A c-12 a b B+5 A b^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{4 \left (b^2-4 a c\right )}-\frac {x^5 \left (-2 a B-\left (x^2 (b B-2 A c)\right )+A b\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\) |
-1/4*(x^5*(A*b - 2*a*B - (b*B - 2*A*c)*x^2))/((b^2 - 4*a*c)*(a + b*x^2 + c *x^4)^2) + (-1/2*(x^3*(5*A*b^2 - 12*a*b*B + 4*a*A*c - (b^2*B - 12*A*b*c + 20*a*B*c)*x^2))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (-(((b^2*B - 12*A*b* c + 20*a*B*c)*x)/c) + (((b^3*B + 3*A*b^2*c - 16*a*b*B*c + 12*a*A*c^2 - (b^ 4*B + 3*A*b^3*c - 18*a*b^2*B*c + 36*a*A*b*c^2 - 40*a^2*B*c^2)/Sqrt[b^2 - 4 *a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*S qrt[c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + ((b^3*B + 3*A*b^2*c - 16*a*b*B*c + 1 2*a*A*c^2 + (b^4*B + 3*A*b^3*c - 18*a*b^2*B*c + 36*a*A*b*c^2 - 40*a^2*B*c^ 2)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c ]]])/(Sqrt[2]*Sqrt[c]*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/c)/(2*(b^2 - 4*a*c)))/ (4*(b^2 - 4*a*c))
3.2.33.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) Int[1/( b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q)) Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*( x_)^4)^(p_.), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(a + b*x^2 + c*x^4)^(p + 1) *((b*d - 2*a*e - (b*e - 2*c*d)*x^2)/(2*(p + 1)*(b^2 - 4*a*c))), x] - Simp[f ^2/(2*(p + 1)*(b^2 - 4*a*c)) Int[(f*x)^(m - 2)*(a + b*x^2 + c*x^4)^(p + 1 )*Simp[(m - 1)*(b*d - 2*a*e) - (4*p + 4 + m + 1)*(b*e - 2*c*d)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && GtQ[m, 1] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])
Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*( x_)^4)^(p_), x_Symbol] :> Simp[e*f*(f*x)^(m - 1)*((a + b*x^2 + c*x^4)^(p + 1)/(c*(m + 4*p + 3))), x] - Simp[f^2/(c*(m + 4*p + 3)) Int[(f*x)^(m - 2)* (a + b*x^2 + c*x^4)^p*Simp[a*e*(m - 1) + (b*e*(m + 2*p + 1) - c*d*(m + 4*p + 3))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && NeQ[b^2 - 4*a*c , 0] && GtQ[m, 1] && NeQ[m + 4*p + 3, 0] && IntegerQ[2*p] && (IntegerQ[p] | | IntegerQ[m])
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.16 (sec) , antiderivative size = 375, normalized size of antiderivative = 0.81
| method | result | size |
| risch | \(\frac {\frac {\left (12 A a \,c^{2}+3 A \,b^{2} c -16 B a b c +B \,b^{3}\right ) x^{7}}{128 a^{2} c^{2}-64 a \,b^{2} c +8 b^{4}}+\frac {\left (16 A a b \,c^{2}+5 A \,b^{3} c -36 B \,a^{2} c^{2}-5 B a \,b^{2} c -B \,b^{4}\right ) x^{5}}{8 c \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {a \left (4 A a \,c^{2}-19 A \,b^{2} c +28 B a b c +2 B \,b^{3}\right ) x^{3}}{8 c \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {a^{2} \left (12 A b c -20 B a c -B \,b^{2}\right ) x}{8 c \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2}}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{4}+\textit {\_Z}^{2} b +a \right )}{\sum }\frac {\left (\frac {\left (12 A a \,c^{2}+3 A \,b^{2} c -16 B a b c +B \,b^{3}\right ) \textit {\_R}^{2}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}-\frac {a \left (12 A b c -20 B a c -B \,b^{2}\right )}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}\right ) \ln \left (x -\textit {\_R} \right )}{2 c \,\textit {\_R}^{3}+\textit {\_R} b}}{16 c}\) | \(375\) |
| default | \(\frac {\frac {\left (12 A a \,c^{2}+3 A \,b^{2} c -16 B a b c +B \,b^{3}\right ) x^{7}}{128 a^{2} c^{2}-64 a \,b^{2} c +8 b^{4}}+\frac {\left (16 A a b \,c^{2}+5 A \,b^{3} c -36 B \,a^{2} c^{2}-5 B a \,b^{2} c -B \,b^{4}\right ) x^{5}}{8 c \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {a \left (4 A a \,c^{2}-19 A \,b^{2} c +28 B a b c +2 B \,b^{3}\right ) x^{3}}{8 c \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {a^{2} \left (12 A b c -20 B a c -B \,b^{2}\right ) x}{8 c \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2}}+\frac {\frac {\left (12 A a \,c^{2} \sqrt {-4 a c +b^{2}}+3 A \,b^{2} c \sqrt {-4 a c +b^{2}}+36 A a b \,c^{2}+3 A \,b^{3} c -16 B a b c \sqrt {-4 a c +b^{2}}+B \,b^{3} \sqrt {-4 a c +b^{2}}-40 B \,a^{2} c^{2}-18 B a \,b^{2} c +B \,b^{4}\right ) \sqrt {2}\, \arctan \left (\frac {c x \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}-\frac {\left (12 A a \,c^{2} \sqrt {-4 a c +b^{2}}+3 A \,b^{2} c \sqrt {-4 a c +b^{2}}-36 A a b \,c^{2}-3 A \,b^{3} c -16 B a b c \sqrt {-4 a c +b^{2}}+B \,b^{3} \sqrt {-4 a c +b^{2}}+40 B \,a^{2} c^{2}+18 B a \,b^{2} c -B \,b^{4}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c x \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}}{32 a^{2} c^{2}-16 a \,b^{2} c +2 b^{4}}\) | \(591\) |
(1/8*(12*A*a*c^2+3*A*b^2*c-16*B*a*b*c+B*b^3)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^ 7+1/8*(16*A*a*b*c^2+5*A*b^3*c-36*B*a^2*c^2-5*B*a*b^2*c-B*b^4)/c/(16*a^2*c^ 2-8*a*b^2*c+b^4)*x^5-1/8/c*a*(4*A*a*c^2-19*A*b^2*c+28*B*a*b*c+2*B*b^3)/(16 *a^2*c^2-8*a*b^2*c+b^4)*x^3+1/8*a^2*(12*A*b*c-20*B*a*c-B*b^2)/c/(16*a^2*c^ 2-8*a*b^2*c+b^4)*x)/(c*x^4+b*x^2+a)^2+1/16/c*sum(((12*A*a*c^2+3*A*b^2*c-16 *B*a*b*c+B*b^3)/(16*a^2*c^2-8*a*b^2*c+b^4)*_R^2-a*(12*A*b*c-20*B*a*c-B*b^2 )/(16*a^2*c^2-8*a*b^2*c+b^4))/(2*_R^3*c+_R*b)*ln(x-_R),_R=RootOf(_Z^4*c+_Z ^2*b+a))
Leaf count of result is larger than twice the leaf count of optimal. 7060 vs. \(2 (414) = 828\).
Time = 6.01 (sec) , antiderivative size = 7060, normalized size of antiderivative = 15.31 \[ \int \frac {x^6 \left (A+B x^2\right )}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {x^6 \left (A+B x^2\right )}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {x^6 \left (A+B x^2\right )}{\left (a+b x^2+c x^4\right )^3} \, dx=\int { \frac {{\left (B x^{2} + A\right )} x^{6}}{{\left (c x^{4} + b x^{2} + a\right )}^{3}} \,d x } \]
1/8*((B*b^3*c + 12*A*a*c^3 - (16*B*a*b - 3*A*b^2)*c^2)*x^7 - (B*b^4 + 4*(9 *B*a^2 - 4*A*a*b)*c^2 + 5*(B*a*b^2 - A*b^3)*c)*x^5 - (2*B*a*b^3 + 4*A*a^2* c^2 + (28*B*a^2*b - 19*A*a*b^2)*c)*x^3 - (B*a^2*b^2 + 4*(5*B*a^3 - 3*A*a^2 *b)*c)*x)/((b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*x^8 + a^2*b^4*c - 8*a^3*b^ 2*c^2 + 16*a^4*c^3 + 2*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*x^6 + (b^6*c - 6*a*b^4*c^2 + 32*a^3*c^4)*x^4 + 2*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c ^3)*x^2) + 1/8*integrate((B*a*b^2 + (B*b^3 + 12*A*a*c^2 - (16*B*a*b - 3*A* b^2)*c)*x^2 + 4*(5*B*a^2 - 3*A*a*b)*c)/(c*x^4 + b*x^2 + a), x)/(b^4*c - 8* a*b^2*c^2 + 16*a^2*c^3)
Leaf count of result is larger than twice the leaf count of optimal. 7578 vs. \(2 (414) = 828\).
Time = 2.35 (sec) , antiderivative size = 7578, normalized size of antiderivative = 16.44 \[ \int \frac {x^6 \left (A+B x^2\right )}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Too large to display} \]
-1/64*(3*(2*b^4*c^3 - 32*a^2*c^5 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sq rt(b^2 - 4*a*c)*c)*b^4*c + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^3*c^2 + 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*c^3 + 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a* c)*c)*a*b*c^3 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)* b^2*c^3 - 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*c^ 4 - 2*(b^2 - 4*a*c)*b^2*c^3 - 8*(b^2 - 4*a*c)*a*c^4)*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)^2*A + (2*b^5*c^2 - 40*a*b^3*c^3 + 128*a^2*b*c^4 - sqrt(2)*sq rt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^5 + 20*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3*c + 2*sqrt(2)*sqrt(b^2 - 4* a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^4*c - 64*sqrt(2)*sqrt(b^2 - 4*a*c)* sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b*c^2 - 32*sqrt(2)*sqrt(b^2 - 4*a*c)*s qrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c^2 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt( b*c + sqrt(b^2 - 4*a*c)*c)*b^3*c^2 + 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^3 - 2*(b^2 - 4*a*c)*b^3*c^2 + 32*(b^2 - 4*a* c)*a*b*c^3)*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)^2*B + 24*(sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^7*c^3 - 12*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c) *c)*a^2*b^5*c^4 - 2*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^6*c^4 - 2* a*b^7*c^4 + 48*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^3*c^5 + 16*sq rt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^4*c^5 + sqrt(2)*sqrt(b*c + ...
Time = 9.74 (sec) , antiderivative size = 19041, normalized size of antiderivative = 41.30 \[ \int \frac {x^6 \left (A+B x^2\right )}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Too large to display} \]
atan(((((5242880*B*a^7*c^8 + 3072*A*a*b^11*c^3 - 3145728*A*a^6*b*c^8 - 256 *B*a*b^12*c^2 - 61440*A*a^2*b^9*c^4 + 491520*A*a^3*b^7*c^5 - 1966080*A*a^4 *b^5*c^6 + 3932160*A*a^5*b^3*c^7 + 61440*B*a^3*b^8*c^4 - 655360*B*a^4*b^6* c^5 + 2949120*B*a^5*b^4*c^6 - 6291456*B*a^6*b^2*c^7)/(512*(b^12*c + 4096*a ^6*c^7 - 24*a*b^10*c^2 + 240*a^2*b^8*c^3 - 1280*a^3*b^6*c^4 + 3840*a^4*b^4 *c^5 - 6144*a^5*b^2*c^6)) - (x*(-(B^2*b^17 + 9*A^2*b^15*c^2 + 9*A^2*c^2*(- (4*a*c - b^2)^15)^(1/2) + B^2*b^2*(-(4*a*c - b^2)^15)^(1/2) + 6*A*B*b^16*c - 5040*A^2*a^2*b^11*c^4 + 37440*A^2*a^3*b^9*c^5 - 103680*A^2*a^4*b^7*c^6 - 9216*A^2*a^5*b^5*c^7 + 552960*A^2*a^6*b^3*c^8 + 1140*B^2*a^2*b^13*c^2 - 10160*B^2*a^3*b^11*c^3 + 34880*B^2*a^4*b^9*c^4 + 43776*B^2*a^5*b^7*c^5 - 6 80960*B^2*a^6*b^5*c^6 + 1863680*B^2*a^7*b^3*c^7 + 983040*A*B*a^8*c^9 - 55* B^2*a*b^15*c - 25*B^2*a*c*(-(4*a*c - b^2)^15)^(1/2) + 180*A^2*a*b^13*c^3 - 737280*A^2*a^7*b*c^9 - 1720320*B^2*a^8*b*c^8 + 240*A*B*a^2*b^12*c^3 + 240 00*A*B*a^3*b^10*c^4 - 241920*A*B*a^4*b^8*c^5 + 992256*A*B*a^5*b^6*c^6 - 17 81760*A*B*a^6*b^4*c^7 + 737280*A*B*a^7*b^2*c^8 + 6*A*B*b*c*(-(4*a*c - b^2) ^15)^(1/2) - 180*A*B*a*b^14*c^2)/(512*(1048576*a^10*c^13 + b^20*c^3 - 40*a *b^18*c^4 + 720*a^2*b^16*c^5 - 7680*a^3*b^14*c^6 + 53760*a^4*b^12*c^7 - 25 8048*a^5*b^10*c^8 + 860160*a^6*b^8*c^9 - 1966080*a^7*b^6*c^10 + 2949120*a^ 8*b^4*c^11 - 2621440*a^9*b^2*c^12)))^(1/2)*(256*b^11*c^3 - 5120*a*b^9*c^4 - 262144*a^5*b*c^8 + 40960*a^2*b^7*c^5 - 163840*a^3*b^5*c^6 + 327680*a^...