Integrand size = 25, antiderivative size = 554 \[ \int \frac {x^8 \left (A+B x^2\right )}{\left (a+b x^2+c x^4\right )^3} \, dx=-\frac {\left (3 b^3 B+A b^2 c-24 a b B c+20 a A c^2\right ) x}{8 c^2 \left (b^2-4 a c\right )^2}+\frac {\left (b^2 B+12 A b c-28 a B c\right ) x^3}{8 c \left (b^2-4 a c\right )^2}-\frac {x^7 \left (A b-2 a B-(b B-2 A c) x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {x^5 \left (7 A b^2-12 a b B-4 a A c+\left (b^2 B+12 A b c-28 a B c\right ) x^2\right )}{8 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {\left (3 b^4 B+A b^3 c-27 a b^2 B c-16 a A b c^2+84 a^2 B c^2-\frac {3 b^5 B+A b^4 c-33 a b^3 B c-18 a A b^2 c^2+132 a^2 b B c^2-40 a^2 A c^3}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{8 \sqrt {2} c^{5/2} \left (b^2-4 a c\right )^2 \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\left (3 b^4 B+A b^3 c-27 a b^2 B c-16 a A b c^2+84 a^2 B c^2+\frac {3 b^5 B+A b^4 c-33 a b^3 B c-18 a A b^2 c^2+132 a^2 b B c^2-40 a^2 A c^3}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{8 \sqrt {2} c^{5/2} \left (b^2-4 a c\right )^2 \sqrt {b+\sqrt {b^2-4 a c}}} \]
-1/8*(20*A*a*c^2+A*b^2*c-24*B*a*b*c+3*B*b^3)*x/c^2/(-4*a*c+b^2)^2+1/8*(12* A*b*c-28*B*a*c+B*b^2)*x^3/c/(-4*a*c+b^2)^2-1/4*x^7*(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^5*(7*A*b^2-12*a*b*B-4*A*a*c+(12 *A*b*c-28*B*a*c+B*b^2)*x^2)/(-4*a*c+b^2)^2/(c*x^4+b*x^2+a)+1/16*arctan(x*2 ^(1/2)*c^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2))*(3*B*b^4+A*b^3*c-27*B*a*b^2*c -16*A*a*b*c^2+84*B*a^2*c^2+(40*A*a^2*c^3+18*A*a*b^2*c^2-A*b^4*c-132*B*a^2* b*c^2+33*B*a*b^3*c-3*B*b^5)/(-4*a*c+b^2)^(1/2))/c^(5/2)/(-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))*(3*B*b^4+A*b^3*c-27*B*a*b^2*c-16*A*a*b*c^2+84*B*a^2*c^ 2+(-40*A*a^2*c^3-18*A*a*b^2*c^2+A*b^4*c+132*B*a^2*b*c^2-33*B*a*b^3*c+3*B*b ^5)/(-4*a*c+b^2)^(1/2))/c^(5/2)/(-4*a*c+b^2)^2*2^(1/2)/(b+(-4*a*c+b^2)^(1/ 2))^(1/2)
Time = 1.52 (sec) , antiderivative size = 644, normalized size of antiderivative = 1.16 \[ \int \frac {x^8 \left (A+B x^2\right )}{\left (a+b x^2+c x^4\right )^3} \, dx=\frac {\frac {2 x \left (2 b^5 B-b^4 c \left (2 A+5 B x^2\right )-4 a^2 c^3 \left (9 A+11 B x^2\right )+a b^2 c^2 \left (11 A+37 B x^2\right )+16 a b c^2 \left (3 a B-A c x^2\right )+b^3 c \left (-17 a B+A c x^2\right )\right )}{\left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac {4 x \left (b^3 (b B-A c) x^2+a^2 c \left (-3 b B+2 c \left (A+B x^2\right )\right )+a b \left (b^2 B+3 A c^2 x^2-b c \left (A+4 B x^2\right )\right )\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {\sqrt {2} \sqrt {c} \left (-3 b^5 B+b^3 c \left (33 a B+A \sqrt {b^2-4 a c}\right )-4 a b c^2 \left (33 a B+4 A \sqrt {b^2-4 a c}\right )+9 a b^2 c \left (2 A c-3 B \sqrt {b^2-4 a c}\right )+b^4 \left (-A c+3 B \sqrt {b^2-4 a c}\right )+4 a^2 c^2 \left (10 A c+21 B \sqrt {b^2-4 a c}\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{5/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {2} \sqrt {c} \left (3 b^5 B+4 a b c^2 \left (33 a B-4 A \sqrt {b^2-4 a c}\right )+b^4 \left (A c+3 B \sqrt {b^2-4 a c}\right )-9 a b^2 c \left (2 A c+3 B \sqrt {b^2-4 a c}\right )+4 a^2 c^2 \left (-10 A c+21 B \sqrt {b^2-4 a c}\right )+b^3 \left (-33 a B c+A c \sqrt {b^2-4 a c}\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{5/2} \sqrt {b+\sqrt {b^2-4 a c}}}}{16 c^3} \]
((2*x*(2*b^5*B - b^4*c*(2*A + 5*B*x^2) - 4*a^2*c^3*(9*A + 11*B*x^2) + a*b^ 2*c^2*(11*A + 37*B*x^2) + 16*a*b*c^2*(3*a*B - A*c*x^2) + b^3*c*(-17*a*B + A*c*x^2)))/((b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) - (4*x*(b^3*(b*B - A*c)*x ^2 + a^2*c*(-3*b*B + 2*c*(A + B*x^2)) + a*b*(b^2*B + 3*A*c^2*x^2 - b*c*(A + 4*B*x^2))))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (Sqrt[2]*Sqrt[c]*(-3 *b^5*B + b^3*c*(33*a*B + A*Sqrt[b^2 - 4*a*c]) - 4*a*b*c^2*(33*a*B + 4*A*Sq rt[b^2 - 4*a*c]) + 9*a*b^2*c*(2*A*c - 3*B*Sqrt[b^2 - 4*a*c]) + b^4*(-(A*c) + 3*B*Sqrt[b^2 - 4*a*c]) + 4*a^2*c^2*(10*A*c + 21*B*Sqrt[b^2 - 4*a*c]))*A rcTan[(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]*(3*b^5*B + 4*a*b*c^2*(3 3*a*B - 4*A*Sqrt[b^2 - 4*a*c]) + b^4*(A*c + 3*B*Sqrt[b^2 - 4*a*c]) - 9*a*b ^2*c*(2*A*c + 3*B*Sqrt[b^2 - 4*a*c]) + 4*a^2*c^2*(-10*A*c + 21*B*Sqrt[b^2 - 4*a*c]) + b^3*(-33*a*B*c + A*c*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]]))/(16*c^3)
Time = 1.05 (sec) , antiderivative size = 544, normalized size of antiderivative = 0.98, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {1598, 1598, 1602, 27, 1602, 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.
| \(\displaystyle \int \frac {x^8 \left (A+B x^2\right )}{\left (a+b x^2+c x^4\right )^3} \, dx\) |
| \(\Big \downarrow \) 1598 |
| \(\displaystyle \frac {\int \frac {x^6 \left (7 (A b-2 a B)-(b B-2 A c) x^2\right )}{\left (c x^4+b x^2+a\right )^2}dx}{4 \left (b^2-4 a c\right )}-\frac {x^7 \left (-2 a B-\left (x^2 (b B-2 A c)\right )+A b\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\) |
| \(\Big \downarrow \) 1598 |
| \(\displaystyle \frac {\frac {\int \frac {x^4 \left (3 \left (B b^2+12 A c b-28 a B c\right ) x^2+5 \left (7 A b^2-12 a B b-4 a A c\right )\right )}{c x^4+b x^2+a}dx}{2 \left (b^2-4 a c\right )}-\frac {x^5 \left (x^2 \left (-28 a B c+12 A b c+b^2 B\right )-4 a A c-12 a b B+7 A b^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{4 \left (b^2-4 a c\right )}-\frac {x^7 \left (-2 a B-\left (x^2 (b B-2 A c)\right )+A b\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\) |
| \(\Big \downarrow \) 1602 |
| \(\displaystyle \frac {\frac {\frac {x^3 \left (-28 a B c+12 A b c+b^2 B\right )}{c}-\frac {\int \frac {3 x^2 \left (\left (3 B b^3+A c b^2-24 a B c b+20 a A c^2\right ) x^2+3 a \left (B b^2+12 A c b-28 a B c\right )\right )}{c x^4+b x^2+a}dx}{3 c}}{2 \left (b^2-4 a c\right )}-\frac {x^5 \left (x^2 \left (-28 a B c+12 A b c+b^2 B\right )-4 a A c-12 a b B+7 A b^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{4 \left (b^2-4 a c\right )}-\frac {x^7 \left (-2 a B-\left (x^2 (b B-2 A c)\right )+A b\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {x^3 \left (-28 a B c+12 A b c+b^2 B\right )}{c}-\frac {\int \frac {x^2 \left (\left (3 B b^3+A c b^2-24 a B c b+20 a A c^2\right ) x^2+3 a \left (B b^2+12 A c b-28 a B c\right )\right )}{c x^4+b x^2+a}dx}{c}}{2 \left (b^2-4 a c\right )}-\frac {x^5 \left (x^2 \left (-28 a B c+12 A b c+b^2 B\right )-4 a A c-12 a b B+7 A b^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{4 \left (b^2-4 a c\right )}-\frac {x^7 \left (-2 a B-\left (x^2 (b B-2 A c)\right )+A b\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\) |
| \(\Big \downarrow \) 1602 |
| \(\displaystyle \frac {\frac {\frac {x^3 \left (-28 a B c+12 A b c+b^2 B\right )}{c}-\frac {\frac {x \left (20 a A c^2-24 a b B c+A b^2 c+3 b^3 B\right )}{c}-\frac {\int \frac {\left (3 B b^4+A c b^3-27 a B c b^2-16 a A c^2 b+84 a^2 B c^2\right ) x^2+a \left (3 B b^3+A c b^2-24 a B c b+20 a A c^2\right )}{c x^4+b x^2+a}dx}{c}}{c}}{2 \left (b^2-4 a c\right )}-\frac {x^5 \left (x^2 \left (-28 a B c+12 A b c+b^2 B\right )-4 a A c-12 a b B+7 A b^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{4 \left (b^2-4 a c\right )}-\frac {x^7 \left (-2 a B-\left (x^2 (b B-2 A c)\right )+A b\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {\frac {\frac {x^3 \left (-28 a B c+12 A b c+b^2 B\right )}{c}-\frac {\frac {x \left (20 a A c^2-24 a b B c+A b^2 c+3 b^3 B\right )}{c}-\frac {\frac {1}{2} \left (-\frac {-40 a^2 A c^3+132 a^2 b B c^2-18 a A b^2 c^2-33 a b^3 B c+A b^4 c+3 b^5 B}{\sqrt {b^2-4 a c}}+84 a^2 B c^2-16 a A b c^2-27 a b^2 B c+A b^3 c+3 b^4 B\right ) \int \frac {1}{c x^2+\frac {1}{2} \left (b-\sqrt {b^2-4 a c}\right )}dx+\frac {1}{2} \left (\frac {-40 a^2 A c^3+132 a^2 b B c^2-18 a A b^2 c^2-33 a b^3 B c+A b^4 c+3 b^5 B}{\sqrt {b^2-4 a c}}+84 a^2 B c^2-16 a A b c^2-27 a b^2 B c+A b^3 c+3 b^4 B\right ) \int \frac {1}{c x^2+\frac {1}{2} \left (b+\sqrt {b^2-4 a c}\right )}dx}{c}}{c}}{2 \left (b^2-4 a c\right )}-\frac {x^5 \left (x^2 \left (-28 a B c+12 A b c+b^2 B\right )-4 a A c-12 a b B+7 A b^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{4 \left (b^2-4 a c\right )}-\frac {x^7 \left (-2 a B-\left (x^2 (b B-2 A c)\right )+A b\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {\frac {x^3 \left (-28 a B c+12 A b c+b^2 B\right )}{c}-\frac {\frac {x \left (20 a A c^2-24 a b B c+A b^2 c+3 b^3 B\right )}{c}-\frac {\frac {\left (-\frac {-40 a^2 A c^3+132 a^2 b B c^2-18 a A b^2 c^2-33 a b^3 B c+A b^4 c+3 b^5 B}{\sqrt {b^2-4 a c}}+84 a^2 B c^2-16 a A b c^2-27 a b^2 B c+A b^3 c+3 b^4 B\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} \sqrt {c} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\left (\frac {-40 a^2 A c^3+132 a^2 b B c^2-18 a A b^2 c^2-33 a b^3 B c+A b^4 c+3 b^5 B}{\sqrt {b^2-4 a c}}+84 a^2 B c^2-16 a A b c^2-27 a b^2 B c+A b^3 c+3 b^4 B\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {2} \sqrt {c} \sqrt {\sqrt {b^2-4 a c}+b}}}{c}}{c}}{2 \left (b^2-4 a c\right )}-\frac {x^5 \left (x^2 \left (-28 a B c+12 A b c+b^2 B\right )-4 a A c-12 a b B+7 A b^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{4 \left (b^2-4 a c\right )}-\frac {x^7 \left (-2 a B-\left (x^2 (b B-2 A c)\right )+A b\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\) |
-1/4*(x^7*(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^5*(7*A*b^2 - 12*a*b*B - 4*a*A*c + (b^2*B + 12*A*b*c - 28*a*B*c)*x^2))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (((b^2*B + 12*A*b*c - 28*a*B*c)*x^3)/c - (((3*b^3*B + A*b^2*c - 24*a*b*B*c + 20*a*A*c^2)*x)/c - (((3*b^4*B + A*b^3*c - 27*a*b^2*B*c - 16*a*A*b*c^2 + 84*a^2*B*c^2 - (3*b ^5*B + A*b^4*c - 33*a*b^3*B*c - 18*a*A*b^2*c^2 + 132*a^2*b*B*c^2 - 40*a^2* A*c^3)/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]]) + ((3*b^4*B + A*b^3 *c - 27*a*b^2*B*c - 16*a*A*b*c^2 + 84*a^2*B*c^2 + (3*b^5*B + A*b^4*c - 33* a*b^3*B*c - 18*a*A*b^2*c^2 + 132*a^2*b*B*c^2 - 40*a^2*A*c^3)/Sqrt[b^2 - 4* a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sq rt[c]*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/c)/c)/(2*(b^2 - 4*a*c)))/(4*(b^2 - 4*a *c))
3.2.32.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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 = 0.17 (sec) , antiderivative size = 445, normalized size of antiderivative = 0.80
| method | result | size |
| risch | \(\frac {-\frac {\left (16 A a b \,c^{2}-A \,b^{3} c +44 B \,a^{2} c^{2}-37 B a \,b^{2} c +5 B \,b^{4}\right ) x^{7}}{8 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) c}-\frac {\left (36 A \,a^{2} c^{3}+5 A a \,b^{2} c^{2}+A \,b^{4} c -4 B \,a^{2} b \,c^{2}-20 B a \,b^{3} c +3 B \,b^{5}\right ) x^{5}}{8 c^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {a \left (28 A a b \,c^{2}+2 A \,b^{3} c +28 B \,a^{2} c^{2}-49 B a \,b^{2} c +6 B \,b^{4}\right ) x^{3}}{8 c^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {a^{2} \left (20 A a \,c^{2}+A \,b^{2} c -24 B a b c +3 B \,b^{3}\right ) x}{8 c^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2}}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{4}+\textit {\_Z}^{2} b +a \right )}{\sum }\frac {\left (-\frac {\left (16 A a b \,c^{2}-A \,b^{3} c -84 B \,a^{2} c^{2}+27 B a \,b^{2} c -3 B \,b^{4}\right ) \textit {\_R}^{2}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {a \left (20 A a \,c^{2}+A \,b^{2} c -24 B a b c +3 B \,b^{3}\right )}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}\right ) \ln \left (x -\textit {\_R} \right )}{2 c \,\textit {\_R}^{3}+\textit {\_R} b}}{16 c^{2}}\) | \(445\) |
| default | \(\frac {-\frac {\left (16 A a b \,c^{2}-A \,b^{3} c +44 B \,a^{2} c^{2}-37 B a \,b^{2} c +5 B \,b^{4}\right ) x^{7}}{8 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) c}-\frac {\left (36 A \,a^{2} c^{3}+5 A a \,b^{2} c^{2}+A \,b^{4} c -4 B \,a^{2} b \,c^{2}-20 B a \,b^{3} c +3 B \,b^{5}\right ) x^{5}}{8 c^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {a \left (28 A a b \,c^{2}+2 A \,b^{3} c +28 B \,a^{2} c^{2}-49 B a \,b^{2} c +6 B \,b^{4}\right ) x^{3}}{8 c^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {a^{2} \left (20 A a \,c^{2}+A \,b^{2} c -24 B a b c +3 B \,b^{3}\right ) x}{8 c^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2}}+\frac {\frac {\left (-16 A a b \,c^{2} \sqrt {-4 a c +b^{2}}+A \,b^{3} c \sqrt {-4 a c +b^{2}}-40 A \,a^{2} c^{3}-18 A a \,b^{2} c^{2}+A \,b^{4} c +84 B \,a^{2} c^{2} \sqrt {-4 a c +b^{2}}-27 B a \,b^{2} c \sqrt {-4 a c +b^{2}}+3 B \,b^{4} \sqrt {-4 a c +b^{2}}+132 B \,a^{2} b \,c^{2}-33 B a \,b^{3} c +3 B \,b^{5}\right ) \sqrt {2}\, \arctan \left (\frac {c x \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}-\frac {\left (-16 A a b \,c^{2} \sqrt {-4 a c +b^{2}}+A \,b^{3} c \sqrt {-4 a c +b^{2}}+40 A \,a^{2} c^{3}+18 A a \,b^{2} c^{2}-A \,b^{4} c +84 B \,a^{2} c^{2} \sqrt {-4 a c +b^{2}}-27 B a \,b^{2} c \sqrt {-4 a c +b^{2}}+3 B \,b^{4} \sqrt {-4 a c +b^{2}}-132 B \,a^{2} b \,c^{2}+33 B a \,b^{3} c -3 B \,b^{5}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c x \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}}{2 c \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}\) | \(710\) |
(-1/8*(16*A*a*b*c^2-A*b^3*c+44*B*a^2*c^2-37*B*a*b^2*c+5*B*b^4)/(16*a^2*c^2 -8*a*b^2*c+b^4)/c*x^7-1/8*(36*A*a^2*c^3+5*A*a*b^2*c^2+A*b^4*c-4*B*a^2*b*c^ 2-20*B*a*b^3*c+3*B*b^5)/c^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^5-1/8*a/c^2*(28*A *a*b*c^2+2*A*b^3*c+28*B*a^2*c^2-49*B*a*b^2*c+6*B*b^4)/(16*a^2*c^2-8*a*b^2* c+b^4)*x^3-1/8*a^2*(20*A*a*c^2+A*b^2*c-24*B*a*b*c+3*B*b^3)/c^2/(16*a^2*c^2 -8*a*b^2*c+b^4)*x)/(c*x^4+b*x^2+a)^2+1/16/c^2*sum((-(16*A*a*b*c^2-A*b^3*c- 84*B*a^2*c^2+27*B*a*b^2*c-3*B*b^4)/(16*a^2*c^2-8*a*b^2*c+b^4)*_R^2+a*(20*A *a*c^2+A*b^2*c-24*B*a*b*c+3*B*b^3)/(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. 9636 vs. \(2 (507) = 1014\).
Time = 18.80 (sec) , antiderivative size = 9636, normalized size of antiderivative = 17.39 \[ \int \frac {x^8 \left (A+B x^2\right )}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {x^8 \left (A+B x^2\right )}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {x^8 \left (A+B x^2\right )}{\left (a+b x^2+c x^4\right )^3} \, dx=\int { \frac {{\left (B x^{2} + A\right )} x^{8}}{{\left (c x^{4} + b x^{2} + a\right )}^{3}} \,d x } \]
-1/8*((5*B*b^4*c + 4*(11*B*a^2 + 4*A*a*b)*c^3 - (37*B*a*b^2 + A*b^3)*c^2)* x^7 + (3*B*b^5 + 36*A*a^2*c^3 - (4*B*a^2*b - 5*A*a*b^2)*c^2 - (20*B*a*b^3 - A*b^4)*c)*x^5 + (6*B*a*b^4 + 28*(B*a^3 + A*a^2*b)*c^2 - (49*B*a^2*b^2 - 2*A*a*b^3)*c)*x^3 + (3*B*a^2*b^3 + 20*A*a^3*c^2 - (24*B*a^3*b - A*a^2*b^2) *c)*x)/((b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*x^8 + a^2*b^4*c^2 - 8*a^3*b^2 *c^3 + 16*a^4*c^4 + 2*(b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*x^6 + (b^6*c^ 2 - 6*a*b^4*c^3 + 32*a^3*c^5)*x^4 + 2*(a*b^5*c^2 - 8*a^2*b^3*c^3 + 16*a^3* b*c^4)*x^2) - 1/8*integrate(-(3*B*a*b^3 + 20*A*a^2*c^2 + (3*B*b^4 + 4*(21* B*a^2 - 4*A*a*b)*c^2 - (27*B*a*b^2 - A*b^3)*c)*x^2 - (24*B*a^2*b - A*a*b^2 )*c)/(c*x^4 + b*x^2 + a), x)/(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)
Leaf count of result is larger than twice the leaf count of optimal. 3987 vs. \(2 (507) = 1014\).
Time = 2.02 (sec) , antiderivative size = 3987, normalized size of antiderivative = 7.20 \[ \int \frac {x^8 \left (A+B x^2\right )}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Too large to display} \]
1/32*((sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^6*c + 12*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^4*c^2 - 2*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c) *c)*b^5*c^2 - 2*b^6*c^2 - 144*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2* b^2*c^3 - 32*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3*c^3 + sqrt(2)*s qrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^4*c^3 - 24*a*b^4*c^3 - 2*b^5*c^3 + 320*sq rt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*c^4 + 160*sqrt(2)*sqrt(b*c + sqr t(b^2 - 4*a*c)*c)*a^2*b*c^4 + 16*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a *b^2*c^4 + 288*a^2*b^2*c^4 + 112*a*b^3*c^4 - 80*sqrt(2)*sqrt(b*c + sqrt(b^ 2 - 4*a*c)*c)*a^2*c^5 - 640*a^3*c^5 - 416*a^2*b*c^5 + sqrt(2)*sqrt(b^2 - 4 *a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^5*c - 56*sqrt(2)*sqrt(b^2 - 4*a*c) *sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3*c^2 - 2*sqrt(2)*sqrt(b^2 - 4*a*c)*s qrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^4*c^2 + 208*sqrt(2)*sqrt(b^2 - 4*a*c)*sqr t(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b*c^3 + 104*sqrt(2)*sqrt(b^2 - 4*a*c)*sqr t(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c^3 + sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b* c + sqrt(b^2 - 4*a*c)*c)*b^3*c^3 - 52*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^4 + 2*(b^2 - 4*a*c)*b^4*c^2 + 32*(b^2 - 4*a*c) *a*b^2*c^3 + 2*(b^2 - 4*a*c)*b^3*c^3 - 160*(b^2 - 4*a*c)*a^2*c^4 - 104*(b^ 2 - 4*a*c)*a*b*c^4)*A + 3*(sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^7 - 1 6*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^5*c - 2*sqrt(2)*sqrt(b*c + s qrt(b^2 - 4*a*c)*c)*b^6*c - 2*b^7*c + 80*sqrt(2)*sqrt(b*c + sqrt(b^2 - ...
Time = 10.32 (sec) , antiderivative size = 22911, normalized size of antiderivative = 41.36 \[ \int \frac {x^8 \left (A+B x^2\right )}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Too large to display} \]
- ((x^5*(3*B*b^5 + 36*A*a^2*c^3 + A*b^4*c - 20*B*a*b^3*c + 5*A*a*b^2*c^2 - 4*B*a^2*b*c^2))/(8*c^2*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (x^7*(5*B*b^4 + 44*B*a^2*c^2 - A*b^3*c + 16*A*a*b*c^2 - 37*B*a*b^2*c))/(8*c*(b^4 + 16*a^2* c^2 - 8*a*b^2*c)) + (x^3*(28*B*a^3*c^2 + 6*B*a*b^4 + 2*A*a*b^3*c + 28*A*a^ 2*b*c^2 - 49*B*a^2*b^2*c))/(8*c^2*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (a^2*x *(3*B*b^3 + 20*A*a*c^2 + A*b^2*c - 24*B*a*b*c))/(8*c^2*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)))/(x^4*(2*a*c + b^2) + a^2 + c^2*x^8 + 2*a*b*x^2 + 2*b*c*x^6) - atan(((((256*A*a*b^12*c^4 - 5242880*A*a^7*c^10 + 768*B*a*b^13*c^3 + 6291 456*B*a^7*b*c^9 - 61440*A*a^3*b^8*c^6 + 655360*A*a^4*b^6*c^7 - 2949120*A*a ^5*b^4*c^8 + 6291456*A*a^6*b^2*c^9 - 21504*B*a^2*b^11*c^4 + 245760*B*a^3*b ^9*c^5 - 1474560*B*a^4*b^7*c^6 + 4915200*B*a^5*b^5*c^7 - 8650752*B*a^6*b^3 *c^8)/(512*(4096*a^6*c^9 + b^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5 - 12 80*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)) - (x*(-(9*B^2*b^19 + A^2*b^17*c^2 + 9*B^2*b^4*(-(4*a*c - b^2)^15)^(1/2) + 6*A*B*b^18*c + 1140 *A^2*a^2*b^13*c^4 - 10160*A^2*a^3*b^11*c^5 + 34880*A^2*a^4*b^9*c^6 + 43776 *A^2*a^5*b^7*c^7 - 680960*A^2*a^6*b^5*c^8 + 1863680*A^2*a^7*b^3*c^9 + 6921 *B^2*a^2*b^15*c^2 - 77580*B^2*a^3*b^13*c^3 + 570960*B^2*a^4*b^11*c^4 - 285 1776*B^2*a^5*b^9*c^5 + 9628416*B^2*a^6*b^7*c^6 - 21095424*B^2*a^7*b^5*c^7 + 27095040*B^2*a^8*b^3*c^8 + A^2*b^2*c^2*(-(4*a*c - b^2)^15)^(1/2) + 441*B ^2*a^2*c^2*(-(4*a*c - b^2)^15)^(1/2) + 6881280*A*B*a^9*c^10 - 369*B^2*a...