Integrand size = 25, antiderivative size = 365 \[ \int \frac {x^{11} \left (A+B x^2\right )}{\left (a+b x^2+c x^4\right )^3} \, dx=\frac {\left (3 b^4 B-A b^3 c-21 a b^2 B c+7 a A b c^2+30 a^2 B c^2\right ) x^2}{2 c^3 \left (b^2-4 a c\right )^2}-\frac {x^8 \left (a (b B-2 A c)+\left (b^2 B-A b c-2 a B c\right ) x^2\right )}{4 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {x^4 \left (a \left (3 b^3 B-A b^2 c-18 a b B c+16 a A c^2\right )+\left (3 b^4 B-A b^3 c-20 a b^2 B c+10 a A b c^2+20 a^2 B c^2\right ) x^2\right )}{4 c^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac {\left (3 b^6 B-A b^5 c-30 a b^4 B c+10 a A b^3 c^2+90 a^2 b^2 B c^2-30 a^2 A b c^3-60 a^3 B c^3\right ) \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 c^4 \left (b^2-4 a c\right )^{5/2}}-\frac {(3 b B-A c) \log \left (a+b x^2+c x^4\right )}{4 c^4} \]
1/2*(7*A*a*b*c^2-A*b^3*c+30*B*a^2*c^2-21*B*a*b^2*c+3*B*b^4)*x^2/c^3/(-4*a* c+b^2)^2-1/4*x^8*(a*(-2*A*c+B*b)+(-A*b*c-2*B*a*c+B*b^2)*x^2)/c/(-4*a*c+b^2 )/(c*x^4+b*x^2+a)^2-1/4*x^4*(a*(16*A*a*c^2-A*b^2*c-18*B*a*b*c+3*B*b^3)+(10 *A*a*b*c^2-A*b^3*c+20*B*a^2*c^2-20*B*a*b^2*c+3*B*b^4)*x^2)/c^2/(-4*a*c+b^2 )^2/(c*x^4+b*x^2+a)-1/2*(-30*A*a^2*b*c^3+10*A*a*b^3*c^2-A*b^5*c-60*B*a^3*c ^3+90*B*a^2*b^2*c^2-30*B*a*b^4*c+3*B*b^6)*arctanh((2*c*x^2+b)/(-4*a*c+b^2) ^(1/2))/c^4/(-4*a*c+b^2)^(5/2)-1/4*(-A*c+3*B*b)*ln(c*x^4+b*x^2+a)/c^4
Time = 0.43 (sec) , antiderivative size = 435, normalized size of antiderivative = 1.19 \[ \int \frac {x^{11} \left (A+B x^2\right )}{\left (a+b x^2+c x^4\right )^3} \, dx=\frac {2 B c^2 x^2+\frac {b^7 B-b^6 c \left (A+6 B x^2\right )+4 a^3 c^4 \left (8 A+9 B x^2\right )-3 a^2 b^2 c^3 \left (13 A+34 B x^2\right )+a b^4 c^2 \left (11 A+48 B x^2\right )+a b^3 c^2 \left (61 a B-30 A c x^2\right )+2 b^5 c \left (-7 a B+2 A c x^2\right )+2 a^2 b c^3 \left (-39 a B+25 A c x^2\right )}{\left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {b^5 (-b B+A c) x^2+a^3 c^2 \left (-5 b B+2 c \left (A+B x^2\right )\right )+a b^3 \left (-b^2 B-5 A c^2 x^2+b c \left (A+6 B x^2\right )\right )+a^2 b c \left (5 b^2 B+5 A c^2 x^2-b c \left (4 A+9 B x^2\right )\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {2 c \left (-3 b^6 B+A b^5 c+30 a b^4 B c-10 a A b^3 c^2-90 a^2 b^2 B c^2+30 a^2 A b c^3+60 a^3 B c^3\right ) \arctan \left (\frac {b+2 c x^2}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{5/2}}+c (-3 b B+A c) \log \left (a+b x^2+c x^4\right )}{4 c^5} \]
(2*B*c^2*x^2 + (b^7*B - b^6*c*(A + 6*B*x^2) + 4*a^3*c^4*(8*A + 9*B*x^2) - 3*a^2*b^2*c^3*(13*A + 34*B*x^2) + a*b^4*c^2*(11*A + 48*B*x^2) + a*b^3*c^2* (61*a*B - 30*A*c*x^2) + 2*b^5*c*(-7*a*B + 2*A*c*x^2) + 2*a^2*b*c^3*(-39*a* B + 25*A*c*x^2))/((b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (b^5*(-(b*B) + A* c)*x^2 + a^3*c^2*(-5*b*B + 2*c*(A + B*x^2)) + a*b^3*(-(b^2*B) - 5*A*c^2*x^ 2 + b*c*(A + 6*B*x^2)) + a^2*b*c*(5*b^2*B + 5*A*c^2*x^2 - b*c*(4*A + 9*B*x ^2)))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) - (2*c*(-3*b^6*B + A*b^5*c + 3 0*a*b^4*B*c - 10*a*A*b^3*c^2 - 90*a^2*b^2*B*c^2 + 30*a^2*A*b*c^3 + 60*a^3* B*c^3)*ArcTan[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(5/2) + c* (-3*b*B + A*c)*Log[a + b*x^2 + c*x^4])/(4*c^5)
Time = 1.09 (sec) , antiderivative size = 391, normalized size of antiderivative = 1.07, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {1578, 1233, 1233, 27, 1200, 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.
| \(\displaystyle \int \frac {x^{11} \left (A+B x^2\right )}{\left (a+b x^2+c x^4\right )^3} \, dx\) |
| \(\Big \downarrow \) 1578 |
| \(\displaystyle \frac {1}{2} \int \frac {x^{10} \left (B x^2+A\right )}{\left (c x^4+b x^2+a\right )^3}dx^2\) |
| \(\Big \downarrow \) 1233 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int \frac {x^6 \left (\left (3 B b^2-A c b-10 a B c\right ) x^2+4 a (b B-2 A c)\right )}{\left (c x^4+b x^2+a\right )^2}dx^2}{2 c \left (b^2-4 a c\right )}-\frac {x^8 \left (x^2 \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\) |
| \(\Big \downarrow \) 1233 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {\int \frac {2 x^2 \left (\left (3 B b^4-A c b^3-21 a B c b^2+7 a A c^2 b+30 a^2 B c^2\right ) x^2+a \left (3 B b^3-A c b^2-18 a B c b+16 a A c^2\right )\right )}{c x^4+b x^2+a}dx^2}{c \left (b^2-4 a c\right )}-\frac {x^4 \left (x^2 \left (20 a^2 B c^2+10 a A b c^2-20 a b^2 B c-A b^3 c+3 b^4 B\right )+a \left (16 a A c^2-18 a b B c-A b^2 c+3 b^3 B\right )\right )}{c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{2 c \left (b^2-4 a c\right )}-\frac {x^8 \left (x^2 \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {2 \int \frac {x^2 \left (\left (3 B b^4-A c b^3-21 a B c b^2+7 a A c^2 b+30 a^2 B c^2\right ) x^2+a \left (3 B b^3-A c b^2-18 a B c b+16 a A c^2\right )\right )}{c x^4+b x^2+a}dx^2}{c \left (b^2-4 a c\right )}-\frac {x^4 \left (x^2 \left (20 a^2 B c^2+10 a A b c^2-20 a b^2 B c-A b^3 c+3 b^4 B\right )+a \left (16 a A c^2-18 a b B c-A b^2 c+3 b^3 B\right )\right )}{c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{2 c \left (b^2-4 a c\right )}-\frac {x^8 \left (x^2 \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\) |
| \(\Big \downarrow \) 1200 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {2 \int \left (-3 B \left (-\frac {b^4}{c}+7 a b^2-10 a^2 c\right )-A \left (b^3-7 a b c\right )-\frac {\left (b^2-4 a c\right )^2 (3 b B-A c) x^2+a \left (3 B b^4-A c b^3-21 a B c b^2+7 a A c^2 b+30 a^2 B c^2\right )}{c \left (c x^4+b x^2+a\right )}\right )dx^2}{c \left (b^2-4 a c\right )}-\frac {x^4 \left (x^2 \left (20 a^2 B c^2+10 a A b c^2-20 a b^2 B c-A b^3 c+3 b^4 B\right )+a \left (16 a A c^2-18 a b B c-A b^2 c+3 b^3 B\right )\right )}{c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{2 c \left (b^2-4 a c\right )}-\frac {x^8 \left (x^2 \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {2 \left (-x^2 \left (3 B \left (-10 a^2 c+7 a b^2-\frac {b^4}{c}\right )+A \left (b^3-7 a b c\right )\right )-\frac {\left (-60 a^3 B c^3-30 a^2 A b c^3+90 a^2 b^2 B c^2+10 a A b^3 c^2-30 a b^4 B c-A b^5 c+3 b^6 B\right ) \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{c^2 \sqrt {b^2-4 a c}}-\frac {\left (b^2-4 a c\right )^2 (3 b B-A c) \log \left (a+b x^2+c x^4\right )}{2 c^2}\right )}{c \left (b^2-4 a c\right )}-\frac {x^4 \left (x^2 \left (20 a^2 B c^2+10 a A b c^2-20 a b^2 B c-A b^3 c+3 b^4 B\right )+a \left (16 a A c^2-18 a b B c-A b^2 c+3 b^3 B\right )\right )}{c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{2 c \left (b^2-4 a c\right )}-\frac {x^8 \left (x^2 \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\) |
(-1/2*(x^8*(a*(b*B - 2*A*c) + (b^2*B - A*b*c - 2*a*B*c)*x^2))/(c*(b^2 - 4* a*c)*(a + b*x^2 + c*x^4)^2) + (-((x^4*(a*(3*b^3*B - A*b^2*c - 18*a*b*B*c + 16*a*A*c^2) + (3*b^4*B - A*b^3*c - 20*a*b^2*B*c + 10*a*A*b*c^2 + 20*a^2*B *c^2)*x^2))/(c*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4))) + (2*(-((3*B*(7*a*b^2 - b^4/c - 10*a^2*c) + A*(b^3 - 7*a*b*c))*x^2) - ((3*b^6*B - A*b^5*c - 30*a* b^4*B*c + 10*a*A*b^3*c^2 + 90*a^2*b^2*B*c^2 - 30*a^2*A*b*c^3 - 60*a^3*B*c^ 3)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(c^2*Sqrt[b^2 - 4*a*c]) - ((b ^2 - 4*a*c)^2*(3*b*B - A*c)*Log[a + b*x^2 + c*x^4])/(2*c^2)))/(c*(b^2 - 4* a*c)))/(2*c*(b^2 - 4*a*c)))/2
3.2.24.3.1 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[(((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.))/((a_.) + (b_.)* (x_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*((f + g* x)^n/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && In tegersQ[n]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(d + e*x)^(m - 1))*(a + b*x + c*x^2) ^(p + 1)*((2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g - c *(b*e*f + b*d*g + 2*a*e*g))*x)/(c*(p + 1)*(b^2 - 4*a*c))), x] - Simp[1/(c*( p + 1)*(b^2 - 4*a*c)) Int[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Sim p[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2*a*e*(e*f *(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*( m + p + 1) + 2*c^2*d*f*(m + 2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2* p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && LtQ[p, -1] && GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, b, c, d, e, f, g]) | | !ILtQ[m + 2*p + 3, 0])
Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_ )^4)^(p_.), x_Symbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && Int egerQ[(m - 1)/2]
Time = 0.31 (sec) , antiderivative size = 624, normalized size of antiderivative = 1.71
| method | result | size |
| default | \(\frac {B \,x^{2}}{2 c^{3}}+\frac {\frac {\frac {\left (25 A \,a^{2} b \,c^{3}-15 A a \,b^{3} c^{2}+2 A \,b^{5} c +18 B \,a^{3} c^{3}-51 B \,a^{2} b^{2} c^{2}+24 B a \,b^{4} c -3 B \,b^{6}\right ) x^{6}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {\left (32 A \,a^{3} c^{4}+11 A \,a^{2} b^{2} c^{3}-19 A a \,b^{4} c^{2}+3 A \,b^{6} c -42 B \,a^{3} b \,c^{3}-41 B \,a^{2} b^{3} c^{2}+34 b^{5} c a B -5 b^{7} B \right ) x^{4}}{2 c \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {a \left (31 A \,a^{2} b \,c^{3}-22 A a \,b^{3} c^{2}+3 A \,b^{5} c +14 B \,a^{3} c^{3}-71 B \,a^{2} b^{2} c^{2}+38 B a \,b^{4} c -5 B \,b^{6}\right ) x^{2}}{c \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {a^{2} \left (24 A \,a^{2} c^{3}-21 A a \,b^{2} c^{2}+3 A \,b^{4} c -58 B \,a^{2} b \,c^{2}+36 B a \,b^{3} c -5 B \,b^{5}\right )}{2 c \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2}}+\frac {\frac {\left (16 A \,a^{2} c^{3}-8 A a \,b^{2} c^{2}+A \,b^{4} c -48 B \,a^{2} b \,c^{2}+24 B a \,b^{3} c -3 B \,b^{5}\right ) \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{2 c}+\frac {2 \left (-7 A \,a^{2} b \,c^{2}+A a \,b^{3} c -30 a^{3} B \,c^{2}+21 B \,a^{2} b^{2} c -3 B a \,b^{4}-\frac {\left (16 A \,a^{2} c^{3}-8 A a \,b^{2} c^{2}+A \,b^{4} c -48 B \,a^{2} b \,c^{2}+24 B a \,b^{3} c -3 B \,b^{5}\right ) b}{2 c}\right ) \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}}{2 c^{3}}\) | \(624\) |
| risch | \(\text {Expression too large to display}\) | \(5515\) |
1/2*B*x^2/c^3+1/2/c^3*(((25*A*a^2*b*c^3-15*A*a*b^3*c^2+2*A*b^5*c+18*B*a^3* c^3-51*B*a^2*b^2*c^2+24*B*a*b^4*c-3*B*b^6)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^6+ 1/2*(32*A*a^3*c^4+11*A*a^2*b^2*c^3-19*A*a*b^4*c^2+3*A*b^6*c-42*B*a^3*b*c^3 -41*B*a^2*b^3*c^2+34*B*a*b^5*c-5*B*b^7)/c/(16*a^2*c^2-8*a*b^2*c+b^4)*x^4+a *(31*A*a^2*b*c^3-22*A*a*b^3*c^2+3*A*b^5*c+14*B*a^3*c^3-71*B*a^2*b^2*c^2+38 *B*a*b^4*c-5*B*b^6)/c/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2+1/2*a^2*(24*A*a^2*c^3 -21*A*a*b^2*c^2+3*A*b^4*c-58*B*a^2*b*c^2+36*B*a*b^3*c-5*B*b^5)/c/(16*a^2*c ^2-8*a*b^2*c+b^4))/(c*x^4+b*x^2+a)^2+1/(16*a^2*c^2-8*a*b^2*c+b^4)*(1/2*(16 *A*a^2*c^3-8*A*a*b^2*c^2+A*b^4*c-48*B*a^2*b*c^2+24*B*a*b^3*c-3*B*b^5)/c*ln (c*x^4+b*x^2+a)+2*(-7*A*a^2*b*c^2+A*a*b^3*c-30*a^3*B*c^2+21*B*a^2*b^2*c-3* B*a*b^4-1/2*(16*A*a^2*c^3-8*A*a*b^2*c^2+A*b^4*c-48*B*a^2*b*c^2+24*B*a*b^3* c-3*B*b^5)*b/c)/(4*a*c-b^2)^(1/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 1586 vs. \(2 (351) = 702\).
Time = 0.53 (sec) , antiderivative size = 3196, normalized size of antiderivative = 8.76 \[ \int \frac {x^{11} \left (A+B x^2\right )}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Too large to display} \]
[1/4*(2*(B*b^6*c^3 - 12*B*a*b^4*c^4 + 48*B*a^2*b^2*c^5 - 64*B*a^3*c^6)*x^1 0 - 5*B*a^2*b^7 - 96*A*a^5*c^4 + 4*(B*b^7*c^2 - 12*B*a*b^5*c^3 + 48*B*a^2* b^3*c^4 - 64*B*a^3*b*c^5)*x^8 - 2*(2*B*b^8*c + 100*(2*B*a^4 + A*a^3*b)*c^5 - (254*B*a^3*b^2 + 85*A*a^2*b^3)*c^4 + (123*B*a^2*b^4 + 23*A*a*b^5)*c^3 - 2*(13*B*a*b^6 + A*b^7)*c^2)*x^6 - (5*B*b^9 + 128*A*a^4*c^5 + 4*(22*B*a^4* b + 3*A*a^3*b^2)*c^4 - (314*B*a^3*b^3 + 87*A*a^2*b^4)*c^3 + (225*B*a^2*b^5 + 31*A*a*b^6)*c^2 - (58*B*a*b^7 + 3*A*b^8)*c)*x^4 + 4*(58*B*a^5*b + 27*A* a^4*b^2)*c^3 - (202*B*a^4*b^3 + 33*A*a^3*b^4)*c^2 - 2*(5*B*a*b^8 + 4*(30*B *a^5 + 31*A*a^4*b)*c^4 - (346*B*a^4*b^2 + 119*A*a^3*b^3)*c^3 + (235*B*a^3* b^4 + 34*A*a^2*b^5)*c^2 - (59*B*a^2*b^6 + 3*A*a*b^7)*c)*x^2 - (3*B*a^2*b^6 + (3*B*b^6*c^2 - 30*(2*B*a^3 + A*a^2*b)*c^5 + 10*(9*B*a^2*b^2 + A*a*b^3)* c^4 - (30*B*a*b^4 + A*b^5)*c^3)*x^8 + 2*(3*B*b^7*c - 30*(2*B*a^3*b + A*a^2 *b^2)*c^4 + 10*(9*B*a^2*b^3 + A*a*b^4)*c^3 - (30*B*a*b^5 + A*b^6)*c^2)*x^6 + (3*B*b^8 - 60*(2*B*a^4 + A*a^3*b)*c^4 + 10*(12*B*a^3*b^2 - A*a^2*b^3)*c ^3 + 2*(15*B*a^2*b^4 + 4*A*a*b^5)*c^2 - (24*B*a*b^6 + A*b^7)*c)*x^4 - 30*( 2*B*a^5 + A*a^4*b)*c^3 + 10*(9*B*a^4*b^2 + A*a^3*b^3)*c^2 + 2*(3*B*a*b^7 - 30*(2*B*a^4*b + A*a^3*b^2)*c^3 + 10*(9*B*a^3*b^3 + A*a^2*b^4)*c^2 - (30*B *a^2*b^5 + A*a*b^6)*c)*x^2 - (30*B*a^3*b^4 + A*a^2*b^5)*c)*sqrt(b^2 - 4*a* c)*log((2*c^2*x^4 + 2*b*c*x^2 + b^2 - 2*a*c + (2*c*x^2 + b)*sqrt(b^2 - 4*a *c))/(c*x^4 + b*x^2 + a)) + (56*B*a^3*b^5 + 3*A*a^2*b^6)*c - (3*B*a^2*b...
Timed out. \[ \int \frac {x^{11} \left (A+B x^2\right )}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {x^{11} \left (A+B x^2\right )}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for more deta
Time = 1.42 (sec) , antiderivative size = 598, normalized size of antiderivative = 1.64 \[ \int \frac {x^{11} \left (A+B x^2\right )}{\left (a+b x^2+c x^4\right )^3} \, dx=\frac {{\left (3 \, B b^{6} - 30 \, B a b^{4} c - A b^{5} c + 90 \, B a^{2} b^{2} c^{2} + 10 \, A a b^{3} c^{2} - 60 \, B a^{3} c^{3} - 30 \, A a^{2} b c^{3}\right )} \arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{2 \, {\left (b^{4} c^{4} - 8 \, a b^{2} c^{5} + 16 \, a^{2} c^{6}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {B x^{2}}{2 \, c^{3}} + \frac {9 \, B b^{5} c^{2} x^{8} - 72 \, B a b^{3} c^{3} x^{8} - 3 \, A b^{4} c^{3} x^{8} + 144 \, B a^{2} b c^{4} x^{8} + 24 \, A a b^{2} c^{4} x^{8} - 48 \, A a^{2} c^{5} x^{8} + 6 \, B b^{6} c x^{6} - 48 \, B a b^{4} c^{2} x^{6} + 2 \, A b^{5} c^{2} x^{6} + 84 \, B a^{2} b^{2} c^{3} x^{6} - 12 \, A a b^{3} c^{3} x^{6} + 72 \, B a^{3} c^{4} x^{6} + 4 \, A a^{2} b c^{4} x^{6} - B b^{7} x^{4} + 14 \, B a b^{5} c x^{4} + 3 \, A b^{6} c x^{4} - 82 \, B a^{2} b^{3} c^{2} x^{4} - 20 \, A a b^{4} c^{2} x^{4} + 204 \, B a^{3} b c^{3} x^{4} + 22 \, A a^{2} b^{2} c^{3} x^{4} - 32 \, A a^{3} c^{4} x^{4} - 2 \, B a b^{6} x^{2} + 8 \, B a^{2} b^{4} c x^{2} + 6 \, A a b^{5} c x^{2} + 4 \, B a^{3} b^{2} c^{2} x^{2} - 40 \, A a^{2} b^{3} c^{2} x^{2} + 56 \, B a^{4} c^{3} x^{2} + 28 \, A a^{3} b c^{3} x^{2} - B a^{2} b^{5} + 3 \, A a^{2} b^{4} c + 28 \, B a^{4} b c^{2} - 18 \, A a^{3} b^{2} c^{2}}{8 \, {\left (b^{4} c^{4} - 8 \, a b^{2} c^{5} + 16 \, a^{2} c^{6}\right )} {\left (c x^{4} + b x^{2} + a\right )}^{2}} - \frac {{\left (3 \, B b - A c\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, c^{4}} \]
1/2*(3*B*b^6 - 30*B*a*b^4*c - A*b^5*c + 90*B*a^2*b^2*c^2 + 10*A*a*b^3*c^2 - 60*B*a^3*c^3 - 30*A*a^2*b*c^3)*arctan((2*c*x^2 + b)/sqrt(-b^2 + 4*a*c))/ ((b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*sqrt(-b^2 + 4*a*c)) + 1/2*B*x^2/c^3 + 1/8*(9*B*b^5*c^2*x^8 - 72*B*a*b^3*c^3*x^8 - 3*A*b^4*c^3*x^8 + 144*B*a^2* b*c^4*x^8 + 24*A*a*b^2*c^4*x^8 - 48*A*a^2*c^5*x^8 + 6*B*b^6*c*x^6 - 48*B*a *b^4*c^2*x^6 + 2*A*b^5*c^2*x^6 + 84*B*a^2*b^2*c^3*x^6 - 12*A*a*b^3*c^3*x^6 + 72*B*a^3*c^4*x^6 + 4*A*a^2*b*c^4*x^6 - B*b^7*x^4 + 14*B*a*b^5*c*x^4 + 3 *A*b^6*c*x^4 - 82*B*a^2*b^3*c^2*x^4 - 20*A*a*b^4*c^2*x^4 + 204*B*a^3*b*c^3 *x^4 + 22*A*a^2*b^2*c^3*x^4 - 32*A*a^3*c^4*x^4 - 2*B*a*b^6*x^2 + 8*B*a^2*b ^4*c*x^2 + 6*A*a*b^5*c*x^2 + 4*B*a^3*b^2*c^2*x^2 - 40*A*a^2*b^3*c^2*x^2 + 56*B*a^4*c^3*x^2 + 28*A*a^3*b*c^3*x^2 - B*a^2*b^5 + 3*A*a^2*b^4*c + 28*B*a ^4*b*c^2 - 18*A*a^3*b^2*c^2)/((b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*(c*x^4 + b*x^2 + a)^2) - 1/4*(3*B*b - A*c)*log(c*x^4 + b*x^2 + a)/c^4
Time = 10.38 (sec) , antiderivative size = 4501, normalized size of antiderivative = 12.33 \[ \int \frac {x^{11} \left (A+B x^2\right )}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Too large to display} \]
((x^6*(18*B*a^3*c^3 - 3*B*b^6 + 2*A*b^5*c + 24*B*a*b^4*c - 15*A*a*b^3*c^2 + 25*A*a^2*b*c^3 - 51*B*a^2*b^2*c^2))/(2*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (a*(24*A*a^3*c^3 - 5*B*a*b^5 + 3*A*a*b^4*c + 36*B*a^2*b^3*c - 58*B*a^3*b* c^2 - 21*A*a^2*b^2*c^2))/(4*c*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (x^2*(14*B *a^4*c^3 - 5*B*a*b^6 + 3*A*a*b^5*c + 31*A*a^3*b*c^3 + 38*B*a^2*b^4*c - 22* A*a^2*b^3*c^2 - 71*B*a^3*b^2*c^2))/(2*c*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) - (x^4*(5*B*b^7 - 32*A*a^3*c^4 - 3*A*b^6*c - 34*B*a*b^5*c + 19*A*a*b^4*c^2 + 42*B*a^3*b*c^3 - 11*A*a^2*b^2*c^3 + 41*B*a^2*b^3*c^2))/(4*c*(b^4 + 16*a^2 *c^2 - 8*a*b^2*c)))/(a^2*c^3 + c^5*x^8 + x^4*(2*a*c^4 + b^2*c^3) + 2*b*c^4 *x^6 + 2*a*b*c^3*x^2) + (B*x^2)/(2*c^3) + (log(((a*(A*c - 3*B*b)^2)/c^6 - (((8*a*(A*c - 3*B*b))/c^2 - (2*(2*a + b*x^2)*(A*c - 3*B*b + c^4*(-(60*B*a^ 3*c^3 - 3*B*b^6 + A*b^5*c + 30*B*a*b^4*c - 10*A*a*b^3*c^2 + 30*A*a^2*b*c^3 - 90*B*a^2*b^2*c^2)^2/(c^8*(4*a*c - b^2)^5))^(1/2)))/c^2 + (2*x^2*(60*B*a ^3*c^3 - 9*B*b^6 + 3*A*b^5*c + 78*B*a*b^4*c - 26*A*a*b^3*c^2 + 62*A*a^2*b* c^3 - 186*B*a^2*b^2*c^2))/(c^2*(4*a*c - b^2)^2))*(A*c - 3*B*b + c^4*(-(60* B*a^3*c^3 - 3*B*b^6 + A*b^5*c + 30*B*a*b^4*c - 10*A*a*b^3*c^2 + 30*A*a^2*b *c^3 - 90*B*a^2*b^2*c^2)^2/(c^8*(4*a*c - b^2)^5))^(1/2)))/(4*c^4) + (x^2*( A*c - 3*B*b)*(30*B*a^3*c^3 - 3*B*b^6 + A*b^5*c + 27*B*a*b^4*c - 9*A*a*b^3* c^2 + 23*A*a^2*b*c^3 - 69*B*a^2*b^2*c^2))/(c^6*(4*a*c - b^2)^2))*((a*(A*c - 3*B*b)^2)/c^6 + (((2*(2*a + b*x^2)*(3*B*b - A*c + c^4*(-(60*B*a^3*c^3...