Integrand size = 25, antiderivative size = 254 \[ \int \frac {x^9 \left (A+B x^2\right )}{\left (a+b x^2+c x^4\right )^3} \, dx=-\frac {x^6 \left (a (b B-2 A c)+\left (b^2 B-A b c-2 a B c\right ) x^2\right )}{4 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {x^2 \left (2 a \left (b^3 B-7 a b B c+6 a A c^2\right )+\left (2 b^4 B-15 a b^2 B c+6 a A b c^2+16 a^2 B c^2\right ) x^2\right )}{4 c^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {\left (b^5 B-10 a b^3 B c+30 a^2 b B c^2-12 a^2 A c^3\right ) \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 c^3 \left (b^2-4 a c\right )^{5/2}}+\frac {B \log \left (a+b x^2+c x^4\right )}{4 c^3} \]
-1/4*x^6*(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^2*(2*a*(6*A*a*c^2-7*B*a*b*c+B*b^3)+(6*A*a*b*c^2+16*B*a^2 *c^2-15*B*a*b^2*c+2*B*b^4)*x^2)/c^2/(-4*a*c+b^2)^2/(c*x^4+b*x^2+a)+1/2*(-1 2*A*a^2*c^3+30*B*a^2*b*c^2-10*B*a*b^3*c+B*b^5)*arctanh((2*c*x^2+b)/(-4*a*c +b^2)^(1/2))/c^3/(-4*a*c+b^2)^(5/2)+1/4*B*ln(c*x^4+b*x^2+a)/c^3
Time = 0.30 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.39 \[ \int \frac {x^9 \left (A+B x^2\right )}{\left (a+b x^2+c x^4\right )^3} \, dx=\frac {\frac {-b^6 B+b^5 c \left (A+4 B x^2\right )-2 a b^3 c^2 \left (4 A+15 B x^2\right )+2 a^2 b c^3 \left (11 A+25 B x^2\right )+4 a^2 c^3 \left (8 a B-5 A c x^2\right )+b^4 c \left (11 a B-2 A c x^2\right )+a b^2 c^2 \left (-39 a B+16 A c x^2\right )}{\left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {2 a^3 B c^2+b^4 (b B-A c) x^2+a b^2 \left (b^2 B+4 A c^2 x^2-b c \left (A+5 B x^2\right )\right )+a^2 c \left (-4 b^2 B-2 A c^2 x^2+b c \left (3 A+5 B x^2\right )\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {2 c \left (b^5 B-10 a b^3 B c+30 a^2 b B c^2-12 a^2 A c^3\right ) \arctan \left (\frac {b+2 c x^2}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{5/2}}+B c \log \left (a+b x^2+c x^4\right )}{4 c^4} \]
((-(b^6*B) + b^5*c*(A + 4*B*x^2) - 2*a*b^3*c^2*(4*A + 15*B*x^2) + 2*a^2*b* c^3*(11*A + 25*B*x^2) + 4*a^2*c^3*(8*a*B - 5*A*c*x^2) + b^4*c*(11*a*B - 2* A*c*x^2) + a*b^2*c^2*(-39*a*B + 16*A*c*x^2))/((b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (2*a^3*B*c^2 + b^4*(b*B - A*c)*x^2 + a*b^2*(b^2*B + 4*A*c^2*x^2 - b*c*(A + 5*B*x^2)) + a^2*c*(-4*b^2*B - 2*A*c^2*x^2 + b*c*(3*A + 5*B*x^2 )))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) - (2*c*(b^5*B - 10*a*b^3*B*c + 3 0*a^2*b*B*c^2 - 12*a^2*A*c^3)*ArcTan[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]])/(- b^2 + 4*a*c)^(5/2) + B*c*Log[a + b*x^2 + c*x^4])/(4*c^4)
Time = 0.57 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.17, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {1578, 1233, 1233, 27, 1142, 1083, 219, 1103}
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.
| \(\displaystyle \int \frac {x^9 \left (A+B x^2\right )}{\left (a+b x^2+c x^4\right )^3} \, dx\) |
| \(\Big \downarrow \) 1578 |
| \(\displaystyle \frac {1}{2} \int \frac {x^8 \left (B x^2+A\right )}{\left (c x^4+b x^2+a\right )^3}dx^2\) |
| \(\Big \downarrow \) 1233 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int \frac {x^4 \left (2 B \left (b^2-4 a c\right ) x^2+3 a (b B-2 A c)\right )}{\left (c x^4+b x^2+a\right )^2}dx^2}{2 c \left (b^2-4 a c\right )}-\frac {x^6 \left (x^2 \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\) |
| \(\Big \downarrow \) 1233 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {\int \frac {2 \left (B \left (b^2-4 a c\right )^2 x^2+a \left (B b^3-7 a B c b+6 a A c^2\right )\right )}{c x^4+b x^2+a}dx^2}{c \left (b^2-4 a c\right )}-\frac {x^2 \left (x^2 \left (16 a^2 B c^2+6 a A b c^2-15 a b^2 B c+2 b^4 B\right )+2 a \left (6 a A c^2-7 a b B c+b^3 B\right )\right )}{c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{2 c \left (b^2-4 a c\right )}-\frac {x^6 \left (x^2 \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {2 \int \frac {B \left (b^2-4 a c\right )^2 x^2+a \left (B b^3-7 a B c b+6 a A c^2\right )}{c x^4+b x^2+a}dx^2}{c \left (b^2-4 a c\right )}-\frac {x^2 \left (x^2 \left (16 a^2 B c^2+6 a A b c^2-15 a b^2 B c+2 b^4 B\right )+2 a \left (6 a A c^2-7 a b B c+b^3 B\right )\right )}{c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{2 c \left (b^2-4 a c\right )}-\frac {x^6 \left (x^2 \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {2 \left (\frac {B \left (b^2-4 a c\right )^2 \int \frac {2 c x^2+b}{c x^4+b x^2+a}dx^2}{2 c}-\frac {\left (-12 a^2 A c^3+30 a^2 b B c^2-10 a b^3 B c+b^5 B\right ) \int \frac {1}{c x^4+b x^2+a}dx^2}{2 c}\right )}{c \left (b^2-4 a c\right )}-\frac {x^2 \left (x^2 \left (16 a^2 B c^2+6 a A b c^2-15 a b^2 B c+2 b^4 B\right )+2 a \left (6 a A c^2-7 a b B c+b^3 B\right )\right )}{c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{2 c \left (b^2-4 a c\right )}-\frac {x^6 \left (x^2 \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {2 \left (\frac {\left (-12 a^2 A c^3+30 a^2 b B c^2-10 a b^3 B c+b^5 B\right ) \int \frac {1}{-x^4+b^2-4 a c}d\left (2 c x^2+b\right )}{c}+\frac {B \left (b^2-4 a c\right )^2 \int \frac {2 c x^2+b}{c x^4+b x^2+a}dx^2}{2 c}\right )}{c \left (b^2-4 a c\right )}-\frac {x^2 \left (x^2 \left (16 a^2 B c^2+6 a A b c^2-15 a b^2 B c+2 b^4 B\right )+2 a \left (6 a A c^2-7 a b B c+b^3 B\right )\right )}{c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{2 c \left (b^2-4 a c\right )}-\frac {x^6 \left (x^2 \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {2 \left (\frac {B \left (b^2-4 a c\right )^2 \int \frac {2 c x^2+b}{c x^4+b x^2+a}dx^2}{2 c}+\frac {\left (-12 a^2 A c^3+30 a^2 b B c^2-10 a b^3 B c+b^5 B\right ) \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{c \sqrt {b^2-4 a c}}\right )}{c \left (b^2-4 a c\right )}-\frac {x^2 \left (x^2 \left (16 a^2 B c^2+6 a A b c^2-15 a b^2 B c+2 b^4 B\right )+2 a \left (6 a A c^2-7 a b B c+b^3 B\right )\right )}{c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{2 c \left (b^2-4 a c\right )}-\frac {x^6 \left (x^2 \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {2 \left (\frac {\left (-12 a^2 A c^3+30 a^2 b B c^2-10 a b^3 B c+b^5 B\right ) \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{c \sqrt {b^2-4 a c}}+\frac {B \left (b^2-4 a c\right )^2 \log \left (a+b x^2+c x^4\right )}{2 c}\right )}{c \left (b^2-4 a c\right )}-\frac {x^2 \left (x^2 \left (16 a^2 B c^2+6 a A b c^2-15 a b^2 B c+2 b^4 B\right )+2 a \left (6 a A c^2-7 a b B c+b^3 B\right )\right )}{c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{2 c \left (b^2-4 a c\right )}-\frac {x^6 \left (x^2 \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\) |
(-1/2*(x^6*(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^2*(2*a*(b^3*B - 7*a*b*B*c + 6*a*A*c^2) + (2*b^4*B - 15*a*b^2*B*c + 6*a*A*b*c^2 + 16*a^2*B*c^2)*x^2))/(c*(b^2 - 4 *a*c)*(a + b*x^2 + c*x^4))) + (2*(((b^5*B - 10*a*b^3*B*c + 30*a^2*b*B*c^2 - 12*a^2*A*c^3)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(c*Sqrt[b^2 - 4* a*c]) + (B*(b^2 - 4*a*c)^2*Log[a + b*x^2 + c*x^4])/(2*c)))/(c*(b^2 - 4*a*c )))/(2*c*(b^2 - 4*a*c)))/2
3.2.25.3.1 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[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(494\) vs. \(2(242)=484\).
Time = 0.23 (sec) , antiderivative size = 495, normalized size of antiderivative = 1.95
| method | result | size |
| default | \(\frac {-\frac {\left (10 A \,a^{2} c^{3}-8 A a \,b^{2} c^{2}+A \,b^{4} c -25 B \,a^{2} b \,c^{2}+15 B a \,b^{3} c -2 B \,b^{5}\right ) x^{6}}{c^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {\left (2 A \,a^{2} b \,c^{3}+8 A a \,b^{3} c^{2}-A \,b^{5} c +32 B \,a^{3} c^{3}+11 B \,a^{2} b^{2} c^{2}-19 B a \,b^{4} c +3 B \,b^{6}\right ) x^{4}}{2 c^{3} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {a \left (6 A \,a^{2} c^{3}-10 A a \,b^{2} c^{2}+A \,b^{4} c -31 B \,a^{2} b \,c^{2}+22 B a \,b^{3} c -3 B \,b^{5}\right ) x^{2}}{c^{3} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {a^{2} \left (10 A a b \,c^{2}-A \,b^{3} c +24 B \,a^{2} c^{2}-21 B a \,b^{2} c +3 B \,b^{4}\right )}{2 c^{3} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{2 \left (c \,x^{4}+b \,x^{2}+a \right )^{2}}+\frac {\frac {\left (16 B \,a^{2} c^{2}-8 B a \,b^{2} c +B \,b^{4}\right ) \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{2 c}+\frac {2 \left (6 A \,a^{2} c^{2}-7 a^{2} b B c +B a \,b^{3}-\frac {\left (16 B \,a^{2} c^{2}-8 B a \,b^{2} c +B \,b^{4}\right ) b}{2 c}\right ) \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{2 c^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}\) | \(495\) |
| risch | \(\text {Expression too large to display}\) | \(2336\) |
1/2*(-1/c^2*(10*A*a^2*c^3-8*A*a*b^2*c^2+A*b^4*c-25*B*a^2*b*c^2+15*B*a*b^3* c-2*B*b^5)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^6+1/2*(2*A*a^2*b*c^3+8*A*a*b^3*c^2 -A*b^5*c+32*B*a^3*c^3+11*B*a^2*b^2*c^2-19*B*a*b^4*c+3*B*b^6)/c^3/(16*a^2*c ^2-8*a*b^2*c+b^4)*x^4-a*(6*A*a^2*c^3-10*A*a*b^2*c^2+A*b^4*c-31*B*a^2*b*c^2 +22*B*a*b^3*c-3*B*b^5)/c^3/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2+1/2*a^2*(10*A*a* b*c^2-A*b^3*c+24*B*a^2*c^2-21*B*a*b^2*c+3*B*b^4)/c^3/(16*a^2*c^2-8*a*b^2*c +b^4))/(c*x^4+b*x^2+a)^2+1/2/c^2/(16*a^2*c^2-8*a*b^2*c+b^4)*(1/2*(16*B*a^2 *c^2-8*B*a*b^2*c+B*b^4)/c*ln(c*x^4+b*x^2+a)+2*(6*A*a^2*c^2-7*a^2*b*B*c+B*a *b^3-1/2*(16*B*a^2*c^2-8*B*a*b^2*c+B*b^4)*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. 1071 vs. \(2 (242) = 484\).
Time = 0.40 (sec) , antiderivative size = 2167, normalized size of antiderivative = 8.53 \[ \int \frac {x^9 \left (A+B x^2\right )}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Too large to display} \]
[1/4*(3*B*a^2*b^6 + 2*(2*B*b^7*c + 40*A*a^3*c^5 - 2*(50*B*a^3*b + 21*A*a^2 *b^2)*c^4 + (85*B*a^2*b^3 + 12*A*a*b^4)*c^3 - (23*B*a*b^5 + A*b^6)*c^2)*x^ 6 + (3*B*b^8 - 8*(16*B*a^4 + A*a^3*b)*c^4 - 6*(2*B*a^3*b^2 + 5*A*a^2*b^3)* c^3 + 3*(29*B*a^2*b^4 + 4*A*a*b^5)*c^2 - (31*B*a*b^6 + A*b^7)*c)*x^4 - 8*( 12*B*a^5 + 5*A*a^4*b)*c^3 + 2*(54*B*a^4*b^2 + 7*A*a^3*b^3)*c^2 + 2*(3*B*a* b^7 + 24*A*a^4*c^4 - 2*(62*B*a^4*b + 23*A*a^3*b^2)*c^3 + 7*(17*B*a^3*b^3 + 2*A*a^2*b^4)*c^2 - (34*B*a^2*b^5 + A*a*b^6)*c)*x^2 - ((B*b^5*c^2 - 10*B*a *b^3*c^3 + 30*B*a^2*b*c^4 - 12*A*a^2*c^5)*x^8 + B*a^2*b^5 - 10*B*a^3*b^3*c + 30*B*a^4*b*c^2 - 12*A*a^4*c^3 + 2*(B*b^6*c - 10*B*a*b^4*c^2 + 30*B*a^2* b^2*c^3 - 12*A*a^2*b*c^4)*x^6 + (B*b^7 - 8*B*a*b^5*c + 10*B*a^2*b^3*c^2 - 24*A*a^3*c^4 + 12*(5*B*a^3*b - A*a^2*b^2)*c^3)*x^4 + 2*(B*a*b^6 - 10*B*a^2 *b^4*c + 30*B*a^3*b^2*c^2 - 12*A*a^3*b*c^3)*x^2)*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)) - (33*B*a^3*b^4 + A*a^2*b^5)*c + (B*a^2*b^6 - 12*B*a^3*b^4 *c + 48*B*a^4*b^2*c^2 - 64*B*a^5*c^3 + (B*b^6*c^2 - 12*B*a*b^4*c^3 + 48*B* a^2*b^2*c^4 - 64*B*a^3*c^5)*x^8 + 2*(B*b^7*c - 12*B*a*b^5*c^2 + 48*B*a^2*b ^3*c^3 - 64*B*a^3*b*c^4)*x^6 + (B*b^8 - 10*B*a*b^6*c + 24*B*a^2*b^4*c^2 + 32*B*a^3*b^2*c^3 - 128*B*a^4*c^4)*x^4 + 2*(B*a*b^7 - 12*B*a^2*b^5*c + 48*B *a^3*b^3*c^2 - 64*B*a^4*b*c^3)*x^2)*log(c*x^4 + b*x^2 + a))/(a^2*b^6*c^3 - 12*a^3*b^4*c^4 + 48*a^4*b^2*c^5 - 64*a^5*c^6 + (b^6*c^5 - 12*a*b^4*c^6...
Timed out. \[ \int \frac {x^9 \left (A+B x^2\right )}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {x^9 \left (A+B x^2\right )}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Exception raised: ValueError} \]
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 = 1.52 (sec) , antiderivative size = 466, normalized size of antiderivative = 1.83 \[ \int \frac {x^9 \left (A+B x^2\right )}{\left (a+b x^2+c x^4\right )^3} \, dx=-\frac {{\left (B b^{5} - 10 \, B a b^{3} c + 30 \, B a^{2} b c^{2} - 12 \, A a^{2} c^{3}\right )} \arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{2 \, {\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {B \log \left (c x^{4} + b x^{2} + a\right )}{4 \, c^{3}} - \frac {3 \, B b^{4} c^{2} x^{8} - 24 \, B a b^{2} c^{3} x^{8} + 48 \, B a^{2} c^{4} x^{8} - 2 \, B b^{5} c x^{6} + 12 \, B a b^{3} c^{2} x^{6} + 4 \, A b^{4} c^{2} x^{6} - 4 \, B a^{2} b c^{3} x^{6} - 32 \, A a b^{2} c^{3} x^{6} + 40 \, A a^{2} c^{4} x^{6} - 3 \, B b^{6} x^{4} + 20 \, B a b^{4} c x^{4} + 2 \, A b^{5} c x^{4} - 22 \, B a^{2} b^{2} c^{2} x^{4} - 16 \, A a b^{3} c^{2} x^{4} + 32 \, B a^{3} c^{3} x^{4} - 4 \, A a^{2} b c^{3} x^{4} - 6 \, B a b^{5} x^{2} + 40 \, B a^{2} b^{3} c x^{2} + 4 \, A a b^{4} c x^{2} - 28 \, B a^{3} b c^{2} x^{2} - 40 \, A a^{2} b^{2} c^{2} x^{2} + 24 \, A a^{3} c^{3} x^{2} - 3 \, B a^{2} b^{4} + 18 \, B a^{3} b^{2} c + 2 \, A a^{2} b^{3} c - 20 \, A a^{3} b c^{2}}{8 \, {\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} {\left (c x^{4} + b x^{2} + a\right )}^{2}} \]
-1/2*(B*b^5 - 10*B*a*b^3*c + 30*B*a^2*b*c^2 - 12*A*a^2*c^3)*arctan((2*c*x^ 2 + b)/sqrt(-b^2 + 4*a*c))/((b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*sqrt(-b^2 + 4*a*c)) + 1/4*B*log(c*x^4 + b*x^2 + a)/c^3 - 1/8*(3*B*b^4*c^2*x^8 - 24* B*a*b^2*c^3*x^8 + 48*B*a^2*c^4*x^8 - 2*B*b^5*c*x^6 + 12*B*a*b^3*c^2*x^6 + 4*A*b^4*c^2*x^6 - 4*B*a^2*b*c^3*x^6 - 32*A*a*b^2*c^3*x^6 + 40*A*a^2*c^4*x^ 6 - 3*B*b^6*x^4 + 20*B*a*b^4*c*x^4 + 2*A*b^5*c*x^4 - 22*B*a^2*b^2*c^2*x^4 - 16*A*a*b^3*c^2*x^4 + 32*B*a^3*c^3*x^4 - 4*A*a^2*b*c^3*x^4 - 6*B*a*b^5*x^ 2 + 40*B*a^2*b^3*c*x^2 + 4*A*a*b^4*c*x^2 - 28*B*a^3*b*c^2*x^2 - 40*A*a^2*b ^2*c^2*x^2 + 24*A*a^3*c^3*x^2 - 3*B*a^2*b^4 + 18*B*a^3*b^2*c + 2*A*a^2*b^3 *c - 20*A*a^3*b*c^2)/((b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*(c*x^4 + b*x^2 + a)^2)
Time = 10.74 (sec) , antiderivative size = 3062, normalized size of antiderivative = 12.06 \[ \int \frac {x^9 \left (A+B x^2\right )}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Too large to display} \]
((x^4*(3*B*b^6 + 32*B*a^3*c^3 - A*b^5*c - 19*B*a*b^4*c + 8*A*a*b^3*c^2 + 2 *A*a^2*b*c^3 + 11*B*a^2*b^2*c^2))/(4*c^3*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (x^6*(2*B*b^5 - 10*A*a^2*c^3 - A*b^4*c - 15*B*a*b^3*c + 8*A*a*b^2*c^2 + 2 5*B*a^2*b*c^2))/(2*c^2*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (a*(24*B*a^3*c^2 + 3*B*a*b^4 - A*a*b^3*c + 10*A*a^2*b*c^2 - 21*B*a^2*b^2*c))/(4*c^3*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) - (x^2*(6*A*a^3*c^3 - 3*B*a*b^5 + A*a*b^4*c + 22* B*a^2*b^3*c - 31*B*a^3*b*c^2 - 10*A*a^2*b^2*c^2))/(2*c^3*(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 ) - (log(((B^2*a)/c^4 - ((B + c^3*(-(B*b^5 - 12*A*a^2*c^3 - 10*B*a*b^3*c + 30*B*a^2*b*c^2)^2/(c^6*(4*a*c - b^2)^5))^(1/2))*((8*B*a)/c - (2*(B + c^3* (-(B*b^5 - 12*A*a^2*c^3 - 10*B*a*b^3*c + 30*B*a^2*b*c^2)^2/(c^6*(4*a*c - b ^2)^5))^(1/2))*(2*a + b*x^2))/c + (2*x^2*(3*B*b^5 - 12*A*a^2*c^3 - 26*B*a* b^3*c + 62*B*a^2*b*c^2))/(c*(4*a*c - b^2)^2)))/(4*c^3) + (B*x^2*(B*b^5 - 6 *A*a^2*c^3 - 9*B*a*b^3*c + 23*B*a^2*b*c^2))/(c^4*(4*a*c - b^2)^2))*((B^2*a )/c^4 - ((B - c^3*(-(B*b^5 - 12*A*a^2*c^3 - 10*B*a*b^3*c + 30*B*a^2*b*c^2) ^2/(c^6*(4*a*c - b^2)^5))^(1/2))*((8*B*a)/c - (2*(B - c^3*(-(B*b^5 - 12*A* a^2*c^3 - 10*B*a*b^3*c + 30*B*a^2*b*c^2)^2/(c^6*(4*a*c - b^2)^5))^(1/2))*( 2*a + b*x^2))/c + (2*x^2*(3*B*b^5 - 12*A*a^2*c^3 - 26*B*a*b^3*c + 62*B*a^2 *b*c^2))/(c*(4*a*c - b^2)^2)))/(4*c^3) + (B*x^2*(B*b^5 - 6*A*a^2*c^3 - 9*B *a*b^3*c + 23*B*a^2*b*c^2))/(c^4*(4*a*c - b^2)^2)))*(2*B*b^10 - 2048*B*...