Integrand size = 25, antiderivative size = 363 \[ \int \frac {A+B x^2}{x^3 \left (a+b x^2+c x^4\right )^3} \, dx=\frac {a b B \left (b^2-7 a c\right )-3 A \left (b^4-7 a b^2 c+10 a^2 c^2\right )}{2 a^3 \left (b^2-4 a c\right )^2 x^2}-\frac {a b B-A \left (b^2-2 a c\right )-(A b-2 a B) c x^2}{4 a \left (b^2-4 a c\right ) x^2 \left (a+b x^2+c x^4\right )^2}-\frac {a b B \left (b^2-10 a c\right )-A \left (3 b^4-20 a b^2 c+20 a^2 c^2\right )+c \left (a B \left (b^2-16 a c\right )-3 A \left (b^3-6 a b c\right )\right ) x^2}{4 a^2 \left (b^2-4 a c\right )^2 x^2 \left (a+b x^2+c x^4\right )}+\frac {\left (a b B \left (b^4-10 a b^2 c+30 a^2 c^2\right )-3 A \left (b^6-10 a b^4 c+30 a^2 b^2 c^2-20 a^3 c^3\right )\right ) \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 a^4 \left (b^2-4 a c\right )^{5/2}}-\frac {(3 A b-a B) \log (x)}{a^4}+\frac {(3 A b-a B) \log \left (a+b x^2+c x^4\right )}{4 a^4} \]
1/2*(a*b*B*(-7*a*c+b^2)-3*A*(10*a^2*c^2-7*a*b^2*c+b^4))/a^3/(-4*a*c+b^2)^2 /x^2+1/4*(-a*b*B+A*(-2*a*c+b^2)+(A*b-2*B*a)*c*x^2)/a/(-4*a*c+b^2)/x^2/(c*x ^4+b*x^2+a)^2+1/4*(-a*b*B*(-10*a*c+b^2)+A*(20*a^2*c^2-20*a*b^2*c+3*b^4)-c* (a*B*(-16*a*c+b^2)-3*A*(-6*a*b*c+b^3))*x^2)/a^2/(-4*a*c+b^2)^2/x^2/(c*x^4+ b*x^2+a)+1/2*(a*b*B*(30*a^2*c^2-10*a*b^2*c+b^4)-3*A*(-20*a^3*c^3+30*a^2*b^ 2*c^2-10*a*b^4*c+b^6))*arctanh((2*c*x^2+b)/(-4*a*c+b^2)^(1/2))/a^4/(-4*a*c +b^2)^(5/2)-(3*A*b-B*a)*ln(x)/a^4+1/4*(3*A*b-B*a)*ln(c*x^4+b*x^2+a)/a^4
Time = 0.90 (sec) , antiderivative size = 642, normalized size of antiderivative = 1.77 \[ \int \frac {A+B x^2}{x^3 \left (a+b x^2+c x^4\right )^3} \, dx=\frac {-\frac {2 a A}{x^2}-\frac {a^2 \left (a B \left (-b^2+2 a c-b c x^2\right )+A \left (b^3-3 a b c+b^2 c x^2-2 a c^2 x^2\right )\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {a \left (a B \left (2 b^4-15 a b^2 c+16 a^2 c^2+2 b^3 c x^2-14 a b c^2 x^2\right )-A \left (4 b^5-29 a b^3 c+46 a^2 b c^2+4 b^4 c x^2-26 a b^2 c^2 x^2+28 a^2 c^3 x^2\right )\right )}{\left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+4 (-3 A b+a B) \log (x)+\frac {\left (-a B \left (b^5-10 a b^3 c+30 a^2 b c^2+b^4 \sqrt {b^2-4 a c}-8 a b^2 c \sqrt {b^2-4 a c}+16 a^2 c^2 \sqrt {b^2-4 a c}\right )+3 A \left (b^6-10 a b^4 c+30 a^2 b^2 c^2-20 a^3 c^3+b^5 \sqrt {b^2-4 a c}-8 a b^3 c \sqrt {b^2-4 a c}+16 a^2 b c^2 \sqrt {b^2-4 a c}\right )\right ) \log \left (b-\sqrt {b^2-4 a c}+2 c x^2\right )}{\left (b^2-4 a c\right )^{5/2}}+\frac {\left (a B \left (b^5-10 a b^3 c+30 a^2 b c^2-b^4 \sqrt {b^2-4 a c}+8 a b^2 c \sqrt {b^2-4 a c}-16 a^2 c^2 \sqrt {b^2-4 a c}\right )+3 A \left (-b^6+10 a b^4 c-30 a^2 b^2 c^2+20 a^3 c^3+b^5 \sqrt {b^2-4 a c}-8 a b^3 c \sqrt {b^2-4 a c}+16 a^2 b c^2 \sqrt {b^2-4 a c}\right )\right ) \log \left (b+\sqrt {b^2-4 a c}+2 c x^2\right )}{\left (b^2-4 a c\right )^{5/2}}}{4 a^4} \]
((-2*a*A)/x^2 - (a^2*(a*B*(-b^2 + 2*a*c - b*c*x^2) + A*(b^3 - 3*a*b*c + b^ 2*c*x^2 - 2*a*c^2*x^2)))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (a*(a*B*( 2*b^4 - 15*a*b^2*c + 16*a^2*c^2 + 2*b^3*c*x^2 - 14*a*b*c^2*x^2) - A*(4*b^5 - 29*a*b^3*c + 46*a^2*b*c^2 + 4*b^4*c*x^2 - 26*a*b^2*c^2*x^2 + 28*a^2*c^3 *x^2)))/((b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + 4*(-3*A*b + a*B)*Log[x] + ((-(a*B*(b^5 - 10*a*b^3*c + 30*a^2*b*c^2 + b^4*Sqrt[b^2 - 4*a*c] - 8*a*b^2 *c*Sqrt[b^2 - 4*a*c] + 16*a^2*c^2*Sqrt[b^2 - 4*a*c])) + 3*A*(b^6 - 10*a*b^ 4*c + 30*a^2*b^2*c^2 - 20*a^3*c^3 + b^5*Sqrt[b^2 - 4*a*c] - 8*a*b^3*c*Sqrt [b^2 - 4*a*c] + 16*a^2*b*c^2*Sqrt[b^2 - 4*a*c]))*Log[b - Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(5/2) + ((a*B*(b^5 - 10*a*b^3*c + 30*a^2*b*c^2 - b^4*Sqrt[b^2 - 4*a*c] + 8*a*b^2*c*Sqrt[b^2 - 4*a*c] - 16*a^2*c^2*Sqrt[b^ 2 - 4*a*c]) + 3*A*(-b^6 + 10*a*b^4*c - 30*a^2*b^2*c^2 + 20*a^3*c^3 + b^5*S qrt[b^2 - 4*a*c] - 8*a*b^3*c*Sqrt[b^2 - 4*a*c] + 16*a^2*b*c^2*Sqrt[b^2 - 4 *a*c]))*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(5/2))/(4*a^4)
Time = 0.84 (sec) , antiderivative size = 405, normalized size of antiderivative = 1.12, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {1578, 1235, 25, 1235, 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.
| \(\displaystyle \int \frac {A+B x^2}{x^3 \left (a+b x^2+c x^4\right )^3} \, dx\) |
| \(\Big \downarrow \) 1578 |
| \(\displaystyle \frac {1}{2} \int \frac {B x^2+A}{x^4 \left (c x^4+b x^2+a\right )^3}dx^2\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle \frac {1}{2} \left (\frac {c x^2 (A b-2 a B)-2 a A c-a b B+A b^2}{2 a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {\int -\frac {3 A b^2-a B b+4 (A b-2 a B) c x^2-10 a A c}{x^4 \left (c x^4+b x^2+a\right )^2}dx^2}{2 a \left (b^2-4 a c\right )}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int \frac {3 A b^2-a B b+4 (A b-2 a B) c x^2-10 a A c}{x^4 \left (c x^4+b x^2+a\right )^2}dx^2}{2 a \left (b^2-4 a c\right )}+\frac {c x^2 (A b-2 a B)-2 a A c-a b B+A b^2}{2 a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle \frac {1}{2} \left (\frac {-\frac {\int \frac {2 \left (c \left (a B \left (b^2-16 a c\right )-3 A \left (b^3-6 a b c\right )\right ) x^2+a b B \left (b^2-7 a c\right )-3 A \left (b^4-7 a c b^2+10 a^2 c^2\right )\right )}{x^4 \left (c x^4+b x^2+a\right )}dx^2}{a \left (b^2-4 a c\right )}-\frac {-A \left (20 a^2 c^2-20 a b^2 c+3 b^4\right )+c x^2 \left (a B \left (b^2-16 a c\right )-3 A \left (b^3-6 a b c\right )\right )+a b B \left (b^2-10 a c\right )}{a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{2 a \left (b^2-4 a c\right )}+\frac {c x^2 (A b-2 a B)-2 a A c-a b B+A b^2}{2 a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (\frac {-\frac {2 \int \frac {c \left (a B \left (b^2-16 a c\right )-3 A \left (b^3-6 a b c\right )\right ) x^2+a b B \left (b^2-7 a c\right )-3 A \left (b^4-7 a c b^2+10 a^2 c^2\right )}{x^4 \left (c x^4+b x^2+a\right )}dx^2}{a \left (b^2-4 a c\right )}-\frac {-A \left (20 a^2 c^2-20 a b^2 c+3 b^4\right )+c x^2 \left (a B \left (b^2-16 a c\right )-3 A \left (b^3-6 a b c\right )\right )+a b B \left (b^2-10 a c\right )}{a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{2 a \left (b^2-4 a c\right )}+\frac {c x^2 (A b-2 a B)-2 a A c-a b B+A b^2}{2 a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\) |
| \(\Big \downarrow \) 1200 |
| \(\displaystyle \frac {1}{2} \left (\frac {-\frac {2 \int \left (-\frac {(a B-3 A b) \left (4 a c-b^2\right )^2}{a^2 x^2}+\frac {-\left ((3 A b-a B) c \left (b^2-4 a c\right )^2 x^2\right )+a b B \left (b^4-9 a c b^2+23 a^2 c^2\right )-3 A \left (b^6-9 a c b^4+23 a^2 c^2 b^2-10 a^3 c^3\right )}{a^2 \left (c x^4+b x^2+a\right )}+\frac {a b B \left (b^2-7 a c\right )-3 A \left (b^4-7 a c b^2+10 a^2 c^2\right )}{a x^4}\right )dx^2}{a \left (b^2-4 a c\right )}-\frac {-A \left (20 a^2 c^2-20 a b^2 c+3 b^4\right )+c x^2 \left (a B \left (b^2-16 a c\right )-3 A \left (b^3-6 a b c\right )\right )+a b B \left (b^2-10 a c\right )}{a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{2 a \left (b^2-4 a c\right )}+\frac {c x^2 (A b-2 a B)-2 a A c-a b B+A b^2}{2 a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (\frac {-\frac {-A \left (20 a^2 c^2-20 a b^2 c+3 b^4\right )+c x^2 \left (a B \left (b^2-16 a c\right )-3 A \left (b^3-6 a b c\right )\right )+a b B \left (b^2-10 a c\right )}{a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {2 \left (\frac {\log \left (x^2\right ) \left (b^2-4 a c\right )^2 (3 A b-a B)}{a^2}-\frac {\left (b^2-4 a c\right )^2 (3 A b-a B) \log \left (a+b x^2+c x^4\right )}{2 a^2}-\frac {a b B \left (b^2-7 a c\right )-3 A \left (10 a^2 c^2-7 a b^2 c+b^4\right )}{a x^2}-\frac {\left (a b B \left (30 a^2 c^2-10 a b^2 c+b^4\right )-3 A \left (-20 a^3 c^3+30 a^2 b^2 c^2-10 a b^4 c+b^6\right )\right ) \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{a^2 \sqrt {b^2-4 a c}}\right )}{a \left (b^2-4 a c\right )}}{2 a \left (b^2-4 a c\right )}+\frac {c x^2 (A b-2 a B)-2 a A c-a b B+A b^2}{2 a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\) |
((A*b^2 - a*b*B - 2*a*A*c + (A*b - 2*a*B)*c*x^2)/(2*a*(b^2 - 4*a*c)*x^2*(a + b*x^2 + c*x^4)^2) + (-((a*b*B*(b^2 - 10*a*c) - A*(3*b^4 - 20*a*b^2*c + 20*a^2*c^2) + c*(a*B*(b^2 - 16*a*c) - 3*A*(b^3 - 6*a*b*c))*x^2)/(a*(b^2 - 4*a*c)*x^2*(a + b*x^2 + c*x^4))) - (2*(-((a*b*B*(b^2 - 7*a*c) - 3*A*(b^4 - 7*a*b^2*c + 10*a^2*c^2))/(a*x^2)) - ((a*b*B*(b^4 - 10*a*b^2*c + 30*a^2*c^ 2) - 3*A*(b^6 - 10*a*b^4*c + 30*a^2*b^2*c^2 - 20*a^3*c^3))*ArcTanh[(b + 2* c*x^2)/Sqrt[b^2 - 4*a*c]])/(a^2*Sqrt[b^2 - 4*a*c]) + ((3*A*b - a*B)*(b^2 - 4*a*c)^2*Log[x^2])/a^2 - ((3*A*b - a*B)*(b^2 - 4*a*c)^2*Log[a + b*x^2 + c *x^4])/(2*a^2)))/(a*(b^2 - 4*a*c)))/(2*a*(b^2 - 4*a*c)))/2
3.2.31.3.1 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)*(f*(b*c*d - b^2*e + 2 *a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^m *(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d* m + b*e*m) - b*d*(3*c*d - b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p] )
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 = 0.28 (sec) , antiderivative size = 616, normalized size of antiderivative = 1.70
| method | result | size |
| default | \(-\frac {A}{2 a^{3} x^{2}}+\frac {\left (-3 A b +B a \right ) \ln \left (x \right )}{a^{4}}-\frac {\frac {\frac {a \,c^{2} \left (14 A \,a^{2} c^{2}-13 A a \,b^{2} c +2 A \,b^{4}+7 a^{2} b B c -B a \,b^{3}\right ) x^{6}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {a c \left (74 A \,a^{2} b \,c^{2}-55 A a \,b^{3} c +8 A \,b^{5}-16 a^{3} B \,c^{2}+29 B \,a^{2} b^{2} c -4 B a \,b^{4}\right ) x^{4}}{32 a^{2} c^{2}-16 a \,b^{2} c +2 b^{4}}+\frac {a \left (18 A \,a^{3} c^{3}+7 A \,a^{2} b^{2} c^{2}-12 A a \,b^{4} c +2 b^{6} A +B \,a^{3} b \,c^{2}+6 B \,a^{2} b^{3} c -B a \,b^{5}\right ) x^{2}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {a^{2} \left (58 A \,a^{2} b \,c^{2}-36 A a \,b^{3} c +5 A \,b^{5}-24 a^{3} B \,c^{2}+21 B \,a^{2} b^{2} c -3 B a \,b^{4}\right )}{32 a^{2} c^{2}-16 a \,b^{2} c +2 b^{4}}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2}}+\frac {\frac {\left (-48 A \,a^{2} b \,c^{3}+24 A a \,b^{3} c^{2}-3 A \,b^{5} c +16 B \,a^{3} c^{3}-8 B \,a^{2} b^{2} c^{2}+B a \,b^{4} c \right ) \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{2 c}+\frac {2 \left (30 A \,a^{3} c^{3}-69 A \,a^{2} b^{2} c^{2}+27 A a \,b^{4} c -3 b^{6} A +23 B \,a^{3} b \,c^{2}-9 B \,a^{2} b^{3} c +B a \,b^{5}-\frac {\left (-48 A \,a^{2} b \,c^{3}+24 A a \,b^{3} c^{2}-3 A \,b^{5} c +16 B \,a^{3} c^{3}-8 B \,a^{2} b^{2} c^{2}+B a \,b^{4} c \right ) b}{2 c}\right ) \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}}{2 a^{4}}\) | \(616\) |
| risch | \(\text {Expression too large to display}\) | \(1345\) |
-1/2*A/a^3/x^2+(-3*A*b+B*a)/a^4*ln(x)-1/2/a^4*((a*c^2*(14*A*a^2*c^2-13*A*a *b^2*c+2*A*b^4+7*B*a^2*b*c-B*a*b^3)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^6+1/2*a*c *(74*A*a^2*b*c^2-55*A*a*b^3*c+8*A*b^5-16*B*a^3*c^2+29*B*a^2*b^2*c-4*B*a*b^ 4)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^4+a*(18*A*a^3*c^3+7*A*a^2*b^2*c^2-12*A*a*b ^4*c+2*A*b^6+B*a^3*b*c^2+6*B*a^2*b^3*c-B*a*b^5)/(16*a^2*c^2-8*a*b^2*c+b^4) *x^2+1/2*a^2*(58*A*a^2*b*c^2-36*A*a*b^3*c+5*A*b^5-24*B*a^3*c^2+21*B*a^2*b^ 2*c-3*B*a*b^4)/(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*(-48*A*a^2*b*c^3+24*A*a*b^3*c^2-3*A*b^5*c+16*B*a^3*c^ 3-8*B*a^2*b^2*c^2+B*a*b^4*c)/c*ln(c*x^4+b*x^2+a)+2*(30*A*a^3*c^3-69*A*a^2* b^2*c^2+27*A*a*b^4*c-3*b^6*A+23*B*a^3*b*c^2-9*B*a^2*b^3*c+B*a*b^5-1/2*(-48 *A*a^2*b*c^3+24*A*a*b^3*c^2-3*A*b^5*c+16*B*a^3*c^3-8*B*a^2*b^2*c^2+B*a*b^4 *c)*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. 1966 vs. \(2 (346) = 692\).
Time = 3.98 (sec) , antiderivative size = 3956, normalized size of antiderivative = 10.90 \[ \int \frac {A+B x^2}{x^3 \left (a+b x^2+c x^4\right )^3} \, dx=\text {Too large to display} \]
[-1/4*(2*A*a^3*b^6 - 24*A*a^4*b^4*c + 96*A*a^5*b^2*c^2 - 128*A*a^6*c^3 - 2 *(120*A*a^4*c^5 + 2*(14*B*a^4*b - 57*A*a^3*b^2)*c^4 - 11*(B*a^3*b^3 - 3*A* a^2*b^4)*c^3 + (B*a^2*b^5 - 3*A*a*b^6)*c^2)*x^8 + (8*(8*B*a^5 - 69*A*a^4*b )*c^4 - 6*(22*B*a^4*b^2 - 81*A*a^3*b^3)*c^3 + 45*(B*a^3*b^4 - 3*A*a^2*b^5) *c^2 - 4*(B*a^2*b^6 - 3*A*a*b^7)*c)*x^6 - 2*(B*a^2*b^7 - 3*A*a*b^8 + 200*A *a^5*c^4 + 2*(2*B*a^5*b - 11*A*a^4*b^2)*c^3 + (23*B*a^4*b^3 - 79*A*a^3*b^4 )*c^2 - 10*(B*a^3*b^5 - 3*A*a^2*b^6)*c)*x^4 - (3*B*a^3*b^6 - 9*A*a^2*b^7 - 8*(12*B*a^6 - 61*A*a^5*b)*c^3 + 2*(54*B*a^5*b^2 - 197*A*a^4*b^3)*c^2 - (3 3*B*a^4*b^4 - 104*A*a^3*b^5)*c)*x^2 - ((60*A*a^3*c^5 + 30*(B*a^3*b - 3*A*a ^2*b^2)*c^4 - 10*(B*a^2*b^3 - 3*A*a*b^4)*c^3 + (B*a*b^5 - 3*A*b^6)*c^2)*x^ 10 + 2*(60*A*a^3*b*c^4 + 30*(B*a^3*b^2 - 3*A*a^2*b^3)*c^3 - 10*(B*a^2*b^4 - 3*A*a*b^5)*c^2 + (B*a*b^6 - 3*A*b^7)*c)*x^8 + (B*a*b^7 - 3*A*b^8 + 120*A *a^4*c^4 + 60*(B*a^4*b - 2*A*a^3*b^2)*c^3 + 10*(B*a^3*b^3 - 3*A*a^2*b^4)*c ^2 - 8*(B*a^2*b^5 - 3*A*a*b^6)*c)*x^6 + 2*(B*a^2*b^6 - 3*A*a*b^7 + 60*A*a^ 4*b*c^3 + 30*(B*a^4*b^2 - 3*A*a^3*b^3)*c^2 - 10*(B*a^3*b^4 - 3*A*a^2*b^5)* c)*x^4 + (B*a^3*b^5 - 3*A*a^2*b^6 + 60*A*a^5*c^3 + 30*(B*a^5*b - 3*A*a^4*b ^2)*c^2 - 10*(B*a^4*b^3 - 3*A*a^3*b^4)*c)*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)) - ((64*(B*a^4 - 3*A*a^3*b)*c^5 - 48*(B*a^3*b^2 - 3*A*a^2*b^3 )*c^4 + 12*(B*a^2*b^4 - 3*A*a*b^5)*c^3 - (B*a*b^6 - 3*A*b^7)*c^2)*x^10 ...
Timed out. \[ \int \frac {A+B x^2}{x^3 \left (a+b x^2+c x^4\right )^3} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {A+B x^2}{x^3 \left (a+b x^2+c x^4\right )^3} \, dx=\text {Exception raised: ValueError} \]
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 = 1.43 (sec) , antiderivative size = 648, normalized size of antiderivative = 1.79 \[ \int \frac {A+B x^2}{x^3 \left (a+b x^2+c x^4\right )^3} \, dx=-\frac {{\left (B a b^{5} - 3 \, A b^{6} - 10 \, B a^{2} b^{3} c + 30 \, A a b^{4} c + 30 \, B a^{3} b c^{2} - 90 \, A a^{2} b^{2} c^{2} + 60 \, A a^{3} c^{3}\right )} \arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{2 \, {\left (a^{4} b^{4} - 8 \, a^{5} b^{2} c + 16 \, a^{6} c^{2}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {3 \, B a b^{4} c^{2} x^{8} - 9 \, A b^{5} c^{2} x^{8} - 24 \, B a^{2} b^{2} c^{3} x^{8} + 72 \, A a b^{3} c^{3} x^{8} + 48 \, B a^{3} c^{4} x^{8} - 144 \, A a^{2} b c^{4} x^{8} + 6 \, B a b^{5} c x^{6} - 18 \, A b^{6} c x^{6} - 44 \, B a^{2} b^{3} c^{2} x^{6} + 136 \, A a b^{4} c^{2} x^{6} + 68 \, B a^{3} b c^{3} x^{6} - 236 \, A a^{2} b^{2} c^{3} x^{6} - 56 \, A a^{3} c^{4} x^{6} + 3 \, B a b^{6} x^{4} - 9 \, A b^{7} x^{4} - 10 \, B a^{2} b^{4} c x^{4} + 38 \, A a b^{5} c x^{4} - 58 \, B a^{3} b^{2} c^{2} x^{4} + 110 \, A a^{2} b^{3} c^{2} x^{4} + 128 \, B a^{4} c^{3} x^{4} - 436 \, A a^{3} b c^{3} x^{4} + 10 \, B a^{2} b^{5} x^{2} - 26 \, A a b^{6} x^{2} - 72 \, B a^{3} b^{3} c x^{2} + 192 \, A a^{2} b^{4} c x^{2} + 92 \, B a^{4} b c^{2} x^{2} - 316 \, A a^{3} b^{2} c^{2} x^{2} - 72 \, A a^{4} c^{3} x^{2} + 9 \, B a^{3} b^{4} - 19 \, A a^{2} b^{5} - 66 \, B a^{4} b^{2} c + 144 \, A a^{3} b^{3} c + 96 \, B a^{5} c^{2} - 260 \, A a^{4} b c^{2}}{8 \, {\left (a^{4} b^{4} - 8 \, a^{5} b^{2} c + 16 \, a^{6} c^{2}\right )} {\left (c x^{4} + b x^{2} + a\right )}^{2}} - \frac {{\left (B a - 3 \, A b\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, a^{4}} + \frac {{\left (B a - 3 \, A b\right )} \log \left (x^{2}\right )}{2 \, a^{4}} - \frac {B a x^{2} - 3 \, A b x^{2} + A a}{2 \, a^{4} x^{2}} \]
-1/2*(B*a*b^5 - 3*A*b^6 - 10*B*a^2*b^3*c + 30*A*a*b^4*c + 30*B*a^3*b*c^2 - 90*A*a^2*b^2*c^2 + 60*A*a^3*c^3)*arctan((2*c*x^2 + b)/sqrt(-b^2 + 4*a*c)) /((a^4*b^4 - 8*a^5*b^2*c + 16*a^6*c^2)*sqrt(-b^2 + 4*a*c)) + 1/8*(3*B*a*b^ 4*c^2*x^8 - 9*A*b^5*c^2*x^8 - 24*B*a^2*b^2*c^3*x^8 + 72*A*a*b^3*c^3*x^8 + 48*B*a^3*c^4*x^8 - 144*A*a^2*b*c^4*x^8 + 6*B*a*b^5*c*x^6 - 18*A*b^6*c*x^6 - 44*B*a^2*b^3*c^2*x^6 + 136*A*a*b^4*c^2*x^6 + 68*B*a^3*b*c^3*x^6 - 236*A* a^2*b^2*c^3*x^6 - 56*A*a^3*c^4*x^6 + 3*B*a*b^6*x^4 - 9*A*b^7*x^4 - 10*B*a^ 2*b^4*c*x^4 + 38*A*a*b^5*c*x^4 - 58*B*a^3*b^2*c^2*x^4 + 110*A*a^2*b^3*c^2* x^4 + 128*B*a^4*c^3*x^4 - 436*A*a^3*b*c^3*x^4 + 10*B*a^2*b^5*x^2 - 26*A*a* b^6*x^2 - 72*B*a^3*b^3*c*x^2 + 192*A*a^2*b^4*c*x^2 + 92*B*a^4*b*c^2*x^2 - 316*A*a^3*b^2*c^2*x^2 - 72*A*a^4*c^3*x^2 + 9*B*a^3*b^4 - 19*A*a^2*b^5 - 66 *B*a^4*b^2*c + 144*A*a^3*b^3*c + 96*B*a^5*c^2 - 260*A*a^4*b*c^2)/((a^4*b^4 - 8*a^5*b^2*c + 16*a^6*c^2)*(c*x^4 + b*x^2 + a)^2) - 1/4*(B*a - 3*A*b)*lo g(c*x^4 + b*x^2 + a)/a^4 + 1/2*(B*a - 3*A*b)*log(x^2)/a^4 - 1/2*(B*a*x^2 - 3*A*b*x^2 + A*a)/(a^4*x^2)
Time = 18.35 (sec) , antiderivative size = 16265, normalized size of antiderivative = 44.81 \[ \int \frac {A+B x^2}{x^3 \left (a+b x^2+c x^4\right )^3} \, dx=\text {Too large to display} \]
(log(((c^5*x^2*(3*A*b^4 + 30*A*a^2*c^2 - B*a*b^3 - 21*A*a*b^2*c + 7*B*a^2* b*c)^3)/(a^9*(4*a*c - b^2)^6) - ((B*a - 3*A*b + a^4*(-(60*A*a^3*c^3 - 3*A* b^6 + B*a*b^5 + 30*A*a*b^4*c - 10*B*a^2*b^3*c + 30*B*a^3*b*c^2 - 90*A*a^2* b^2*c^2)^2/(a^8*(4*a*c - b^2)^5))^(1/2))*(((B*a - 3*A*b + a^4*(-(60*A*a^3* c^3 - 3*A*b^6 + B*a*b^5 + 30*A*a*b^4*c - 10*B*a^2*b^3*c + 30*B*a^3*b*c^2 - 90*A*a^2*b^2*c^2)^2/(a^8*(4*a*c - b^2)^5))^(1/2))*((4*b*c^2*(30*A*a^3*c^3 - 3*A*b^6 + B*a*b^5 + 27*A*a*b^4*c - 9*B*a^2*b^3*c + 23*B*a^3*b*c^2 - 69* A*a^2*b^2*c^2))/(a^3*(4*a*c - b^2)^2) + (2*c^3*x^2*(B*a*b^5 - 300*A*a^3*c^ 3 - 3*A*b^6 + 6*A*a*b^4*c - 2*B*a^2*b^3*c + 10*B*a^3*b*c^2 + 90*A*a^2*b^2* c^2))/(a^3*(4*a*c - b^2)^2) + (b*c^2*(B*a - 3*A*b + a^4*(-(60*A*a^3*c^3 - 3*A*b^6 + B*a*b^5 + 30*A*a*b^4*c - 10*B*a^2*b^3*c + 30*B*a^3*b*c^2 - 90*A* a^2*b^2*c^2)^2/(a^8*(4*a*c - b^2)^5))^(1/2))*(a*b + 3*b^2*x^2 - 10*a*c*x^2 ))/a^4))/(4*a^4) + (c^3*(900*A^2*a^5*c^5 - 36*A^2*b^10 - 4*B^2*a^2*b^8 + 2 4*A*B*a*b^9 - 3078*A^2*a^2*b^6*c^2 + 7533*A^2*a^3*b^4*c^3 - 7020*A^2*a^4*b ^2*c^4 - 302*B^2*a^4*b^4*c^2 + 497*B^2*a^5*b^2*c^3 + 549*A^2*a*b^8*c + 61* B^2*a^3*b^6*c + 1932*A*B*a^3*b^5*c^2 - 4002*A*B*a^4*b^3*c^3 - 366*A*B*a^2* b^7*c + 2340*A*B*a^5*b*c^4))/(a^6*(4*a*c - b^2)^4) - (c^4*x^2*(54*A^2*b^9 + 6*B^2*a^2*b^7 - 36*A*B*a*b^8 + 4311*A^2*a^2*b^5*c^2 - 9900*A^2*a^3*b^3*c ^3 + 409*B^2*a^4*b^3*c^2 - 2400*A*B*a^5*c^4 - 801*A^2*a*b^7*c + 8100*A^2*a ^4*b*c^4 - 89*B^2*a^3*b^5*c - 560*B^2*a^5*b*c^3 - 2664*A*B*a^3*b^4*c^2 ...