Integrand size = 25, antiderivative size = 223 \[ \int \frac {A+B x^2}{x^3 \left (a+b x^2+c x^4\right )^2} \, dx=-\frac {2 A b^2-a b B-6 a A c}{2 a^2 \left (b^2-4 a c\right ) x^2}-\frac {a b B-A \left (b^2-2 a c\right )-(A b-2 a B) c x^2}{2 a \left (b^2-4 a c\right ) x^2 \left (a+b x^2+c x^4\right )}+\frac {\left (a b B \left (b^2-6 a c\right )-2 A \left (b^4-6 a b^2 c+6 a^2 c^2\right )\right ) \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 a^3 \left (b^2-4 a c\right )^{3/2}}-\frac {(2 A b-a B) \log (x)}{a^3}+\frac {(2 A b-a B) \log \left (a+b x^2+c x^4\right )}{4 a^3} \]
1/2*(6*A*a*c-2*A*b^2+B*a*b)/a^2/(-4*a*c+b^2)/x^2+1/2*(-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)+1/2*(a*b*B*(-6*a*c +b^2)-2*A*(6*a^2*c^2-6*a*b^2*c+b^4))*arctanh((2*c*x^2+b)/(-4*a*c+b^2)^(1/2 ))/a^3/(-4*a*c+b^2)^(3/2)-(2*A*b-B*a)*ln(x)/a^3+1/4*(2*A*b-B*a)*ln(c*x^4+b *x^2+a)/a^3
Time = 0.34 (sec) , antiderivative size = 379, normalized size of antiderivative = 1.70 \[ \int \frac {A+B x^2}{x^3 \left (a+b x^2+c x^4\right )^2} \, dx=\frac {-\frac {2 a A}{x^2}-\frac {2 a \left (a B \left (-b^2+2 a c-b c x^2\right )+A \left (b^3-3 a b c+b^2 c x^2-2 a c^2 x^2\right )\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+4 (-2 A b+a B) \log (x)+\frac {\left (a B \left (-b^3+6 a b c-b^2 \sqrt {b^2-4 a c}+4 a c \sqrt {b^2-4 a c}\right )+2 A \left (b^4-6 a b^2 c+6 a^2 c^2+b^3 \sqrt {b^2-4 a c}-4 a b c \sqrt {b^2-4 a c}\right )\right ) \log \left (b-\sqrt {b^2-4 a c}+2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}+\frac {\left (a B \left (b^3-6 a b c-b^2 \sqrt {b^2-4 a c}+4 a c \sqrt {b^2-4 a c}\right )+2 A \left (-b^4+6 a b^2 c-6 a^2 c^2+b^3 \sqrt {b^2-4 a c}-4 a b c \sqrt {b^2-4 a c}\right )\right ) \log \left (b+\sqrt {b^2-4 a c}+2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}}{4 a^3} \]
((-2*a*A)/x^2 - (2*a*(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)) + 4*(-2*A*b + a*B)*Log[x] + ((a*B*(-b^3 + 6*a*b*c - b^2*Sqrt[b^2 - 4*a*c] + 4*a*c*Sqrt[ b^2 - 4*a*c]) + 2*A*(b^4 - 6*a*b^2*c + 6*a^2*c^2 + b^3*Sqrt[b^2 - 4*a*c] - 4*a*b*c*Sqrt[b^2 - 4*a*c]))*Log[b - Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(3/2) + ((a*B*(b^3 - 6*a*b*c - b^2*Sqrt[b^2 - 4*a*c] + 4*a*c*Sqrt[b ^2 - 4*a*c]) + 2*A*(-b^4 + 6*a*b^2*c - 6*a^2*c^2 + b^3*Sqrt[b^2 - 4*a*c] - 4*a*b*c*Sqrt[b^2 - 4*a*c]))*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(3/2))/(4*a^3)
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.
| \(\displaystyle \int \frac {A+B x^2}{x^3 \left (a+b x^2+c x^4\right )^2} \, dx\) |
| \(\Big \downarrow \) 1578 |
| \(\displaystyle \frac {1}{2} \int \frac {B x^2+A}{x^4 \left (c x^4+b x^2+a\right )^2}dx^2\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle \frac {1}{2} \left (\frac {c x^2 (A b-2 a B)-2 a A c-a b B+A b^2}{a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\int -\frac {2 A b^2-a B b+2 (A b-2 a B) c x^2-6 a A c}{x^4 \left (c x^4+b x^2+a\right )}dx^2}{a \left (b^2-4 a c\right )}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int -\frac {-2 (A b-2 a B) c x^2+a b B-2 A \left (b^2-3 a c\right )}{x^4 \left (c x^4+b x^2+a\right )}dx^2}{a \left (b^2-4 a c\right )}+\frac {c x^2 (A b-2 a B)-2 a A c-a b B+A b^2}{a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (\frac {c x^2 (A b-2 a B)-2 a A c-a b B+A b^2}{a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\int -\frac {2 A b^2-a B b+2 (A b-2 a B) c x^2-6 a A c}{x^4 \left (c x^4+b x^2+a\right )}dx^2}{a \left (b^2-4 a c\right )}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int -\frac {-2 (A b-2 a B) c x^2+a b B-2 A \left (b^2-3 a c\right )}{x^4 \left (c x^4+b x^2+a\right )}dx^2}{a \left (b^2-4 a c\right )}+\frac {c x^2 (A b-2 a B)-2 a A c-a b B+A b^2}{a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (\frac {c x^2 (A b-2 a B)-2 a A c-a b B+A b^2}{a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\int -\frac {2 A b^2-a B b+2 (A b-2 a B) c x^2-6 a A c}{x^4 \left (c x^4+b x^2+a\right )}dx^2}{a \left (b^2-4 a c\right )}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int -\frac {-2 (A b-2 a B) c x^2+a b B-2 A \left (b^2-3 a c\right )}{x^4 \left (c x^4+b x^2+a\right )}dx^2}{a \left (b^2-4 a c\right )}+\frac {c x^2 (A b-2 a B)-2 a A c-a b B+A b^2}{a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (\frac {c x^2 (A b-2 a B)-2 a A c-a b B+A b^2}{a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\int -\frac {2 A b^2-a B b+2 (A b-2 a B) c x^2-6 a A c}{x^4 \left (c x^4+b x^2+a\right )}dx^2}{a \left (b^2-4 a c\right )}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int -\frac {-2 (A b-2 a B) c x^2+a b B-2 A \left (b^2-3 a c\right )}{x^4 \left (c x^4+b x^2+a\right )}dx^2}{a \left (b^2-4 a c\right )}+\frac {c x^2 (A b-2 a B)-2 a A c-a b B+A b^2}{a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (\frac {c x^2 (A b-2 a B)-2 a A c-a b B+A b^2}{a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\int -\frac {2 A b^2-a B b+2 (A b-2 a B) c x^2-6 a A c}{x^4 \left (c x^4+b x^2+a\right )}dx^2}{a \left (b^2-4 a c\right )}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int -\frac {-2 (A b-2 a B) c x^2+a b B-2 A \left (b^2-3 a c\right )}{x^4 \left (c x^4+b x^2+a\right )}dx^2}{a \left (b^2-4 a c\right )}+\frac {c x^2 (A b-2 a B)-2 a A c-a b B+A b^2}{a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (\frac {c x^2 (A b-2 a B)-2 a A c-a b B+A b^2}{a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\int -\frac {2 A b^2-a B b+2 (A b-2 a B) c x^2-6 a A c}{x^4 \left (c x^4+b x^2+a\right )}dx^2}{a \left (b^2-4 a c\right )}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int -\frac {-2 (A b-2 a B) c x^2+a b B-2 A \left (b^2-3 a c\right )}{x^4 \left (c x^4+b x^2+a\right )}dx^2}{a \left (b^2-4 a c\right )}+\frac {c x^2 (A b-2 a B)-2 a A c-a b B+A b^2}{a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (\frac {c x^2 (A b-2 a B)-2 a A c-a b B+A b^2}{a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\int -\frac {2 A b^2-a B b+2 (A b-2 a B) c x^2-6 a A c}{x^4 \left (c x^4+b x^2+a\right )}dx^2}{a \left (b^2-4 a c\right )}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int -\frac {-2 (A b-2 a B) c x^2+a b B-2 A \left (b^2-3 a c\right )}{x^4 \left (c x^4+b x^2+a\right )}dx^2}{a \left (b^2-4 a c\right )}+\frac {c x^2 (A b-2 a B)-2 a A c-a b B+A b^2}{a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (\frac {c x^2 (A b-2 a B)-2 a A c-a b B+A b^2}{a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\int -\frac {2 A b^2-a B b+2 (A b-2 a B) c x^2-6 a A c}{x^4 \left (c x^4+b x^2+a\right )}dx^2}{a \left (b^2-4 a c\right )}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int -\frac {-2 (A b-2 a B) c x^2+a b B-2 A \left (b^2-3 a c\right )}{x^4 \left (c x^4+b x^2+a\right )}dx^2}{a \left (b^2-4 a c\right )}+\frac {c x^2 (A b-2 a B)-2 a A c-a b B+A b^2}{a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (\frac {c x^2 (A b-2 a B)-2 a A c-a b B+A b^2}{a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\int -\frac {2 A b^2-a B b+2 (A b-2 a B) c x^2-6 a A c}{x^4 \left (c x^4+b x^2+a\right )}dx^2}{a \left (b^2-4 a c\right )}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int -\frac {-2 (A b-2 a B) c x^2+a b B-2 A \left (b^2-3 a c\right )}{x^4 \left (c x^4+b x^2+a\right )}dx^2}{a \left (b^2-4 a c\right )}+\frac {c x^2 (A b-2 a B)-2 a A c-a b B+A b^2}{a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (\frac {c x^2 (A b-2 a B)-2 a A c-a b B+A b^2}{a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\int -\frac {2 A b^2-a B b+2 (A b-2 a B) c x^2-6 a A c}{x^4 \left (c x^4+b x^2+a\right )}dx^2}{a \left (b^2-4 a c\right )}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int -\frac {-2 (A b-2 a B) c x^2+a b B-2 A \left (b^2-3 a c\right )}{x^4 \left (c x^4+b x^2+a\right )}dx^2}{a \left (b^2-4 a c\right )}+\frac {c x^2 (A b-2 a B)-2 a A c-a b B+A b^2}{a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (\frac {c x^2 (A b-2 a B)-2 a A c-a b B+A b^2}{a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\int -\frac {2 A b^2-a B b+2 (A b-2 a B) c x^2-6 a A c}{x^4 \left (c x^4+b x^2+a\right )}dx^2}{a \left (b^2-4 a c\right )}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int -\frac {-2 (A b-2 a B) c x^2+a b B-2 A \left (b^2-3 a c\right )}{x^4 \left (c x^4+b x^2+a\right )}dx^2}{a \left (b^2-4 a c\right )}+\frac {c x^2 (A b-2 a B)-2 a A c-a b B+A b^2}{a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (\frac {c x^2 (A b-2 a B)-2 a A c-a b B+A b^2}{a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\int -\frac {2 A b^2-a B b+2 (A b-2 a B) c x^2-6 a A c}{x^4 \left (c x^4+b x^2+a\right )}dx^2}{a \left (b^2-4 a c\right )}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int -\frac {-2 (A b-2 a B) c x^2+a b B-2 A \left (b^2-3 a c\right )}{x^4 \left (c x^4+b x^2+a\right )}dx^2}{a \left (b^2-4 a c\right )}+\frac {c x^2 (A b-2 a B)-2 a A c-a b B+A b^2}{a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (\frac {c x^2 (A b-2 a B)-2 a A c-a b B+A b^2}{a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\int -\frac {2 A b^2-a B b+2 (A b-2 a B) c x^2-6 a A c}{x^4 \left (c x^4+b x^2+a\right )}dx^2}{a \left (b^2-4 a c\right )}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int -\frac {-2 (A b-2 a B) c x^2+a b B-2 A \left (b^2-3 a c\right )}{x^4 \left (c x^4+b x^2+a\right )}dx^2}{a \left (b^2-4 a c\right )}+\frac {c x^2 (A b-2 a B)-2 a A c-a b B+A b^2}{a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (\frac {c x^2 (A b-2 a B)-2 a A c-a b B+A b^2}{a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\int -\frac {2 A b^2-a B b+2 (A b-2 a B) c x^2-6 a A c}{x^4 \left (c x^4+b x^2+a\right )}dx^2}{a \left (b^2-4 a c\right )}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int -\frac {-2 (A b-2 a B) c x^2+a b B-2 A \left (b^2-3 a c\right )}{x^4 \left (c x^4+b x^2+a\right )}dx^2}{a \left (b^2-4 a c\right )}+\frac {c x^2 (A b-2 a B)-2 a A c-a b B+A b^2}{a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (\frac {c x^2 (A b-2 a B)-2 a A c-a b B+A b^2}{a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\int -\frac {2 A b^2-a B b+2 (A b-2 a B) c x^2-6 a A c}{x^4 \left (c x^4+b x^2+a\right )}dx^2}{a \left (b^2-4 a c\right )}\right )\) |
3.2.17.3.1 Defintions of rubi rules used
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 = 0.16 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.35
| method | result | size |
| default | \(-\frac {A}{2 a^{2} x^{2}}+\frac {\left (-2 A b +B a \right ) \ln \left (x \right )}{a^{3}}-\frac {\frac {\frac {a c \left (2 A a c -A \,b^{2}+a b B \right ) x^{2}}{4 a c -b^{2}}+\frac {a \left (3 A a b c -A \,b^{3}-2 a^{2} B c +B a \,b^{2}\right )}{4 a c -b^{2}}}{c \,x^{4}+b \,x^{2}+a}+\frac {\frac {\left (-8 A a b \,c^{2}+2 A \,b^{3} c +4 B \,a^{2} c^{2}-B a \,b^{2} c \right ) \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{2 c}+\frac {2 \left (6 A \,a^{2} c^{2}-10 A a \,b^{2} c +2 A \,b^{4}+5 a^{2} b B c -B a \,b^{3}-\frac {\left (-8 A a b \,c^{2}+2 A \,b^{3} c +4 B \,a^{2} c^{2}-B a \,b^{2} c \right ) b}{2 c}\right ) \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{4 a c -b^{2}}}{2 a^{3}}\) | \(300\) |
| risch | \(\frac {-\frac {c \left (6 A a c -2 A \,b^{2}+a b B \right ) x^{4}}{2 a^{2} \left (4 a c -b^{2}\right )}-\frac {\left (7 A a b c -2 A \,b^{3}-2 a^{2} B c +B a \,b^{2}\right ) x^{2}}{2 \left (4 a c -b^{2}\right ) a^{2}}-\frac {A}{2 a}}{x^{2} \left (c \,x^{4}+b \,x^{2}+a \right )}-\frac {2 \ln \left (x \right ) A b}{a^{3}}+\frac {\ln \left (x \right ) B}{a^{2}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (64 a^{6} c^{3}-48 b^{2} a^{5} c^{2}+12 a^{4} b^{4} c -a^{3} b^{6}\right ) \textit {\_Z}^{2}+\left (-128 b \,c^{3} a^{3} A +96 b^{3} c^{2} a^{2} A -24 b^{5} c a A +2 b^{7} A +64 a^{4} c^{3} B -48 a^{3} b^{2} c^{2} B +12 a^{2} b^{4} c B -a \,b^{6} B \right ) \textit {\_Z} +36 a \,c^{4} A^{2}-8 b^{2} c^{3} A^{2}-28 a b \,c^{3} A B +6 b^{3} c^{2} A B +16 a^{2} c^{3} B^{2}-3 b^{2} a \,c^{2} B^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (-160 a^{7} c^{3}+128 a^{6} b^{2} c^{2}-34 a^{5} b^{4} c +3 a^{4} b^{6}\right ) \textit {\_R}^{2}+\left (136 A \,a^{4} b \,c^{3}-66 A \,a^{3} b^{3} c^{2}+8 A \,a^{2} b^{5} c -80 B \,a^{5} c^{3}+36 B \,a^{4} b^{2} c^{2}-4 B \,a^{3} b^{4} c \right ) \textit {\_R} -72 A^{2} a^{2} c^{4}+48 A^{2} a \,b^{2} c^{3}-8 A^{2} b^{4} c^{2}-24 A B \,a^{2} b \,c^{3}+8 A B a \,b^{3} c^{2}-2 B^{2} a^{2} b^{2} c^{2}\right ) x^{2}+\left (16 a^{7} b \,c^{2}-8 a^{6} b^{3} c +a^{5} b^{5}\right ) \textit {\_R}^{2}+\left (-24 A \,a^{5} c^{3}+78 A \,a^{4} b^{2} c^{2}-34 A \,a^{3} b^{4} c +4 A \,a^{2} b^{6}-36 B \,a^{5} b \,c^{2}+17 B \,a^{4} b^{3} c -2 B \,a^{3} b^{5}\right ) \textit {\_R} -96 A^{2} a^{2} b \,c^{3}+56 A^{2} a \,b^{3} c^{2}-8 A^{2} b^{5} c +48 A B \,a^{3} c^{3}-44 A B \,a^{2} b^{2} c^{2}+8 A B a \,b^{4} c +8 B^{2} a^{3} b \,c^{2}-2 B^{2} a^{2} b^{3} c \right )\right )}{2}\) | \(690\) |
-1/2*A/a^2/x^2+(-2*A*b+B*a)/a^3*ln(x)-1/2/a^3*((a*c*(2*A*a*c-A*b^2+B*a*b)/ (4*a*c-b^2)*x^2+a*(3*A*a*b*c-A*b^3-2*B*a^2*c+B*a*b^2)/(4*a*c-b^2))/(c*x^4+ b*x^2+a)+1/(4*a*c-b^2)*(1/2*(-8*A*a*b*c^2+2*A*b^3*c+4*B*a^2*c^2-B*a*b^2*c) /c*ln(c*x^4+b*x^2+a)+2*(6*A*a^2*c^2-10*A*a*b^2*c+2*A*b^4+5*a^2*b*B*c-B*a*b ^3-1/2*(-8*A*a*b*c^2+2*A*b^3*c+4*B*a^2*c^2-B*a*b^2*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. 806 vs. \(2 (207) = 414\).
Time = 1.15 (sec) , antiderivative size = 1635, normalized size of antiderivative = 7.33 \[ \int \frac {A+B x^2}{x^3 \left (a+b x^2+c x^4\right )^2} \, dx=\text {Too large to display} \]
[-1/4*(2*A*a^2*b^4 - 16*A*a^3*b^2*c + 32*A*a^4*c^2 + 2*(24*A*a^3*c^3 + 2*( 2*B*a^3*b - 7*A*a^2*b^2)*c^2 - (B*a^2*b^3 - 2*A*a*b^4)*c)*x^4 - 2*(B*a^2*b ^4 - 2*A*a*b^5 + 4*(2*B*a^4 - 7*A*a^3*b)*c^2 - 3*(2*B*a^3*b^2 - 5*A*a^2*b^ 3)*c)*x^2 + ((12*A*a^2*c^3 + 6*(B*a^2*b - 2*A*a*b^2)*c^2 - (B*a*b^3 - 2*A* b^4)*c)*x^6 - (B*a*b^4 - 2*A*b^5 - 12*A*a^2*b*c^2 - 6*(B*a^2*b^2 - 2*A*a*b ^3)*c)*x^4 - (B*a^2*b^3 - 2*A*a*b^4 - 12*A*a^3*c^2 - 6*(B*a^3*b - 2*A*a^2* b^2)*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)) + ((16*(B*a^3 - 2*A* a^2*b)*c^3 - 8*(B*a^2*b^2 - 2*A*a*b^3)*c^2 + (B*a*b^4 - 2*A*b^5)*c)*x^6 + (B*a*b^5 - 2*A*b^6 + 16*(B*a^3*b - 2*A*a^2*b^2)*c^2 - 8*(B*a^2*b^3 - 2*A*a *b^4)*c)*x^4 + (B*a^2*b^4 - 2*A*a*b^5 + 16*(B*a^4 - 2*A*a^3*b)*c^2 - 8*(B* a^3*b^2 - 2*A*a^2*b^3)*c)*x^2)*log(c*x^4 + b*x^2 + a) - 4*((16*(B*a^3 - 2* A*a^2*b)*c^3 - 8*(B*a^2*b^2 - 2*A*a*b^3)*c^2 + (B*a*b^4 - 2*A*b^5)*c)*x^6 + (B*a*b^5 - 2*A*b^6 + 16*(B*a^3*b - 2*A*a^2*b^2)*c^2 - 8*(B*a^2*b^3 - 2*A *a*b^4)*c)*x^4 + (B*a^2*b^4 - 2*A*a*b^5 + 16*(B*a^4 - 2*A*a^3*b)*c^2 - 8*( B*a^3*b^2 - 2*A*a^2*b^3)*c)*x^2)*log(x))/((a^3*b^4*c - 8*a^4*b^2*c^2 + 16* a^5*c^3)*x^6 + (a^3*b^5 - 8*a^4*b^3*c + 16*a^5*b*c^2)*x^4 + (a^4*b^4 - 8*a ^5*b^2*c + 16*a^6*c^2)*x^2), -1/4*(2*A*a^2*b^4 - 16*A*a^3*b^2*c + 32*A*a^4 *c^2 + 2*(24*A*a^3*c^3 + 2*(2*B*a^3*b - 7*A*a^2*b^2)*c^2 - (B*a^2*b^3 - 2* A*a*b^4)*c)*x^4 - 2*(B*a^2*b^4 - 2*A*a*b^5 + 4*(2*B*a^4 - 7*A*a^3*b)*c^...
Timed out. \[ \int \frac {A+B x^2}{x^3 \left (a+b x^2+c x^4\right )^2} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {A+B x^2}{x^3 \left (a+b x^2+c x^4\right )^2} \, dx=\text {Exception raised: ValueError} \]
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 = 0.56 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.12 \[ \int \frac {A+B x^2}{x^3 \left (a+b x^2+c x^4\right )^2} \, dx=-\frac {{\left (B a b^{3} - 2 \, A b^{4} - 6 \, B a^{2} b c + 12 \, A a b^{2} c - 12 \, A a^{2} c^{2}\right )} \arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{2 \, {\left (a^{3} b^{2} - 4 \, a^{4} c\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {B a b c x^{4} - 2 \, A b^{2} c x^{4} + 6 \, A a c^{2} x^{4} + B a b^{2} x^{2} - 2 \, A b^{3} x^{2} - 2 \, B a^{2} c x^{2} + 7 \, A a b c x^{2} - A a b^{2} + 4 \, A a^{2} c}{2 \, {\left (c x^{6} + b x^{4} + a x^{2}\right )} {\left (a^{2} b^{2} - 4 \, a^{3} c\right )}} - \frac {{\left (B a - 2 \, A b\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, a^{3}} + \frac {{\left (B a - 2 \, A b\right )} \log \left (x^{2}\right )}{2 \, a^{3}} \]
-1/2*(B*a*b^3 - 2*A*b^4 - 6*B*a^2*b*c + 12*A*a*b^2*c - 12*A*a^2*c^2)*arcta n((2*c*x^2 + b)/sqrt(-b^2 + 4*a*c))/((a^3*b^2 - 4*a^4*c)*sqrt(-b^2 + 4*a*c )) + 1/2*(B*a*b*c*x^4 - 2*A*b^2*c*x^4 + 6*A*a*c^2*x^4 + B*a*b^2*x^2 - 2*A* b^3*x^2 - 2*B*a^2*c*x^2 + 7*A*a*b*c*x^2 - A*a*b^2 + 4*A*a^2*c)/((c*x^6 + b *x^4 + a*x^2)*(a^2*b^2 - 4*a^3*c)) - 1/4*(B*a - 2*A*b)*log(c*x^4 + b*x^2 + a)/a^3 + 1/2*(B*a - 2*A*b)*log(x^2)/a^3
Time = 13.33 (sec) , antiderivative size = 10034, normalized size of antiderivative = 45.00 \[ \int \frac {A+B x^2}{x^3 \left (a+b x^2+c x^4\right )^2} \, dx=\text {Too large to display} \]
(log(((c^4*(2*A*b - B*a)*(6*A*a*c - 2*A*b^2 + B*a*b)^2)/(a^6*(4*a*c - b^2) ^2) - ((((B*a - 2*A*b + a^3*(-(2*A*b^4 + 12*A*a^2*c^2 - B*a*b^3 - 12*A*a*b ^2*c + 6*B*a^2*b*c)^2/(a^6*(4*a*c - b^2)^3))^(1/2))*((b*c^2*(B*a - 2*A*b + a^3*(-(2*A*b^4 + 12*A*a^2*c^2 - B*a*b^3 - 12*A*a*b^2*c + 6*B*a^2*b*c)^2/( a^6*(4*a*c - b^2)^3))^(1/2))*(a*b + 3*b^2*x^2 - 10*a*c*x^2))/a^3 + (4*b*c^ 2*(2*A*b^4 + 6*A*a^2*c^2 - B*a*b^3 - 10*A*a*b^2*c + 5*B*a^2*b*c))/(a^2*(4* a*c - b^2)) + (2*c^3*x^2*(2*A*b^4 - 60*A*a^2*c^2 - B*a*b^3 + 4*A*a*b^2*c + 10*B*a^2*b*c))/(a^2*(4*a*c - b^2))))/(4*a^3) + (c^3*(36*A^2*a^3*c^3 - 16* A^2*b^6 - 4*B^2*a^2*b^4 + 16*A*B*a*b^5 - 216*A^2*a^2*b^2*c^2 + 116*A^2*a*b ^4*c + 17*B^2*a^3*b^2*c - 92*A*B*a^2*b^3*c + 108*A*B*a^3*b*c^2))/(a^4*(4*a *c - b^2)^2) - (2*c^4*x^2*(12*A^2*b^5 + 3*B^2*a^2*b^3 - 12*A*B*a*b^4 - 60* A*B*a^3*c^2 - 82*A^2*a*b^3*c - 10*B^2*a^3*b*c + 138*A^2*a^2*b*c^2 + 61*A*B *a^2*b^2*c))/(a^4*(4*a*c - b^2)^2))*(B*a - 2*A*b + a^3*(-(2*A*b^4 + 12*A*a ^2*c^2 - B*a*b^3 - 12*A*a*b^2*c + 6*B*a^2*b*c)^2/(a^6*(4*a*c - b^2)^3))^(1 /2)))/(4*a^3) + (c^5*x^2*(6*A*a*c - 2*A*b^2 + B*a*b)^3)/(a^6*(4*a*c - b^2) ^3))*((c^4*(2*A*b - B*a)*(6*A*a*c - 2*A*b^2 + B*a*b)^2)/(a^6*(4*a*c - b^2) ^2) - ((((2*A*b - B*a + a^3*(-(2*A*b^4 + 12*A*a^2*c^2 - B*a*b^3 - 12*A*a*b ^2*c + 6*B*a^2*b*c)^2/(a^6*(4*a*c - b^2)^3))^(1/2))*((4*b*c^2*(2*A*b^4 + 6 *A*a^2*c^2 - B*a*b^3 - 10*A*a*b^2*c + 5*B*a^2*b*c))/(a^2*(4*a*c - b^2)) - (b*c^2*(2*A*b - B*a + a^3*(-(2*A*b^4 + 12*A*a^2*c^2 - B*a*b^3 - 12*A*a*...