Integrand size = 28, antiderivative size = 514 \[ \int \frac {A+B x+C x^2}{x^2 \left (a+b x^2+c x^4\right )^2} \, dx=-\frac {3 A b^2-10 a A c-a b C}{2 a^2 \left (b^2-4 a c\right ) x}+\frac {B \left (b^2-2 a c+b c x^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {A \left (b^2-2 a c\right )-a b C+c (A b-2 a C) x^2}{2 a \left (b^2-4 a c\right ) x \left (a+b x^2+c x^4\right )}-\frac {\sqrt {c} \left (A \left (3 b^3-16 a b c+3 b^2 \sqrt {b^2-4 a c}-10 a c \sqrt {b^2-4 a c}\right )-a \left (b^2-12 a c+b \sqrt {b^2-4 a c}\right ) C\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {c} \left (3 A b^2-10 a A c-a b C-\frac {A \left (3 b^3-16 a b c\right )-a \left (b^2-12 a c\right ) C}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a^2 \left (b^2-4 a c\right ) \sqrt {b+\sqrt {b^2-4 a c}}}+\frac {b B \left (b^2-6 a c\right ) \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 a^2 \left (b^2-4 a c\right )^{3/2}}+\frac {B \log (x)}{a^2}-\frac {B \log \left (a+b x^2+c x^4\right )}{4 a^2} \]
1/2*(10*A*a*c-3*A*b^2+C*a*b)/a^2/(-4*a*c+b^2)/x+1/2*B*(b*c*x^2-2*a*c+b^2)/ a/(-4*a*c+b^2)/(c*x^4+b*x^2+a)+1/2*(A*(-2*a*c+b^2)-a*b*C+c*(A*b-2*C*a)*x^2 )/a/(-4*a*c+b^2)/x/(c*x^4+b*x^2+a)+1/2*b*B*(-6*a*c+b^2)*arctanh((2*c*x^2+b )/(-4*a*c+b^2)^(1/2))/a^2/(-4*a*c+b^2)^(3/2)+B*ln(x)/a^2-1/4*B*ln(c*x^4+b* x^2+a)/a^2-1/4*arctan(x*2^(1/2)*c^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2))*c^(1 /2)*(-a*C*(b^2-12*a*c+b*(-4*a*c+b^2)^(1/2))+A*(3*b^3-16*a*b*c+3*b^2*(-4*a* c+b^2)^(1/2)-10*a*c*(-4*a*c+b^2)^(1/2)))/a^2/(-4*a*c+b^2)^(3/2)*2^(1/2)/(b -(-4*a*c+b^2)^(1/2))^(1/2)-1/4*arctan(x*2^(1/2)*c^(1/2)/(b+(-4*a*c+b^2)^(1 /2))^(1/2))*c^(1/2)*(3*A*b^2-10*a*A*c-a*b*C+(-A*(-16*a*b*c+3*b^3)+a*(-12*a *c+b^2)*C)/(-4*a*c+b^2)^(1/2))/a^2/(-4*a*c+b^2)*2^(1/2)/(b+(-4*a*c+b^2)^(1 /2))^(1/2)
Time = 1.19 (sec) , antiderivative size = 559, normalized size of antiderivative = 1.09 \[ \int \frac {A+B x+C x^2}{x^2 \left (a+b x^2+c x^4\right )^2} \, dx=\frac {-\frac {4 A}{x}+\frac {-4 a^2 c (B+C x)-2 A b^2 x \left (b+c x^2\right )+2 a \left (2 A c^2 x^3+b^2 (B+C x)+b c x (3 A+x (B+C x))\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {\sqrt {2} \sqrt {c} \left (A \left (-3 b^3+16 a b c-3 b^2 \sqrt {b^2-4 a c}+10 a c \sqrt {b^2-4 a c}\right )+a \left (b^2-12 a c+b \sqrt {b^2-4 a c}\right ) C\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {2} \sqrt {c} \left (A \left (3 b^3-16 a b c-3 b^2 \sqrt {b^2-4 a c}+10 a c \sqrt {b^2-4 a c}\right )+a \left (-b^2+12 a c+b \sqrt {b^2-4 a c}\right ) C\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt {b+\sqrt {b^2-4 a c}}}+4 B \log (x)-\frac {B \left (b^3-6 a b c+b^2 \sqrt {b^2-4 a c}-4 a c \sqrt {b^2-4 a c}\right ) \log \left (-b+\sqrt {b^2-4 a c}-2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac {B \left (-b^3+6 a b c+b^2 \sqrt {b^2-4 a c}-4 a c \sqrt {b^2-4 a c}\right ) \log \left (b+\sqrt {b^2-4 a c}+2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}}{4 a^2} \]
((-4*A)/x + (-4*a^2*c*(B + C*x) - 2*A*b^2*x*(b + c*x^2) + 2*a*(2*A*c^2*x^3 + b^2*(B + C*x) + b*c*x*(3*A + x*(B + C*x))))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (Sqrt[2]*Sqrt[c]*(A*(-3*b^3 + 16*a*b*c - 3*b^2*Sqrt[b^2 - 4*a*c ] + 10*a*c*Sqrt[b^2 - 4*a*c]) + a*(b^2 - 12*a*c + b*Sqrt[b^2 - 4*a*c])*C)* ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3 /2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*Sqrt[c]*(A*(3*b^3 - 16*a*b*c - 3*b^2*Sqrt[b^2 - 4*a*c] + 10*a*c*Sqrt[b^2 - 4*a*c]) + a*(-b^2 + 12*a*c + b*Sqrt[b^2 - 4*a*c])*C)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a *c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]) + 4*B*Log[x] - (B *(b^3 - 6*a*b*c + b^2*Sqrt[b^2 - 4*a*c] - 4*a*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) - (B*(-b^3 + 6*a*b*c + b^2*Sqrt[b^2 - 4*a*c] - 4*a*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^2)
Time = 1.13 (sec) , antiderivative size = 534, normalized size of antiderivative = 1.04, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {2193, 27, 1434, 1165, 25, 1200, 1600, 25, 1604, 1480, 218, 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.
| \(\displaystyle \int \frac {A+B x+C x^2}{x^2 \left (a+b x^2+c x^4\right )^2} \, dx\) |
| \(\Big \downarrow \) 2193 |
| \(\displaystyle \int \frac {C x^2+A}{x^2 \left (c x^4+b x^2+a\right )^2}dx+\int \frac {B}{x \left (c x^4+b x^2+a\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {C x^2+A}{x^2 \left (c x^4+b x^2+a\right )^2}dx+B \int \frac {1}{x \left (c x^4+b x^2+a\right )^2}dx\) |
| \(\Big \downarrow \) 1434 |
| \(\displaystyle \int \frac {C x^2+A}{x^2 \left (c x^4+b x^2+a\right )^2}dx+\frac {1}{2} B \int \frac {1}{x^2 \left (c x^4+b x^2+a\right )^2}dx^2\) |
| \(\Big \downarrow \) 1165 |
| \(\displaystyle \int \frac {C x^2+A}{x^2 \left (c x^4+b x^2+a\right )^2}dx+\frac {1}{2} B \left (\frac {-2 a c+b^2+b c x^2}{a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\int -\frac {b^2+c x^2 b-4 a c}{x^2 \left (c x^4+b x^2+a\right )}dx^2}{a \left (b^2-4 a c\right )}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int \frac {C x^2+A}{x^2 \left (c x^4+b x^2+a\right )^2}dx+\frac {1}{2} B \left (\frac {\int \frac {b^2+c x^2 b-4 a c}{x^2 \left (c x^4+b x^2+a\right )}dx^2}{a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x^2}{a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )\) |
| \(\Big \downarrow \) 1200 |
| \(\displaystyle \int \frac {C x^2+A}{x^2 \left (c x^4+b x^2+a\right )^2}dx+\frac {1}{2} B \left (\frac {\int \left (\frac {b^2-4 a c}{a x^2}+\frac {-c \left (b^2-4 a c\right ) x^2-b \left (b^2-5 a c\right )}{a \left (c x^4+b x^2+a\right )}\right )dx^2}{a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x^2}{a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )\) |
| \(\Big \downarrow \) 1600 |
| \(\displaystyle -\frac {\int -\frac {3 A b^2-a C b+3 c (A b-2 a C) x^2-10 a A c}{x^2 \left (c x^4+b x^2+a\right )}dx}{2 a \left (b^2-4 a c\right )}+\frac {1}{2} B \left (\frac {\int \left (\frac {b^2-4 a c}{a x^2}+\frac {-c \left (b^2-4 a c\right ) x^2-b \left (b^2-5 a c\right )}{a \left (c x^4+b x^2+a\right )}\right )dx^2}{a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x^2}{a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )+\frac {A \left (b^2-2 a c\right )+c x^2 (A b-2 a C)-a b C}{2 a x \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {3 A b^2-a C b+3 c (A b-2 a C) x^2-10 a A c}{x^2 \left (c x^4+b x^2+a\right )}dx}{2 a \left (b^2-4 a c\right )}+\frac {1}{2} B \left (\frac {\int \left (\frac {b^2-4 a c}{a x^2}+\frac {-c \left (b^2-4 a c\right ) x^2-b \left (b^2-5 a c\right )}{a \left (c x^4+b x^2+a\right )}\right )dx^2}{a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x^2}{a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )+\frac {A \left (b^2-2 a c\right )+c x^2 (A b-2 a C)-a b C}{2 a x \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\) |
| \(\Big \downarrow \) 1604 |
| \(\displaystyle \frac {-\frac {\int \frac {c \left (3 A b^2-a C b-10 a A c\right ) x^2+A \left (3 b^3-13 a b c\right )-a \left (b^2-6 a c\right ) C}{c x^4+b x^2+a}dx}{a}-\frac {-10 a A c-a b C+3 A b^2}{a x}}{2 a \left (b^2-4 a c\right )}+\frac {1}{2} B \left (\frac {\int \left (\frac {b^2-4 a c}{a x^2}+\frac {-c \left (b^2-4 a c\right ) x^2-b \left (b^2-5 a c\right )}{a \left (c x^4+b x^2+a\right )}\right )dx^2}{a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x^2}{a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )+\frac {A \left (b^2-2 a c\right )+c x^2 (A b-2 a C)-a b C}{2 a x \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {-\frac {\frac {1}{2} c \left (\frac {A \left (3 b^3-16 a b c\right )-a C \left (b^2-12 a c\right )}{\sqrt {b^2-4 a c}}-10 a A c-a b C+3 A b^2\right ) \int \frac {1}{c x^2+\frac {1}{2} \left (b-\sqrt {b^2-4 a c}\right )}dx-\frac {c \left (A \left (-3 b^2 \sqrt {b^2-4 a c}+10 a c \sqrt {b^2-4 a c}-16 a b c+3 b^3\right )-a C \left (-b \sqrt {b^2-4 a c}-12 a c+b^2\right )\right ) \int \frac {1}{c x^2+\frac {1}{2} \left (b+\sqrt {b^2-4 a c}\right )}dx}{2 \sqrt {b^2-4 a c}}}{a}-\frac {-10 a A c-a b C+3 A b^2}{a x}}{2 a \left (b^2-4 a c\right )}+\frac {1}{2} B \left (\frac {\int \left (\frac {b^2-4 a c}{a x^2}+\frac {-c \left (b^2-4 a c\right ) x^2-b \left (b^2-5 a c\right )}{a \left (c x^4+b x^2+a\right )}\right )dx^2}{a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x^2}{a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )+\frac {A \left (b^2-2 a c\right )+c x^2 (A b-2 a C)-a b C}{2 a x \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {1}{2} B \left (\frac {\int \left (\frac {b^2-4 a c}{a x^2}+\frac {-c \left (b^2-4 a c\right ) x^2-b \left (b^2-5 a c\right )}{a \left (c x^4+b x^2+a\right )}\right )dx^2}{a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x^2}{a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )+\frac {-\frac {\frac {\sqrt {c} \left (\frac {A \left (3 b^3-16 a b c\right )-a C \left (b^2-12 a c\right )}{\sqrt {b^2-4 a c}}-10 a A c-a b C+3 A b^2\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {c} \left (A \left (-3 b^2 \sqrt {b^2-4 a c}+10 a c \sqrt {b^2-4 a c}-16 a b c+3 b^3\right )-a C \left (-b \sqrt {b^2-4 a c}-12 a c+b^2\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {2} \sqrt {b^2-4 a c} \sqrt {\sqrt {b^2-4 a c}+b}}}{a}-\frac {-10 a A c-a b C+3 A b^2}{a x}}{2 a \left (b^2-4 a c\right )}+\frac {A \left (b^2-2 a c\right )+c x^2 (A b-2 a C)-a b C}{2 a x \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {\frac {\sqrt {c} \left (\frac {A \left (3 b^3-16 a b c\right )-a C \left (b^2-12 a c\right )}{\sqrt {b^2-4 a c}}-10 a A c-a b C+3 A b^2\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {c} \left (A \left (-3 b^2 \sqrt {b^2-4 a c}+10 a c \sqrt {b^2-4 a c}-16 a b c+3 b^3\right )-a C \left (-b \sqrt {b^2-4 a c}-12 a c+b^2\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {2} \sqrt {b^2-4 a c} \sqrt {\sqrt {b^2-4 a c}+b}}}{a}-\frac {-10 a A c-a b C+3 A b^2}{a x}}{2 a \left (b^2-4 a c\right )}+\frac {A \left (b^2-2 a c\right )+c x^2 (A b-2 a C)-a b C}{2 a x \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {1}{2} B \left (\frac {\frac {b \left (b^2-6 a c\right ) \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{a \sqrt {b^2-4 a c}}+\frac {\log \left (x^2\right ) \left (b^2-4 a c\right )}{a}-\frac {\left (b^2-4 a c\right ) \log \left (a+b x^2+c x^4\right )}{2 a}}{a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x^2}{a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )\) |
(A*(b^2 - 2*a*c) - a*b*C + c*(A*b - 2*a*C)*x^2)/(2*a*(b^2 - 4*a*c)*x*(a + b*x^2 + c*x^4)) + (-((3*A*b^2 - 10*a*A*c - a*b*C)/(a*x)) - ((Sqrt[c]*(3*A* b^2 - 10*a*A*c - a*b*C + (A*(3*b^3 - 16*a*b*c) - a*(b^2 - 12*a*c)*C)/Sqrt[ b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sq rt[2]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (Sqrt[c]*(A*(3*b^3 - 16*a*b*c - 3*b^2 *Sqrt[b^2 - 4*a*c] + 10*a*c*Sqrt[b^2 - 4*a*c]) - a*(b^2 - 12*a*c - b*Sqrt[ b^2 - 4*a*c])*C)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/ (Sqrt[2]*Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/a)/(2*a*(b^2 - 4* a*c)) + (B*((b^2 - 2*a*c + b*c*x^2)/(a*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + ((b*(b^2 - 6*a*c)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(a*Sqrt[b^2 - 4*a*c]) + ((b^2 - 4*a*c)*Log[x^2])/a - ((b^2 - 4*a*c)*Log[a + b*x^2 + c* x^4])/(2*a))/(a*(b^2 - 4*a*c))))/2
3.1.35.3.1 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_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(d + e*x)^(m + 1)*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*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*Simp[b*c*d*e*(2*p - m + 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && LtQ[p, -1] && IntQuadraticQ[a, b, c, d, e, m, p, 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[(x_)^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp [1/2 Subst[Int[x^((m - 1)/2)*(a + b*x + c*x^2)^p, x], x, x^2], x] /; Free Q[{a, b, c, p}, x] && IntegerQ[(m - 1)/2]
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*x)^(m + 1))*(a + b*x^2 + c*x^4)^(p + 1) *((d*(b^2 - 2*a*c) - a*b*e + (b*d - 2*a*e)*c*x^2)/(2*a*f*(p + 1)*(b^2 - 4*a *c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c)) Int[(f*x)^m*(a + b*x^2 + c *x^4)^(p + 1)*Simp[d*(b^2*(m + 2*(p + 1) + 1) - 2*a*c*(m + 4*(p + 1) + 1)) - a*b*e*(m + 1) + c*(m + 2*(2*p + 3) + 1)*(b*d - 2*a*e)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && Int egerQ[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[d*(f*x)^(m + 1)*((a + b*x^2 + c*x^4)^(p + 1) /(a*f*(m + 1))), x] + Simp[1/(a*f^2*(m + 1)) Int[(f*x)^(m + 2)*(a + b*x^2 + c*x^4)^p*Simp[a*e*(m + 1) - b*d*(m + 2*p + 3) - c*d*(m + 4*p + 5)*x^2, x ], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[ m, -1] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])
Int[(Pq_)*((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_S ymbol] :> Module[{q = Expon[Pq, x], k}, Int[Sum[Coeff[Pq, x, 2*k]*x^(2*k), {k, 0, q/2 + 1}]*(d*x)^m*(a + b*x^2 + c*x^4)^p, x] + Simp[1/d Int[Sum[Coe ff[Pq, x, 2*k + 1]*x^(2*k), {k, 0, (q + 1)/2}]*(d*x)^(m + 1)*(a + b*x^2 + c *x^4)^p, x], x]] /; FreeQ[{a, b, c, d, m, p}, x] && PolyQ[Pq, x] && !PolyQ [Pq, x^2]
Time = 0.23 (sec) , antiderivative size = 667, normalized size of antiderivative = 1.30
| method | result | size |
| default | \(-\frac {A}{a^{2} x}+\frac {\ln \left (x \right ) B}{a^{2}}-\frac {\frac {\frac {c \left (2 A a c -A \,b^{2}+a b C \right ) x^{3}}{8 a c -2 b^{2}}+\frac {x^{2} B a b c}{8 a c -2 b^{2}}+\frac {\left (3 A a b c -A \,b^{3}-2 a^{2} c C +C a \,b^{2}\right ) x}{8 a c -2 b^{2}}-\frac {a B \left (2 a c -b^{2}\right )}{2 \left (4 a c -b^{2}\right )}}{c \,x^{4}+b \,x^{2}+a}+\frac {2 c \left (\frac {\frac {\left (12 B a b c \sqrt {-4 a c +b^{2}}-2 B \,b^{3} \sqrt {-4 a c +b^{2}}+32 B \,a^{2} c^{2}-16 B a \,b^{2} c +2 B \,b^{4}\right ) \ln \left (2 c \,x^{2}+\sqrt {-4 a c +b^{2}}+b \right )}{4 c}+\frac {\left (16 A a b c \sqrt {-4 a c +b^{2}}-3 A \,b^{3} \sqrt {-4 a c +b^{2}}+40 A \,a^{2} c^{2}-22 A a \,b^{2} c +3 A \,b^{4}-12 C \sqrt {-4 a c +b^{2}}\, a^{2} c +C \sqrt {-4 a c +b^{2}}\, a \,b^{2}+4 C \,a^{2} b c -C a \,b^{3}\right ) \sqrt {2}\, \arctan \left (\frac {c x \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{2 \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}}{16 a c -4 b^{2}}+\frac {-\frac {\left (12 B a b c \sqrt {-4 a c +b^{2}}-2 B \,b^{3} \sqrt {-4 a c +b^{2}}-32 B \,a^{2} c^{2}+16 B a \,b^{2} c -2 B \,b^{4}\right ) \ln \left (-2 c \,x^{2}+\sqrt {-4 a c +b^{2}}-b \right )}{4 c}+\frac {\left (16 A a b c \sqrt {-4 a c +b^{2}}-3 A \,b^{3} \sqrt {-4 a c +b^{2}}-40 A \,a^{2} c^{2}+22 A a \,b^{2} c -3 A \,b^{4}-12 C \sqrt {-4 a c +b^{2}}\, a^{2} c +C \sqrt {-4 a c +b^{2}}\, a \,b^{2}-4 C \,a^{2} b c +C a \,b^{3}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c x \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{2 \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}}{16 a c -4 b^{2}}\right )}{4 a c -b^{2}}}{a^{2}}\) | \(667\) |
| risch | \(\text {Expression too large to display}\) | \(3288\) |
-A/a^2/x+ln(x)/a^2*B-1/a^2*((1/2*c*(2*A*a*c-A*b^2+C*a*b)/(4*a*c-b^2)*x^3+1 /2/(4*a*c-b^2)*x^2*B*a*b*c+1/2*(3*A*a*b*c-A*b^3-2*C*a^2*c+C*a*b^2)/(4*a*c- b^2)*x-1/2*a*B*(2*a*c-b^2)/(4*a*c-b^2))/(c*x^4+b*x^2+a)+2/(4*a*c-b^2)*c*(1 /(16*a*c-4*b^2)*(1/4*(12*B*a*b*c*(-4*a*c+b^2)^(1/2)-2*B*b^3*(-4*a*c+b^2)^( 1/2)+32*B*a^2*c^2-16*B*a*b^2*c+2*B*b^4)/c*ln(2*c*x^2+(-4*a*c+b^2)^(1/2)+b) +1/2*(16*A*a*b*c*(-4*a*c+b^2)^(1/2)-3*A*b^3*(-4*a*c+b^2)^(1/2)+40*A*a^2*c^ 2-22*A*a*b^2*c+3*A*b^4-12*C*(-4*a*c+b^2)^(1/2)*a^2*c+C*(-4*a*c+b^2)^(1/2)* a*b^2+4*C*a^2*b*c-C*a*b^3)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan (c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)))+1/(16*a*c-4*b^2)*(-1/4*(12 *B*a*b*c*(-4*a*c+b^2)^(1/2)-2*B*b^3*(-4*a*c+b^2)^(1/2)-32*B*a^2*c^2+16*B*a *b^2*c-2*B*b^4)/c*ln(-2*c*x^2+(-4*a*c+b^2)^(1/2)-b)+1/2*(16*A*a*b*c*(-4*a* c+b^2)^(1/2)-3*A*b^3*(-4*a*c+b^2)^(1/2)-40*A*a^2*c^2+22*A*a*b^2*c-3*A*b^4- 12*C*(-4*a*c+b^2)^(1/2)*a^2*c+C*(-4*a*c+b^2)^(1/2)*a*b^2-4*C*a^2*b*c+C*a*b ^3)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(c*x*2^(1/2)/((-b+(-4 *a*c+b^2)^(1/2))*c)^(1/2)))))
Timed out. \[ \int \frac {A+B x+C x^2}{x^2 \left (a+b x^2+c x^4\right )^2} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {A+B x+C x^2}{x^2 \left (a+b x^2+c x^4\right )^2} \, dx=\text {Timed out} \]
\[ \int \frac {A+B x+C x^2}{x^2 \left (a+b x^2+c x^4\right )^2} \, dx=\int { \frac {C x^{2} + B x + A}{{\left (c x^{4} + b x^{2} + a\right )}^{2} x^{2}} \,d x } \]
1/2*(B*a*b*c*x^3 + (10*A*a*c^2 + (C*a*b - 3*A*b^2)*c)*x^4 - 2*A*a*b^2 + 8* A*a^2*c + (C*a*b^2 - 3*A*b^3 - (2*C*a^2 - 11*A*a*b)*c)*x^2 + (B*a*b^2 - 2* B*a^2*c)*x)/((a^2*b^2*c - 4*a^3*c^2)*x^5 + (a^2*b^3 - 4*a^3*b*c)*x^3 + (a^ 3*b^2 - 4*a^4*c)*x) + 1/2*integrate((C*a*b^2 - 3*A*b^3 - 2*(B*b^2*c - 4*B* a*c^2)*x^3 + (10*A*a*c^2 + (C*a*b - 3*A*b^2)*c)*x^2 - (6*C*a^2 - 13*A*a*b) *c - 2*(B*b^3 - 5*B*a*b*c)*x)/(c*x^4 + b*x^2 + a), x)/(a^2*b^2 - 4*a^3*c) + B*log(x)/a^2
Leaf count of result is larger than twice the leaf count of optimal. 9013 vs. \(2 (453) = 906\).
Time = 2.08 (sec) , antiderivative size = 9013, normalized size of antiderivative = 17.54 \[ \int \frac {A+B x+C x^2}{x^2 \left (a+b x^2+c x^4\right )^2} \, dx=\text {Too large to display} \]
-1/4*B*log(abs(c*x^4 + b*x^2 + a))/a^2 + B*log(abs(x))/a^2 + 1/2*(C*a*b*c* x^4 - 3*A*b^2*c*x^4 + 10*A*a*c^2*x^4 + B*a*b*c*x^3 + C*a*b^2*x^2 - 3*A*b^3 *x^2 - 2*C*a^2*c*x^2 + 11*A*a*b*c*x^2 + B*a*b^2*x - 2*B*a^2*c*x - 2*A*a*b^ 2 + 8*A*a^2*c)/((c*x^5 + b*x^3 + a*x)*(a^2*b^2 - 4*a^3*c)) + 1/16*((a^4*b^ 4*c - 8*a^5*b^2*c^2 + 16*a^6*c^3)^2*(6*b^4*c^2 - 44*a*b^2*c^3 + 80*a^2*c^4 - 3*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^4 + 22*sq rt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c + 6*sqrt(2 )*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^3*c - 40*sqrt(2)*sqr t(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*c^2 - 20*sqrt(2)*sqrt(b ^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^2 - 3*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^2*c^2 + 10*sqrt(2)*sqrt(b^2 - 4* a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*c^3 - 6*(b^2 - 4*a*c)*b^2*c^2 + 20* (b^2 - 4*a*c)*a*c^3)*A - (a^4*b^4*c - 8*a^5*b^2*c^2 + 16*a^6*c^3)^2*(2*a*b ^3*c^2 - 8*a^2*b*c^3 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a *c)*c)*a*b^3 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c) *a^2*b*c + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b ^2*c - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^2 - 2*(b^2 - 4*a*c)*a*b*c^2)*C - 2*(3*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c) *a^4*b^9*c - 49*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*b^7*c^2 - 6*sq rt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^8*c^2 - 6*a^4*b^9*c^2 + 300...
Time = 8.87 (sec) , antiderivative size = 8684, normalized size of antiderivative = 16.89 \[ \int \frac {A+B x+C x^2}{x^2 \left (a+b x^2+c x^4\right )^2} \, dx=\text {Too large to display} \]
symsum(log(root(1572864*a^10*b^2*c^5*z^4 - 983040*a^9*b^4*c^4*z^4 + 327680 *a^8*b^6*c^3*z^4 - 61440*a^7*b^8*c^2*z^4 + 6144*a^6*b^10*c*z^4 - 1048576*a ^11*c^6*z^4 - 256*a^5*b^12*z^4 + 1572864*B*a^8*b^2*c^5*z^3 - 983040*B*a^7* b^4*c^4*z^3 + 327680*B*a^6*b^6*c^3*z^3 - 61440*B*a^5*b^8*c^2*z^3 + 6144*B* a^4*b^10*c*z^3 - 1048576*B*a^9*c^6*z^3 - 256*B*a^3*b^12*z^3 - 2432*A*C*a^2 *b^10*c*z^2 - 491520*A*C*a^6*b^2*c^5*z^2 + 358400*A*C*a^5*b^4*c^4*z^2 - 12 9024*A*C*a^4*b^6*c^3*z^2 + 24768*A*C*a^3*b^8*c^2*z^2 + 96*A*C*a*b^12*z^2 + 61440*C^2*a^7*b*c^5*z^2 + 432*C^2*a^3*b^9*c*z^2 + 1536*B^2*a^2*b^10*c*z^2 - 430080*A^2*a^6*b*c^6*z^2 + 3408*A^2*a*b^11*c*z^2 + 245760*A*C*a^7*c^6*z ^2 - 61440*C^2*a^6*b^3*c^4*z^2 + 24064*C^2*a^5*b^5*c^3*z^2 - 4608*C^2*a^4* b^7*c^2*z^2 + 516096*B^2*a^6*b^2*c^5*z^2 - 288768*B^2*a^5*b^4*c^4*z^2 + 88 576*B^2*a^4*b^6*c^3*z^2 - 15744*B^2*a^3*b^8*c^2*z^2 + 716800*A^2*a^5*b^3*c ^5*z^2 - 483840*A^2*a^4*b^5*c^4*z^2 + 170496*A^2*a^3*b^7*c^3*z^2 - 33232*A ^2*a^2*b^9*c^2*z^2 - 64*B^2*a*b^12*z^2 - 393216*B^2*a^7*c^6*z^2 - 16*C^2*a ^2*b^11*z^2 - 144*A^2*b^13*z^2 - 110592*A*B*C*a^4*b^2*c^5*z + 36864*A*B*C* a^3*b^4*c^4*z - 5376*A*B*C*a^2*b^6*c^3*z + 288*A*B*C*a*b^8*c^2*z + 3072*B* C^2*a^5*b*c^5*z - 138240*A^2*B*a^4*b*c^6*z + 7344*A^2*B*a*b^7*c^3*z + 1228 80*A*B*C*a^5*c^6*z - 2304*B*C^2*a^4*b^3*c^4*z + 576*B*C^2*a^3*b^5*c^3*z - 48*B*C^2*a^2*b^7*c^2*z + 131328*A^2*B*a^3*b^3*c^5*z - 46656*A^2*B*a^2*b^5* c^4*z + 61440*B^3*a^4*b^2*c^5*z - 21504*B^3*a^3*b^4*c^4*z + 3328*B^3*a^...