Integrand size = 18, antiderivative size = 425 \[ \int \frac {1}{x^2 \left (a+b x^2+c x^4\right )^3} \, dx=-\frac {3 \left (5 b^2-12 a c\right ) \left (b^2-5 a c\right )}{8 a^3 \left (b^2-4 a c\right )^2 x}+\frac {b^2-2 a c+b c x^2}{4 a \left (b^2-4 a c\right ) x \left (a+b x^2+c x^4\right )^2}+\frac {5 b^4-35 a b^2 c+36 a^2 c^2+b c \left (5 b^2-32 a c\right ) x^2}{8 a^2 \left (b^2-4 a c\right )^2 x \left (a+b x^2+c x^4\right )}-\frac {3 \sqrt {c} \left (\left (5 b^2-12 a c\right ) \left (b^2-5 a c\right )+\frac {b \left (5 b^4-47 a b^2 c+124 a^2 c^2\right )}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{8 \sqrt {2} a^3 \left (b^2-4 a c\right )^2 \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {3 \sqrt {c} \left (\left (5 b^2-12 a c\right ) \left (b^2-5 a c\right )-\frac {5 b^5-47 a b^3 c+124 a^2 b c^2}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{8 \sqrt {2} a^3 \left (b^2-4 a c\right )^2 \sqrt {b+\sqrt {b^2-4 a c}}} \] Output:
-3/8*(-12*a*c+5*b^2)*(-5*a*c+b^2)/a^3/(-4*a*c+b^2)^2/x+1/4*(b*c*x^2-2*a*c+ b^2)/a/(-4*a*c+b^2)/x/(c*x^4+b*x^2+a)^2+1/8*(5*b^4-35*a*b^2*c+36*c^2*a^2+b *c*(-32*a*c+5*b^2)*x^2)/a^2/(-4*a*c+b^2)^2/x/(c*x^4+b*x^2+a)-3/16*c^(1/2)* ((-12*a*c+5*b^2)*(-5*a*c+b^2)+b*(124*a^2*c^2-47*a*b^2*c+5*b^4)/(-4*a*c+b^2 )^(1/2))*arctan(2^(1/2)*c^(1/2)*x/(b-(-4*a*c+b^2)^(1/2))^(1/2))*2^(1/2)/a^ 3/(-4*a*c+b^2)^2/(b-(-4*a*c+b^2)^(1/2))^(1/2)-3/16*c^(1/2)*((-12*a*c+5*b^2 )*(-5*a*c+b^2)-(124*a^2*b*c^2-47*a*b^3*c+5*b^5)/(-4*a*c+b^2)^(1/2))*arctan (2^(1/2)*c^(1/2)*x/(b+(-4*a*c+b^2)^(1/2))^(1/2))*2^(1/2)/a^3/(-4*a*c+b^2)^ 2/(b+(-4*a*c+b^2)^(1/2))^(1/2)
Time = 1.17 (sec) , antiderivative size = 454, normalized size of antiderivative = 1.07 \[ \int \frac {1}{x^2 \left (a+b x^2+c x^4\right )^3} \, dx=-\frac {\frac {16}{x}+\frac {4 a x \left (b^3-3 a b c+b^2 c x^2-2 a c^2 x^2\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {2 x \left (7 b^5-52 a b^3 c+84 a^2 b c^2+7 b^4 c x^2-47 a b^2 c^2 x^2+52 a^2 c^3 x^2\right )}{\left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {3 \sqrt {2} \sqrt {c} \left (5 b^5-47 a b^3 c+124 a^2 b c^2+5 b^4 \sqrt {b^2-4 a c}-37 a b^2 c \sqrt {b^2-4 a c}+60 a^2 c^2 \sqrt {b^2-4 a c}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{5/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {3 \sqrt {2} \sqrt {c} \left (-5 b^5+47 a b^3 c-124 a^2 b c^2+5 b^4 \sqrt {b^2-4 a c}-37 a b^2 c \sqrt {b^2-4 a c}+60 a^2 c^2 \sqrt {b^2-4 a c}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{5/2} \sqrt {b+\sqrt {b^2-4 a c}}}}{16 a^3} \] Input:
Integrate[1/(x^2*(a + b*x^2 + c*x^4)^3),x]
Output:
-1/16*(16/x + (4*a*x*(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) + (2*x*(7*b^5 - 52*a*b^3*c + 84*a^2*b*c^2 + 7* b^4*c*x^2 - 47*a*b^2*c^2*x^2 + 52*a^2*c^3*x^2))/((b^2 - 4*a*c)^2*(a + b*x^ 2 + c*x^4)) + (3*Sqrt[2]*Sqrt[c]*(5*b^5 - 47*a*b^3*c + 124*a^2*b*c^2 + 5*b ^4*Sqrt[b^2 - 4*a*c] - 37*a*b^2*c*Sqrt[b^2 - 4*a*c] + 60*a^2*c^2*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (3*Sqrt[2]*Sqrt[c]*(-5*b^5 + 47*a*b^3*c - 124*a^2*b*c^2 + 5*b^4*Sqrt[b^2 - 4*a*c] - 37*a*b^2*c*Sqrt[b^2 - 4*a*c] + 60*a^2*c^2*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[ b + Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(5/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]) )/a^3
Time = 1.26 (sec) , antiderivative size = 423, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {1441, 25, 1600, 27, 1604, 1480, 218}
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.
| \(\displaystyle \int \frac {1}{x^2 \left (a+b x^2+c x^4\right )^3} \, dx\) |
| \(\Big \downarrow \) 1441 |
| \(\displaystyle \frac {-2 a c+b^2+b c x^2}{4 a x \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {\int -\frac {5 b^2+7 c x^2 b-18 a c}{x^2 \left (c x^4+b x^2+a\right )^2}dx}{4 a \left (b^2-4 a c\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {5 b^2+7 c x^2 b-18 a c}{x^2 \left (c x^4+b x^2+a\right )^2}dx}{4 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x^2}{4 a x \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\) |
| \(\Big \downarrow \) 1600 |
| \(\displaystyle \frac {\frac {36 a^2 c^2+b c x^2 \left (5 b^2-32 a c\right )-35 a b^2 c+5 b^4}{2 a x \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\int -\frac {3 \left (b c \left (5 b^2-32 a c\right ) x^2+\left (5 b^2-12 a c\right ) \left (b^2-5 a c\right )\right )}{x^2 \left (c x^4+b x^2+a\right )}dx}{2 a \left (b^2-4 a c\right )}}{4 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x^2}{4 a x \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {3 \int \frac {b c \left (5 b^2-32 a c\right ) x^2+\left (5 b^2-12 a c\right ) \left (b^2-5 a c\right )}{x^2 \left (c x^4+b x^2+a\right )}dx}{2 a \left (b^2-4 a c\right )}+\frac {36 a^2 c^2+b c x^2 \left (5 b^2-32 a c\right )-35 a b^2 c+5 b^4}{2 a x \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{4 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x^2}{4 a x \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\) |
| \(\Big \downarrow \) 1604 |
| \(\displaystyle \frac {\frac {3 \left (-\frac {\int \frac {c \left (5 b^2-12 a c\right ) \left (b^2-5 a c\right ) x^2+b \left (5 b^4-42 a c b^2+92 a^2 c^2\right )}{c x^4+b x^2+a}dx}{a}-\frac {\left (5 b^2-12 a c\right ) \left (b^2-5 a c\right )}{a x}\right )}{2 a \left (b^2-4 a c\right )}+\frac {36 a^2 c^2+b c x^2 \left (5 b^2-32 a c\right )-35 a b^2 c+5 b^4}{2 a x \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{4 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x^2}{4 a x \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {\frac {3 \left (-\frac {\frac {1}{2} c \left (\frac {b \left (124 a^2 c^2-47 a b^2 c+5 b^4\right )}{\sqrt {b^2-4 a c}}+\left (5 b^2-12 a c\right ) \left (b^2-5 a c\right )\right ) \int \frac {1}{c x^2+\frac {1}{2} \left (b-\sqrt {b^2-4 a c}\right )}dx+\frac {1}{2} c \left (\left (5 b^2-12 a c\right ) \left (b^2-5 a c\right )-\frac {b \left (124 a^2 c^2-47 a b^2 c+5 b^4\right )}{\sqrt {b^2-4 a c}}\right ) \int \frac {1}{c x^2+\frac {1}{2} \left (b+\sqrt {b^2-4 a c}\right )}dx}{a}-\frac {\left (5 b^2-12 a c\right ) \left (b^2-5 a c\right )}{a x}\right )}{2 a \left (b^2-4 a c\right )}+\frac {36 a^2 c^2+b c x^2 \left (5 b^2-32 a c\right )-35 a b^2 c+5 b^4}{2 a x \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{4 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x^2}{4 a x \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {3 \left (-\frac {\frac {\sqrt {c} \left (\frac {b \left (124 a^2 c^2-47 a b^2 c+5 b^4\right )}{\sqrt {b^2-4 a c}}+\left (5 b^2-12 a c\right ) \left (b^2-5 a c\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {c} \left (\left (5 b^2-12 a c\right ) \left (b^2-5 a c\right )-\frac {b \left (124 a^2 c^2-47 a b^2 c+5 b^4\right )}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {2} \sqrt {\sqrt {b^2-4 a c}+b}}}{a}-\frac {\left (5 b^2-12 a c\right ) \left (b^2-5 a c\right )}{a x}\right )}{2 a \left (b^2-4 a c\right )}+\frac {36 a^2 c^2+b c x^2 \left (5 b^2-32 a c\right )-35 a b^2 c+5 b^4}{2 a x \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{4 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x^2}{4 a x \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\) |
Input:
Int[1/(x^2*(a + b*x^2 + c*x^4)^3),x]
Output:
(b^2 - 2*a*c + b*c*x^2)/(4*a*(b^2 - 4*a*c)*x*(a + b*x^2 + c*x^4)^2) + ((5* b^4 - 35*a*b^2*c + 36*a^2*c^2 + b*c*(5*b^2 - 32*a*c)*x^2)/(2*a*(b^2 - 4*a* c)*x*(a + b*x^2 + c*x^4)) + (3*(-(((5*b^2 - 12*a*c)*(b^2 - 5*a*c))/(a*x)) - ((Sqrt[c]*((5*b^2 - 12*a*c)*(b^2 - 5*a*c) + (b*(5*b^4 - 47*a*b^2*c + 124 *a^2*c^2))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[c]*((5*b^2 - 12 *a*c)*(b^2 - 5*a*c) - (b*(5*b^4 - 47*a*b^2*c + 124*a^2*c^2))/Sqrt[b^2 - 4* a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sq rt[b + Sqrt[b^2 - 4*a*c]]))/a))/(2*a*(b^2 - 4*a*c)))/(4*a*(b^2 - 4*a*c))
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_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(-(d*x)^(m + 1))*(b^2 - 2*a*c + b*c*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2*a*d*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c)) Int[(d*x)^m*(a + b*x^2 + c*x^4)^(p + 1)*Simp[b^2*(m + 2*p + 3) - 2*a*c*(m + 4*p + 5) + b*c*(m + 4*p + 7)*x^2, x], x], x] /; FreeQ[{a, b, c, d, m}, x ] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])
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])
Time = 0.24 (sec) , antiderivative size = 517, normalized size of antiderivative = 1.22
| method | result | size |
| default | \(-\frac {\frac {\frac {c^{2} \left (52 a^{2} c^{2}-47 a \,b^{2} c +7 b^{4}\right ) x^{7}}{128 a^{2} c^{2}-64 a \,b^{2} c +8 b^{4}}+\frac {c b \left (136 a^{2} c^{2}-99 a \,b^{2} c +14 b^{4}\right ) x^{5}}{128 a^{2} c^{2}-64 a \,b^{2} c +8 b^{4}}+\frac {\left (68 a^{3} c^{3}+25 a^{2} b^{2} c^{2}-43 a \,b^{4} c +7 b^{6}\right ) x^{3}}{128 a^{2} c^{2}-64 a \,b^{2} c +8 b^{4}}+\frac {3 a b \left (36 a^{2} c^{2}-22 a \,b^{2} c +3 b^{4}\right ) x}{8 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2}}+\frac {3 c \left (\frac {\left (60 a^{2} c^{2} \sqrt {-4 a c +b^{2}}-37 a \,b^{2} c \sqrt {-4 a c +b^{2}}+5 b^{4} \sqrt {-4 a c +b^{2}}-124 a^{2} b \,c^{2}+47 a \,b^{3} c -5 b^{5}\right ) \sqrt {2}\, \arctan \left (\frac {c x \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}-\frac {\left (60 a^{2} c^{2} \sqrt {-4 a c +b^{2}}-37 a \,b^{2} c \sqrt {-4 a c +b^{2}}+5 b^{4} \sqrt {-4 a c +b^{2}}+124 a^{2} b \,c^{2}-47 a \,b^{3} c +5 b^{5}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c x \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{a^{3}}-\frac {1}{a^{3} x}\) | \(517\) |
| risch | \(\frac {-\frac {3 c^{2} \left (60 a^{2} c^{2}-37 a \,b^{2} c +5 b^{4}\right ) x^{8}}{8 a^{3} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {c b \left (392 a^{2} c^{2}-227 a \,b^{2} c +30 b^{4}\right ) x^{6}}{8 a^{3} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {\left (324 a^{3} c^{3}+25 a^{2} b^{2} c^{2}-91 a \,b^{4} c +15 b^{6}\right ) x^{4}}{8 a^{3} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {b \left (364 a^{2} c^{2}-194 a \,b^{2} c +25 b^{4}\right ) x^{2}}{8 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a^{2}}-\frac {1}{a}}{x \left (c \,x^{4}+b \,x^{2}+a \right )^{2}}+\frac {3 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (1048576 a^{17} c^{10}-2621440 a^{16} b^{2} c^{9}+2949120 a^{15} b^{4} c^{8}-1966080 a^{14} b^{6} c^{7}+860160 a^{13} b^{8} c^{6}-258048 a^{12} b^{10} c^{5}+53760 a^{11} b^{12} c^{4}-7680 a^{10} b^{14} c^{3}+720 a^{9} b^{16} c^{2}-40 a^{8} b^{18} c +a^{7} b^{20}\right ) \textit {\_Z}^{4}+\left (18923520 a^{10} b \,c^{10}-52039680 a^{9} b^{3} c^{9}+62684160 a^{8} b^{5} c^{8}-43904256 a^{7} b^{7} c^{7}+19905600 a^{6} b^{9} c^{6}-6126640 a^{5} b^{11} c^{5}+1299860 a^{4} b^{13} c^{4}-188095 a^{3} b^{15} c^{3}+17794 a^{2} b^{17} c^{2}-995 a \,b^{19} c +25 b^{21}\right ) \textit {\_Z}^{2}+12960000 a^{4} c^{11}-10771200 a^{3} b^{2} c^{10}+3426016 a^{2} b^{4} c^{9}-493680 a \,b^{6} c^{8}+27225 b^{8} c^{7}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (2621440 a^{17} c^{10}-6684672 a^{16} b^{2} c^{9}+7667712 a^{15} b^{4} c^{8}-5210112 a^{14} b^{6} c^{7}+2322432 a^{13} b^{8} c^{6}-709632 a^{12} b^{10} c^{5}+150528 a^{11} b^{12} c^{4}-21888 a^{10} b^{14} c^{3}+2088 a^{9} b^{16} c^{2}-118 a^{8} b^{18} c +3 a^{7} b^{20}\right ) \textit {\_R}^{4}+\left (40734720 a^{10} b \,c^{10}-109361152 a^{9} b^{3} c^{9}+129620736 a^{8} b^{5} c^{8}-89776128 a^{7} b^{7} c^{7}+40383216 a^{6} b^{9} c^{6}-12360216 a^{5} b^{11} c^{5}+2612269 a^{4} b^{13} c^{4}-377035 a^{3} b^{15} c^{3}+35613 a^{2} b^{17} c^{2}-1990 a \,b^{19} c +50 b^{21}\right ) \textit {\_R}^{2}+25920000 a^{4} c^{11}-21542400 a^{3} b^{2} c^{10}+6852032 a^{2} b^{4} c^{9}-987360 a \,b^{6} c^{8}+54450 b^{8} c^{7}\right ) x +\left (983040 a^{14} c^{10}-3833856 a^{13} b^{2} c^{9}+5758976 a^{12} b^{4} c^{8}-4741120 a^{11} b^{6} c^{7}+2444288 a^{10} b^{8} c^{6}-837760 a^{9} b^{10} c^{5}+195104 a^{8} b^{12} c^{4}-30664 a^{7} b^{14} c^{3}+3125 a^{6} b^{16} c^{2}-187 a^{5} b^{18} c +5 a^{4} b^{20}\right ) \textit {\_R}^{3}+\left (1843200 a^{7} b \,c^{10}-1975552 a^{6} b^{3} c^{9}+846336 a^{5} b^{5} c^{8}-181152 a^{4} b^{7} c^{7}+19360 a^{3} b^{9} c^{6}-825 a^{2} b^{11} c^{5}\right ) \textit {\_R} \right )\right )}{16}\) | \(999\) |
Input:
int(1/x^2/(c*x^4+b*x^2+a)^3,x,method=_RETURNVERBOSE)
Output:
-1/a^3*((1/8*c^2*(52*a^2*c^2-47*a*b^2*c+7*b^4)/(16*a^2*c^2-8*a*b^2*c+b^4)* x^7+1/8*c*b*(136*a^2*c^2-99*a*b^2*c+14*b^4)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^5 +1/8*(68*a^3*c^3+25*a^2*b^2*c^2-43*a*b^4*c+7*b^6)/(16*a^2*c^2-8*a*b^2*c+b^ 4)*x^3+3/8*a*b*(36*a^2*c^2-22*a*b^2*c+3*b^4)/(16*a^2*c^2-8*a*b^2*c+b^4)*x) /(c*x^4+b*x^2+a)^2+3/2/(16*a^2*c^2-8*a*b^2*c+b^4)*c*(1/8*(60*a^2*c^2*(-4*a *c+b^2)^(1/2)-37*a*b^2*c*(-4*a*c+b^2)^(1/2)+5*b^4*(-4*a*c+b^2)^(1/2)-124*a ^2*b*c^2+47*a*b^3*c-5*b^5)/(-4*a*c+b^2)^(1/2)*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))-1/8*(60* a^2*c^2*(-4*a*c+b^2)^(1/2)-37*a*b^2*c*(-4*a*c+b^2)^(1/2)+5*b^4*(-4*a*c+b^2 )^(1/2)+124*a^2*b*c^2-47*a*b^3*c+5*b^5)/(-4*a*c+b^2)^(1/2)*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))))-1/a^3/x
Leaf count of result is larger than twice the leaf count of optimal. 4924 vs. \(2 (379) = 758\).
Time = 1.10 (sec) , antiderivative size = 4924, normalized size of antiderivative = 11.59 \[ \int \frac {1}{x^2 \left (a+b x^2+c x^4\right )^3} \, dx=\text {Too large to display} \] Input:
integrate(1/x^2/(c*x^4+b*x^2+a)^3,x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {1}{x^2 \left (a+b x^2+c x^4\right )^3} \, dx=\text {Timed out} \] Input:
integrate(1/x**2/(c*x**4+b*x**2+a)**3,x)
Output:
Timed out
\[ \int \frac {1}{x^2 \left (a+b x^2+c x^4\right )^3} \, dx=\int { \frac {1}{{\left (c x^{4} + b x^{2} + a\right )}^{3} x^{2}} \,d x } \] Input:
integrate(1/x^2/(c*x^4+b*x^2+a)^3,x, algorithm="maxima")
Output:
-1/8*(3*(5*b^4*c^2 - 37*a*b^2*c^3 + 60*a^2*c^4)*x^8 + (30*b^5*c - 227*a*b^ 3*c^2 + 392*a^2*b*c^3)*x^6 + 8*a^2*b^4 - 64*a^3*b^2*c + 128*a^4*c^2 + (15* b^6 - 91*a*b^4*c + 25*a^2*b^2*c^2 + 324*a^3*c^3)*x^4 + (25*a*b^5 - 194*a^2 *b^3*c + 364*a^3*b*c^2)*x^2)/((a^3*b^4*c^2 - 8*a^4*b^2*c^3 + 16*a^5*c^4)*x ^9 + 2*(a^3*b^5*c - 8*a^4*b^3*c^2 + 16*a^5*b*c^3)*x^7 + (a^3*b^6 - 6*a^4*b ^4*c + 32*a^6*c^3)*x^5 + 2*(a^4*b^5 - 8*a^5*b^3*c + 16*a^6*b*c^2)*x^3 + (a ^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)*x) - 3/8*integrate((5*b^5 - 42*a*b^3*c + 92*a^2*b*c^2 + (5*b^4*c - 37*a*b^2*c^2 + 60*a^2*c^3)*x^2)/(c*x^4 + b*x^2 + a), x)/(a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)
Leaf count of result is larger than twice the leaf count of optimal. 5273 vs. \(2 (379) = 758\).
Time = 2.03 (sec) , antiderivative size = 5273, normalized size of antiderivative = 12.41 \[ \int \frac {1}{x^2 \left (a+b x^2+c x^4\right )^3} \, dx=\text {Too large to display} \] Input:
integrate(1/x^2/(c*x^4+b*x^2+a)^3,x, algorithm="giac")
Output:
-3/64*(10*a^6*b^14*c^2 - 254*a^7*b^12*c^3 + 2712*a^8*b^10*c^4 - 15552*a^9* b^8*c^5 + 50432*a^10*b^6*c^6 - 87552*a^11*b^4*c^7 + 63488*a^12*b^2*c^8 - 5 *sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^6*b^14 + 127* sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^7*b^12*c + 10* sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^6*b^13*c - 135 6*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^8*b^10*c^2 - 214*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^7*b^11*c^ 2 - 5*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^6*b^12*c ^2 + 7776*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^9*b^ 8*c^3 + 1856*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^8 *b^9*c^3 + 107*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a ^7*b^10*c^3 - 25216*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c) *c)*a^10*b^6*c^4 - 8128*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4* a*c)*c)*a^9*b^7*c^4 - 928*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^8*b^8*c^4 + 43776*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^ 2 - 4*a*c)*c)*a^11*b^4*c^5 + 17920*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sq rt(b^2 - 4*a*c)*c)*a^10*b^5*c^5 + 4064*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^9*b^6*c^5 - 31744*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt( b*c + sqrt(b^2 - 4*a*c)*c)*a^12*b^2*c^6 - 15872*sqrt(2)*sqrt(b^2 - 4*a*c)* sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^11*b^3*c^6 - 8960*sqrt(2)*sqrt(b^2 - ...
Time = 23.51 (sec) , antiderivative size = 12130, normalized size of antiderivative = 28.54 \[ \int \frac {1}{x^2 \left (a+b x^2+c x^4\right )^3} \, dx=\text {Too large to display} \] Input:
int(1/(x^2*(a + b*x^2 + c*x^4)^3),x)
Output:
- atan(((x*(271790899200*a^20*c^14 - 230400*a^9*b^22*c^3 + 9861120*a^10*b^ 20*c^4 - 191038464*a^11*b^18*c^5 + 2207803392*a^12*b^16*c^6 - 16878108672* a^13*b^14*c^7 + 89374851072*a^14*b^12*c^8 - 333226967040*a^15*b^10*c^9 + 8 69815812096*a^16*b^8*c^10 - 1543847804928*a^17*b^6*c^11 + 1747313491968*a^ 18*b^4*c^12 - 1101055131648*a^19*b^2*c^13) + (-(9*(25*b^21 - 25*b^6*(-(4*a *c - b^2)^15)^(1/2) + 18923520*a^10*b*c^10 + 17794*a^2*b^17*c^2 - 188095*a ^3*b^15*c^3 + 1299860*a^4*b^13*c^4 - 6126640*a^5*b^11*c^5 + 19905600*a^6*b ^9*c^6 - 43904256*a^7*b^7*c^7 + 62684160*a^8*b^5*c^8 - 52039680*a^9*b^3*c^ 9 + 225*a^3*c^3*(-(4*a*c - b^2)^15)^(1/2) - 995*a*b^19*c - 694*a^2*b^2*c^2 *(-(4*a*c - b^2)^15)^(1/2) + 245*a*b^4*c*(-(4*a*c - b^2)^15)^(1/2)))/(512* (a^7*b^20 + 1048576*a^17*c^10 - 40*a^8*b^18*c + 720*a^9*b^16*c^2 - 7680*a^ 10*b^14*c^3 + 53760*a^11*b^12*c^4 - 258048*a^12*b^10*c^5 + 860160*a^13*b^8 *c^6 - 1966080*a^14*b^6*c^7 + 2949120*a^15*b^4*c^8 - 2621440*a^16*b^2*c^9) ))^(1/2)*(245760*a^12*b^23*c^2 - 1185410973696*a^23*b*c^13 - 10911744*a^13 *b^21*c^3 + 220397568*a^14*b^19*c^4 - 2673082368*a^15*b^17*c^5 + 216300257 28*a^16*b^15*c^6 - 122607894528*a^17*b^13*c^7 + 496773365760*a^18*b^11*c^8 - 1438679826432*a^19*b^9*c^9 + 2918430277632*a^20*b^7*c^10 - 394922242867 2*a^21*b^5*c^11 + 3208340570112*a^22*b^3*c^12 + x*(-(9*(25*b^21 - 25*b^6*( -(4*a*c - b^2)^15)^(1/2) + 18923520*a^10*b*c^10 + 17794*a^2*b^17*c^2 - 188 095*a^3*b^15*c^3 + 1299860*a^4*b^13*c^4 - 6126640*a^5*b^11*c^5 + 199056...
Time = 3.17 (sec) , antiderivative size = 7940, normalized size of antiderivative = 18.68 \[ \int \frac {1}{x^2 \left (a+b x^2+c x^4\right )^3} \, dx =\text {Too large to display} \] Input:
int(1/x^2/(c*x^4+b*x^2+a)^3,x)
Output:
(720*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b))*a**5*c**3*x - 996*sqrt(a)*sqr t(2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/ sqrt(2*sqrt(c)*sqrt(a) + b))*a**4*b**2*c**2*x + 1440*sqrt(a)*sqrt(2*sqrt(c )*sqrt(a) + b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqr t(c)*sqrt(a) + b))*a**4*b*c**3*x**3 + 1440*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt( a) + b))*a**4*c**4*x**5 + 312*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + b)*atan((sq rt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b))*a**3 *b**4*c*x - 1992*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2*sqrt(c)* sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b))*a**3*b**3*c**2*x* *3 - 1272*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b))*a**3*b**2*c**3*x**5 + 14 40*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b))*a**3*b*c**4*x**7 + 720*sqrt(a)* sqrt(2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)* x)/sqrt(2*sqrt(c)*sqrt(a) + b))*a**3*c**5*x**9 - 30*sqrt(a)*sqrt(2*sqrt(c) *sqrt(a) + b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt (c)*sqrt(a) + b))*a**2*b**6*x + 624*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + b)*at an((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) +...