Integrand size = 18, antiderivative size = 298 \[ \int \frac {x^6}{\left (a+b x^2+c x^4\right )^3} \, dx=\frac {x^3 \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {3 x \left (4 a b+\left (b^2+4 a c\right ) x^2\right )}{8 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {3 \left (b^2+4 a c-\frac {b \left (b^2+12 a c\right )}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{8 \sqrt {2} \sqrt {c} \left (b^2-4 a c\right )^2 \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {3 \left (b^2+4 a c+\frac {b \left (b^2+12 a c\right )}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{8 \sqrt {2} \sqrt {c} \left (b^2-4 a c\right )^2 \sqrt {b+\sqrt {b^2-4 a c}}} \] Output:
1/4*x^3*(b*x^2+2*a)/(-4*a*c+b^2)/(c*x^4+b*x^2+a)^2+3/8*x*(4*a*b+(4*a*c+b^2 )*x^2)/(-4*a*c+b^2)^2/(c*x^4+b*x^2+a)+3/16*(b^2+4*a*c-b*(12*a*c+b^2)/(-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)/c^(1/2)/(-4*a*c+b^2)^2/(b-(-4*a*c+b^2)^(1/2))^(1/2)+3/16*(b^2+4*a*c+b* (12*a*c+b^2)/(-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)/c^(1/2)/(-4*a*c+b^2)^2/(b+(-4*a*c+b^2)^(1/2))^(1/2)
Time = 0.55 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.15 \[ \int \frac {x^6}{\left (a+b x^2+c x^4\right )^3} \, dx=\frac {\frac {4 b^3 x+8 a b c x+6 b^2 c x^3+24 a c^2 x^3}{\left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac {4 \left (b^2 x^3+a x \left (b-2 c x^2\right )\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {3 \sqrt {2} \sqrt {c} \left (-b^3-12 a b c+b^2 \sqrt {b^2-4 a c}+4 a c \sqrt {b^2-4 a c}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{5/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {3 \sqrt {2} \sqrt {c} \left (b^3+12 a b c+b^2 \sqrt {b^2-4 a c}+4 a c \sqrt {b^2-4 a c}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{5/2} \sqrt {b+\sqrt {b^2-4 a c}}}}{16 c} \] Input:
Integrate[x^6/(a + b*x^2 + c*x^4)^3,x]
Output:
((4*b^3*x + 8*a*b*c*x + 6*b^2*c*x^3 + 24*a*c^2*x^3)/((b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) - (4*(b^2*x^3 + a*x*(b - 2*c*x^2)))/((b^2 - 4*a*c)*(a + b* x^2 + c*x^4)^2) + (3*Sqrt[2]*Sqrt[c]*(-b^3 - 12*a*b*c + b^2*Sqrt[b^2 - 4*a *c] + 4*a*c*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]*(b^3 + 12*a*b*c + b^2*Sqrt[b^2 - 4*a*c] + 4*a*c*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]]))/(16*c)
Time = 0.84 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.02, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {1440, 27, 1598, 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.
| \(\displaystyle \int \frac {x^6}{\left (a+b x^2+c x^4\right )^3} \, dx\) |
| \(\Big \downarrow \) 1440 |
| \(\displaystyle \frac {x^3 \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {\int \frac {3 x^2 \left (2 a-b x^2\right )}{\left (c x^4+b x^2+a\right )^2}dx}{4 \left (b^2-4 a c\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^3 \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {3 \int \frac {x^2 \left (2 a-b x^2\right )}{\left (c x^4+b x^2+a\right )^2}dx}{4 \left (b^2-4 a c\right )}\) |
| \(\Big \downarrow \) 1598 |
| \(\displaystyle \frac {x^3 \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {3 \left (\frac {\int \frac {4 a b-\left (b^2+4 a c\right ) x^2}{c x^4+b x^2+a}dx}{2 \left (b^2-4 a c\right )}-\frac {x \left (x^2 \left (4 a c+b^2\right )+4 a b\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )}{4 \left (b^2-4 a c\right )}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {x^3 \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {3 \left (\frac {-\frac {1}{2} \left (\frac {b \left (12 a c+b^2\right )}{\sqrt {b^2-4 a c}}+4 a c+b^2\right ) \int \frac {1}{c x^2+\frac {1}{2} \left (b+\sqrt {b^2-4 a c}\right )}dx-\frac {1}{2} \left (-\frac {12 a b c+b^3}{\sqrt {b^2-4 a c}}+4 a c+b^2\right ) \int \frac {1}{c x^2+\frac {1}{2} \left (b-\sqrt {b^2-4 a c}\right )}dx}{2 \left (b^2-4 a c\right )}-\frac {x \left (x^2 \left (4 a c+b^2\right )+4 a b\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )}{4 \left (b^2-4 a c\right )}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {x^3 \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {3 \left (\frac {-\frac {\left (\frac {b \left (12 a c+b^2\right )}{\sqrt {b^2-4 a c}}+4 a c+b^2\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {2} \sqrt {c} \sqrt {\sqrt {b^2-4 a c}+b}}-\frac {\left (-\frac {12 a b c+b^3}{\sqrt {b^2-4 a c}}+4 a c+b^2\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} \sqrt {c} \sqrt {b-\sqrt {b^2-4 a c}}}}{2 \left (b^2-4 a c\right )}-\frac {x \left (x^2 \left (4 a c+b^2\right )+4 a b\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )}{4 \left (b^2-4 a c\right )}\) |
Input:
Int[x^6/(a + b*x^2 + c*x^4)^3,x]
Output:
(x^3*(2*a + b*x^2))/(4*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) - (3*(-1/2*(x* (4*a*b + (b^2 + 4*a*c)*x^2))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (-(((b^ 2 + 4*a*c - (b^3 + 12*a*b*c)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x) /Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[c]*Sqrt[b - Sqrt[b^2 - 4*a*c] ])) - ((b^2 + 4*a*c + (b*(b^2 + 12*a*c))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2 ]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[c]*Sqrt[b + Sqrt[ b^2 - 4*a*c]]))/(2*(b^2 - 4*a*c))))/(4*(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^3)*(d*x)^(m - 3)*(2*a + b*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2* (p + 1)*(b^2 - 4*a*c))), x] + Simp[d^4/(2*(p + 1)*(b^2 - 4*a*c)) Int[(d*x )^(m - 4)*(2*a*(m - 3) + b*(m + 4*p + 3)*x^2)*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && Gt Q[m, 3] && 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*(f*x)^(m - 1)*(a + b*x^2 + c*x^4)^(p + 1) *((b*d - 2*a*e - (b*e - 2*c*d)*x^2)/(2*(p + 1)*(b^2 - 4*a*c))), x] - Simp[f ^2/(2*(p + 1)*(b^2 - 4*a*c)) Int[(f*x)^(m - 2)*(a + b*x^2 + c*x^4)^(p + 1 )*Simp[(m - 1)*(b*d - 2*a*e) - (4*p + 4 + m + 1)*(b*e - 2*c*d)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && GtQ[m, 1] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.16 (sec) , antiderivative size = 251, normalized size of antiderivative = 0.84
| method | result | size |
| risch | \(\frac {\frac {3 c \left (4 a c +b^{2}\right ) x^{7}}{8 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {b \left (16 a c +5 b^{2}\right ) x^{5}}{128 a^{2} c^{2}-64 a \,b^{2} c +8 b^{4}}-\frac {\left (4 a c -19 b^{2}\right ) a \,x^{3}}{8 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {3 a^{2} b x}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2}}+\frac {3 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4} c +\textit {\_Z}^{2} b +a \right )}{\sum }\frac {\left (\frac {\left (4 a c +b^{2}\right ) \textit {\_R}^{2}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}-\frac {4 a b}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{3} c +\textit {\_R} b}\right )}{16}\) | \(251\) |
| default | \(\frac {\frac {3 c \left (4 a c +b^{2}\right ) x^{7}}{8 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {b \left (16 a c +5 b^{2}\right ) x^{5}}{128 a^{2} c^{2}-64 a \,b^{2} c +8 b^{4}}-\frac {\left (4 a c -19 b^{2}\right ) a \,x^{3}}{8 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {3 a^{2} b x}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2}}+\frac {3 c \left (\frac {\left (4 a c \sqrt {-4 a c +b^{2}}+b^{2} \sqrt {-4 a c +b^{2}}+12 a b c +b^{3}\right ) \sqrt {2}\, \arctan \left (\frac {c x \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}-\frac {\left (4 a c \sqrt {-4 a c +b^{2}}+b^{2} \sqrt {-4 a c +b^{2}}-12 a b c -b^{3}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c x \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}\) | \(374\) |
Input:
int(x^6/(c*x^4+b*x^2+a)^3,x,method=_RETURNVERBOSE)
Output:
(3/8*c*(4*a*c+b^2)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^7+1/8*b*(16*a*c+5*b^2)/(16 *a^2*c^2-8*a*b^2*c+b^4)*x^5-1/8*(4*a*c-19*b^2)*a/(16*a^2*c^2-8*a*b^2*c+b^4 )*x^3+3/2*a^2*b/(16*a^2*c^2-8*a*b^2*c+b^4)*x)/(c*x^4+b*x^2+a)^2+3/16*sum(( (4*a*c+b^2)/(16*a^2*c^2-8*a*b^2*c+b^4)*_R^2-4*a*b/(16*a^2*c^2-8*a*b^2*c+b^ 4))/(2*_R^3*c+_R*b)*ln(x-_R),_R=RootOf(_Z^4*c+_Z^2*b+a))
Leaf count of result is larger than twice the leaf count of optimal. 3128 vs. \(2 (254) = 508\).
Time = 0.15 (sec) , antiderivative size = 3128, normalized size of antiderivative = 10.50 \[ \int \frac {x^6}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Too large to display} \] Input:
integrate(x^6/(c*x^4+b*x^2+a)^3,x, algorithm="fricas")
Output:
Too large to include
Leaf count of result is larger than twice the leaf count of optimal. 627 vs. \(2 (292) = 584\).
Time = 10.74 (sec) , antiderivative size = 627, normalized size of antiderivative = 2.10 \[ \int \frac {x^6}{\left (a+b x^2+c x^4\right )^3} \, dx=\frac {12 a^{2} b x + x^{7} \cdot \left (12 a c^{2} + 3 b^{2} c\right ) + x^{5} \cdot \left (16 a b c + 5 b^{3}\right ) + x^{3} \left (- 4 a^{2} c + 19 a b^{2}\right )}{128 a^{4} c^{2} - 64 a^{3} b^{2} c + 8 a^{2} b^{4} + x^{8} \cdot \left (128 a^{2} c^{4} - 64 a b^{2} c^{3} + 8 b^{4} c^{2}\right ) + x^{6} \cdot \left (256 a^{2} b c^{3} - 128 a b^{3} c^{2} + 16 b^{5} c\right ) + x^{4} \cdot \left (256 a^{3} c^{3} - 48 a b^{4} c + 8 b^{6}\right ) + x^{2} \cdot \left (256 a^{3} b c^{2} - 128 a^{2} b^{3} c + 16 a b^{5}\right )} + \operatorname {RootSum} {\left (t^{4} \cdot \left (68719476736 a^{10} c^{11} - 171798691840 a^{9} b^{2} c^{10} + 193273528320 a^{8} b^{4} c^{9} - 128849018880 a^{7} b^{6} c^{8} + 56371445760 a^{6} b^{8} c^{7} - 16911433728 a^{5} b^{10} c^{6} + 3523215360 a^{4} b^{12} c^{5} - 503316480 a^{3} b^{14} c^{4} + 47185920 a^{2} b^{16} c^{3} - 2621440 a b^{18} c^{2} + 65536 b^{20} c\right ) + t^{2} \left (- 188743680 a^{7} b c^{7} + 141557760 a^{6} b^{3} c^{6} - 2359296 a^{5} b^{5} c^{5} - 26542080 a^{4} b^{7} c^{4} + 9584640 a^{3} b^{9} c^{3} - 1290240 a^{2} b^{11} c^{2} + 46080 a b^{13} c + 2304 b^{15}\right ) + 20736 a^{5} c^{4} + 103680 a^{4} b^{2} c^{3} + 142560 a^{3} b^{4} c^{2} + 32400 a^{2} b^{6} c + 2025 a b^{8}, \left ( t \mapsto t \log {\left (x + \frac {33554432 t^{3} a^{6} c^{7} - 16777216 t^{3} a^{5} b^{2} c^{6} - 10485760 t^{3} a^{4} b^{4} c^{5} + 10485760 t^{3} a^{3} b^{6} c^{4} - 3276800 t^{3} a^{2} b^{8} c^{3} + 458752 t^{3} a b^{10} c^{2} - 24576 t^{3} b^{12} c - 64512 t a^{3} b c^{3} - 43776 t a^{2} b^{3} c^{2} - 21312 t a b^{5} c - 144 t b^{7}}{432 a^{2} c^{2} + 1080 a b^{2} c + 135 b^{4}} \right )} \right )\right )} \] Input:
integrate(x**6/(c*x**4+b*x**2+a)**3,x)
Output:
(12*a**2*b*x + x**7*(12*a*c**2 + 3*b**2*c) + x**5*(16*a*b*c + 5*b**3) + x* *3*(-4*a**2*c + 19*a*b**2))/(128*a**4*c**2 - 64*a**3*b**2*c + 8*a**2*b**4 + x**8*(128*a**2*c**4 - 64*a*b**2*c**3 + 8*b**4*c**2) + x**6*(256*a**2*b*c **3 - 128*a*b**3*c**2 + 16*b**5*c) + x**4*(256*a**3*c**3 - 48*a*b**4*c + 8 *b**6) + x**2*(256*a**3*b*c**2 - 128*a**2*b**3*c + 16*a*b**5)) + RootSum(_ t**4*(68719476736*a**10*c**11 - 171798691840*a**9*b**2*c**10 + 19327352832 0*a**8*b**4*c**9 - 128849018880*a**7*b**6*c**8 + 56371445760*a**6*b**8*c** 7 - 16911433728*a**5*b**10*c**6 + 3523215360*a**4*b**12*c**5 - 503316480*a **3*b**14*c**4 + 47185920*a**2*b**16*c**3 - 2621440*a*b**18*c**2 + 65536*b **20*c) + _t**2*(-188743680*a**7*b*c**7 + 141557760*a**6*b**3*c**6 - 23592 96*a**5*b**5*c**5 - 26542080*a**4*b**7*c**4 + 9584640*a**3*b**9*c**3 - 129 0240*a**2*b**11*c**2 + 46080*a*b**13*c + 2304*b**15) + 20736*a**5*c**4 + 1 03680*a**4*b**2*c**3 + 142560*a**3*b**4*c**2 + 32400*a**2*b**6*c + 2025*a* b**8, Lambda(_t, _t*log(x + (33554432*_t**3*a**6*c**7 - 16777216*_t**3*a** 5*b**2*c**6 - 10485760*_t**3*a**4*b**4*c**5 + 10485760*_t**3*a**3*b**6*c** 4 - 3276800*_t**3*a**2*b**8*c**3 + 458752*_t**3*a*b**10*c**2 - 24576*_t**3 *b**12*c - 64512*_t*a**3*b*c**3 - 43776*_t*a**2*b**3*c**2 - 21312*_t*a*b** 5*c - 144*_t*b**7)/(432*a**2*c**2 + 1080*a*b**2*c + 135*b**4))))
\[ \int \frac {x^6}{\left (a+b x^2+c x^4\right )^3} \, dx=\int { \frac {x^{6}}{{\left (c x^{4} + b x^{2} + a\right )}^{3}} \,d x } \] Input:
integrate(x^6/(c*x^4+b*x^2+a)^3,x, algorithm="maxima")
Output:
1/8*(3*(b^2*c + 4*a*c^2)*x^7 + (5*b^3 + 16*a*b*c)*x^5 + 12*a^2*b*x + (19*a *b^2 - 4*a^2*c)*x^3)/((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*x^8 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^6 + a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2 + ( b^6 - 6*a*b^4*c + 32*a^3*c^3)*x^4 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2) *x^2) + 3/8*integrate(((b^2 + 4*a*c)*x^2 - 4*a*b)/(c*x^4 + b*x^2 + a), x)/ (b^4 - 8*a*b^2*c + 16*a^2*c^2)
Leaf count of result is larger than twice the leaf count of optimal. 1750 vs. \(2 (254) = 508\).
Time = 1.67 (sec) , antiderivative size = 1750, normalized size of antiderivative = 5.87 \[ \int \frac {x^6}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Too large to display} \] Input:
integrate(x^6/(c*x^4+b*x^2+a)^3,x, algorithm="giac")
Output:
-3/16*(2*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^5 - 16*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3*c - 4*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c )*b^4*c - 4*b^5*c + 32*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b*c^2 + 16*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c^2 + 2*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^3*c^2 + 32*a*b^3*c^2 + 6*b^4*c^2 - 8*sqrt(2)*sqr t(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^3 - 64*a^2*b*c^3 - 16*a*b^2*c^3 - 32*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 + 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c + 6*sq rt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^3*c + 16*sqrt(2) *sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*c^2 + 8*sqrt(2)*sqr t(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 - 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*c^3 + 4*(b^2 - 4*a*c)*b^3*c - 16* (b^2 - 4*a*c)*a*b*c^2 - 6*(b^2 - 4*a*c)*b^2*c^2 - 8*(b^2 - 4*a*c)*a*c^3)*a rctan(2*sqrt(1/2)*x/sqrt((b^5 - 8*a*b^3*c + 16*a^2*b*c^2 + sqrt((b^5 - 8*a *b^3*c + 16*a^2*b*c^2)^2 - 4*(a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2)*(b^4*c - 8 *a*b^2*c^2 + 16*a^2*c^3)))/(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)))/((b^8 - 16 *a*b^6*c - 2*b^7*c + 96*a^2*b^4*c^2 + 24*a*b^5*c^2 + b^6*c^2 - 256*a^3*b^2 *c^3 - 96*a^2*b^3*c^3 - 12*a*b^4*c^3 + 256*a^4*c^4 + 128*a^3*b*c^4 + 48*a^ 2*b^2*c^4 - 64*a^3*c^5)*abs(c)) - 3/16*(2*sqrt(2)*sqrt(b*c - sqrt(b^2 -...
Time = 18.06 (sec) , antiderivative size = 8521, normalized size of antiderivative = 28.59 \[ \int \frac {x^6}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Too large to display} \] Input:
int(x^6/(a + b*x^2 + c*x^4)^3,x)
Output:
atan(((((3*(1024*a*b^11*c^2 - 1048576*a^6*b*c^7 - 20480*a^2*b^9*c^3 + 1638 40*a^3*b^7*c^4 - 655360*a^4*b^5*c^5 + 1310720*a^5*b^3*c^6))/(512*(b^12 + 4 096*a^6*c^6 + 240*a^2*b^8*c^2 - 1280*a^3*b^6*c^3 + 3840*a^4*b^4*c^4 - 6144 *a^5*b^2*c^5 - 24*a*b^10*c)) - (x*(-(9*(b^15 + (-(4*a*c - b^2)^15)^(1/2) - 81920*a^7*b*c^7 - 560*a^2*b^11*c^2 + 4160*a^3*b^9*c^3 - 11520*a^4*b^7*c^4 - 1024*a^5*b^5*c^5 + 61440*a^6*b^3*c^6 + 20*a*b^13*c))/(512*(b^20*c + 104 8576*a^10*c^11 - 40*a*b^18*c^2 + 720*a^2*b^16*c^3 - 7680*a^3*b^14*c^4 + 53 760*a^4*b^12*c^5 - 258048*a^5*b^10*c^6 + 860160*a^6*b^8*c^7 - 1966080*a^7* b^6*c^8 + 2949120*a^8*b^4*c^9 - 2621440*a^9*b^2*c^10)))^(1/2)*(256*b^11*c^ 2 - 5120*a*b^9*c^3 - 262144*a^5*b*c^7 + 40960*a^2*b^7*c^4 - 163840*a^3*b^5 *c^5 + 327680*a^4*b^3*c^6))/(32*(b^8 + 256*a^4*c^4 + 96*a^2*b^4*c^2 - 256* a^3*b^2*c^3 - 16*a*b^6*c)))*(-(9*(b^15 + (-(4*a*c - b^2)^15)^(1/2) - 81920 *a^7*b*c^7 - 560*a^2*b^11*c^2 + 4160*a^3*b^9*c^3 - 11520*a^4*b^7*c^4 - 102 4*a^5*b^5*c^5 + 61440*a^6*b^3*c^6 + 20*a*b^13*c))/(512*(b^20*c + 1048576*a ^10*c^11 - 40*a*b^18*c^2 + 720*a^2*b^16*c^3 - 7680*a^3*b^14*c^4 + 53760*a^ 4*b^12*c^5 - 258048*a^5*b^10*c^6 + 860160*a^6*b^8*c^7 - 1966080*a^7*b^6*c^ 8 + 2949120*a^8*b^4*c^9 - 2621440*a^9*b^2*c^10)))^(1/2) - (x*(9*b^6*c - 28 8*a^3*c^4 + 126*a*b^4*c^2 + 576*a^2*b^2*c^3))/(32*(b^8 + 256*a^4*c^4 + 96* a^2*b^4*c^2 - 256*a^3*b^2*c^3 - 16*a*b^6*c)))*(-(9*(b^15 + (-(4*a*c - b^2) ^15)^(1/2) - 81920*a^7*b*c^7 - 560*a^2*b^11*c^2 + 4160*a^3*b^9*c^3 - 11...
Time = 4.53 (sec) , antiderivative size = 4644, normalized size of antiderivative = 15.58 \[ \int \frac {x^6}{\left (a+b x^2+c x^4\right )^3} \, dx =\text {Too large to display} \] Input:
int(x^6/(c*x^4+b*x^2+a)^3,x)
Output:
( - 48*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**2 - 36*sqrt(a)*sqrt (2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/s qrt(2*sqrt(c)*sqrt(a) + b))*a**2*b**2*c - 96*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)*sqr t(a) + b))*a**2*b*c**2*x**2 - 96*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*c**3*x**4 - 72*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*b**3*c*x**2 - 120*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*b**2*c**2*x**4 - 96*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*b*c**3*x**6 - 48*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*c**4*x**8 - 36*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + b)*ata n((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b)) *b**4*c*x**4 - 72*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))*b**3*c**2*x**6 - 36*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))*b**2*c**3*x**8 + 72*sqrt(c)...