Integrand size = 18, antiderivative size = 289 \[ \int \frac {x^4}{\left (a+b x^2+c x^4\right )^3} \, dx=\frac {x \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {x \left (7 b^2-4 a c+12 b c x^2\right )}{8 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {3 \sqrt {c} \left (3 b^2+4 a c-2 b \sqrt {b^2-4 a c}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{4 \sqrt {2} \left (b^2-4 a c\right )^{5/2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {3 \sqrt {c} \left (3 b^2+4 a c+2 b \sqrt {b^2-4 a c}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{4 \sqrt {2} \left (b^2-4 a c\right )^{5/2} \sqrt {b+\sqrt {b^2-4 a c}}} \] Output:
1/4*x*(b*x^2+2*a)/(-4*a*c+b^2)/(c*x^4+b*x^2+a)^2-1/8*x*(12*b*c*x^2-4*a*c+7 *b^2)/(-4*a*c+b^2)^2/(c*x^4+b*x^2+a)+3/8*c^(1/2)*(3*b^2+4*a*c-2*b*(-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) /(-4*a*c+b^2)^(5/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2)-3/8*c^(1/2)*(3*b^2+4*a*c+ 2*b*(-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)/(-4*a*c+b^2)^(5/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2)
Time = 0.48 (sec) , antiderivative size = 285, normalized size of antiderivative = 0.99 \[ \int \frac {x^4}{\left (a+b x^2+c x^4\right )^3} \, dx=\frac {1}{8} \left (\frac {2 \left (2 a x+b x^3\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {-7 b^2 x+4 a c x-12 b c x^3}{\left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {3 \sqrt {2} \sqrt {c} \left (3 b^2+4 a c-2 b \sqrt {b^2-4 a c}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{5/2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {3 \sqrt {2} \sqrt {c} \left (3 b^2+4 a c+2 b \sqrt {b^2-4 a c}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{5/2} \sqrt {b+\sqrt {b^2-4 a c}}}\right ) \] Input:
Integrate[x^4/(a + b*x^2 + c*x^4)^3,x]
Output:
((2*(2*a*x + b*x^3))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (-7*b^2*x + 4 *a*c*x - 12*b*c*x^3)/((b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (3*Sqrt[2]*Sq rt[c]*(3*b^2 + 4*a*c - 2*b*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/S qrt[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]*(3*b^2 + 4*a*c + 2*b*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]]))/8
Time = 0.79 (sec) , antiderivative size = 296, 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, 1492, 27, 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.
| \(\displaystyle \int \frac {x^4}{\left (a+b x^2+c x^4\right )^3} \, dx\) |
| \(\Big \downarrow \) 1440 |
| \(\displaystyle \frac {x \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {\int \frac {2 a-5 b x^2}{\left (c x^4+b x^2+a\right )^2}dx}{4 \left (b^2-4 a c\right )}\) |
| \(\Big \downarrow \) 1492 |
| \(\displaystyle \frac {x \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {\frac {x \left (-4 a c+7 b^2+12 b c x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\int \frac {3 a \left (b^2-4 c x^2 b+4 a c\right )}{c x^4+b x^2+a}dx}{2 a \left (b^2-4 a c\right )}}{4 \left (b^2-4 a c\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {\frac {x \left (-4 a c+7 b^2+12 b c x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {3 \int \frac {b^2-4 c x^2 b+4 a c}{c x^4+b x^2+a}dx}{2 \left (b^2-4 a c\right )}}{4 \left (b^2-4 a c\right )}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {x \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {\frac {x \left (-4 a c+7 b^2+12 b c x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {3 \left (-c \left (2 b-\frac {4 a c+3 b^2}{\sqrt {b^2-4 a c}}\right ) \int \frac {1}{c x^2+\frac {1}{2} \left (b-\sqrt {b^2-4 a c}\right )}dx-c \left (\frac {4 a c+3 b^2}{\sqrt {b^2-4 a c}}+2 b\right ) \int \frac {1}{c x^2+\frac {1}{2} \left (b+\sqrt {b^2-4 a c}\right )}dx\right )}{2 \left (b^2-4 a c\right )}}{4 \left (b^2-4 a c\right )}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {x \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {\frac {x \left (-4 a c+7 b^2+12 b c x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {3 \left (-\frac {\sqrt {2} \sqrt {c} \left (2 b-\frac {4 a c+3 b^2}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} \sqrt {c} \left (\frac {4 a c+3 b^2}{\sqrt {b^2-4 a c}}+2 b\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{2 \left (b^2-4 a c\right )}}{4 \left (b^2-4 a c\right )}\) |
Input:
Int[x^4/(a + b*x^2 + c*x^4)^3,x]
Output:
(x*(2*a + b*x^2))/(4*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) - ((x*(7*b^2 - 4 *a*c + 12*b*c*x^2))/(2*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - (3*(-((Sqrt[2] *Sqrt[c]*(2*b - (3*b^2 + 4*a*c)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c] *x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (Sqrt[2]* Sqrt[c]*(2*b + (3*b^2 + 4*a*c)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]* x)/Sqrt[b + Sqrt[b^2 - 4*a*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[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symb ol] :> Simp[x*(a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c)) Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 7)*(d*b - 2*a*e)*c*x^2, x]*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && IntegerQ[2*p]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.17 (sec) , antiderivative size = 251, normalized size of antiderivative = 0.87
| method | result | size |
| risch | \(\frac {-\frac {3 b \,c^{2} x^{7}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {c \left (4 a c -19 b^{2}\right ) x^{5}}{128 a^{2} c^{2}-64 a \,b^{2} c +8 b^{4}}-\frac {b \left (16 a c +5 b^{2}\right ) x^{3}}{8 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {3 a \left (4 a c +b^{2}\right ) x}{8 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2}}+\frac {3 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4} c +\textit {\_Z}^{2} b +a \right )}{\sum }\frac {\left (-\frac {4 b c \,\textit {\_R}^{2}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {4 a c +b^{2}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{3} c +\textit {\_R} b}\right )}{16}\) | \(251\) |
| default | \(\frac {-\frac {3 b \,c^{2} x^{7}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {c \left (4 a c -19 b^{2}\right ) x^{5}}{128 a^{2} c^{2}-64 a \,b^{2} c +8 b^{4}}-\frac {b \left (16 a c +5 b^{2}\right ) x^{3}}{8 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {3 a \left (4 a c +b^{2}\right ) x}{8 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2}}+\frac {3 c \left (\frac {\left (-2 b \sqrt {-4 a c +b^{2}}-4 a c -3 b^{2}\right ) \sqrt {2}\, \arctan \left (\frac {c x \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{4 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}-\frac {\left (3 b^{2}+4 a c -2 b \sqrt {-4 a c +b^{2}}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c x \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{4 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}\) | \(338\) |
Input:
int(x^4/(c*x^4+b*x^2+a)^3,x,method=_RETURNVERBOSE)
Output:
(-3/2*b*c^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^7+1/8*c*(4*a*c-19*b^2)/(16*a^2*c^ 2-8*a*b^2*c+b^4)*x^5-1/8*b*(16*a*c+5*b^2)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3-3 /8*a*(4*a*c+b^2)/(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*b*c/(16*a^2*c^2-8*a*b^2*c+b^4)*_R^2+(4*a*c+b^2)/(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. \(2 (241) = 482\).
Time = 0.17 (sec) , antiderivative size = 3128, normalized size of antiderivative = 10.82 \[ \int \frac {x^4}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Too large to display} \] Input:
integrate(x^4/(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. 648 vs. \(2 (274) = 548\).
Time = 107.16 (sec) , antiderivative size = 648, normalized size of antiderivative = 2.24 \[ \int \frac {x^4}{\left (a+b x^2+c x^4\right )^3} \, dx=\frac {- 12 b c^{2} x^{7} + x^{5} \cdot \left (4 a c^{2} - 19 b^{2} c\right ) + x^{3} \left (- 16 a b c - 5 b^{3}\right ) + x \left (- 12 a^{2} c - 3 a b^{2}\right )}{128 a^{4} c^{2} - 64 a^{3} b^{2} c + 8 a^{2} b^{4} + x^{8} \cdot \left (128 a^{2} c^{4} - 64 a b^{2} c^{3} + 8 b^{4} c^{2}\right ) + x^{6} \cdot \left (256 a^{2} b c^{3} - 128 a b^{3} c^{2} + 16 b^{5} c\right ) + x^{4} \cdot \left (256 a^{3} c^{3} - 48 a b^{4} c + 8 b^{6}\right ) + x^{2} \cdot \left (256 a^{3} b c^{2} - 128 a^{2} b^{3} c + 16 a b^{5}\right )} + \operatorname {RootSum} {\left (t^{4} \cdot \left (68719476736 a^{11} c^{10} - 171798691840 a^{10} b^{2} c^{9} + 193273528320 a^{9} b^{4} c^{8} - 128849018880 a^{8} b^{6} c^{7} + 56371445760 a^{7} b^{8} c^{6} - 16911433728 a^{6} b^{10} c^{5} + 3523215360 a^{5} b^{12} c^{4} - 503316480 a^{4} b^{14} c^{3} + 47185920 a^{3} b^{16} c^{2} - 2621440 a^{2} b^{18} c + 65536 a b^{20}\right ) + t^{2} \left (- 188743680 a^{7} b c^{7} + 141557760 a^{6} b^{3} c^{6} - 2359296 a^{5} b^{5} c^{5} - 26542080 a^{4} b^{7} c^{4} + 9584640 a^{3} b^{9} c^{3} - 1290240 a^{2} b^{11} c^{2} + 46080 a b^{13} c + 2304 b^{15}\right ) + 20736 a^{4} c^{5} + 103680 a^{3} b^{2} c^{4} + 142560 a^{2} b^{4} c^{3} + 32400 a b^{6} c^{2} + 2025 b^{8} c, \left ( t \mapsto t \log {\left (x + \frac {50331648 t^{3} a^{7} b c^{6} - 58720256 t^{3} a^{6} b^{3} c^{5} + 26214400 t^{3} a^{5} b^{5} c^{4} - 5242880 t^{3} a^{4} b^{7} c^{3} + 327680 t^{3} a^{3} b^{9} c^{2} + 32768 t^{3} a^{2} b^{11} c - 4096 t^{3} a b^{13} + 18432 t a^{4} c^{4} - 78336 t a^{3} b^{2} c^{3} - 40320 t a^{2} b^{4} c^{2} - 3168 t a b^{6} c - 144 t b^{8}}{432 a^{2} c^{3} + 1080 a b^{2} c^{2} + 135 b^{4} c} \right )} \right )\right )} \] Input:
integrate(x**4/(c*x**4+b*x**2+a)**3,x)
Output:
(-12*b*c**2*x**7 + x**5*(4*a*c**2 - 19*b**2*c) + x**3*(-16*a*b*c - 5*b**3) + x*(-12*a**2*c - 3*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**11*c**10 - 171798691840*a**10*b**2*c**9 + 193273528 320*a**9*b**4*c**8 - 128849018880*a**8*b**6*c**7 + 56371445760*a**7*b**8*c **6 - 16911433728*a**6*b**10*c**5 + 3523215360*a**5*b**12*c**4 - 503316480 *a**4*b**14*c**3 + 47185920*a**3*b**16*c**2 - 2621440*a**2*b**18*c + 65536 *a*b**20) + _t**2*(-188743680*a**7*b*c**7 + 141557760*a**6*b**3*c**6 - 235 9296*a**5*b**5*c**5 - 26542080*a**4*b**7*c**4 + 9584640*a**3*b**9*c**3 - 1 290240*a**2*b**11*c**2 + 46080*a*b**13*c + 2304*b**15) + 20736*a**4*c**5 + 103680*a**3*b**2*c**4 + 142560*a**2*b**4*c**3 + 32400*a*b**6*c**2 + 2025* b**8*c, Lambda(_t, _t*log(x + (50331648*_t**3*a**7*b*c**6 - 58720256*_t**3 *a**6*b**3*c**5 + 26214400*_t**3*a**5*b**5*c**4 - 5242880*_t**3*a**4*b**7* c**3 + 327680*_t**3*a**3*b**9*c**2 + 32768*_t**3*a**2*b**11*c - 4096*_t**3 *a*b**13 + 18432*_t*a**4*c**4 - 78336*_t*a**3*b**2*c**3 - 40320*_t*a**2*b* *4*c**2 - 3168*_t*a*b**6*c - 144*_t*b**8)/(432*a**2*c**3 + 1080*a*b**2*c** 2 + 135*b**4*c))))
\[ \int \frac {x^4}{\left (a+b x^2+c x^4\right )^3} \, dx=\int { \frac {x^{4}}{{\left (c x^{4} + b x^{2} + a\right )}^{3}} \,d x } \] Input:
integrate(x^4/(c*x^4+b*x^2+a)^3,x, algorithm="maxima")
Output:
-1/8*(12*b*c^2*x^7 + (19*b^2*c - 4*a*c^2)*x^5 + (5*b^3 + 16*a*b*c)*x^3 + 3 *(a*b^2 + 4*a^2*c)*x)/((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((4*b*c*x^2 - b^2 - 4*a*c)/(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. 1861 vs. \(2 (241) = 482\).
Time = 1.53 (sec) , antiderivative size = 1861, normalized size of antiderivative = 6.44 \[ \int \frac {x^4}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Too large to display} \] Input:
integrate(x^4/(c*x^4+b*x^2+a)^3,x, algorithm="giac")
Output:
3/32*(sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^6 - 4*sqrt(2)*sqrt(b*c + s qrt(b^2 - 4*a*c)*c)*a*b^4*c - 2*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^ 5*c - 2*b^6*c - 16*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^2 + s qrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^4*c^2 + 8*a*b^4*c^2 + 2*b^5*c^2 + 64*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*c^3 + 32*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b*c^3 + 32*a^2*b^2*c^3 + 16*a*b^3*c^3 - 16*sqrt (2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*c^4 - 128*a^3*c^4 - 96*a^2*b*c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^5 - 8*sqrt(2) *sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3*c + 2*sqrt(2)*sqr t(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^4*c + 48*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b*c^2 + 24*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c^2 - sqrt(2)*sqrt(b^2 - 4* a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^3*c^2 - 12*sqrt(2)*sqrt(b^2 - 4*a*c )*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^3 + 2*(b^2 - 4*a*c)*b^4*c - 2*(b^2 - 4*a*c)*b^3*c^2 - 32*(b^2 - 4*a*c)*a^2*c^3 - 24*(b^2 - 4*a*c)*a*b*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)))/((a*b^8 - 16*a^2*b^6*c - 2*a*b^7*c + 96*a^3*b^4*c^2 + 24*a^2*b^5*c^2 + a*b^6*c^2 - 2 56*a^4*b^2*c^3 - 96*a^3*b^3*c^3 - 12*a^2*b^4*c^3 + 256*a^5*c^4 + 128*a^...
Time = 22.00 (sec) , antiderivative size = 8397, normalized size of antiderivative = 29.06 \[ \int \frac {x^4}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Too large to display} \] Input:
int(x^4/(a + b*x^2 + c*x^4)^3,x)
Output:
atan(((((3*(262144*a^6*c^8 - 64*b^12*c^2 + 1024*a*b^10*c^3 - 5120*a^2*b^8* c^4 + 81920*a^4*b^4*c^6 - 262144*a^5*b^2*c^7))/(128*(b^12 + 4096*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*((-(4*a*c - b^2)^15)^(1/2) - b^15 + 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*(a*b^20 + 1048576*a^11*c^10 - 40*a^2*b^18*c + 720*a^3*b^16*c^2 - 7680*a^4*b^14*c^3 + 53760*a^5*b^12*c ^4 - 258048*a^6*b^10*c^5 + 860160*a^7*b^8*c^6 - 1966080*a^8*b^6*c^7 + 2949 120*a^9*b^4*c^8 - 2621440*a^10*b^2*c^9)))^(1/2)*(128*b^11*c^2 - 2560*a*b^9 *c^3 - 131072*a^5*b*c^7 + 20480*a^2*b^7*c^4 - 81920*a^3*b^5*c^5 + 163840*a ^4*b^3*c^6))/(16*(b^8 + 256*a^4*c^4 + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 - 1 6*a*b^6*c)))*((9*((-(4*a*c - b^2)^15)^(1/2) - b^15 + 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*(a*b^20 + 1048576*a^11*c^10 - 40*a^ 2*b^18*c + 720*a^3*b^16*c^2 - 7680*a^4*b^14*c^3 + 53760*a^5*b^12*c^4 - 258 048*a^6*b^10*c^5 + 860160*a^7*b^8*c^6 - 1966080*a^8*b^6*c^7 + 2949120*a^9* b^4*c^8 - 2621440*a^10*b^2*c^9)))^(1/2) - (x*(144*a^2*c^5 + 117*b^4*c^3 + 72*a*b^2*c^4))/(16*(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*((-(4*a*c - b^2)^15)^(1/2) - b^15 + 81920*a^7*b*c^7 + 5 60*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...
Time = 4.35 (sec) , antiderivative size = 4644, normalized size of antiderivative = 16.07 \[ \int \frac {x^4}{\left (a+b x^2+c x^4\right )^3} \, dx =\text {Too large to display} \] Input:
int(x^4/(c*x^4+b*x^2+a)^3,x)
Output:
(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**3*b*c + 6*sqrt(a)*sqrt(2*sq rt(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**3 + 144*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**2*c*x**2 + 144*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 *c**2*x**4 + 12*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2*sqrt(c)*s qrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b))*a*b**4*x**2 + 84*s qrt(a)*sqrt(2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*s qrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b))*a*b**3*c*x**4 + 144*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**6 + 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)*s qrt(a) + b))*a*b*c**3*x**8 + 6*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + b)*atan((s qrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b))*b** 5*x**4 + 12*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**4*c*x**6 + 6*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**8 - 48*sqrt(c)*sqrt(2*sq...