\(\int \frac {x^6}{a+f x^2+c x^4} \, dx\) [784]

Optimal result

Integrand size = 18, antiderivative size = 304 \[ \int \frac {x^6}{a+f x^2+c x^4} \, dx=-\frac {f x}{c^2}+\frac {x^3}{3 c}+\frac {\left (a c-\sqrt {a} \sqrt {c} f-f^2\right ) \arctan \left (\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f}-2 \sqrt {c} x}{\sqrt {2 \sqrt {a} \sqrt {c}+f}}\right )}{2 c^{5/2} \sqrt {2 \sqrt {a} \sqrt {c}+f}}-\frac {\left (a c-\sqrt {a} \sqrt {c} f-f^2\right ) \arctan \left (\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f}+2 \sqrt {c} x}{\sqrt {2 \sqrt {a} \sqrt {c}+f}}\right )}{2 c^{5/2} \sqrt {2 \sqrt {a} \sqrt {c}+f}}+\frac {\left (a c+\sqrt {a} \sqrt {c} f-f^2\right ) \text {arctanh}\left (\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {a}+\sqrt {c} x^2}\right )}{2 c^{5/2} \sqrt {2 \sqrt {a} \sqrt {c}-f}} \] Output:

-f*x/c^2+1/3*x^3/c+1/2*(a*c-a^(1/2)*c^(1/2)*f-f^2)*arctan(((2*a^(1/2)*c^(1 
/2)-f)^(1/2)-2*c^(1/2)*x)/(2*a^(1/2)*c^(1/2)+f)^(1/2))/c^(5/2)/(2*a^(1/2)* 
c^(1/2)+f)^(1/2)-1/2*(a*c-a^(1/2)*c^(1/2)*f-f^2)*arctan(((2*a^(1/2)*c^(1/2 
)-f)^(1/2)+2*c^(1/2)*x)/(2*a^(1/2)*c^(1/2)+f)^(1/2))/c^(5/2)/(2*a^(1/2)*c^ 
(1/2)+f)^(1/2)+1/2*(a*c+a^(1/2)*c^(1/2)*f-f^2)*arctanh((2*a^(1/2)*c^(1/2)- 
f)^(1/2)*x/(a^(1/2)+c^(1/2)*x^2))/c^(5/2)/(2*a^(1/2)*c^(1/2)-f)^(1/2)
 

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 250, normalized size of antiderivative = 0.82 \[ \int \frac {x^6}{a+f x^2+c x^4} \, dx=-\frac {f x}{c^2}+\frac {x^3}{3 c}+\frac {\left (3 a c f-f^3-a c \sqrt {-4 a c+f^2}+f^2 \sqrt {-4 a c+f^2}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {f-\sqrt {-4 a c+f^2}}}\right )}{\sqrt {2} c^{5/2} \sqrt {-4 a c+f^2} \sqrt {f-\sqrt {-4 a c+f^2}}}+\frac {\left (-3 a c f+f^3-a c \sqrt {-4 a c+f^2}+f^2 \sqrt {-4 a c+f^2}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {f+\sqrt {-4 a c+f^2}}}\right )}{\sqrt {2} c^{5/2} \sqrt {-4 a c+f^2} \sqrt {f+\sqrt {-4 a c+f^2}}} \] Input:

Integrate[x^6/(a + f*x^2 + c*x^4),x]
 

Output:

-((f*x)/c^2) + x^3/(3*c) + ((3*a*c*f - f^3 - a*c*Sqrt[-4*a*c + f^2] + f^2* 
Sqrt[-4*a*c + f^2])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[f - Sqrt[-4*a*c + f^2] 
]])/(Sqrt[2]*c^(5/2)*Sqrt[-4*a*c + f^2]*Sqrt[f - Sqrt[-4*a*c + f^2]]) + (( 
-3*a*c*f + f^3 - a*c*Sqrt[-4*a*c + f^2] + f^2*Sqrt[-4*a*c + f^2])*ArcTan[( 
Sqrt[2]*Sqrt[c]*x)/Sqrt[f + Sqrt[-4*a*c + f^2]]])/(Sqrt[2]*c^(5/2)*Sqrt[-4 
*a*c + f^2]*Sqrt[f + Sqrt[-4*a*c + f^2]])
 

Rubi [A] (verified)

Time = 1.32 (sec) , antiderivative size = 451, normalized size of antiderivative = 1.48, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.611, Rules used = {1442, 27, 1602, 1483, 27, 1142, 25, 27, 1083, 217, 1103}

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.

\(\displaystyle \int \frac {x^6}{a+c x^4+f x^2} \, dx\)

\(\Big \downarrow \) 1442

\(\displaystyle \frac {x^3}{3 c}-\frac {\int \frac {3 x^2 \left (f x^2+a\right )}{c x^4+f x^2+a}dx}{3 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {x^3}{3 c}-\frac {\int \frac {x^2 \left (f x^2+a\right )}{c x^4+f x^2+a}dx}{c}\)

\(\Big \downarrow \) 1602

\(\displaystyle \frac {x^3}{3 c}-\frac {\frac {f x}{c}-\frac {\int \frac {a f-\left (a c-f^2\right ) x^2}{c x^4+f x^2+a}dx}{c}}{c}\)

\(\Big \downarrow \) 1483

\(\displaystyle \frac {x^3}{3 c}-\frac {\frac {f x}{c}-\frac {\frac {\int \frac {\sqrt {a} \left (\sqrt {a} \sqrt {2 \sqrt {a} \sqrt {c}-f} f-\sqrt {c} \left (\sqrt {a} f+\frac {a c-f^2}{\sqrt {c}}\right ) x\right )}{\sqrt {c} \left (x^2-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx}{2 \sqrt {a} \sqrt {2 \sqrt {a} \sqrt {c}-f}}+\frac {\int \frac {\sqrt {a} \left (\sqrt {a} \sqrt {2 \sqrt {a} \sqrt {c}-f} f+\sqrt {c} \left (\sqrt {a} f+\frac {a c-f^2}{\sqrt {c}}\right ) x\right )}{\sqrt {c} \left (x^2+\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx}{2 \sqrt {a} \sqrt {2 \sqrt {a} \sqrt {c}-f}}}{c}}{c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {x^3}{3 c}-\frac {\frac {f x}{c}-\frac {\frac {\int \frac {\sqrt {a} \sqrt {2 \sqrt {a} \sqrt {c}-f} f-\left (-f^2+\sqrt {a} \sqrt {c} f+a c\right ) x}{x^2-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-f}}+\frac {\int \frac {\sqrt {a} \sqrt {2 \sqrt {a} \sqrt {c}-f} f+\left (-f^2+\sqrt {a} \sqrt {c} f+a c\right ) x}{x^2+\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-f}}}{c}}{c}\)

\(\Big \downarrow \) 1142

\(\displaystyle \frac {x^3}{3 c}-\frac {\frac {f x}{c}-\frac {\frac {-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} \left (-\sqrt {a} \sqrt {c} f+a c-f^2\right ) \int \frac {1}{x^2-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {c}}-\frac {1}{2} \left (\sqrt {a} \sqrt {c} f+a c-f^2\right ) \int -\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f}-2 \sqrt {c} x}{\sqrt {c} \left (x^2-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx}{2 \sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-f}}+\frac {\frac {1}{2} \left (\sqrt {a} \sqrt {c} f+a c-f^2\right ) \int \frac {2 \sqrt {c} x+\sqrt {2 \sqrt {a} \sqrt {c}-f}}{\sqrt {c} \left (x^2+\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} \left (-\sqrt {a} \sqrt {c} f+a c-f^2\right ) \int \frac {1}{x^2+\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {c}}}{2 \sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-f}}}{c}}{c}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {x^3}{3 c}-\frac {\frac {f x}{c}-\frac {\frac {\frac {1}{2} \left (\sqrt {a} \sqrt {c} f+a c-f^2\right ) \int \frac {\sqrt {2 \sqrt {a} \sqrt {c}-f}-2 \sqrt {c} x}{\sqrt {c} \left (x^2-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} \left (-\sqrt {a} \sqrt {c} f+a c-f^2\right ) \int \frac {1}{x^2-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {c}}}{2 \sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-f}}+\frac {\frac {1}{2} \left (\sqrt {a} \sqrt {c} f+a c-f^2\right ) \int \frac {2 \sqrt {c} x+\sqrt {2 \sqrt {a} \sqrt {c}-f}}{\sqrt {c} \left (x^2+\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} \left (-\sqrt {a} \sqrt {c} f+a c-f^2\right ) \int \frac {1}{x^2+\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {c}}}{2 \sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-f}}}{c}}{c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {x^3}{3 c}-\frac {\frac {f x}{c}-\frac {\frac {\frac {\left (\sqrt {a} \sqrt {c} f+a c-f^2\right ) \int \frac {\sqrt {2 \sqrt {a} \sqrt {c}-f}-2 \sqrt {c} x}{x^2-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {c}}-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} \left (-\sqrt {a} \sqrt {c} f+a c-f^2\right ) \int \frac {1}{x^2-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {c}}}{2 \sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-f}}+\frac {\frac {\left (\sqrt {a} \sqrt {c} f+a c-f^2\right ) \int \frac {2 \sqrt {c} x+\sqrt {2 \sqrt {a} \sqrt {c}-f}}{x^2+\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {c}}-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} \left (-\sqrt {a} \sqrt {c} f+a c-f^2\right ) \int \frac {1}{x^2+\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {c}}}{2 \sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-f}}}{c}}{c}\)

\(\Big \downarrow \) 1083

\(\displaystyle \frac {x^3}{3 c}-\frac {\frac {f x}{c}-\frac {\frac {\frac {\left (\sqrt {a} \sqrt {c} f+a c-f^2\right ) \int \frac {\sqrt {2 \sqrt {a} \sqrt {c}-f}-2 \sqrt {c} x}{x^2-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {c}}+\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} \left (-\sqrt {a} \sqrt {c} f+a c-f^2\right ) \int \frac {1}{-\left (2 x-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f}}{\sqrt {c}}\right )^2-\frac {f+2 \sqrt {a} \sqrt {c}}{c}}d\left (2 x-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f}}{\sqrt {c}}\right )}{\sqrt {c}}}{2 \sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-f}}+\frac {\frac {\left (\sqrt {a} \sqrt {c} f+a c-f^2\right ) \int \frac {2 \sqrt {c} x+\sqrt {2 \sqrt {a} \sqrt {c}-f}}{x^2+\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {c}}+\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} \left (-\sqrt {a} \sqrt {c} f+a c-f^2\right ) \int \frac {1}{-\left (2 x+\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f}}{\sqrt {c}}\right )^2-\frac {f+2 \sqrt {a} \sqrt {c}}{c}}d\left (2 x+\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f}}{\sqrt {c}}\right )}{\sqrt {c}}}{2 \sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-f}}}{c}}{c}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {x^3}{3 c}-\frac {\frac {f x}{c}-\frac {\frac {\frac {\left (\sqrt {a} \sqrt {c} f+a c-f^2\right ) \int \frac {\sqrt {2 \sqrt {a} \sqrt {c}-f}-2 \sqrt {c} x}{x^2-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {c}}-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} \left (-\sqrt {a} \sqrt {c} f+a c-f^2\right ) \arctan \left (\frac {\sqrt {c} \left (2 x-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f}}{\sqrt {c}}\right )}{\sqrt {2 \sqrt {a} \sqrt {c}+f}}\right )}{\sqrt {2 \sqrt {a} \sqrt {c}+f}}}{2 \sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-f}}+\frac {\frac {\left (\sqrt {a} \sqrt {c} f+a c-f^2\right ) \int \frac {2 \sqrt {c} x+\sqrt {2 \sqrt {a} \sqrt {c}-f}}{x^2+\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {c}}-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} \left (-\sqrt {a} \sqrt {c} f+a c-f^2\right ) \arctan \left (\frac {\sqrt {c} \left (\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f}}{\sqrt {c}}+2 x\right )}{\sqrt {2 \sqrt {a} \sqrt {c}+f}}\right )}{\sqrt {2 \sqrt {a} \sqrt {c}+f}}}{2 \sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-f}}}{c}}{c}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {x^3}{3 c}-\frac {\frac {f x}{c}-\frac {\frac {-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} \left (-\sqrt {a} \sqrt {c} f+a c-f^2\right ) \arctan \left (\frac {\sqrt {c} \left (2 x-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f}}{\sqrt {c}}\right )}{\sqrt {2 \sqrt {a} \sqrt {c}+f}}\right )}{\sqrt {2 \sqrt {a} \sqrt {c}+f}}-\frac {1}{2} \left (\sqrt {a} \sqrt {c} f+a c-f^2\right ) \log \left (-x \sqrt {2 \sqrt {a} \sqrt {c}-f}+\sqrt {a}+\sqrt {c} x^2\right )}{2 \sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-f}}+\frac {\frac {1}{2} \left (\sqrt {a} \sqrt {c} f+a c-f^2\right ) \log \left (x \sqrt {2 \sqrt {a} \sqrt {c}-f}+\sqrt {a}+\sqrt {c} x^2\right )-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} \left (-\sqrt {a} \sqrt {c} f+a c-f^2\right ) \arctan \left (\frac {\sqrt {c} \left (\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f}}{\sqrt {c}}+2 x\right )}{\sqrt {2 \sqrt {a} \sqrt {c}+f}}\right )}{\sqrt {2 \sqrt {a} \sqrt {c}+f}}}{2 \sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-f}}}{c}}{c}\)

Input:

Int[x^6/(a + f*x^2 + c*x^4),x]
 

Output:

x^3/(3*c) - ((f*x)/c - ((-((Sqrt[2*Sqrt[a]*Sqrt[c] - f]*(a*c - Sqrt[a]*Sqr 
t[c]*f - f^2)*ArcTan[(Sqrt[c]*(-(Sqrt[2*Sqrt[a]*Sqrt[c] - f]/Sqrt[c]) + 2* 
x))/Sqrt[2*Sqrt[a]*Sqrt[c] + f]])/Sqrt[2*Sqrt[a]*Sqrt[c] + f]) - ((a*c + S 
qrt[a]*Sqrt[c]*f - f^2)*Log[Sqrt[a] - Sqrt[2*Sqrt[a]*Sqrt[c] - f]*x + Sqrt 
[c]*x^2])/2)/(2*Sqrt[c]*Sqrt[2*Sqrt[a]*Sqrt[c] - f]) + (-((Sqrt[2*Sqrt[a]* 
Sqrt[c] - f]*(a*c - Sqrt[a]*Sqrt[c]*f - f^2)*ArcTan[(Sqrt[c]*(Sqrt[2*Sqrt[ 
a]*Sqrt[c] - f]/Sqrt[c] + 2*x))/Sqrt[2*Sqrt[a]*Sqrt[c] + f]])/Sqrt[2*Sqrt[ 
a]*Sqrt[c] + f]) + ((a*c + Sqrt[a]*Sqrt[c]*f - f^2)*Log[Sqrt[a] + Sqrt[2*S 
qrt[a]*Sqrt[c] - f]*x + Sqrt[c]*x^2])/2)/(2*Sqrt[c]*Sqrt[2*Sqrt[a]*Sqrt[c] 
 - f]))/c)/c
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 1083

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2   Subst[I 
nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, 
x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1142

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[(2*c*d - b*e)/(2*c)   Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) 
Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
 

rule 1442

Int[((d_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] 
:> Simp[d^3*(d*x)^(m - 3)*((a + b*x^2 + c*x^4)^(p + 1)/(c*(m + 4*p + 1))), 
x] - Simp[d^4/(c*(m + 4*p + 1))   Int[(d*x)^(m - 4)*Simp[a*(m - 3) + b*(m + 
 2*p - 1)*x^2, x]*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, p}, x 
] && NeQ[b^2 - 4*a*c, 0] && GtQ[m, 3] && NeQ[m + 4*p + 1, 0] && IntegerQ[2* 
p] && (IntegerQ[p] || IntegerQ[m])
 

rule 1483

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : 
> With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Simp[1/(2*c*q*r)   In 
t[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Simp[1/(2*c*q*r)   Int[(d*r 
 + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && N 
eQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]
 

rule 1602

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*( 
x_)^4)^(p_), x_Symbol] :> Simp[e*f*(f*x)^(m - 1)*((a + b*x^2 + c*x^4)^(p + 
1)/(c*(m + 4*p + 3))), x] - Simp[f^2/(c*(m + 4*p + 3))   Int[(f*x)^(m - 2)* 
(a + b*x^2 + c*x^4)^p*Simp[a*e*(m - 1) + (b*e*(m + 2*p + 1) - c*d*(m + 4*p 
+ 3))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && NeQ[b^2 - 4*a*c 
, 0] && GtQ[m, 1] && NeQ[m + 4*p + 3, 0] && IntegerQ[2*p] && (IntegerQ[p] | 
| IntegerQ[m])
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.11 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.24

method	result	size
risch	\(\frac {x^{3}}{3 c}-\frac {f x}{c^{2}}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4} c +f \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\left (-a c +f^{2}\right ) \textit {\_R}^{2}+a f \right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{3} c +\textit {\_R} f}}{2 c^{2}}\)	\(73\)
default	\(\frac {\frac {1}{3} c \,x^{3}-f x}{c^{2}}+\frac {\frac {\left (-a c \sqrt {-4 a c +f^{2}}+f^{2} \sqrt {-4 a c +f^{2}}-3 a c f +f^{3}\right ) \sqrt {2}\, \arctan \left (\frac {c x \sqrt {2}}{\sqrt {\left (\sqrt {-4 a c +f^{2}}+f \right ) c}}\right )}{2 c \sqrt {-4 a c +f^{2}}\, \sqrt {\left (\sqrt {-4 a c +f^{2}}+f \right ) c}}-\frac {\left (-a c \sqrt {-4 a c +f^{2}}+f^{2} \sqrt {-4 a c +f^{2}}+3 a c f -f^{3}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c x \sqrt {2}}{\sqrt {\left (\sqrt {-4 a c +f^{2}}-f \right ) c}}\right )}{2 c \sqrt {-4 a c +f^{2}}\, \sqrt {\left (\sqrt {-4 a c +f^{2}}-f \right ) c}}}{c}\)	\(217\)

Input:

int(x^6/(c*x^4+f*x^2+a),x,method=_RETURNVERBOSE)
 

Output:

1/3*x^3/c-f*x/c^2+1/2/c^2*sum(((-a*c+f^2)*_R^2+a*f)/(2*_R^3*c+_R*f)*ln(x-_ 
R),_R=RootOf(_Z^4*c+_Z^2*f+a))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1604 vs. \(2 (225) = 450\).

Time = 0.12 (sec) , antiderivative size = 1604, normalized size of antiderivative = 5.28 \[ \int \frac {x^6}{a+f x^2+c x^4} \, dx=\text {Too large to display} \] Input:

integrate(x^6/(c*x^4+f*x^2+a),x, algorithm="fricas")
 

Output:

1/6*(2*c*x^3 + 3*sqrt(1/2)*c^2*sqrt((5*a^2*c^2*f - 5*a*c*f^3 + f^5 + (4*a* 
c^6 - c^5*f^2)*sqrt(-(a^4*c^4 - 6*a^3*c^3*f^2 + 11*a^2*c^2*f^4 - 6*a*c*f^6 
 + f^8)/(4*a*c^11 - c^10*f^2)))/(4*a*c^6 - c^5*f^2))*log(2*(a^4*c^2 - 3*a^ 
3*c*f^2 + a^2*f^4)*x + sqrt(1/2)*(4*a^3*c^3*f - 13*a^2*c^2*f^3 + 7*a*c*f^5 
 - f^7 - (8*a^2*c^7 - 6*a*c^6*f^2 + c^5*f^4)*sqrt(-(a^4*c^4 - 6*a^3*c^3*f^ 
2 + 11*a^2*c^2*f^4 - 6*a*c*f^6 + f^8)/(4*a*c^11 - c^10*f^2)))*sqrt((5*a^2* 
c^2*f - 5*a*c*f^3 + f^5 + (4*a*c^6 - c^5*f^2)*sqrt(-(a^4*c^4 - 6*a^3*c^3*f 
^2 + 11*a^2*c^2*f^4 - 6*a*c*f^6 + f^8)/(4*a*c^11 - c^10*f^2)))/(4*a*c^6 - 
c^5*f^2))) - 3*sqrt(1/2)*c^2*sqrt((5*a^2*c^2*f - 5*a*c*f^3 + f^5 + (4*a*c^ 
6 - c^5*f^2)*sqrt(-(a^4*c^4 - 6*a^3*c^3*f^2 + 11*a^2*c^2*f^4 - 6*a*c*f^6 + 
 f^8)/(4*a*c^11 - c^10*f^2)))/(4*a*c^6 - c^5*f^2))*log(2*(a^4*c^2 - 3*a^3* 
c*f^2 + a^2*f^4)*x - sqrt(1/2)*(4*a^3*c^3*f - 13*a^2*c^2*f^3 + 7*a*c*f^5 - 
 f^7 - (8*a^2*c^7 - 6*a*c^6*f^2 + c^5*f^4)*sqrt(-(a^4*c^4 - 6*a^3*c^3*f^2 
+ 11*a^2*c^2*f^4 - 6*a*c*f^6 + f^8)/(4*a*c^11 - c^10*f^2)))*sqrt((5*a^2*c^ 
2*f - 5*a*c*f^3 + f^5 + (4*a*c^6 - c^5*f^2)*sqrt(-(a^4*c^4 - 6*a^3*c^3*f^2 
 + 11*a^2*c^2*f^4 - 6*a*c*f^6 + f^8)/(4*a*c^11 - c^10*f^2)))/(4*a*c^6 - c^ 
5*f^2))) + 3*sqrt(1/2)*c^2*sqrt((5*a^2*c^2*f - 5*a*c*f^3 + f^5 - (4*a*c^6 
- c^5*f^2)*sqrt(-(a^4*c^4 - 6*a^3*c^3*f^2 + 11*a^2*c^2*f^4 - 6*a*c*f^6 + f 
^8)/(4*a*c^11 - c^10*f^2)))/(4*a*c^6 - c^5*f^2))*log(2*(a^4*c^2 - 3*a^3*c* 
f^2 + a^2*f^4)*x + sqrt(1/2)*(4*a^3*c^3*f - 13*a^2*c^2*f^3 + 7*a*c*f^5 ...
 

Sympy [A] (verification not implemented)

Time = 1.90 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.64 \[ \int \frac {x^6}{a+f x^2+c x^4} \, dx=\operatorname {RootSum} {\left (t^{4} \cdot \left (256 a^{2} c^{7} - 128 a c^{6} f^{2} + 16 c^{5} f^{4}\right ) + t^{2} \left (- 80 a^{3} c^{3} f + 100 a^{2} c^{2} f^{3} - 36 a c f^{5} + 4 f^{7}\right ) + a^{5}, \left ( t \mapsto t \log {\left (x + \frac {- 64 t^{3} a^{2} c^{7} + 48 t^{3} a c^{6} f^{2} - 8 t^{3} c^{5} f^{4} + 14 t a^{3} c^{3} f - 28 t a^{2} c^{2} f^{3} + 14 t a c f^{5} - 2 t f^{7}}{a^{4} c^{2} - 3 a^{3} c f^{2} + a^{2} f^{4}} \right )} \right )\right )} + \frac {x^{3}}{3 c} - \frac {f x}{c^{2}} \] Input:

integrate(x**6/(c*x**4+f*x**2+a),x)
 

Output:

RootSum(_t**4*(256*a**2*c**7 - 128*a*c**6*f**2 + 16*c**5*f**4) + _t**2*(-8 
0*a**3*c**3*f + 100*a**2*c**2*f**3 - 36*a*c*f**5 + 4*f**7) + a**5, Lambda( 
_t, _t*log(x + (-64*_t**3*a**2*c**7 + 48*_t**3*a*c**6*f**2 - 8*_t**3*c**5* 
f**4 + 14*_t*a**3*c**3*f - 28*_t*a**2*c**2*f**3 + 14*_t*a*c*f**5 - 2*_t*f* 
*7)/(a**4*c**2 - 3*a**3*c*f**2 + a**2*f**4)))) + x**3/(3*c) - f*x/c**2
 

Maxima [F]

\[ \int \frac {x^6}{a+f x^2+c x^4} \, dx=\int { \frac {x^{6}}{c x^{4} + f x^{2} + a} \,d x } \] Input:

integrate(x^6/(c*x^4+f*x^2+a),x, algorithm="maxima")
 

Output:

1/3*(c*x^3 - 3*f*x)/c^2 + integrate(-((a*c - f^2)*x^2 - a*f)/(c*x^4 + f*x^ 
2 + a), x)/c^2
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2479 vs. \(2 (225) = 450\).

Time = 0.61 (sec) , antiderivative size = 2479, normalized size of antiderivative = 8.15 \[ \int \frac {x^6}{a+f x^2+c x^4} \, dx=\text {Too large to display} \] Input:

integrate(x^6/(c*x^4+f*x^2+a),x, algorithm="giac")
 

Output:

1/8*(24*a^2*c^6*f^2 - 14*a*c^5*f^4 + 2*c^4*f^6 - 12*sqrt(2)*sqrt(-4*a*c + 
f^2)*sqrt(c*f + sqrt(-4*a*c + f^2)*c)*a^2*c^4*f^2 + 3*sqrt(2)*sqrt(-4*a*c 
+ f^2)*sqrt(c*f + sqrt(-4*a*c + f^2)*c)*a*c^5*f^2 - 6*sqrt(2)*sqrt(-4*a*c 
+ f^2)*sqrt(c*f + sqrt(-4*a*c + f^2)*c)*a*c^4*f^3 + 7*sqrt(2)*sqrt(-4*a*c 
+ f^2)*sqrt(c*f + sqrt(-4*a*c + f^2)*c)*a*c^3*f^4 - sqrt(2)*sqrt(-4*a*c + 
f^2)*sqrt(c*f + sqrt(-4*a*c + f^2)*c)*c^4*f^4 + 2*sqrt(2)*sqrt(-4*a*c + f^ 
2)*sqrt(c*f + sqrt(-4*a*c + f^2)*c)*c^3*f^5 - sqrt(2)*sqrt(-4*a*c + f^2)*s 
qrt(c*f + sqrt(-4*a*c + f^2)*c)*c^2*f^6 - 6*(4*a*c - f^2)*a*c^5*f^2 + 2*(4 
*a*c - f^2)*c^4*f^4 + (32*a^3*c^5 - 48*a^2*c^4*f^2 + 18*a*c^3*f^4 - 2*c^2* 
f^6 - 16*sqrt(2)*sqrt(-4*a*c + f^2)*sqrt(c*f + sqrt(-4*a*c + f^2)*c)*a^3*c 
^3 + 4*sqrt(2)*sqrt(-4*a*c + f^2)*sqrt(c*f + sqrt(-4*a*c + f^2)*c)*a^2*c^4 
 - 8*sqrt(2)*sqrt(-4*a*c + f^2)*sqrt(c*f + sqrt(-4*a*c + f^2)*c)*a^2*c^3*f 
 + 24*sqrt(2)*sqrt(-4*a*c + f^2)*sqrt(c*f + sqrt(-4*a*c + f^2)*c)*a^2*c^2* 
f^2 - 5*sqrt(2)*sqrt(-4*a*c + f^2)*sqrt(c*f + sqrt(-4*a*c + f^2)*c)*a*c^3* 
f^2 + 10*sqrt(2)*sqrt(-4*a*c + f^2)*sqrt(c*f + sqrt(-4*a*c + f^2)*c)*a*c^2 
*f^3 - 9*sqrt(2)*sqrt(-4*a*c + f^2)*sqrt(c*f + sqrt(-4*a*c + f^2)*c)*a*c*f 
^4 + sqrt(2)*sqrt(-4*a*c + f^2)*sqrt(c*f + sqrt(-4*a*c + f^2)*c)*c^2*f^4 - 
 2*sqrt(2)*sqrt(-4*a*c + f^2)*sqrt(c*f + sqrt(-4*a*c + f^2)*c)*c*f^5 + sqr 
t(2)*sqrt(-4*a*c + f^2)*sqrt(c*f + sqrt(-4*a*c + f^2)*c)*f^6 - 8*(4*a*c - 
f^2)*a^2*c^4 + 10*(4*a*c - f^2)*a*c^3*f^2 - 2*(4*a*c - f^2)*c^2*f^4)*c^...
 

Mupad [B] (verification not implemented)

Time = 0.66 (sec) , antiderivative size = 4127, normalized size of antiderivative = 13.58 \[ \int \frac {x^6}{a+f x^2+c x^4} \, dx=\text {Too large to display} \] Input:

int(x^6/(a + c*x^4 + f*x^2),x)
 

Output:

x^3/(3*c) - atan(((((4*a*c^3*f^3 - 16*a^2*c^4*f)/c^3 - (2*x*(4*c^5*f^3 - 1 
6*a*c^6*f)*(-(f^7 + f^4*(-(4*a*c - f^2)^3)^(1/2) - 20*a^3*c^3*f + 25*a^2*c 
^2*f^3 + a^2*c^2*(-(4*a*c - f^2)^3)^(1/2) - 9*a*c*f^5 - 3*a*c*f^2*(-(4*a*c 
 - f^2)^3)^(1/2))/(8*(16*a^2*c^7 + c^5*f^4 - 8*a*c^6*f^2)))^(1/2))/c^3)*(- 
(f^7 + f^4*(-(4*a*c - f^2)^3)^(1/2) - 20*a^3*c^3*f + 25*a^2*c^2*f^3 + a^2* 
c^2*(-(4*a*c - f^2)^3)^(1/2) - 9*a*c*f^5 - 3*a*c*f^2*(-(4*a*c - f^2)^3)^(1 
/2))/(8*(16*a^2*c^7 + c^5*f^4 - 8*a*c^6*f^2)))^(1/2) - (2*x*(f^6 - 2*a^3*c 
^3 + 9*a^2*c^2*f^2 - 6*a*c*f^4))/c^3)*(-(f^7 + f^4*(-(4*a*c - f^2)^3)^(1/2 
) - 20*a^3*c^3*f + 25*a^2*c^2*f^3 + a^2*c^2*(-(4*a*c - f^2)^3)^(1/2) - 9*a 
*c*f^5 - 3*a*c*f^2*(-(4*a*c - f^2)^3)^(1/2))/(8*(16*a^2*c^7 + c^5*f^4 - 8* 
a*c^6*f^2)))^(1/2)*1i - (((4*a*c^3*f^3 - 16*a^2*c^4*f)/c^3 + (2*x*(4*c^5*f 
^3 - 16*a*c^6*f)*(-(f^7 + f^4*(-(4*a*c - f^2)^3)^(1/2) - 20*a^3*c^3*f + 25 
*a^2*c^2*f^3 + a^2*c^2*(-(4*a*c - f^2)^3)^(1/2) - 9*a*c*f^5 - 3*a*c*f^2*(- 
(4*a*c - f^2)^3)^(1/2))/(8*(16*a^2*c^7 + c^5*f^4 - 8*a*c^6*f^2)))^(1/2))/c 
^3)*(-(f^7 + f^4*(-(4*a*c - f^2)^3)^(1/2) - 20*a^3*c^3*f + 25*a^2*c^2*f^3 
+ a^2*c^2*(-(4*a*c - f^2)^3)^(1/2) - 9*a*c*f^5 - 3*a*c*f^2*(-(4*a*c - f^2) 
^3)^(1/2))/(8*(16*a^2*c^7 + c^5*f^4 - 8*a*c^6*f^2)))^(1/2) + (2*x*(f^6 - 2 
*a^3*c^3 + 9*a^2*c^2*f^2 - 6*a*c*f^4))/c^3)*(-(f^7 + f^4*(-(4*a*c - f^2)^3 
)^(1/2) - 20*a^3*c^3*f + 25*a^2*c^2*f^3 + a^2*c^2*(-(4*a*c - f^2)^3)^(1/2) 
 - 9*a*c*f^5 - 3*a*c*f^2*(-(4*a*c - f^2)^3)^(1/2))/(8*(16*a^2*c^7 + c^5...
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 761, normalized size of antiderivative = 2.50 \[ \int \frac {x^6}{a+f x^2+c x^4} \, dx =\text {Too large to display} \] Input:

int(x^6/(c*x^4+f*x^2+a),x)
 

Output:

(12*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + f)*atan((sqrt(2*sqrt(c)*sqrt(a) - f) 
- 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + f))*a*c**2 - 6*sqrt(a)*sqrt(2*sqrt 
(c)*sqrt(a) + f)*atan((sqrt(2*sqrt(c)*sqrt(a) - f) - 2*sqrt(c)*x)/sqrt(2*s 
qrt(c)*sqrt(a) + f))*c*f**2 - 18*sqrt(c)*sqrt(2*sqrt(c)*sqrt(a) + f)*atan( 
(sqrt(2*sqrt(c)*sqrt(a) - f) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + f))*a 
*c*f + 6*sqrt(c)*sqrt(2*sqrt(c)*sqrt(a) + f)*atan((sqrt(2*sqrt(c)*sqrt(a) 
- f) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + f))*f**3 - 12*sqrt(a)*sqrt(2* 
sqrt(c)*sqrt(a) + f)*atan((sqrt(2*sqrt(c)*sqrt(a) - f) + 2*sqrt(c)*x)/sqrt 
(2*sqrt(c)*sqrt(a) + f))*a*c**2 + 6*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + f)*at 
an((sqrt(2*sqrt(c)*sqrt(a) - f) + 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + f) 
)*c*f**2 + 18*sqrt(c)*sqrt(2*sqrt(c)*sqrt(a) + f)*atan((sqrt(2*sqrt(c)*sqr 
t(a) - f) + 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + f))*a*c*f - 6*sqrt(c)*sq 
rt(2*sqrt(c)*sqrt(a) + f)*atan((sqrt(2*sqrt(c)*sqrt(a) - f) + 2*sqrt(c)*x) 
/sqrt(2*sqrt(c)*sqrt(a) + f))*f**3 - 6*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) - f) 
*log( - sqrt(2*sqrt(c)*sqrt(a) - f)*x + sqrt(a) + sqrt(c)*x**2)*a*c**2 + 3 
*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) - f)*log( - sqrt(2*sqrt(c)*sqrt(a) - f)*x 
+ sqrt(a) + sqrt(c)*x**2)*c*f**2 + 6*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) - f)*l 
og(sqrt(2*sqrt(c)*sqrt(a) - f)*x + sqrt(a) + sqrt(c)*x**2)*a*c**2 - 3*sqrt 
(a)*sqrt(2*sqrt(c)*sqrt(a) - f)*log(sqrt(2*sqrt(c)*sqrt(a) - f)*x + sqrt(a 
) + sqrt(c)*x**2)*c*f**2 - 9*sqrt(c)*sqrt(2*sqrt(c)*sqrt(a) - f)*log( -...