\(\int \frac {x^4}{a+f x^2+c x^4} \, dx\) [785]

Optimal result

Integrand size = 18, antiderivative size = 268 \[ \int \frac {x^4}{a+f x^2+c x^4} \, dx=\frac {x}{c}+\frac {\left (\sqrt {a} \sqrt {c}+f\right ) \arctan \left (\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f}-2 \sqrt {c} x}{\sqrt {2 \sqrt {a} \sqrt {c}+f}}\right )}{2 c^{3/2} \sqrt {2 \sqrt {a} \sqrt {c}+f}}-\frac {\left (\sqrt {a} \sqrt {c}+f\right ) \arctan \left (\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f}+2 \sqrt {c} x}{\sqrt {2 \sqrt {a} \sqrt {c}+f}}\right )}{2 c^{3/2} \sqrt {2 \sqrt {a} \sqrt {c}+f}}-\frac {\left (\sqrt {a} \sqrt {c}-f\right ) \text {arctanh}\left (\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {a}+\sqrt {c} x^2}\right )}{2 c^{3/2} \sqrt {2 \sqrt {a} \sqrt {c}-f}} \] Output:

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

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.75 \[ \int \frac {x^4}{a+f x^2+c x^4} \, dx=\frac {x}{c}-\frac {\left (2 a c-f^2+f \sqrt {-4 a c+f^2}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {f-\sqrt {-4 a c+f^2}}}\right )}{\sqrt {2} c^{3/2} \sqrt {-4 a c+f^2} \sqrt {f-\sqrt {-4 a c+f^2}}}-\frac {\left (-2 a c+f^2+f \sqrt {-4 a c+f^2}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {f+\sqrt {-4 a c+f^2}}}\right )}{\sqrt {2} c^{3/2} \sqrt {-4 a c+f^2} \sqrt {f+\sqrt {-4 a c+f^2}}} \] Input:

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

Output:

x/c - ((2*a*c - f^2 + f*Sqrt[-4*a*c + f^2])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqr 
t[f - Sqrt[-4*a*c + f^2]]])/(Sqrt[2]*c^(3/2)*Sqrt[-4*a*c + f^2]*Sqrt[f - S 
qrt[-4*a*c + f^2]]) - ((-2*a*c + f^2 + f*Sqrt[-4*a*c + f^2])*ArcTan[(Sqrt[ 
2]*Sqrt[c]*x)/Sqrt[f + Sqrt[-4*a*c + f^2]]])/(Sqrt[2]*c^(3/2)*Sqrt[-4*a*c 
+ f^2]*Sqrt[f + Sqrt[-4*a*c + f^2]])
 

Rubi [A] (verified)

Time = 1.07 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.50, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1442, 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.

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

\(\Big \downarrow \) 1442

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

\(\Big \downarrow \) 1483

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1142

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1083

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 1103

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

Input:

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

Output:

x/c - (((Sqrt[2*Sqrt[a]*Sqrt[c] - f]*(Sqrt[a]*Sqrt[c] + f)*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] - ((Sqrt[a]*Sqrt[c] - f)*Log[Sqrt[a] - Sq 
rt[2*Sqrt[a]*Sqrt[c] - f]*x + Sqrt[c]*x^2])/2)/(2*Sqrt[c]*Sqrt[2*Sqrt[a]*S 
qrt[c] - f]) + ((Sqrt[2*Sqrt[a]*Sqrt[c] - f]*(Sqrt[a]*Sqrt[c] + f)*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] + ((Sqrt[a]*Sqrt[c] - f)*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]))/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]
 

Maple [C] (verified)

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

Time = 0.10 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.21

method	result	size
risch	\(\frac {x}{c}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4} c +f \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (-\textit {\_R}^{2} f -a \right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{3} c +\textit {\_R} f}}{2 c}\)	\(57\)
default	\(\frac {x}{c}+\frac {\left (-f \sqrt {-4 a c +f^{2}}+2 a c -f^{2}\right ) \sqrt {2}\, \arctan \left (\frac {c x \sqrt {2}}{\sqrt {\left (\sqrt {-4 a c +f^{2}}+f \right ) c}}\right )}{2 c \sqrt {-4 a c +f^{2}}\, \sqrt {\left (\sqrt {-4 a c +f^{2}}+f \right ) c}}-\frac {\left (-f \sqrt {-4 a c +f^{2}}-2 a c +f^{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c x \sqrt {2}}{\sqrt {\left (\sqrt {-4 a c +f^{2}}-f \right ) c}}\right )}{2 c \sqrt {-4 a c +f^{2}}\, \sqrt {\left (\sqrt {-4 a c +f^{2}}-f \right ) c}}\)	\(169\)

Input:

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

Output:

x/c+1/2/c*sum((-_R^2*f-a)/(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. 1119 vs. \(2 (191) = 382\).

Time = 0.11 (sec) , antiderivative size = 1119, normalized size of antiderivative = 4.18 \[ \int \frac {x^4}{a+f x^2+c x^4} \, dx =\text {Too large to display} \] Input:

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

Output:

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

Sympy [A] (verification not implemented)

Time = 1.08 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.48 \[ \int \frac {x^4}{a+f x^2+c x^4} \, dx=\operatorname {RootSum} {\left (t^{4} \cdot \left (256 a^{2} c^{5} - 128 a c^{4} f^{2} + 16 c^{3} f^{4}\right ) + t^{2} \cdot \left (48 a^{2} c^{2} f - 28 a c f^{3} + 4 f^{5}\right ) + a^{3}, \left ( t \mapsto t \log {\left (x + \frac {32 t^{3} a c^{4} f - 8 t^{3} c^{3} f^{3} - 4 t a^{2} c^{2} + 8 t a c f^{2} - 2 t f^{4}}{a^{2} c - a f^{2}} \right )} \right )\right )} + \frac {x}{c} \] Input:

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

Output:

RootSum(_t**4*(256*a**2*c**5 - 128*a*c**4*f**2 + 16*c**3*f**4) + _t**2*(48 
*a**2*c**2*f - 28*a*c*f**3 + 4*f**5) + a**3, Lambda(_t, _t*log(x + (32*_t* 
*3*a*c**4*f - 8*_t**3*c**3*f**3 - 4*_t*a**2*c**2 + 8*_t*a*c*f**2 - 2*_t*f* 
*4)/(a**2*c - a*f**2)))) + x/c
 

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2133 vs. \(2 (191) = 382\).

Time = 0.59 (sec) , antiderivative size = 2133, normalized size of antiderivative = 7.96 \[ \int \frac {x^4}{a+f x^2+c x^4} \, dx=\text {Too large to display} \] Input:

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

Output:

x/c - 1/8*(16*a^2*c^6*f - 12*a*c^5*f^3 + 2*c^4*f^5 - 8*sqrt(2)*sqrt(-4*a*c 
 + f^2)*sqrt(c*f + sqrt(-4*a*c + f^2)*c)*a^2*c^4*f + 2*sqrt(2)*sqrt(-4*a*c 
 + f^2)*sqrt(c*f + sqrt(-4*a*c + f^2)*c)*a*c^5*f - 4*sqrt(2)*sqrt(-4*a*c + 
 f^2)*sqrt(c*f + sqrt(-4*a*c + f^2)*c)*a*c^4*f^2 + 6*sqrt(2)*sqrt(-4*a*c + 
 f^2)*sqrt(c*f + sqrt(-4*a*c + f^2)*c)*a*c^3*f^3 - sqrt(2)*sqrt(-4*a*c + f 
^2)*sqrt(c*f + sqrt(-4*a*c + f^2)*c)*c^4*f^3 + 2*sqrt(2)*sqrt(-4*a*c + f^2 
)*sqrt(c*f + sqrt(-4*a*c + f^2)*c)*c^3*f^4 - sqrt(2)*sqrt(-4*a*c + f^2)*sq 
rt(c*f + sqrt(-4*a*c + f^2)*c)*c^2*f^5 - 4*(4*a*c - f^2)*a*c^5*f + 2*(4*a* 
c - f^2)*c^4*f^3 - (32*a^2*c^4*f - 16*a*c^3*f^3 + 2*c^2*f^5 - 16*sqrt(2)*s 
qrt(-4*a*c + f^2)*sqrt(c*f + sqrt(-4*a*c + f^2)*c)*a^2*c^2*f + 4*sqrt(2)*s 
qrt(-4*a*c + f^2)*sqrt(c*f + sqrt(-4*a*c + f^2)*c)*a*c^3*f - 8*sqrt(2)*sqr 
t(-4*a*c + f^2)*sqrt(c*f + sqrt(-4*a*c + f^2)*c)*a*c^2*f^2 + 8*sqrt(2)*sqr 
t(-4*a*c + f^2)*sqrt(c*f + sqrt(-4*a*c + f^2)*c)*a*c*f^3 - sqrt(2)*sqrt(-4 
*a*c + f^2)*sqrt(c*f + sqrt(-4*a*c + f^2)*c)*c^2*f^3 + 2*sqrt(2)*sqrt(-4*a 
*c + f^2)*sqrt(c*f + sqrt(-4*a*c + f^2)*c)*c*f^4 - sqrt(2)*sqrt(-4*a*c + f 
^2)*sqrt(c*f + sqrt(-4*a*c + f^2)*c)*f^5 - 8*(4*a*c - f^2)*a*c^3*f + 2*(4* 
a*c - f^2)*c^2*f^3)*c^2 + 2*(16*sqrt(2)*sqrt(c*f + sqrt(-4*a*c + f^2)*c)*a 
^3*c^4 - 4*sqrt(2)*sqrt(c*f + sqrt(-4*a*c + f^2)*c)*a^2*c^5 - 32*a^3*c^5 + 
 8*sqrt(2)*sqrt(c*f + sqrt(-4*a*c + f^2)*c)*a^2*c^4*f - 8*sqrt(2)*sqrt(c*f 
 + sqrt(-4*a*c + f^2)*c)*a^2*c^3*f^2 + sqrt(2)*sqrt(c*f + sqrt(-4*a*c +...
 

Mupad [B] (verification not implemented)

Time = 18.53 (sec) , antiderivative size = 3026, normalized size of antiderivative = 11.29 \[ \int \frac {x^4}{a+f x^2+c x^4} \, dx=\text {Too large to display} \] Input:

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

Output:

x/c - atan(((((16*a^2*c^3 - 4*a*c^2*f^2)/c - (2*x*(4*c^3*f^3 - 16*a*c^4*f) 
*(-(f^5 + f^2*(-(4*a*c - f^2)^3)^(1/2) + 12*a^2*c^2*f - 7*a*c*f^3 - a*c*(- 
(4*a*c - f^2)^3)^(1/2))/(8*(16*a^2*c^5 + c^3*f^4 - 8*a*c^4*f^2)))^(1/2))/c 
)*(-(f^5 + f^2*(-(4*a*c - f^2)^3)^(1/2) + 12*a^2*c^2*f - 7*a*c*f^3 - a*c*( 
-(4*a*c - f^2)^3)^(1/2))/(8*(16*a^2*c^5 + c^3*f^4 - 8*a*c^4*f^2)))^(1/2) - 
 (2*x*(f^4 + 2*a^2*c^2 - 4*a*c*f^2))/c)*(-(f^5 + f^2*(-(4*a*c - f^2)^3)^(1 
/2) + 12*a^2*c^2*f - 7*a*c*f^3 - a*c*(-(4*a*c - f^2)^3)^(1/2))/(8*(16*a^2* 
c^5 + c^3*f^4 - 8*a*c^4*f^2)))^(1/2)*1i - (((16*a^2*c^3 - 4*a*c^2*f^2)/c + 
 (2*x*(4*c^3*f^3 - 16*a*c^4*f)*(-(f^5 + f^2*(-(4*a*c - f^2)^3)^(1/2) + 12* 
a^2*c^2*f - 7*a*c*f^3 - a*c*(-(4*a*c - f^2)^3)^(1/2))/(8*(16*a^2*c^5 + c^3 
*f^4 - 8*a*c^4*f^2)))^(1/2))/c)*(-(f^5 + f^2*(-(4*a*c - f^2)^3)^(1/2) + 12 
*a^2*c^2*f - 7*a*c*f^3 - a*c*(-(4*a*c - f^2)^3)^(1/2))/(8*(16*a^2*c^5 + c^ 
3*f^4 - 8*a*c^4*f^2)))^(1/2) + (2*x*(f^4 + 2*a^2*c^2 - 4*a*c*f^2))/c)*(-(f 
^5 + f^2*(-(4*a*c - f^2)^3)^(1/2) + 12*a^2*c^2*f - 7*a*c*f^3 - a*c*(-(4*a* 
c - f^2)^3)^(1/2))/(8*(16*a^2*c^5 + c^3*f^4 - 8*a*c^4*f^2)))^(1/2)*1i)/((( 
(16*a^2*c^3 - 4*a*c^2*f^2)/c - (2*x*(4*c^3*f^3 - 16*a*c^4*f)*(-(f^5 + f^2* 
(-(4*a*c - f^2)^3)^(1/2) + 12*a^2*c^2*f - 7*a*c*f^3 - a*c*(-(4*a*c - f^2)^ 
3)^(1/2))/(8*(16*a^2*c^5 + c^3*f^4 - 8*a*c^4*f^2)))^(1/2))/c)*(-(f^5 + f^2 
*(-(4*a*c - f^2)^3)^(1/2) + 12*a^2*c^2*f - 7*a*c*f^3 - a*c*(-(4*a*c - f^2) 
^3)^(1/2))/(8*(16*a^2*c^5 + c^3*f^4 - 8*a*c^4*f^2)))^(1/2) - (2*x*(f^4 ...
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 547, normalized size of antiderivative = 2.04 \[ \int \frac {x^4}{a+f x^2+c x^4} \, dx=\frac {2 \sqrt {a}\, \sqrt {2 \sqrt {c}\, \sqrt {a}+f}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}-f}-2 \sqrt {c}\, x}{\sqrt {2 \sqrt {c}\, \sqrt {a}+f}}\right ) c f +4 \sqrt {c}\, \sqrt {2 \sqrt {c}\, \sqrt {a}+f}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}-f}-2 \sqrt {c}\, x}{\sqrt {2 \sqrt {c}\, \sqrt {a}+f}}\right ) a c -2 \sqrt {c}\, \sqrt {2 \sqrt {c}\, \sqrt {a}+f}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}-f}-2 \sqrt {c}\, x}{\sqrt {2 \sqrt {c}\, \sqrt {a}+f}}\right ) f^{2}-2 \sqrt {a}\, \sqrt {2 \sqrt {c}\, \sqrt {a}+f}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}-f}+2 \sqrt {c}\, x}{\sqrt {2 \sqrt {c}\, \sqrt {a}+f}}\right ) c f -4 \sqrt {c}\, \sqrt {2 \sqrt {c}\, \sqrt {a}+f}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}-f}+2 \sqrt {c}\, x}{\sqrt {2 \sqrt {c}\, \sqrt {a}+f}}\right ) a c +2 \sqrt {c}\, \sqrt {2 \sqrt {c}\, \sqrt {a}+f}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}-f}+2 \sqrt {c}\, x}{\sqrt {2 \sqrt {c}\, \sqrt {a}+f}}\right ) f^{2}-\sqrt {a}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-f}\, \mathrm {log}\left (-\sqrt {2 \sqrt {c}\, \sqrt {a}-f}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) c f +\sqrt {a}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-f}\, \mathrm {log}\left (\sqrt {2 \sqrt {c}\, \sqrt {a}-f}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) c f +2 \sqrt {c}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-f}\, \mathrm {log}\left (-\sqrt {2 \sqrt {c}\, \sqrt {a}-f}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) a c -\sqrt {c}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-f}\, \mathrm {log}\left (-\sqrt {2 \sqrt {c}\, \sqrt {a}-f}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) f^{2}-2 \sqrt {c}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-f}\, \mathrm {log}\left (\sqrt {2 \sqrt {c}\, \sqrt {a}-f}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) a c +\sqrt {c}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-f}\, \mathrm {log}\left (\sqrt {2 \sqrt {c}\, \sqrt {a}-f}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) f^{2}+16 a \,c^{2} x -4 c \,f^{2} x}{4 c^{2} \left (4 a c -f^{2}\right )} \] Input:

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

Output:

(2*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))*c*f + 4*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 - 2*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**2 - 2* 
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))*c*f - 4*sqrt(c)*sqrt(2*sqrt(c)*sqr 
t(a) + f)*atan((sqrt(2*sqrt(c)*sqrt(a) - f) + 2*sqrt(c)*x)/sqrt(2*sqrt(c)* 
sqrt(a) + f))*a*c + 2*sqrt(c)*sqrt(2*sqrt(c)*sqrt(a) + f)*atan((sqrt(2*sqr 
t(c)*sqrt(a) - f) + 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + f))*f**2 - 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 + sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) - f)*log(sqrt(2*s 
qrt(c)*sqrt(a) - f)*x + sqrt(a) + sqrt(c)*x**2)*c*f + 2*sqrt(c)*sqrt(2*sqr 
t(c)*sqrt(a) - f)*log( - sqrt(2*sqrt(c)*sqrt(a) - f)*x + sqrt(a) + sqrt(c) 
*x**2)*a*c - sqrt(c)*sqrt(2*sqrt(c)*sqrt(a) - f)*log( - sqrt(2*sqrt(c)*sqr 
t(a) - f)*x + sqrt(a) + sqrt(c)*x**2)*f**2 - 2*sqrt(c)*sqrt(2*sqrt(c)*sqrt 
(a) - f)*log(sqrt(2*sqrt(c)*sqrt(a) - f)*x + sqrt(a) + sqrt(c)*x**2)*a*c + 
 sqrt(c)*sqrt(2*sqrt(c)*sqrt(a) - f)*log(sqrt(2*sqrt(c)*sqrt(a) - f)*x + s 
qrt(a) + sqrt(c)*x**2)*f**2 + 16*a*c**2*x - 4*c*f**2*x)/(4*c**2*(4*a*c - f 
**2))