\(\int \frac {x^4}{a-b x^2+c x^4} \, dx\) [822]

Optimal result

Integrand size = 19, antiderivative size = 179 \[ \int \frac {x^4}{a-b x^2+c x^4} \, dx=\frac {x}{c}-\frac {\left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} c^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\left (b+\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} c^{3/2} \sqrt {b+\sqrt {b^2-4 a c}}} \] Output:

x/c-1/2*(b-(-2*a*c+b^2)/(-4*a*c+b^2)^(1/2))*arctanh(2^(1/2)*c^(1/2)*x/(b-( 
-4*a*c+b^2)^(1/2))^(1/2))*2^(1/2)/c^(3/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2)-1/2 
*(b+(-2*a*c+b^2)/(-4*a*c+b^2)^(1/2))*arctanh(2^(1/2)*c^(1/2)*x/(b+(-4*a*c+ 
b^2)^(1/2))^(1/2))*2^(1/2)/c^(3/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2)
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.16 \[ \int \frac {x^4}{a-b x^2+c x^4} \, dx=\frac {x}{c}+\frac {\left (b^2-2 a c+b \sqrt {b^2-4 a c}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {-b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} c^{3/2} \sqrt {b^2-4 a c} \sqrt {-b-\sqrt {b^2-4 a c}}}+\frac {\left (-b^2+2 a c+b \sqrt {b^2-4 a c}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {-b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} c^{3/2} \sqrt {b^2-4 a c} \sqrt {-b+\sqrt {b^2-4 a c}}} \] Input:

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

Output:

x/c + ((b^2 - 2*a*c + b*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt 
[-b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*c^(3/2)*Sqrt[b^2 - 4*a*c]*Sqrt[-b - Sq 
rt[b^2 - 4*a*c]]) + ((-b^2 + 2*a*c + b*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]* 
Sqrt[c]*x)/Sqrt[-b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*c^(3/2)*Sqrt[b^2 - 4*a* 
c]*Sqrt[-b + Sqrt[b^2 - 4*a*c]])
 

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.02, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {1442, 1480, 221}

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.

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

\(\Big \downarrow \) 1442

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

\(\Big \downarrow \) 1480

\(\displaystyle \frac {x}{c}-\frac {-\frac {1}{2} \left (\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}+b\right ) \int \frac {1}{c x^2+\frac {1}{2} \left (-b-\sqrt {b^2-4 a c}\right )}dx-\frac {1}{2} \left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \int \frac {1}{c x^2+\frac {1}{2} \left (\sqrt {b^2-4 a c}-b\right )}dx}{c}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {x}{c}-\frac {\frac {\left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} \sqrt {c} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\left (\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}+b\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {2} \sqrt {c} \sqrt {\sqrt {b^2-4 a c}+b}}}{c}\)

Input:

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

Output:

x/c - (((b - (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*ArcTanh[(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 + (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*ArcTanh[(Sqrt[2]*Sqrt[c]*x)/Sqr 
t[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[c]*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/ 
c
 

Defintions of rubi rules used

rule 221

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

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 1480

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]
 

Maple [C] (verified)

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

Time = 0.10 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.32

method	result	size
risch	\(\frac {x}{c}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{4}-\textit {\_Z}^{2} b +a \right )}{\sum }\frac {\left (\textit {\_R}^{2} b -a \right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{3} c -\textit {\_R} b}}{2 c}\)	\(58\)
default	\(\frac {x}{c}-\frac {\left (b \sqrt {-4 a c +b^{2}}-2 a c +b^{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c x \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{2 \sqrt {-4 a c +b^{2}}\, c \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\left (b \sqrt {-4 a c +b^{2}}+2 a c -b^{2}\right ) \sqrt {2}\, \arctan \left (\frac {c x \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{2 \sqrt {-4 a c +b^{2}}\, c \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\)	\(167\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1051 vs. \(2 (143) = 286\).

Time = 0.10 (sec) , antiderivative size = 1051, normalized size of antiderivative = 5.87 \[ \int \frac {x^4}{a-b x^2+c x^4} \, dx =\text {Too large to display} \] Input:

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

Output:

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

Sympy [A] (verification not implemented)

Time = 1.12 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.72 \[ \int \frac {x^4}{a-b x^2+c x^4} \, dx=\operatorname {RootSum} {\left (t^{4} \cdot \left (256 a^{2} c^{5} - 128 a b^{2} c^{4} + 16 b^{4} c^{3}\right ) + t^{2} \left (- 48 a^{2} b c^{2} + 28 a b^{3} c - 4 b^{5}\right ) + a^{3}, \left ( t \mapsto t \log {\left (x + \frac {- 32 t^{3} a b c^{4} + 8 t^{3} b^{3} c^{3} - 4 t a^{2} c^{2} + 8 t a b^{2} c - 2 t b^{4}}{a^{2} c - a b^{2}} \right )} \right )\right )} + \frac {x}{c} \] Input:

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2153 vs. \(2 (143) = 286\).

Time = 0.57 (sec) , antiderivative size = 2153, normalized size of antiderivative = 12.03 \[ \int \frac {x^4}{a-b x^2+c x^4} \, dx=\text {Too large to display} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 18.44 (sec) , antiderivative size = 3000, normalized size of antiderivative = 16.76 \[ \int \frac {x^4}{a-b x^2+c x^4} \, dx=\text {Too large to display} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 535, normalized size of antiderivative = 2.99 \[ \int \frac {x^4}{a-b x^2+c x^4} \, dx=\frac {-2 \sqrt {a}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-b}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}+b}-2 \sqrt {c}\, x}{\sqrt {2 \sqrt {c}\, \sqrt {a}-b}}\right ) b c +4 \sqrt {c}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-b}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}+b}-2 \sqrt {c}\, x}{\sqrt {2 \sqrt {c}\, \sqrt {a}-b}}\right ) a c -2 \sqrt {c}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-b}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}+b}-2 \sqrt {c}\, x}{\sqrt {2 \sqrt {c}\, \sqrt {a}-b}}\right ) b^{2}+2 \sqrt {a}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-b}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}+b}+2 \sqrt {c}\, x}{\sqrt {2 \sqrt {c}\, \sqrt {a}-b}}\right ) b c -4 \sqrt {c}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-b}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}+b}+2 \sqrt {c}\, x}{\sqrt {2 \sqrt {c}\, \sqrt {a}-b}}\right ) a c +2 \sqrt {c}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-b}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}+b}+2 \sqrt {c}\, x}{\sqrt {2 \sqrt {c}\, \sqrt {a}-b}}\right ) b^{2}+\sqrt {a}\, \sqrt {2 \sqrt {c}\, \sqrt {a}+b}\, \mathrm {log}\left (-\sqrt {2 \sqrt {c}\, \sqrt {a}+b}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) b c -\sqrt {a}\, \sqrt {2 \sqrt {c}\, \sqrt {a}+b}\, \mathrm {log}\left (\sqrt {2 \sqrt {c}\, \sqrt {a}+b}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) b c +2 \sqrt {c}\, \sqrt {2 \sqrt {c}\, \sqrt {a}+b}\, \mathrm {log}\left (-\sqrt {2 \sqrt {c}\, \sqrt {a}+b}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) a c -\sqrt {c}\, \sqrt {2 \sqrt {c}\, \sqrt {a}+b}\, \mathrm {log}\left (-\sqrt {2 \sqrt {c}\, \sqrt {a}+b}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) b^{2}-2 \sqrt {c}\, \sqrt {2 \sqrt {c}\, \sqrt {a}+b}\, \mathrm {log}\left (\sqrt {2 \sqrt {c}\, \sqrt {a}+b}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) a c +\sqrt {c}\, \sqrt {2 \sqrt {c}\, \sqrt {a}+b}\, \mathrm {log}\left (\sqrt {2 \sqrt {c}\, \sqrt {a}+b}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) b^{2}+16 a \,c^{2} x -4 b^{2} c x}{4 c^{2} \left (4 a c -b^{2}\right )} \] Input:

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

Output:

( - 2*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*c + 4*sqrt(c)*sqrt(2*sqrt( 
c)*sqrt(a) - b)*atan((sqrt(2*sqrt(c)*sqrt(a) + b) - 2*sqrt(c)*x)/sqrt(2*sq 
rt(c)*sqrt(a) - b))*a*c - 2*sqrt(c)*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 + 
 2*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*c - 4*sqrt(c)*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 + 2*sqrt(c)*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 + sq 
rt(a)*sqrt(2*sqrt(c)*sqrt(a) + b)*log( - sqrt(2*sqrt(c)*sqrt(a) + b)*x + s 
qrt(a) + sqrt(c)*x**2)*b*c - sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + b)*log(sqrt( 
2*sqrt(c)*sqrt(a) + b)*x + sqrt(a) + sqrt(c)*x**2)*b*c + 2*sqrt(c)*sqrt(2* 
sqrt(c)*sqrt(a) + b)*log( - sqrt(2*sqrt(c)*sqrt(a) + b)*x + sqrt(a) + sqrt 
(c)*x**2)*a*c - sqrt(c)*sqrt(2*sqrt(c)*sqrt(a) + b)*log( - sqrt(2*sqrt(c)* 
sqrt(a) + b)*x + sqrt(a) + sqrt(c)*x**2)*b**2 - 2*sqrt(c)*sqrt(2*sqrt(c)*s 
qrt(a) + b)*log(sqrt(2*sqrt(c)*sqrt(a) + b)*x + sqrt(a) + sqrt(c)*x**2)*a* 
c + sqrt(c)*sqrt(2*sqrt(c)*sqrt(a) + b)*log(sqrt(2*sqrt(c)*sqrt(a) + b)*x 
+ sqrt(a) + sqrt(c)*x**2)*b**2 + 16*a*c**2*x - 4*b**2*c*x)/(4*c**2*(4*a*c 
- b**2))