Integrand size = 18, antiderivative size = 87 \[ \int \frac {x^5}{a+f x^2+c x^4} \, dx=\frac {x^2}{2 c}-\frac {\left (2 a c-f^2\right ) \arctan \left (\frac {f+2 c x^2}{\sqrt {4 a c-f^2}}\right )}{2 c^2 \sqrt {4 a c-f^2}}-\frac {f \log \left (a+f x^2+c x^4\right )}{4 c^2} \] Output:
1/2*x^2/c-1/2*(2*a*c-f^2)*arctan((2*c*x^2+f)/(4*a*c-f^2)^(1/2))/c^2/(4*a*c -f^2)^(1/2)-1/4*f*ln(c*x^4+f*x^2+a)/c^2
Time = 0.03 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.90 \[ \int \frac {x^5}{a+f x^2+c x^4} \, dx=\frac {2 c x^2+\frac {2 \left (-2 a c+f^2\right ) \arctan \left (\frac {f+2 c x^2}{\sqrt {4 a c-f^2}}\right )}{\sqrt {4 a c-f^2}}-f \log \left (a+f x^2+c x^4\right )}{4 c^2} \] Input:
Integrate[x^5/(a + f*x^2 + c*x^4),x]
Output:
(2*c*x^2 + (2*(-2*a*c + f^2)*ArcTan[(f + 2*c*x^2)/Sqrt[4*a*c - f^2]])/Sqrt [4*a*c - f^2] - f*Log[a + f*x^2 + c*x^4])/(4*c^2)
Time = 0.42 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1434, 1143, 2009}
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.
| \(\displaystyle \int \frac {x^5}{a+c x^4+f x^2} \, dx\) |
| \(\Big \downarrow \) 1434 |
| \(\displaystyle \frac {1}{2} \int \frac {x^4}{c x^4+f x^2+a}dx^2\) |
| \(\Big \downarrow \) 1143 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {1}{c}-\frac {f x^2+a}{c \left (c x^4+f x^2+a\right )}\right )dx^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\left (2 a c-f^2\right ) \arctan \left (\frac {2 c x^2+f}{\sqrt {4 a c-f^2}}\right )}{c^2 \sqrt {4 a c-f^2}}-\frac {f \log \left (a+c x^4+f x^2\right )}{2 c^2}+\frac {x^2}{c}\right )\) |
Input:
Int[x^5/(a + f*x^2 + c*x^4),x]
Output:
(x^2/c - ((2*a*c - f^2)*ArcTan[(f + 2*c*x^2)/Sqrt[4*a*c - f^2]])/(c^2*Sqrt [4*a*c - f^2]) - (f*Log[a + f*x^2 + c*x^4])/(2*c^2))/2
Int[((d_.) + (e_.)*(x_))^(m_)/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[m, 1]
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp [1/2 Subst[Int[x^((m - 1)/2)*(a + b*x + c*x^2)^p, x], x, x^2], x] /; Free Q[{a, b, c, p}, x] && IntegerQ[(m - 1)/2]
Time = 0.12 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.95
| method | result | size |
| default | \(\frac {x^{2}}{2 c}+\frac {-\frac {f \ln \left (c \,x^{4}+f \,x^{2}+a \right )}{2 c}+\frac {2 \left (-a +\frac {f^{2}}{2 c}\right ) \arctan \left (\frac {2 c \,x^{2}+f}{\sqrt {4 a c -f^{2}}}\right )}{\sqrt {4 a c -f^{2}}}}{2 c}\) | \(83\) |
| risch | \(\frac {x^{2}}{2 c}-\frac {\ln \left (\left (-8 a^{2} c^{2}+6 a c \,f^{2}-f^{4}+\sqrt {-\left (4 a c -f^{2}\right ) \left (2 a c -f^{2}\right )^{2}}\, f \right ) x^{2}+2 \sqrt {-\left (4 a c -f^{2}\right ) \left (2 a c -f^{2}\right )^{2}}\, a \right ) a f}{c \left (4 a c -f^{2}\right )}+\frac {\ln \left (\left (-8 a^{2} c^{2}+6 a c \,f^{2}-f^{4}+\sqrt {-\left (4 a c -f^{2}\right ) \left (2 a c -f^{2}\right )^{2}}\, f \right ) x^{2}+2 \sqrt {-\left (4 a c -f^{2}\right ) \left (2 a c -f^{2}\right )^{2}}\, a \right ) f^{3}}{4 c^{2} \left (4 a c -f^{2}\right )}+\frac {\ln \left (\left (-8 a^{2} c^{2}+6 a c \,f^{2}-f^{4}+\sqrt {-\left (4 a c -f^{2}\right ) \left (2 a c -f^{2}\right )^{2}}\, f \right ) x^{2}+2 \sqrt {-\left (4 a c -f^{2}\right ) \left (2 a c -f^{2}\right )^{2}}\, a \right ) \sqrt {-\left (4 a c -f^{2}\right ) \left (2 a c -f^{2}\right )^{2}}}{4 c^{2} \left (4 a c -f^{2}\right )}-\frac {\ln \left (\left (-8 a^{2} c^{2}+6 a c \,f^{2}-f^{4}-\sqrt {-\left (4 a c -f^{2}\right ) \left (2 a c -f^{2}\right )^{2}}\, f \right ) x^{2}-2 \sqrt {-\left (4 a c -f^{2}\right ) \left (2 a c -f^{2}\right )^{2}}\, a \right ) a f}{c \left (4 a c -f^{2}\right )}+\frac {\ln \left (\left (-8 a^{2} c^{2}+6 a c \,f^{2}-f^{4}-\sqrt {-\left (4 a c -f^{2}\right ) \left (2 a c -f^{2}\right )^{2}}\, f \right ) x^{2}-2 \sqrt {-\left (4 a c -f^{2}\right ) \left (2 a c -f^{2}\right )^{2}}\, a \right ) f^{3}}{4 c^{2} \left (4 a c -f^{2}\right )}-\frac {\ln \left (\left (-8 a^{2} c^{2}+6 a c \,f^{2}-f^{4}-\sqrt {-\left (4 a c -f^{2}\right ) \left (2 a c -f^{2}\right )^{2}}\, f \right ) x^{2}-2 \sqrt {-\left (4 a c -f^{2}\right ) \left (2 a c -f^{2}\right )^{2}}\, a \right ) \sqrt {-\left (4 a c -f^{2}\right ) \left (2 a c -f^{2}\right )^{2}}}{4 c^{2} \left (4 a c -f^{2}\right )}\) | \(681\) |
Input:
int(x^5/(c*x^4+f*x^2+a),x,method=_RETURNVERBOSE)
Output:
1/2*x^2/c+1/2/c*(-1/2*f/c*ln(c*x^4+f*x^2+a)+2*(-a+1/2*f^2/c)/(4*a*c-f^2)^( 1/2)*arctan((2*c*x^2+f)/(4*a*c-f^2)^(1/2)))
Time = 0.08 (sec) , antiderivative size = 256, normalized size of antiderivative = 2.94 \[ \int \frac {x^5}{a+f x^2+c x^4} \, dx=\left [\frac {2 \, {\left (4 \, a c^{2} - c f^{2}\right )} x^{2} + {\left (2 \, a c - f^{2}\right )} \sqrt {-4 \, a c + f^{2}} \log \left (\frac {2 \, c^{2} x^{4} + 2 \, c f x^{2} - 2 \, a c + f^{2} - {\left (2 \, c x^{2} + f\right )} \sqrt {-4 \, a c + f^{2}}}{c x^{4} + f x^{2} + a}\right ) - {\left (4 \, a c f - f^{3}\right )} \log \left (c x^{4} + f x^{2} + a\right )}{4 \, {\left (4 \, a c^{3} - c^{2} f^{2}\right )}}, \frac {2 \, {\left (4 \, a c^{2} - c f^{2}\right )} x^{2} + 2 \, \sqrt {4 \, a c - f^{2}} {\left (2 \, a c - f^{2}\right )} \arctan \left (-\frac {2 \, c x^{2} + f}{\sqrt {4 \, a c - f^{2}}}\right ) - {\left (4 \, a c f - f^{3}\right )} \log \left (c x^{4} + f x^{2} + a\right )}{4 \, {\left (4 \, a c^{3} - c^{2} f^{2}\right )}}\right ] \] Input:
integrate(x^5/(c*x^4+f*x^2+a),x, algorithm="fricas")
Output:
[1/4*(2*(4*a*c^2 - c*f^2)*x^2 + (2*a*c - f^2)*sqrt(-4*a*c + f^2)*log((2*c^ 2*x^4 + 2*c*f*x^2 - 2*a*c + f^2 - (2*c*x^2 + f)*sqrt(-4*a*c + f^2))/(c*x^4 + f*x^2 + a)) - (4*a*c*f - f^3)*log(c*x^4 + f*x^2 + a))/(4*a*c^3 - c^2*f^ 2), 1/4*(2*(4*a*c^2 - c*f^2)*x^2 + 2*sqrt(4*a*c - f^2)*(2*a*c - f^2)*arcta n(-(2*c*x^2 + f)/sqrt(4*a*c - f^2)) - (4*a*c*f - f^3)*log(c*x^4 + f*x^2 + a))/(4*a*c^3 - c^2*f^2)]
Leaf count of result is larger than twice the leaf count of optimal. 316 vs. \(2 (71) = 142\).
Time = 1.04 (sec) , antiderivative size = 316, normalized size of antiderivative = 3.63 \[ \int \frac {x^5}{a+f x^2+c x^4} \, dx=\left (- \frac {f}{4 c^{2}} - \frac {\sqrt {- 4 a c + f^{2}} \cdot \left (2 a c - f^{2}\right )}{4 c^{2} \cdot \left (4 a c - f^{2}\right )}\right ) \log {\left (x^{2} + \frac {- 8 a c^{2} \left (- \frac {f}{4 c^{2}} - \frac {\sqrt {- 4 a c + f^{2}} \cdot \left (2 a c - f^{2}\right )}{4 c^{2} \cdot \left (4 a c - f^{2}\right )}\right ) - a f + 2 c f^{2} \left (- \frac {f}{4 c^{2}} - \frac {\sqrt {- 4 a c + f^{2}} \cdot \left (2 a c - f^{2}\right )}{4 c^{2} \cdot \left (4 a c - f^{2}\right )}\right )}{2 a c - f^{2}} \right )} + \left (- \frac {f}{4 c^{2}} + \frac {\sqrt {- 4 a c + f^{2}} \cdot \left (2 a c - f^{2}\right )}{4 c^{2} \cdot \left (4 a c - f^{2}\right )}\right ) \log {\left (x^{2} + \frac {- 8 a c^{2} \left (- \frac {f}{4 c^{2}} + \frac {\sqrt {- 4 a c + f^{2}} \cdot \left (2 a c - f^{2}\right )}{4 c^{2} \cdot \left (4 a c - f^{2}\right )}\right ) - a f + 2 c f^{2} \left (- \frac {f}{4 c^{2}} + \frac {\sqrt {- 4 a c + f^{2}} \cdot \left (2 a c - f^{2}\right )}{4 c^{2} \cdot \left (4 a c - f^{2}\right )}\right )}{2 a c - f^{2}} \right )} + \frac {x^{2}}{2 c} \] Input:
integrate(x**5/(c*x**4+f*x**2+a),x)
Output:
(-f/(4*c**2) - sqrt(-4*a*c + f**2)*(2*a*c - f**2)/(4*c**2*(4*a*c - f**2))) *log(x**2 + (-8*a*c**2*(-f/(4*c**2) - sqrt(-4*a*c + f**2)*(2*a*c - f**2)/( 4*c**2*(4*a*c - f**2))) - a*f + 2*c*f**2*(-f/(4*c**2) - sqrt(-4*a*c + f**2 )*(2*a*c - f**2)/(4*c**2*(4*a*c - f**2))))/(2*a*c - f**2)) + (-f/(4*c**2) + sqrt(-4*a*c + f**2)*(2*a*c - f**2)/(4*c**2*(4*a*c - f**2)))*log(x**2 + ( -8*a*c**2*(-f/(4*c**2) + sqrt(-4*a*c + f**2)*(2*a*c - f**2)/(4*c**2*(4*a*c - f**2))) - a*f + 2*c*f**2*(-f/(4*c**2) + sqrt(-4*a*c + f**2)*(2*a*c - f* *2)/(4*c**2*(4*a*c - f**2))))/(2*a*c - f**2)) + x**2/(2*c)
Exception generated. \[ \int \frac {x^5}{a+f x^2+c x^4} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(x^5/(c*x^4+f*x^2+a),x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(f^2-4*a*c>0)', see `assume?` for more deta
Time = 0.39 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.89 \[ \int \frac {x^5}{a+f x^2+c x^4} \, dx=\frac {x^{2}}{2 \, c} - \frac {f \log \left (c x^{4} + f x^{2} + a\right )}{4 \, c^{2}} - \frac {{\left (2 \, a c - f^{2}\right )} \arctan \left (\frac {2 \, c x^{2} + f}{\sqrt {4 \, a c - f^{2}}}\right )}{2 \, \sqrt {4 \, a c - f^{2}} c^{2}} \] Input:
integrate(x^5/(c*x^4+f*x^2+a),x, algorithm="giac")
Output:
1/2*x^2/c - 1/4*f*log(c*x^4 + f*x^2 + a)/c^2 - 1/2*(2*a*c - f^2)*arctan((2 *c*x^2 + f)/sqrt(4*a*c - f^2))/(sqrt(4*a*c - f^2)*c^2)
Time = 0.52 (sec) , antiderivative size = 655, normalized size of antiderivative = 7.53 \[ \int \frac {x^5}{a+f x^2+c x^4} \, dx=\frac {x^2}{2\,c}+\frac {\ln \left (c\,x^4+f\,x^2+a\right )\,\left (2\,f^3-8\,a\,c\,f\right )}{2\,\left (16\,a\,c^3-4\,c^2\,f^2\right )}+\frac {\mathrm {atan}\left (\frac {2\,c^2\,\left (4\,a\,c-f^2\right )\,\left (\frac {\frac {\left (8\,a\,f+\frac {8\,a\,c^2\,\left (2\,f^3-8\,a\,c\,f\right )}{16\,a\,c^3-4\,c^2\,f^2}\right )\,\left (2\,a\,c-f^2\right )}{8\,c^2\,\sqrt {4\,a\,c-f^2}}+\frac {a\,\left (2\,f^3-8\,a\,c\,f\right )\,\left (2\,a\,c-f^2\right )}{\sqrt {4\,a\,c-f^2}\,\left (16\,a\,c^3-4\,c^2\,f^2\right )}}{a}-x^2\,\left (\frac {\frac {\left (\frac {4\,a\,c^3-6\,c^2\,f^2}{c^2}-\frac {4\,c^2\,f\,\left (2\,f^3-8\,a\,c\,f\right )}{16\,a\,c^3-4\,c^2\,f^2}\right )\,\left (2\,a\,c-f^2\right )}{8\,c^2\,\sqrt {4\,a\,c-f^2}}-\frac {f\,\left (2\,f^3-8\,a\,c\,f\right )\,\left (2\,a\,c-f^2\right )}{2\,\sqrt {4\,a\,c-f^2}\,\left (16\,a\,c^3-4\,c^2\,f^2\right )}}{a}+\frac {f\,\left (\frac {\left (\frac {4\,a\,c^3-6\,c^2\,f^2}{c^2}-\frac {4\,c^2\,f\,\left (2\,f^3-8\,a\,c\,f\right )}{16\,a\,c^3-4\,c^2\,f^2}\right )\,\left (2\,f^3-8\,a\,c\,f\right )}{2\,\left (16\,a\,c^3-4\,c^2\,f^2\right )}-\frac {f^3-a\,c\,f}{c^2}+\frac {f\,{\left (2\,a\,c-f^2\right )}^2}{2\,c^2\,\left (4\,a\,c-f^2\right )}\right )}{2\,a\,\sqrt {4\,a\,c-f^2}}\right )+\frac {f\,\left (\frac {a\,f^2}{c^2}+\frac {\left (2\,f^3-8\,a\,c\,f\right )\,\left (8\,a\,f+\frac {8\,a\,c^2\,\left (2\,f^3-8\,a\,c\,f\right )}{16\,a\,c^3-4\,c^2\,f^2}\right )}{2\,\left (16\,a\,c^3-4\,c^2\,f^2\right )}-\frac {a\,{\left (2\,a\,c-f^2\right )}^2}{c^2\,\left (4\,a\,c-f^2\right )}\right )}{2\,a\,\sqrt {4\,a\,c-f^2}}\right )}{4\,a^2\,c^2-4\,a\,c\,f^2+f^4}\right )\,\left (2\,a\,c-f^2\right )}{2\,c^2\,\sqrt {4\,a\,c-f^2}} \] Input:
int(x^5/(a + c*x^4 + f*x^2),x)
Output:
x^2/(2*c) + (log(a + c*x^4 + f*x^2)*(2*f^3 - 8*a*c*f))/(2*(16*a*c^3 - 4*c^ 2*f^2)) + (atan((2*c^2*(4*a*c - f^2)*((((8*a*f + (8*a*c^2*(2*f^3 - 8*a*c*f ))/(16*a*c^3 - 4*c^2*f^2))*(2*a*c - f^2))/(8*c^2*(4*a*c - f^2)^(1/2)) + (a *(2*f^3 - 8*a*c*f)*(2*a*c - f^2))/((4*a*c - f^2)^(1/2)*(16*a*c^3 - 4*c^2*f ^2)))/a - x^2*(((((4*a*c^3 - 6*c^2*f^2)/c^2 - (4*c^2*f*(2*f^3 - 8*a*c*f))/ (16*a*c^3 - 4*c^2*f^2))*(2*a*c - f^2))/(8*c^2*(4*a*c - f^2)^(1/2)) - (f*(2 *f^3 - 8*a*c*f)*(2*a*c - f^2))/(2*(4*a*c - f^2)^(1/2)*(16*a*c^3 - 4*c^2*f^ 2)))/a + (f*((((4*a*c^3 - 6*c^2*f^2)/c^2 - (4*c^2*f*(2*f^3 - 8*a*c*f))/(16 *a*c^3 - 4*c^2*f^2))*(2*f^3 - 8*a*c*f))/(2*(16*a*c^3 - 4*c^2*f^2)) - (f^3 - a*c*f)/c^2 + (f*(2*a*c - f^2)^2)/(2*c^2*(4*a*c - f^2))))/(2*a*(4*a*c - f ^2)^(1/2))) + (f*((a*f^2)/c^2 + ((2*f^3 - 8*a*c*f)*(8*a*f + (8*a*c^2*(2*f^ 3 - 8*a*c*f))/(16*a*c^3 - 4*c^2*f^2)))/(2*(16*a*c^3 - 4*c^2*f^2)) - (a*(2* a*c - f^2)^2)/(c^2*(4*a*c - f^2))))/(2*a*(4*a*c - f^2)^(1/2))))/(f^4 + 4*a ^2*c^2 - 4*a*c*f^2))*(2*a*c - f^2))/(2*c^2*(4*a*c - f^2)^(1/2))
Time = 0.27 (sec) , antiderivative size = 366, normalized size of antiderivative = 4.21 \[ \int \frac {x^5}{a+f x^2+c x^4} \, dx=\frac {4 \sqrt {2 \sqrt {c}\, \sqrt {a}+f}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-f}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}-f}-2 \sqrt {c}\, x}{\sqrt {2 \sqrt {c}\, \sqrt {a}+f}}\right ) a c -2 \sqrt {2 \sqrt {c}\, \sqrt {a}+f}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-f}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}-f}-2 \sqrt {c}\, x}{\sqrt {2 \sqrt {c}\, \sqrt {a}+f}}\right ) f^{2}+4 \sqrt {2 \sqrt {c}\, \sqrt {a}+f}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-f}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}-f}+2 \sqrt {c}\, x}{\sqrt {2 \sqrt {c}\, \sqrt {a}+f}}\right ) a c -2 \sqrt {2 \sqrt {c}\, \sqrt {a}+f}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-f}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}-f}+2 \sqrt {c}\, x}{\sqrt {2 \sqrt {c}\, \sqrt {a}+f}}\right ) f^{2}-4 \,\mathrm {log}\left (-\sqrt {2 \sqrt {c}\, \sqrt {a}-f}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) a c f +\mathrm {log}\left (-\sqrt {2 \sqrt {c}\, \sqrt {a}-f}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) f^{3}-4 \,\mathrm {log}\left (\sqrt {2 \sqrt {c}\, \sqrt {a}-f}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) a c f +\mathrm {log}\left (\sqrt {2 \sqrt {c}\, \sqrt {a}-f}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) f^{3}+8 a \,c^{2} x^{2}-2 c \,f^{2} x^{2}}{4 c^{2} \left (4 a c -f^{2}\right )} \] Input:
int(x^5/(c*x^4+f*x^2+a),x)
Output:
(4*sqrt(2*sqrt(c)*sqrt(a) + f)*sqrt(2*sqrt(c)*sqrt(a) - f)*atan((sqrt(2*sq rt(c)*sqrt(a) - f) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + f))*a*c - 2*sqr t(2*sqrt(c)*sqrt(a) + f)*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 + 4*sqrt(2*s qrt(c)*sqrt(a) + f)*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(2*sqrt(c) *sqrt(a) + f)*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 - 4*log( - sqrt(2*sqrt( c)*sqrt(a) - f)*x + sqrt(a) + sqrt(c)*x**2)*a*c*f + log( - sqrt(2*sqrt(c)* sqrt(a) - f)*x + sqrt(a) + sqrt(c)*x**2)*f**3 - 4*log(sqrt(2*sqrt(c)*sqrt( a) - f)*x + sqrt(a) + sqrt(c)*x**2)*a*c*f + log(sqrt(2*sqrt(c)*sqrt(a) - f )*x + sqrt(a) + sqrt(c)*x**2)*f**3 + 8*a*c**2*x**2 - 2*c*f**2*x**2)/(4*c** 2*(4*a*c - f**2))