Integrand size = 18, antiderivative size = 107 \[ \int \frac {x^7}{a+f x^2+c x^4} \, dx=-\frac {f x^2}{2 c^2}+\frac {x^4}{4 c}+\frac {f \left (3 a c-f^2\right ) \arctan \left (\frac {f+2 c x^2}{\sqrt {4 a c-f^2}}\right )}{2 c^3 \sqrt {4 a c-f^2}}-\frac {\left (a c-f^2\right ) \log \left (a+f x^2+c x^4\right )}{4 c^3} \] Output:
-1/2*f*x^2/c^2+1/4*x^4/c+1/2*f*(3*a*c-f^2)*arctan((2*c*x^2+f)/(4*a*c-f^2)^ (1/2))/c^3/(4*a*c-f^2)^(1/2)-1/4*(a*c-f^2)*ln(c*x^4+f*x^2+a)/c^3
Time = 0.06 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.87 \[ \int \frac {x^7}{a+f x^2+c x^4} \, dx=\frac {c x^2 \left (-2 f+c x^2\right )-\frac {2 f \left (-3 a c+f^2\right ) \arctan \left (\frac {f+2 c x^2}{\sqrt {4 a c-f^2}}\right )}{\sqrt {4 a c-f^2}}+\left (-a c+f^2\right ) \log \left (a+f x^2+c x^4\right )}{4 c^3} \] Input:
Integrate[x^7/(a + f*x^2 + c*x^4),x]
Output:
(c*x^2*(-2*f + c*x^2) - (2*f*(-3*a*c + f^2)*ArcTan[(f + 2*c*x^2)/Sqrt[4*a* c - f^2]])/Sqrt[4*a*c - f^2] + (-(a*c) + f^2)*Log[a + f*x^2 + c*x^4])/(4*c ^3)
Time = 0.54 (sec) , antiderivative size = 106, 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.
| \(\displaystyle \int \frac {x^7}{a+c x^4+f x^2} \, dx\) |
| \(\Big \downarrow \) 1434 |
| \(\displaystyle \frac {1}{2} \int \frac {x^6}{c x^4+f x^2+a}dx^2\) |
| \(\Big \downarrow \) 1143 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {x^2}{c}-\frac {f}{c^2}+\frac {a f-\left (a c-f^2\right ) x^2}{c^2 \left (c x^4+f x^2+a\right )}\right )dx^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (\frac {f \left (3 a c-f^2\right ) \arctan \left (\frac {2 c x^2+f}{\sqrt {4 a c-f^2}}\right )}{c^3 \sqrt {4 a c-f^2}}-\frac {\left (a c-f^2\right ) \log \left (a+c x^4+f x^2\right )}{2 c^3}-\frac {f x^2}{c^2}+\frac {x^4}{2 c}\right )\) |
Input:
Int[x^7/(a + f*x^2 + c*x^4),x]
Output:
(-((f*x^2)/c^2) + x^4/(2*c) + (f*(3*a*c - f^2)*ArcTan[(f + 2*c*x^2)/Sqrt[4 *a*c - f^2]])/(c^3*Sqrt[4*a*c - f^2]) - ((a*c - f^2)*Log[a + f*x^2 + c*x^4 ])/(2*c^3))/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 = 0.17 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.99
| method | result | size |
| default | \(\frac {\frac {1}{2} c \,x^{4}-f \,x^{2}}{2 c^{2}}+\frac {\frac {\left (-a c +f^{2}\right ) \ln \left (c \,x^{4}+f \,x^{2}+a \right )}{2 c}+\frac {2 \left (a f -\frac {\left (-a c +f^{2}\right ) f}{2 c}\right ) \arctan \left (\frac {2 c \,x^{2}+f}{\sqrt {4 a c -f^{2}}}\right )}{\sqrt {4 a c -f^{2}}}}{2 c^{2}}\) | \(106\) |
| risch | \(\frac {x^{4}}{4 c}-\frac {f \,x^{2}}{2 c^{2}}+\frac {f^{2}}{4 c^{3}}-\frac {\ln \left (\left (12 a^{2} c^{2} f -7 a c \,f^{3}+f^{5}+\sqrt {-f^{2} \left (4 a c -f^{2}\right ) \left (3 a c -f^{2}\right )^{2}}\, f \right ) x^{2}+2 \sqrt {-f^{2} \left (4 a c -f^{2}\right ) \left (3 a c -f^{2}\right )^{2}}\, a \right ) a^{2}}{c \left (4 a c -f^{2}\right )}+\frac {5 \ln \left (\left (12 a^{2} c^{2} f -7 a c \,f^{3}+f^{5}+\sqrt {-f^{2} \left (4 a c -f^{2}\right ) \left (3 a c -f^{2}\right )^{2}}\, f \right ) x^{2}+2 \sqrt {-f^{2} \left (4 a c -f^{2}\right ) \left (3 a c -f^{2}\right )^{2}}\, a \right ) a \,f^{2}}{4 c^{2} \left (4 a c -f^{2}\right )}-\frac {\ln \left (\left (12 a^{2} c^{2} f -7 a c \,f^{3}+f^{5}+\sqrt {-f^{2} \left (4 a c -f^{2}\right ) \left (3 a c -f^{2}\right )^{2}}\, f \right ) x^{2}+2 \sqrt {-f^{2} \left (4 a c -f^{2}\right ) \left (3 a c -f^{2}\right )^{2}}\, a \right ) f^{4}}{4 c^{3} \left (4 a c -f^{2}\right )}+\frac {\ln \left (\left (12 a^{2} c^{2} f -7 a c \,f^{3}+f^{5}+\sqrt {-f^{2} \left (4 a c -f^{2}\right ) \left (3 a c -f^{2}\right )^{2}}\, f \right ) x^{2}+2 \sqrt {-f^{2} \left (4 a c -f^{2}\right ) \left (3 a c -f^{2}\right )^{2}}\, a \right ) \sqrt {-f^{2} \left (4 a c -f^{2}\right ) \left (3 a c -f^{2}\right )^{2}}}{4 c^{3} \left (4 a c -f^{2}\right )}-\frac {\ln \left (\left (12 a^{2} c^{2} f -7 a c \,f^{3}+f^{5}-\sqrt {-f^{2} \left (4 a c -f^{2}\right ) \left (3 a c -f^{2}\right )^{2}}\, f \right ) x^{2}-2 \sqrt {-f^{2} \left (4 a c -f^{2}\right ) \left (3 a c -f^{2}\right )^{2}}\, a \right ) a^{2}}{c \left (4 a c -f^{2}\right )}+\frac {5 \ln \left (\left (12 a^{2} c^{2} f -7 a c \,f^{3}+f^{5}-\sqrt {-f^{2} \left (4 a c -f^{2}\right ) \left (3 a c -f^{2}\right )^{2}}\, f \right ) x^{2}-2 \sqrt {-f^{2} \left (4 a c -f^{2}\right ) \left (3 a c -f^{2}\right )^{2}}\, a \right ) a \,f^{2}}{4 c^{2} \left (4 a c -f^{2}\right )}-\frac {\ln \left (\left (12 a^{2} c^{2} f -7 a c \,f^{3}+f^{5}-\sqrt {-f^{2} \left (4 a c -f^{2}\right ) \left (3 a c -f^{2}\right )^{2}}\, f \right ) x^{2}-2 \sqrt {-f^{2} \left (4 a c -f^{2}\right ) \left (3 a c -f^{2}\right )^{2}}\, a \right ) f^{4}}{4 c^{3} \left (4 a c -f^{2}\right )}-\frac {\ln \left (\left (12 a^{2} c^{2} f -7 a c \,f^{3}+f^{5}-\sqrt {-f^{2} \left (4 a c -f^{2}\right ) \left (3 a c -f^{2}\right )^{2}}\, f \right ) x^{2}-2 \sqrt {-f^{2} \left (4 a c -f^{2}\right ) \left (3 a c -f^{2}\right )^{2}}\, a \right ) \sqrt {-f^{2} \left (4 a c -f^{2}\right ) \left (3 a c -f^{2}\right )^{2}}}{4 c^{3} \left (4 a c -f^{2}\right )}\) | \(957\) |
Input:
int(x^7/(c*x^4+f*x^2+a),x,method=_RETURNVERBOSE)
Output:
1/2/c^2*(1/2*c*x^4-f*x^2)+1/2/c^2*(1/2*(-a*c+f^2)/c*ln(c*x^4+f*x^2+a)+2*(a *f-1/2*(-a*c+f^2)*f/c)/(4*a*c-f^2)^(1/2)*arctan((2*c*x^2+f)/(4*a*c-f^2)^(1 /2)))
Time = 0.09 (sec) , antiderivative size = 313, normalized size of antiderivative = 2.93 \[ \int \frac {x^7}{a+f x^2+c x^4} \, dx=\left [\frac {{\left (4 \, a c^{3} - c^{2} f^{2}\right )} x^{4} - 2 \, {\left (4 \, a c^{2} f - c f^{3}\right )} x^{2} + {\left (3 \, a c f - f^{3}\right )} \sqrt {-4 \, a c + f^{2}} \log \left (\frac {2 \, c^{2} x^{4} + 2 \, c f x^{2} - 2 \, a c + f^{2} + {\left (2 \, c x^{2} + f\right )} \sqrt {-4 \, a c + f^{2}}}{c x^{4} + f x^{2} + a}\right ) - {\left (4 \, a^{2} c^{2} - 5 \, a c f^{2} + f^{4}\right )} \log \left (c x^{4} + f x^{2} + a\right )}{4 \, {\left (4 \, a c^{4} - c^{3} f^{2}\right )}}, \frac {{\left (4 \, a c^{3} - c^{2} f^{2}\right )} x^{4} - 2 \, {\left (4 \, a c^{2} f - c f^{3}\right )} x^{2} - 2 \, {\left (3 \, a c f - f^{3}\right )} \sqrt {4 \, a c - f^{2}} \arctan \left (-\frac {2 \, c x^{2} + f}{\sqrt {4 \, a c - f^{2}}}\right ) - {\left (4 \, a^{2} c^{2} - 5 \, a c f^{2} + f^{4}\right )} \log \left (c x^{4} + f x^{2} + a\right )}{4 \, {\left (4 \, a c^{4} - c^{3} f^{2}\right )}}\right ] \] Input:
integrate(x^7/(c*x^4+f*x^2+a),x, algorithm="fricas")
Output:
[1/4*((4*a*c^3 - c^2*f^2)*x^4 - 2*(4*a*c^2*f - c*f^3)*x^2 + (3*a*c*f - f^3 )*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^2*c^2 - 5*a*c*f^2 + f^ 4)*log(c*x^4 + f*x^2 + a))/(4*a*c^4 - c^3*f^2), 1/4*((4*a*c^3 - c^2*f^2)*x ^4 - 2*(4*a*c^2*f - c*f^3)*x^2 - 2*(3*a*c*f - f^3)*sqrt(4*a*c - f^2)*arcta n(-(2*c*x^2 + f)/sqrt(4*a*c - f^2)) - (4*a^2*c^2 - 5*a*c*f^2 + f^4)*log(c* x^4 + f*x^2 + a))/(4*a*c^4 - c^3*f^2)]
Leaf count of result is larger than twice the leaf count of optimal. 391 vs. \(2 (92) = 184\).
Time = 1.42 (sec) , antiderivative size = 391, normalized size of antiderivative = 3.65 \[ \int \frac {x^7}{a+f x^2+c x^4} \, dx=\left (- \frac {f \sqrt {- 4 a c + f^{2}} \cdot \left (3 a c - f^{2}\right )}{4 c^{3} \cdot \left (4 a c - f^{2}\right )} - \frac {a c - f^{2}}{4 c^{3}}\right ) \log {\left (x^{2} + \frac {2 a^{2} c + 8 a c^{3} \left (- \frac {f \sqrt {- 4 a c + f^{2}} \cdot \left (3 a c - f^{2}\right )}{4 c^{3} \cdot \left (4 a c - f^{2}\right )} - \frac {a c - f^{2}}{4 c^{3}}\right ) - a f^{2} - 2 c^{2} f^{2} \left (- \frac {f \sqrt {- 4 a c + f^{2}} \cdot \left (3 a c - f^{2}\right )}{4 c^{3} \cdot \left (4 a c - f^{2}\right )} - \frac {a c - f^{2}}{4 c^{3}}\right )}{3 a c f - f^{3}} \right )} + \left (\frac {f \sqrt {- 4 a c + f^{2}} \cdot \left (3 a c - f^{2}\right )}{4 c^{3} \cdot \left (4 a c - f^{2}\right )} - \frac {a c - f^{2}}{4 c^{3}}\right ) \log {\left (x^{2} + \frac {2 a^{2} c + 8 a c^{3} \left (\frac {f \sqrt {- 4 a c + f^{2}} \cdot \left (3 a c - f^{2}\right )}{4 c^{3} \cdot \left (4 a c - f^{2}\right )} - \frac {a c - f^{2}}{4 c^{3}}\right ) - a f^{2} - 2 c^{2} f^{2} \left (\frac {f \sqrt {- 4 a c + f^{2}} \cdot \left (3 a c - f^{2}\right )}{4 c^{3} \cdot \left (4 a c - f^{2}\right )} - \frac {a c - f^{2}}{4 c^{3}}\right )}{3 a c f - f^{3}} \right )} + \frac {x^{4}}{4 c} - \frac {f x^{2}}{2 c^{2}} \] Input:
integrate(x**7/(c*x**4+f*x**2+a),x)
Output:
(-f*sqrt(-4*a*c + f**2)*(3*a*c - f**2)/(4*c**3*(4*a*c - f**2)) - (a*c - f* *2)/(4*c**3))*log(x**2 + (2*a**2*c + 8*a*c**3*(-f*sqrt(-4*a*c + f**2)*(3*a *c - f**2)/(4*c**3*(4*a*c - f**2)) - (a*c - f**2)/(4*c**3)) - a*f**2 - 2*c **2*f**2*(-f*sqrt(-4*a*c + f**2)*(3*a*c - f**2)/(4*c**3*(4*a*c - f**2)) - (a*c - f**2)/(4*c**3)))/(3*a*c*f - f**3)) + (f*sqrt(-4*a*c + f**2)*(3*a*c - f**2)/(4*c**3*(4*a*c - f**2)) - (a*c - f**2)/(4*c**3))*log(x**2 + (2*a** 2*c + 8*a*c**3*(f*sqrt(-4*a*c + f**2)*(3*a*c - f**2)/(4*c**3*(4*a*c - f**2 )) - (a*c - f**2)/(4*c**3)) - a*f**2 - 2*c**2*f**2*(f*sqrt(-4*a*c + f**2)* (3*a*c - f**2)/(4*c**3*(4*a*c - f**2)) - (a*c - f**2)/(4*c**3)))/(3*a*c*f - f**3)) + x**4/(4*c) - f*x**2/(2*c**2)
Exception generated. \[ \int \frac {x^7}{a+f x^2+c x^4} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(x^7/(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 = 0.36 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.89 \[ \int \frac {x^7}{a+f x^2+c x^4} \, dx=\frac {c x^{4} - 2 \, f x^{2}}{4 \, c^{2}} - \frac {{\left (a c - f^{2}\right )} \log \left (c x^{4} + f x^{2} + a\right )}{4 \, c^{3}} + \frac {{\left (3 \, a c f - f^{3}\right )} \arctan \left (\frac {2 \, c x^{2} + f}{\sqrt {4 \, a c - f^{2}}}\right )}{2 \, \sqrt {4 \, a c - f^{2}} c^{3}} \] Input:
integrate(x^7/(c*x^4+f*x^2+a),x, algorithm="giac")
Output:
1/4*(c*x^4 - 2*f*x^2)/c^2 - 1/4*(a*c - f^2)*log(c*x^4 + f*x^2 + a)/c^3 + 1 /2*(3*a*c*f - f^3)*arctan((2*c*x^2 + f)/sqrt(4*a*c - f^2))/(sqrt(4*a*c - f ^2)*c^3)
Time = 17.65 (sec) , antiderivative size = 842, normalized size of antiderivative = 7.87 \[ \int \frac {x^7}{a+f x^2+c x^4} \, dx=\frac {x^4}{4\,c}-\frac {\ln \left (c\,x^4+f\,x^2+a\right )\,\left (8\,a^2\,c^2-10\,a\,c\,f^2+2\,f^4\right )}{2\,\left (16\,a\,c^4-4\,c^3\,f^2\right )}-\frac {f\,x^2}{2\,c^2}+\frac {f\,\mathrm {atan}\left (\frac {2\,c^4\,\left (4\,a\,c-f^2\right )\,\left (\frac {\frac {f\,\left (\frac {8\,a^2\,c^4-8\,a\,c^3\,f^2}{c^4}-\frac {8\,a\,c^2\,\left (8\,a^2\,c^2-10\,a\,c\,f^2+2\,f^4\right )}{16\,a\,c^4-4\,c^3\,f^2}\right )\,\left (3\,a\,c-f^2\right )}{8\,c^3\,\sqrt {4\,a\,c-f^2}}-\frac {a\,f\,\left (3\,a\,c-f^2\right )\,\left (8\,a^2\,c^2-10\,a\,c\,f^2+2\,f^4\right )}{c\,\sqrt {4\,a\,c-f^2}\,\left (16\,a\,c^4-4\,c^3\,f^2\right )}}{a}-x^2\,\left (\frac {\frac {f\,\left (3\,a\,c-f^2\right )\,\left (\frac {6\,c^3\,f^3-10\,a\,c^4\,f}{c^4}+\frac {4\,c^2\,f\,\left (8\,a^2\,c^2-10\,a\,c\,f^2+2\,f^4\right )}{16\,a\,c^4-4\,c^3\,f^2}\right )}{8\,c^3\,\sqrt {4\,a\,c-f^2}}+\frac {f^2\,\left (3\,a\,c-f^2\right )\,\left (8\,a^2\,c^2-10\,a\,c\,f^2+2\,f^4\right )}{2\,c\,\sqrt {4\,a\,c-f^2}\,\left (16\,a\,c^4-4\,c^3\,f^2\right )}}{a}+\frac {f\,\left (\frac {2\,a^2\,c^2\,f-3\,a\,c\,f^3+f^5}{c^4}+\frac {\left (\frac {6\,c^3\,f^3-10\,a\,c^4\,f}{c^4}+\frac {4\,c^2\,f\,\left (8\,a^2\,c^2-10\,a\,c\,f^2+2\,f^4\right )}{16\,a\,c^4-4\,c^3\,f^2}\right )\,\left (8\,a^2\,c^2-10\,a\,c\,f^2+2\,f^4\right )}{2\,\left (16\,a\,c^4-4\,c^3\,f^2\right )}-\frac {f^3\,{\left (3\,a\,c-f^2\right )}^2}{2\,c^4\,\left (4\,a\,c-f^2\right )}\right )}{2\,a\,\sqrt {4\,a\,c-f^2}}\right )+\frac {f\,\left (\frac {\left (\frac {8\,a^2\,c^4-8\,a\,c^3\,f^2}{c^4}-\frac {8\,a\,c^2\,\left (8\,a^2\,c^2-10\,a\,c\,f^2+2\,f^4\right )}{16\,a\,c^4-4\,c^3\,f^2}\right )\,\left (8\,a^2\,c^2-10\,a\,c\,f^2+2\,f^4\right )}{2\,\left (16\,a\,c^4-4\,c^3\,f^2\right )}-\frac {a^3\,c^2-2\,a^2\,c\,f^2+a\,f^4}{c^4}+\frac {a\,f^2\,{\left (3\,a\,c-f^2\right )}^2}{c^4\,\left (4\,a\,c-f^2\right )}\right )}{2\,a\,\sqrt {4\,a\,c-f^2}}\right )}{9\,a^2\,c^2\,f^2-6\,a\,c\,f^4+f^6}\right )\,\left (3\,a\,c-f^2\right )}{2\,c^3\,\sqrt {4\,a\,c-f^2}} \] Input:
int(x^7/(a + c*x^4 + f*x^2),x)
Output:
x^4/(4*c) - (log(a + c*x^4 + f*x^2)*(2*f^4 + 8*a^2*c^2 - 10*a*c*f^2))/(2*( 16*a*c^4 - 4*c^3*f^2)) - (f*x^2)/(2*c^2) + (f*atan((2*c^4*(4*a*c - f^2)*(( (f*((8*a^2*c^4 - 8*a*c^3*f^2)/c^4 - (8*a*c^2*(2*f^4 + 8*a^2*c^2 - 10*a*c*f ^2))/(16*a*c^4 - 4*c^3*f^2))*(3*a*c - f^2))/(8*c^3*(4*a*c - f^2)^(1/2)) - (a*f*(3*a*c - f^2)*(2*f^4 + 8*a^2*c^2 - 10*a*c*f^2))/(c*(4*a*c - f^2)^(1/2 )*(16*a*c^4 - 4*c^3*f^2)))/a - x^2*(((f*(3*a*c - f^2)*((6*c^3*f^3 - 10*a*c ^4*f)/c^4 + (4*c^2*f*(2*f^4 + 8*a^2*c^2 - 10*a*c*f^2))/(16*a*c^4 - 4*c^3*f ^2)))/(8*c^3*(4*a*c - f^2)^(1/2)) + (f^2*(3*a*c - f^2)*(2*f^4 + 8*a^2*c^2 - 10*a*c*f^2))/(2*c*(4*a*c - f^2)^(1/2)*(16*a*c^4 - 4*c^3*f^2)))/a + (f*(( f^5 + 2*a^2*c^2*f - 3*a*c*f^3)/c^4 + (((6*c^3*f^3 - 10*a*c^4*f)/c^4 + (4*c ^2*f*(2*f^4 + 8*a^2*c^2 - 10*a*c*f^2))/(16*a*c^4 - 4*c^3*f^2))*(2*f^4 + 8* a^2*c^2 - 10*a*c*f^2))/(2*(16*a*c^4 - 4*c^3*f^2)) - (f^3*(3*a*c - f^2)^2)/ (2*c^4*(4*a*c - f^2))))/(2*a*(4*a*c - f^2)^(1/2))) + (f*((((8*a^2*c^4 - 8* a*c^3*f^2)/c^4 - (8*a*c^2*(2*f^4 + 8*a^2*c^2 - 10*a*c*f^2))/(16*a*c^4 - 4* c^3*f^2))*(2*f^4 + 8*a^2*c^2 - 10*a*c*f^2))/(2*(16*a*c^4 - 4*c^3*f^2)) - ( a*f^4 + a^3*c^2 - 2*a^2*c*f^2)/c^4 + (a*f^2*(3*a*c - f^2)^2)/(c^4*(4*a*c - f^2))))/(2*a*(4*a*c - f^2)^(1/2))))/(f^6 + 9*a^2*c^2*f^2 - 6*a*c*f^4))*(3 *a*c - f^2))/(2*c^3*(4*a*c - f^2)^(1/2))
Time = 0.23 (sec) , antiderivative size = 458, normalized size of antiderivative = 4.28 \[ \int \frac {x^7}{a+f x^2+c x^4} \, dx=\frac {-6 \sqrt {2 \sqrt {c}\, \sqrt {a}+f}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-f}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}-f}-2 \sqrt {c}\, x}{\sqrt {2 \sqrt {c}\, \sqrt {a}+f}}\right ) a c f +2 \sqrt {2 \sqrt {c}\, \sqrt {a}+f}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-f}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}-f}-2 \sqrt {c}\, x}{\sqrt {2 \sqrt {c}\, \sqrt {a}+f}}\right ) f^{3}-6 \sqrt {2 \sqrt {c}\, \sqrt {a}+f}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-f}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}-f}+2 \sqrt {c}\, x}{\sqrt {2 \sqrt {c}\, \sqrt {a}+f}}\right ) a c f +2 \sqrt {2 \sqrt {c}\, \sqrt {a}+f}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-f}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}-f}+2 \sqrt {c}\, x}{\sqrt {2 \sqrt {c}\, \sqrt {a}+f}}\right ) f^{3}-4 \,\mathrm {log}\left (-\sqrt {2 \sqrt {c}\, \sqrt {a}-f}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) a^{2} c^{2}+5 \,\mathrm {log}\left (-\sqrt {2 \sqrt {c}\, \sqrt {a}-f}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) a c \,f^{2}-\mathrm {log}\left (-\sqrt {2 \sqrt {c}\, \sqrt {a}-f}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) f^{4}-4 \,\mathrm {log}\left (\sqrt {2 \sqrt {c}\, \sqrt {a}-f}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) a^{2} c^{2}+5 \,\mathrm {log}\left (\sqrt {2 \sqrt {c}\, \sqrt {a}-f}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) a c \,f^{2}-\mathrm {log}\left (\sqrt {2 \sqrt {c}\, \sqrt {a}-f}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) f^{4}+4 a \,c^{3} x^{4}-8 a \,c^{2} f \,x^{2}-c^{2} f^{2} x^{4}+2 c \,f^{3} x^{2}}{4 c^{3} \left (4 a c -f^{2}\right )} \] Input:
int(x^7/(c*x^4+f*x^2+a),x)
Output:
( - 6*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))*a*c*f + 2*sqrt(2*sqrt(c)*sqrt(a) + f)*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**3 - 6*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))*a*c*f + 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**3 - 4*log( - sqrt( 2*sqrt(c)*sqrt(a) - f)*x + sqrt(a) + sqrt(c)*x**2)*a**2*c**2 + 5*log( - sq rt(2*sqrt(c)*sqrt(a) - f)*x + sqrt(a) + sqrt(c)*x**2)*a*c*f**2 - log( - sq rt(2*sqrt(c)*sqrt(a) - f)*x + sqrt(a) + sqrt(c)*x**2)*f**4 - 4*log(sqrt(2* sqrt(c)*sqrt(a) - f)*x + sqrt(a) + sqrt(c)*x**2)*a**2*c**2 + 5*log(sqrt(2* sqrt(c)*sqrt(a) - f)*x + sqrt(a) + sqrt(c)*x**2)*a*c*f**2 - log(sqrt(2*sqr t(c)*sqrt(a) - f)*x + sqrt(a) + sqrt(c)*x**2)*f**4 + 4*a*c**3*x**4 - 8*a*c **2*f*x**2 - c**2*f**2*x**4 + 2*c*f**3*x**2)/(4*c**3*(4*a*c - f**2))