Integrand size = 18, antiderivative size = 114 \[ \int \frac {1}{x^5 \left (a+b x^2+c x^4\right )} \, dx=-\frac {1}{4 a x^4}+\frac {b}{2 a^2 x^2}+\frac {b \left (b^2-3 a c\right ) \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 a^3 \sqrt {b^2-4 a c}}+\frac {\left (b^2-a c\right ) \log (x)}{a^3}-\frac {\left (b^2-a c\right ) \log \left (a+b x^2+c x^4\right )}{4 a^3} \] Output:
-1/4/a/x^4+1/2*b/a^2/x^2+1/2*b*(-3*a*c+b^2)*arctanh((2*c*x^2+b)/(-4*a*c+b^ 2)^(1/2))/a^3/(-4*a*c+b^2)^(1/2)+(-a*c+b^2)*ln(x)/a^3-1/4*(-a*c+b^2)*ln(c* x^4+b*x^2+a)/a^3
Time = 0.15 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.65 \[ \int \frac {1}{x^5 \left (a+b x^2+c x^4\right )} \, dx=\frac {-\frac {a^2}{x^4}+\frac {2 a b}{x^2}+4 \left (b^2-a c\right ) \log (x)-\frac {\left (b^3-3 a b c+b^2 \sqrt {b^2-4 a c}-a c \sqrt {b^2-4 a c}\right ) \log \left (b-\sqrt {b^2-4 a c}+2 c x^2\right )}{\sqrt {b^2-4 a c}}+\frac {\left (b^3-3 a b c-b^2 \sqrt {b^2-4 a c}+a c \sqrt {b^2-4 a c}\right ) \log \left (b+\sqrt {b^2-4 a c}+2 c x^2\right )}{\sqrt {b^2-4 a c}}}{4 a^3} \] Input:
Integrate[1/(x^5*(a + b*x^2 + c*x^4)),x]
Output:
(-(a^2/x^4) + (2*a*b)/x^2 + 4*(b^2 - a*c)*Log[x] - ((b^3 - 3*a*b*c + b^2*S qrt[b^2 - 4*a*c] - a*c*Sqrt[b^2 - 4*a*c])*Log[b - Sqrt[b^2 - 4*a*c] + 2*c* x^2])/Sqrt[b^2 - 4*a*c] + ((b^3 - 3*a*b*c - b^2*Sqrt[b^2 - 4*a*c] + a*c*Sq rt[b^2 - 4*a*c])*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/Sqrt[b^2 - 4*a*c])/ (4*a^3)
Time = 0.57 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.08, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {1434, 1145, 25, 1200, 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.
| \(\displaystyle \int \frac {1}{x^5 \left (a+b x^2+c x^4\right )} \, dx\) |
| \(\Big \downarrow \) 1434 |
| \(\displaystyle \frac {1}{2} \int \frac {1}{x^6 \left (c x^4+b x^2+a\right )}dx^2\) |
| \(\Big \downarrow \) 1145 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int -\frac {c x^2+b}{x^4 \left (c x^4+b x^2+a\right )}dx^2}{a}-\frac {1}{2 a x^4}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\int \frac {c x^2+b}{x^4 \left (c x^4+b x^2+a\right )}dx^2}{a}-\frac {1}{2 a x^4}\right )\) |
| \(\Big \downarrow \) 1200 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\int \left (\frac {b}{a x^4}+\frac {c \left (b^2-a c\right ) x^2+b \left (b^2-2 a c\right )}{a^2 \left (c x^4+b x^2+a\right )}+\frac {a c-b^2}{a^2 x^2}\right )dx^2}{a}-\frac {1}{2 a x^4}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (-\frac {-\frac {b \left (b^2-3 a c\right ) \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{a^2 \sqrt {b^2-4 a c}}-\frac {\log \left (x^2\right ) \left (b^2-a c\right )}{a^2}+\frac {\left (b^2-a c\right ) \log \left (a+b x^2+c x^4\right )}{2 a^2}-\frac {b}{a x^2}}{a}-\frac {1}{2 a x^4}\right )\) |
Input:
Int[1/(x^5*(a + b*x^2 + c*x^4)),x]
Output:
(-1/2*1/(a*x^4) - (-(b/(a*x^2)) - (b*(b^2 - 3*a*c)*ArcTanh[(b + 2*c*x^2)/S qrt[b^2 - 4*a*c]])/(a^2*Sqrt[b^2 - 4*a*c]) - ((b^2 - a*c)*Log[x^2])/a^2 + ((b^2 - a*c)*Log[a + b*x^2 + c*x^4])/(2*a^2))/a)/2
Int[((d_.) + (e_.)*(x_))^(m_)/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[e*((d + e*x)^(m + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp [1/(c*d^2 - b*d*e + a*e^2) Int[(d + e*x)^(m + 1)*(Simp[c*d - b*e - c*e*x, x]/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && ILtQ[m, -1]
Int[(((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.))/((a_.) + (b_.)* (x_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*((f + g* x)^n/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && In tegersQ[n]
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 = 134, normalized size of antiderivative = 1.18
| method | result | size |
| default | \(-\frac {1}{4 a \,x^{4}}+\frac {\left (-a c +b^{2}\right ) \ln \left (x \right )}{a^{3}}+\frac {b}{2 a^{2} x^{2}}+\frac {\frac {\left (c^{2} a -b^{2} c \right ) \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{2 c}+\frac {2 \left (2 a b c -b^{3}-\frac {\left (c^{2} a -b^{2} c \right ) b}{2 c}\right ) \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{2 a^{3}}\) | \(134\) |
| risch | \(\frac {\frac {b \,x^{2}}{2 a^{2}}-\frac {1}{4 a}}{x^{4}}-\frac {\ln \left (x \right ) c}{a^{2}}+\frac {\ln \left (x \right ) b^{2}}{a^{3}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (4 c \,a^{4}-a^{3} b^{2}\right ) \textit {\_Z}^{2}+\left (-4 a^{2} c^{2}+5 a \,b^{2} c -b^{4}\right ) \textit {\_Z} +c^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (10 c \,a^{5}-3 a^{4} b^{2}\right ) \textit {\_R}^{2}+\left (-5 a^{3} c^{2}+4 a^{2} b^{2} c \right ) \textit {\_R} +2 b^{2} c^{2}\right ) x^{2}-a^{5} b \,\textit {\_R}^{2}+\left (-3 c b \,a^{3}+2 a^{2} b^{3}\right ) \textit {\_R} -2 a b \,c^{2}+2 b^{3} c \right )\right )}{2}\) | \(186\) |
Input:
int(1/x^5/(c*x^4+b*x^2+a),x,method=_RETURNVERBOSE)
Output:
-1/4/a/x^4+(-a*c+b^2)*ln(x)/a^3+1/2*b/a^2/x^2+1/2/a^3*(1/2*(a*c^2-b^2*c)/c *ln(c*x^4+b*x^2+a)+2*(2*a*b*c-b^3-1/2*(a*c^2-b^2*c)*b/c)/(4*a*c-b^2)^(1/2) *arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2)))
Time = 0.10 (sec) , antiderivative size = 374, normalized size of antiderivative = 3.28 \[ \int \frac {1}{x^5 \left (a+b x^2+c x^4\right )} \, dx=\left [-\frac {{\left (b^{3} - 3 \, a b c\right )} \sqrt {b^{2} - 4 \, a c} x^{4} \log \left (\frac {2 \, c^{2} x^{4} + 2 \, b c x^{2} + b^{2} - 2 \, a c - {\left (2 \, c x^{2} + b\right )} \sqrt {b^{2} - 4 \, a c}}{c x^{4} + b x^{2} + a}\right ) + {\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} x^{4} \log \left (c x^{4} + b x^{2} + a\right ) - 4 \, {\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} x^{4} \log \left (x\right ) + a^{2} b^{2} - 4 \, a^{3} c - 2 \, {\left (a b^{3} - 4 \, a^{2} b c\right )} x^{2}}{4 \, {\left (a^{3} b^{2} - 4 \, a^{4} c\right )} x^{4}}, \frac {2 \, {\left (b^{3} - 3 \, a b c\right )} \sqrt {-b^{2} + 4 \, a c} x^{4} \arctan \left (-\frac {{\left (2 \, c x^{2} + b\right )} \sqrt {-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) - {\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} x^{4} \log \left (c x^{4} + b x^{2} + a\right ) + 4 \, {\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} x^{4} \log \left (x\right ) - a^{2} b^{2} + 4 \, a^{3} c + 2 \, {\left (a b^{3} - 4 \, a^{2} b c\right )} x^{2}}{4 \, {\left (a^{3} b^{2} - 4 \, a^{4} c\right )} x^{4}}\right ] \] Input:
integrate(1/x^5/(c*x^4+b*x^2+a),x, algorithm="fricas")
Output:
[-1/4*((b^3 - 3*a*b*c)*sqrt(b^2 - 4*a*c)*x^4*log((2*c^2*x^4 + 2*b*c*x^2 + b^2 - 2*a*c - (2*c*x^2 + b)*sqrt(b^2 - 4*a*c))/(c*x^4 + b*x^2 + a)) + (b^4 - 5*a*b^2*c + 4*a^2*c^2)*x^4*log(c*x^4 + b*x^2 + a) - 4*(b^4 - 5*a*b^2*c + 4*a^2*c^2)*x^4*log(x) + a^2*b^2 - 4*a^3*c - 2*(a*b^3 - 4*a^2*b*c)*x^2)/( (a^3*b^2 - 4*a^4*c)*x^4), 1/4*(2*(b^3 - 3*a*b*c)*sqrt(-b^2 + 4*a*c)*x^4*ar ctan(-(2*c*x^2 + b)*sqrt(-b^2 + 4*a*c)/(b^2 - 4*a*c)) - (b^4 - 5*a*b^2*c + 4*a^2*c^2)*x^4*log(c*x^4 + b*x^2 + a) + 4*(b^4 - 5*a*b^2*c + 4*a^2*c^2)*x ^4*log(x) - a^2*b^2 + 4*a^3*c + 2*(a*b^3 - 4*a^2*b*c)*x^2)/((a^3*b^2 - 4*a ^4*c)*x^4)]
Leaf count of result is larger than twice the leaf count of optimal. 423 vs. \(2 (107) = 214\).
Time = 175.64 (sec) , antiderivative size = 423, normalized size of antiderivative = 3.71 \[ \int \frac {1}{x^5 \left (a+b x^2+c x^4\right )} \, dx=\left (- \frac {b \sqrt {- 4 a c + b^{2}} \cdot \left (3 a c - b^{2}\right )}{4 a^{3} \cdot \left (4 a c - b^{2}\right )} + \frac {a c - b^{2}}{4 a^{3}}\right ) \log {\left (x^{2} + \frac {8 a^{4} c \left (- \frac {b \sqrt {- 4 a c + b^{2}} \cdot \left (3 a c - b^{2}\right )}{4 a^{3} \cdot \left (4 a c - b^{2}\right )} + \frac {a c - b^{2}}{4 a^{3}}\right ) - 2 a^{3} b^{2} \left (- \frac {b \sqrt {- 4 a c + b^{2}} \cdot \left (3 a c - b^{2}\right )}{4 a^{3} \cdot \left (4 a c - b^{2}\right )} + \frac {a c - b^{2}}{4 a^{3}}\right ) - 2 a^{2} c^{2} + 4 a b^{2} c - b^{4}}{3 a b c^{2} - b^{3} c} \right )} + \left (\frac {b \sqrt {- 4 a c + b^{2}} \cdot \left (3 a c - b^{2}\right )}{4 a^{3} \cdot \left (4 a c - b^{2}\right )} + \frac {a c - b^{2}}{4 a^{3}}\right ) \log {\left (x^{2} + \frac {8 a^{4} c \left (\frac {b \sqrt {- 4 a c + b^{2}} \cdot \left (3 a c - b^{2}\right )}{4 a^{3} \cdot \left (4 a c - b^{2}\right )} + \frac {a c - b^{2}}{4 a^{3}}\right ) - 2 a^{3} b^{2} \left (\frac {b \sqrt {- 4 a c + b^{2}} \cdot \left (3 a c - b^{2}\right )}{4 a^{3} \cdot \left (4 a c - b^{2}\right )} + \frac {a c - b^{2}}{4 a^{3}}\right ) - 2 a^{2} c^{2} + 4 a b^{2} c - b^{4}}{3 a b c^{2} - b^{3} c} \right )} + \frac {- a + 2 b x^{2}}{4 a^{2} x^{4}} - \frac {\left (a c - b^{2}\right ) \log {\left (x \right )}}{a^{3}} \] Input:
integrate(1/x**5/(c*x**4+b*x**2+a),x)
Output:
(-b*sqrt(-4*a*c + b**2)*(3*a*c - b**2)/(4*a**3*(4*a*c - b**2)) + (a*c - b* *2)/(4*a**3))*log(x**2 + (8*a**4*c*(-b*sqrt(-4*a*c + b**2)*(3*a*c - b**2)/ (4*a**3*(4*a*c - b**2)) + (a*c - b**2)/(4*a**3)) - 2*a**3*b**2*(-b*sqrt(-4 *a*c + b**2)*(3*a*c - b**2)/(4*a**3*(4*a*c - b**2)) + (a*c - b**2)/(4*a**3 )) - 2*a**2*c**2 + 4*a*b**2*c - b**4)/(3*a*b*c**2 - b**3*c)) + (b*sqrt(-4* a*c + b**2)*(3*a*c - b**2)/(4*a**3*(4*a*c - b**2)) + (a*c - b**2)/(4*a**3) )*log(x**2 + (8*a**4*c*(b*sqrt(-4*a*c + b**2)*(3*a*c - b**2)/(4*a**3*(4*a* c - b**2)) + (a*c - b**2)/(4*a**3)) - 2*a**3*b**2*(b*sqrt(-4*a*c + b**2)*( 3*a*c - b**2)/(4*a**3*(4*a*c - b**2)) + (a*c - b**2)/(4*a**3)) - 2*a**2*c* *2 + 4*a*b**2*c - b**4)/(3*a*b*c**2 - b**3*c)) + (-a + 2*b*x**2)/(4*a**2*x **4) - (a*c - b**2)*log(x)/a**3
Exception generated. \[ \int \frac {1}{x^5 \left (a+b x^2+c x^4\right )} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(1/x^5/(c*x^4+b*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(4*a*c-b^2>0)', see `assume?` for more deta
Time = 0.39 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.11 \[ \int \frac {1}{x^5 \left (a+b x^2+c x^4\right )} \, dx=-\frac {{\left (b^{2} - a c\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, a^{3}} + \frac {{\left (b^{2} - a c\right )} \log \left (x^{2}\right )}{2 \, a^{3}} - \frac {{\left (b^{3} - 3 \, a b c\right )} \arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{2 \, \sqrt {-b^{2} + 4 \, a c} a^{3}} - \frac {3 \, b^{2} x^{4} - 3 \, a c x^{4} - 2 \, a b x^{2} + a^{2}}{4 \, a^{3} x^{4}} \] Input:
integrate(1/x^5/(c*x^4+b*x^2+a),x, algorithm="giac")
Output:
-1/4*(b^2 - a*c)*log(c*x^4 + b*x^2 + a)/a^3 + 1/2*(b^2 - a*c)*log(x^2)/a^3 - 1/2*(b^3 - 3*a*b*c)*arctan((2*c*x^2 + b)/sqrt(-b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)*a^3) - 1/4*(3*b^2*x^4 - 3*a*c*x^4 - 2*a*b*x^2 + a^2)/(a^3*x^4)
Time = 19.41 (sec) , antiderivative size = 2451, normalized size of antiderivative = 21.50 \[ \int \frac {1}{x^5 \left (a+b x^2+c x^4\right )} \, dx=\text {Too large to display} \] Input:
int(1/(x^5*(a + b*x^2 + c*x^4)),x)
Output:
(log(a + b*x^2 + c*x^4)*(2*b^4 + 8*a^2*c^2 - 10*a*b^2*c))/(2*(16*a^4*c - 4 *a^3*b^2)) - (1/(4*a) - (b*x^2)/(2*a^2))/x^4 - (log(x)*(a*c - b^2))/a^3 + (b*atan((2*a^6*(4*a*c - b^2)*((((b*(3*a*c - b^2)*((4*a^4*b^4*c^2 - 8*a^5*b ^2*c^3)/a^6 - (2*a*b^2*c^2*(2*b^4 + 8*a^2*c^2 - 10*a*b^2*c))/(16*a^4*c - 4 *a^3*b^2)))/(4*a^3*(4*a*c - b^2)^(1/2)) - (b^3*c^2*(3*a*c - b^2)*(2*b^4 + 8*a^2*c^2 - 10*a*b^2*c))/(2*a^2*(4*a*c - b^2)^(1/2)*(16*a^4*c - 4*a^3*b^2) ))*(2*b^4 + 8*a^2*c^2 - 10*a*b^2*c))/(2*(16*a^4*c - 4*a^3*b^2)) + (b^5*c^2 *(3*a*c - b^2)^3)/(16*a^8*(4*a*c - b^2)^(3/2)) + (b*(3*a*c - b^2)*((4*a^2* b^4*c^3 - 5*a^3*b^2*c^4)/a^6 + (((4*a^4*b^4*c^2 - 8*a^5*b^2*c^3)/a^6 - (2* a*b^2*c^2*(2*b^4 + 8*a^2*c^2 - 10*a*b^2*c))/(16*a^4*c - 4*a^3*b^2))*(2*b^4 + 8*a^2*c^2 - 10*a*b^2*c))/(2*(16*a^4*c - 4*a^3*b^2))))/(4*a^3*(4*a*c - b ^2)^(1/2)))*(3*b^6 - 10*a^3*c^3 + 27*a^2*b^2*c^2 - 18*a*b^4*c))/(c^2*(b^6* c^2 - 6*a*b^4*c^3 + 9*a^2*b^2*c^4)*(6*b^6 - 25*a^3*c^3 + 54*a^2*b^2*c^2 - 36*a*b^4*c)) - (16*a^9*x^2*((3*b*(b^4 + 3*a^2*c^2 - 4*a*b^2*c)*((((5*a^3*b *c^5 - 6*a^2*b^3*c^4)/a^6 - (((10*a^5*b*c^4 + 2*a^4*b^3*c^3)/a^6 + ((40*a^ 7*b*c^3 - 12*a^6*b^3*c^2)*(2*b^4 + 8*a^2*c^2 - 10*a*b^2*c))/(2*a^6*(16*a^4 *c - 4*a^3*b^2)))*(2*b^4 + 8*a^2*c^2 - 10*a*b^2*c))/(2*(16*a^4*c - 4*a^3*b ^2)))*(2*b^4 + 8*a^2*c^2 - 10*a*b^2*c))/(2*(16*a^4*c - 4*a^3*b^2)) - (b^3* c^5)/a^6 + (b*(3*a*c - b^2)*((b*((10*a^5*b*c^4 + 2*a^4*b^3*c^3)/a^6 + ((40 *a^7*b*c^3 - 12*a^6*b^3*c^2)*(2*b^4 + 8*a^2*c^2 - 10*a*b^2*c))/(2*a^6*(...
Time = 0.18 (sec) , antiderivative size = 517, normalized size of antiderivative = 4.54 \[ \int \frac {1}{x^5 \left (a+b x^2+c x^4\right )} \, dx=\frac {-6 \sqrt {2 \sqrt {c}\, \sqrt {a}+b}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-b}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}-b}-2 \sqrt {c}\, x}{\sqrt {2 \sqrt {c}\, \sqrt {a}+b}}\right ) a b c \,x^{4}+2 \sqrt {2 \sqrt {c}\, \sqrt {a}+b}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-b}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}-b}-2 \sqrt {c}\, x}{\sqrt {2 \sqrt {c}\, \sqrt {a}+b}}\right ) b^{3} x^{4}-6 \sqrt {2 \sqrt {c}\, \sqrt {a}+b}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-b}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}-b}+2 \sqrt {c}\, x}{\sqrt {2 \sqrt {c}\, \sqrt {a}+b}}\right ) a b c \,x^{4}+2 \sqrt {2 \sqrt {c}\, \sqrt {a}+b}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-b}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}-b}+2 \sqrt {c}\, x}{\sqrt {2 \sqrt {c}\, \sqrt {a}+b}}\right ) b^{3} x^{4}+4 \,\mathrm {log}\left (-\sqrt {2 \sqrt {c}\, \sqrt {a}-b}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) a^{2} c^{2} x^{4}-5 \,\mathrm {log}\left (-\sqrt {2 \sqrt {c}\, \sqrt {a}-b}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) a \,b^{2} c \,x^{4}+\mathrm {log}\left (-\sqrt {2 \sqrt {c}\, \sqrt {a}-b}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) b^{4} x^{4}+4 \,\mathrm {log}\left (\sqrt {2 \sqrt {c}\, \sqrt {a}-b}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) a^{2} c^{2} x^{4}-5 \,\mathrm {log}\left (\sqrt {2 \sqrt {c}\, \sqrt {a}-b}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) a \,b^{2} c \,x^{4}+\mathrm {log}\left (\sqrt {2 \sqrt {c}\, \sqrt {a}-b}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) b^{4} x^{4}-16 \,\mathrm {log}\left (x \right ) a^{2} c^{2} x^{4}+20 \,\mathrm {log}\left (x \right ) a \,b^{2} c \,x^{4}-4 \,\mathrm {log}\left (x \right ) b^{4} x^{4}-4 a^{3} c +a^{2} b^{2}+8 a^{2} b c \,x^{2}-2 a \,b^{3} x^{2}}{4 a^{3} x^{4} \left (4 a c -b^{2}\right )} \] Input:
int(1/x^5/(c*x^4+b*x^2+a),x)
Output:
( - 6*sqrt(2*sqrt(c)*sqrt(a) + b)*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*b*c*x* *4 + 2*sqrt(2*sqrt(c)*sqrt(a) + b)*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**3*x* *4 - 6*sqrt(2*sqrt(c)*sqrt(a) + b)*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*b*c*x **4 + 2*sqrt(2*sqrt(c)*sqrt(a) + b)*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**3*x **4 + 4*log( - sqrt(2*sqrt(c)*sqrt(a) - b)*x + sqrt(a) + sqrt(c)*x**2)*a** 2*c**2*x**4 - 5*log( - sqrt(2*sqrt(c)*sqrt(a) - b)*x + sqrt(a) + sqrt(c)*x **2)*a*b**2*c*x**4 + log( - sqrt(2*sqrt(c)*sqrt(a) - b)*x + sqrt(a) + sqrt (c)*x**2)*b**4*x**4 + 4*log(sqrt(2*sqrt(c)*sqrt(a) - b)*x + sqrt(a) + sqrt (c)*x**2)*a**2*c**2*x**4 - 5*log(sqrt(2*sqrt(c)*sqrt(a) - b)*x + sqrt(a) + sqrt(c)*x**2)*a*b**2*c*x**4 + log(sqrt(2*sqrt(c)*sqrt(a) - b)*x + sqrt(a) + sqrt(c)*x**2)*b**4*x**4 - 16*log(x)*a**2*c**2*x**4 + 20*log(x)*a*b**2*c *x**4 - 4*log(x)*b**4*x**4 - 4*a**3*c + a**2*b**2 + 8*a**2*b*c*x**2 - 2*a* b**3*x**2)/(4*a**3*x**4*(4*a*c - b**2))