Integrand size = 18, antiderivative size = 150 \[ \int \frac {x}{a x+b x^3+c x^5} \, dx=\frac {\sqrt {2} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b+\sqrt {b^2-4 a c}}} \] Output:
2^(1/2)*c^(1/2)*arctan(2^(1/2)*c^(1/2)*x/(b-(-4*a*c+b^2)^(1/2))^(1/2))/(-4 *a*c+b^2)^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2)-2^(1/2)*c^(1/2)*arctan(2^(1/2 )*c^(1/2)*x/(b+(-4*a*c+b^2)^(1/2))^(1/2))/(-4*a*c+b^2)^(1/2)/(b+(-4*a*c+b^ 2)^(1/2))^(1/2)
Time = 0.04 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.86 \[ \int \frac {x}{a x+b x^3+c x^5} \, dx=\frac {\sqrt {2} \sqrt {c} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c}} \] Input:
Integrate[x/(a*x + b*x^3 + c*x^5),x]
Output:
(Sqrt[2]*Sqrt[c]*(ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]]/ Sqrt[b - Sqrt[b^2 - 4*a*c]] - ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]]/Sqrt[b + Sqrt[b^2 - 4*a*c]]))/Sqrt[b^2 - 4*a*c]
Time = 0.23 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {9, 1406, 218}
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.
| \(\displaystyle \int \frac {x}{a x+b x^3+c x^5} \, dx\) |
| \(\Big \downarrow \) 9 |
| \(\displaystyle \int \frac {1}{a+b x^2+c x^4}dx\) |
| \(\Big \downarrow \) 1406 |
| \(\displaystyle \frac {c \int \frac {1}{c x^2+\frac {1}{2} \left (b-\sqrt {b^2-4 a c}\right )}dx}{\sqrt {b^2-4 a c}}-\frac {c \int \frac {1}{c x^2+\frac {1}{2} \left (b+\sqrt {b^2-4 a c}\right )}dx}{\sqrt {b^2-4 a c}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\sqrt {2} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {b^2-4 a c} \sqrt {\sqrt {b^2-4 a c}+b}}\) |
Input:
Int[x/(a*x + b*x^3 + c*x^5),x]
Output:
(Sqrt[2]*Sqrt[c]*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/ (Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (Sqrt[2]*Sqrt[c]*ArcTan[ (Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[ b + Sqrt[b^2 - 4*a*c]])
Int[(u_.)*(Px_)^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[1/e^(p*r) Int[u*(e*x)^(m + p*r)*ExpandToSum[Px/x^r, x]^p, x], x] /; IGtQ[r, 0]] /; FreeQ[{e, m}, x] && PolyQ[Px, x] && IntegerQ[p] && !MonomialQ[Px, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[b^ 2 - 4*a*c, 2]}, Simp[c/q Int[1/(b/2 - q/2 + c*x^2), x], x] - Simp[c/q I nt[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c , 0] && PosQ[b^2 - 4*a*c]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.08 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.25
| method | result | size |
| risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{4}+\textit {\_Z}^{2} b +a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{3} c +\textit {\_R} b}\right )}{2}\) | \(38\) |
| default | \(4 c \left (-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {c x \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{4 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}-\frac {\sqrt {2}\, \arctan \left (\frac {c x \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{4 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )\) | \(117\) |
Input:
int(x/(c*x^5+b*x^3+a*x),x,method=_RETURNVERBOSE)
Output:
1/2*sum(1/(2*_R^3*c+_R*b)*ln(x-_R),_R=RootOf(_Z^4*c+_Z^2*b+a))
Leaf count of result is larger than twice the leaf count of optimal. 613 vs. \(2 (114) = 228\).
Time = 0.09 (sec) , antiderivative size = 613, normalized size of antiderivative = 4.09 \[ \int \frac {x}{a x+b x^3+c x^5} \, dx=-\frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {-\frac {b + \frac {a b^{2} - 4 \, a^{2} c}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}}{a b^{2} - 4 \, a^{2} c}} \log \left (2 \, c x + \sqrt {\frac {1}{2}} {\left (b^{2} - 4 \, a c - \frac {a b^{3} - 4 \, a^{2} b c}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}\right )} \sqrt {-\frac {b + \frac {a b^{2} - 4 \, a^{2} c}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}}{a b^{2} - 4 \, a^{2} c}}\right ) + \frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {-\frac {b + \frac {a b^{2} - 4 \, a^{2} c}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}}{a b^{2} - 4 \, a^{2} c}} \log \left (2 \, c x - \sqrt {\frac {1}{2}} {\left (b^{2} - 4 \, a c - \frac {a b^{3} - 4 \, a^{2} b c}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}\right )} \sqrt {-\frac {b + \frac {a b^{2} - 4 \, a^{2} c}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}}{a b^{2} - 4 \, a^{2} c}}\right ) - \frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {-\frac {b - \frac {a b^{2} - 4 \, a^{2} c}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}}{a b^{2} - 4 \, a^{2} c}} \log \left (2 \, c x + \sqrt {\frac {1}{2}} {\left (b^{2} - 4 \, a c + \frac {a b^{3} - 4 \, a^{2} b c}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}\right )} \sqrt {-\frac {b - \frac {a b^{2} - 4 \, a^{2} c}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}}{a b^{2} - 4 \, a^{2} c}}\right ) + \frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {-\frac {b - \frac {a b^{2} - 4 \, a^{2} c}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}}{a b^{2} - 4 \, a^{2} c}} \log \left (2 \, c x - \sqrt {\frac {1}{2}} {\left (b^{2} - 4 \, a c + \frac {a b^{3} - 4 \, a^{2} b c}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}\right )} \sqrt {-\frac {b - \frac {a b^{2} - 4 \, a^{2} c}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}}{a b^{2} - 4 \, a^{2} c}}\right ) \] Input:
integrate(x/(c*x^5+b*x^3+a*x),x, algorithm="fricas")
Output:
-1/2*sqrt(1/2)*sqrt(-(b + (a*b^2 - 4*a^2*c)/sqrt(a^2*b^2 - 4*a^3*c))/(a*b^ 2 - 4*a^2*c))*log(2*c*x + sqrt(1/2)*(b^2 - 4*a*c - (a*b^3 - 4*a^2*b*c)/sqr t(a^2*b^2 - 4*a^3*c))*sqrt(-(b + (a*b^2 - 4*a^2*c)/sqrt(a^2*b^2 - 4*a^3*c) )/(a*b^2 - 4*a^2*c))) + 1/2*sqrt(1/2)*sqrt(-(b + (a*b^2 - 4*a^2*c)/sqrt(a^ 2*b^2 - 4*a^3*c))/(a*b^2 - 4*a^2*c))*log(2*c*x - sqrt(1/2)*(b^2 - 4*a*c - (a*b^3 - 4*a^2*b*c)/sqrt(a^2*b^2 - 4*a^3*c))*sqrt(-(b + (a*b^2 - 4*a^2*c)/ sqrt(a^2*b^2 - 4*a^3*c))/(a*b^2 - 4*a^2*c))) - 1/2*sqrt(1/2)*sqrt(-(b - (a *b^2 - 4*a^2*c)/sqrt(a^2*b^2 - 4*a^3*c))/(a*b^2 - 4*a^2*c))*log(2*c*x + sq rt(1/2)*(b^2 - 4*a*c + (a*b^3 - 4*a^2*b*c)/sqrt(a^2*b^2 - 4*a^3*c))*sqrt(- (b - (a*b^2 - 4*a^2*c)/sqrt(a^2*b^2 - 4*a^3*c))/(a*b^2 - 4*a^2*c))) + 1/2* sqrt(1/2)*sqrt(-(b - (a*b^2 - 4*a^2*c)/sqrt(a^2*b^2 - 4*a^3*c))/(a*b^2 - 4 *a^2*c))*log(2*c*x - sqrt(1/2)*(b^2 - 4*a*c + (a*b^3 - 4*a^2*b*c)/sqrt(a^2 *b^2 - 4*a^3*c))*sqrt(-(b - (a*b^2 - 4*a^2*c)/sqrt(a^2*b^2 - 4*a^3*c))/(a* b^2 - 4*a^2*c)))
Time = 0.78 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.58 \[ \int \frac {x}{a x+b x^3+c x^5} \, dx=\operatorname {RootSum} {\left (t^{4} \cdot \left (256 a^{3} c^{2} - 128 a^{2} b^{2} c + 16 a b^{4}\right ) + t^{2} \left (- 16 a b c + 4 b^{3}\right ) + c, \left ( t \mapsto t \log {\left (x + \frac {32 t^{3} a^{2} b c - 8 t^{3} a b^{3} + 4 t a c - 2 t b^{2}}{c} \right )} \right )\right )} \] Input:
integrate(x/(c*x**5+b*x**3+a*x),x)
Output:
RootSum(_t**4*(256*a**3*c**2 - 128*a**2*b**2*c + 16*a*b**4) + _t**2*(-16*a *b*c + 4*b**3) + c, Lambda(_t, _t*log(x + (32*_t**3*a**2*b*c - 8*_t**3*a*b **3 + 4*_t*a*c - 2*_t*b**2)/c)))
\[ \int \frac {x}{a x+b x^3+c x^5} \, dx=\int { \frac {x}{c x^{5} + b x^{3} + a x} \,d x } \] Input:
integrate(x/(c*x^5+b*x^3+a*x),x, algorithm="maxima")
Output:
integrate(x/(c*x^5 + b*x^3 + a*x), x)
Leaf count of result is larger than twice the leaf count of optimal. 1024 vs. \(2 (114) = 228\).
Time = 0.40 (sec) , antiderivative size = 1024, normalized size of antiderivative = 6.83 \[ \int \frac {x}{a x+b x^3+c x^5} \, dx =\text {Too large to display} \] Input:
integrate(x/(c*x^5+b*x^3+a*x),x, algorithm="giac")
Output:
1/4*(sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^4 - 8*sqrt(2)*sqrt(b*c + sq rt(b^2 - 4*a*c)*c)*a*b^2*c - 2*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^3 *c - 2*b^4*c + 16*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*c^2 + 8*sqrt (2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^2 + sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^2*c^2 + 16*a*b^2*c^2 - 2*b^3*c^2 - 4*sqrt(2)*sqrt(b*c + sqrt (b^2 - 4*a*c)*c)*a*c^3 - 32*a^2*c^3 + 8*a*b*c^3 + sqrt(2)*sqrt(b^2 - 4*a*c )*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^3 - 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b *c + sqrt(b^2 - 4*a*c)*c)*a*b*c - 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + s qrt(b^2 - 4*a*c)*c)*b^2*c + sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b*c^2 + 2*(b^2 - 4*a*c)*b^2*c - 8*(b^2 - 4*a*c)*a*c^2 + 2*(b^2 - 4*a*c)*b*c^2)*arctan(2*sqrt(1/2)*x/sqrt((b + sqrt(b^2 - 4*a*c))/c))/((a *b^4 - 8*a^2*b^2*c - 2*a*b^3*c + 16*a^3*c^2 + 8*a^2*b*c^2 + a*b^2*c^2 - 4* a^2*c^3)*abs(c)) + 1/4*(sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^4 - 8*sq rt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^2*c - 2*sqrt(2)*sqrt(b*c - sqrt( b^2 - 4*a*c)*c)*b^3*c + 2*b^4*c + 16*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)* c)*a^2*c^2 + 8*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b*c^2 + sqrt(2)*s qrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^2*c^2 - 16*a*b^2*c^2 - 2*b^3*c^2 - 4*sqrt (2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*c^3 + 32*a^2*c^3 + 8*a*b*c^3 + sqrt( 2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^3 - 4*sqrt(2)*sqrt( b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b*c - 2*sqrt(2)*sqrt(b^2...
Time = 12.66 (sec) , antiderivative size = 763, normalized size of antiderivative = 5.09 \[ \int \frac {x}{a x+b x^3+c x^5} \, dx=-\mathrm {atan}\left (\frac {b^4\,x\,1{}\mathrm {i}+b\,x\,\sqrt {-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}\,1{}\mathrm {i}+a^2\,c^2\,x\,16{}\mathrm {i}-a\,b^2\,c\,x\,8{}\mathrm {i}}{4\,a\,b^4\,\sqrt {-\frac {b^3+\sqrt {-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}-4\,a\,b\,c}{128\,a^3\,c^2-64\,a^2\,b^2\,c+8\,a\,b^4}}+64\,a^3\,c^2\,\sqrt {-\frac {b^3+\sqrt {-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}-4\,a\,b\,c}{128\,a^3\,c^2-64\,a^2\,b^2\,c+8\,a\,b^4}}-32\,a^2\,b^2\,c\,\sqrt {-\frac {b^3+\sqrt {-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}-4\,a\,b\,c}{128\,a^3\,c^2-64\,a^2\,b^2\,c+8\,a\,b^4}}}\right )\,\sqrt {-\frac {b^3+\sqrt {-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}-4\,a\,b\,c}{128\,a^3\,c^2-64\,a^2\,b^2\,c+8\,a\,b^4}}\,2{}\mathrm {i}-\mathrm {atan}\left (\frac {b^4\,x\,1{}\mathrm {i}-b\,x\,\sqrt {-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}\,1{}\mathrm {i}+a^2\,c^2\,x\,16{}\mathrm {i}-a\,b^2\,c\,x\,8{}\mathrm {i}}{4\,a\,b^4\,\sqrt {\frac {\sqrt {-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}-b^3+4\,a\,b\,c}{128\,a^3\,c^2-64\,a^2\,b^2\,c+8\,a\,b^4}}+64\,a^3\,c^2\,\sqrt {\frac {\sqrt {-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}-b^3+4\,a\,b\,c}{128\,a^3\,c^2-64\,a^2\,b^2\,c+8\,a\,b^4}}-32\,a^2\,b^2\,c\,\sqrt {\frac {\sqrt {-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}-b^3+4\,a\,b\,c}{128\,a^3\,c^2-64\,a^2\,b^2\,c+8\,a\,b^4}}}\right )\,\sqrt {\frac {\sqrt {-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}-b^3+4\,a\,b\,c}{128\,a^3\,c^2-64\,a^2\,b^2\,c+8\,a\,b^4}}\,2{}\mathrm {i} \] Input:
int(x/(a*x + b*x^3 + c*x^5),x)
Output:
- atan((b^4*x*1i + b*x*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)^(1 /2)*1i + a^2*c^2*x*16i - a*b^2*c*x*8i)/(4*a*b^4*(-(b^3 + (b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)^(1/2) - 4*a*b*c)/(8*a*b^4 + 128*a^3*c^2 - 64*a^2*b^2*c))^(1/2) + 64*a^3*c^2*(-(b^3 + (b^6 - 64*a^3*c^3 + 48*a^2*b^2* c^2 - 12*a*b^4*c)^(1/2) - 4*a*b*c)/(8*a*b^4 + 128*a^3*c^2 - 64*a^2*b^2*c)) ^(1/2) - 32*a^2*b^2*c*(-(b^3 + (b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b ^4*c)^(1/2) - 4*a*b*c)/(8*a*b^4 + 128*a^3*c^2 - 64*a^2*b^2*c))^(1/2)))*(-( b^3 + (b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)^(1/2) - 4*a*b*c)/(8 *a*b^4 + 128*a^3*c^2 - 64*a^2*b^2*c))^(1/2)*2i - atan((b^4*x*1i - b*x*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)^(1/2)*1i + a^2*c^2*x*16i - a* b^2*c*x*8i)/(4*a*b^4*(((b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)^(1 /2) - b^3 + 4*a*b*c)/(8*a*b^4 + 128*a^3*c^2 - 64*a^2*b^2*c))^(1/2) + 64*a^ 3*c^2*(((b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)^(1/2) - b^3 + 4*a *b*c)/(8*a*b^4 + 128*a^3*c^2 - 64*a^2*b^2*c))^(1/2) - 32*a^2*b^2*c*(((b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)^(1/2) - b^3 + 4*a*b*c)/(8*a*b^ 4 + 128*a^3*c^2 - 64*a^2*b^2*c))^(1/2)))*(((b^6 - 64*a^3*c^3 + 48*a^2*b^2* c^2 - 12*a*b^4*c)^(1/2) - b^3 + 4*a*b*c)/(8*a*b^4 + 128*a^3*c^2 - 64*a^2*b ^2*c))^(1/2)*2i
Time = 0.21 (sec) , antiderivative size = 351, normalized size of antiderivative = 2.34 \[ \int \frac {x}{a x+b x^3+c x^5} \, dx=\frac {2 \sqrt {a}\, \sqrt {2 \sqrt {c}\, \sqrt {a}+b}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}-b}-2 \sqrt {c}\, x}{\sqrt {2 \sqrt {c}\, \sqrt {a}+b}}\right ) b -4 \sqrt {c}\, \sqrt {2 \sqrt {c}\, \sqrt {a}+b}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}-b}-2 \sqrt {c}\, x}{\sqrt {2 \sqrt {c}\, \sqrt {a}+b}}\right ) a -2 \sqrt {a}\, \sqrt {2 \sqrt {c}\, \sqrt {a}+b}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}-b}+2 \sqrt {c}\, x}{\sqrt {2 \sqrt {c}\, \sqrt {a}+b}}\right ) b +4 \sqrt {c}\, \sqrt {2 \sqrt {c}\, \sqrt {a}+b}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}-b}+2 \sqrt {c}\, x}{\sqrt {2 \sqrt {c}\, \sqrt {a}+b}}\right ) a -\sqrt {a}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-b}\, \mathrm {log}\left (-\sqrt {2 \sqrt {c}\, \sqrt {a}-b}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) b +\sqrt {a}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-b}\, \mathrm {log}\left (\sqrt {2 \sqrt {c}\, \sqrt {a}-b}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) b -2 \sqrt {c}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-b}\, \mathrm {log}\left (-\sqrt {2 \sqrt {c}\, \sqrt {a}-b}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) a +2 \sqrt {c}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-b}\, \mathrm {log}\left (\sqrt {2 \sqrt {c}\, \sqrt {a}-b}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) a}{4 a \left (4 a c -b^{2}\right )} \] Input:
int(x/(c*x^5+b*x^3+a*x),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 - 4*sqrt(c)*sqrt(2*sqrt(c)*sq rt(a) + b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c) *sqrt(a) + b))*a - 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 + 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 - sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) - b)*l og( - sqrt(2*sqrt(c)*sqrt(a) - b)*x + sqrt(a) + sqrt(c)*x**2)*b + 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 - 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 + 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 )/(4*a*(4*a*c - b**2))