\(\int \frac {x^{11}}{(a x+b x^3+c x^5)^2} \, dx\) [36]

Optimal result

Integrand size = 20, antiderivative size = 166 \[ \int \frac {x^{11}}{\left (a x+b x^3+c x^5\right )^2} \, dx=\frac {\left (b^2-3 a c\right ) x^2}{c^2 \left (b^2-4 a c\right )}-\frac {b x^4}{2 c \left (b^2-4 a c\right )}+\frac {x^6 \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\left (b^4-6 a b^2 c+6 a^2 c^2\right ) \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{c^3 \left (b^2-4 a c\right )^{3/2}}-\frac {b \log \left (a+b x^2+c x^4\right )}{2 c^3} \] Output:

(-3*a*c+b^2)*x^2/c^2/(-4*a*c+b^2)-1/2*b*x^4/c/(-4*a*c+b^2)+1/2*x^6*(b*x^2+ 
2*a)/(-4*a*c+b^2)/(c*x^4+b*x^2+a)-(6*a^2*c^2-6*a*b^2*c+b^4)*arctanh((2*c*x 
^2+b)/(-4*a*c+b^2)^(1/2))/c^3/(-4*a*c+b^2)^(3/2)-1/2*b*ln(c*x^4+b*x^2+a)/c 
^3
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.91 \[ \int \frac {x^{11}}{\left (a x+b x^3+c x^5\right )^2} \, dx=\frac {c x^2+\frac {-b^4 x^2-a b^2 \left (b-4 c x^2\right )+a^2 c \left (3 b-2 c x^2\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {2 \left (b^4-6 a b^2 c+6 a^2 c^2\right ) \arctan \left (\frac {b+2 c x^2}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{3/2}}-b \log \left (a+b x^2+c x^4\right )}{2 c^3} \] Input:

Integrate[x^11/(a*x + b*x^3 + c*x^5)^2,x]
 

Output:

(c*x^2 + (-(b^4*x^2) - a*b^2*(b - 4*c*x^2) + a^2*c*(3*b - 2*c*x^2))/((b^2 
- 4*a*c)*(a + b*x^2 + c*x^4)) - (2*(b^4 - 6*a*b^2*c + 6*a^2*c^2)*ArcTan[(b 
 + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2) - b*Log[a + b*x^2 + 
c*x^4])/(2*c^3)
 

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {9, 1434, 1164, 27, 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.

\(\displaystyle \int \frac {x^{11}}{\left (a x+b x^3+c x^5\right )^2} \, dx\)

\(\Big \downarrow \) 9

\(\displaystyle \int \frac {x^9}{\left (a+b x^2+c x^4\right )^2}dx\)

\(\Big \downarrow \) 1434

\(\displaystyle \frac {1}{2} \int \frac {x^8}{\left (c x^4+b x^2+a\right )^2}dx^2\)

\(\Big \downarrow \) 1164

\(\displaystyle \frac {1}{2} \left (\frac {x^6 \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\int \frac {2 x^4 \left (b x^2+3 a\right )}{c x^4+b x^2+a}dx^2}{b^2-4 a c}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \left (\frac {x^6 \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {2 \int \frac {x^4 \left (b x^2+3 a\right )}{c x^4+b x^2+a}dx^2}{b^2-4 a c}\right )\)

\(\Big \downarrow \) 1200

\(\displaystyle \frac {1}{2} \left (\frac {x^6 \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {2 \int \left (\frac {b x^2}{c}-\frac {b^2-3 a c}{c^2}+\frac {b \left (b^2-4 a c\right ) x^2+a \left (b^2-3 a c\right )}{c^2 \left (c x^4+b x^2+a\right )}\right )dx^2}{b^2-4 a c}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} \left (\frac {x^6 \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {2 \left (\frac {\left (6 a^2 c^2-6 a b^2 c+b^4\right ) \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{c^3 \sqrt {b^2-4 a c}}+\frac {b \left (b^2-4 a c\right ) \log \left (a+b x^2+c x^4\right )}{2 c^3}-\frac {x^2 \left (b^2-3 a c\right )}{c^2}+\frac {b x^4}{2 c}\right )}{b^2-4 a c}\right )\)

Input:

Int[x^11/(a*x + b*x^3 + c*x^5)^2,x]
 

Output:

((x^6*(2*a + b*x^2))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - (2*(-(((b^2 - 3 
*a*c)*x^2)/c^2) + (b*x^4)/(2*c) + ((b^4 - 6*a*b^2*c + 6*a^2*c^2)*ArcTanh[( 
b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(c^3*Sqrt[b^2 - 4*a*c]) + (b*(b^2 - 4*a*c 
)*Log[a + b*x^2 + c*x^4])/(2*c^3)))/(b^2 - 4*a*c))/2
 

Defintions of rubi rules used

rule 9

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]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 1164

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(d + e*x)^(m - 1)*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b*x 
+ c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a* 
c))   Int[(d + e*x)^(m - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2* 
c*d^2*(2*p + 3) + e*(b*e - 2*d*c)*(m + 2*p + 2)*x, x]*(a + b*x + c*x^2)^(p 
+ 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[p, -1] && GtQ[m, 1] && Int 
QuadraticQ[a, b, c, d, e, m, p, x]
 

rule 1200

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]
 

rule 1434

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]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 0.16 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.26

method	result	size
default	\(\frac {x^{2}}{2 c^{2}}-\frac {\frac {-\frac {\left (2 a^{2} c^{2}-4 a \,b^{2} c +b^{4}\right ) x^{2}}{c \left (4 a c -b^{2}\right )}+\frac {b a \left (3 a c -b^{2}\right )}{c \left (4 a c -b^{2}\right )}}{c \,x^{4}+b \,x^{2}+a}+\frac {\frac {\left (4 a b c -b^{3}\right ) \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{c}+\frac {4 \left (3 a^{2} c -b^{2} a -\frac {\left (4 a b c -b^{3}\right ) b}{2 c}\right ) \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{4 a c -b^{2}}}{2 c^{2}}\)	\(209\)
risch	\(\text {Expression too large to display}\)	\(1217\)

Input:

int(x^11/(c*x^5+b*x^3+a*x)^2,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 422 vs. \(2 (156) = 312\).

Time = 0.10 (sec) , antiderivative size = 868, normalized size of antiderivative = 5.23 \[ \int \frac {x^{11}}{\left (a x+b x^3+c x^5\right )^2} \, dx =\text {Too large to display} \] Input:

integrate(x^11/(c*x^5+b*x^3+a*x)^2,x, algorithm="fricas")
 

Output:

[1/2*((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*x^6 - a*b^5 + 7*a^2*b^3*c - 12* 
a^3*b*c^2 + (b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^4 - (b^6 - 9*a*b^4*c + 
26*a^2*b^2*c^2 - 24*a^3*c^3)*x^2 - (a*b^4 - 6*a^2*b^2*c + 6*a^3*c^2 + (b^4 
*c - 6*a*b^2*c^2 + 6*a^2*c^3)*x^4 + (b^5 - 6*a*b^3*c + 6*a^2*b*c^2)*x^2)*s 
qrt(b^2 - 4*a*c)*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)) - (a*b^5 - 8*a^2*b^3*c + 16*a^3*b* 
c^2 + (b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^4 + (b^6 - 8*a*b^4*c + 16*a^2 
*b^2*c^2)*x^2)*log(c*x^4 + b*x^2 + a))/(a*b^4*c^3 - 8*a^2*b^2*c^4 + 16*a^3 
*c^5 + (b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*x^4 + (b^5*c^3 - 8*a*b^3*c^4 + 
 16*a^2*b*c^5)*x^2), 1/2*((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*x^6 - a*b^5 
 + 7*a^2*b^3*c - 12*a^3*b*c^2 + (b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^4 - 
 (b^6 - 9*a*b^4*c + 26*a^2*b^2*c^2 - 24*a^3*c^3)*x^2 - 2*(a*b^4 - 6*a^2*b^ 
2*c + 6*a^3*c^2 + (b^4*c - 6*a*b^2*c^2 + 6*a^2*c^3)*x^4 + (b^5 - 6*a*b^3*c 
 + 6*a^2*b*c^2)*x^2)*sqrt(-b^2 + 4*a*c)*arctan(-(2*c*x^2 + b)*sqrt(-b^2 + 
4*a*c)/(b^2 - 4*a*c)) - (a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2 + (b^5*c - 8*a 
*b^3*c^2 + 16*a^2*b*c^3)*x^4 + (b^6 - 8*a*b^4*c + 16*a^2*b^2*c^2)*x^2)*log 
(c*x^4 + b*x^2 + a))/(a*b^4*c^3 - 8*a^2*b^2*c^4 + 16*a^3*c^5 + (b^4*c^4 - 
8*a*b^2*c^5 + 16*a^2*c^6)*x^4 + (b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*x^2 
)]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 877 vs. \(2 (151) = 302\).

Time = 97.80 (sec) , antiderivative size = 877, normalized size of antiderivative = 5.28 \[ \int \frac {x^{11}}{\left (a x+b x^3+c x^5\right )^2} \, dx =\text {Too large to display} \] Input:

integrate(x**11/(c*x**5+b*x**3+a*x)**2,x)
 

Output:

(-b/(2*c**3) - sqrt(-(4*a*c - b**2)**3)*(6*a**2*c**2 - 6*a*b**2*c + b**4)/ 
(2*c**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)))*log(x**2 
 + (-5*a**2*b*c - 16*a**2*c**4*(-b/(2*c**3) - sqrt(-(4*a*c - b**2)**3)*(6* 
a**2*c**2 - 6*a*b**2*c + b**4)/(2*c**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 
 12*a*b**4*c - b**6))) + a*b**3 + 8*a*b**2*c**3*(-b/(2*c**3) - sqrt(-(4*a* 
c - b**2)**3)*(6*a**2*c**2 - 6*a*b**2*c + b**4)/(2*c**3*(64*a**3*c**3 - 48 
*a**2*b**2*c**2 + 12*a*b**4*c - b**6))) - b**4*c**2*(-b/(2*c**3) - sqrt(-( 
4*a*c - b**2)**3)*(6*a**2*c**2 - 6*a*b**2*c + b**4)/(2*c**3*(64*a**3*c**3 
- 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))))/(6*a**2*c**2 - 6*a*b**2*c + b 
**4)) + (-b/(2*c**3) + sqrt(-(4*a*c - b**2)**3)*(6*a**2*c**2 - 6*a*b**2*c 
+ b**4)/(2*c**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)))* 
log(x**2 + (-5*a**2*b*c - 16*a**2*c**4*(-b/(2*c**3) + sqrt(-(4*a*c - b**2) 
**3)*(6*a**2*c**2 - 6*a*b**2*c + b**4)/(2*c**3*(64*a**3*c**3 - 48*a**2*b** 
2*c**2 + 12*a*b**4*c - b**6))) + a*b**3 + 8*a*b**2*c**3*(-b/(2*c**3) + sqr 
t(-(4*a*c - b**2)**3)*(6*a**2*c**2 - 6*a*b**2*c + b**4)/(2*c**3*(64*a**3*c 
**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))) - b**4*c**2*(-b/(2*c**3) + 
 sqrt(-(4*a*c - b**2)**3)*(6*a**2*c**2 - 6*a*b**2*c + b**4)/(2*c**3*(64*a* 
*3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))))/(6*a**2*c**2 - 6*a*b* 
*2*c + b**4)) + (-3*a**2*b*c + a*b**3 + x**2*(2*a**2*c**2 - 4*a*b**2*c + b 
**4))/(8*a**2*c**4 - 2*a*b**2*c**3 + x**4*(8*a*c**5 - 2*b**2*c**4) + x*...
 

Maxima [F]

\[ \int \frac {x^{11}}{\left (a x+b x^3+c x^5\right )^2} \, dx=\int { \frac {x^{11}}{{\left (c x^{5} + b x^{3} + a x\right )}^{2}} \,d x } \] Input:

integrate(x^11/(c*x^5+b*x^3+a*x)^2,x, algorithm="maxima")
                                                                                    
                                                                                    
 

Output:

-1/2*(a*b^3 - 3*a^2*b*c + (b^4 - 4*a*b^2*c + 2*a^2*c^2)*x^2)/(a*b^2*c^3 - 
4*a^2*c^4 + (b^2*c^4 - 4*a*c^5)*x^4 + (b^3*c^3 - 4*a*b*c^4)*x^2) + 1/2*x^2 
/c^2 + 2*integrate(-((b^3 - 4*a*b*c)*x^3 + (a*b^2 - 3*a^2*c)*x)/(c*x^4 + b 
*x^2 + a), x)/(b^2*c^2 - 4*a*c^3)
 

Giac [A] (verification not implemented)

Time = 0.34 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.97 \[ \int \frac {x^{11}}{\left (a x+b x^3+c x^5\right )^2} \, dx=\frac {{\left (b^{4} - 6 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} \arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} c^{3} - 4 \, a c^{4}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {x^{2}}{2 \, c^{2}} + \frac {b^{3} x^{4} - 4 \, a b c x^{4} - 2 \, a^{2} c x^{2} - a^{2} b}{2 \, {\left (c x^{4} + b x^{2} + a\right )} {\left (b^{2} c^{2} - 4 \, a c^{3}\right )}} - \frac {b \log \left (c x^{4} + b x^{2} + a\right )}{2 \, c^{3}} \] Input:

integrate(x^11/(c*x^5+b*x^3+a*x)^2,x, algorithm="giac")
 

Output:

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

Mupad [B] (verification not implemented)

Time = 12.26 (sec) , antiderivative size = 1473, normalized size of antiderivative = 8.87 \[ \int \frac {x^{11}}{\left (a x+b x^3+c x^5\right )^2} \, dx=\text {Too large to display} \] Input:

int(x^11/(a*x + b*x^3 + c*x^5)^2,x)
 

Output:

((a*(b^3 - 3*a*b*c))/(2*c*(4*a*c - b^2)) + (x^2*(b^4 + 2*a^2*c^2 - 4*a*b^2 
*c))/(2*c*(4*a*c - b^2)))/(a*c^2 + c^3*x^4 + b*c^2*x^2) + x^2/(2*c^2) + (l 
og(a + b*x^2 + c*x^4)*(b^7 - 64*a^3*b*c^3 + 48*a^2*b^3*c^2 - 12*a*b^5*c))/ 
(2*(64*a^3*c^6 - b^6*c^3 + 12*a*b^4*c^4 - 48*a^2*b^2*c^5)) + (atan(((4*a*c 
^5*(4*a*c - b^2)^3 - b^2*c^4*(4*a*c - b^2)^3)*(((((16*a*b)/c + (8*a*c^2*(b 
^7 - 64*a^3*b*c^3 + 48*a^2*b^3*c^2 - 12*a*b^5*c))/(64*a^3*c^6 - b^6*c^3 + 
12*a*b^4*c^4 - 48*a^2*b^2*c^5))*(b^4 + 6*a^2*c^2 - 6*a*b^2*c))/(2*c^3*(4*a 
*c - b^2)^(3/2)) + (4*a*(b^4 + 6*a^2*c^2 - 6*a*b^2*c)*(b^7 - 64*a^3*b*c^3 
+ 48*a^2*b^3*c^2 - 12*a*b^5*c))/(c*(4*a*c - b^2)^(3/2)*(64*a^3*c^6 - b^6*c 
^3 + 12*a*b^4*c^4 - 48*a^2*b^2*c^5)))/(2*a*(4*a*c - b^2)) - x^2*(((((4*(6* 
a^2*c^5 + 3*b^4*c^3 - 14*a*b^2*c^4))/(4*a*c^5 - b^2*c^4) + (2*(2*b^3*c^6 - 
 8*a*b*c^7)*(b^7 - 64*a^3*b*c^3 + 48*a^2*b^3*c^2 - 12*a*b^5*c))/((4*a*c^5 
- b^2*c^4)*(64*a^3*c^6 - b^6*c^3 + 12*a*b^4*c^4 - 48*a^2*b^2*c^5)))*(b^4 + 
 6*a^2*c^2 - 6*a*b^2*c))/(2*c^3*(4*a*c - b^2)^(3/2)) + ((2*b^3*c^6 - 8*a*b 
*c^7)*(b^4 + 6*a^2*c^2 - 6*a*b^2*c)*(b^7 - 64*a^3*b*c^3 + 48*a^2*b^3*c^2 - 
 12*a*b^5*c))/(c^3*(4*a*c - b^2)^(3/2)*(4*a*c^5 - b^2*c^4)*(64*a^3*c^6 - b 
^6*c^3 + 12*a*b^4*c^4 - 48*a^2*b^2*c^5)))/(2*a*(4*a*c - b^2)) + (b*((4*(b^ 
5 + 3*a^2*b*c^2 - 5*a*b^3*c))/(4*a*c^5 - b^2*c^4) + (((4*(6*a^2*c^5 + 3*b^ 
4*c^3 - 14*a*b^2*c^4))/(4*a*c^5 - b^2*c^4) + (2*(2*b^3*c^6 - 8*a*b*c^7)*(b 
^7 - 64*a^3*b*c^3 + 48*a^2*b^3*c^2 - 12*a*b^5*c))/((4*a*c^5 - b^2*c^4)*...
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 1889, normalized size of antiderivative = 11.38 \[ \int \frac {x^{11}}{\left (a x+b x^3+c x^5\right )^2} \, dx =\text {Too large to display} \] Input:

int(x^11/(c*x^5+b*x^3+a*x)^2,x)
 

Output:

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