\(\int \frac {1}{(a x+b x^3+c x^5)^2} \, dx\) [49]

Optimal result

Integrand size = 16, antiderivative size = 308 \[ \int \frac {1}{\left (a x+b x^3+c x^5\right )^2} \, dx=-\frac {3 b^2-10 a c}{2 a^2 \left (b^2-4 a c\right ) x}+\frac {b^2-2 a c+b c x^2}{2 a \left (b^2-4 a c\right ) x \left (a+b x^2+c x^4\right )}-\frac {\sqrt {c} \left (3 b^3-16 a b c+\left (3 b^2-10 a c\right ) \sqrt {b^2-4 a c}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {c} \left (3 b^3-16 a b c-\left (3 b^2-10 a c\right ) \sqrt {b^2-4 a c}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt {b+\sqrt {b^2-4 a c}}} \] Output:

-1/2*(-10*a*c+3*b^2)/a^2/(-4*a*c+b^2)/x+1/2*(b*c*x^2-2*a*c+b^2)/a/(-4*a*c+ 
b^2)/x/(c*x^4+b*x^2+a)-1/4*c^(1/2)*(3*b^3-16*a*b*c+(-10*a*c+3*b^2)*(-4*a*c 
+b^2)^(1/2))*arctan(2^(1/2)*c^(1/2)*x/(b-(-4*a*c+b^2)^(1/2))^(1/2))*2^(1/2 
)/a^2/(-4*a*c+b^2)^(3/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2)+1/4*c^(1/2)*(3*b^3-1 
6*a*b*c-(-10*a*c+3*b^2)*(-4*a*c+b^2)^(1/2))*arctan(2^(1/2)*c^(1/2)*x/(b+(- 
4*a*c+b^2)^(1/2))^(1/2))*2^(1/2)/a^2/(-4*a*c+b^2)^(3/2)/(b+(-4*a*c+b^2)^(1 
/2))^(1/2)
 

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 302, normalized size of antiderivative = 0.98 \[ \int \frac {1}{\left (a x+b x^3+c x^5\right )^2} \, dx=\frac {-\frac {4}{x}-\frac {2 x \left (b^3-3 a b c+b^2 c x^2-2 a c^2 x^2\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {\sqrt {2} \sqrt {c} \left (-3 b^3+16 a b c-3 b^2 \sqrt {b^2-4 a c}+10 a c \sqrt {b^2-4 a c}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {2} \sqrt {c} \left (3 b^3-16 a b c-3 b^2 \sqrt {b^2-4 a c}+10 a c \sqrt {b^2-4 a c}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt {b+\sqrt {b^2-4 a c}}}}{4 a^2} \] Input:

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

Output:

(-4/x - (2*x*(b^3 - 3*a*b*c + b^2*c*x^2 - 2*a*c^2*x^2))/((b^2 - 4*a*c)*(a 
+ b*x^2 + c*x^4)) + (Sqrt[2]*Sqrt[c]*(-3*b^3 + 16*a*b*c - 3*b^2*Sqrt[b^2 - 
 4*a*c] + 10*a*c*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sq 
rt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sq 
rt[2]*Sqrt[c]*(3*b^3 - 16*a*b*c - 3*b^2*Sqrt[b^2 - 4*a*c] + 10*a*c*Sqrt[b^ 
2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/((b^2 
 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(4*a^2)
 

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {1949, 1441, 25, 1604, 1480, 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.

\(\displaystyle \int \frac {1}{\left (a x+b x^3+c x^5\right )^2} \, dx\)

\(\Big \downarrow \) 1949

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

\(\Big \downarrow \) 1441

\(\displaystyle \frac {-2 a c+b^2+b c x^2}{2 a x \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\int -\frac {3 b^2+3 c x^2 b-10 a c}{x^2 \left (c x^4+b x^2+a\right )}dx}{2 a \left (b^2-4 a c\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {3 b^2+3 c x^2 b-10 a c}{x^2 \left (c x^4+b x^2+a\right )}dx}{2 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x^2}{2 a x \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\)

\(\Big \downarrow \) 1604

\(\displaystyle \frac {-\frac {\int \frac {c \left (3 b^2-10 a c\right ) x^2+b \left (3 b^2-13 a c\right )}{c x^4+b x^2+a}dx}{a}-\frac {3 b^2-10 a c}{a x}}{2 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x^2}{2 a x \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\)

\(\Big \downarrow \) 1480

\(\displaystyle \frac {-\frac {\frac {1}{2} c \left (-\frac {16 a b c}{\sqrt {b^2-4 a c}}+\frac {3 b^3}{\sqrt {b^2-4 a c}}-10 a c+3 b^2\right ) \int \frac {1}{c x^2+\frac {1}{2} \left (b-\sqrt {b^2-4 a c}\right )}dx-\frac {c \left (-\left (3 b^2-10 a c\right ) \sqrt {b^2-4 a c}-16 a b c+3 b^3\right ) \int \frac {1}{c x^2+\frac {1}{2} \left (b+\sqrt {b^2-4 a c}\right )}dx}{2 \sqrt {b^2-4 a c}}}{a}-\frac {3 b^2-10 a c}{a x}}{2 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x^2}{2 a x \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {-\frac {\frac {\sqrt {c} \left (-\frac {16 a b c}{\sqrt {b^2-4 a c}}+\frac {3 b^3}{\sqrt {b^2-4 a c}}-10 a c+3 b^2\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {c} \left (-\left (3 b^2-10 a c\right ) \sqrt {b^2-4 a c}-16 a b c+3 b^3\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {2} \sqrt {b^2-4 a c} \sqrt {\sqrt {b^2-4 a c}+b}}}{a}-\frac {3 b^2-10 a c}{a x}}{2 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x^2}{2 a x \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\)

Input:

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

Output:

(b^2 - 2*a*c + b*c*x^2)/(2*a*(b^2 - 4*a*c)*x*(a + b*x^2 + c*x^4)) + (-((3* 
b^2 - 10*a*c)/(a*x)) - ((Sqrt[c]*(3*b^2 - 10*a*c + (3*b^3)/Sqrt[b^2 - 4*a* 
c] - (16*a*b*c)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqr 
t[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (Sqrt[c]*(3*b^3 
- 16*a*b*c - (3*b^2 - 10*a*c)*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x 
)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b 
^2 - 4*a*c]]))/a)/(2*a*(b^2 - 4*a*c))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 218

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]
 

rule 1441

Int[((d_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] 
:> Simp[(-(d*x)^(m + 1))*(b^2 - 2*a*c + b*c*x^2)*((a + b*x^2 + c*x^4)^(p + 
1)/(2*a*d*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c)) 
  Int[(d*x)^m*(a + b*x^2 + c*x^4)^(p + 1)*Simp[b^2*(m + 2*p + 3) - 2*a*c*(m 
 + 4*p + 5) + b*c*(m + 4*p + 7)*x^2, x], x], x] /; FreeQ[{a, b, c, d, m}, x 
] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IntegerQ[2*p] && (IntegerQ[p] || 
IntegerQ[m])
 

rule 1480

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : 
> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q))   Int[1/( 
b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q))   Int[1/(b/2 
+ q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] 
 && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
 

rule 1604

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*( 
x_)^4)^(p_), x_Symbol] :> Simp[d*(f*x)^(m + 1)*((a + b*x^2 + c*x^4)^(p + 1) 
/(a*f*(m + 1))), x] + Simp[1/(a*f^2*(m + 1))   Int[(f*x)^(m + 2)*(a + b*x^2 
 + c*x^4)^p*Simp[a*e*(m + 1) - b*d*(m + 2*p + 3) - c*d*(m + 4*p + 5)*x^2, x 
], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[ 
m, -1] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])
 

rule 1949

Int[((b_.)*(x_)^(n_.) + (a_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(p_), x_Symbol 
] :> Int[x^(p*q)*(a + b*x^(n - q) + c*x^(2*(n - q)))^p, x] /; FreeQ[{a, b, 
c, n, q}, x] && EqQ[r, 2*n - q] && PosQ[n - q] && IntegerQ[p]
 

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 294, normalized size of antiderivative = 0.95

method	result	size
default	\(-\frac {\frac {\frac {c \left (2 a c -b^{2}\right ) x^{3}}{8 a c -2 b^{2}}+\frac {b \left (3 a c -b^{2}\right ) x}{8 a c -2 b^{2}}}{c \,x^{4}+b \,x^{2}+a}+\frac {2 c \left (-\frac {\left (10 a c \sqrt {-4 a c +b^{2}}-3 b^{2} \sqrt {-4 a c +b^{2}}+16 a b c -3 b^{3}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c x \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\left (10 a c \sqrt {-4 a c +b^{2}}-3 b^{2} \sqrt {-4 a c +b^{2}}-16 a b c +3 b^{3}\right ) \sqrt {2}\, \arctan \left (\frac {c x \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{4 a c -b^{2}}}{a^{2}}-\frac {1}{a^{2} x}\)	\(294\)
risch	\(\frac {-\frac {c \left (10 a c -3 b^{2}\right ) x^{4}}{2 a^{2} \left (4 a c -b^{2}\right )}-\frac {b \left (11 a c -3 b^{2}\right ) x^{2}}{2 \left (4 a c -b^{2}\right ) a^{2}}-\frac {1}{a}}{x \left (c \,x^{4}+b \,x^{2}+a \right )}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (4096 a^{11} c^{6}-6144 a^{10} b^{2} c^{5}+3840 a^{9} b^{4} c^{4}-1280 a^{8} b^{6} c^{3}+240 a^{7} b^{8} c^{2}-24 a^{6} b^{10} c +a^{5} b^{12}\right ) \textit {\_Z}^{4}+\left (26880 c^{6} a^{6} b -44800 c^{5} a^{5} b^{3}+30240 c^{4} a^{4} b^{5}-10656 c^{3} a^{3} b^{7}+2077 c^{2} a^{2} b^{9}-213 c a \,b^{11}+9 b^{13}\right ) \textit {\_Z}^{2}+10000 a^{2} c^{7}-4200 a \,b^{2} c^{6}+441 b^{4} c^{5}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (10240 a^{11} c^{6}-15872 a^{10} b^{2} c^{5}+10240 a^{9} b^{4} c^{4}-3520 a^{8} b^{6} c^{3}+680 a^{7} b^{8} c^{2}-70 a^{6} b^{10} c +3 a^{5} b^{12}\right ) \textit {\_R}^{4}+\left (57280 c^{6} a^{6} b -92752 c^{5} a^{5} b^{3}+61540 c^{4} a^{4} b^{5}-21471 c^{3} a^{3} b^{7}+4163 c^{2} a^{2} b^{9}-426 c a \,b^{11}+18 b^{13}\right ) \textit {\_R}^{2}+20000 a^{2} c^{7}-8400 a \,b^{2} c^{6}+882 b^{4} c^{5}\right ) x +\left (2560 a^{9} c^{6}-6656 a^{8} b^{2} c^{5}+5824 a^{7} b^{4} c^{4}-2464 a^{6} b^{6} c^{3}+554 a^{5} b^{8} c^{2}-64 a^{4} b^{10} c +3 a^{3} b^{12}\right ) \textit {\_R}^{3}+\left (1200 a^{4} b \,c^{6}-552 a^{3} b^{3} c^{5}+63 a^{2} b^{5} c^{4}\right ) \textit {\_R} \right )\right )}{4}\)	\(547\)

Input:

int(1/(c*x^5+b*x^3+a*x)^2,x,method=_RETURNVERBOSE)
 

Output:

-1/a^2*((1/2*c*(2*a*c-b^2)/(4*a*c-b^2)*x^3+1/2*b*(3*a*c-b^2)/(4*a*c-b^2)*x 
)/(c*x^4+b*x^2+a)+2/(4*a*c-b^2)*c*(-1/8*(10*a*c*(-4*a*c+b^2)^(1/2)-3*b^2*( 
-4*a*c+b^2)^(1/2)+16*a*b*c-3*b^3)/(-4*a*c+b^2)^(1/2)*2^(1/2)/((-b+(-4*a*c+ 
b^2)^(1/2))*c)^(1/2)*arctanh(c*x*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2) 
)+1/8*(10*a*c*(-4*a*c+b^2)^(1/2)-3*b^2*(-4*a*c+b^2)^(1/2)-16*a*b*c+3*b^3)/ 
(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^( 
1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))))-1/a^2/x
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2912 vs. \(2 (260) = 520\).

Time = 0.39 (sec) , antiderivative size = 2912, normalized size of antiderivative = 9.45 \[ \int \frac {1}{\left (a x+b x^3+c x^5\right )^2} \, dx=\text {Too large to display} \] Input:

integrate(1/(c*x^5+b*x^3+a*x)^2,x, algorithm="fricas")
 

Output:

-1/4*(2*(3*b^2*c - 10*a*c^2)*x^4 + 4*a*b^2 - 16*a^2*c + 2*(3*b^3 - 11*a*b* 
c)*x^2 - sqrt(1/2)*((a^2*b^2*c - 4*a^3*c^2)*x^5 + (a^2*b^3 - 4*a^3*b*c)*x^ 
3 + (a^3*b^2 - 4*a^4*c)*x)*sqrt(-(9*b^7 - 105*a*b^5*c + 385*a^2*b^3*c^2 - 
420*a^3*b*c^3 + (a^5*b^6 - 12*a^6*b^4*c + 48*a^7*b^2*c^2 - 64*a^8*c^3)*sqr 
t((81*b^8 - 918*a*b^6*c + 3051*a^2*b^4*c^2 - 2550*a^3*b^2*c^3 + 625*a^4*c^ 
4)/(a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)))/(a^5*b^6 - 
 12*a^6*b^4*c + 48*a^7*b^2*c^2 - 64*a^8*c^3))*log(-(189*b^6*c^3 - 1971*a*b 
^4*c^4 + 5625*a^2*b^2*c^5 - 2500*a^3*c^6)*x + 1/2*sqrt(1/2)*(27*b^11 - 486 
*a*b^9*c + 3330*a^2*b^7*c^2 - 10549*a^3*b^5*c^3 + 14408*a^4*b^3*c^4 - 5200 
*a^5*b*c^5 - (3*a^5*b^10 - 55*a^6*b^8*c + 392*a^7*b^6*c^2 - 1344*a^8*b^4*c 
^3 + 2176*a^9*b^2*c^4 - 1280*a^10*c^5)*sqrt((81*b^8 - 918*a*b^6*c + 3051*a 
^2*b^4*c^2 - 2550*a^3*b^2*c^3 + 625*a^4*c^4)/(a^10*b^6 - 12*a^11*b^4*c + 4 
8*a^12*b^2*c^2 - 64*a^13*c^3)))*sqrt(-(9*b^7 - 105*a*b^5*c + 385*a^2*b^3*c 
^2 - 420*a^3*b*c^3 + (a^5*b^6 - 12*a^6*b^4*c + 48*a^7*b^2*c^2 - 64*a^8*c^3 
)*sqrt((81*b^8 - 918*a*b^6*c + 3051*a^2*b^4*c^2 - 2550*a^3*b^2*c^3 + 625*a 
^4*c^4)/(a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)))/(a^5* 
b^6 - 12*a^6*b^4*c + 48*a^7*b^2*c^2 - 64*a^8*c^3))) + sqrt(1/2)*((a^2*b^2* 
c - 4*a^3*c^2)*x^5 + (a^2*b^3 - 4*a^3*b*c)*x^3 + (a^3*b^2 - 4*a^4*c)*x)*sq 
rt(-(9*b^7 - 105*a*b^5*c + 385*a^2*b^3*c^2 - 420*a^3*b*c^3 + (a^5*b^6 - 12 
*a^6*b^4*c + 48*a^7*b^2*c^2 - 64*a^8*c^3)*sqrt((81*b^8 - 918*a*b^6*c + ...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{\left (a x+b x^3+c x^5\right )^2} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F]

\[ \int \frac {1}{\left (a x+b x^3+c x^5\right )^2} \, dx=\int { \frac {1}{{\left (c x^{5} + b x^{3} + a x\right )}^{2}} \,d x } \] Input:

integrate(1/(c*x^5+b*x^3+a*x)^2,x, algorithm="maxima")
 

Output:

-1/2*((3*b^2*c - 10*a*c^2)*x^4 + 2*a*b^2 - 8*a^2*c + (3*b^3 - 11*a*b*c)*x^ 
2)/((a^2*b^2*c - 4*a^3*c^2)*x^5 + (a^2*b^3 - 4*a^3*b*c)*x^3 + (a^3*b^2 - 4 
*a^4*c)*x) + 1/2*integrate(-(3*b^3 - 13*a*b*c + (3*b^2*c - 10*a*c^2)*x^2)/ 
(c*x^4 + b*x^2 + a), x)/(a^2*b^2 - 4*a^3*c)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3087 vs. \(2 (260) = 520\).

Time = 0.48 (sec) , antiderivative size = 3087, normalized size of antiderivative = 10.02 \[ \int \frac {1}{\left (a x+b x^3+c x^5\right )^2} \, dx=\text {Too large to display} \] Input:

integrate(1/(c*x^5+b*x^3+a*x)^2,x, algorithm="giac")
 

Output:

-1/2*(3*b^2*c*x^4 - 10*a*c^2*x^4 + 3*b^3*x^2 - 11*a*b*c*x^2 + 2*a*b^2 - 8* 
a^2*c)/((c*x^5 + b*x^3 + a*x)*(a^2*b^2 - 4*a^3*c)) + 1/16*(6*a^4*b^8*c^2 - 
 80*a^5*b^6*c^3 + 352*a^6*b^4*c^4 - 512*a^7*b^2*c^5 - 3*sqrt(2)*sqrt(b^2 - 
 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^8 + 40*sqrt(2)*sqrt(b^2 - 4* 
a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*b^6*c + 6*sqrt(2)*sqrt(b^2 - 4*a* 
c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^7*c - 176*sqrt(2)*sqrt(b^2 - 4*a* 
c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^6*b^4*c^2 - 56*sqrt(2)*sqrt(b^2 - 4*a 
*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*b^5*c^2 - 3*sqrt(2)*sqrt(b^2 - 4*a 
*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^6*c^2 + 256*sqrt(2)*sqrt(b^2 - 4 
*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^7*b^2*c^3 + 128*sqrt(2)*sqrt(b^2 - 
 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^6*b^3*c^3 + 28*sqrt(2)*sqrt(b^2 
- 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*b^4*c^3 - 64*sqrt(2)*sqrt(b^2 
 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^6*b^2*c^4 - 6*(b^2 - 4*a*c)*a^ 
4*b^6*c^2 + 56*(b^2 - 4*a*c)*a^5*b^4*c^3 - 128*(b^2 - 4*a*c)*a^6*b^2*c^4 + 
 (6*b^4*c^2 - 44*a*b^2*c^3 + 80*a^2*c^4 - 3*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt 
(b*c + sqrt(b^2 - 4*a*c)*c)*b^4 + 22*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + 
sqrt(b^2 - 4*a*c)*c)*a*b^2*c + 6*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt 
(b^2 - 4*a*c)*c)*b^3*c - 40*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 
- 4*a*c)*c)*a^2*c^2 - 20*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4 
*a*c)*c)*a*b*c^2 - 3*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*...
 

Mupad [B] (verification not implemented)

Time = 13.83 (sec) , antiderivative size = 7555, normalized size of antiderivative = 24.53 \[ \int \frac {1}{\left (a x+b x^3+c x^5\right )^2} \, dx=\text {Too large to display} \] Input:

int(1/(a*x + b*x^3 + c*x^5)^2,x)
 

Output:

- atan((((-(9*b^13 - 9*b^4*(-(4*a*c - b^2)^9)^(1/2) + 26880*a^6*b*c^6 + 20 
77*a^2*b^9*c^2 - 10656*a^3*b^7*c^3 + 30240*a^4*b^5*c^4 - 44800*a^5*b^3*c^5 
 - 25*a^2*c^2*(-(4*a*c - b^2)^9)^(1/2) - 213*a*b^11*c + 51*a*b^2*c*(-(4*a* 
c - b^2)^9)^(1/2))/(32*(a^5*b^12 + 4096*a^11*c^6 - 24*a^6*b^10*c + 240*a^7 
*b^8*c^2 - 1280*a^8*b^6*c^3 + 3840*a^9*b^4*c^4 - 6144*a^10*b^2*c^5)))^(1/2 
)*(851968*a^14*b*c^8 + 192*a^8*b^13*c^2 - 4672*a^9*b^11*c^3 + 47360*a^10*b 
^9*c^4 - 256000*a^11*b^7*c^5 + 778240*a^12*b^5*c^6 - 1261568*a^13*b^3*c^7 
+ x*(-(9*b^13 - 9*b^4*(-(4*a*c - b^2)^9)^(1/2) + 26880*a^6*b*c^6 + 2077*a^ 
2*b^9*c^2 - 10656*a^3*b^7*c^3 + 30240*a^4*b^5*c^4 - 44800*a^5*b^3*c^5 - 25 
*a^2*c^2*(-(4*a*c - b^2)^9)^(1/2) - 213*a*b^11*c + 51*a*b^2*c*(-(4*a*c - b 
^2)^9)^(1/2))/(32*(a^5*b^12 + 4096*a^11*c^6 - 24*a^6*b^10*c + 240*a^7*b^8* 
c^2 - 1280*a^8*b^6*c^3 + 3840*a^9*b^4*c^4 - 6144*a^10*b^2*c^5)))^(1/2)*(10 
48576*a^16*b*c^8 + 256*a^10*b^13*c^2 - 6144*a^11*b^11*c^3 + 61440*a^12*b^9 
*c^4 - 327680*a^13*b^7*c^5 + 983040*a^14*b^5*c^6 - 1572864*a^15*b^3*c^7)) 
+ x*(204800*a^12*c^9 + 144*a^6*b^12*c^3 - 3264*a^7*b^10*c^4 + 30112*a^8*b^ 
8*c^5 - 143360*a^9*b^6*c^6 + 365568*a^10*b^4*c^7 - 458752*a^11*b^2*c^8))*( 
-(9*b^13 - 9*b^4*(-(4*a*c - b^2)^9)^(1/2) + 26880*a^6*b*c^6 + 2077*a^2*b^9 
*c^2 - 10656*a^3*b^7*c^3 + 30240*a^4*b^5*c^4 - 44800*a^5*b^3*c^5 - 25*a^2* 
c^2*(-(4*a*c - b^2)^9)^(1/2) - 213*a*b^11*c + 51*a*b^2*c*(-(4*a*c - b^2)^9 
)^(1/2))/(32*(a^5*b^12 + 4096*a^11*c^6 - 24*a^6*b^10*c + 240*a^7*b^8*c^...
 

Reduce [B] (verification not implemented)

Time = 1.22 (sec) , antiderivative size = 3104, normalized size of antiderivative = 10.08 \[ \int \frac {1}{\left (a x+b x^3+c x^5\right )^2} \, dx =\text {Too large to display} \] Input:

int(1/(c*x^5+b*x^3+a*x)^2,x)
 

Output:

(40*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))*a**3*c**2*x - 38*sqrt(a)*sqrt( 
2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sq 
rt(2*sqrt(c)*sqrt(a) + b))*a**2*b**2*c*x + 40*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)*sq 
rt(a) + b))*a**2*b*c**2*x**3 + 40*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))* 
a**2*c**3*x**5 + 6*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))*a*b**4*x - 38*s 
qrt(a)*sqrt(2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*s 
qrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b))*a*b**3*c*x**3 - 38*sqrt(a)*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))*a*b**2*c**2*x**5 + 6*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)*sqr 
t(a) + b))*b**5*x**3 + 6*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*c*x* 
*5 + 32*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**3*b*c*x - 6*sqrt(c)*sqr 
t(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*x + 32*sqrt(c)*sqrt(2*sqrt(c)*sq...