\(\int \frac {1}{x^2 (a x+b x^3+c x^5)^2} \, dx\) [50]

Optimal result

Integrand size = 20, antiderivative size = 361 \[ \int \frac {1}{x^2 \left (a x+b x^3+c x^5\right )^2} \, dx=-\frac {5 b^2-14 a c}{6 a^2 \left (b^2-4 a c\right ) x^3}+\frac {b \left (5 b^2-19 a c\right )}{2 a^3 \left (b^2-4 a c\right ) x}+\frac {b^2-2 a c+b c x^2}{2 a \left (b^2-4 a c\right ) x^3 \left (a+b x^2+c x^4\right )}+\frac {\sqrt {c} \left (5 b^4-29 a b^2 c+28 a^2 c^2+b \left (5 b^2-19 a c\right ) \sqrt {b^2-4 a c}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a^3 \left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {c} \left (5 b^4-29 a b^2 c+28 a^2 c^2-b \left (5 b^2-19 a c\right ) \sqrt {b^2-4 a c}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a^3 \left (b^2-4 a c\right )^{3/2} \sqrt {b+\sqrt {b^2-4 a c}}} \] Output:

-1/6*(-14*a*c+5*b^2)/a^2/(-4*a*c+b^2)/x^3+1/2*b*(-19*a*c+5*b^2)/a^3/(-4*a* 
c+b^2)/x+1/2*(b*c*x^2-2*a*c+b^2)/a/(-4*a*c+b^2)/x^3/(c*x^4+b*x^2+a)+1/4*c^ 
(1/2)*(5*b^4-29*a*b^2*c+28*c^2*a^2+b*(-19*a*c+5*b^2)*(-4*a*c+b^2)^(1/2))*a 
rctan(2^(1/2)*c^(1/2)*x/(b-(-4*a*c+b^2)^(1/2))^(1/2))*2^(1/2)/a^3/(-4*a*c+ 
b^2)^(3/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2)-1/4*c^(1/2)*(5*b^4-29*a*b^2*c+28*c 
^2*a^2-b*(-19*a*c+5*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^3/(-4*a*c+b^2)^(3/2)/(b+(-4*a*c+b^2)^( 
1/2))^(1/2)
 

Mathematica [A] (verified)

Time = 0.40 (sec) , antiderivative size = 344, normalized size of antiderivative = 0.95 \[ \int \frac {1}{x^2 \left (a x+b x^3+c x^5\right )^2} \, dx=\frac {-\frac {4 a}{x^3}+\frac {24 b}{x}+\frac {6 x \left (b^4-4 a b^2 c+2 a^2 c^2+b^3 c x^2-3 a b c^2 x^2\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {3 \sqrt {2} \sqrt {c} \left (5 b^4-29 a b^2 c+28 a^2 c^2+5 b^3 \sqrt {b^2-4 a c}-19 a b c \sqrt {b^2-4 a c}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {3 \sqrt {2} \sqrt {c} \left (-5 b^4+29 a b^2 c-28 a^2 c^2+5 b^3 \sqrt {b^2-4 a c}-19 a b c \sqrt {b^2-4 a c}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt {b+\sqrt {b^2-4 a c}}}}{12 a^3} \] Input:

Integrate[1/(x^2*(a*x + b*x^3 + c*x^5)^2),x]
 

Output:

((-4*a)/x^3 + (24*b)/x + (6*x*(b^4 - 4*a*b^2*c + 2*a^2*c^2 + b^3*c*x^2 - 3 
*a*b*c^2*x^2))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (3*Sqrt[2]*Sqrt[c]*(5 
*b^4 - 29*a*b^2*c + 28*a^2*c^2 + 5*b^3*Sqrt[b^2 - 4*a*c] - 19*a*b*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]]) + (3*Sqrt[2]*Sqrt[c]*(-5*b^4 
 + 29*a*b^2*c - 28*a^2*c^2 + 5*b^3*Sqrt[b^2 - 4*a*c] - 19*a*b*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]]))/(12*a^3)
 

Rubi [A] (verified)

Time = 1.15 (sec) , antiderivative size = 358, normalized size of antiderivative = 0.99, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {9, 1441, 25, 1604, 27, 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.

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

\(\Big \downarrow \) 9

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

\(\Big \downarrow \) 1441

\(\displaystyle \frac {-2 a c+b^2+b c x^2}{2 a x^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\int -\frac {5 b^2+5 c x^2 b-14 a c}{x^4 \left (c x^4+b x^2+a\right )}dx}{2 a \left (b^2-4 a c\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {5 b^2+5 c x^2 b-14 a c}{x^4 \left (c x^4+b x^2+a\right )}dx}{2 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x^2}{2 a x^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\)

\(\Big \downarrow \) 1604

\(\displaystyle \frac {-\frac {\int \frac {3 \left (c \left (5 b^2-14 a c\right ) x^2+b \left (5 b^2-19 a c\right )\right )}{x^2 \left (c x^4+b x^2+a\right )}dx}{3 a}-\frac {5 b^2-14 a c}{3 a x^3}}{2 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x^2}{2 a x^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {\int \frac {c \left (5 b^2-14 a c\right ) x^2+b \left (5 b^2-19 a c\right )}{x^2 \left (c x^4+b x^2+a\right )}dx}{a}-\frac {5 b^2-14 a c}{3 a x^3}}{2 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x^2}{2 a x^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\)

\(\Big \downarrow \) 1604

\(\displaystyle \frac {-\frac {-\frac {\int \frac {5 b^4-24 a c b^2+c \left (5 b^2-19 a c\right ) x^2 b+14 a^2 c^2}{c x^4+b x^2+a}dx}{a}-\frac {b \left (5 b^2-19 a c\right )}{a x}}{a}-\frac {5 b^2-14 a c}{3 a x^3}}{2 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x^2}{2 a x^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\)

\(\Big \downarrow \) 1480

\(\displaystyle \frac {-\frac {-\frac {\frac {c \left (28 a^2 c^2-29 a b^2 c+b \left (5 b^2-19 a c\right ) \sqrt {b^2-4 a c}+5 b^4\right ) \int \frac {1}{c x^2+\frac {1}{2} \left (b-\sqrt {b^2-4 a c}\right )}dx}{2 \sqrt {b^2-4 a c}}-\frac {c \left (28 a^2 c^2-29 a b^2 c-b \left (5 b^2-19 a c\right ) \sqrt {b^2-4 a c}+5 b^4\right ) \int \frac {1}{c x^2+\frac {1}{2} \left (b+\sqrt {b^2-4 a c}\right )}dx}{2 \sqrt {b^2-4 a c}}}{a}-\frac {b \left (5 b^2-19 a c\right )}{a x}}{a}-\frac {5 b^2-14 a c}{3 a x^3}}{2 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x^2}{2 a x^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {-\frac {-\frac {\frac {\sqrt {c} \left (28 a^2 c^2-29 a b^2 c+b \left (5 b^2-19 a c\right ) \sqrt {b^2-4 a c}+5 b^4\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} \sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {c} \left (28 a^2 c^2-29 a b^2 c-b \left (5 b^2-19 a c\right ) \sqrt {b^2-4 a c}+5 b^4\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {2} \sqrt {b^2-4 a c} \sqrt {\sqrt {b^2-4 a c}+b}}}{a}-\frac {b \left (5 b^2-19 a c\right )}{a x}}{a}-\frac {5 b^2-14 a c}{3 a x^3}}{2 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x^2}{2 a x^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\)

Input:

Int[1/(x^2*(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^3*(a + b*x^2 + c*x^4)) + (-1/ 
3*(5*b^2 - 14*a*c)/(a*x^3) - (-((b*(5*b^2 - 19*a*c))/(a*x)) - ((Sqrt[c]*(5 
*b^4 - 29*a*b^2*c + 28*a^2*c^2 + b*(5*b^2 - 19*a*c)*Sqrt[b^2 - 4*a*c])*Arc 
Tan[(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]]) - (Sqrt[c]*(5*b^4 - 29*a*b^2*c + 28*a^ 
2*c^2 - b*(5*b^2 - 19*a*c)*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/S 
qrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 
- 4*a*c]]))/a)/a)/(2*a*(b^2 - 4*a*c))
 

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 25

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

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 334, normalized size of antiderivative = 0.93

method	result	size
default	\(-\frac {\frac {-\frac {b c \left (3 a c -b^{2}\right ) x^{3}}{2 \left (4 a c -b^{2}\right )}+\frac {\left (2 a^{2} c^{2}-4 a \,b^{2} c +b^{4}\right ) x}{8 a c -2 b^{2}}}{c \,x^{4}+b \,x^{2}+a}+\frac {2 c \left (-\frac {\left (-19 a b c \sqrt {-4 a c +b^{2}}+5 b^{3} \sqrt {-4 a c +b^{2}}+28 a^{2} c^{2}-29 a \,b^{2} c +5 b^{4}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c x \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\left (-19 a b c \sqrt {-4 a c +b^{2}}+5 b^{3} \sqrt {-4 a c +b^{2}}-28 a^{2} c^{2}+29 a \,b^{2} c -5 b^{4}\right ) \sqrt {2}\, \arctan \left (\frac {c x \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{4 a c -b^{2}}}{a^{3}}-\frac {1}{3 a^{2} x^{3}}+\frac {2 b}{a^{3} x}\)	\(334\)
risch	\(\frac {\frac {c b \left (19 a c -5 b^{2}\right ) x^{6}}{2 \left (4 a c -b^{2}\right ) a^{3}}-\frac {\left (14 a^{2} c^{2}-62 a \,b^{2} c +15 b^{4}\right ) x^{4}}{6 a^{3} \left (4 a c -b^{2}\right )}+\frac {5 b \,x^{2}}{3 a^{2}}-\frac {1}{3 a}}{x^{3} \left (c \,x^{4}+b \,x^{2}+a \right )}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (4096 a^{13} c^{6}-6144 a^{12} b^{2} c^{5}+3840 a^{11} b^{4} c^{4}-1280 a^{10} b^{6} c^{3}+240 a^{9} b^{8} c^{2}-24 a^{8} b^{10} c +a^{7} b^{12}\right ) \textit {\_Z}^{4}+\left (-80640 c^{7} a^{7} b +215040 c^{6} a^{6} b^{3}-219744 c^{5} a^{5} b^{5}+116928 c^{4} a^{4} b^{7}-35767 c^{3} a^{3} b^{9}+6366 c^{2} a^{2} b^{11}-615 c a \,b^{13}+25 b^{15}\right ) \textit {\_Z}^{2}+38416 a^{2} c^{9}-17640 a \,b^{2} c^{8}+2025 b^{4} c^{7}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (10240 a^{13} c^{6}-15872 a^{12} b^{2} c^{5}+10240 a^{11} b^{4} c^{4}-3520 a^{10} b^{6} c^{3}+680 a^{9} b^{8} c^{2}-70 a^{8} b^{10} c +3 a^{7} b^{12}\right ) \textit {\_R}^{4}+\left (-175168 c^{7} a^{7} b +451120 c^{6} a^{6} b^{3}-451644 c^{5} a^{5} b^{5}+237257 c^{4} a^{4} b^{7}-71999 c^{3} a^{3} b^{9}+12757 c^{2} a^{2} b^{11}-1230 c a \,b^{13}+50 b^{15}\right ) \textit {\_R}^{2}+76832 a^{2} c^{9}-35280 a \,b^{2} c^{8}+4050 b^{4} c^{7}\right ) x +\left (-8448 a^{10} b \,c^{6}+15872 a^{9} b^{3} c^{5}-11872 a^{8} b^{5} c^{4}+4592 a^{7} b^{7} c^{3}-977 a^{6} b^{9} c^{2}+109 a^{5} b^{11} c -5 a^{4} b^{13}\right ) \textit {\_R}^{3}+\left (10976 a^{5} c^{8}-9184 a^{4} b^{2} c^{7}+2510 a^{3} b^{4} c^{6}-225 a^{2} b^{6} c^{5}\right ) \textit {\_R} \right )\right )}{4}\)	\(600\)

Input:

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

Output:

-1/a^3*((-1/2*b*c*(3*a*c-b^2)/(4*a*c-b^2)*x^3+1/2*(2*a^2*c^2-4*a*b^2*c+b^4 
)/(4*a*c-b^2)*x)/(c*x^4+b*x^2+a)+2/(4*a*c-b^2)*c*(-1/8*(-19*a*b*c*(-4*a*c+ 
b^2)^(1/2)+5*b^3*(-4*a*c+b^2)^(1/2)+28*a^2*c^2-29*a*b^2*c+5*b^4)/(-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*(-19*a*b*c*(-4*a*c+b^2)^(1/2)+5*b^3*( 
-4*a*c+b^2)^(1/2)-28*a^2*c^2+29*a*b^2*c-5*b^4)/(-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))))-1/3/a^2/x^3+2/a^3*b/x
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3435 vs. \(2 (311) = 622\).

Time = 0.51 (sec) , antiderivative size = 3435, normalized size of antiderivative = 9.52 \[ \int \frac {1}{x^2 \left (a x+b x^3+c x^5\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

1/6*(3*(5*b^3*c - 19*a*b*c^2)*x^6 + (15*b^4 - 62*a*b^2*c + 14*a^2*c^2)*x^4 
 - 2*a^2*b^2 + 8*a^3*c + 10*(a*b^3 - 4*a^2*b*c)*x^2)/((a^3*b^2*c - 4*a^4*c 
^2)*x^7 + (a^3*b^3 - 4*a^4*b*c)*x^5 + (a^4*b^2 - 4*a^5*c)*x^3) - 1/2*integ 
rate(-(5*b^4 - 24*a*b^2*c + 14*a^2*c^2 + (5*b^3*c - 19*a*b*c^2)*x^2)/(c*x^ 
4 + b*x^2 + a), x)/(a^3*b^2 - 4*a^4*c)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3651 vs. \(2 (311) = 622\).

Time = 0.77 (sec) , antiderivative size = 3651, normalized size of antiderivative = 10.11 \[ \int \frac {1}{x^2 \left (a x+b x^3+c x^5\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

1/2*(b^3*c*x^3 - 3*a*b*c^2*x^3 + b^4*x - 4*a*b^2*c*x + 2*a^2*c^2*x)/((a^3* 
b^2 - 4*a^4*c)*(c*x^4 + b*x^2 + a)) + 1/16*(10*a^6*b^9*c^2 - 138*a^7*b^7*c 
^3 + 680*a^8*b^5*c^4 - 1376*a^9*b^3*c^5 + 896*a^10*b*c^6 - 5*sqrt(2)*sqrt( 
b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^6*b^9 + 69*sqrt(2)*sqrt(b^2 
 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^7*b^7*c + 10*sqrt(2)*sqrt(b^2 
- 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^6*b^8*c - 340*sqrt(2)*sqrt(b^2 
- 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^8*b^5*c^2 - 98*sqrt(2)*sqrt(b^2 
 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^7*b^6*c^2 - 5*sqrt(2)*sqrt(b^2 
 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^6*b^7*c^2 + 688*sqrt(2)*sqrt(b 
^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^9*b^3*c^3 + 288*sqrt(2)*sqrt 
(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^8*b^4*c^3 + 49*sqrt(2)*sqr 
t(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^7*b^5*c^3 - 448*sqrt(2)*s 
qrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^10*b*c^4 - 224*sqrt(2)* 
sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^9*b^2*c^4 - 144*sqrt(2 
)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^8*b^3*c^4 + 112*sqrt 
(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^9*b*c^5 - 10*(b^2 
- 4*a*c)*a^6*b^7*c^2 + 98*(b^2 - 4*a*c)*a^7*b^5*c^3 - 288*(b^2 - 4*a*c)*a^ 
8*b^3*c^4 + 224*(b^2 - 4*a*c)*a^9*b*c^5 + (10*b^5*c^2 - 78*a*b^3*c^3 + 152 
*a^2*b*c^4 - 5*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b 
^5 + 39*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3...
 

Mupad [B] (verification not implemented)

Time = 13.95 (sec) , antiderivative size = 8739, normalized size of antiderivative = 24.21 \[ \int \frac {1}{x^2 \left (a x+b x^3+c x^5\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

atan((((-(25*b^15 - 25*b^6*(-(4*a*c - b^2)^9)^(1/2) - 80640*a^7*b*c^7 + 63 
66*a^2*b^11*c^2 - 35767*a^3*b^9*c^3 + 116928*a^4*b^7*c^4 - 219744*a^5*b^5* 
c^5 + 215040*a^6*b^3*c^6 + 49*a^3*c^3*(-(4*a*c - b^2)^9)^(1/2) - 615*a*b^1 
3*c - 246*a^2*b^2*c^2*(-(4*a*c - b^2)^9)^(1/2) + 165*a*b^4*c*(-(4*a*c - b^ 
2)^9)^(1/2))/(32*(a^7*b^12 + 4096*a^13*c^6 - 24*a^8*b^10*c + 240*a^9*b^8*c 
^2 - 1280*a^10*b^6*c^3 + 3840*a^11*b^4*c^4 - 6144*a^12*b^2*c^5)))^(1/2)*(3 
20*a^12*b^14*c^2 - 917504*a^19*c^9 - 7936*a^13*b^12*c^3 + 82816*a^14*b^10* 
c^4 - 468480*a^15*b^8*c^5 + 1536000*a^16*b^6*c^6 - 2867200*a^17*b^4*c^7 + 
2719744*a^18*b^2*c^8 + x*(-(25*b^15 - 25*b^6*(-(4*a*c - b^2)^9)^(1/2) - 80 
640*a^7*b*c^7 + 6366*a^2*b^11*c^2 - 35767*a^3*b^9*c^3 + 116928*a^4*b^7*c^4 
 - 219744*a^5*b^5*c^5 + 215040*a^6*b^3*c^6 + 49*a^3*c^3*(-(4*a*c - b^2)^9) 
^(1/2) - 615*a*b^13*c - 246*a^2*b^2*c^2*(-(4*a*c - b^2)^9)^(1/2) + 165*a*b 
^4*c*(-(4*a*c - b^2)^9)^(1/2))/(32*(a^7*b^12 + 4096*a^13*c^6 - 24*a^8*b^10 
*c + 240*a^9*b^8*c^2 - 1280*a^10*b^6*c^3 + 3840*a^11*b^4*c^4 - 6144*a^12*b 
^2*c^5)))^(1/2)*(1048576*a^21*b*c^8 + 256*a^15*b^13*c^2 - 6144*a^16*b^11*c 
^3 + 61440*a^17*b^9*c^4 - 327680*a^18*b^7*c^5 + 983040*a^19*b^5*c^6 - 1572 
864*a^20*b^3*c^7)) - x*(401408*a^16*c^10 - 400*a^9*b^14*c^3 + 9440*a^10*b^ 
12*c^4 - 92816*a^11*b^10*c^5 + 488096*a^12*b^8*c^6 - 1458688*a^13*b^6*c^7 
+ 2401280*a^14*b^4*c^8 - 1871872*a^15*b^2*c^9))*(-(25*b^15 - 25*b^6*(-(4*a 
*c - b^2)^9)^(1/2) - 80640*a^7*b*c^7 + 6366*a^2*b^11*c^2 - 35767*a^3*b^...
 

Reduce [B] (verification not implemented)

Time = 1.85 (sec) , antiderivative size = 3826, normalized size of antiderivative = 10.60 \[ \int \frac {1}{x^2 \left (a x+b x^3+c x^5\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 312*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*b*c**2*x**3 + 204*sqr 
t(a)*sqrt(2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqr 
t(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b))*a**2*b**3*c*x**3 - 312*sqrt(a)*sqrt(2 
*sqrt(c)*sqrt(a) + b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqr 
t(2*sqrt(c)*sqrt(a) + b))*a**2*b**2*c**2*x**5 - 312*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*b*c**3*x**7 - 30*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**5*x**3 + 204*sqrt(a)*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*b**4*c*x 
**5 + 204*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**3*c**2*x**7 - 30*sq 
rt(a)*sqrt(2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sq 
rt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b))*b**6*x**5 - 30*sqrt(a)*sqrt(2*sqrt(c 
)*sqrt(a) + b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqr 
t(c)*sqrt(a) + b))*b**5*c*x**7 + 168*sqrt(c)*sqrt(2*sqrt(c)*sqrt(a) + b)*a 
tan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b 
))*a**4*c**2*x**3 - 174*sqrt(c)*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*...