\(\int \frac {1}{x (a+b x^2+c x^4)^3} \, dx\) [808]

Optimal result

Integrand size = 18, antiderivative size = 200 \[ \int \frac {1}{x \left (a+b x^2+c x^4\right )^3} \, dx=\frac {b^2-2 a c+b c x^2}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {2 b^4-15 a b^2 c+16 a^2 c^2+2 b c \left (b^2-7 a c\right ) x^2}{4 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {b \left (b^4-10 a b^2 c+30 a^2 c^2\right ) \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 a^3 \left (b^2-4 a c\right )^{5/2}}+\frac {\log (x)}{a^3}-\frac {\log \left (a+b x^2+c x^4\right )}{4 a^3} \] Output:

1/4*(b*c*x^2-2*a*c+b^2)/a/(-4*a*c+b^2)/(c*x^4+b*x^2+a)^2+1/4*(2*b^4-15*a*b 
^2*c+16*c^2*a^2+2*b*c*(-7*a*c+b^2)*x^2)/a^2/(-4*a*c+b^2)^2/(c*x^4+b*x^2+a) 
+1/2*b*(30*a^2*c^2-10*a*b^2*c+b^4)*arctanh((2*c*x^2+b)/(-4*a*c+b^2)^(1/2)) 
/a^3/(-4*a*c+b^2)^(5/2)+ln(x)/a^3-1/4*ln(c*x^4+b*x^2+a)/a^3
 

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.71 \[ \int \frac {1}{x \left (a+b x^2+c x^4\right )^3} \, dx=\frac {\frac {a^2 \left (b^2-2 a c+b c x^2\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {a \left (2 b^4-15 a b^2 c+16 a^2 c^2+2 b^3 c x^2-14 a b c^2 x^2\right )}{\left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+4 \log (x)-\frac {\left (b^5-10 a b^3 c+30 a^2 b c^2+b^4 \sqrt {b^2-4 a c}-8 a b^2 c \sqrt {b^2-4 a c}+16 a^2 c^2 \sqrt {b^2-4 a c}\right ) \log \left (b-\sqrt {b^2-4 a c}+2 c x^2\right )}{\left (b^2-4 a c\right )^{5/2}}+\frac {\left (b^5-10 a b^3 c+30 a^2 b c^2-b^4 \sqrt {b^2-4 a c}+8 a b^2 c \sqrt {b^2-4 a c}-16 a^2 c^2 \sqrt {b^2-4 a c}\right ) \log \left (b+\sqrt {b^2-4 a c}+2 c x^2\right )}{\left (b^2-4 a c\right )^{5/2}}}{4 a^3} \] Input:

Integrate[1/(x*(a + b*x^2 + c*x^4)^3),x]
 

Output:

((a^2*(b^2 - 2*a*c + b*c*x^2))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (a* 
(2*b^4 - 15*a*b^2*c + 16*a^2*c^2 + 2*b^3*c*x^2 - 14*a*b*c^2*x^2))/((b^2 - 
4*a*c)^2*(a + b*x^2 + c*x^4)) + 4*Log[x] - ((b^5 - 10*a*b^3*c + 30*a^2*b*c 
^2 + b^4*Sqrt[b^2 - 4*a*c] - 8*a*b^2*c*Sqrt[b^2 - 4*a*c] + 16*a^2*c^2*Sqrt 
[b^2 - 4*a*c])*Log[b - Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(5/2) + 
 ((b^5 - 10*a*b^3*c + 30*a^2*b*c^2 - b^4*Sqrt[b^2 - 4*a*c] + 8*a*b^2*c*Sqr 
t[b^2 - 4*a*c] - 16*a^2*c^2*Sqrt[b^2 - 4*a*c])*Log[b + Sqrt[b^2 - 4*a*c] + 
 2*c*x^2])/(b^2 - 4*a*c)^(5/2))/(4*a^3)
 

Rubi [A] (verified)

Time = 0.89 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.27, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {1434, 1165, 25, 1235, 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.

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

\(\Big \downarrow \) 1434

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

\(\Big \downarrow \) 1165

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1235

\(\displaystyle \frac {1}{2} \left (\frac {\frac {16 a^2 c^2+2 b c x^2 \left (b^2-7 a c\right )-15 a b^2 c+2 b^4}{a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\int -\frac {2 \left (\left (b^2-4 a c\right )^2+b c \left (b^2-7 a c\right ) x^2\right )}{x^2 \left (c x^4+b x^2+a\right )}dx^2}{a \left (b^2-4 a c\right )}}{2 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x^2}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \left (\frac {\frac {2 \int \frac {\left (b^2-4 a c\right )^2+b c \left (b^2-7 a c\right ) x^2}{x^2 \left (c x^4+b x^2+a\right )}dx^2}{a \left (b^2-4 a c\right )}+\frac {16 a^2 c^2+2 b c x^2 \left (b^2-7 a c\right )-15 a b^2 c+2 b^4}{a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{2 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x^2}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\)

\(\Big \downarrow \) 1200

\(\displaystyle \frac {1}{2} \left (\frac {\frac {2 \int \left (\frac {\left (4 a c-b^2\right )^2}{a x^2}+\frac {-c \left (b^2-4 a c\right )^2 x^2-b \left (b^4-9 a c b^2+23 a^2 c^2\right )}{a \left (c x^4+b x^2+a\right )}\right )dx^2}{a \left (b^2-4 a c\right )}+\frac {16 a^2 c^2+2 b c x^2 \left (b^2-7 a c\right )-15 a b^2 c+2 b^4}{a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{2 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x^2}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} \left (\frac {\frac {2 \left (\frac {b \left (30 a^2 c^2-10 a b^2 c+b^4\right ) \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{a \sqrt {b^2-4 a c}}+\frac {\log \left (x^2\right ) \left (b^2-4 a c\right )^2}{a}-\frac {\left (b^2-4 a c\right )^2 \log \left (a+b x^2+c x^4\right )}{2 a}\right )}{a \left (b^2-4 a c\right )}+\frac {16 a^2 c^2+2 b c x^2 \left (b^2-7 a c\right )-15 a b^2 c+2 b^4}{a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{2 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x^2}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\)

Input:

Int[1/(x*(a + b*x^2 + c*x^4)^3),x]
 

Output:

((b^2 - 2*a*c + b*c*x^2)/(2*a*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + ((2*b 
^4 - 15*a*b^2*c + 16*a^2*c^2 + 2*b*c*(b^2 - 7*a*c)*x^2)/(a*(b^2 - 4*a*c)*( 
a + b*x^2 + c*x^4)) + (2*((b*(b^4 - 10*a*b^2*c + 30*a^2*c^2)*ArcTanh[(b + 
2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(a*Sqrt[b^2 - 4*a*c]) + ((b^2 - 4*a*c)^2*Log[ 
x^2])/a - ((b^2 - 4*a*c)^2*Log[a + b*x^2 + c*x^4])/(2*a)))/(a*(b^2 - 4*a*c 
)))/(2*a*(b^2 - 4*a*c)))/2
 

Defintions of rubi rules used

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 1165

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(d + e*x)^(m + 1)*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e) 
*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^ 
2))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))   Int[(d 
+ e*x)^m*Simp[b*c*d*e*(2*p - m + 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p 
+ 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x, x]*(a + 
 b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && LtQ[p, -1] 
 && IntQuadraticQ[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 1235

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

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 = 0.21 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.80

method	result	size
default	\(-\frac {\frac {\frac {a b \,c^{2} \left (7 a c -b^{2}\right ) x^{6}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}-\frac {c a \left (16 a^{2} c^{2}-29 a \,b^{2} c +4 b^{4}\right ) x^{4}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {a b \left (a^{2} c^{2}+6 a \,b^{2} c -b^{4}\right ) x^{2}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}-\frac {3 a^{2} \left (8 a^{2} c^{2}-7 a \,b^{2} c +b^{4}\right )}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2}}+\frac {\frac {\left (16 a^{2} c^{3}-8 a \,b^{2} c^{2}+b^{4} c \right ) \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{2 c}+\frac {2 \left (23 a^{2} b \,c^{2}-9 a \,b^{3} c +b^{5}-\frac {\left (16 a^{2} c^{3}-8 a \,b^{2} c^{2}+b^{4} c \right ) b}{2 c}\right ) \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}}{2 a^{3}}+\frac {\ln \left (x \right )}{a^{3}}\)	\(360\)
risch	\(\frac {-\frac {b \,c^{2} \left (7 a c -b^{2}\right ) x^{6}}{2 a^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {c \left (16 a^{2} c^{2}-29 a \,b^{2} c +4 b^{4}\right ) x^{4}}{4 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a^{2}}-\frac {b \left (a^{2} c^{2}+6 a \,b^{2} c -b^{4}\right ) x^{2}}{2 a^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {6 a^{2} c^{2}-\frac {21}{4} a \,b^{2} c +\frac {3}{4} b^{4}}{a \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2}}+\frac {\ln \left (x \right )}{a^{3}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (1024 a^{8} c^{5}-1280 a^{7} b^{2} c^{4}+640 c^{3} b^{4} a^{6}-160 a^{5} b^{6} c^{2}+20 a^{4} b^{8} c -b^{10} a^{3}\right ) \textit {\_Z}^{2}+\left (1024 c^{5} a^{5}-1280 a^{4} b^{2} c^{4}+640 a^{3} b^{4} c^{3}-160 a^{2} b^{6} c^{2}+20 a \,b^{8} c -b^{10}\right ) \textit {\_Z} +256 a^{2} c^{5}-95 a \,b^{2} c^{4}+10 b^{4} c^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (2560 a^{9} c^{5}-3328 c^{4} b^{2} a^{8}+1728 a^{7} b^{4} c^{3}-448 a^{6} b^{6} c^{2}+58 a^{5} b^{8} c -3 a^{4} b^{10}\right ) \textit {\_R}^{2}+\left (1280 a^{6} c^{5}-1168 c^{4} b^{2} a^{5}+408 a^{4} b^{4} c^{3}-65 a^{3} b^{6} c^{2}+4 a^{2} b^{8} c \right ) \textit {\_R} +98 a^{2} b^{2} c^{4}-28 a \,b^{4} c^{3}+2 b^{6} c^{2}\right ) x^{2}+\left (-256 a^{9} b \,c^{4}+256 a^{8} b^{3} c^{3}-96 a^{7} b^{5} c^{2}+16 a^{6} b^{7} c -a^{5} b^{9}\right ) \textit {\_R}^{2}+\left (624 a^{6} b \,c^{4}-584 a^{5} b^{3} c^{3}+207 a^{4} b^{5} c^{2}-33 a^{3} b^{7} c +2 a^{2} b^{9}\right ) \textit {\_R} -224 a^{3} b \,c^{4}+144 a^{2} b^{3} c^{3}-30 a \,b^{5} c^{2}+2 b^{7} c \right )\right )}{2}\)	\(656\)

Input:

int(1/x/(c*x^4+b*x^2+a)^3,x,method=_RETURNVERBOSE)
 

Output:

-1/2/a^3*((a*b*c^2*(7*a*c-b^2)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^6-1/2*c*a*(16* 
a^2*c^2-29*a*b^2*c+4*b^4)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^4+a*b*(a^2*c^2+6*a* 
b^2*c-b^4)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2-3/2*a^2*(8*a^2*c^2-7*a*b^2*c+b^4 
)/(16*a^2*c^2-8*a*b^2*c+b^4))/(c*x^4+b*x^2+a)^2+1/(16*a^2*c^2-8*a*b^2*c+b^ 
4)*(1/2*(16*a^2*c^3-8*a*b^2*c^2+b^4*c)/c*ln(c*x^4+b*x^2+a)+2*(23*a^2*b*c^2 
-9*a*b^3*c+b^5-1/2*(16*a^2*c^3-8*a*b^2*c^2+b^4*c)*b/c)/(4*a*c-b^2)^(1/2)*a 
rctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))))+ln(x)/a^3
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 997 vs. \(2 (188) = 376\).

Time = 0.37 (sec) , antiderivative size = 2017, normalized size of antiderivative = 10.08 \[ \int \frac {1}{x \left (a+b x^2+c x^4\right )^3} \, dx=\text {Too large to display} \] Input:

integrate(1/x/(c*x^4+b*x^2+a)^3,x, algorithm="fricas")
 

Output:

[1/4*(3*a^2*b^6 - 33*a^3*b^4*c + 108*a^4*b^2*c^2 - 96*a^5*c^3 + 2*(a*b^5*c 
^2 - 11*a^2*b^3*c^3 + 28*a^3*b*c^4)*x^6 + (4*a*b^6*c - 45*a^2*b^4*c^2 + 13 
2*a^3*b^2*c^3 - 64*a^4*c^4)*x^4 + 2*(a*b^7 - 10*a^2*b^5*c + 23*a^3*b^3*c^2 
 + 4*a^4*b*c^3)*x^2 + ((b^5*c^2 - 10*a*b^3*c^3 + 30*a^2*b*c^4)*x^8 + a^2*b 
^5 - 10*a^3*b^3*c + 30*a^4*b*c^2 + 2*(b^6*c - 10*a*b^4*c^2 + 30*a^2*b^2*c^ 
3)*x^6 + (b^7 - 8*a*b^5*c + 10*a^2*b^3*c^2 + 60*a^3*b*c^3)*x^4 + 2*(a*b^6 
- 10*a^2*b^4*c + 30*a^3*b^2*c^2)*x^2)*sqrt(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)) - ((b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*x^8 + a^2*b 
^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3 + 2*(b^7*c - 12*a*b^5*c^2 
+ 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*x^6 + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 
+ 32*a^3*b^2*c^3 - 128*a^4*c^4)*x^4 + 2*(a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3 
*c^2 - 64*a^4*b*c^3)*x^2)*log(c*x^4 + b*x^2 + a) + 4*((b^6*c^2 - 12*a*b^4* 
c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*x^8 + a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b 
^2*c^2 - 64*a^5*c^3 + 2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b* 
c^4)*x^6 + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c 
^4)*x^4 + 2*(a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^4*b*c^3)*x^2)*lo 
g(x))/(a^5*b^6 - 12*a^6*b^4*c + 48*a^7*b^2*c^2 - 64*a^8*c^3 + (a^3*b^6*c^2 
 - 12*a^4*b^4*c^3 + 48*a^5*b^2*c^4 - 64*a^6*c^5)*x^8 + 2*(a^3*b^7*c - 12*a 
^4*b^5*c^2 + 48*a^5*b^3*c^3 - 64*a^6*b*c^4)*x^6 + (a^3*b^8 - 10*a^4*b^6...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{x \left (a+b x^2+c x^4\right )^3} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{x \left (a+b x^2+c x^4\right )^3} \, dx=\text {Exception raised: ValueError} \] Input:

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

Giac [A] (verification not implemented)

Time = 1.10 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.62 \[ \int \frac {1}{x \left (a+b x^2+c x^4\right )^3} \, dx=-\frac {{\left (b^{5} - 10 \, a b^{3} c + 30 \, a^{2} b c^{2}\right )} \arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{2 \, {\left (a^{3} b^{4} - 8 \, a^{4} b^{2} c + 16 \, a^{5} c^{2}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {3 \, b^{4} c^{2} x^{8} - 24 \, a b^{2} c^{3} x^{8} + 48 \, a^{2} c^{4} x^{8} + 6 \, b^{5} c x^{6} - 44 \, a b^{3} c^{2} x^{6} + 68 \, a^{2} b c^{3} x^{6} + 3 \, b^{6} x^{4} - 10 \, a b^{4} c x^{4} - 58 \, a^{2} b^{2} c^{2} x^{4} + 128 \, a^{3} c^{3} x^{4} + 10 \, a b^{5} x^{2} - 72 \, a^{2} b^{3} c x^{2} + 92 \, a^{3} b c^{2} x^{2} + 9 \, a^{2} b^{4} - 66 \, a^{3} b^{2} c + 96 \, a^{4} c^{2}}{8 \, {\left (a^{3} b^{4} - 8 \, a^{4} b^{2} c + 16 \, a^{5} c^{2}\right )} {\left (c x^{4} + b x^{2} + a\right )}^{2}} - \frac {\log \left (c x^{4} + b x^{2} + a\right )}{4 \, a^{3}} + \frac {\log \left (x^{2}\right )}{2 \, a^{3}} \] Input:

integrate(1/x/(c*x^4+b*x^2+a)^3,x, algorithm="giac")
 

Output:

-1/2*(b^5 - 10*a*b^3*c + 30*a^2*b*c^2)*arctan((2*c*x^2 + b)/sqrt(-b^2 + 4* 
a*c))/((a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*sqrt(-b^2 + 4*a*c)) + 1/8*(3*b 
^4*c^2*x^8 - 24*a*b^2*c^3*x^8 + 48*a^2*c^4*x^8 + 6*b^5*c*x^6 - 44*a*b^3*c^ 
2*x^6 + 68*a^2*b*c^3*x^6 + 3*b^6*x^4 - 10*a*b^4*c*x^4 - 58*a^2*b^2*c^2*x^4 
 + 128*a^3*c^3*x^4 + 10*a*b^5*x^2 - 72*a^2*b^3*c*x^2 + 92*a^3*b*c^2*x^2 + 
9*a^2*b^4 - 66*a^3*b^2*c + 96*a^4*c^2)/((a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^ 
2)*(c*x^4 + b*x^2 + a)^2) - 1/4*log(c*x^4 + b*x^2 + a)/a^3 + 1/2*log(x^2)/ 
a^3
 

Mupad [B] (verification not implemented)

Time = 25.32 (sec) , antiderivative size = 9339, normalized size of antiderivative = 46.70 \[ \int \frac {1}{x \left (a+b x^2+c x^4\right )^3} \, dx=\text {Too large to display} \] Input:

int(1/(x*(a + b*x^2 + c*x^4)^3),x)
                                                                                    
                                                                                    
 

Output:

log(x)/a^3 + ((3*(b^4 + 8*a^2*c^2 - 7*a*b^2*c))/(4*a*(b^4 + 16*a^2*c^2 - 8 
*a*b^2*c)) + (x^4*(4*b^4*c + 16*a^2*c^3 - 29*a*b^2*c^2))/(4*a^2*(b^4 + 16* 
a^2*c^2 - 8*a*b^2*c)) - (b*x^2*(a^2*c^2 - b^4 + 6*a*b^2*c))/(2*a^2*(b^4 + 
16*a^2*c^2 - 8*a*b^2*c)) - (b*c^2*x^6*(7*a*c - b^2))/(2*a^2*(b^4 + 16*a^2* 
c^2 - 8*a*b^2*c)))/(x^4*(2*a*c + b^2) + a^2 + c^2*x^8 + 2*a*b*x^2 + 2*b*c* 
x^6) - (log((((a^3*(-(b^2*(b^4 + 30*a^2*c^2 - 10*a*b^2*c)^2)/(a^6*(4*a*c - 
 b^2)^5))^(1/2) + 1)*((b^2*c^3*(4*b^6 - 497*a^3*c^3 + 302*a^2*b^2*c^2 - 61 
*a*b^4*c))/(a^4*(4*a*c - b^2)^4) - ((a^3*(-(b^2*(b^4 + 30*a^2*c^2 - 10*a*b 
^2*c)^2)/(a^6*(4*a*c - b^2)^5))^(1/2) + 1)*((4*b^2*c^2*(b^4 + 23*a^2*c^2 - 
 9*a*b^2*c))/(a^2*(4*a*c - b^2)^2) + (b*c^2*(a^3*(-(b^2*(b^4 + 30*a^2*c^2 
- 10*a*b^2*c)^2)/(a^6*(4*a*c - b^2)^5))^(1/2) + 1)*(a*b + 3*b^2*x^2 - 10*a 
*c*x^2))/a^3 + (2*b*c^3*x^2*(b^4 + 10*a^2*c^2 - 2*a*b^2*c))/(a^2*(4*a*c - 
b^2)^2)))/(4*a^3) + (b*c^4*x^2*(6*b^6 - 560*a^3*c^3 + 409*a^2*b^2*c^2 - 89 
*a*b^4*c))/(a^4*(4*a*c - b^2)^4)))/(4*a^3) - (b^2*c^4*(7*a*c - b^2)^2)/(a^ 
6*(4*a*c - b^2)^4) + (b^3*c^5*x^2*(7*a*c - b^2)^3)/(a^6*(4*a*c - b^2)^6))* 
(((a^3*(-(b^2*(b^4 + 30*a^2*c^2 - 10*a*b^2*c)^2)/(a^6*(4*a*c - b^2)^5))^(1 
/2) - 1)*(((a^3*(-(b^2*(b^4 + 30*a^2*c^2 - 10*a*b^2*c)^2)/(a^6*(4*a*c - b^ 
2)^5))^(1/2) - 1)*((4*b^2*c^2*(b^4 + 23*a^2*c^2 - 9*a*b^2*c))/(a^2*(4*a*c 
- b^2)^2) - (b*c^2*(a^3*(-(b^2*(b^4 + 30*a^2*c^2 - 10*a*b^2*c)^2)/(a^6*(4* 
a*c - b^2)^5))^(1/2) - 1)*(a*b + 3*b^2*x^2 - 10*a*c*x^2))/a^3 + (2*b*c^...
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 4090, normalized size of antiderivative = 20.45 \[ \int \frac {1}{x \left (a+b x^2+c x^4\right )^3} \, dx =\text {Too large to display} \] Input:

int(1/x/(c*x^4+b*x^2+a)^3,x)
 

Output:

(60*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**4*b*c** 
2 - 20*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**3*b* 
*3*c + 120*sqrt(2*sqrt(c)*sqrt(a) + b)*sqrt(2*sqrt(c)*sqrt(a) - b)*atan((s 
qrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b))*a** 
3*b**2*c**2*x**2 + 120*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**3*b*c**3*x**4 + 2*sqrt(2*sqrt(c)*sqrt(a) + b)*sqrt(2*sqrt(c)*s 
qrt(a) - b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c 
)*sqrt(a) + b))*a**2*b**5 - 40*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**4*c*x**2 + 20*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**2*x**4 + 120*sqrt(2*sqrt(c)*sqrt(a) 
+ b)*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**2*c**3*x**6 + 60*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**4*x**8 + 4*sqrt( 
2*sqrt(c)*sqrt(a) + b)*sqrt(2*sqrt(c)*sqrt(a) - b)*atan((sqrt(2*sqrt(c)...