\(\int \frac {1}{x^3 (a+b x^2+c x^4)^3} \, dx\) [809]

Optimal result

Integrand size = 18, antiderivative size = 255 \[ \int \frac {1}{x^3 \left (a+b x^2+c x^4\right )^3} \, dx=-\frac {3 \left (b^2-5 a c\right ) \left (b^2-2 a c\right )}{2 a^3 \left (b^2-4 a c\right )^2 x^2}+\frac {b^2-2 a c+b c x^2}{4 a \left (b^2-4 a c\right ) x^2 \left (a+b x^2+c x^4\right )^2}+\frac {3 b^4-20 a b^2 c+20 a^2 c^2+3 b c \left (b^2-6 a c\right ) x^2}{4 a^2 \left (b^2-4 a c\right )^2 x^2 \left (a+b x^2+c x^4\right )}-\frac {3 \left (b^6-10 a b^4 c+30 a^2 b^2 c^2-20 a^3 c^3\right ) \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 a^4 \left (b^2-4 a c\right )^{5/2}}-\frac {3 b \log (x)}{a^4}+\frac {3 b \log \left (a+b x^2+c x^4\right )}{4 a^4} \] Output:

-3/2*(-5*a*c+b^2)*(-2*a*c+b^2)/a^3/(-4*a*c+b^2)^2/x^2+1/4*(b*c*x^2-2*a*c+b 
^2)/a/(-4*a*c+b^2)/x^2/(c*x^4+b*x^2+a)^2+1/4*(3*b^4-20*a*b^2*c+20*c^2*a^2+ 
3*b*c*(-6*a*c+b^2)*x^2)/a^2/(-4*a*c+b^2)^2/x^2/(c*x^4+b*x^2+a)-3/2*(-20*a^ 
3*c^3+30*a^2*b^2*c^2-10*a*b^4*c+b^6)*arctanh((2*c*x^2+b)/(-4*a*c+b^2)^(1/2 
))/a^4/(-4*a*c+b^2)^(5/2)-3*b*ln(x)/a^4+3/4*b*ln(c*x^4+b*x^2+a)/a^4
 

Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.58 \[ \int \frac {1}{x^3 \left (a+b x^2+c x^4\right )^3} \, dx=\frac {-\frac {2 a}{x^2}+\frac {a^2 \left (b^3-3 a b c+b^2 c x^2-2 a c^2 x^2\right )}{\left (-b^2+4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {a \left (4 b^5-29 a b^3 c+46 a^2 b c^2+4 b^4 c x^2-26 a b^2 c^2 x^2+28 a^2 c^3 x^2\right )}{\left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-12 b \log (x)+\frac {3 \left (b^6-10 a b^4 c+30 a^2 b^2 c^2-20 a^3 c^3+b^5 \sqrt {b^2-4 a c}-8 a b^3 c \sqrt {b^2-4 a c}+16 a^2 b c^2 \sqrt {b^2-4 a c}\right ) \log \left (b-\sqrt {b^2-4 a c}+2 c x^2\right )}{\left (b^2-4 a c\right )^{5/2}}+\frac {3 \left (-b^6+10 a b^4 c-30 a^2 b^2 c^2+20 a^3 c^3+b^5 \sqrt {b^2-4 a c}-8 a b^3 c \sqrt {b^2-4 a c}+16 a^2 b c^2 \sqrt {b^2-4 a c}\right ) \log \left (b+\sqrt {b^2-4 a c}+2 c x^2\right )}{\left (b^2-4 a c\right )^{5/2}}}{4 a^4} \] Input:

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

Output:

((-2*a)/x^2 + (a^2*(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)^2) - (a*(4*b^5 - 29*a*b^3*c + 46*a^2*b*c^2 + 4*b^4 
*c*x^2 - 26*a*b^2*c^2*x^2 + 28*a^2*c^3*x^2))/((b^2 - 4*a*c)^2*(a + b*x^2 + 
 c*x^4)) - 12*b*Log[x] + (3*(b^6 - 10*a*b^4*c + 30*a^2*b^2*c^2 - 20*a^3*c^ 
3 + b^5*Sqrt[b^2 - 4*a*c] - 8*a*b^3*c*Sqrt[b^2 - 4*a*c] + 16*a^2*b*c^2*Sqr 
t[b^2 - 4*a*c])*Log[b - Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(5/2) 
+ (3*(-b^6 + 10*a*b^4*c - 30*a^2*b^2*c^2 + 20*a^3*c^3 + b^5*Sqrt[b^2 - 4*a 
*c] - 8*a*b^3*c*Sqrt[b^2 - 4*a*c] + 16*a^2*b*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^4)
 

Rubi [A] (verified)

Time = 1.01 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.17, 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.

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

\(\Big \downarrow \) 1434

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

\(\Big \downarrow \) 1165

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1235

\(\displaystyle \frac {1}{2} \left (\frac {\frac {20 a^2 c^2+3 b c x^2 \left (b^2-6 a c\right )-20 a b^2 c+3 b^4}{a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\int -\frac {6 \left (b c \left (b^2-6 a c\right ) x^2+\left (b^2-5 a c\right ) \left (b^2-2 a c\right )\right )}{x^4 \left (c x^4+b x^2+a\right )}dx^2}{a \left (b^2-4 a c\right )}}{2 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x^2}{2 a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \left (\frac {\frac {6 \int \frac {b c \left (b^2-6 a c\right ) x^2+\left (b^2-5 a c\right ) \left (b^2-2 a c\right )}{x^4 \left (c x^4+b x^2+a\right )}dx^2}{a \left (b^2-4 a c\right )}+\frac {20 a^2 c^2+3 b c x^2 \left (b^2-6 a c\right )-20 a b^2 c+3 b^4}{a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{2 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x^2}{2 a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\)

\(\Big \downarrow \) 1200

\(\displaystyle \frac {1}{2} \left (\frac {\frac {6 \int \left (-\frac {b \left (4 a c-b^2\right )^2}{a^2 x^2}+\frac {b^6-9 a c b^4+23 a^2 c^2 b^2+c \left (b^2-4 a c\right )^2 x^2 b-10 a^3 c^3}{a^2 \left (c x^4+b x^2+a\right )}+\frac {\left (b^2-5 a c\right ) \left (b^2-2 a c\right )}{a x^4}\right )dx^2}{a \left (b^2-4 a c\right )}+\frac {20 a^2 c^2+3 b c x^2 \left (b^2-6 a c\right )-20 a b^2 c+3 b^4}{a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{2 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x^2}{2 a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} \left (\frac {\frac {20 a^2 c^2+3 b c x^2 \left (b^2-6 a c\right )-20 a b^2 c+3 b^4}{a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {6 \left (-\frac {b \log \left (x^2\right ) \left (b^2-4 a c\right )^2}{a^2}+\frac {b \left (b^2-4 a c\right )^2 \log \left (a+b x^2+c x^4\right )}{2 a^2}-\frac {\left (-20 a^3 c^3+30 a^2 b^2 c^2-10 a b^4 c+b^6\right ) \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{a^2 \sqrt {b^2-4 a c}}-\frac {\left (b^2-5 a c\right ) \left (b^2-2 a c\right )}{a x^2}\right )}{a \left (b^2-4 a c\right )}}{2 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x^2}{2 a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\)

Input:

Int[1/(x^3*(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)*x^2*(a + b*x^2 + c*x^4)^2) + ( 
(3*b^4 - 20*a*b^2*c + 20*a^2*c^2 + 3*b*c*(b^2 - 6*a*c)*x^2)/(a*(b^2 - 4*a* 
c)*x^2*(a + b*x^2 + c*x^4)) + (6*(-(((b^2 - 5*a*c)*(b^2 - 2*a*c))/(a*x^2)) 
 - ((b^6 - 10*a*b^4*c + 30*a^2*b^2*c^2 - 20*a^3*c^3)*ArcTanh[(b + 2*c*x^2) 
/Sqrt[b^2 - 4*a*c]])/(a^2*Sqrt[b^2 - 4*a*c]) - (b*(b^2 - 4*a*c)^2*Log[x^2] 
)/a^2 + (b*(b^2 - 4*a*c)^2*Log[a + b*x^2 + c*x^4])/(2*a^2)))/(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 = 0.23 (sec) , antiderivative size = 412, normalized size of antiderivative = 1.62

method	result	size
default	\(-\frac {\frac {\frac {c^{2} a \left (14 a^{2} c^{2}-13 a \,b^{2} c +2 b^{4}\right ) x^{6}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {a b c \left (74 a^{2} c^{2}-55 a \,b^{2} c +8 b^{4}\right ) x^{4}}{32 a^{2} c^{2}-16 a \,b^{2} c +2 b^{4}}+\frac {a \left (18 a^{3} c^{3}+7 a^{2} b^{2} c^{2}-12 a \,b^{4} c +2 b^{6}\right ) x^{2}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {a^{2} b \left (58 a^{2} c^{2}-36 a \,b^{2} c +5 b^{4}\right )}{32 a^{2} c^{2}-16 a \,b^{2} c +2 b^{4}}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2}}+\frac {\frac {3 \left (-16 a^{2} b \,c^{3}+8 a \,b^{3} c^{2}-b^{5} c \right ) \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{2 c}+\frac {6 \left (10 a^{3} c^{3}-23 a^{2} b^{2} c^{2}+9 a \,b^{4} c -b^{6}-\frac {\left (-16 a^{2} b \,c^{3}+8 a \,b^{3} c^{2}-b^{5} c \right ) b}{2 c}\right ) \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}}{2 a^{4}}-\frac {1}{2 a^{3} x^{2}}-\frac {3 b \ln \left (x \right )}{a^{4}}\)	\(412\)
risch	\(\frac {-\frac {3 c^{2} \left (10 a^{2} c^{2}-7 a \,b^{2} c +b^{4}\right ) x^{8}}{2 a^{3} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {3 b c \left (46 a^{2} c^{2}-29 a \,b^{2} c +4 b^{4}\right ) x^{6}}{4 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a^{3}}-\frac {\left (50 a^{3} c^{3}+7 a^{2} b^{2} c^{2}-18 a \,b^{4} c +3 b^{6}\right ) x^{4}}{2 a^{3} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {b \left (122 a^{2} c^{2}-68 a \,b^{2} c +9 b^{4}\right ) x^{2}}{4 a^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {1}{2 a}}{x^{2} \left (c \,x^{4}+b \,x^{2}+a \right )^{2}}-\frac {3 b \ln \left (x \right )}{a^{4}}+\frac {3 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (1024 a^{9} c^{5}-1280 c^{4} b^{2} a^{8}+640 a^{7} b^{4} c^{3}-160 a^{6} b^{6} c^{2}+20 a^{5} b^{8} c -a^{4} b^{10}\right ) \textit {\_Z}^{2}+\left (-1024 a^{5} b \,c^{5}+1280 a^{4} b^{3} c^{4}-640 a^{3} b^{5} c^{3}+160 a^{2} b^{7} c^{2}-20 a \,b^{9} c +b^{11}\right ) \textit {\_Z} +100 a^{2} c^{6}-44 a \,b^{2} c^{5}+5 b^{4} c^{4}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (2560 a^{11} c^{5}-3328 a^{10} b^{2} c^{4}+1728 a^{9} b^{4} c^{3}-448 a^{8} b^{6} c^{2}+58 a^{7} b^{8} c -3 b^{10} a^{6}\right ) \textit {\_R}^{2}+\left (-1120 a^{7} b \,c^{5}+1088 a^{6} b^{3} c^{4}-398 a^{5} b^{5} c^{3}+65 a^{4} b^{7} c^{2}-4 a^{3} b^{9} c \right ) \textit {\_R} +200 a^{4} c^{6}-280 a^{3} b^{2} c^{5}+138 a^{2} b^{4} c^{4}-28 a \,b^{6} c^{3}+2 b^{8} c^{2}\right ) x^{2}+\left (-256 a^{11} b \,c^{4}+256 a^{10} b^{3} c^{3}-96 a^{9} b^{5} c^{2}+16 a^{8} b^{7} c -a^{7} b^{9}\right ) \textit {\_R}^{2}+\left (160 a^{8} c^{5}-704 a^{7} b^{2} c^{4}+594 c^{3} b^{4} a^{6}-207 a^{5} b^{6} c^{2}+33 a^{4} b^{8} c -2 b^{10} a^{3}\right ) \textit {\_R} +320 a^{4} b \,c^{5}-384 a^{3} b^{3} c^{4}+164 a^{2} b^{5} c^{3}-30 a \,b^{7} c^{2}+2 b^{9} c \right )\right )}{2}\)	\(732\)

Input:

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

Output:

-1/2/a^4*((c^2*a*(14*a^2*c^2-13*a*b^2*c+2*b^4)/(16*a^2*c^2-8*a*b^2*c+b^4)* 
x^6+1/2*a*b*c*(74*a^2*c^2-55*a*b^2*c+8*b^4)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^4 
+a*(18*a^3*c^3+7*a^2*b^2*c^2-12*a*b^4*c+2*b^6)/(16*a^2*c^2-8*a*b^2*c+b^4)* 
x^2+1/2*a^2*b*(58*a^2*c^2-36*a*b^2*c+5*b^4)/(16*a^2*c^2-8*a*b^2*c+b^4))/(c 
*x^4+b*x^2+a)^2+3/(16*a^2*c^2-8*a*b^2*c+b^4)*(1/2*(-16*a^2*b*c^3+8*a*b^3*c 
^2-b^5*c)/c*ln(c*x^4+b*x^2+a)+2*(10*a^3*c^3-23*a^2*b^2*c^2+9*a*b^4*c-b^6-1 
/2*(-16*a^2*b*c^3+8*a*b^3*c^2-b^5*c)*b/c)/(4*a*c-b^2)^(1/2)*arctan((2*c*x^ 
2+b)/(4*a*c-b^2)^(1/2))))-1/2/a^3/x^2-3*b*ln(x)/a^4
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1144 vs. \(2 (241) = 482\).

Time = 0.57 (sec) , antiderivative size = 2312, normalized size of antiderivative = 9.07 \[ \int \frac {1}{x^3 \left (a+b x^2+c x^4\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/4*(2*a^3*b^6 - 24*a^4*b^4*c + 96*a^5*b^2*c^2 - 128*a^6*c^3 + 6*(a*b^6* 
c^2 - 11*a^2*b^4*c^3 + 38*a^3*b^2*c^4 - 40*a^4*c^5)*x^8 + 3*(4*a*b^7*c - 4 
5*a^2*b^5*c^2 + 162*a^3*b^3*c^3 - 184*a^4*b*c^4)*x^6 + 2*(3*a*b^8 - 30*a^2 
*b^6*c + 79*a^3*b^4*c^2 + 22*a^4*b^2*c^3 - 200*a^5*c^4)*x^4 + (9*a^2*b^7 - 
 104*a^3*b^5*c + 394*a^4*b^3*c^2 - 488*a^5*b*c^3)*x^2 + 3*((b^6*c^2 - 10*a 
*b^4*c^3 + 30*a^2*b^2*c^4 - 20*a^3*c^5)*x^10 + 2*(b^7*c - 10*a*b^5*c^2 + 3 
0*a^2*b^3*c^3 - 20*a^3*b*c^4)*x^8 + (b^8 - 8*a*b^6*c + 10*a^2*b^4*c^2 + 40 
*a^3*b^2*c^3 - 40*a^4*c^4)*x^6 + 2*(a*b^7 - 10*a^2*b^5*c + 30*a^3*b^3*c^2 
- 20*a^4*b*c^3)*x^4 + (a^2*b^6 - 10*a^3*b^4*c + 30*a^4*b^2*c^2 - 20*a^5*c^ 
3)*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)) - 3*((b^7*c^2 - 12*a*b^5* 
c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*x^10 + 2*(b^8*c - 12*a*b^6*c^2 + 48*a 
^2*b^4*c^3 - 64*a^3*b^2*c^4)*x^8 + (b^9 - 10*a*b^7*c + 24*a^2*b^5*c^2 + 32 
*a^3*b^3*c^3 - 128*a^4*b*c^4)*x^6 + 2*(a*b^8 - 12*a^2*b^6*c + 48*a^3*b^4*c 
^2 - 64*a^4*b^2*c^3)*x^4 + (a^2*b^7 - 12*a^3*b^5*c + 48*a^4*b^3*c^2 - 64*a 
^5*b*c^3)*x^2)*log(c*x^4 + b*x^2 + a) + 12*((b^7*c^2 - 12*a*b^5*c^3 + 48*a 
^2*b^3*c^4 - 64*a^3*b*c^5)*x^10 + 2*(b^8*c - 12*a*b^6*c^2 + 48*a^2*b^4*c^3 
 - 64*a^3*b^2*c^4)*x^8 + (b^9 - 10*a*b^7*c + 24*a^2*b^5*c^2 + 32*a^3*b^3*c 
^3 - 128*a^4*b*c^4)*x^6 + 2*(a*b^8 - 12*a^2*b^6*c + 48*a^3*b^4*c^2 - 64*a^ 
4*b^2*c^3)*x^4 + (a^2*b^7 - 12*a^3*b^5*c + 48*a^4*b^3*c^2 - 64*a^5*b*c^...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate(1/x^3/(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 = 1.08 (sec) , antiderivative size = 382, normalized size of antiderivative = 1.50 \[ \int \frac {1}{x^3 \left (a+b x^2+c x^4\right )^3} \, dx=\frac {3 \, {\left (b^{6} - 10 \, a b^{4} c + 30 \, a^{2} b^{2} c^{2} - 20 \, a^{3} c^{3}\right )} \arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{2 \, {\left (a^{4} b^{4} - 8 \, a^{5} b^{2} c + 16 \, a^{6} c^{2}\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {9 \, b^{5} c^{2} x^{8} - 72 \, a b^{3} c^{3} x^{8} + 144 \, a^{2} b c^{4} x^{8} + 18 \, b^{6} c x^{6} - 136 \, a b^{4} c^{2} x^{6} + 236 \, a^{2} b^{2} c^{3} x^{6} + 56 \, a^{3} c^{4} x^{6} + 9 \, b^{7} x^{4} - 38 \, a b^{5} c x^{4} - 110 \, a^{2} b^{3} c^{2} x^{4} + 436 \, a^{3} b c^{3} x^{4} + 26 \, a b^{6} x^{2} - 192 \, a^{2} b^{4} c x^{2} + 316 \, a^{3} b^{2} c^{2} x^{2} + 72 \, a^{4} c^{3} x^{2} + 19 \, a^{2} b^{5} - 144 \, a^{3} b^{3} c + 260 \, a^{4} b c^{2}}{8 \, {\left (a^{4} b^{4} - 8 \, a^{5} b^{2} c + 16 \, a^{6} c^{2}\right )} {\left (c x^{4} + b x^{2} + a\right )}^{2}} + \frac {3 \, b \log \left (c x^{4} + b x^{2} + a\right )}{4 \, a^{4}} - \frac {3 \, b \log \left (x^{2}\right )}{2 \, a^{4}} + \frac {3 \, b x^{2} - a}{2 \, a^{4} x^{2}} \] Input:

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

Output:

3/2*(b^6 - 10*a*b^4*c + 30*a^2*b^2*c^2 - 20*a^3*c^3)*arctan((2*c*x^2 + b)/ 
sqrt(-b^2 + 4*a*c))/((a^4*b^4 - 8*a^5*b^2*c + 16*a^6*c^2)*sqrt(-b^2 + 4*a* 
c)) - 1/8*(9*b^5*c^2*x^8 - 72*a*b^3*c^3*x^8 + 144*a^2*b*c^4*x^8 + 18*b^6*c 
*x^6 - 136*a*b^4*c^2*x^6 + 236*a^2*b^2*c^3*x^6 + 56*a^3*c^4*x^6 + 9*b^7*x^ 
4 - 38*a*b^5*c*x^4 - 110*a^2*b^3*c^2*x^4 + 436*a^3*b*c^3*x^4 + 26*a*b^6*x^ 
2 - 192*a^2*b^4*c*x^2 + 316*a^3*b^2*c^2*x^2 + 72*a^4*c^3*x^2 + 19*a^2*b^5 
- 144*a^3*b^3*c + 260*a^4*b*c^2)/((a^4*b^4 - 8*a^5*b^2*c + 16*a^6*c^2)*(c* 
x^4 + b*x^2 + a)^2) + 3/4*b*log(c*x^4 + b*x^2 + a)/a^4 - 3/2*b*log(x^2)/a^ 
4 + 1/2*(3*b*x^2 - a)/(a^4*x^2)
 

Mupad [B] (verification not implemented)

Time = 26.21 (sec) , antiderivative size = 10074, normalized size of antiderivative = 39.51 \[ \int \frac {1}{x^3 \left (a+b x^2+c x^4\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

(log(((27*c^5*x^2*(b^4 + 10*a^2*c^2 - 7*a*b^2*c)^3)/(a^9*(4*a*c - b^2)^6) 
- ((3*b - 3*a^4*(-(b^6 - 20*a^3*c^3 + 30*a^2*b^2*c^2 - 10*a*b^4*c)^2/(a^8* 
(4*a*c - b^2)^5))^(1/2))*((9*c^3*(4*b^10 - 100*a^5*c^5 + 342*a^2*b^6*c^2 - 
 837*a^3*b^4*c^3 + 780*a^4*b^2*c^4 - 61*a*b^8*c))/(a^6*(4*a*c - b^2)^4) - 
((3*b - 3*a^4*(-(b^6 - 20*a^3*c^3 + 30*a^2*b^2*c^2 - 10*a*b^4*c)^2/(a^8*(4 
*a*c - b^2)^5))^(1/2))*((6*c^3*x^2*(b^6 + 100*a^3*c^3 - 30*a^2*b^2*c^2 - 2 
*a*b^4*c))/(a^3*(4*a*c - b^2)^2) + (b*c^2*(3*b - 3*a^4*(-(b^6 - 20*a^3*c^3 
 + 30*a^2*b^2*c^2 - 10*a*b^4*c)^2/(a^8*(4*a*c - b^2)^5))^(1/2))*(a*b + 3*b 
^2*x^2 - 10*a*c*x^2))/a^4 + (12*b*c^2*(b^6 - 10*a^3*c^3 + 23*a^2*b^2*c^2 - 
 9*a*b^4*c))/(a^3*(4*a*c - b^2)^2)))/(4*a^4) + (9*b*c^4*x^2*(6*b^8 + 900*a 
^4*c^4 + 479*a^2*b^4*c^2 - 1100*a^3*b^2*c^3 - 89*a*b^6*c))/(a^6*(4*a*c - b 
^2)^4)))/(4*a^4) + (27*b*c^4*(b^4 + 10*a^2*c^2 - 7*a*b^2*c)^2)/(a^9*(4*a*c 
 - b^2)^4))*((27*c^5*x^2*(b^4 + 10*a^2*c^2 - 7*a*b^2*c)^3)/(a^9*(4*a*c - b 
^2)^6) - ((3*b + 3*a^4*(-(b^6 - 20*a^3*c^3 + 30*a^2*b^2*c^2 - 10*a*b^4*c)^ 
2/(a^8*(4*a*c - b^2)^5))^(1/2))*((9*c^3*(4*b^10 - 100*a^5*c^5 + 342*a^2*b^ 
6*c^2 - 837*a^3*b^4*c^3 + 780*a^4*b^2*c^4 - 61*a*b^8*c))/(a^6*(4*a*c - b^2 
)^4) - ((3*b + 3*a^4*(-(b^6 - 20*a^3*c^3 + 30*a^2*b^2*c^2 - 10*a*b^4*c)^2/ 
(a^8*(4*a*c - b^2)^5))^(1/2))*((6*c^3*x^2*(b^6 + 100*a^3*c^3 - 30*a^2*b^2* 
c^2 - 2*a*b^4*c))/(a^3*(4*a*c - b^2)^2) + (b*c^2*(3*b + 3*a^4*(-(b^6 - 20* 
a^3*c^3 + 30*a^2*b^2*c^2 - 10*a*b^4*c)^2/(a^8*(4*a*c - b^2)^5))^(1/2))*...
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 4941, normalized size of antiderivative = 19.38 \[ \int \frac {1}{x^3 \left (a+b x^2+c x^4\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(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**5*b*c* 
*3*x**2 - 180*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**4*b**3*c**2*x**2 + 240*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**4*b**2*c**3*x**4 + 240*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))*a**4*b*c**4*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**3*b**5*c*x**2 - 360*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**4*c**2*x**4 - 240*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**3*x**6 + 
 240*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**2 
*c**4*x**8 + 120*sqrt(2*sqrt(c)*sqrt(a) + b)*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**3*b*c**5*x**10 - 6*sqrt(2*sqrt(c)*sqrt(a) + b)*sqrt(2*sqrt(c)*sqr...