\(\int \frac {1}{x (a+b x+c x^2)^4} \, dx\) [264]

Optimal result

Integrand size = 16, antiderivative size = 282 \[ \int \frac {1}{x \left (a+b x+c x^2\right )^4} \, dx=\frac {b^2-2 a c+b c x}{3 a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac {3 b^4-23 a b^2 c+24 a^2 c^2+b c \left (3 b^2-22 a c\right ) x}{6 a^2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}+\frac {2 b^6-23 a b^4 c+86 a^2 b^2 c^2-64 a^3 c^3+2 b c \left (b^4-11 a b^2 c+38 a^2 c^2\right ) x}{2 a^3 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}+\frac {b \left (b^6-14 a b^4 c+70 a^2 b^2 c^2-140 a^3 c^3\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{a^4 \left (b^2-4 a c\right )^{7/2}}+\frac {\log (x)}{a^4}-\frac {\log \left (a+b x+c x^2\right )}{2 a^4} \] Output:

1/3*(b*c*x-2*a*c+b^2)/a/(-4*a*c+b^2)/(c*x^2+b*x+a)^3+1/6*(3*b^4-23*a*b^2*c 
+24*a^2*c^2+b*c*(-22*a*c+3*b^2)*x)/a^2/(-4*a*c+b^2)^2/(c*x^2+b*x+a)^2+1/2* 
(2*b^6-23*a*b^4*c+86*a^2*b^2*c^2-64*a^3*c^3+2*b*c*(38*a^2*c^2-11*a*b^2*c+b 
^4)*x)/a^3/(-4*a*c+b^2)^3/(c*x^2+b*x+a)+b*(-140*a^3*c^3+70*a^2*b^2*c^2-14* 
a*b^4*c+b^6)*arctanh((2*c*x+b)/(-4*a*c+b^2)^(1/2))/a^4/(-4*a*c+b^2)^(7/2)+ 
ln(x)/a^4-1/2*ln(c*x^2+b*x+a)/a^4
 

Mathematica [A] (verified)

Time = 0.46 (sec) , antiderivative size = 278, normalized size of antiderivative = 0.99 \[ \int \frac {1}{x \left (a+b x+c x^2\right )^4} \, dx=\frac {-\frac {2 a^3 \left (-b^2+2 a c-b c x\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))^3}+\frac {a^2 \left (3 b^4-23 a b^2 c+24 a^2 c^2+3 b^3 c x-22 a b c^2 x\right )}{\left (b^2-4 a c\right )^2 (a+x (b+c x))^2}-\frac {3 a \left (-2 b^6+23 a b^4 c-86 a^2 b^2 c^2+64 a^3 c^3-2 b^5 c x+22 a b^3 c^2 x-76 a^2 b c^3 x\right )}{\left (b^2-4 a c\right )^3 (a+x (b+c x))}+\frac {6 b \left (b^6-14 a b^4 c+70 a^2 b^2 c^2-140 a^3 c^3\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{7/2}}+6 \log (x)-3 \log (a+x (b+c x))}{6 a^4} \] Input:

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

Output:

((-2*a^3*(-b^2 + 2*a*c - b*c*x))/((b^2 - 4*a*c)*(a + x*(b + c*x))^3) + (a^ 
2*(3*b^4 - 23*a*b^2*c + 24*a^2*c^2 + 3*b^3*c*x - 22*a*b*c^2*x))/((b^2 - 4* 
a*c)^2*(a + x*(b + c*x))^2) - (3*a*(-2*b^6 + 23*a*b^4*c - 86*a^2*b^2*c^2 + 
 64*a^3*c^3 - 2*b^5*c*x + 22*a*b^3*c^2*x - 76*a^2*b*c^3*x))/((b^2 - 4*a*c) 
^3*(a + x*(b + c*x))) + (6*b*(b^6 - 14*a*b^4*c + 70*a^2*b^2*c^2 - 140*a^3* 
c^3)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(7/2) + 6*Log[ 
x] - 3*Log[a + x*(b + c*x)])/(6*a^4)
 

Rubi [A] (verified)

Time = 1.14 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.24, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1165, 25, 1235, 27, 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.

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

\(\Big \downarrow \) 1165

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1235

\(\displaystyle \frac {\frac {24 a^2 c^2+b c x \left (3 b^2-22 a c\right )-23 a b^2 c+3 b^4}{2 a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {\int -\frac {3 \left (2 \left (b^2-4 a c\right )^2+b c \left (3 b^2-22 a c\right ) x\right )}{x \left (c x^2+b x+a\right )^2}dx}{2 a \left (b^2-4 a c\right )}}{3 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x}{3 a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {3 \int \frac {2 \left (b^2-4 a c\right )^2+b c \left (3 b^2-22 a c\right ) x}{x \left (c x^2+b x+a\right )^2}dx}{2 a \left (b^2-4 a c\right )}+\frac {24 a^2 c^2+b c x \left (3 b^2-22 a c\right )-23 a b^2 c+3 b^4}{2 a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}}{3 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x}{3 a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}\)

\(\Big \downarrow \) 1235

\(\displaystyle \frac {\frac {3 \left (\frac {-64 a^3 c^3+86 a^2 b^2 c^2+2 b c x \left (38 a^2 c^2-11 a b^2 c+b^4\right )-23 a b^4 c+2 b^6}{a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {\int -\frac {2 \left (\left (b^2-4 a c\right )^3+b c \left (b^4-11 a c b^2+38 a^2 c^2\right ) x\right )}{x \left (c x^2+b x+a\right )}dx}{a \left (b^2-4 a c\right )}\right )}{2 a \left (b^2-4 a c\right )}+\frac {24 a^2 c^2+b c x \left (3 b^2-22 a c\right )-23 a b^2 c+3 b^4}{2 a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}}{3 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x}{3 a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {3 \left (\frac {2 \int \frac {\left (b^2-4 a c\right )^3+b c \left (b^4-11 a c b^2+38 a^2 c^2\right ) x}{x \left (c x^2+b x+a\right )}dx}{a \left (b^2-4 a c\right )}+\frac {-64 a^3 c^3+86 a^2 b^2 c^2+2 b c x \left (38 a^2 c^2-11 a b^2 c+b^4\right )-23 a b^4 c+2 b^6}{a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\right )}{2 a \left (b^2-4 a c\right )}+\frac {24 a^2 c^2+b c x \left (3 b^2-22 a c\right )-23 a b^2 c+3 b^4}{2 a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}}{3 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x}{3 a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}\)

\(\Big \downarrow \) 1200

\(\displaystyle \frac {\frac {3 \left (\frac {2 \int \left (\frac {-c x \left (b^2-4 a c\right )^3-b \left (b^2-6 a c\right ) \left (b^4-7 a c b^2+17 a^2 c^2\right )}{a \left (c x^2+b x+a\right )}-\frac {\left (4 a c-b^2\right )^3}{a x}\right )dx}{a \left (b^2-4 a c\right )}+\frac {-64 a^3 c^3+86 a^2 b^2 c^2+2 b c x \left (38 a^2 c^2-11 a b^2 c+b^4\right )-23 a b^4 c+2 b^6}{a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\right )}{2 a \left (b^2-4 a c\right )}+\frac {24 a^2 c^2+b c x \left (3 b^2-22 a c\right )-23 a b^2 c+3 b^4}{2 a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}}{3 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x}{3 a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {24 a^2 c^2+b c x \left (3 b^2-22 a c\right )-23 a b^2 c+3 b^4}{2 a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {3 \left (\frac {2 \left (\frac {b \left (-140 a^3 c^3+70 a^2 b^2 c^2-14 a b^4 c+b^6\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{a \sqrt {b^2-4 a c}}-\frac {\left (b^2-4 a c\right )^3 \log \left (a+b x+c x^2\right )}{2 a}+\frac {\log (x) \left (b^2-4 a c\right )^3}{a}\right )}{a \left (b^2-4 a c\right )}+\frac {-64 a^3 c^3+86 a^2 b^2 c^2+2 b c x \left (38 a^2 c^2-11 a b^2 c+b^4\right )-23 a b^4 c+2 b^6}{a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\right )}{2 a \left (b^2-4 a c\right )}}{3 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x}{3 a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}\)

Input:

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

Output:

(b^2 - 2*a*c + b*c*x)/(3*a*(b^2 - 4*a*c)*(a + b*x + c*x^2)^3) + ((3*b^4 - 
23*a*b^2*c + 24*a^2*c^2 + b*c*(3*b^2 - 22*a*c)*x)/(2*a*(b^2 - 4*a*c)*(a + 
b*x + c*x^2)^2) + (3*((2*b^6 - 23*a*b^4*c + 86*a^2*b^2*c^2 - 64*a^3*c^3 + 
2*b*c*(b^4 - 11*a*b^2*c + 38*a^2*c^2)*x)/(a*(b^2 - 4*a*c)*(a + b*x + c*x^2 
)) + (2*((b*(b^6 - 14*a*b^4*c + 70*a^2*b^2*c^2 - 140*a^3*c^3)*ArcTanh[(b + 
 2*c*x)/Sqrt[b^2 - 4*a*c]])/(a*Sqrt[b^2 - 4*a*c]) + ((b^2 - 4*a*c)^3*Log[x 
])/a - ((b^2 - 4*a*c)^3*Log[a + b*x + c*x^2])/(2*a)))/(a*(b^2 - 4*a*c))))/ 
(2*a*(b^2 - 4*a*c)))/(3*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 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 2009

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(660\) vs. \(2(270)=540\).

Time = 0.83 (sec) , antiderivative size = 661, normalized size of antiderivative = 2.34

method	result	size
default	\(\frac {\ln \left (x \right )}{a^{4}}-\frac {\frac {\frac {b \,c^{3} a \left (38 a^{2} c^{2}-11 c a \,b^{2}+b^{4}\right ) x^{5}}{64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}-\frac {c^{2} a \left (64 a^{3} c^{3}-238 a^{2} b^{2} c^{2}+67 a \,b^{4} c -6 b^{6}\right ) x^{4}}{2 \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}+\frac {b a c \left (160 a^{3} c^{3}+578 a^{2} b^{2} c^{2}-189 a \,b^{4} c +18 b^{6}\right ) x^{3}}{384 a^{3} c^{3}-288 a^{2} b^{2} c^{2}+72 a \,b^{4} c -6 b^{6}}-\frac {a \left (160 a^{4} c^{4}-328 a^{3} b^{2} c^{3}+27 a^{2} b^{4} c^{2}+13 a \,b^{6} c -2 b^{8}\right ) x^{2}}{2 \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}-\frac {a^{2} b \left (44 a^{3} c^{3}-172 a^{2} b^{2} c^{2}+54 a \,b^{4} c -5 b^{6}\right ) x}{2 \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}-\frac {\left (352 a^{3} c^{3}-438 a^{2} b^{2} c^{2}+124 a \,b^{4} c -11 b^{6}\right ) a^{3}}{6 \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}}{\left (c \,x^{2}+b x +a \right )^{3}}+\frac {\frac {\left (64 a^{3} c^{4}-48 a^{2} b^{2} c^{3}+12 a \,b^{4} c^{2}-b^{6} c \right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (102 a^{3} b \,c^{3}-59 a^{2} b^{3} c^{2}+13 a \,b^{5} c -b^{7}-\frac {\left (64 a^{3} c^{4}-48 a^{2} b^{2} c^{3}+12 a \,b^{4} c^{2}-b^{6} c \right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}}{a^{4}}\)	\(661\)
risch	\(\text {Expression too large to display}\)	\(9189\)

Input:

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

Output:

ln(x)/a^4-1/a^4*((b*c^3*a*(38*a^2*c^2-11*a*b^2*c+b^4)/(64*a^3*c^3-48*a^2*b 
^2*c^2+12*a*b^4*c-b^6)*x^5-1/2*c^2*a*(64*a^3*c^3-238*a^2*b^2*c^2+67*a*b^4* 
c-6*b^6)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x^4+1/6*b*a*c*(160*a^3 
*c^3+578*a^2*b^2*c^2-189*a*b^4*c+18*b^6)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b 
^4*c-b^6)*x^3-1/2*a*(160*a^4*c^4-328*a^3*b^2*c^3+27*a^2*b^4*c^2+13*a*b^6*c 
-2*b^8)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x^2-1/2*a^2*b*(44*a^3*c 
^3-172*a^2*b^2*c^2+54*a*b^4*c-5*b^6)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c 
-b^6)*x-1/6*(352*a^3*c^3-438*a^2*b^2*c^2+124*a*b^4*c-11*b^6)*a^3/(64*a^3*c 
^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6))/(c*x^2+b*x+a)^3+1/(64*a^3*c^3-48*a^2*b^ 
2*c^2+12*a*b^4*c-b^6)*(1/2*(64*a^3*c^4-48*a^2*b^2*c^3+12*a*b^4*c^2-b^6*c)/ 
c*ln(c*x^2+b*x+a)+2*(102*a^3*b*c^3-59*a^2*b^3*c^2+13*a*b^5*c-b^7-1/2*(64*a 
^3*c^4-48*a^2*b^2*c^3+12*a*b^4*c^2-b^6*c)*b/c)/(4*a*c-b^2)^(1/2)*arctan((2 
*c*x+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. 1764 vs. \(2 (270) = 540\).

Time = 0.95 (sec) , antiderivative size = 3548, normalized size of antiderivative = 12.58 \[ \int \frac {1}{x \left (a+b x+c x^2\right )^4} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate(1/x/(c*x^2+b*x+a)^4,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 = 0.18 (sec) , antiderivative size = 405, normalized size of antiderivative = 1.44 \[ \int \frac {1}{x \left (a+b x+c x^2\right )^4} \, dx=-\frac {{\left (b^{7} - 14 \, a b^{5} c + 70 \, a^{2} b^{3} c^{2} - 140 \, a^{3} b c^{3}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (a^{4} b^{6} - 12 \, a^{5} b^{4} c + 48 \, a^{6} b^{2} c^{2} - 64 \, a^{7} c^{3}\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {\log \left (c x^{2} + b x + a\right )}{2 \, a^{4}} + \frac {\log \left ({\left | x \right |}\right )}{a^{4}} + \frac {11 \, a^{3} b^{6} - 124 \, a^{4} b^{4} c + 438 \, a^{5} b^{2} c^{2} - 352 \, a^{6} c^{3} + 6 \, {\left (a b^{5} c^{3} - 11 \, a^{2} b^{3} c^{4} + 38 \, a^{3} b c^{5}\right )} x^{5} + 3 \, {\left (6 \, a b^{6} c^{2} - 67 \, a^{2} b^{4} c^{3} + 238 \, a^{3} b^{2} c^{4} - 64 \, a^{4} c^{5}\right )} x^{4} + {\left (18 \, a b^{7} c - 189 \, a^{2} b^{5} c^{2} + 578 \, a^{3} b^{3} c^{3} + 160 \, a^{4} b c^{4}\right )} x^{3} + 3 \, {\left (2 \, a b^{8} - 13 \, a^{2} b^{6} c - 27 \, a^{3} b^{4} c^{2} + 328 \, a^{4} b^{2} c^{3} - 160 \, a^{5} c^{4}\right )} x^{2} + 3 \, {\left (5 \, a^{2} b^{7} - 54 \, a^{3} b^{5} c + 172 \, a^{4} b^{3} c^{2} - 44 \, a^{5} b c^{3}\right )} x}{6 \, {\left (c x^{2} + b x + a\right )}^{3} {\left (b^{2} - 4 \, a c\right )}^{3} a^{4}} \] Input:

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

Output:

-(b^7 - 14*a*b^5*c + 70*a^2*b^3*c^2 - 140*a^3*b*c^3)*arctan((2*c*x + b)/sq 
rt(-b^2 + 4*a*c))/((a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)* 
sqrt(-b^2 + 4*a*c)) - 1/2*log(c*x^2 + b*x + a)/a^4 + log(abs(x))/a^4 + 1/6 
*(11*a^3*b^6 - 124*a^4*b^4*c + 438*a^5*b^2*c^2 - 352*a^6*c^3 + 6*(a*b^5*c^ 
3 - 11*a^2*b^3*c^4 + 38*a^3*b*c^5)*x^5 + 3*(6*a*b^6*c^2 - 67*a^2*b^4*c^3 + 
 238*a^3*b^2*c^4 - 64*a^4*c^5)*x^4 + (18*a*b^7*c - 189*a^2*b^5*c^2 + 578*a 
^3*b^3*c^3 + 160*a^4*b*c^4)*x^3 + 3*(2*a*b^8 - 13*a^2*b^6*c - 27*a^3*b^4*c 
^2 + 328*a^4*b^2*c^3 - 160*a^5*c^4)*x^2 + 3*(5*a^2*b^7 - 54*a^3*b^5*c + 17 
2*a^4*b^3*c^2 - 44*a^5*b*c^3)*x)/((c*x^2 + b*x + a)^3*(b^2 - 4*a*c)^3*a^4)
 

Mupad [B] (verification not implemented)

Time = 11.04 (sec) , antiderivative size = 1680, normalized size of antiderivative = 5.96 \[ \int \frac {1}{x \left (a+b x+c x^2\right )^4} \, dx=\text {Too large to display} \] Input:

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

Output:

log(x)/a^4 + ((11*b^6 - 352*a^3*c^3 + 438*a^2*b^2*c^2 - 124*a*b^4*c)/(6*a* 
(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)) - (x^2*(160*a^4*c^4 - 2* 
b^8 + 27*a^2*b^4*c^2 - 328*a^3*b^2*c^3 + 13*a*b^6*c))/(2*a^3*(b^6 - 64*a^3 
*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)) + (x*(5*b^7 - 44*a^3*b*c^3 + 172*a^2* 
b^3*c^2 - 54*a*b^5*c))/(2*a^2*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^ 
4*c)) - (x^4*(64*a^3*c^5 - 6*b^6*c^2 + 67*a*b^4*c^3 - 238*a^2*b^2*c^4))/(2 
*a^3*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)) + (x^3*(18*b^7*c - 
189*a*b^5*c^2 + 160*a^3*b*c^4 + 578*a^2*b^3*c^3))/(6*a^3*(b^6 - 64*a^3*c^3 
 + 48*a^2*b^2*c^2 - 12*a*b^4*c)) + (b*c^2*x^5*(b^4*c + 38*a^2*c^3 - 11*a*b 
^2*c^2))/(a^3*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)))/(x^2*(3*a 
*b^2 + 3*a^2*c) + x^4*(3*a*c^2 + 3*b^2*c) + a^3 + x^3*(b^3 + 6*a*b*c) + c^ 
3*x^6 + 3*b*c^2*x^5 + 3*a^2*b*x) + (log(2*a*b^14 + 2*b^15*x - 24576*a^8*c^ 
7 + 2*a*b^7*(-(4*a*c - b^2)^7)^(1/2) - 55*a^2*b^12*c + 2*b^8*x*(-(4*a*c - 
b^2)^7)^(1/2) + 647*a^3*b^10*c^2 - 4218*a^4*b^8*c^3 + 16408*a^5*b^6*c^4 - 
37856*a^6*b^4*c^5 + 47488*a^7*b^2*c^6 - 25*a^2*b^5*c*(-(4*a*c - b^2)^7)^(1 
/2) - 166*a^4*b*c^3*(-(4*a*c - b^2)^7)^(1/2) + 673*a^2*b^11*c^2*x - 4504*a 
^3*b^9*c^3*x + 18124*a^4*b^7*c^4*x - 43792*a^5*b^5*c^5*x + 58688*a^6*b^3*c 
^6*x + 192*a^4*c^4*x*(-(4*a*c - b^2)^7)^(1/2) - 56*a*b^13*c*x + 107*a^3*b^ 
3*c^2*(-(4*a*c - b^2)^7)^(1/2) - 33536*a^7*b*c^7*x - 28*a*b^6*c*x*(-(4*a*c 
 - b^2)^7)^(1/2) + 143*a^2*b^4*c^2*x*(-(4*a*c - b^2)^7)^(1/2) - 310*a^3...
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 3378, normalized size of antiderivative = 11.98 \[ \int \frac {1}{x \left (a+b x+c x^2\right )^4} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 840*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**6*b*c** 
3 + 420*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**5*b**3* 
c**2 - 2520*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**5*b 
**2*c**3*x - 2520*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))* 
a**5*b*c**4*x**2 - 84*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b** 
2))*a**4*b**5*c + 1260*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b* 
*2))*a**4*b**4*c**2*x - 1260*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a* 
c - b**2))*a**4*b**3*c**3*x**2 - 5040*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/ 
sqrt(4*a*c - b**2))*a**4*b**2*c**4*x**3 - 2520*sqrt(4*a*c - b**2)*atan((b 
+ 2*c*x)/sqrt(4*a*c - b**2))*a**4*b*c**5*x**4 + 6*sqrt(4*a*c - b**2)*atan( 
(b + 2*c*x)/sqrt(4*a*c - b**2))*a**3*b**7 - 252*sqrt(4*a*c - b**2)*atan((b 
 + 2*c*x)/sqrt(4*a*c - b**2))*a**3*b**6*c*x + 1008*sqrt(4*a*c - b**2)*atan 
((b + 2*c*x)/sqrt(4*a*c - b**2))*a**3*b**5*c**2*x**2 + 1680*sqrt(4*a*c - b 
**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**3*b**4*c**3*x**3 - 1260*sqrt( 
4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**3*b**3*c**4*x**4 - 2 
520*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**3*b**2*c**5 
*x**5 - 840*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**3*b 
*c**6*x**6 + 18*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a* 
*2*b**8*x - 234*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a* 
*2*b**7*c*x**2 - 84*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b*...