\(\int \frac {1}{x^3 (a+b x+c x^2)^3} \, dx\) [254]

Optimal result

Integrand size = 16, antiderivative size = 280 \[ \int \frac {1}{x^3 \left (a+b x+c x^2\right )^3} \, dx=-\frac {1}{2 a^3 x^2}+\frac {3 b}{a^4 x}+\frac {b^4-4 a b^2 c+2 a^2 c^2+b c \left (b^2-3 a c\right ) x}{2 a^3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {6 b^6-47 a b^4 c+97 a^2 b^2 c^2-32 a^3 c^3+6 b c \left (b^4-7 a b^2 c+11 a^2 c^2\right ) x}{2 a^4 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac {3 b \left (2 b^6-21 a b^4 c+70 a^2 b^2 c^2-70 a^3 c^3\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{a^5 \left (b^2-4 a c\right )^{5/2}}+\frac {3 \left (2 b^2-a c\right ) \log (x)}{a^5}-\frac {3 \left (2 b^2-a c\right ) \log \left (a+b x+c x^2\right )}{2 a^5} \] Output:

-1/2/a^3/x^2+3*b/a^4/x+1/2*(b^4-4*a*b^2*c+2*a^2*c^2+b*c*(-3*a*c+b^2)*x)/a^ 
3/(-4*a*c+b^2)/(c*x^2+b*x+a)^2+1/2*(6*b^6-47*a*b^4*c+97*a^2*b^2*c^2-32*a^3 
*c^3+6*b*c*(11*a^2*c^2-7*a*b^2*c+b^4)*x)/a^4/(-4*a*c+b^2)^2/(c*x^2+b*x+a)+ 
3*b*(-70*a^3*c^3+70*a^2*b^2*c^2-21*a*b^4*c+2*b^6)*arctanh((2*c*x+b)/(-4*a* 
c+b^2)^(1/2))/a^5/(-4*a*c+b^2)^(5/2)+3*(-a*c+2*b^2)*ln(x)/a^5-3/2*(-a*c+2* 
b^2)*ln(c*x^2+b*x+a)/a^5
 

Mathematica [A] (verified)

Time = 0.39 (sec) , antiderivative size = 269, normalized size of antiderivative = 0.96 \[ \int \frac {1}{x^3 \left (a+b x+c x^2\right )^3} \, dx=\frac {-\frac {a^2}{x^2}+\frac {6 a b}{x}+\frac {a^2 \left (b^4-4 a b^2 c+2 a^2 c^2+b^3 c x-3 a b c^2 x\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))^2}+\frac {a \left (6 b^6-47 a b^4 c+97 a^2 b^2 c^2-32 a^3 c^3+6 b^5 c x-42 a b^3 c^2 x+66 a^2 b c^3 x\right )}{\left (b^2-4 a c\right )^2 (a+x (b+c x))}-\frac {6 b \left (2 b^6-21 a b^4 c+70 a^2 b^2 c^2-70 a^3 c^3\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{5/2}}+6 \left (2 b^2-a c\right ) \log (x)+3 \left (-2 b^2+a c\right ) \log (a+x (b+c x))}{2 a^5} \] Input:

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

Output:

(-(a^2/x^2) + (6*a*b)/x + (a^2*(b^4 - 4*a*b^2*c + 2*a^2*c^2 + b^3*c*x - 3* 
a*b*c^2*x))/((b^2 - 4*a*c)*(a + x*(b + c*x))^2) + (a*(6*b^6 - 47*a*b^4*c + 
 97*a^2*b^2*c^2 - 32*a^3*c^3 + 6*b^5*c*x - 42*a*b^3*c^2*x + 66*a^2*b*c^3*x 
))/((b^2 - 4*a*c)^2*(a + x*(b + c*x))) - (6*b*(2*b^6 - 21*a*b^4*c + 70*a^2 
*b^2*c^2 - 70*a^3*c^3)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a 
*c)^(5/2) + 6*(2*b^2 - a*c)*Log[x] + 3*(-2*b^2 + a*c)*Log[a + x*(b + c*x)] 
)/(2*a^5)
 

Rubi [A] (verified)

Time = 1.11 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.19, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {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.

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

\(\Big \downarrow \) 1165

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1235

\(\displaystyle \frac {\frac {24 a^2 c^2+2 b c x \left (2 b^2-11 a c\right )-25 a b^2 c+4 b^4}{a x^2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {\int -\frac {6 \left (2 b^4-13 a c b^2+c \left (2 b^2-11 a c\right ) x b+16 a^2 c^2\right )}{x^3 \left (c x^2+b x+a\right )}dx}{a \left (b^2-4 a c\right )}}{2 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x}{2 a x^2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {6 \int \frac {2 b^4-13 a c b^2+c \left (2 b^2-11 a c\right ) x b+16 a^2 c^2}{x^3 \left (c x^2+b x+a\right )}dx}{a \left (b^2-4 a c\right )}+\frac {24 a^2 c^2+2 b c x \left (2 b^2-11 a c\right )-25 a b^2 c+4 b^4}{a x^2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}}{2 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x}{2 a x^2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 1200

\(\displaystyle \frac {\frac {6 \int \left (-\frac {\left (a c-2 b^2\right ) \left (4 a c-b^2\right )^2}{a^3 x}+\frac {-c \left (2 b^2-a c\right ) x \left (b^2-4 a c\right )^2-b \left (2 b^6-19 a c b^4+55 a^2 c^2 b^2-43 a^3 c^3\right )}{a^3 \left (c x^2+b x+a\right )}+\frac {b \left (2 b^2-9 a c\right ) \left (3 a c-b^2\right )}{a^2 x^2}+\frac {2 b^4-13 a c b^2+16 a^2 c^2}{a x^3}\right )dx}{a \left (b^2-4 a c\right )}+\frac {24 a^2 c^2+2 b c x \left (2 b^2-11 a c\right )-25 a b^2 c+4 b^4}{a x^2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}}{2 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x}{2 a x^2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {24 a^2 c^2+2 b c x \left (2 b^2-11 a c\right )-25 a b^2 c+4 b^4}{a x^2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {6 \left (-\frac {\left (2 b^2-a c\right ) \left (b^2-4 a c\right )^2 \log \left (a+b x+c x^2\right )}{2 a^3}+\frac {\log (x) \left (2 b^2-a c\right ) \left (b^2-4 a c\right )^2}{a^3}+\frac {b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right )}{a^2 x}-\frac {16 a^2 c^2-13 a b^2 c+2 b^4}{2 a x^2}+\frac {b \left (-70 a^3 c^3+70 a^2 b^2 c^2-21 a b^4 c+2 b^6\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{a^3 \sqrt {b^2-4 a c}}\right )}{a \left (b^2-4 a c\right )}}{2 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x}{2 a x^2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\)

Input:

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

Output:

(b^2 - 2*a*c + b*c*x)/(2*a*(b^2 - 4*a*c)*x^2*(a + b*x + c*x^2)^2) + ((4*b^ 
4 - 25*a*b^2*c + 24*a^2*c^2 + 2*b*c*(2*b^2 - 11*a*c)*x)/(a*(b^2 - 4*a*c)*x 
^2*(a + b*x + c*x^2)) + (6*(-1/2*(2*b^4 - 13*a*b^2*c + 16*a^2*c^2)/(a*x^2) 
 + (b*(2*b^2 - 9*a*c)*(b^2 - 3*a*c))/(a^2*x) + (b*(2*b^6 - 21*a*b^4*c + 70 
*a^2*b^2*c^2 - 70*a^3*c^3)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(a^3*Sq 
rt[b^2 - 4*a*c]) + ((b^2 - 4*a*c)^2*(2*b^2 - a*c)*Log[x])/a^3 - ((b^2 - 4* 
a*c)^2*(2*b^2 - a*c)*Log[a + b*x + c*x^2])/(2*a^3)))/(a*(b^2 - 4*a*c)))/(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 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 [A] (verified)

Time = 0.80 (sec) , antiderivative size = 462, normalized size of antiderivative = 1.65

method	result	size
default	\(-\frac {1}{2 a^{3} x^{2}}+\frac {\left (-3 a c +6 b^{2}\right ) \ln \left (x \right )}{a^{5}}+\frac {3 b}{a^{4} x}+\frac {\frac {\frac {3 a b \,c^{2} \left (11 a^{2} c^{2}-7 c a \,b^{2}+b^{4}\right ) x^{3}}{16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}}-\frac {a c \left (32 a^{3} c^{3}-163 a^{2} b^{2} c^{2}+89 a \,b^{4} c -12 b^{6}\right ) x^{2}}{2 \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right )}+\frac {a b \left (23 a^{3} c^{3}+24 a^{2} b^{2} c^{2}-20 a \,b^{4} c +3 b^{6}\right ) x}{16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}}-\frac {a^{2} \left (40 a^{3} c^{3}-115 a^{2} b^{2} c^{2}+55 a \,b^{4} c -7 b^{6}\right )}{2 \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right )}}{\left (c \,x^{2}+b x +a \right )^{2}}+\frac {\frac {3 \left (16 a^{3} c^{4}-40 a^{2} b^{2} c^{3}+17 a \,b^{4} c^{2}-2 b^{6} c \right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {6 \left (43 a^{3} b \,c^{3}-55 a^{2} b^{3} c^{2}+19 a \,b^{5} c -2 b^{7}-\frac {\left (16 a^{3} c^{4}-40 a^{2} b^{2} c^{3}+17 a \,b^{4} c^{2}-2 b^{6} c \right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}}}{a^{5}}\)	\(462\)
risch	\(\text {Expression too large to display}\)	\(9013\)

Input:

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

Output:

-1/2/a^3/x^2+(-3*a*c+6*b^2)/a^5*ln(x)+3*b/a^4/x+1/a^5*((3*a*b*c^2*(11*a^2* 
c^2-7*a*b^2*c+b^4)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3-1/2*a*c*(32*a^3*c^3-163* 
a^2*b^2*c^2+89*a*b^4*c-12*b^6)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2+a*b*(23*a^3* 
c^3+24*a^2*b^2*c^2-20*a*b^4*c+3*b^6)/(16*a^2*c^2-8*a*b^2*c+b^4)*x-1/2*a^2* 
(40*a^3*c^3-115*a^2*b^2*c^2+55*a*b^4*c-7*b^6)/(16*a^2*c^2-8*a*b^2*c+b^4))/ 
(c*x^2+b*x+a)^2+3/(16*a^2*c^2-8*a*b^2*c+b^4)*(1/2*(16*a^3*c^4-40*a^2*b^2*c 
^3+17*a*b^4*c^2-2*b^6*c)/c*ln(c*x^2+b*x+a)+2*(43*a^3*b*c^3-55*a^2*b^3*c^2+ 
19*a*b^5*c-2*b^7-1/2*(16*a^3*c^4-40*a^2*b^2*c^3+17*a*b^4*c^2-2*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. 1324 vs. \(2 (268) = 536\).

Time = 1.09 (sec) , antiderivative size = 2669, normalized size of antiderivative = 9.53 \[ \int \frac {1}{x^3 \left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/2*(a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3 - 6*(2*a*b^7*c 
^2 - 23*a^2*b^5*c^3 + 87*a^3*b^3*c^4 - 108*a^4*b*c^5)*x^5 - 3*(8*a*b^8*c - 
 94*a^2*b^6*c^2 + 369*a^3*b^4*c^3 - 500*a^4*b^2*c^4 + 64*a^5*c^5)*x^4 - 2* 
(6*a*b^9 - 63*a^2*b^7*c + 188*a^3*b^5*c^2 - 25*a^4*b^3*c^3 - 412*a^5*b*c^4 
)*x^3 - (18*a^2*b^8 - 217*a^3*b^6*c + 887*a^4*b^4*c^2 - 1300*a^5*b^2*c^3 + 
 288*a^6*c^4)*x^2 + 3*((2*b^7*c^2 - 21*a*b^5*c^3 + 70*a^2*b^3*c^4 - 70*a^3 
*b*c^5)*x^6 + 2*(2*b^8*c - 21*a*b^6*c^2 + 70*a^2*b^4*c^3 - 70*a^3*b^2*c^4) 
*x^5 + (2*b^9 - 17*a*b^7*c + 28*a^2*b^5*c^2 + 70*a^3*b^3*c^3 - 140*a^4*b*c 
^4)*x^4 + 2*(2*a*b^8 - 21*a^2*b^6*c + 70*a^3*b^4*c^2 - 70*a^4*b^2*c^3)*x^3 
 + (2*a^2*b^7 - 21*a^3*b^5*c + 70*a^4*b^3*c^2 - 70*a^5*b*c^3)*x^2)*sqrt(b^ 
2 - 4*a*c)*log((2*c^2*x^2 + 2*b*c*x + b^2 - 2*a*c - sqrt(b^2 - 4*a*c)*(2*c 
*x + b))/(c*x^2 + b*x + a)) - 4*(a^3*b^7 - 12*a^4*b^5*c + 48*a^5*b^3*c^2 - 
 64*a^6*b*c^3)*x + 3*((2*b^8*c^2 - 25*a*b^6*c^3 + 108*a^2*b^4*c^4 - 176*a^ 
3*b^2*c^5 + 64*a^4*c^6)*x^6 + 2*(2*b^9*c - 25*a*b^7*c^2 + 108*a^2*b^5*c^3 
- 176*a^3*b^3*c^4 + 64*a^4*b*c^5)*x^5 + (2*b^10 - 21*a*b^8*c + 58*a^2*b^6* 
c^2 + 40*a^3*b^4*c^3 - 288*a^4*b^2*c^4 + 128*a^5*c^5)*x^4 + 2*(2*a*b^9 - 2 
5*a^2*b^7*c + 108*a^3*b^5*c^2 - 176*a^4*b^3*c^3 + 64*a^5*b*c^4)*x^3 + (2*a 
^2*b^8 - 25*a^3*b^6*c + 108*a^4*b^4*c^2 - 176*a^5*b^2*c^3 + 64*a^6*c^4)*x^ 
2)*log(c*x^2 + b*x + a) - 6*((2*b^8*c^2 - 25*a*b^6*c^3 + 108*a^2*b^4*c^4 - 
 176*a^3*b^2*c^5 + 64*a^4*c^6)*x^6 + 2*(2*b^9*c - 25*a*b^7*c^2 + 108*a^...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate(1/x^3/(c*x^2+b*x+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 = 0.19 (sec) , antiderivative size = 410, normalized size of antiderivative = 1.46 \[ \int \frac {1}{x^3 \left (a+b x+c x^2\right )^3} \, dx=-\frac {3 \, {\left (2 \, b^{7} - 21 \, a b^{5} c + 70 \, a^{2} b^{3} c^{2} - 70 \, a^{3} b c^{3}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (a^{5} b^{4} - 8 \, a^{6} b^{2} c + 16 \, a^{7} c^{2}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {12 \, b^{5} c^{2} x^{5} - 90 \, a b^{3} c^{3} x^{5} + 162 \, a^{2} b c^{4} x^{5} + 24 \, b^{6} c x^{4} - 186 \, a b^{4} c^{2} x^{4} + 363 \, a^{2} b^{2} c^{3} x^{4} - 48 \, a^{3} c^{4} x^{4} + 12 \, b^{7} x^{3} - 78 \, a b^{5} c x^{3} + 64 \, a^{2} b^{3} c^{2} x^{3} + 206 \, a^{3} b c^{3} x^{3} + 18 \, a b^{6} x^{2} - 145 \, a^{2} b^{4} c x^{2} + 307 \, a^{3} b^{2} c^{2} x^{2} - 72 \, a^{4} c^{3} x^{2} + 4 \, a^{2} b^{5} x - 32 \, a^{3} b^{3} c x + 64 \, a^{4} b c^{2} x - a^{3} b^{4} + 8 \, a^{4} b^{2} c - 16 \, a^{5} c^{2}}{2 \, {\left (a^{4} b^{4} - 8 \, a^{5} b^{2} c + 16 \, a^{6} c^{2}\right )} {\left (c x^{3} + b x^{2} + a x\right )}^{2}} - \frac {3 \, {\left (2 \, b^{2} - a c\right )} \log \left (c x^{2} + b x + a\right )}{2 \, a^{5}} + \frac {3 \, {\left (2 \, b^{2} - a c\right )} \log \left ({\left | x \right |}\right )}{a^{5}} \] Input:

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

Output:

-3*(2*b^7 - 21*a*b^5*c + 70*a^2*b^3*c^2 - 70*a^3*b*c^3)*arctan((2*c*x + b) 
/sqrt(-b^2 + 4*a*c))/((a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)*sqrt(-b^2 + 4*a 
*c)) + 1/2*(12*b^5*c^2*x^5 - 90*a*b^3*c^3*x^5 + 162*a^2*b*c^4*x^5 + 24*b^6 
*c*x^4 - 186*a*b^4*c^2*x^4 + 363*a^2*b^2*c^3*x^4 - 48*a^3*c^4*x^4 + 12*b^7 
*x^3 - 78*a*b^5*c*x^3 + 64*a^2*b^3*c^2*x^3 + 206*a^3*b*c^3*x^3 + 18*a*b^6* 
x^2 - 145*a^2*b^4*c*x^2 + 307*a^3*b^2*c^2*x^2 - 72*a^4*c^3*x^2 + 4*a^2*b^5 
*x - 32*a^3*b^3*c*x + 64*a^4*b*c^2*x - a^3*b^4 + 8*a^4*b^2*c - 16*a^5*c^2) 
/((a^4*b^4 - 8*a^5*b^2*c + 16*a^6*c^2)*(c*x^3 + b*x^2 + a*x)^2) - 3/2*(2*b 
^2 - a*c)*log(c*x^2 + b*x + a)/a^5 + 3*(2*b^2 - a*c)*log(abs(x))/a^5
 

Mupad [B] (verification not implemented)

Time = 10.81 (sec) , antiderivative size = 1404, normalized size of antiderivative = 5.01 \[ \int \frac {1}{x^3 \left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

((2*b*x)/a^2 - 1/(2*a) + (x^2*(18*b^6 - 72*a^3*c^3 + 307*a^2*b^2*c^2 - 145 
*a*b^4*c))/(2*a^3*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (3*x^4*(8*b^6*c - 16*a 
^3*c^4 - 62*a*b^4*c^2 + 121*a^2*b^2*c^3))/(2*a^4*(b^4 + 16*a^2*c^2 - 8*a*b 
^2*c)) + (b*x^3*(6*b^6 + 103*a^3*c^3 + 32*a^2*b^2*c^2 - 39*a*b^4*c))/(a^4* 
(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (3*b*c^2*x^5*(2*b^4 + 27*a^2*c^2 - 15*a* 
b^2*c))/(a^4*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)))/(x^4*(2*a*c + b^2) + a^2*x^2 
 + c^2*x^6 + 2*a*b*x^3 + 2*b*c*x^5) - (3*log(x)*(a*c - 2*b^2))/a^5 + (3*lo 
g(4*a*b^12 + 4*b^13*x + 1536*a^7*c^6 - 4*a*b^7*(-(4*a*c - b^2)^5)^(1/2) - 
80*a^2*b^10*c - 4*b^8*x*(-(4*a*c - b^2)^5)^(1/2) + 645*a^3*b^8*c^2 - 2643* 
a^4*b^6*c^3 + 5640*a^5*b^4*c^4 - 5552*a^6*b^2*c^5 + 36*a^2*b^5*c*(-(4*a*c 
- b^2)^5)^(1/2) + 59*a^4*b*c^3*(-(4*a*c - b^2)^5)^(1/2) + 682*a^2*b^9*c^2* 
x - 2913*a^3*b^7*c^3*x + 6606*a^4*b^5*c^4*x - 7232*a^5*b^3*c^5*x - 48*a^4* 
c^4*x*(-(4*a*c - b^2)^5)^(1/2) - 82*a*b^11*c*x - 95*a^3*b^3*c^2*(-(4*a*c - 
 b^2)^5)^(1/2) + 2656*a^6*b*c^6*x + 42*a*b^6*c*x*(-(4*a*c - b^2)^5)^(1/2) 
- 146*a^2*b^4*c^2*x*(-(4*a*c - b^2)^5)^(1/2) + 179*a^3*b^2*c^3*x*(-(4*a*c 
- b^2)^5)^(1/2))*(2*b^12 + 1024*a^6*c^6 - 2*b^7*(-(4*a*c - b^2)^5)^(1/2) + 
 340*a^2*b^8*c^2 - 1440*a^3*b^6*c^3 + 3200*a^4*b^4*c^4 - 3328*a^5*b^2*c^5 
- 41*a*b^10*c + 70*a^3*b*c^3*(-(4*a*c - b^2)^5)^(1/2) - 70*a^2*b^3*c^2*(-( 
4*a*c - b^2)^5)^(1/2) + 21*a*b^5*c*(-(4*a*c - b^2)^5)^(1/2)))/(2*a^5*(4*a* 
c - b^2)^5) + (3*log(4*a*b^12 + 4*b^13*x + 1536*a^7*c^6 + 4*a*b^7*(-(4*...
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 2450, normalized size of antiderivative = 8.75 \[ \int \frac {1}{x^3 \left (a+b x+c x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(420*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**5*b*c**3*x 
**2 - 420*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**4*b** 
3*c**2*x**2 + 840*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))* 
a**4*b**2*c**3*x**3 + 840*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - 
 b**2))*a**4*b*c**4*x**4 + 126*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4* 
a*c - b**2))*a**3*b**5*c*x**2 - 840*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sq 
rt(4*a*c - b**2))*a**3*b**4*c**2*x**3 - 420*sqrt(4*a*c - b**2)*atan((b + 2 
*c*x)/sqrt(4*a*c - b**2))*a**3*b**3*c**3*x**4 + 840*sqrt(4*a*c - b**2)*ata 
n((b + 2*c*x)/sqrt(4*a*c - b**2))*a**3*b**2*c**4*x**5 + 420*sqrt(4*a*c - b 
**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**3*b*c**5*x**6 - 12*sqrt(4*a*c 
 - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b**7*x**2 + 252*sqrt(4* 
a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b**6*c*x**3 - 168*sq 
rt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b**5*c**2*x**4 
- 840*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b**4*c* 
*3*x**5 - 420*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2 
*b**3*c**4*x**6 - 24*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2 
))*a*b**8*x**3 + 102*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2 
))*a*b**7*c*x**4 + 252*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b* 
*2))*a*b**6*c**2*x**5 + 126*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c 
 - b**2))*a*b**5*c**3*x**6 - 12*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqr...