\(\int \frac {x^2}{(a+b x+c x^2)^4} \, dx\) [261]

Optimal result

Integrand size = 16, antiderivative size = 165 \[ \int \frac {x^2}{\left (a+b x+c x^2\right )^4} \, dx=-\frac {a b+\left (b^2-2 a c\right ) x}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac {\left (b^2+a c\right ) (b+2 c x)}{3 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}-\frac {2 \left (b^2+a c\right ) (b+2 c x)}{\left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}+\frac {8 c \left (b^2+a c\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/2}} \] Output:

-1/3*(a*b+(-2*a*c+b^2)*x)/c/(-4*a*c+b^2)/(c*x^2+b*x+a)^3+1/3*(a*c+b^2)*(2* 
c*x+b)/c/(-4*a*c+b^2)^2/(c*x^2+b*x+a)^2-2*(a*c+b^2)*(2*c*x+b)/(-4*a*c+b^2) 
^3/(c*x^2+b*x+a)+8*c*(a*c+b^2)*arctanh((2*c*x+b)/(-4*a*c+b^2)^(1/2))/(-4*a 
*c+b^2)^(7/2)
 

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.01 \[ \int \frac {x^2}{\left (a+b x+c x^2\right )^4} \, dx=\frac {1}{3} \left (\frac {b^2 x+a (b-2 c x)}{c \left (-b^2+4 a c\right ) (a+x (b+c x))^3}+\frac {\left (b^2+a c\right ) (b+2 c x)}{c \left (b^2-4 a c\right )^2 (a+x (b+c x))^2}-\frac {6 \left (b^2+a c\right ) (b+2 c x)}{\left (b^2-4 a c\right )^3 (a+x (b+c x))}+\frac {24 c \left (b^2+a c\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{7/2}}\right ) \] Input:

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

Output:

((b^2*x + a*(b - 2*c*x))/(c*(-b^2 + 4*a*c)*(a + x*(b + c*x))^3) + ((b^2 + 
a*c)*(b + 2*c*x))/(c*(b^2 - 4*a*c)^2*(a + x*(b + c*x))^2) - (6*(b^2 + a*c) 
*(b + 2*c*x))/((b^2 - 4*a*c)^3*(a + x*(b + c*x))) + (24*c*(b^2 + a*c)*ArcT 
an[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(7/2))/3
 

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.07, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {1164, 27, 1159, 1086, 1083, 219}

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.

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

\(\Big \downarrow \) 1164

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1159

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

\(\Big \downarrow \) 1086

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

\(\Big \downarrow \) 1083

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

\(\Big \downarrow \) 219

\(\displaystyle \frac {x (2 a+b x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}-\frac {2 \left (-\frac {3 \left (a c+b^2\right ) \left (\frac {4 c \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac {b+2 c x}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\right )}{b^2-4 a c}-\frac {2 x \left (a c+b^2\right )+5 a b}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\right )}{3 \left (b^2-4 a c\right )}\)

Input:

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

Output:

(x*(2*a + b*x))/(3*(b^2 - 4*a*c)*(a + b*x + c*x^2)^3) - (2*(-1/2*(5*a*b + 
2*(b^2 + a*c)*x)/((b^2 - 4*a*c)*(a + b*x + c*x^2)^2) - (3*(b^2 + a*c)*(-(( 
b + 2*c*x)/((b^2 - 4*a*c)*(a + b*x + c*x^2))) + (4*c*ArcTanh[(b + 2*c*x)/S 
qrt[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(3/2)))/(b^2 - 4*a*c)))/(3*(b^2 - 4*a*c))
 

Defintions of rubi rules used

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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1083

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2   Subst[I 
nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, 
x]
 

rule 1086

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) 
*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] - Simp[2*c*((2*p + 
 3)/((p + 1)*(b^2 - 4*a*c)))   Int[(a + b*x + c*x^2)^(p + 1), x], x] /; Fre 
eQ[{a, b, c}, x] && ILtQ[p, -1]
 

rule 1159

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

rule 1164

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(408\) vs. \(2(157)=314\).

Time = 0.81 (sec) , antiderivative size = 409, normalized size of antiderivative = 2.48

method	result	size
default	\(\frac {\frac {4 c^{3} \left (a c +b^{2}\right ) x^{5}}{64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}+\frac {10 c^{2} \left (a c +b^{2}\right ) b \,x^{4}}{64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}+\frac {2 \left (16 a c +11 b^{2}\right ) c \left (a c +b^{2}\right ) x^{3}}{3 \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}+\frac {b \left (16 a c +b^{2}\right ) \left (a c +b^{2}\right ) x^{2}}{64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}-\frac {a \left (4 a^{2} c^{2}-22 c a \,b^{2}-b^{4}\right ) x}{64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}+\frac {a^{2} b \left (26 a c +b^{2}\right )}{192 a^{3} c^{3}-144 a^{2} b^{2} c^{2}+36 a \,b^{4} c -3 b^{6}}}{\left (c \,x^{2}+b x +a \right )^{3}}+\frac {8 c \left (a c +b^{2}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) \sqrt {4 a c -b^{2}}}\)	\(409\)
risch	\(\frac {\frac {4 c^{3} \left (a c +b^{2}\right ) x^{5}}{64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}+\frac {10 c^{2} \left (a c +b^{2}\right ) b \,x^{4}}{64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}+\frac {2 \left (16 a c +11 b^{2}\right ) c \left (a c +b^{2}\right ) x^{3}}{3 \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}+\frac {b \left (16 a c +b^{2}\right ) \left (a c +b^{2}\right ) x^{2}}{64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}-\frac {a \left (4 a^{2} c^{2}-22 c a \,b^{2}-b^{4}\right ) x}{64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}+\frac {a^{2} b \left (26 a c +b^{2}\right )}{192 a^{3} c^{3}-144 a^{2} b^{2} c^{2}+36 a \,b^{4} c -3 b^{6}}}{\left (c \,x^{2}+b x +a \right )^{3}}+\frac {4 c^{2} \ln \left (\left (-128 a^{3} c^{4}+96 a^{2} b^{2} c^{3}-24 a \,b^{4} c^{2}+2 b^{6} c \right ) x +\left (-4 a c +b^{2}\right )^{\frac {7}{2}}-64 a^{3} b \,c^{3}+48 a^{2} b^{3} c^{2}-12 a \,b^{5} c +b^{7}\right ) a}{\left (-4 a c +b^{2}\right )^{\frac {7}{2}}}+\frac {4 c \ln \left (\left (-128 a^{3} c^{4}+96 a^{2} b^{2} c^{3}-24 a \,b^{4} c^{2}+2 b^{6} c \right ) x +\left (-4 a c +b^{2}\right )^{\frac {7}{2}}-64 a^{3} b \,c^{3}+48 a^{2} b^{3} c^{2}-12 a \,b^{5} c +b^{7}\right ) b^{2}}{\left (-4 a c +b^{2}\right )^{\frac {7}{2}}}-\frac {4 c^{2} \ln \left (\left (128 a^{3} c^{4}-96 a^{2} b^{2} c^{3}+24 a \,b^{4} c^{2}-2 b^{6} c \right ) x +\left (-4 a c +b^{2}\right )^{\frac {7}{2}}+64 a^{3} b \,c^{3}-48 a^{2} b^{3} c^{2}+12 a \,b^{5} c -b^{7}\right ) a}{\left (-4 a c +b^{2}\right )^{\frac {7}{2}}}-\frac {4 c \ln \left (\left (128 a^{3} c^{4}-96 a^{2} b^{2} c^{3}+24 a \,b^{4} c^{2}-2 b^{6} c \right ) x +\left (-4 a c +b^{2}\right )^{\frac {7}{2}}+64 a^{3} b \,c^{3}-48 a^{2} b^{3} c^{2}+12 a \,b^{5} c -b^{7}\right ) b^{2}}{\left (-4 a c +b^{2}\right )^{\frac {7}{2}}}\)	\(717\)

Input:

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

Output:

(4*c^3*(a*c+b^2)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x^5+10*c^2*(a* 
c+b^2)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*b*x^4+2/3*(16*a*c+11*b^2 
)*c*(a*c+b^2)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x^3+b*(16*a*c+b^2 
)*(a*c+b^2)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x^2-a*(4*a^2*c^2-22 
*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+1/3*a^2*b*(26*a 
*c+b^2)/(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+8*c*(a 
*c+b^2)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)/(4*a*c-b^2)^(1/2)*arcta 
n((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. 756 vs. \(2 (157) = 314\).

Time = 0.12 (sec) , antiderivative size = 1533, normalized size of antiderivative = 9.29 \[ \int \frac {x^2}{\left (a+b x+c x^2\right )^4} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/3*(a^2*b^5 + 22*a^3*b^3*c - 104*a^4*b*c^2 + 12*(b^4*c^3 - 3*a*b^2*c^4 
- 4*a^2*c^5)*x^5 + 30*(b^5*c^2 - 3*a*b^3*c^3 - 4*a^2*b*c^4)*x^4 + 2*(11*b^ 
6*c - 17*a*b^4*c^2 - 92*a^2*b^2*c^3 - 64*a^3*c^4)*x^3 + 3*(b^7 + 13*a*b^5* 
c - 52*a^2*b^3*c^2 - 64*a^3*b*c^3)*x^2 + 12*((b^2*c^4 + a*c^5)*x^6 + a^3*b 
^2*c + a^4*c^2 + 3*(b^3*c^3 + a*b*c^4)*x^5 + 3*(b^4*c^2 + 2*a*b^2*c^3 + a^ 
2*c^4)*x^4 + (b^5*c + 7*a*b^3*c^2 + 6*a^2*b*c^3)*x^3 + 3*(a*b^4*c + 2*a^2* 
b^2*c^2 + a^3*c^3)*x^2 + 3*(a^2*b^3*c + a^3*b*c^2)*x)*sqrt(b^2 - 4*a*c)*lo 
g((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)) + 3*(a*b^6 + 18*a^2*b^4*c - 92*a^3*b^2*c^2 + 16*a^4*c^3)*x) 
/(a^3*b^8 - 16*a^4*b^6*c + 96*a^5*b^4*c^2 - 256*a^6*b^2*c^3 + 256*a^7*c^4 
+ (b^8*c^3 - 16*a*b^6*c^4 + 96*a^2*b^4*c^5 - 256*a^3*b^2*c^6 + 256*a^4*c^7 
)*x^6 + 3*(b^9*c^2 - 16*a*b^7*c^3 + 96*a^2*b^5*c^4 - 256*a^3*b^3*c^5 + 256 
*a^4*b*c^6)*x^5 + 3*(b^10*c - 15*a*b^8*c^2 + 80*a^2*b^6*c^3 - 160*a^3*b^4* 
c^4 + 256*a^5*c^6)*x^4 + (b^11 - 10*a*b^9*c + 320*a^3*b^5*c^3 - 1280*a^4*b 
^3*c^4 + 1536*a^5*b*c^5)*x^3 + 3*(a*b^10 - 15*a^2*b^8*c + 80*a^3*b^6*c^2 - 
 160*a^4*b^4*c^3 + 256*a^6*c^5)*x^2 + 3*(a^2*b^9 - 16*a^3*b^7*c + 96*a^4*b 
^5*c^2 - 256*a^5*b^3*c^3 + 256*a^6*b*c^4)*x), -1/3*(a^2*b^5 + 22*a^3*b^3*c 
 - 104*a^4*b*c^2 + 12*(b^4*c^3 - 3*a*b^2*c^4 - 4*a^2*c^5)*x^5 + 30*(b^5*c^ 
2 - 3*a*b^3*c^3 - 4*a^2*b*c^4)*x^4 + 2*(11*b^6*c - 17*a*b^4*c^2 - 92*a^2*b 
^2*c^3 - 64*a^3*c^4)*x^3 + 3*(b^7 + 13*a*b^5*c - 52*a^2*b^3*c^2 - 64*a^...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 920 vs. \(2 (160) = 320\).

Time = 1.06 (sec) , antiderivative size = 920, normalized size of antiderivative = 5.58 \[ \int \frac {x^2}{\left (a+b x+c x^2\right )^4} \, dx =\text {Too large to display} \] Input:

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

Output:

-4*c*sqrt(-1/(4*a*c - b**2)**7)*(a*c + b**2)*log(x + (-1024*a**4*c**5*sqrt 
(-1/(4*a*c - b**2)**7)*(a*c + b**2) + 1024*a**3*b**2*c**4*sqrt(-1/(4*a*c - 
 b**2)**7)*(a*c + b**2) - 384*a**2*b**4*c**3*sqrt(-1/(4*a*c - b**2)**7)*(a 
*c + b**2) + 64*a*b**6*c**2*sqrt(-1/(4*a*c - b**2)**7)*(a*c + b**2) + 4*a* 
b*c**2 - 4*b**8*c*sqrt(-1/(4*a*c - b**2)**7)*(a*c + b**2) + 4*b**3*c)/(8*a 
*c**3 + 8*b**2*c**2)) + 4*c*sqrt(-1/(4*a*c - b**2)**7)*(a*c + b**2)*log(x 
+ (1024*a**4*c**5*sqrt(-1/(4*a*c - b**2)**7)*(a*c + b**2) - 1024*a**3*b**2 
*c**4*sqrt(-1/(4*a*c - b**2)**7)*(a*c + b**2) + 384*a**2*b**4*c**3*sqrt(-1 
/(4*a*c - b**2)**7)*(a*c + b**2) - 64*a*b**6*c**2*sqrt(-1/(4*a*c - b**2)** 
7)*(a*c + b**2) + 4*a*b*c**2 + 4*b**8*c*sqrt(-1/(4*a*c - b**2)**7)*(a*c + 
b**2) + 4*b**3*c)/(8*a*c**3 + 8*b**2*c**2)) + (26*a**3*b*c + a**2*b**3 + x 
**5*(12*a*c**4 + 12*b**2*c**3) + x**4*(30*a*b*c**3 + 30*b**3*c**2) + x**3* 
(32*a**2*c**3 + 54*a*b**2*c**2 + 22*b**4*c) + x**2*(48*a**2*b*c**2 + 51*a* 
b**3*c + 3*b**5) + x*(-12*a**3*c**2 + 66*a**2*b**2*c + 3*a*b**4))/(192*a** 
6*c**3 - 144*a**5*b**2*c**2 + 36*a**4*b**4*c - 3*a**3*b**6 + x**6*(192*a** 
3*c**6 - 144*a**2*b**2*c**5 + 36*a*b**4*c**4 - 3*b**6*c**3) + x**5*(576*a* 
*3*b*c**5 - 432*a**2*b**3*c**4 + 108*a*b**5*c**3 - 9*b**7*c**2) + x**4*(57 
6*a**4*c**5 + 144*a**3*b**2*c**4 - 324*a**2*b**4*c**3 + 99*a*b**6*c**2 - 9 
*b**8*c) + x**3*(1152*a**4*b*c**4 - 672*a**3*b**3*c**3 + 72*a**2*b**5*c**2 
 + 18*a*b**7*c - 3*b**9) + x**2*(576*a**5*c**4 + 144*a**4*b**2*c**3 - 3...
 

Maxima [F(-2)]

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

integrate(x^2/(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 = 0.16 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.62 \[ \int \frac {x^2}{\left (a+b x+c x^2\right )^4} \, dx=-\frac {8 \, {\left (b^{2} c + a c^{2}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {12 \, b^{2} c^{3} x^{5} + 12 \, a c^{4} x^{5} + 30 \, b^{3} c^{2} x^{4} + 30 \, a b c^{3} x^{4} + 22 \, b^{4} c x^{3} + 54 \, a b^{2} c^{2} x^{3} + 32 \, a^{2} c^{3} x^{3} + 3 \, b^{5} x^{2} + 51 \, a b^{3} c x^{2} + 48 \, a^{2} b c^{2} x^{2} + 3 \, a b^{4} x + 66 \, a^{2} b^{2} c x - 12 \, a^{3} c^{2} x + a^{2} b^{3} + 26 \, a^{3} b c}{3 \, {\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} {\left (c x^{2} + b x + a\right )}^{3}} \] Input:

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

Output:

-8*(b^2*c + a*c^2)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^6 - 12*a*b^4 
*c + 48*a^2*b^2*c^2 - 64*a^3*c^3)*sqrt(-b^2 + 4*a*c)) - 1/3*(12*b^2*c^3*x^ 
5 + 12*a*c^4*x^5 + 30*b^3*c^2*x^4 + 30*a*b*c^3*x^4 + 22*b^4*c*x^3 + 54*a*b 
^2*c^2*x^3 + 32*a^2*c^3*x^3 + 3*b^5*x^2 + 51*a*b^3*c*x^2 + 48*a^2*b*c^2*x^ 
2 + 3*a*b^4*x + 66*a^2*b^2*c*x - 12*a^3*c^2*x + a^2*b^3 + 26*a^3*b*c)/((b^ 
6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3)*(c*x^2 + b*x + a)^3)
 

Mupad [B] (verification not implemented)

Time = 9.31 (sec) , antiderivative size = 574, normalized size of antiderivative = 3.48 \[ \int \frac {x^2}{\left (a+b x+c x^2\right )^4} \, dx=-\frac {\frac {26\,c\,a^3\,b+a^2\,b^3}{3\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}+\frac {2\,x^3\,\left (16\,a^2\,c^3+27\,a\,b^2\,c^2+11\,b^4\,c\right )}{3\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}+\frac {a\,x\,\left (-4\,a^2\,c^2+22\,a\,b^2\,c+b^4\right )}{-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}+\frac {x^2\,\left (b^2+a\,c\right )\,\left (b^3+16\,a\,c\,b\right )}{-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}+\frac {4\,c^3\,x^5\,\left (b^2+a\,c\right )}{-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}+\frac {10\,b\,c^2\,x^4\,\left (b^2+a\,c\right )}{-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}}{x^2\,\left (3\,c\,a^2+3\,a\,b^2\right )+x^4\,\left (3\,b^2\,c+3\,a\,c^2\right )+a^3+x^3\,\left (b^3+6\,a\,c\,b\right )+c^3\,x^6+3\,b\,c^2\,x^5+3\,a^2\,b\,x}-\frac {8\,c\,\mathrm {atan}\left (\frac {\left (\frac {8\,c^2\,x\,\left (b^2+a\,c\right )}{{\left (4\,a\,c-b^2\right )}^{7/2}}+\frac {4\,c\,\left (b^2+a\,c\right )\,\left (-64\,a^3\,b\,c^3+48\,a^2\,b^3\,c^2-12\,a\,b^5\,c+b^7\right )}{{\left (4\,a\,c-b^2\right )}^{7/2}\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}\right )\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}{4\,b^2\,c+4\,a\,c^2}\right )\,\left (b^2+a\,c\right )}{{\left (4\,a\,c-b^2\right )}^{7/2}} \] Input:

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

Output:

- ((a^2*b^3 + 26*a^3*b*c)/(3*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4 
*c)) + (2*x^3*(11*b^4*c + 16*a^2*c^3 + 27*a*b^2*c^2))/(3*(b^6 - 64*a^3*c^3 
 + 48*a^2*b^2*c^2 - 12*a*b^4*c)) + (a*x*(b^4 - 4*a^2*c^2 + 22*a*b^2*c))/(b 
^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c) + (x^2*(a*c + b^2)*(b^3 + 1 
6*a*b*c))/(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c) + (4*c^3*x^5*(a 
*c + b^2))/(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c) + (10*b*c^2*x^ 
4*(a*c + b^2))/(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) - (8*c*atan((((8*c^2*x*(a*c + b^2))/(4*a* 
c - b^2)^(7/2) + (4*c*(a*c + b^2)*(b^7 - 64*a^3*b*c^3 + 48*a^2*b^3*c^2 - 1 
2*a*b^5*c))/((4*a*c - b^2)^(7/2)*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a 
*b^4*c)))*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c))/(4*a*c^2 + 4*b 
^2*c))*(a*c + b^2))/(4*a*c - b^2)^(7/2)
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 1404, normalized size of antiderivative = 8.51 \[ \int \frac {x^2}{\left (a+b x+c x^2\right )^4} \, dx =\text {Too large to display} \] Input:

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

Output:

(24*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**4*b*c**2 + 
24*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**3*b**3*c + 7 
2*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**3*b**2*c**2*x 
 + 72*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**3*b*c**3* 
x**2 + 72*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b** 
4*c*x + 144*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b 
**3*c**2*x**2 + 144*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2) 
)*a**2*b**2*c**3*x**3 + 72*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c 
- b**2))*a**2*b*c**4*x**4 + 72*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4* 
a*c - b**2))*a*b**5*c*x**2 + 168*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt( 
4*a*c - b**2))*a*b**4*c**2*x**3 + 144*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/ 
sqrt(4*a*c - b**2))*a*b**3*c**3*x**4 + 72*sqrt(4*a*c - b**2)*atan((b + 2*c 
*x)/sqrt(4*a*c - b**2))*a*b**2*c**4*x**5 + 24*sqrt(4*a*c - b**2)*atan((b + 
 2*c*x)/sqrt(4*a*c - b**2))*a*b*c**5*x**6 + 24*sqrt(4*a*c - b**2)*atan((b 
+ 2*c*x)/sqrt(4*a*c - b**2))*b**6*c*x**3 + 72*sqrt(4*a*c - b**2)*atan((b + 
 2*c*x)/sqrt(4*a*c - b**2))*b**5*c**2*x**4 + 72*sqrt(4*a*c - b**2)*atan((b 
 + 2*c*x)/sqrt(4*a*c - b**2))*b**4*c**3*x**5 + 24*sqrt(4*a*c - b**2)*atan( 
(b + 2*c*x)/sqrt(4*a*c - b**2))*b**3*c**4*x**6 - 16*a**5*c**3 + 92*a**4*b* 
*2*c**2 - 96*a**4*b*c**3*x - 48*a**4*c**4*x**2 - 18*a**3*b**4*c + 240*a**3 
*b**3*c**2*x + 108*a**3*b**2*c**3*x**2 + 32*a**3*b*c**4*x**3 - 48*a**3*...