\(\int \frac {1}{(a+b x+c x^2)^4} \, dx\) [263]

Optimal result

Integrand size = 12, antiderivative size = 136 \[ \int \frac {1}{\left (a+b x+c x^2\right )^4} \, dx=-\frac {b+2 c x}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac {5 c (b+2 c x)}{3 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}-\frac {10 c^2 (b+2 c x)}{\left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}+\frac {40 c^3 \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/2}} \] Output:

-1/3*(2*c*x+b)/(-4*a*c+b^2)/(c*x^2+b*x+a)^3+5/3*c*(2*c*x+b)/(-4*a*c+b^2)^2 
/(c*x^2+b*x+a)^2-10*c^2*(2*c*x+b)/(-4*a*c+b^2)^3/(c*x^2+b*x+a)+40*c^3*arct 
anh((2*c*x+b)/(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(7/2)
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.99 \[ \int \frac {1}{\left (a+b x+c x^2\right )^4} \, dx=-\frac {\frac {\left (b^2-4 a c\right )^2 (b+2 c x)}{(a+x (b+c x))^3}-\frac {5 c \left (b^2-4 a c\right ) (b+2 c x)}{(a+x (b+c x))^2}+\frac {30 c^2 (b+2 c x)}{a+x (b+c x)}+\frac {120 c^3 \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}}{3 \left (b^2-4 a c\right )^3} \] Input:

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

Output:

-1/3*(((b^2 - 4*a*c)^2*(b + 2*c*x))/(a + x*(b + c*x))^3 - (5*c*(b^2 - 4*a* 
c)*(b + 2*c*x))/(a + x*(b + c*x))^2 + (30*c^2*(b + 2*c*x))/(a + x*(b + c*x 
)) + (120*c^3*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c])/ 
(b^2 - 4*a*c)^3
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.18, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {1086, 1086, 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.

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

\(\Big \downarrow \) 1086

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

\(\Big \downarrow \) 1086

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

\(\Big \downarrow \) 1086

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

\(\Big \downarrow \) 1083

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

\(\Big \downarrow \) 219

\(\displaystyle -\frac {10 c \left (-\frac {3 c \left (\frac {4 c \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac {b+2 c x}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\right )}{b^2-4 a c}-\frac {b+2 c x}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\right )}{3 \left (b^2-4 a c\right )}-\frac {b+2 c x}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}\)

Input:

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

Output:

-1/3*(b + 2*c*x)/((b^2 - 4*a*c)*(a + b*x + c*x^2)^3) - (10*c*(-1/2*(b + 2* 
c*x)/((b^2 - 4*a*c)*(a + b*x + c*x^2)^2) - (3*c*(-((b + 2*c*x)/((b^2 - 4*a 
*c)*(a + b*x + c*x^2))) + (4*c*ArcTanh[(b + 2*c*x)/Sqrt[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 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]
 

Maple [A] (verified)

Time = 0.76 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.21

method	result	size
default	\(\frac {2 c x +b}{3 \left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{3}}+\frac {10 c \left (\frac {2 c x +b}{2 \left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{2}}+\frac {3 c \left (\frac {2 c x +b}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )}+\frac {4 c \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}}}\right )}{4 a c -b^{2}}\right )}{3 \left (4 a c -b^{2}\right )}\)	\(164\)
risch	\(\frac {\frac {20 c^{5} x^{5}}{64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}+\frac {50 b \,c^{4} x^{4}}{64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}+\frac {10 \left (16 a c +11 b^{2}\right ) c^{3} x^{3}}{3 \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}+\frac {5 b \left (16 a c +b^{2}\right ) c^{2} x^{2}}{64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}+\frac {c \left (44 a^{2} c^{2}+18 c a \,b^{2}-b^{4}\right ) x}{64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}+\frac {b \left (66 a^{2} c^{2}-13 c a \,b^{2}+b^{4}\right )}{192 a^{3} c^{3}-144 a^{2} b^{2} c^{2}+36 a \,b^{4} c -3 b^{6}}}{\left (c \,x^{2}+b x +a \right )^{3}}+\frac {20 c^{3} \ln \left (\left (-128 a^{3} c^{4}+96 a^{2} b^{2} c^{3}-24 a \,b^{4} c^{2}+2 b^{6} c \right ) x +\left (-4 a c +b^{2}\right )^{\frac {7}{2}}-64 a^{3} b \,c^{3}+48 a^{2} b^{3} c^{2}-12 a \,b^{5} c +b^{7}\right )}{\left (-4 a c +b^{2}\right )^{\frac {7}{2}}}-\frac {20 c^{3} \ln \left (\left (128 a^{3} c^{4}-96 a^{2} b^{2} c^{3}+24 a \,b^{4} c^{2}-2 b^{6} c \right ) x +\left (-4 a c +b^{2}\right )^{\frac {7}{2}}+64 a^{3} b \,c^{3}-48 a^{2} b^{3} c^{2}+12 a \,b^{5} c -b^{7}\right )}{\left (-4 a c +b^{2}\right )^{\frac {7}{2}}}\)	\(508\)

Input:

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

Output:

1/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^3+10/3*c/(4*a*c-b^2)*(1/2*(2*c*x+b 
)/(4*a*c-b^2)/(c*x^2+b*x+a)^2+3*c/(4*a*c-b^2)*((2*c*x+b)/(4*a*c-b^2)/(c*x^ 
2+b*x+a)+4*c/(4*a*c-b^2)^(3/2)*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. 658 vs. \(2 (128) = 256\).

Time = 0.10 (sec) , antiderivative size = 1337, normalized size of antiderivative = 9.83 \[ \int \frac {1}{\left (a+b x+c x^2\right )^4} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/3*(b^7 - 17*a*b^5*c + 118*a^2*b^3*c^2 - 264*a^3*b*c^3 + 60*(b^2*c^5 - 
4*a*c^6)*x^5 + 150*(b^3*c^4 - 4*a*b*c^5)*x^4 + 10*(11*b^4*c^3 - 28*a*b^2*c 
^4 - 64*a^2*c^5)*x^3 + 15*(b^5*c^2 + 12*a*b^3*c^3 - 64*a^2*b*c^4)*x^2 + 60 
*(c^6*x^6 + 3*b*c^5*x^5 + 3*a^2*b*c^3*x + a^3*c^3 + 3*(b^2*c^4 + a*c^5)*x^ 
4 + (b^3*c^3 + 6*a*b*c^4)*x^3 + 3*(a*b^2*c^3 + a^2*c^4)*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)) - 3*(b^6*c - 22*a*b^4*c^2 + 28*a^2*b^2*c^3 + 176*a^3 
*c^4)*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 - 12 
80*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*(b^7 - 17*a*b^5 
*c + 118*a^2*b^3*c^2 - 264*a^3*b*c^3 + 60*(b^2*c^5 - 4*a*c^6)*x^5 + 150*(b 
^3*c^4 - 4*a*b*c^5)*x^4 + 10*(11*b^4*c^3 - 28*a*b^2*c^4 - 64*a^2*c^5)*x^3 
+ 15*(b^5*c^2 + 12*a*b^3*c^3 - 64*a^2*b*c^4)*x^2 - 120*(c^6*x^6 + 3*b*c^5* 
x^5 + 3*a^2*b*c^3*x + a^3*c^3 + 3*(b^2*c^4 + a*c^5)*x^4 + (b^3*c^3 + 6*a*b 
*c^4)*x^3 + 3*(a*b^2*c^3 + a^2*c^4)*x^2)*sqrt(-b^2 + 4*a*c)*arctan(-sqr...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 777 vs. \(2 (134) = 268\).

Time = 0.90 (sec) , antiderivative size = 777, normalized size of antiderivative = 5.71 \[ \int \frac {1}{\left (a+b x+c x^2\right )^4} \, dx=- 20 c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \log {\left (x + \frac {- 5120 a^{4} c^{7} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} + 5120 a^{3} b^{2} c^{6} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} - 1920 a^{2} b^{4} c^{5} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} + 320 a b^{6} c^{4} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} - 20 b^{8} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} + 20 b c^{3}}{40 c^{4}} \right )} + 20 c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \log {\left (x + \frac {5120 a^{4} c^{7} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} - 5120 a^{3} b^{2} c^{6} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} + 1920 a^{2} b^{4} c^{5} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} - 320 a b^{6} c^{4} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} + 20 b^{8} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} + 20 b c^{3}}{40 c^{4}} \right )} + \frac {66 a^{2} b c^{2} - 13 a b^{3} c + b^{5} + 150 b c^{4} x^{4} + 60 c^{5} x^{5} + x^{3} \cdot \left (160 a c^{4} + 110 b^{2} c^{3}\right ) + x^{2} \cdot \left (240 a b c^{3} + 15 b^{3} c^{2}\right ) + x \left (132 a^{2} c^{3} + 54 a b^{2} c^{2} - 3 b^{4} c\right )}{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} \cdot \left (192 a^{3} c^{6} - 144 a^{2} b^{2} c^{5} + 36 a b^{4} c^{4} - 3 b^{6} c^{3}\right ) + x^{5} \cdot \left (576 a^{3} b c^{5} - 432 a^{2} b^{3} c^{4} + 108 a b^{5} c^{3} - 9 b^{7} c^{2}\right ) + x^{4} \cdot \left (576 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\right ) + x^{3} \cdot \left (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}\right ) + x^{2} \cdot \left (576 a^{5} c^{4} + 144 a^{4} b^{2} c^{3} - 324 a^{3} b^{4} c^{2} + 99 a^{2} b^{6} c - 9 a b^{8}\right ) + x \left (576 a^{5} b c^{3} - 432 a^{4} b^{3} c^{2} + 108 a^{3} b^{5} c - 9 a^{2} b^{7}\right )} \] Input:

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

Output:

-20*c**3*sqrt(-1/(4*a*c - b**2)**7)*log(x + (-5120*a**4*c**7*sqrt(-1/(4*a* 
c - b**2)**7) + 5120*a**3*b**2*c**6*sqrt(-1/(4*a*c - b**2)**7) - 1920*a**2 
*b**4*c**5*sqrt(-1/(4*a*c - b**2)**7) + 320*a*b**6*c**4*sqrt(-1/(4*a*c - b 
**2)**7) - 20*b**8*c**3*sqrt(-1/(4*a*c - b**2)**7) + 20*b*c**3)/(40*c**4)) 
 + 20*c**3*sqrt(-1/(4*a*c - b**2)**7)*log(x + (5120*a**4*c**7*sqrt(-1/(4*a 
*c - b**2)**7) - 5120*a**3*b**2*c**6*sqrt(-1/(4*a*c - b**2)**7) + 1920*a** 
2*b**4*c**5*sqrt(-1/(4*a*c - b**2)**7) - 320*a*b**6*c**4*sqrt(-1/(4*a*c - 
b**2)**7) + 20*b**8*c**3*sqrt(-1/(4*a*c - b**2)**7) + 20*b*c**3)/(40*c**4) 
) + (66*a**2*b*c**2 - 13*a*b**3*c + b**5 + 150*b*c**4*x**4 + 60*c**5*x**5 
+ x**3*(160*a*c**4 + 110*b**2*c**3) + x**2*(240*a*b*c**3 + 15*b**3*c**2) + 
 x*(132*a**2*c**3 + 54*a*b**2*c**2 - 3*b**4*c))/(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*(576*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*(1 
152*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 - 324*a**3*b**4*c**2 + 9 
9*a**2*b**6*c - 9*a*b**8) + x*(576*a**5*b*c**3 - 432*a**4*b**3*c**2 + 108* 
a**3*b**5*c - 9*a**2*b**7))
 

Maxima [F(-2)]

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

integrate(1/(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 = 220, normalized size of antiderivative = 1.62 \[ \int \frac {1}{\left (a+b x+c x^2\right )^4} \, dx=-\frac {40 \, c^{3} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {60 \, c^{5} x^{5} + 150 \, b c^{4} x^{4} + 110 \, b^{2} c^{3} x^{3} + 160 \, a c^{4} x^{3} + 15 \, b^{3} c^{2} x^{2} + 240 \, a b c^{3} x^{2} - 3 \, b^{4} c x + 54 \, a b^{2} c^{2} x + 132 \, a^{2} c^{3} x + b^{5} - 13 \, a b^{3} c + 66 \, a^{2} b c^{2}}{3 \, {\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} {\left (c x^{2} + b x + a\right )}^{3}} \] Input:

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

Output:

-40*c^3*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*(60*c^5*x^5 + 150*b*c^4*x 
^4 + 110*b^2*c^3*x^3 + 160*a*c^4*x^3 + 15*b^3*c^2*x^2 + 240*a*b*c^3*x^2 - 
3*b^4*c*x + 54*a*b^2*c^2*x + 132*a^2*c^3*x + b^5 - 13*a*b^3*c + 66*a^2*b*c 
^2)/((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 [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b x+c x^2\right )^4} \, dx=\left \{\begin {array}{cl} \frac {20\,\left (\frac {b}{2}+c\,x\right )\,\left (\frac {c^2}{6\,{\left (4\,a\,c-b^2\right )}^2\,{\left (c\,x^2+b\,x+a\right )}^2}+\frac {c^3}{{\left (4\,a\,c-b^2\right )}^3\,\left (c\,x^2+b\,x+a\right )}+\frac {c}{30\,\left (4\,a\,c-b^2\right )\,{\left (c\,x^2+b\,x+a\right )}^3}\right )}{c}-\frac {20\,c^3\,\ln \left (\frac {\frac {b}{2}-\sqrt {\frac {b^2}{4}-a\,c}+c\,x}{\frac {b}{2}+\sqrt {\frac {b^2}{4}-a\,c}+c\,x}\right )}{{\left (b^2-4\,a\,c\right )}^{7/2}} & \text {\ if\ \ }0<b^2-4\,a\,c\\ \frac {20\,\left (\frac {b}{2}+c\,x\right )\,\left (\frac {c^2}{6\,{\left (4\,a\,c-b^2\right )}^2\,{\left (c\,x^2+b\,x+a\right )}^2}+\frac {c^3}{{\left (4\,a\,c-b^2\right )}^3\,\left (c\,x^2+b\,x+a\right )}+\frac {c}{30\,\left (4\,a\,c-b^2\right )\,{\left (c\,x^2+b\,x+a\right )}^3}\right )}{c}+\frac {20\,c^3\,\mathrm {atan}\left (\frac {\frac {b}{2}+c\,x}{\sqrt {a\,c-\frac {b^2}{4}}}\right )}{\sqrt {a\,c-\frac {b^2}{4}}\,{\left (4\,a\,c-b^2\right )}^3} & \text {\ if\ \ }b^2-4\,a\,c<0\\ \int \frac {1}{{\left (c\,x^2+b\,x+a\right )}^4} \,d x & \text {\ if\ \ }b^2-4\,a\,c\notin \mathbb {R}\vee b^2=4\,a\,c \end {array}\right . \] Input:

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

Output:

piecewise(0 < - 4*a*c + b^2, - (20*c^3*log((b/2 - (- a*c + b^2/4)^(1/2) + 
c*x)/(b/2 + (- a*c + b^2/4)^(1/2) + c*x)))/(- 4*a*c + b^2)^(7/2) + (20*(b/ 
2 + c*x)*(c^2/(6*(4*a*c - b^2)^2*(a + b*x + c*x^2)^2) + c^3/((4*a*c - b^2) 
^3*(a + b*x + c*x^2)) + c/(30*(4*a*c - b^2)*(a + b*x + c*x^2)^3)))/c, - 4* 
a*c + b^2 < 0, (20*(b/2 + c*x)*(c^2/(6*(4*a*c - b^2)^2*(a + b*x + c*x^2)^2 
) + c^3/((4*a*c - b^2)^3*(a + b*x + c*x^2)) + c/(30*(4*a*c - b^2)*(a + b*x 
 + c*x^2)^3)))/c + (20*c^3*atan((b/2 + c*x)/(a*c - b^2/4)^(1/2)))/((a*c - 
b^2/4)^(1/2)*(4*a*c - b^2)^3), ~in(- 4*a*c + b^2, 'real') | b^2 == 4*a*c, 
int(1/(a + b*x + c*x^2)^4, x))
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 1056, normalized size of antiderivative = 7.76 \[ \int \frac {1}{\left (a+b x+c x^2\right )^4} \, dx =\text {Too large to display} \] Input:

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

Output:

(120*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**3*b*c**3 + 
 360*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b**2*c** 
3*x + 360*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b*c 
**4*x**2 + 360*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b 
**3*c**3*x**2 + 720*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2) 
)*a*b**2*c**4*x**3 + 360*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - 
b**2))*a*b*c**5*x**4 + 120*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c 
- b**2))*b**4*c**3*x**3 + 360*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a 
*c - b**2))*b**3*c**4*x**4 + 360*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt( 
4*a*c - b**2))*b**2*c**5*x**5 + 120*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sq 
rt(4*a*c - b**2))*b*c**6*x**6 - 80*a**4*c**4 + 284*a**3*b**2*c**3 + 288*a* 
*3*b*c**4*x - 240*a**3*c**5*x**2 - 118*a**2*b**4*c**2 + 144*a**2*b**3*c**3 
*x + 780*a**2*b**2*c**4*x**2 + 160*a**2*b*c**5*x**3 - 240*a**2*c**6*x**4 + 
 17*a*b**6*c - 66*a*b**5*c**2*x - 120*a*b**4*c**3*x**2 + 320*a*b**3*c**4*x 
**3 + 420*a*b**2*c**5*x**4 - 80*a*c**7*x**6 - b**8 + 3*b**7*c*x - 15*b**6* 
c**2*x**2 - 90*b**5*c**3*x**3 - 90*b**4*c**4*x**4 + 20*b**2*c**6*x**6)/(3* 
b*(256*a**7*c**4 - 256*a**6*b**2*c**3 + 768*a**6*b*c**4*x + 768*a**6*c**5* 
x**2 + 96*a**5*b**4*c**2 - 768*a**5*b**3*c**3*x + 1536*a**5*b*c**5*x**3 + 
768*a**5*c**6*x**4 - 16*a**4*b**6*c + 288*a**4*b**5*c**2*x - 480*a**4*b**4 
*c**3*x**2 - 1280*a**4*b**3*c**4*x**3 + 768*a**4*b*c**6*x**5 + 256*a**4...