\(\int \frac {(d+e x)^2}{(a+b x+c x^2)^4} \, dx\) [494]

Optimal result

Integrand size = 20, antiderivative size = 266 \[ \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^4} \, dx=-\frac {(b+2 c x) (d+e x)^2}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac {(d+e x) \left (5 b c d-b^2 e-6 a c e+5 c (2 c d-b e) x\right )}{3 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}+\frac {20 b^2 c d e+40 a c^2 d e-b^3 e^2-2 b c \left (15 c d^2+13 a e^2\right )-12 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) x}{3 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}+\frac {8 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/2}} \] Output:

-1/3*(2*c*x+b)*(e*x+d)^2/(-4*a*c+b^2)/(c*x^2+b*x+a)^3+1/3*(e*x+d)*(5*b*c*d 
-b^2*e-6*a*c*e+5*c*(-b*e+2*c*d)*x)/(-4*a*c+b^2)^2/(c*x^2+b*x+a)^2+1/3*(20* 
b^2*c*d*e+40*a*c^2*d*e-b^3*e^2-2*b*c*(13*a*e^2+15*c*d^2)-12*c*(5*c^2*d^2+b 
^2*e^2-c*e*(-a*e+5*b*d))*x)/(-4*a*c+b^2)^3/(c*x^2+b*x+a)+8*c*(5*c^2*d^2+b^ 
2*e^2-c*e*(-a*e+5*b*d))*arctanh((2*c*x+b)/(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2) 
^(7/2)
 

Mathematica [A] (verified)

Time = 0.37 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.97 \[ \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^4} \, dx=\frac {1}{3} \left (\frac {\left (5 c^2 d^2+b^2 e^2+c e (-5 b d+a e)\right ) (b+2 c x)}{c \left (b^2-4 a c\right )^2 (a+x (b+c x))^2}-\frac {6 \left (5 c^2 d^2+b^2 e^2+c e (-5 b d+a e)\right ) (b+2 c x)}{\left (b^2-4 a c\right )^3 (a+x (b+c x))}+\frac {a b e^2+2 c^2 d^2 x+b^2 e^2 x+b c d (d-2 e x)-2 a c e (2 d+e x)}{c \left (-b^2+4 a c\right ) (a+x (b+c x))^3}+\frac {24 c \left (5 c^2 d^2+b^2 e^2+c e (-5 b d+a e)\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{7/2}}\right ) \] Input:

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

Output:

(((5*c^2*d^2 + b^2*e^2 + c*e*(-5*b*d + a*e))*(b + 2*c*x))/(c*(b^2 - 4*a*c) 
^2*(a + x*(b + c*x))^2) - (6*(5*c^2*d^2 + b^2*e^2 + c*e*(-5*b*d + a*e))*(b 
 + 2*c*x))/((b^2 - 4*a*c)^3*(a + x*(b + c*x))) + (a*b*e^2 + 2*c^2*d^2*x + 
b^2*e^2*x + b*c*d*(d - 2*e*x) - 2*a*c*e*(2*d + e*x))/(c*(-b^2 + 4*a*c)*(a 
+ x*(b + c*x))^3) + (24*c*(5*c^2*d^2 + b^2*e^2 + c*e*(-5*b*d + a*e))*ArcTa 
n[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(7/2))/3
 

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 259, normalized size of antiderivative = 0.97, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, 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.

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

\(\Big \downarrow \) 1164

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1159

\(\displaystyle -\frac {2 \left (\frac {-2 x \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )-5 b \left (a e^2+c d^2\right )+8 a c d e+3 b^2 d e}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {3 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right ) \int \frac {1}{\left (c x^2+b x+a\right )^2}dx}{b^2-4 a c}\right )}{3 \left (b^2-4 a c\right )}-\frac {(d+e x) (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}\)

\(\Big \downarrow \) 1086

\(\displaystyle -\frac {2 \left (\frac {-2 x \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )-5 b \left (a e^2+c d^2\right )+8 a c d e+3 b^2 d e}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {3 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right ) \left (-\frac {2 c \int \frac {1}{c x^2+b x+a}dx}{b^2-4 a c}-\frac {b+2 c x}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\right )}{b^2-4 a c}\right )}{3 \left (b^2-4 a c\right )}-\frac {(d+e x) (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}\)

\(\Big \downarrow \) 1083

\(\displaystyle -\frac {2 \left (\frac {-2 x \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )-5 b \left (a e^2+c d^2\right )+8 a c d e+3 b^2 d e}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {3 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right ) \left (\frac {4 c \int \frac {1}{b^2-(b+2 c x)^2-4 a c}d(b+2 c x)}{b^2-4 a c}-\frac {b+2 c x}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\right )}{b^2-4 a c}\right )}{3 \left (b^2-4 a c\right )}-\frac {(d+e x) (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {2 \left (\frac {-2 x \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )-5 b \left (a e^2+c d^2\right )+8 a c d e+3 b^2 d e}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {3 \left (\frac {4 c \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac {b+2 c x}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{b^2-4 a c}\right )}{3 \left (b^2-4 a c\right )}-\frac {(d+e x) (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}\)

Input:

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

Output:

-1/3*((d + e*x)*(b*d - 2*a*e + (2*c*d - b*e)*x))/((b^2 - 4*a*c)*(a + b*x + 
 c*x^2)^3) - (2*((3*b^2*d*e + 8*a*c*d*e - 5*b*(c*d^2 + a*e^2) - 2*(5*c^2*d 
^2 + b^2*e^2 - c*e*(5*b*d - a*e))*x)/(2*(b^2 - 4*a*c)*(a + b*x + c*x^2)^2) 
 - (3*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*(-((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 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. \(648\) vs. \(2(256)=512\).

Time = 0.94 (sec) , antiderivative size = 649, normalized size of antiderivative = 2.44

method	result	size
default	\(\frac {\frac {4 c^{3} \left (a c \,e^{2}+b^{2} e^{2}-5 b c d e +5 c^{2} d^{2}\right ) x^{5}}{64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}+\frac {10 c^{2} \left (a c \,e^{2}+b^{2} e^{2}-5 b c d e +5 c^{2} d^{2}\right ) b \,x^{4}}{64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}+\frac {2 \left (16 a c +11 b^{2}\right ) c \left (a c \,e^{2}+b^{2} e^{2}-5 b c d e +5 c^{2} d^{2}\right ) x^{3}}{3 \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}+\frac {b \left (16 a c +b^{2}\right ) \left (a c \,e^{2}+b^{2} e^{2}-5 b c d e +5 c^{2} d^{2}\right ) x^{2}}{64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}-\frac {\left (4 a^{3} c^{2} e^{2}-22 a^{2} b^{2} c \,e^{2}+44 a^{2} b \,c^{2} d e -44 c^{3} d^{2} a^{2}-a \,b^{4} e^{2}+18 a \,b^{3} c d e -18 a \,b^{2} c^{2} d^{2}-b^{5} d e +b^{4} c \,d^{2}\right ) x}{64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}+\frac {26 a^{3} b c \,e^{2}-64 a^{3} c^{2} d e +a^{2} b^{3} e^{2}-18 a^{2} b^{2} c d e +66 a^{2} b \,c^{2} d^{2}+a \,b^{4} d e -13 c \,d^{2} a \,b^{3}+b^{5} d^{2}}{192 a^{3} c^{3}-144 a^{2} b^{2} c^{2}+36 a \,b^{4} c -3 b^{6}}}{\left (c \,x^{2}+b x +a \right )^{3}}+\frac {8 c \left (a c \,e^{2}+b^{2} e^{2}-5 b c d e +5 c^{2} d^{2}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) \sqrt {4 a c -b^{2}}}\)	\(649\)
risch	\(\text {Expression too large to display}\)	\(1340\)

Input:

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

Output:

(4*c^3*(a*c*e^2+b^2*e^2-5*b*c*d*e+5*c^2*d^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*e^2+b^2*e^2-5*b*c*d*e+5*c^2*d^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*e^2+b^2*e 
^2-5*b*c*d*e+5*c^2*d^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*e^2+b^2*e^2-5*b*c*d*e+5*c^2*d^2)/(64*a^3*c^3-48*a^2*b^2*c 
^2+12*a*b^4*c-b^6)*x^2-(4*a^3*c^2*e^2-22*a^2*b^2*c*e^2+44*a^2*b*c^2*d*e-44 
*a^2*c^3*d^2-a*b^4*e^2+18*a*b^3*c*d*e-18*a*b^2*c^2*d^2-b^5*d*e+b^4*c*d^2)/ 
(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x+1/3*(26*a^3*b*c*e^2-64*a^3*c^ 
2*d*e+a^2*b^3*e^2-18*a^2*b^2*c*d*e+66*a^2*b*c^2*d^2+a*b^4*d*e-13*a*b^3*c*d 
^2+b^5*d^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*e^2+b^2*e^2-5*b*c*d*e+5*c^2*d^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)*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. 1370 vs. \(2 (256) = 512\).

Time = 0.12 (sec) , antiderivative size = 2761, normalized size of antiderivative = 10.38 \[ \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^4} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/3*(12*(5*(b^2*c^5 - 4*a*c^6)*d^2 - 5*(b^3*c^4 - 4*a*b*c^5)*d*e + (b^4* 
c^3 - 3*a*b^2*c^4 - 4*a^2*c^5)*e^2)*x^5 + 30*(5*(b^3*c^4 - 4*a*b*c^5)*d^2 
- 5*(b^4*c^3 - 4*a*b^2*c^4)*d*e + (b^5*c^2 - 3*a*b^3*c^3 - 4*a^2*b*c^4)*e^ 
2)*x^4 + 2*(5*(11*b^4*c^3 - 28*a*b^2*c^4 - 64*a^2*c^5)*d^2 - 5*(11*b^5*c^2 
 - 28*a*b^3*c^3 - 64*a^2*b*c^4)*d*e + (11*b^6*c - 17*a*b^4*c^2 - 92*a^2*b^ 
2*c^3 - 64*a^3*c^4)*e^2)*x^3 + (b^7 - 17*a*b^5*c + 118*a^2*b^3*c^2 - 264*a 
^3*b*c^3)*d^2 + (a*b^6 - 22*a^2*b^4*c + 8*a^3*b^2*c^2 + 256*a^4*c^3)*d*e + 
 (a^2*b^5 + 22*a^3*b^3*c - 104*a^4*b*c^2)*e^2 + 3*(5*(b^5*c^2 + 12*a*b^3*c 
^3 - 64*a^2*b*c^4)*d^2 - 5*(b^6*c + 12*a*b^4*c^2 - 64*a^2*b^2*c^3)*d*e + ( 
b^7 + 13*a*b^5*c - 52*a^2*b^3*c^2 - 64*a^3*b*c^3)*e^2)*x^2 + 12*(5*a^3*c^3 
*d^2 - 5*a^3*b*c^2*d*e + (5*c^6*d^2 - 5*b*c^5*d*e + (b^2*c^4 + a*c^5)*e^2) 
*x^6 + 3*(5*b*c^5*d^2 - 5*b^2*c^4*d*e + (b^3*c^3 + a*b*c^4)*e^2)*x^5 + 3*( 
5*(b^2*c^4 + a*c^5)*d^2 - 5*(b^3*c^3 + a*b*c^4)*d*e + (b^4*c^2 + 2*a*b^2*c 
^3 + a^2*c^4)*e^2)*x^4 + (5*(b^3*c^3 + 6*a*b*c^4)*d^2 - 5*(b^4*c^2 + 6*a*b 
^2*c^3)*d*e + (b^5*c + 7*a*b^3*c^2 + 6*a^2*b*c^3)*e^2)*x^3 + (a^3*b^2*c + 
a^4*c^2)*e^2 + 3*(5*(a*b^2*c^3 + a^2*c^4)*d^2 - 5*(a*b^3*c^2 + a^2*b*c^3)* 
d*e + (a*b^4*c + 2*a^2*b^2*c^2 + a^3*c^3)*e^2)*x^2 + 3*(5*a^2*b*c^3*d^2 - 
5*a^2*b^2*c^2*d*e + (a^2*b^3*c + a^3*b*c^2)*e^2)*x)*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)...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1635 vs. \(2 (269) = 538\).

Time = 3.57 (sec) , antiderivative size = 1635, normalized size of antiderivative = 6.15 \[ \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^4} \, dx=\text {Too large to display} \] Input:

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

Output:

-4*c*sqrt(-1/(4*a*c - b**2)**7)*(a*c*e**2 + b**2*e**2 - 5*b*c*d*e + 5*c**2 
*d**2)*log(x + (-1024*a**4*c**5*sqrt(-1/(4*a*c - b**2)**7)*(a*c*e**2 + b** 
2*e**2 - 5*b*c*d*e + 5*c**2*d**2) + 1024*a**3*b**2*c**4*sqrt(-1/(4*a*c - b 
**2)**7)*(a*c*e**2 + b**2*e**2 - 5*b*c*d*e + 5*c**2*d**2) - 384*a**2*b**4* 
c**3*sqrt(-1/(4*a*c - b**2)**7)*(a*c*e**2 + b**2*e**2 - 5*b*c*d*e + 5*c**2 
*d**2) + 64*a*b**6*c**2*sqrt(-1/(4*a*c - b**2)**7)*(a*c*e**2 + b**2*e**2 - 
 5*b*c*d*e + 5*c**2*d**2) + 4*a*b*c**2*e**2 - 4*b**8*c*sqrt(-1/(4*a*c - b* 
*2)**7)*(a*c*e**2 + b**2*e**2 - 5*b*c*d*e + 5*c**2*d**2) + 4*b**3*c*e**2 - 
 20*b**2*c**2*d*e + 20*b*c**3*d**2)/(8*a*c**3*e**2 + 8*b**2*c**2*e**2 - 40 
*b*c**3*d*e + 40*c**4*d**2)) + 4*c*sqrt(-1/(4*a*c - b**2)**7)*(a*c*e**2 + 
b**2*e**2 - 5*b*c*d*e + 5*c**2*d**2)*log(x + (1024*a**4*c**5*sqrt(-1/(4*a* 
c - b**2)**7)*(a*c*e**2 + b**2*e**2 - 5*b*c*d*e + 5*c**2*d**2) - 1024*a**3 
*b**2*c**4*sqrt(-1/(4*a*c - b**2)**7)*(a*c*e**2 + b**2*e**2 - 5*b*c*d*e + 
5*c**2*d**2) + 384*a**2*b**4*c**3*sqrt(-1/(4*a*c - b**2)**7)*(a*c*e**2 + b 
**2*e**2 - 5*b*c*d*e + 5*c**2*d**2) - 64*a*b**6*c**2*sqrt(-1/(4*a*c - b**2 
)**7)*(a*c*e**2 + b**2*e**2 - 5*b*c*d*e + 5*c**2*d**2) + 4*a*b*c**2*e**2 + 
 4*b**8*c*sqrt(-1/(4*a*c - b**2)**7)*(a*c*e**2 + b**2*e**2 - 5*b*c*d*e + 5 
*c**2*d**2) + 4*b**3*c*e**2 - 20*b**2*c**2*d*e + 20*b*c**3*d**2)/(8*a*c**3 
*e**2 + 8*b**2*c**2*e**2 - 40*b*c**3*d*e + 40*c**4*d**2)) + (26*a**3*b*c*e 
**2 - 64*a**3*c**2*d*e + a**2*b**3*e**2 - 18*a**2*b**2*c*d*e + 66*a**2*...
 

Maxima [F(-2)]

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

integrate((e*x+d)^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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 607 vs. \(2 (256) = 512\).

Time = 0.39 (sec) , antiderivative size = 607, normalized size of antiderivative = 2.28 \[ \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^4} \, dx=-\frac {8 \, {\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e + b^{2} c e^{2} + a c^{2} e^{2}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {60 \, c^{5} d^{2} x^{5} - 60 \, b c^{4} d e x^{5} + 12 \, b^{2} c^{3} e^{2} x^{5} + 12 \, a c^{4} e^{2} x^{5} + 150 \, b c^{4} d^{2} x^{4} - 150 \, b^{2} c^{3} d e x^{4} + 30 \, b^{3} c^{2} e^{2} x^{4} + 30 \, a b c^{3} e^{2} x^{4} + 110 \, b^{2} c^{3} d^{2} x^{3} + 160 \, a c^{4} d^{2} x^{3} - 110 \, b^{3} c^{2} d e x^{3} - 160 \, a b c^{3} d e x^{3} + 22 \, b^{4} c e^{2} x^{3} + 54 \, a b^{2} c^{2} e^{2} x^{3} + 32 \, a^{2} c^{3} e^{2} x^{3} + 15 \, b^{3} c^{2} d^{2} x^{2} + 240 \, a b c^{3} d^{2} x^{2} - 15 \, b^{4} c d e x^{2} - 240 \, a b^{2} c^{2} d e x^{2} + 3 \, b^{5} e^{2} x^{2} + 51 \, a b^{3} c e^{2} x^{2} + 48 \, a^{2} b c^{2} e^{2} x^{2} - 3 \, b^{4} c d^{2} x + 54 \, a b^{2} c^{2} d^{2} x + 132 \, a^{2} c^{3} d^{2} x + 3 \, b^{5} d e x - 54 \, a b^{3} c d e x - 132 \, a^{2} b c^{2} d e x + 3 \, a b^{4} e^{2} x + 66 \, a^{2} b^{2} c e^{2} x - 12 \, a^{3} c^{2} e^{2} x + b^{5} d^{2} - 13 \, a b^{3} c d^{2} + 66 \, a^{2} b c^{2} d^{2} + a b^{4} d e - 18 \, a^{2} b^{2} c d e - 64 \, a^{3} c^{2} d e + a^{2} b^{3} e^{2} + 26 \, a^{3} b c e^{2}}{3 \, {\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} {\left (c x^{2} + b x + a\right )}^{3}} \] Input:

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

Output:

-8*(5*c^3*d^2 - 5*b*c^2*d*e + b^2*c*e^2 + a*c^2*e^2)*arctan((2*c*x + b)/sq 
rt(-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*d^2*x^5 - 60*b*c^4*d*e*x^5 + 12*b^2*c^3*e^2*x^ 
5 + 12*a*c^4*e^2*x^5 + 150*b*c^4*d^2*x^4 - 150*b^2*c^3*d*e*x^4 + 30*b^3*c^ 
2*e^2*x^4 + 30*a*b*c^3*e^2*x^4 + 110*b^2*c^3*d^2*x^3 + 160*a*c^4*d^2*x^3 - 
 110*b^3*c^2*d*e*x^3 - 160*a*b*c^3*d*e*x^3 + 22*b^4*c*e^2*x^3 + 54*a*b^2*c 
^2*e^2*x^3 + 32*a^2*c^3*e^2*x^3 + 15*b^3*c^2*d^2*x^2 + 240*a*b*c^3*d^2*x^2 
 - 15*b^4*c*d*e*x^2 - 240*a*b^2*c^2*d*e*x^2 + 3*b^5*e^2*x^2 + 51*a*b^3*c*e 
^2*x^2 + 48*a^2*b*c^2*e^2*x^2 - 3*b^4*c*d^2*x + 54*a*b^2*c^2*d^2*x + 132*a 
^2*c^3*d^2*x + 3*b^5*d*e*x - 54*a*b^3*c*d*e*x - 132*a^2*b*c^2*d*e*x + 3*a* 
b^4*e^2*x + 66*a^2*b^2*c*e^2*x - 12*a^3*c^2*e^2*x + b^5*d^2 - 13*a*b^3*c*d 
^2 + 66*a^2*b*c^2*d^2 + a*b^4*d*e - 18*a^2*b^2*c*d*e - 64*a^3*c^2*d*e + a^ 
2*b^3*e^2 + 26*a^3*b*c*e^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 [B] (verification not implemented)

Time = 5.66 (sec) , antiderivative size = 872, normalized size of antiderivative = 3.28 \[ \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^4} \, dx=-\frac {\frac {26\,a^3\,b\,c\,e^2-64\,a^3\,c^2\,d\,e+a^2\,b^3\,e^2-18\,a^2\,b^2\,c\,d\,e+66\,a^2\,b\,c^2\,d^2+a\,b^4\,d\,e-13\,a\,b^3\,c\,d^2+b^5\,d^2}{3\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}+\frac {x\,\left (-4\,a^3\,c^2\,e^2+22\,a^2\,b^2\,c\,e^2-44\,a^2\,b\,c^2\,d\,e+44\,a^2\,c^3\,d^2+a\,b^4\,e^2-18\,a\,b^3\,c\,d\,e+18\,a\,b^2\,c^2\,d^2+b^5\,d\,e-b^4\,c\,d^2\right )}{-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}+\frac {2\,x^3\,\left (11\,b^2\,c+16\,a\,c^2\right )\,\left (b^2\,e^2-5\,b\,c\,d\,e+5\,c^2\,d^2+a\,c\,e^2\right )}{3\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}+\frac {x^2\,\left (b^3+16\,a\,c\,b\right )\,\left (b^2\,e^2-5\,b\,c\,d\,e+5\,c^2\,d^2+a\,c\,e^2\right )}{-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}+\frac {4\,c^3\,x^5\,\left (b^2\,e^2-5\,b\,c\,d\,e+5\,c^2\,d^2+a\,c\,e^2\right )}{-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}+\frac {10\,b\,c^2\,x^4\,\left (b^2\,e^2-5\,b\,c\,d\,e+5\,c^2\,d^2+a\,c\,e^2\right )}{-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}}{x^2\,\left (3\,c\,a^2+3\,a\,b^2\right )+x^4\,\left (3\,b^2\,c+3\,a\,c^2\right )+a^3+x^3\,\left (b^3+6\,a\,c\,b\right )+c^3\,x^6+3\,b\,c^2\,x^5+3\,a^2\,b\,x}-\frac {8\,c\,\mathrm {atan}\left (\frac {\left (\frac {8\,c^2\,x\,\left (b^2\,e^2-5\,b\,c\,d\,e+5\,c^2\,d^2+a\,c\,e^2\right )}{{\left (4\,a\,c-b^2\right )}^{7/2}}+\frac {4\,c\,\left (-64\,a^3\,b\,c^3+48\,a^2\,b^3\,c^2-12\,a\,b^5\,c+b^7\right )\,\left (b^2\,e^2-5\,b\,c\,d\,e+5\,c^2\,d^2+a\,c\,e^2\right )}{{\left (4\,a\,c-b^2\right )}^{7/2}\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}\right )\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}{4\,b^2\,c\,e^2-20\,b\,c^2\,d\,e+20\,c^3\,d^2+4\,a\,c^2\,e^2}\right )\,\left (b^2\,e^2-5\,b\,c\,d\,e+5\,c^2\,d^2+a\,c\,e^2\right )}{{\left (4\,a\,c-b^2\right )}^{7/2}} \] Input:

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

Output:

- ((b^5*d^2 + a^2*b^3*e^2 + 66*a^2*b*c^2*d^2 + a*b^4*d*e - 13*a*b^3*c*d^2 
+ 26*a^3*b*c*e^2 - 64*a^3*c^2*d*e - 18*a^2*b^2*c*d*e)/(3*(b^6 - 64*a^3*c^3 
 + 48*a^2*b^2*c^2 - 12*a*b^4*c)) + (x*(a*b^4*e^2 - b^4*c*d^2 + 44*a^2*c^3* 
d^2 - 4*a^3*c^2*e^2 + b^5*d*e + 18*a*b^2*c^2*d^2 + 22*a^2*b^2*c*e^2 - 44*a 
^2*b*c^2*d*e - 18*a*b^3*c*d*e))/(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a* 
b^4*c) + (2*x^3*(16*a*c^2 + 11*b^2*c)*(b^2*e^2 + 5*c^2*d^2 + a*c*e^2 - 5*b 
*c*d*e))/(3*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)) + (x^2*(b^3 
+ 16*a*b*c)*(b^2*e^2 + 5*c^2*d^2 + a*c*e^2 - 5*b*c*d*e))/(b^6 - 64*a^3*c^3 
 + 48*a^2*b^2*c^2 - 12*a*b^4*c) + (4*c^3*x^5*(b^2*e^2 + 5*c^2*d^2 + a*c*e^ 
2 - 5*b*c*d*e))/(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*(b^2*e^2 + 5*c^2*d^2 + a*c*e^2 - 5*b*c*d*e))/(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*(b^2*e^2 + 5*c^2*d^2 + a*c*e^2 - 5*b*c*d*e))/(4*a*c - b 
^2)^(7/2) + (4*c*(b^7 - 64*a^3*b*c^3 + 48*a^2*b^3*c^2 - 12*a*b^5*c)*(b^2*e 
^2 + 5*c^2*d^2 + a*c*e^2 - 5*b*c*d*e))/((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))/(20*c^3*d^2 + 4*a*c^2*e^2 + 4*b^2*c*e^2 - 20*b*c^2*d*e))*(b^2 
*e^2 + 5*c^2*d^2 + a*c*e^2 - 5*b*c*d*e))/(4*a*c - b^2)^(7/2)
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 2977, normalized size of antiderivative = 11.19 \[ \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^4} \, dx =\text {Too large to display} \] Input:

int((e*x+d)^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*e* 
*2 + 24*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**3*b**3* 
c*e**2 - 120*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**3* 
b**2*c**2*d*e + 72*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2)) 
*a**3*b**2*c**2*e**2*x + 120*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a* 
c - b**2))*a**3*b*c**3*d**2 + 72*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt( 
4*a*c - b**2))*a**3*b*c**3*e**2*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*e**2*x - 360*sqrt(4*a*c - b**2)*atan(( 
b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b**3*c**2*d*e*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*e**2*x**2 + 360*sqr 
t(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b**2*c**3*d**2*x 
 - 360*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b**2*c 
**3*d*e*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*e**2*x**3 + 360*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4 
*a*c - b**2))*a**2*b*c**4*d**2*x**2 + 72*sqrt(4*a*c - b**2)*atan((b + 2*c* 
x)/sqrt(4*a*c - b**2))*a**2*b*c**4*e**2*x**4 + 72*sqrt(4*a*c - b**2)*atan( 
(b + 2*c*x)/sqrt(4*a*c - b**2))*a*b**5*c*e**2*x**2 - 360*sqrt(4*a*c - b**2 
)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b**4*c**2*d*e*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*e**2*x**3 + 36 
0*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b**3*c**3*d...