\(\int \frac {1}{(d+e x)^2 (a+b x+c x^2)^2} \, dx\) [483]

Optimal result

Integrand size = 20, antiderivative size = 345 \[ \int \frac {1}{(d+e x)^2 \left (a+b x+c x^2\right )^2} \, dx=-\frac {e^3}{\left (c d^2-b d e+a e^2\right )^2 (d+e x)}+\frac {2 b^2 c d e-4 a c^2 d e-b^3 e^2-b c \left (c d^2-3 a e^2\right )-c \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) x}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}+\frac {2 \left (2 c^4 d^4-b^4 e^4-4 c^3 d^2 e (b d-3 a e)-6 a c^2 e^3 (2 b d+a e)+2 b^2 c e^3 (b d+3 a e)\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2} \left (c d^2-b d e+a e^2\right )^3}+\frac {2 e^3 (2 c d-b e) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}-\frac {e^3 (2 c d-b e) \log \left (a+b x+c x^2\right )}{\left (c d^2-b d e+a e^2\right )^3} \] Output:

-e^3/(a*e^2-b*d*e+c*d^2)^2/(e*x+d)+(2*b^2*c*d*e-4*a*c^2*d*e-b^3*e^2-b*c*(- 
3*a*e^2+c*d^2)-c*(2*c^2*d^2+b^2*e^2-2*c*e*(a*e+b*d))*x)/(-4*a*c+b^2)/(a*e^ 
2-b*d*e+c*d^2)^2/(c*x^2+b*x+a)+2*(2*c^4*d^4-b^4*e^4-4*c^3*d^2*e*(-3*a*e+b* 
d)-6*a*c^2*e^3*(a*e+2*b*d)+2*b^2*c*e^3*(3*a*e+b*d))*arctanh((2*c*x+b)/(-4* 
a*c+b^2)^(1/2))/(-4*a*c+b^2)^(3/2)/(a*e^2-b*d*e+c*d^2)^3+2*e^3*(-b*e+2*c*d 
)*ln(e*x+d)/(a*e^2-b*d*e+c*d^2)^3-e^3*(-b*e+2*c*d)*ln(c*x^2+b*x+a)/(a*e^2- 
b*d*e+c*d^2)^3
 

Mathematica [A] (verified)

Time = 0.40 (sec) , antiderivative size = 339, normalized size of antiderivative = 0.98 \[ \int \frac {1}{(d+e x)^2 \left (a+b x+c x^2\right )^2} \, dx=-\frac {e^3}{\left (c d^2+e (-b d+a e)\right )^2 (d+e x)}+\frac {-b^3 e^2+b^2 c e (2 d-e x)+b c \left (3 a e^2-c d (d-2 e x)\right )-2 c^2 \left (c d^2 x+a e (2 d-e x)\right )}{\left (b^2-4 a c\right ) \left (c d^2+e (-b d+a e)\right )^2 (a+x (b+c x))}-\frac {2 \left (2 c^4 d^4-b^4 e^4-4 c^3 d^2 e (b d-3 a e)-6 a c^2 e^3 (2 b d+a e)+2 b^2 c e^3 (b d+3 a e)\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{3/2} \left (-c d^2+e (b d-a e)\right )^3}-\frac {2 e^3 (-2 c d+b e) \log (d+e x)}{\left (c d^2+e (-b d+a e)\right )^3}+\frac {e^3 (-2 c d+b e) \log (a+x (b+c x))}{\left (c d^2+e (-b d+a e)\right )^3} \] Input:

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

Output:

-(e^3/((c*d^2 + e*(-(b*d) + a*e))^2*(d + e*x))) + (-(b^3*e^2) + b^2*c*e*(2 
*d - e*x) + b*c*(3*a*e^2 - c*d*(d - 2*e*x)) - 2*c^2*(c*d^2*x + a*e*(2*d - 
e*x)))/((b^2 - 4*a*c)*(c*d^2 + e*(-(b*d) + a*e))^2*(a + x*(b + c*x))) - (2 
*(2*c^4*d^4 - b^4*e^4 - 4*c^3*d^2*e*(b*d - 3*a*e) - 6*a*c^2*e^3*(2*b*d + a 
*e) + 2*b^2*c*e^3*(b*d + 3*a*e))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/( 
(-b^2 + 4*a*c)^(3/2)*(-(c*d^2) + e*(b*d - a*e))^3) - (2*e^3*(-2*c*d + b*e) 
*Log[d + e*x])/(c*d^2 + e*(-(b*d) + a*e))^3 + (e^3*(-2*c*d + b*e)*Log[a + 
x*(b + c*x)])/(c*d^2 + e*(-(b*d) + a*e))^3
 

Rubi [A] (verified)

Time = 0.82 (sec) , antiderivative size = 382, normalized size of antiderivative = 1.11, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1165, 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.

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

\(\Big \downarrow \) 1165

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1200

\(\displaystyle -\frac {2 \int \left (\frac {\left (b^2-4 a c\right ) (b e-2 c d) e^4}{\left (c d^2-b e d+a e^2\right )^2 (d+e x)}+\frac {\left (-c^2 d^2-b^2 e^2+c e (b d+3 a e)\right ) e^2}{\left (c d^2-b e d+a e^2\right ) (d+e x)^2}+\frac {c^4 d^4-2 c^3 e (b d-3 a e) d^2-b^4 e^4-a c^2 e^3 (10 b d+3 a e)+b^2 c e^3 (2 b d+5 a e)+c \left (b^2-4 a c\right ) e^3 (2 c d-b e) x}{\left (c d^2-b e d+a e^2\right )^2 \left (c x^2+b x+a\right )}\right )dx}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d}{\left (b^2-4 a c\right ) (d+e x) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 \left (-\frac {\text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (2 b^2 c e^3 (3 a e+b d)-4 c^3 d^2 e (b d-3 a e)-6 a c^2 e^3 (a e+2 b d)-b^4 e^4+2 c^4 d^4\right )}{\sqrt {b^2-4 a c} \left (a e^2-b d e+c d^2\right )^2}+\frac {e \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )}{(d+e x) \left (a e^2-b d e+c d^2\right )}+\frac {e^3 \left (b^2-4 a c\right ) (2 c d-b e) \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )^2}-\frac {e^3 \left (b^2-4 a c\right ) (2 c d-b e) \log (d+e x)}{\left (a e^2-b d e+c d^2\right )^2}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d}{\left (b^2-4 a c\right ) (d+e x) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}\)

Input:

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

Output:

-((b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x)/((b^2 - 4*a*c)*(c*d^2 - b* 
d*e + a*e^2)*(d + e*x)*(a + b*x + c*x^2))) - (2*((e*(c^2*d^2 + b^2*e^2 - c 
*e*(b*d + 3*a*e)))/((c*d^2 - b*d*e + a*e^2)*(d + e*x)) - ((2*c^4*d^4 - b^4 
*e^4 - 4*c^3*d^2*e*(b*d - 3*a*e) - 6*a*c^2*e^3*(2*b*d + a*e) + 2*b^2*c*e^3 
*(b*d + 3*a*e))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(Sqrt[b^2 - 4*a*c] 
*(c*d^2 - b*d*e + a*e^2)^2) - ((b^2 - 4*a*c)*e^3*(2*c*d - b*e)*Log[d + e*x 
])/(c*d^2 - b*d*e + a*e^2)^2 + ((b^2 - 4*a*c)*e^3*(2*c*d - b*e)*Log[a + b* 
x + c*x^2])/(2*(c*d^2 - b*d*e + a*e^2)^2)))/((b^2 - 4*a*c)*(c*d^2 - b*d*e 
+ a*e^2))
 

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 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 2009

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

Maple [A] (verified)

Time = 1.25 (sec) , antiderivative size = 529, normalized size of antiderivative = 1.53

method	result	size
default	\(-\frac {e^{3}}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \left (e x +d \right )}-\frac {2 e^{3} \left (b e -2 c d \right ) \ln \left (e x +d \right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{3}}-\frac {\frac {\frac {c \left (2 e^{4} a^{2} c -a \,b^{2} e^{4}+b^{3} d \,e^{3}-3 b^{2} c \,d^{2} e^{2}+4 b \,c^{2} d^{3} e -2 d^{4} c^{3}\right ) x}{4 a c -b^{2}}+\frac {3 a^{2} b c \,e^{4}-4 a^{2} c^{2} d \,e^{3}-a \,b^{3} e^{4}-a \,b^{2} c d \,e^{3}+6 a b \,c^{2} d^{2} e^{2}-4 a \,c^{3} d^{3} e +b^{4} d \,e^{3}-3 b^{3} c \,d^{2} e^{2}+3 b^{2} c^{2} d^{3} e -d^{4} b \,c^{3}}{4 a c -b^{2}}}{c \,x^{2}+b x +a}+\frac {\frac {\left (-4 a b \,c^{2} e^{4}+8 a \,c^{3} d \,e^{3}+b^{3} c \,e^{4}-2 b^{2} c^{2} d \,e^{3}\right ) \ln \left (c \,x^{2}+b x +a \right )}{c}+\frac {4 \left (3 e^{4} a^{2} c^{2}-5 a \,b^{2} c \,e^{4}+10 a b \,c^{2} d \,e^{3}-6 d^{2} e^{2} a \,c^{3}+b^{4} e^{4}-2 d \,e^{3} b^{3} c +2 d^{3} e b \,c^{3}-d^{4} c^{4}-\frac {\left (-4 a b \,c^{2} e^{4}+8 a \,c^{3} d \,e^{3}+b^{3} c \,e^{4}-2 b^{2} c^{2} d \,e^{3}\right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{4 a c -b^{2}}}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{3}}\)	\(529\)
risch	\(\text {Expression too large to display}\)	\(56540\)

Input:

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

Output:

-e^3/(a*e^2-b*d*e+c*d^2)^2/(e*x+d)-2*e^3/(a*e^2-b*d*e+c*d^2)^3*(b*e-2*c*d) 
*ln(e*x+d)-1/(a*e^2-b*d*e+c*d^2)^3*((c*(2*a^2*c*e^4-a*b^2*e^4+b^3*d*e^3-3* 
b^2*c*d^2*e^2+4*b*c^2*d^3*e-2*c^3*d^4)/(4*a*c-b^2)*x+(3*a^2*b*c*e^4-4*a^2* 
c^2*d*e^3-a*b^3*e^4-a*b^2*c*d*e^3+6*a*b*c^2*d^2*e^2-4*a*c^3*d^3*e+b^4*d*e^ 
3-3*b^3*c*d^2*e^2+3*b^2*c^2*d^3*e-b*c^3*d^4)/(4*a*c-b^2))/(c*x^2+b*x+a)+2/ 
(4*a*c-b^2)*(1/2*(-4*a*b*c^2*e^4+8*a*c^3*d*e^3+b^3*c*e^4-2*b^2*c^2*d*e^3)/ 
c*ln(c*x^2+b*x+a)+2*(3*e^4*a^2*c^2-5*a*b^2*c*e^4+10*a*b*c^2*d*e^3-6*d^2*e^ 
2*a*c^3+b^4*e^4-2*d*e^3*b^3*c+2*d^3*e*b*c^3-d^4*c^4-1/2*(-4*a*b*c^2*e^4+8* 
a*c^3*d*e^3+b^3*c*e^4-2*b^2*c^2*d*e^3)*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. 2739 vs. \(2 (341) = 682\).

Time = 45.69 (sec) , antiderivative size = 5498, normalized size of antiderivative = 15.94 \[ \int \frac {1}{(d+e x)^2 \left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate(1/(e*x+d)^2/(c*x^2+b*x+a)^2,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. 932 vs. \(2 (341) = 682\).

Time = 0.39 (sec) , antiderivative size = 932, normalized size of antiderivative = 2.70 \[ \int \frac {1}{(d+e x)^2 \left (a+b x+c x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

-e^7/((c^2*d^4*e^4 - 2*b*c*d^3*e^5 + b^2*d^2*e^6 + 2*a*c*d^2*e^6 - 2*a*b*d 
*e^7 + a^2*e^8)*(e*x + d)) - (2*c*d*e^3 - b*e^4)*log(c - 2*c*d/(e*x + d) + 
 c*d^2/(e*x + d)^2 + b*e/(e*x + d) - b*d*e/(e*x + d)^2 + a*e^2/(e*x + d)^2 
)/(c^3*d^6 - 3*b*c^2*d^5*e + 3*b^2*c*d^4*e^2 + 3*a*c^2*d^4*e^2 - b^3*d^3*e 
^3 - 6*a*b*c*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a^2*c*d^2*e^4 - 3*a^2*b*d*e^5 + 
 a^3*e^6) - 2*(2*c^4*d^4*e^2 - 4*b*c^3*d^3*e^3 + 12*a*c^3*d^2*e^4 + 2*b^3* 
c*d*e^5 - 12*a*b*c^2*d*e^5 - b^4*e^6 + 6*a*b^2*c*e^6 - 6*a^2*c^2*e^6)*arct 
an((2*c*d - 2*c*d^2/(e*x + d) - b*e + 2*b*d*e/(e*x + d) - 2*a*e^2/(e*x + d 
))/(sqrt(-b^2 + 4*a*c)*e))/((b^2*c^3*d^6 - 4*a*c^4*d^6 - 3*b^3*c^2*d^5*e + 
 12*a*b*c^3*d^5*e + 3*b^4*c*d^4*e^2 - 9*a*b^2*c^2*d^4*e^2 - 12*a^2*c^3*d^4 
*e^2 - b^5*d^3*e^3 - 2*a*b^3*c*d^3*e^3 + 24*a^2*b*c^2*d^3*e^3 + 3*a*b^4*d^ 
2*e^4 - 9*a^2*b^2*c*d^2*e^4 - 12*a^3*c^2*d^2*e^4 - 3*a^2*b^3*d*e^5 + 12*a^ 
3*b*c*d*e^5 + a^3*b^2*e^6 - 4*a^4*c*e^6)*sqrt(-b^2 + 4*a*c)*e^2) - ((2*c^4 
*d^3*e - 3*b*c^3*d^2*e^2 + 3*b^2*c^2*d*e^3 - 6*a*c^3*d*e^3 - b^3*c*e^4 + 3 
*a*b*c^2*e^4)/(c*d^2 - b*d*e + a*e^2) - (2*c^4*d^4*e^2 - 4*b*c^3*d^3*e^3 + 
 6*b^2*c^2*d^2*e^4 - 12*a*c^3*d^2*e^4 - 4*b^3*c*d*e^5 + 12*a*b*c^2*d*e^5 + 
 b^4*e^6 - 4*a*b^2*c*e^6 + 2*a^2*c^2*e^6)/((c*d^2 - b*d*e + a*e^2)*(e*x + 
d)*e))/((c*d^2 - b*d*e + a*e^2)^2*(b^2 - 4*a*c)*(c - 2*c*d/(e*x + d) + c*d 
^2/(e*x + d)^2 + b*e/(e*x + d) - b*d*e/(e*x + d)^2 + a*e^2/(e*x + d)^2))
 

Mupad [B] (verification not implemented)

Time = 8.15 (sec) , antiderivative size = 2239, normalized size of antiderivative = 6.49 \[ \int \frac {1}{(d+e x)^2 \left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

((a*b^2*e^3 + b*c^2*d^3 - 4*a^2*c*e^3 + b^3*d*e^2 + 4*a*c^2*d^2*e - 2*b^2* 
c*d^2*e - 3*a*b*c*d*e^2)/(4*a*c^3*d^4 + 4*a^3*c*e^4 - a^2*b^2*e^4 - b^2*c^ 
2*d^4 - b^4*d^2*e^2 + 8*a^2*c^2*d^2*e^2 + 2*a*b^3*d*e^3 + 2*b^3*c*d^3*e - 
8*a*b*c^2*d^3*e - 8*a^2*b*c*d*e^3 + 2*a*b^2*c*d^2*e^2) - (2*x^2*(3*a*c^2*e 
^3 - b^2*c*e^3 - c^3*d^2*e + b*c^2*d*e^2))/(4*a*c^3*d^4 + 4*a^3*c*e^4 - a^ 
2*b^2*e^4 - b^2*c^2*d^4 - b^4*d^2*e^2 + 8*a^2*c^2*d^2*e^2 + 2*a*b^3*d*e^3 
+ 2*b^3*c*d^3*e - 8*a*b*c^2*d^3*e - 8*a^2*b*c*d*e^3 + 2*a*b^2*c*d^2*e^2) + 
 (x*(2*b^3*e^3 + 2*c^3*d^3 - 7*a*b*c*e^3 + 2*a*c^2*d*e^2 - b*c^2*d^2*e - b 
^2*c*d*e^2))/(4*a*c^3*d^4 + 4*a^3*c*e^4 - a^2*b^2*e^4 - b^2*c^2*d^4 - b^4* 
d^2*e^2 + 8*a^2*c^2*d^2*e^2 + 2*a*b^3*d*e^3 + 2*b^3*c*d^3*e - 8*a*b*c^2*d^ 
3*e - 8*a^2*b*c*d*e^3 + 2*a*b^2*c*d^2*e^2))/(a*d + x*(a*e + b*d) + x^2*(b* 
e + c*d) + c*e*x^3) - (log(b^3 + (-(4*a*c - b^2)^3)^(1/2) - 4*a*b*c - 8*a* 
c^2*x + 2*b^2*c*x)*(b^7*e^4 - b^4*e^4*(-(4*a*c - b^2)^3)^(1/2) + 2*c^4*d^4 
*(-(4*a*c - b^2)^3)^(1/2) - 64*a^3*b*c^3*e^4 + 128*a^3*c^4*d*e^3 + 48*a^2* 
b^3*c^2*e^4 - 6*a^2*c^2*e^4*(-(4*a*c - b^2)^3)^(1/2) - 12*a*b^5*c*e^4 - 2* 
b^6*c*d*e^3 + 6*a*b^2*c*e^4*(-(4*a*c - b^2)^3)^(1/2) + 24*a*b^4*c^2*d*e^3 
- 4*b*c^3*d^3*e*(-(4*a*c - b^2)^3)^(1/2) + 2*b^3*c*d*e^3*(-(4*a*c - b^2)^3 
)^(1/2) - 96*a^2*b^2*c^3*d*e^3 + 12*a*c^3*d^2*e^2*(-(4*a*c - b^2)^3)^(1/2) 
 - 12*a*b*c^2*d*e^3*(-(4*a*c - b^2)^3)^(1/2)))/(64*a^3*c^6*d^6 - a^3*b^6*e 
^6 + 64*a^6*c^3*e^6 - b^6*c^3*d^6 + b^9*d^3*e^3 + 12*a*b^4*c^4*d^6 + 12...
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 7041, normalized size of antiderivative = 20.41 \[ \int \frac {1}{(d+e x)^2 \left (a+b x+c x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 12*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**3*b*c**2 
*d*e**5 - 12*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**3* 
b*c**2*e**6*x - 12*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2)) 
*a**3*c**3*d**2*e**4 - 12*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - 
 b**2))*a**3*c**3*d*e**5*x + 12*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4 
*a*c - b**2))*a**2*b**3*c*d*e**5 + 12*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/ 
sqrt(4*a*c - b**2))*a**2*b**3*c*e**6*x - 12*sqrt(4*a*c - b**2)*atan((b + 2 
*c*x)/sqrt(4*a*c - b**2))*a**2*b**2*c**2*d**2*e**4 - 24*sqrt(4*a*c - b**2) 
*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b**2*c**2*d*e**5*x - 12*sqrt(4* 
a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b**2*c**2*e**6*x**2 
- 12*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b*c**3*d 
**2*e**4*x - 24*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a* 
*2*b*c**3*d*e**5*x**2 - 12*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c 
- b**2))*a**2*b*c**3*e**6*x**3 + 24*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sq 
rt(4*a*c - b**2))*a**2*c**4*d**4*e**2 + 24*sqrt(4*a*c - b**2)*atan((b + 2* 
c*x)/sqrt(4*a*c - b**2))*a**2*c**4*d**3*e**3*x - 12*sqrt(4*a*c - b**2)*ata 
n((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*c**4*d**2*e**4*x**2 - 12*sqrt(4*a*c 
 - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*c**4*d*e**5*x**3 - 2*sq 
rt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b**5*d*e**5 - 2*sq 
rt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b**5*e**6*x + 2...