\(\int \frac {1}{(d+e x) (a+b x+c x^2)^2} \, dx\) [482]

Optimal result

Integrand size = 20, antiderivative size = 224 \[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^2} \, dx=-\frac {b c d-b^2 e+2 a c e+c (2 c d-b e) x}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )}+\frac {(2 c d-b e) \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2} \left (c d^2-b d e+a e^2\right )^2}+\frac {e^3 \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^2}-\frac {e^3 \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^2} \, dx=\frac {-b^2 e+2 c (a e+c d x)+b c (d-e x)}{\left (b^2-4 a c\right ) \left (-c d^2+e (b d-a e)\right ) (a+x (b+c x))}+\frac {(-2 c d+b e) \left (-2 c^2 d^2+b^2 e^2+2 c e (b d-3 a e)\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{3/2} \left (c d^2+e (-b d+a e)\right )^2}+\frac {e^3 \log (d+e x)}{\left (c d^2+e (-b d+a e)\right )^2}-\frac {e^3 \log (a+x (b+c x))}{2 \left (c d^2+e (-b d+a e)\right )^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.22, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {1165, 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.

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

\(\Big \downarrow \) 1165

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

\(\Big \downarrow \) 1200

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

\(\Big \downarrow \) 2009

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

Input:

Int[1/((d + e*x)*(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)*(a + b*x + c*x^2))) - (-(((2*c*d - b*e)*(2*c^2*d^2 - 2*b*c*d* 
e - b^2*e^2 + 6*a*c*e^2)*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))) - ((b^2 - 4*a*c)*e^3*Log[d + e*x])/(c* 
d^2 - b*d*e + a*e^2) + ((b^2 - 4*a*c)*e^3*Log[a + b*x + c*x^2])/(2*(c*d^2 
- b*d*e + a*e^2)))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))
 

Defintions of rubi rules used

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 = 1.15 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.57

method	result	size
default	\(\frac {e^{3} \ln \left (e x +d \right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{2}}-\frac {\frac {\frac {c \left (a \,e^{3} b -2 a d \,e^{2} c -d \,e^{2} b^{2}+3 d^{2} e b c -2 c^{2} d^{3}\right ) x}{4 a c -b^{2}}-\frac {2 a^{2} c \,e^{3}-b^{2} e^{3} a -b d \,e^{2} a c +2 a \,c^{2} d^{2} e +b^{3} e^{2} d -2 b^{2} c \,d^{2} e +d^{3} b \,c^{2}}{4 a c -b^{2}}}{c \,x^{2}+b x +a}+\frac {\frac {\left (4 a \,c^{2} e^{3}-b^{2} c \,e^{3}\right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (5 a b c \,e^{3}-6 d \,e^{2} a \,c^{2}-b^{3} e^{3}+3 d^{2} e b \,c^{2}-2 d^{3} c^{3}-\frac {\left (4 a \,c^{2} e^{3}-b^{2} c \,e^{3}\right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{4 a c -b^{2}}}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{2}}\)	\(351\)
risch	\(\text {Expression too large to display}\)	\(15758\)

Input:

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

Output:

e^3*ln(e*x+d)/(a*e^2-b*d*e+c*d^2)^2-1/(a*e^2-b*d*e+c*d^2)^2*((c*(a*b*e^3-2 
*a*c*d*e^2-b^2*d*e^2+3*b*c*d^2*e-2*c^2*d^3)/(4*a*c-b^2)*x-(2*a^2*c*e^3-a*b 
^2*e^3-a*b*c*d*e^2+2*a*c^2*d^2*e+b^3*d*e^2-2*b^2*c*d^2*e+b*c^2*d^3)/(4*a*c 
-b^2))/(c*x^2+b*x+a)+1/(4*a*c-b^2)*(1/2*(4*a*c^2*e^3-b^2*c*e^3)/c*ln(c*x^2 
+b*x+a)+2*(5*a*b*c*e^3-6*d*e^2*a*c^2-b^3*e^3+3*d^2*e*b*c^2-2*d^3*c^3-1/2*( 
4*a*c^2*e^3-b^2*c*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. 1054 vs. \(2 (218) = 436\).

Time = 12.55 (sec) , antiderivative size = 2128, normalized size of antiderivative = 9.50 \[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/2*(2*(b^3*c^2 - 4*a*b*c^3)*d^3 - 4*(b^4*c - 5*a*b^2*c^2 + 4*a^2*c^3)*d 
^2*e + 2*(b^5 - 5*a*b^3*c + 4*a^2*b*c^2)*d*e^2 - 2*(a*b^4 - 6*a^2*b^2*c + 
8*a^3*c^2)*e^3 - (4*a*c^3*d^3 - 6*a*b*c^2*d^2*e + 12*a^2*c^2*d*e^2 + (a*b^ 
3 - 6*a^2*b*c)*e^3 + (4*c^4*d^3 - 6*b*c^3*d^2*e + 12*a*c^3*d*e^2 + (b^3*c 
- 6*a*b*c^2)*e^3)*x^2 + (4*b*c^3*d^3 - 6*b^2*c^2*d^2*e + 12*a*b*c^2*d*e^2 
+ (b^4 - 6*a*b^2*c)*e^3)*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)) + 2*(2*(b^2 
*c^3 - 4*a*c^4)*d^3 - 3*(b^3*c^2 - 4*a*b*c^3)*d^2*e + (b^4*c - 2*a*b^2*c^2 
 - 8*a^2*c^3)*d*e^2 - (a*b^3*c - 4*a^2*b*c^2)*e^3)*x + ((b^4*c - 8*a*b^2*c 
^2 + 16*a^2*c^3)*e^3*x^2 + (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*e^3*x + (a*b^4 
 - 8*a^2*b^2*c + 16*a^3*c^2)*e^3)*log(c*x^2 + b*x + a) - 2*((b^4*c - 8*a*b 
^2*c^2 + 16*a^2*c^3)*e^3*x^2 + (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*e^3*x + (a 
*b^4 - 8*a^2*b^2*c + 16*a^3*c^2)*e^3)*log(e*x + d))/((a*b^4*c^2 - 8*a^2*b^ 
2*c^3 + 16*a^3*c^4)*d^4 - 2*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c^3)*d^3*e 
 + (a*b^6 - 6*a^2*b^4*c + 32*a^4*c^3)*d^2*e^2 - 2*(a^2*b^5 - 8*a^3*b^3*c + 
 16*a^4*b*c^2)*d*e^3 + (a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*e^4 + ((b^4*c^ 
3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d^4 - 2*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^ 
4)*d^3*e + (b^6*c - 6*a*b^4*c^2 + 32*a^3*c^4)*d^2*e^2 - 2*(a*b^5*c - 8*a^2 
*b^3*c^2 + 16*a^3*b*c^3)*d*e^3 + (a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3)* 
e^4)*x^2 + ((b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d^4 - 2*(b^6*c - 8*a...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate(1/(e*x+d)/(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. 489 vs. \(2 (218) = 436\).

Time = 0.32 (sec) , antiderivative size = 489, normalized size of antiderivative = 2.18 \[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^2} \, dx=\frac {e^{4} \log \left ({\left | e x + d \right |}\right )}{c^{2} d^{4} e - 2 \, b c d^{3} e^{2} + b^{2} d^{2} e^{3} + 2 \, a c d^{2} e^{3} - 2 \, a b d e^{4} + a^{2} e^{5}} - \frac {e^{3} \log \left (c x^{2} + b x + a\right )}{2 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2} + 2 \, a c d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )}} - \frac {{\left (4 \, c^{3} d^{3} - 6 \, b c^{2} d^{2} e + 12 \, a c^{2} d e^{2} + b^{3} e^{3} - 6 \, a b c e^{3}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} c^{2} d^{4} - 4 \, a c^{3} d^{4} - 2 \, b^{3} c d^{3} e + 8 \, a b c^{2} d^{3} e + b^{4} d^{2} e^{2} - 2 \, a b^{2} c d^{2} e^{2} - 8 \, a^{2} c^{2} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + 8 \, a^{2} b c d e^{3} + a^{2} b^{2} e^{4} - 4 \, a^{3} c e^{4}\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {b c^{2} d^{3} - 2 \, b^{2} c d^{2} e + 2 \, a c^{2} d^{2} e + b^{3} d e^{2} - a b c d e^{2} - a b^{2} e^{3} + 2 \, a^{2} c e^{3} + {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e + b^{2} c d e^{2} + 2 \, a c^{2} d e^{2} - a b c e^{3}\right )} x}{{\left (c d^{2} - b d e + a e^{2}\right )}^{2} {\left (c x^{2} + b x + a\right )} {\left (b^{2} - 4 \, a c\right )}} \] Input:

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

Output:

e^4*log(abs(e*x + d))/(c^2*d^4*e - 2*b*c*d^3*e^2 + b^2*d^2*e^3 + 2*a*c*d^2 
*e^3 - 2*a*b*d*e^4 + a^2*e^5) - 1/2*e^3*log(c*x^2 + b*x + a)/(c^2*d^4 - 2* 
b*c*d^3*e + b^2*d^2*e^2 + 2*a*c*d^2*e^2 - 2*a*b*d*e^3 + a^2*e^4) - (4*c^3* 
d^3 - 6*b*c^2*d^2*e + 12*a*c^2*d*e^2 + b^3*e^3 - 6*a*b*c*e^3)*arctan((2*c* 
x + b)/sqrt(-b^2 + 4*a*c))/((b^2*c^2*d^4 - 4*a*c^3*d^4 - 2*b^3*c*d^3*e + 8 
*a*b*c^2*d^3*e + b^4*d^2*e^2 - 2*a*b^2*c*d^2*e^2 - 8*a^2*c^2*d^2*e^2 - 2*a 
*b^3*d*e^3 + 8*a^2*b*c*d*e^3 + a^2*b^2*e^4 - 4*a^3*c*e^4)*sqrt(-b^2 + 4*a* 
c)) - (b*c^2*d^3 - 2*b^2*c*d^2*e + 2*a*c^2*d^2*e + b^3*d*e^2 - a*b*c*d*e^2 
 - a*b^2*e^3 + 2*a^2*c*e^3 + (2*c^3*d^3 - 3*b*c^2*d^2*e + b^2*c*d*e^2 + 2* 
a*c^2*d*e^2 - a*b*c*e^3)*x)/((c*d^2 - b*d*e + a*e^2)^2*(c*x^2 + b*x + a)*( 
b^2 - 4*a*c))
 

Mupad [B] (verification not implemented)

Time = 8.17 (sec) , antiderivative size = 2953, normalized size of antiderivative = 13.18 \[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

(e^3*log(d + e*x))/(a^2*e^4 + c^2*d^4 + b^2*d^2*e^2 - 2*a*b*d*e^3 - 2*b*c* 
d^3*e + 2*a*c*d^2*e^2) - ((2*a*c*e - b^2*e + b*c*d)/(a*b^2*e^2 - 4*a*c^2*d 
^2 - 4*a^2*c*e^2 + b^2*c*d^2 - b^3*d*e + 4*a*b*c*d*e) + (x*(2*c^2*d - b*c* 
e))/(a*b^2*e^2 - 4*a*c^2*d^2 - 4*a^2*c*e^2 + b^2*c*d^2 - b^3*d*e + 4*a*b*c 
*d*e))/(a + b*x + c*x^2) + (log(96*a^4*c^3*e^5 - 2*b^7*e^5*x - 2*a*b^6*e^5 
 - 2*b^3*c^4*d^5 - 2*c^4*d^5*(-(4*a*c - b^2)^3)^(1/2) + 2*a*b^3*e^5*(-(4*a 
*c - b^2)^3)^(1/2) + 23*a^2*b^4*c*e^5 - 32*a^2*c^5*d^4*e + 3*b^4*c^3*d^4*e 
 + b^6*c*d^2*e^3 - 4*b^2*c^5*d^5*x + 2*b^4*e^5*x*(-(4*a*c - b^2)^3)^(1/2) 
- 84*a^3*b^2*c^2*e^5 - 192*a^3*c^4*d^2*e^3 + 8*a*b*c^5*d^5 + 16*a*c^6*d^5* 
x + 2*a*b^5*c*d*e^4 + 24*a*b^5*c*e^5*x + 4*b^6*c*d*e^4*x + 72*a^2*b^2*c^3* 
d^2*e^3 - 9*a^2*b*c*e^5*(-(4*a*c - b^2)^3)^(1/2) - 4*a*b^2*c^4*d^4*e + 72* 
a^3*b*c^3*d*e^4 + 3*b*c^3*d^4*e*(-(4*a*c - b^2)^3)^(1/2) + 120*a^3*b*c^3*e 
^5*x - 240*a^3*c^4*d*e^4*x + 10*b^3*c^4*d^4*e*x - 4*c^4*d^4*e*x*(-(4*a*c - 
 b^2)^3)^(1/2) - 20*a*b^3*c^3*d^3*e^2 - 10*a*b^4*c^2*d^2*e^3 + 80*a^2*b*c^ 
4*d^3*e^2 - 26*a^2*b^3*c^2*d*e^4 - 4*a*c^3*d^3*e^2*(-(4*a*c - b^2)^3)^(1/2 
) + 30*a^2*c^2*d*e^4*(-(4*a*c - b^2)^3)^(1/2) + b^3*c*d^2*e^3*(-(4*a*c - b 
^2)^3)^(1/2) - 94*a^2*b^3*c^2*e^5*x + 12*a^2*c^2*e^5*x*(-(4*a*c - b^2)^3)^ 
(1/2) + 32*a^2*c^5*d^3*e^2*x - 8*b^4*c^3*d^3*e^2*x + 2*b^5*c^2*d^2*e^3*x - 
 6*a*b*c^2*d^2*e^3*(-(4*a*c - b^2)^3)^(1/2) + 24*a*b^2*c^4*d^3*e^2*x + 4*a 
*b^3*c^3*d^2*e^3*x - 48*a^2*b*c^4*d^2*e^3*x + 204*a^2*b^2*c^3*d*e^4*x -...
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 1941, normalized size of antiderivative = 8.67 \[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^2} \, dx =\text {Too large to display} \] Input:

int(1/(e*x+d)/(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**2*b**2*c 
*e**3 + 24*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b* 
c**2*d*e**2 + 2*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a* 
b**4*e**3 - 12*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b 
**3*c*e**3*x - 12*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))* 
a*b**2*c**2*d**2*e + 24*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b 
**2))*a*b**2*c**2*d*e**2*x - 12*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4 
*a*c - b**2))*a*b**2*c**2*e**3*x**2 + 8*sqrt(4*a*c - b**2)*atan((b + 2*c*x 
)/sqrt(4*a*c - b**2))*a*b*c**3*d**3 + 24*sqrt(4*a*c - b**2)*atan((b + 2*c* 
x)/sqrt(4*a*c - b**2))*a*b*c**3*d*e**2*x**2 + 2*sqrt(4*a*c - b**2)*atan((b 
 + 2*c*x)/sqrt(4*a*c - b**2))*b**5*e**3*x + 2*sqrt(4*a*c - b**2)*atan((b + 
 2*c*x)/sqrt(4*a*c - b**2))*b**4*c*e**3*x**2 - 12*sqrt(4*a*c - b**2)*atan( 
(b + 2*c*x)/sqrt(4*a*c - b**2))*b**3*c**2*d**2*e*x + 8*sqrt(4*a*c - b**2)* 
atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b**2*c**3*d**3*x - 12*sqrt(4*a*c - b* 
*2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b**2*c**3*d**2*e*x**2 + 8*sqrt(4* 
a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b*c**4*d**3*x**2 - 16*log 
(a + b*x + c*x**2)*a**3*b*c**2*e**3 + 8*log(a + b*x + c*x**2)*a**2*b**3*c* 
e**3 - 16*log(a + b*x + c*x**2)*a**2*b**2*c**2*e**3*x - 16*log(a + b*x + c 
*x**2)*a**2*b*c**3*e**3*x**2 - log(a + b*x + c*x**2)*a*b**5*e**3 + 8*log(a 
 + b*x + c*x**2)*a*b**4*c*e**3*x + 8*log(a + b*x + c*x**2)*a*b**3*c**2*...