\(\int \frac {1}{(b d+2 c d x)^2 (a+b x+c x^2)^2} \, dx\) [67]

Optimal result

Integrand size = 24, antiderivative size = 98 \[ \int \frac {1}{(b d+2 c d x)^2 \left (a+b x+c x^2\right )^2} \, dx=-\frac {12 c}{\left (b^2-4 a c\right )^2 d^2 (b+2 c x)}-\frac {1}{\left (b^2-4 a c\right ) d^2 (b+2 c x) \left (a+b x+c x^2\right )}+\frac {12 c \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2} d^2} \] Output:

-12*c/(-4*a*c+b^2)^2/d^2/(2*c*x+b)-1/(-4*a*c+b^2)/d^2/(2*c*x+b)/(c*x^2+b*x 
+a)+12*c*arctanh((2*c*x+b)/(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(5/2)/d^2
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.86 \[ \int \frac {1}{(b d+2 c d x)^2 \left (a+b x+c x^2\right )^2} \, dx=-\frac {\frac {8 c}{b+2 c x}+\frac {b+2 c x}{a+x (b+c x)}+\frac {12 c \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}}{\left (b^2-4 a c\right )^2 d^2} \] Input:

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

Output:

-(((8*c)/(b + 2*c*x) + (b + 2*c*x)/(a + x*(b + c*x)) + (12*c*ArcTan[(b + 2 
*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c])/((b^2 - 4*a*c)^2*d^2))
 

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.09, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {1111, 27, 1117, 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.

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

\(\Big \downarrow \) 1111

\(\displaystyle -\frac {6 c \int \frac {1}{d^2 (b+2 c x)^2 \left (c x^2+b x+a\right )}dx}{b^2-4 a c}-\frac {1}{d^2 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {6 c \int \frac {1}{(b+2 c x)^2 \left (c x^2+b x+a\right )}dx}{d^2 \left (b^2-4 a c\right )}-\frac {1}{d^2 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )}\)

\(\Big \downarrow \) 1117

\(\displaystyle -\frac {6 c \left (\frac {\int \frac {1}{c x^2+b x+a}dx}{b^2-4 a c}+\frac {2}{\left (b^2-4 a c\right ) (b+2 c x)}\right )}{d^2 \left (b^2-4 a c\right )}-\frac {1}{d^2 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )}\)

\(\Big \downarrow \) 1083

\(\displaystyle -\frac {6 c \left (\frac {2}{\left (b^2-4 a c\right ) (b+2 c x)}-\frac {2 \int \frac {1}{b^2-(b+2 c x)^2-4 a c}d(b+2 c x)}{b^2-4 a c}\right )}{d^2 \left (b^2-4 a c\right )}-\frac {1}{d^2 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {6 c \left (\frac {2}{\left (b^2-4 a c\right ) (b+2 c x)}-\frac {2 \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}\right )}{d^2 \left (b^2-4 a c\right )}-\frac {1}{d^2 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )}\)

Input:

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

Output:

-(1/((b^2 - 4*a*c)*d^2*(b + 2*c*x)*(a + b*x + c*x^2))) - (6*c*(2/((b^2 - 4 
*a*c)*(b + 2*c*x)) - (2*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*a 
*c)^(3/2)))/((b^2 - 4*a*c)*d^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 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 1111

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_S 
ymbol] :> Simp[2*c*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(e*(p + 1)* 
(b^2 - 4*a*c))), x] - Simp[2*c*e*((m + 2*p + 3)/(e*(p + 1)*(b^2 - 4*a*c))) 
  Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e 
, m}, x] && EqQ[2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] && LtQ[p, -1] &&  !G 
tQ[m, 1] && RationalQ[m] && IntegerQ[2*p]
 

rule 1117

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_S 
ymbol] :> Simp[-2*b*d*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(d^2*(m 
+ 1)*(b^2 - 4*a*c))), x] + Simp[b^2*((m + 2*p + 3)/(d^2*(m + 1)*(b^2 - 4*a* 
c)))   Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, 
 d, e, p}, x] && EqQ[2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] && LtQ[m, -1] & 
& (IntegerQ[2*p] || (IntegerQ[m] && RationalQ[p]) || IntegerQ[(m + 2*p + 3) 
/2])
 

Maple [A] (verified)

Time = 1.02 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00

method	result	size
default	\(\frac {-\frac {8 c}{\left (4 a c -b^{2}\right )^{2} \left (2 c x +b \right )}-\frac {\frac {2 c x +b}{c \,x^{2}+b x +a}+\frac {12 c \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{\left (4 a c -b^{2}\right )^{2}}}{d^{2}}\)	\(98\)
risch	\(\frac {-\frac {12 c^{2} x^{2}}{16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}}-\frac {12 b c x}{16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}}-\frac {8 a c +b^{2}}{16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}}}{d^{2} \left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )}+6 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (1024 a^{5} c^{5} d^{4}-1280 a^{4} b^{2} c^{4} d^{4}+640 a^{3} b^{4} c^{3} d^{4}-160 a^{2} b^{6} c^{2} d^{4}+20 a \,b^{8} c \,d^{4}-b^{10} d^{4}\right ) \textit {\_Z}^{2}+c^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (6144 a^{5} c^{6} d^{4}-7680 a^{4} b^{2} c^{5} d^{4}+3840 a^{3} b^{4} c^{4} d^{4}-960 a^{2} b^{6} c^{3} d^{4}+120 a \,b^{8} c^{2} d^{4}-6 b^{10} c \,d^{4}\right ) \textit {\_R}^{2}+4 c^{3}\right ) x +\left (3072 a^{5} b \,c^{5} d^{4}-3840 a^{4} b^{3} c^{4} d^{4}+1920 a^{3} b^{5} c^{3} d^{4}-480 a^{2} b^{7} c^{2} d^{4}+60 a \,b^{9} c \,d^{4}-3 b^{11} d^{4}\right ) \textit {\_R}^{2}+\left (64 a^{3} c^{4} d^{2}-48 a^{2} b^{2} c^{3} d^{2}+12 a \,b^{4} c^{2} d^{2}-c \,d^{2} b^{6}\right ) \textit {\_R} +2 b \,c^{2}\right )\right )\)	\(422\)

Input:

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

Output:

1/d^2*(-8/(4*a*c-b^2)^2*c/(2*c*x+b)-1/(4*a*c-b^2)^2*((2*c*x+b)/(c*x^2+b*x+ 
a)+12*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. 312 vs. \(2 (94) = 188\).

Time = 0.10 (sec) , antiderivative size = 644, normalized size of antiderivative = 6.57 \[ \int \frac {1}{(b d+2 c d x)^2 \left (a+b x+c x^2\right )^2} \, dx=\left [-\frac {b^{4} + 4 \, a b^{2} c - 32 \, a^{2} c^{2} + 12 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{2} - 6 \, {\left (2 \, c^{3} x^{3} + 3 \, b c^{2} x^{2} + a b c + {\left (b^{2} c + 2 \, a c^{2}\right )} x\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c + \sqrt {b^{2} - 4 \, a c} {\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) + 12 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} x}{2 \, {\left (b^{6} c^{2} - 12 \, a b^{4} c^{3} + 48 \, a^{2} b^{2} c^{4} - 64 \, a^{3} c^{5}\right )} d^{2} x^{3} + 3 \, {\left (b^{7} c - 12 \, a b^{5} c^{2} + 48 \, a^{2} b^{3} c^{3} - 64 \, a^{3} b c^{4}\right )} d^{2} x^{2} + {\left (b^{8} - 10 \, a b^{6} c + 24 \, a^{2} b^{4} c^{2} + 32 \, a^{3} b^{2} c^{3} - 128 \, a^{4} c^{4}\right )} d^{2} x + {\left (a b^{7} - 12 \, a^{2} b^{5} c + 48 \, a^{3} b^{3} c^{2} - 64 \, a^{4} b c^{3}\right )} d^{2}}, -\frac {b^{4} + 4 \, a b^{2} c - 32 \, a^{2} c^{2} + 12 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{2} - 12 \, {\left (2 \, c^{3} x^{3} + 3 \, b c^{2} x^{2} + a b c + {\left (b^{2} c + 2 \, a c^{2}\right )} x\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) + 12 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} x}{2 \, {\left (b^{6} c^{2} - 12 \, a b^{4} c^{3} + 48 \, a^{2} b^{2} c^{4} - 64 \, a^{3} c^{5}\right )} d^{2} x^{3} + 3 \, {\left (b^{7} c - 12 \, a b^{5} c^{2} + 48 \, a^{2} b^{3} c^{3} - 64 \, a^{3} b c^{4}\right )} d^{2} x^{2} + {\left (b^{8} - 10 \, a b^{6} c + 24 \, a^{2} b^{4} c^{2} + 32 \, a^{3} b^{2} c^{3} - 128 \, a^{4} c^{4}\right )} d^{2} x + {\left (a b^{7} - 12 \, a^{2} b^{5} c + 48 \, a^{3} b^{3} c^{2} - 64 \, a^{4} b c^{3}\right )} d^{2}}\right ] \] Input:

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

Output:

[-(b^4 + 4*a*b^2*c - 32*a^2*c^2 + 12*(b^2*c^2 - 4*a*c^3)*x^2 - 6*(2*c^3*x^ 
3 + 3*b*c^2*x^2 + a*b*c + (b^2*c + 2*a*c^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)) + 12*(b^3*c - 4*a*b*c^2)*x)/(2*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^ 
2*c^4 - 64*a^3*c^5)*d^2*x^3 + 3*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 6 
4*a^3*b*c^4)*d^2*x^2 + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 
 - 128*a^4*c^4)*d^2*x + (a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^4*b* 
c^3)*d^2), -(b^4 + 4*a*b^2*c - 32*a^2*c^2 + 12*(b^2*c^2 - 4*a*c^3)*x^2 - 1 
2*(2*c^3*x^3 + 3*b*c^2*x^2 + a*b*c + (b^2*c + 2*a*c^2)*x)*sqrt(-b^2 + 4*a* 
c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + 12*(b^3*c - 4*a 
*b*c^2)*x)/(2*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^2*x 
^3 + 3*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^2*x^2 + (b 
^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*d^2*x + ( 
a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^4*b*c^3)*d^2)]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 459 vs. \(2 (92) = 184\).

Time = 0.93 (sec) , antiderivative size = 459, normalized size of antiderivative = 4.68 \[ \int \frac {1}{(b d+2 c d x)^2 \left (a+b x+c x^2\right )^2} \, dx=\frac {6 c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \log {\left (x + \frac {- 384 a^{3} c^{4} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} + 288 a^{2} b^{2} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} - 72 a b^{4} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} + 6 b^{6} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} + 6 b c}{12 c^{2}} \right )}}{d^{2}} - \frac {6 c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \log {\left (x + \frac {384 a^{3} c^{4} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} - 288 a^{2} b^{2} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} + 72 a b^{4} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} - 6 b^{6} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} + 6 b c}{12 c^{2}} \right )}}{d^{2}} + \frac {- 8 a c - b^{2} - 12 b c x - 12 c^{2} x^{2}}{16 a^{3} b c^{2} d^{2} - 8 a^{2} b^{3} c d^{2} + a b^{5} d^{2} + x^{3} \cdot \left (32 a^{2} c^{4} d^{2} - 16 a b^{2} c^{3} d^{2} + 2 b^{4} c^{2} d^{2}\right ) + x^{2} \cdot \left (48 a^{2} b c^{3} d^{2} - 24 a b^{3} c^{2} d^{2} + 3 b^{5} c d^{2}\right ) + x \left (32 a^{3} c^{3} d^{2} - 6 a b^{4} c d^{2} + b^{6} d^{2}\right )} \] Input:

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

Output:

6*c*sqrt(-1/(4*a*c - b**2)**5)*log(x + (-384*a**3*c**4*sqrt(-1/(4*a*c - b* 
*2)**5) + 288*a**2*b**2*c**3*sqrt(-1/(4*a*c - b**2)**5) - 72*a*b**4*c**2*s 
qrt(-1/(4*a*c - b**2)**5) + 6*b**6*c*sqrt(-1/(4*a*c - b**2)**5) + 6*b*c)/( 
12*c**2))/d**2 - 6*c*sqrt(-1/(4*a*c - b**2)**5)*log(x + (384*a**3*c**4*sqr 
t(-1/(4*a*c - b**2)**5) - 288*a**2*b**2*c**3*sqrt(-1/(4*a*c - b**2)**5) + 
72*a*b**4*c**2*sqrt(-1/(4*a*c - b**2)**5) - 6*b**6*c*sqrt(-1/(4*a*c - b**2 
)**5) + 6*b*c)/(12*c**2))/d**2 + (-8*a*c - b**2 - 12*b*c*x - 12*c**2*x**2) 
/(16*a**3*b*c**2*d**2 - 8*a**2*b**3*c*d**2 + a*b**5*d**2 + x**3*(32*a**2*c 
**4*d**2 - 16*a*b**2*c**3*d**2 + 2*b**4*c**2*d**2) + x**2*(48*a**2*b*c**3* 
d**2 - 24*a*b**3*c**2*d**2 + 3*b**5*c*d**2) + x*(32*a**3*c**3*d**2 - 6*a*b 
**4*c*d**2 + b**6*d**2))
 

Maxima [F(-2)]

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

integrate(1/(2*c*d*x+b*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. 220 vs. \(2 (94) = 188\).

Time = 0.13 (sec) , antiderivative size = 220, normalized size of antiderivative = 2.24 \[ \int \frac {1}{(b d+2 c d x)^2 \left (a+b x+c x^2\right )^2} \, dx=-\frac {8 \, c^{5} d^{7}}{{\left (b^{4} c^{4} d^{8} - 8 \, a b^{2} c^{5} d^{8} + 16 \, a^{2} c^{6} d^{8}\right )} {\left (2 \, c d x + b d\right )}} - \frac {12 \, c \arctan \left (\frac {\frac {b^{2} d}{2 \, c d x + b d} - \frac {4 \, a c d}{2 \, c d x + b d}}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {-b^{2} + 4 \, a c} d^{2}} + \frac {4 \, c}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} {\left (2 \, c d x + b d\right )} {\left (\frac {b^{2} d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} - \frac {4 \, a c d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} - 1\right )} d} \] Input:

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

Output:

-8*c^5*d^7/((b^4*c^4*d^8 - 8*a*b^2*c^5*d^8 + 16*a^2*c^6*d^8)*(2*c*d*x + b* 
d)) - 12*c*arctan((b^2*d/(2*c*d*x + b*d) - 4*a*c*d/(2*c*d*x + b*d))/sqrt(- 
b^2 + 4*a*c))/((b^4 - 8*a*b^2*c + 16*a^2*c^2)*sqrt(-b^2 + 4*a*c)*d^2) + 4* 
c/((b^4 - 8*a*b^2*c + 16*a^2*c^2)*(2*c*d*x + b*d)*(b^2*d^2/(2*c*d*x + b*d) 
^2 - 4*a*c*d^2/(2*c*d*x + b*d)^2 - 1)*d)
 

Mupad [B] (verification not implemented)

Time = 5.77 (sec) , antiderivative size = 232, normalized size of antiderivative = 2.37 \[ \int \frac {1}{(b d+2 c d x)^2 \left (a+b x+c x^2\right )^2} \, dx=-\frac {\frac {b^2+8\,a\,c}{{\left (4\,a\,c-b^2\right )}^2}+\frac {12\,c^2\,x^2}{{\left (4\,a\,c-b^2\right )}^2}+\frac {12\,b\,c\,x}{{\left (4\,a\,c-b^2\right )}^2}}{x\,\left (b^2\,d^2+2\,a\,c\,d^2\right )+2\,c^2\,d^2\,x^3+a\,b\,d^2+3\,b\,c\,d^2\,x^2}-\frac {12\,c\,\mathrm {atan}\left (\frac {\frac {6\,c\,\left (16\,a^2\,b\,c^2\,d^2-8\,a\,b^3\,c\,d^2+b^5\,d^2\right )}{d^2\,{\left (4\,a\,c-b^2\right )}^{5/2}}+\frac {12\,c^2\,x\,\left (16\,a^2\,c^2\,d^2-8\,a\,b^2\,c\,d^2+b^4\,d^2\right )}{d^2\,{\left (4\,a\,c-b^2\right )}^{5/2}}}{6\,c}\right )}{d^2\,{\left (4\,a\,c-b^2\right )}^{5/2}} \] Input:

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

Output:

- ((8*a*c + b^2)/(4*a*c - b^2)^2 + (12*c^2*x^2)/(4*a*c - b^2)^2 + (12*b*c* 
x)/(4*a*c - b^2)^2)/(x*(b^2*d^2 + 2*a*c*d^2) + 2*c^2*d^2*x^3 + a*b*d^2 + 3 
*b*c*d^2*x^2) - (12*c*atan(((6*c*(b^5*d^2 + 16*a^2*b*c^2*d^2 - 8*a*b^3*c*d 
^2))/(d^2*(4*a*c - b^2)^(5/2)) + (12*c^2*x*(b^4*d^2 + 16*a^2*c^2*d^2 - 8*a 
*b^2*c*d^2))/(d^2*(4*a*c - b^2)^(5/2)))/(6*c)))/(d^2*(4*a*c - b^2)^(5/2))
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 448, normalized size of antiderivative = 4.57 \[ \int \frac {1}{(b d+2 c d x)^2 \left (a+b x+c x^2\right )^2} \, dx=\frac {-12 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) a \,b^{2} c -24 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) a b \,c^{2} x -12 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) b^{3} c x -36 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) b^{2} c^{2} x^{2}-24 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) b \,c^{3} x^{3}-16 a^{2} b \,c^{2}+32 a^{2} c^{3} x -40 a \,b^{2} c^{2} x +32 a \,c^{4} x^{3}+b^{5}+8 b^{4} c x -8 b^{2} c^{3} x^{3}}{b \,d^{2} \left (128 a^{3} c^{5} x^{3}-96 a^{2} b^{2} c^{4} x^{3}+24 a \,b^{4} c^{3} x^{3}-2 b^{6} c^{2} x^{3}+192 a^{3} b \,c^{4} x^{2}-144 a^{2} b^{3} c^{3} x^{2}+36 a \,b^{5} c^{2} x^{2}-3 b^{7} c \,x^{2}+128 a^{4} c^{4} x -32 a^{3} b^{2} c^{3} x -24 a^{2} b^{4} c^{2} x +10 a \,b^{6} c x -b^{8} x +64 a^{4} b \,c^{3}-48 a^{3} b^{3} c^{2}+12 a^{2} b^{5} c -a \,b^{7}\right )} \] Input:

int(1/(2*c*d*x+b*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*b**2*c - 
24*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b*c**2*x - 12 
*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b**3*c*x - 36*sqr 
t(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b**2*c**2*x**2 - 24*s 
qrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b*c**3*x**3 - 16*a* 
*2*b*c**2 + 32*a**2*c**3*x - 40*a*b**2*c**2*x + 32*a*c**4*x**3 + b**5 + 8* 
b**4*c*x - 8*b**2*c**3*x**3)/(b*d**2*(64*a**4*b*c**3 + 128*a**4*c**4*x - 4 
8*a**3*b**3*c**2 - 32*a**3*b**2*c**3*x + 192*a**3*b*c**4*x**2 + 128*a**3*c 
**5*x**3 + 12*a**2*b**5*c - 24*a**2*b**4*c**2*x - 144*a**2*b**3*c**3*x**2 
- 96*a**2*b**2*c**4*x**3 - a*b**7 + 10*a*b**6*c*x + 36*a*b**5*c**2*x**2 + 
24*a*b**4*c**3*x**3 - b**8*x - 3*b**7*c*x**2 - 2*b**6*c**2*x**3))