\(\int \frac {1}{(b d+2 c d x)^2 (a+b x+c x^2)^3} \, dx\) [80]

Optimal result

Integrand size = 24, antiderivative size = 140 \[ \int \frac {1}{(b d+2 c d x)^2 \left (a+b x+c x^2\right )^3} \, dx=\frac {60 c^2}{\left (b^2-4 a c\right )^3 d^2 (b+2 c x)}-\frac {1}{2 \left (b^2-4 a c\right ) d^2 (b+2 c x) \left (a+b x+c x^2\right )^2}+\frac {5 c}{\left (b^2-4 a c\right )^2 d^2 (b+2 c x) \left (a+b x+c x^2\right )}-\frac {60 c^2 \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/2} d^2} \] Output:

60*c^2/(-4*a*c+b^2)^3/d^2/(2*c*x+b)-1/2/(-4*a*c+b^2)/d^2/(2*c*x+b)/(c*x^2+ 
b*x+a)^2+5*c/(-4*a*c+b^2)^2/d^2/(2*c*x+b)/(c*x^2+b*x+a)-60*c^2*arctanh((2* 
c*x+b)/(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(7/2)/d^2
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.85 \[ \int \frac {1}{(b d+2 c d x)^2 \left (a+b x+c x^2\right )^3} \, dx=\frac {\frac {64 c^2}{b+2 c x}-\frac {\left (b^2-4 a c\right ) (b+2 c x)}{(a+x (b+c x))^2}+\frac {14 c (b+2 c x)}{a+x (b+c x)}+\frac {120 c^2 \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}}{2 \left (b^2-4 a c\right )^3 d^2} \] Input:

Integrate[1/((b*d + 2*c*d*x)^2*(a + b*x + c*x^2)^3),x]
 

Output:

((64*c^2)/(b + 2*c*x) - ((b^2 - 4*a*c)*(b + 2*c*x))/(a + x*(b + c*x))^2 + 
(14*c*(b + 2*c*x))/(a + x*(b + c*x)) + (120*c^2*ArcTan[(b + 2*c*x)/Sqrt[-b 
^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c])/(2*(b^2 - 4*a*c)^3*d^2)
 

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.11, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1111, 27, 1111, 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.

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

\(\Big \downarrow \) 1111

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1111

\(\displaystyle -\frac {5 c \left (-\frac {6 c \int \frac {1}{(b+2 c x)^2 \left (c x^2+b x+a\right )}dx}{b^2-4 a c}-\frac {1}{\left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )}\right )}{d^2 \left (b^2-4 a c\right )}-\frac {1}{2 d^2 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 1117

\(\displaystyle -\frac {5 c \left (-\frac {6 c \left (\frac {\int \frac {1}{c x^2+b x+a}dx}{b^2-4 a c}+\frac {2}{\left (b^2-4 a c\right ) (b+2 c x)}\right )}{b^2-4 a c}-\frac {1}{\left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )}\right )}{d^2 \left (b^2-4 a c\right )}-\frac {1}{2 d^2 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 1083

\(\displaystyle -\frac {5 c \left (-\frac {6 c \left (\frac {2}{\left (b^2-4 a c\right ) (b+2 c x)}-\frac {2 \int \frac {1}{b^2-(b+2 c x)^2-4 a c}d(b+2 c x)}{b^2-4 a c}\right )}{b^2-4 a c}-\frac {1}{\left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )}\right )}{d^2 \left (b^2-4 a c\right )}-\frac {1}{2 d^2 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {5 c \left (-\frac {6 c \left (\frac {2}{\left (b^2-4 a c\right ) (b+2 c x)}-\frac {2 \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}\right )}{b^2-4 a c}-\frac {1}{\left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )}\right )}{d^2 \left (b^2-4 a c\right )}-\frac {1}{2 d^2 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^2}\)

Input:

Int[1/((b*d + 2*c*d*x)^2*(a + b*x + c*x^2)^3),x]
 

Output:

-1/2*1/((b^2 - 4*a*c)*d^2*(b + 2*c*x)*(a + b*x + c*x^2)^2) - (5*c*(-(1/((b 
^2 - 4*a*c)*(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)))/((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 = 1.07 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.99

method	result	size
default	\(\frac {-\frac {32 c^{2}}{\left (4 a c -b^{2}\right )^{3} \left (2 c x +b \right )}-\frac {\frac {14 c^{3} x^{3}+21 b \,c^{2} x^{2}+6 c \left (3 a c +b^{2}\right ) x +\frac {b \left (18 a c -b^{2}\right )}{2}}{\left (c \,x^{2}+b x +a \right )^{2}}+\frac {60 c^{2} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{\left (4 a c -b^{2}\right )^{3}}}{d^{2}}\)	\(139\)
risch	\(\frac {-\frac {60 c^{4} x^{4}}{64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}-\frac {120 b \,c^{3} x^{3}}{64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}-\frac {5 c^{2} \left (20 a c +13 b^{2}\right ) x^{2}}{64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}-\frac {5 b c \left (20 a c +b^{2}\right ) x}{64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}-\frac {64 a^{2} c^{2}+18 c a \,b^{2}-b^{4}}{2 \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}}{d^{2} \left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{2}}+30 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (16384 a^{7} c^{7} d^{4}-28672 a^{6} b^{2} c^{6} d^{4}+21504 a^{5} b^{4} c^{5} d^{4}-8960 a^{4} b^{6} c^{4} d^{4}+2240 a^{3} b^{8} c^{3} d^{4}-336 a^{2} b^{10} c^{2} d^{4}+28 a \,b^{12} c \,d^{4}-b^{14} d^{4}\right ) \textit {\_Z}^{2}+c^{4}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (98304 a^{7} c^{8} d^{4}-172032 a^{6} b^{2} c^{7} d^{4}+129024 a^{5} b^{4} c^{6} d^{4}-53760 a^{4} b^{6} c^{5} d^{4}+13440 a^{3} b^{8} c^{4} d^{4}-2016 a^{2} b^{10} c^{3} d^{4}+168 a \,b^{12} c^{2} d^{4}-6 b^{14} c \,d^{4}\right ) \textit {\_R}^{2}+4 c^{5}\right ) x +\left (49152 a^{7} b \,c^{7} d^{4}-86016 a^{6} b^{3} c^{6} d^{4}+64512 a^{5} b^{5} c^{5} d^{4}-26880 a^{4} b^{7} c^{4} d^{4}+6720 a^{3} b^{9} c^{3} d^{4}-1008 a^{2} b^{11} c^{2} d^{4}+84 a \,b^{13} c \,d^{4}-3 b^{15} d^{4}\right ) \textit {\_R}^{2}+\left (256 a^{4} c^{6} d^{2}-256 a^{3} b^{2} c^{5} d^{2}+96 a^{2} b^{4} c^{4} d^{2}-16 a \,b^{6} c^{3} d^{2}+b^{8} c^{2} d^{2}\right ) \textit {\_R} +2 b \,c^{4}\right )\right )\)	\(676\)

Input:

int(1/(2*c*d*x+b*d)^2/(c*x^2+b*x+a)^3,x,method=_RETURNVERBOSE)
 

Output:

1/d^2*(-32/(4*a*c-b^2)^3*c^2/(2*c*x+b)-1/(4*a*c-b^2)^3*((14*c^3*x^3+21*b*c 
^2*x^2+6*c*(3*a*c+b^2)*x+1/2*b*(18*a*c-b^2))/(c*x^2+b*x+a)^2+60*c^2/(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. 575 vs. \(2 (134) = 268\).

Time = 0.10 (sec) , antiderivative size = 1170, normalized size of antiderivative = 8.36 \[ \int \frac {1}{(b d+2 c d x)^2 \left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate(1/(2*c*d*x+b*d)^2/(c*x^2+b*x+a)^3,x, algorithm="fricas")
 

Output:

[-1/2*(b^6 - 22*a*b^4*c + 8*a^2*b^2*c^2 + 256*a^3*c^3 - 120*(b^2*c^4 - 4*a 
*c^5)*x^4 - 240*(b^3*c^3 - 4*a*b*c^4)*x^3 - 10*(13*b^4*c^2 - 32*a*b^2*c^3 
- 80*a^2*c^4)*x^2 + 60*(2*c^5*x^5 + 5*b*c^4*x^4 + a^2*b*c^2 + 4*(b^2*c^3 + 
 a*c^4)*x^3 + (b^3*c^2 + 6*a*b*c^3)*x^2 + 2*(a*b^2*c^2 + a^2*c^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)) - 10*(b^5*c + 16*a*b^3*c^2 - 80*a^2*b*c^3)*x 
)/(2*(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)*d^2*x^5 + 5*(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)*d^2*x^4 + 4*(b^10*c - 15*a*b^8*c^2 + 80*a^2*b^6*c^3 - 1 
60*a^3*b^4*c^4 + 256*a^5*c^6)*d^2*x^3 + (b^11 - 10*a*b^9*c + 320*a^3*b^5*c 
^3 - 1280*a^4*b^3*c^4 + 1536*a^5*b*c^5)*d^2*x^2 + 2*(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)*d^2*x + (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)*d^2), -1/2*(b 
^6 - 22*a*b^4*c + 8*a^2*b^2*c^2 + 256*a^3*c^3 - 120*(b^2*c^4 - 4*a*c^5)*x^ 
4 - 240*(b^3*c^3 - 4*a*b*c^4)*x^3 - 10*(13*b^4*c^2 - 32*a*b^2*c^3 - 80*a^2 
*c^4)*x^2 + 120*(2*c^5*x^5 + 5*b*c^4*x^4 + a^2*b*c^2 + 4*(b^2*c^3 + a*c^4) 
*x^3 + (b^3*c^2 + 6*a*b*c^3)*x^2 + 2*(a*b^2*c^2 + a^2*c^3)*x)*sqrt(-b^2 + 
4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) - 10*(b^5*c + 
 16*a*b^3*c^2 - 80*a^2*b*c^3)*x)/(2*(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)*d^2*x^5 + 5*(b^9*c^2 - 16*a*b^7*c^3...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 804 vs. \(2 (133) = 266\).

Time = 1.61 (sec) , antiderivative size = 804, normalized size of antiderivative = 5.74 \[ \int \frac {1}{(b d+2 c d x)^2 \left (a+b x+c x^2\right )^3} \, dx=\frac {30 c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \log {\left (x + \frac {- 7680 a^{4} c^{6} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} + 7680 a^{3} b^{2} c^{5} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} - 2880 a^{2} b^{4} c^{4} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} + 480 a b^{6} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} - 30 b^{8} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} + 30 b c^{2}}{60 c^{3}} \right )}}{d^{2}} - \frac {30 c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \log {\left (x + \frac {7680 a^{4} c^{6} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} - 7680 a^{3} b^{2} c^{5} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} + 2880 a^{2} b^{4} c^{4} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} - 480 a b^{6} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} + 30 b^{8} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} + 30 b c^{2}}{60 c^{3}} \right )}}{d^{2}} + \frac {- 64 a^{2} c^{2} - 18 a b^{2} c + b^{4} - 240 b c^{3} x^{3} - 120 c^{4} x^{4} + x^{2} \left (- 200 a c^{3} - 130 b^{2} c^{2}\right ) + x \left (- 200 a b c^{2} - 10 b^{3} c\right )}{128 a^{5} b c^{3} d^{2} - 96 a^{4} b^{3} c^{2} d^{2} + 24 a^{3} b^{5} c d^{2} - 2 a^{2} b^{7} d^{2} + x^{5} \cdot \left (256 a^{3} c^{6} d^{2} - 192 a^{2} b^{2} c^{5} d^{2} + 48 a b^{4} c^{4} d^{2} - 4 b^{6} c^{3} d^{2}\right ) + x^{4} \cdot \left (640 a^{3} b c^{5} d^{2} - 480 a^{2} b^{3} c^{4} d^{2} + 120 a b^{5} c^{3} d^{2} - 10 b^{7} c^{2} d^{2}\right ) + x^{3} \cdot \left (512 a^{4} c^{5} d^{2} + 128 a^{3} b^{2} c^{4} d^{2} - 288 a^{2} b^{4} c^{3} d^{2} + 88 a b^{6} c^{2} d^{2} - 8 b^{8} c d^{2}\right ) + x^{2} \cdot \left (768 a^{4} b c^{4} d^{2} - 448 a^{3} b^{3} c^{3} d^{2} + 48 a^{2} b^{5} c^{2} d^{2} + 12 a b^{7} c d^{2} - 2 b^{9} d^{2}\right ) + x \left (256 a^{5} c^{4} d^{2} + 64 a^{4} b^{2} c^{3} d^{2} - 144 a^{3} b^{4} c^{2} d^{2} + 44 a^{2} b^{6} c d^{2} - 4 a b^{8} d^{2}\right )} \] Input:

integrate(1/(2*c*d*x+b*d)**2/(c*x**2+b*x+a)**3,x)
 

Output:

30*c**2*sqrt(-1/(4*a*c - b**2)**7)*log(x + (-7680*a**4*c**6*sqrt(-1/(4*a*c 
 - b**2)**7) + 7680*a**3*b**2*c**5*sqrt(-1/(4*a*c - b**2)**7) - 2880*a**2* 
b**4*c**4*sqrt(-1/(4*a*c - b**2)**7) + 480*a*b**6*c**3*sqrt(-1/(4*a*c - b* 
*2)**7) - 30*b**8*c**2*sqrt(-1/(4*a*c - b**2)**7) + 30*b*c**2)/(60*c**3))/ 
d**2 - 30*c**2*sqrt(-1/(4*a*c - b**2)**7)*log(x + (7680*a**4*c**6*sqrt(-1/ 
(4*a*c - b**2)**7) - 7680*a**3*b**2*c**5*sqrt(-1/(4*a*c - b**2)**7) + 2880 
*a**2*b**4*c**4*sqrt(-1/(4*a*c - b**2)**7) - 480*a*b**6*c**3*sqrt(-1/(4*a* 
c - b**2)**7) + 30*b**8*c**2*sqrt(-1/(4*a*c - b**2)**7) + 30*b*c**2)/(60*c 
**3))/d**2 + (-64*a**2*c**2 - 18*a*b**2*c + b**4 - 240*b*c**3*x**3 - 120*c 
**4*x**4 + x**2*(-200*a*c**3 - 130*b**2*c**2) + x*(-200*a*b*c**2 - 10*b**3 
*c))/(128*a**5*b*c**3*d**2 - 96*a**4*b**3*c**2*d**2 + 24*a**3*b**5*c*d**2 
- 2*a**2*b**7*d**2 + x**5*(256*a**3*c**6*d**2 - 192*a**2*b**2*c**5*d**2 + 
48*a*b**4*c**4*d**2 - 4*b**6*c**3*d**2) + x**4*(640*a**3*b*c**5*d**2 - 480 
*a**2*b**3*c**4*d**2 + 120*a*b**5*c**3*d**2 - 10*b**7*c**2*d**2) + x**3*(5 
12*a**4*c**5*d**2 + 128*a**3*b**2*c**4*d**2 - 288*a**2*b**4*c**3*d**2 + 88 
*a*b**6*c**2*d**2 - 8*b**8*c*d**2) + x**2*(768*a**4*b*c**4*d**2 - 448*a**3 
*b**3*c**3*d**2 + 48*a**2*b**5*c**2*d**2 + 12*a*b**7*c*d**2 - 2*b**9*d**2) 
 + x*(256*a**5*c**4*d**2 + 64*a**4*b**2*c**3*d**2 - 144*a**3*b**4*c**2*d** 
2 + 44*a**2*b**6*c*d**2 - 4*a*b**8*d**2))
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{(b d+2 c d x)^2 \left (a+b x+c x^2\right )^3} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(1/(2*c*d*x+b*d)^2/(c*x^2+b*x+a)^3,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. 302 vs. \(2 (134) = 268\).

Time = 0.11 (sec) , antiderivative size = 302, normalized size of antiderivative = 2.16 \[ \int \frac {1}{(b d+2 c d x)^2 \left (a+b x+c x^2\right )^3} \, dx=\frac {32 \, c^{8} d^{11}}{{\left (b^{6} c^{6} d^{12} - 12 \, a b^{4} c^{7} d^{12} + 48 \, a^{2} b^{2} c^{8} d^{12} - 64 \, a^{3} c^{9} d^{12}\right )} {\left (2 \, c d x + b d\right )}} - \frac {60 \, c^{2} \arctan \left (-\frac {\frac {b^{2} d}{2 \, c d x + b d} - \frac {4 \, a c d}{2 \, c d x + b d}}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt {-b^{2} + 4 \, a c} d^{2}} - \frac {4 \, {\left (\frac {9 \, b^{2} c^{2} d}{{\left (2 \, c d x + b d\right )}^{3}} - \frac {36 \, a c^{3} d}{{\left (2 \, c d x + b d\right )}^{3}} - \frac {7 \, c^{2}}{{\left (2 \, c d x + b d\right )} d}\right )}}{{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} {\left (\frac {b^{2} d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} - \frac {4 \, a c d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} - 1\right )}^{2}} \] Input:

integrate(1/(2*c*d*x+b*d)^2/(c*x^2+b*x+a)^3,x, algorithm="giac")
 

Output:

32*c^8*d^11/((b^6*c^6*d^12 - 12*a*b^4*c^7*d^12 + 48*a^2*b^2*c^8*d^12 - 64* 
a^3*c^9*d^12)*(2*c*d*x + b*d)) - 60*c^2*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^6 - 12*a*b^4*c + 48*a^2*b^ 
2*c^2 - 64*a^3*c^3)*sqrt(-b^2 + 4*a*c)*d^2) - 4*(9*b^2*c^2*d/(2*c*d*x + b* 
d)^3 - 36*a*c^3*d/(2*c*d*x + b*d)^3 - 7*c^2/((2*c*d*x + b*d)*d))/((b^6 - 1 
2*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3)*(b^2*d^2/(2*c*d*x + b*d)^2 - 4*a* 
c*d^2/(2*c*d*x + b*d)^2 - 1)^2)
 

Mupad [B] (verification not implemented)

Time = 5.57 (sec) , antiderivative size = 494, normalized size of antiderivative = 3.53 \[ \int \frac {1}{(b d+2 c d x)^2 \left (a+b x+c x^2\right )^3} \, dx=\frac {60\,c^2\,\mathrm {atan}\left (\frac {\frac {30\,c^2\,\left (-64\,a^3\,b\,c^3\,d^2+48\,a^2\,b^3\,c^2\,d^2-12\,a\,b^5\,c\,d^2+b^7\,d^2\right )}{d^2\,{\left (4\,a\,c-b^2\right )}^{7/2}}+\frac {60\,c^3\,x\,\left (-64\,a^3\,c^3\,d^2+48\,a^2\,b^2\,c^2\,d^2-12\,a\,b^4\,c\,d^2+b^6\,d^2\right )}{d^2\,{\left (4\,a\,c-b^2\right )}^{7/2}}}{30\,c^2}\right )}{d^2\,{\left (4\,a\,c-b^2\right )}^{7/2}}-\frac {\frac {64\,a^2\,c^2+18\,a\,b^2\,c-b^4}{2\,\left (4\,a\,c-b^2\right )\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {60\,c^4\,x^4}{\left (4\,a\,c-b^2\right )\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {5\,c\,x\,\left (b^3+20\,a\,c\,b\right )}{\left (4\,a\,c-b^2\right )\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {5\,c\,x^2\,\left (13\,b^2\,c+20\,a\,c^2\right )}{\left (4\,a\,c-b^2\right )\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {120\,b\,c^3\,x^3}{\left (4\,a\,c-b^2\right )\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}}{x^3\,\left (4\,b^2\,c\,d^2+4\,a\,c^2\,d^2\right )+x\,\left (2\,c\,a^2\,d^2+2\,a\,b^2\,d^2\right )+x^2\,\left (b^3\,d^2+6\,a\,c\,b\,d^2\right )+a^2\,b\,d^2+2\,c^3\,d^2\,x^5+5\,b\,c^2\,d^2\,x^4} \] Input:

int(1/((b*d + 2*c*d*x)^2*(a + b*x + c*x^2)^3),x)
 

Output:

(60*c^2*atan(((30*c^2*(b^7*d^2 - 64*a^3*b*c^3*d^2 + 48*a^2*b^3*c^2*d^2 - 1 
2*a*b^5*c*d^2))/(d^2*(4*a*c - b^2)^(7/2)) + (60*c^3*x*(b^6*d^2 - 64*a^3*c^ 
3*d^2 + 48*a^2*b^2*c^2*d^2 - 12*a*b^4*c*d^2))/(d^2*(4*a*c - b^2)^(7/2)))/( 
30*c^2)))/(d^2*(4*a*c - b^2)^(7/2)) - ((64*a^2*c^2 - b^4 + 18*a*b^2*c)/(2* 
(4*a*c - b^2)*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (60*c^4*x^4)/((4*a*c - b^2 
)*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (5*c*x*(b^3 + 20*a*b*c))/((4*a*c - b^2 
)*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (5*c*x^2*(20*a*c^2 + 13*b^2*c))/((4*a* 
c - b^2)*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (120*b*c^3*x^3)/((4*a*c - b^2)* 
(b^4 + 16*a^2*c^2 - 8*a*b^2*c)))/(x^3*(4*a*c^2*d^2 + 4*b^2*c*d^2) + x*(2*a 
*b^2*d^2 + 2*a^2*c*d^2) + x^2*(b^3*d^2 + 6*a*b*c*d^2) + a^2*b*d^2 + 2*c^3* 
d^2*x^5 + 5*b*c^2*d^2*x^4)
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 893, normalized size of antiderivative = 6.38 \[ \int \frac {1}{(b d+2 c d x)^2 \left (a+b x+c x^2\right )^3} \, dx =\text {Too large to display} \] Input:

int(1/(2*c*d*x+b*d)^2/(c*x^2+b*x+a)^3,x)
 

Output:

( - 120*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b**2* 
c**2 - 240*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b* 
c**3*x - 240*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b** 
3*c**2*x - 720*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b 
**2*c**3*x**2 - 480*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2) 
)*a*b*c**4*x**3 - 120*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b** 
2))*b**4*c**2*x**2 - 480*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - 
b**2))*b**3*c**3*x**3 - 600*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c 
 - b**2))*b**2*c**4*x**4 - 240*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4* 
a*c - b**2))*b*c**5*x**5 - 160*a**3*b*c**3 + 192*a**3*c**4*x - 32*a**2*b** 
3*c**2 - 656*a**2*b**2*c**3*x - 224*a**2*b*c**4*x**2 + 384*a**2*c**5*x**3 
+ 22*a*b**5*c + 112*a*b**4*c**2*x - 368*a*b**3*c**3*x**2 - 672*a*b**2*c**4 
*x**3 + 192*a*c**6*x**5 - b**7 + 10*b**6*c*x + 106*b**5*c**2*x**2 + 144*b* 
*4*c**3*x**3 - 48*b**2*c**5*x**5)/(2*b*d**2*(256*a**6*b*c**4 + 512*a**6*c* 
*5*x - 256*a**5*b**3*c**3 + 1536*a**5*b*c**5*x**2 + 1024*a**5*c**6*x**3 + 
96*a**4*b**5*c**2 - 320*a**4*b**4*c**3*x - 1280*a**4*b**3*c**4*x**2 + 1280 
*a**4*b*c**6*x**4 + 512*a**4*c**7*x**5 - 16*a**3*b**7*c + 160*a**3*b**6*c* 
*2*x + 320*a**3*b**5*c**3*x**2 - 640*a**3*b**4*c**4*x**3 - 1280*a**3*b**3* 
c**5*x**4 - 512*a**3*b**2*c**6*x**5 + a**2*b**9 - 30*a**2*b**8*c*x + 320*a 
**2*b**6*c**3*x**3 + 480*a**2*b**5*c**4*x**4 + 192*a**2*b**4*c**5*x**5 ...