\(\int \frac {1}{(b d+2 c d x)^3 (a+b x+c x^2)^3} \, dx\) [81]

Optimal result

Integrand size = 24, antiderivative size = 154 \[ \int \frac {1}{(b d+2 c d x)^3 \left (a+b x+c x^2\right )^3} \, dx=\frac {48 c^2}{\left (b^2-4 a c\right )^3 d^3 (b+2 c x)^2}-\frac {1}{2 \left (b^2-4 a c\right ) d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^2}+\frac {6 c}{\left (b^2-4 a c\right )^2 d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )}-\frac {96 c^2 \log (b+2 c x)}{\left (b^2-4 a c\right )^4 d^3}+\frac {48 c^2 \log \left (a+b x+c x^2\right )}{\left (b^2-4 a c\right )^4 d^3} \] Output:

48*c^2/(-4*a*c+b^2)^3/d^3/(2*c*x+b)^2-1/2/(-4*a*c+b^2)/d^3/(2*c*x+b)^2/(c* 
x^2+b*x+a)^2+6*c/(-4*a*c+b^2)^2/d^3/(2*c*x+b)^2/(c*x^2+b*x+a)-96*c^2*ln(2* 
c*x+b)/(-4*a*c+b^2)^4/d^3+48*c^2*ln(c*x^2+b*x+a)/(-4*a*c+b^2)^4/d^3
 

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.72 \[ \int \frac {1}{(b d+2 c d x)^3 \left (a+b x+c x^2\right )^3} \, dx=\frac {\frac {32 c^2 \left (b^2-4 a c\right )}{(b+2 c x)^2}-\frac {\left (b^2-4 a c\right )^2}{(a+x (b+c x))^2}+\frac {16 c \left (b^2-4 a c\right )}{a+x (b+c x)}-192 c^2 \log (b+2 c x)+96 c^2 \log (a+x (b+c x))}{2 \left (b^2-4 a c\right )^4 d^3} \] Input:

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

Output:

((32*c^2*(b^2 - 4*a*c))/(b + 2*c*x)^2 - (b^2 - 4*a*c)^2/(a + x*(b + c*x))^ 
2 + (16*c*(b^2 - 4*a*c))/(a + x*(b + c*x)) - 192*c^2*Log[b + 2*c*x] + 96*c 
^2*Log[a + x*(b + c*x)])/(2*(b^2 - 4*a*c)^4*d^3)
 

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.12, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {1111, 27, 1111, 1117, 1105, 16, 1103}

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.

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

\(\Big \downarrow \) 1111

\(\displaystyle -\frac {6 c \int \frac {1}{d^3 (b+2 c x)^3 \left (c x^2+b x+a\right )^2}dx}{b^2-4 a c}-\frac {1}{2 d^3 \left (b^2-4 a c\right ) (b+2 c x)^2 \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {6 c \int \frac {1}{(b+2 c x)^3 \left (c x^2+b x+a\right )^2}dx}{d^3 \left (b^2-4 a c\right )}-\frac {1}{2 d^3 \left (b^2-4 a c\right ) (b+2 c x)^2 \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 1111

\(\displaystyle -\frac {6 c \left (-\frac {8 c \int \frac {1}{(b+2 c x)^3 \left (c x^2+b x+a\right )}dx}{b^2-4 a c}-\frac {1}{\left (b^2-4 a c\right ) (b+2 c x)^2 \left (a+b x+c x^2\right )}\right )}{d^3 \left (b^2-4 a c\right )}-\frac {1}{2 d^3 \left (b^2-4 a c\right ) (b+2 c x)^2 \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 1117

\(\displaystyle -\frac {6 c \left (-\frac {8 c \left (\frac {\int \frac {1}{(b+2 c x) \left (c x^2+b x+a\right )}dx}{b^2-4 a c}+\frac {1}{\left (b^2-4 a c\right ) (b+2 c x)^2}\right )}{b^2-4 a c}-\frac {1}{\left (b^2-4 a c\right ) (b+2 c x)^2 \left (a+b x+c x^2\right )}\right )}{d^3 \left (b^2-4 a c\right )}-\frac {1}{2 d^3 \left (b^2-4 a c\right ) (b+2 c x)^2 \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 1105

\(\displaystyle -\frac {6 c \left (-\frac {8 c \left (\frac {\frac {\int \frac {b+2 c x}{c x^2+b x+a}dx}{b^2-4 a c}-\frac {4 c \int \frac {1}{b+2 c x}dx}{b^2-4 a c}}{b^2-4 a c}+\frac {1}{\left (b^2-4 a c\right ) (b+2 c x)^2}\right )}{b^2-4 a c}-\frac {1}{\left (b^2-4 a c\right ) (b+2 c x)^2 \left (a+b x+c x^2\right )}\right )}{d^3 \left (b^2-4 a c\right )}-\frac {1}{2 d^3 \left (b^2-4 a c\right ) (b+2 c x)^2 \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 16

\(\displaystyle -\frac {6 c \left (-\frac {8 c \left (\frac {\frac {\int \frac {b+2 c x}{c x^2+b x+a}dx}{b^2-4 a c}-\frac {2 \log (b+2 c x)}{b^2-4 a c}}{b^2-4 a c}+\frac {1}{\left (b^2-4 a c\right ) (b+2 c x)^2}\right )}{b^2-4 a c}-\frac {1}{\left (b^2-4 a c\right ) (b+2 c x)^2 \left (a+b x+c x^2\right )}\right )}{d^3 \left (b^2-4 a c\right )}-\frac {1}{2 d^3 \left (b^2-4 a c\right ) (b+2 c x)^2 \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 1103

\(\displaystyle -\frac {1}{2 d^3 \left (b^2-4 a c\right ) (b+2 c x)^2 \left (a+b x+c x^2\right )^2}-\frac {6 c \left (-\frac {1}{\left (b^2-4 a c\right ) (b+2 c x)^2 \left (a+b x+c x^2\right )}-\frac {8 c \left (\frac {\frac {\log \left (a+b x+c x^2\right )}{b^2-4 a c}-\frac {2 \log (b+2 c x)}{b^2-4 a c}}{b^2-4 a c}+\frac {1}{\left (b^2-4 a c\right ) (b+2 c x)^2}\right )}{b^2-4 a c}\right )}{d^3 \left (b^2-4 a c\right )}\)

Input:

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

Output:

-1/2*1/((b^2 - 4*a*c)*d^3*(b + 2*c*x)^2*(a + b*x + c*x^2)^2) - (6*c*(-(1/( 
(b^2 - 4*a*c)*(b + 2*c*x)^2*(a + b*x + c*x^2))) - (8*c*(1/((b^2 - 4*a*c)*( 
b + 2*c*x)^2) + ((-2*Log[b + 2*c*x])/(b^2 - 4*a*c) + Log[a + b*x + c*x^2]/ 
(b^2 - 4*a*c))/(b^2 - 4*a*c)))/(b^2 - 4*a*c)))/((b^2 - 4*a*c)*d^3)
 

Defintions of rubi rules used

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

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 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1105

Int[1/(((d_) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] 
:> Simp[-4*b*(c/(d*(b^2 - 4*a*c)))   Int[1/(b + 2*c*x), x], x] + Simp[b^2/( 
d^2*(b^2 - 4*a*c))   Int[(d + e*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b 
, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
 

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 = 1.03 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.99

method	result	size
default	\(\frac {-\frac {96 c^{2} \ln \left (2 c x +b \right )}{\left (4 a c -b^{2}\right )^{4}}-\frac {16 c^{2}}{\left (4 a c -b^{2}\right )^{3} \left (2 c x +b \right )^{2}}+\frac {\frac {-8 c^{2} \left (4 a c -b^{2}\right ) x^{2}-8 b c \left (4 a c -b^{2}\right ) x -40 a^{2} c^{2}+12 c a \,b^{2}-\frac {b^{4}}{2}}{\left (c \,x^{2}+b x +a \right )^{2}}+48 c^{2} \ln \left (c \,x^{2}+b x +a \right )}{\left (4 a c -b^{2}\right )^{4}}}{d^{3}}\)	\(152\)
risch	\(\frac {-\frac {48 c^{4} x^{4}}{64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}-\frac {96 b \,c^{3} x^{3}}{64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}-\frac {18 c^{2} \left (4 a c +3 b^{2}\right ) x^{2}}{64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}-\frac {6 c b \left (12 a c +b^{2}\right ) x}{64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}-\frac {32 a^{2} c^{2}+20 c a \,b^{2}-b^{4}}{2 \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}}{d^{3} \left (2 c x +b \right )^{2} \left (c \,x^{2}+b x +a \right )^{2}}-\frac {96 c^{2} \ln \left (2 c x +b \right )}{d^{3} \left (256 a^{4} c^{4}-256 a^{3} b^{2} c^{3}+96 a^{2} b^{4} c^{2}-16 a \,b^{6} c +b^{8}\right )}+\frac {48 c^{2} \ln \left (-c \,x^{2}-b x -a \right )}{d^{3} \left (256 a^{4} c^{4}-256 a^{3} b^{2} c^{3}+96 a^{2} b^{4} c^{2}-16 a \,b^{6} c +b^{8}\right )}\)	\(392\)
norman	\(\frac {\frac {-64 a^{2} c^{6}-40 a \,b^{2} c^{5}+2 c^{4} b^{4}}{4 d \,c^{4} \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}-\frac {48 c^{4} x^{4}}{d \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}+\frac {\left (-288 a \,c^{7}-216 c^{6} b^{2}\right ) x^{2}}{4 d \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) c^{4}}+\frac {b \left (-144 a \,c^{6}-12 c^{5} b^{2}\right ) x}{2 d \,c^{4} \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}-\frac {96 c^{3} b \,x^{3}}{d \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}}{d^{2} \left (2 c x +b \right )^{2} \left (c \,x^{2}+b x +a \right )^{2}}-\frac {96 c^{2} \ln \left (2 c x +b \right )}{d^{3} \left (256 a^{4} c^{4}-256 a^{3} b^{2} c^{3}+96 a^{2} b^{4} c^{2}-16 a \,b^{6} c +b^{8}\right )}+\frac {48 c^{2} \ln \left (c \,x^{2}+b x +a \right )}{d^{3} \left (256 a^{4} c^{4}-256 a^{3} b^{2} c^{3}+96 a^{2} b^{4} c^{2}-16 a \,b^{6} c +b^{8}\right )}\)	\(425\)
parallelrisch	\(\frac {48 a \,b^{4} c^{5}-2 b^{6} c^{4}-1536 x^{3} a b \,c^{8}-4608 \ln \left (\frac {b}{2}+c x \right ) x^{5} b \,c^{9}+2304 \ln \left (c \,x^{2}+b x +a \right ) x^{5} b \,c^{9}-3072 \ln \left (\frac {b}{2}+c x \right ) x^{4} a \,c^{9}-4992 \ln \left (\frac {b}{2}+c x \right ) x^{4} b^{2} c^{8}+1536 \ln \left (c \,x^{2}+b x +a \right ) x^{4} a \,c^{9}+2496 \ln \left (c \,x^{2}+b x +a \right ) x^{4} b^{2} c^{8}-2304 \ln \left (\frac {b}{2}+c x \right ) x^{3} b^{3} c^{7}+1152 \ln \left (c \,x^{2}+b x +a \right ) x^{3} b^{3} c^{7}-1536 \ln \left (\frac {b}{2}+c x \right ) x^{2} a^{2} c^{8}-384 \ln \left (\frac {b}{2}+c x \right ) x^{2} b^{4} c^{6}+768 \ln \left (c \,x^{2}+b x +a \right ) x^{2} a^{2} c^{8}+192 \ln \left (c \,x^{2}+b x +a \right ) x^{2} b^{4} c^{6}-384 \ln \left (\frac {b}{2}+c x \right ) a^{2} b^{2} c^{6}+192 \ln \left (c \,x^{2}+b x +a \right ) a^{2} b^{2} c^{6}-576 x^{2} a \,b^{2} c^{7}-1152 x \,a^{2} b \,c^{7}+192 x a \,b^{3} c^{6}-6144 \ln \left (\frac {b}{2}+c x \right ) x^{3} a b \,c^{8}+3072 \ln \left (c \,x^{2}+b x +a \right ) x^{3} a b \,c^{8}-3840 \ln \left (\frac {b}{2}+c x \right ) x^{2} a \,b^{2} c^{7}+1920 \ln \left (c \,x^{2}+b x +a \right ) x^{2} a \,b^{2} c^{7}-1536 \ln \left (\frac {b}{2}+c x \right ) x \,a^{2} b \,c^{7}-768 \ln \left (\frac {b}{2}+c x \right ) x a \,b^{3} c^{6}+768 \ln \left (c \,x^{2}+b x +a \right ) x \,a^{2} b \,c^{7}+384 \ln \left (c \,x^{2}+b x +a \right ) x a \,b^{3} c^{6}-256 a^{3} c^{7}-1152 x^{2} a^{2} c^{8}-768 x^{4} a \,c^{9}+192 x^{4} b^{2} c^{8}-96 a^{2} b^{2} c^{6}-1536 \ln \left (\frac {b}{2}+c x \right ) x^{6} c^{10}+24 x \,b^{5} c^{5}+384 x^{3} b^{3} c^{7}+768 \ln \left (c \,x^{2}+b x +a \right ) x^{6} c^{10}+216 x^{2} b^{4} c^{6}}{4 \left (256 a^{4} c^{4}-256 a^{3} b^{2} c^{3}+96 a^{2} b^{4} c^{2}-16 a \,b^{6} c +b^{8}\right ) \left (c \,x^{2}+b x +a \right )^{2} \left (2 c x +b \right )^{2} c^{4} d^{3}}\)	\(687\)

Input:

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

Output:

1/d^3*(-96*c^2/(4*a*c-b^2)^4*ln(2*c*x+b)-16*c^2/(4*a*c-b^2)^3/(2*c*x+b)^2+ 
1/(4*a*c-b^2)^4*((-8*c^2*(4*a*c-b^2)*x^2-8*b*c*(4*a*c-b^2)*x-40*a^2*c^2+12 
*c*a*b^2-1/2*b^4)/(c*x^2+b*x+a)^2+48*c^2*ln(c*x^2+b*x+a)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 809 vs. \(2 (152) = 304\).

Time = 0.10 (sec) , antiderivative size = 809, normalized size of antiderivative = 5.25 \[ \int \frac {1}{(b d+2 c d x)^3 \left (a+b x+c x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/2*(b^6 - 24*a*b^4*c + 48*a^2*b^2*c^2 + 128*a^3*c^3 - 96*(b^2*c^4 - 4*a* 
c^5)*x^4 - 192*(b^3*c^3 - 4*a*b*c^4)*x^3 - 36*(3*b^4*c^2 - 8*a*b^2*c^3 - 1 
6*a^2*c^4)*x^2 - 12*(b^5*c + 8*a*b^3*c^2 - 48*a^2*b*c^3)*x - 96*(4*c^6*x^6 
 + 12*b*c^5*x^5 + a^2*b^2*c^2 + (13*b^2*c^4 + 8*a*c^5)*x^4 + 2*(3*b^3*c^3 
+ 8*a*b*c^4)*x^3 + (b^4*c^2 + 10*a*b^2*c^3 + 4*a^2*c^4)*x^2 + 2*(a*b^3*c^2 
 + 2*a^2*b*c^3)*x)*log(c*x^2 + b*x + a) + 192*(4*c^6*x^6 + 12*b*c^5*x^5 + 
a^2*b^2*c^2 + (13*b^2*c^4 + 8*a*c^5)*x^4 + 2*(3*b^3*c^3 + 8*a*b*c^4)*x^3 + 
 (b^4*c^2 + 10*a*b^2*c^3 + 4*a^2*c^4)*x^2 + 2*(a*b^3*c^2 + 2*a^2*b*c^3)*x) 
*log(2*c*x + b))/(4*(b^8*c^4 - 16*a*b^6*c^5 + 96*a^2*b^4*c^6 - 256*a^3*b^2 
*c^7 + 256*a^4*c^8)*d^3*x^6 + 12*(b^9*c^3 - 16*a*b^7*c^4 + 96*a^2*b^5*c^5 
- 256*a^3*b^3*c^6 + 256*a^4*b*c^7)*d^3*x^5 + (13*b^10*c^2 - 200*a*b^8*c^3 
+ 1120*a^2*b^6*c^4 - 2560*a^3*b^4*c^5 + 1280*a^4*b^2*c^6 + 2048*a^5*c^7)*d 
^3*x^4 + 2*(3*b^11*c - 40*a*b^9*c^2 + 160*a^2*b^7*c^3 - 1280*a^4*b^3*c^5 + 
 2048*a^5*b*c^6)*d^3*x^3 + (b^12 - 6*a*b^10*c - 60*a^2*b^8*c^2 + 640*a^3*b 
^6*c^3 - 1920*a^4*b^4*c^4 + 1536*a^5*b^2*c^5 + 1024*a^6*c^6)*d^3*x^2 + 2*( 
a*b^11 - 14*a^2*b^9*c + 64*a^3*b^7*c^2 - 64*a^4*b^5*c^3 - 256*a^5*b^3*c^4 
+ 512*a^6*b*c^5)*d^3*x + (a^2*b^10 - 16*a^3*b^8*c + 96*a^4*b^6*c^2 - 256*a 
^5*b^4*c^3 + 256*a^6*b^2*c^4)*d^3)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 600 vs. \(2 (155) = 310\).

Time = 2.38 (sec) , antiderivative size = 600, normalized size of antiderivative = 3.90 \[ \int \frac {1}{(b d+2 c d x)^3 \left (a+b x+c x^2\right )^3} \, dx=- \frac {96 c^{2} \log {\left (\frac {b}{2 c} + x \right )}}{d^{3} \left (4 a c - b^{2}\right )^{4}} + \frac {48 c^{2} \log {\left (\frac {a}{c} + \frac {b x}{c} + x^{2} \right )}}{d^{3} \left (4 a c - b^{2}\right )^{4}} + \frac {- 32 a^{2} c^{2} - 20 a b^{2} c + b^{4} - 192 b c^{3} x^{3} - 96 c^{4} x^{4} + x^{2} \left (- 144 a c^{3} - 108 b^{2} c^{2}\right ) + x \left (- 144 a b c^{2} - 12 b^{3} c\right )}{128 a^{5} b^{2} c^{3} d^{3} - 96 a^{4} b^{4} c^{2} d^{3} + 24 a^{3} b^{6} c d^{3} - 2 a^{2} b^{8} d^{3} + x^{6} \cdot \left (512 a^{3} c^{7} d^{3} - 384 a^{2} b^{2} c^{6} d^{3} + 96 a b^{4} c^{5} d^{3} - 8 b^{6} c^{4} d^{3}\right ) + x^{5} \cdot \left (1536 a^{3} b c^{6} d^{3} - 1152 a^{2} b^{3} c^{5} d^{3} + 288 a b^{5} c^{4} d^{3} - 24 b^{7} c^{3} d^{3}\right ) + x^{4} \cdot \left (1024 a^{4} c^{6} d^{3} + 896 a^{3} b^{2} c^{5} d^{3} - 1056 a^{2} b^{4} c^{4} d^{3} + 296 a b^{6} c^{3} d^{3} - 26 b^{8} c^{2} d^{3}\right ) + x^{3} \cdot \left (2048 a^{4} b c^{5} d^{3} - 768 a^{3} b^{3} c^{4} d^{3} - 192 a^{2} b^{5} c^{3} d^{3} + 112 a b^{7} c^{2} d^{3} - 12 b^{9} c d^{3}\right ) + x^{2} \cdot \left (512 a^{5} c^{5} d^{3} + 896 a^{4} b^{2} c^{4} d^{3} - 736 a^{3} b^{4} c^{3} d^{3} + 136 a^{2} b^{6} c^{2} d^{3} + 4 a b^{8} c d^{3} - 2 b^{10} d^{3}\right ) + x \left (512 a^{5} b c^{4} d^{3} - 128 a^{4} b^{3} c^{3} d^{3} - 96 a^{3} b^{5} c^{2} d^{3} + 40 a^{2} b^{7} c d^{3} - 4 a b^{9} d^{3}\right )} \] Input:

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

Output:

-96*c**2*log(b/(2*c) + x)/(d**3*(4*a*c - b**2)**4) + 48*c**2*log(a/c + b*x 
/c + x**2)/(d**3*(4*a*c - b**2)**4) + (-32*a**2*c**2 - 20*a*b**2*c + b**4 
- 192*b*c**3*x**3 - 96*c**4*x**4 + x**2*(-144*a*c**3 - 108*b**2*c**2) + x* 
(-144*a*b*c**2 - 12*b**3*c))/(128*a**5*b**2*c**3*d**3 - 96*a**4*b**4*c**2* 
d**3 + 24*a**3*b**6*c*d**3 - 2*a**2*b**8*d**3 + x**6*(512*a**3*c**7*d**3 - 
 384*a**2*b**2*c**6*d**3 + 96*a*b**4*c**5*d**3 - 8*b**6*c**4*d**3) + x**5* 
(1536*a**3*b*c**6*d**3 - 1152*a**2*b**3*c**5*d**3 + 288*a*b**5*c**4*d**3 - 
 24*b**7*c**3*d**3) + x**4*(1024*a**4*c**6*d**3 + 896*a**3*b**2*c**5*d**3 
- 1056*a**2*b**4*c**4*d**3 + 296*a*b**6*c**3*d**3 - 26*b**8*c**2*d**3) + x 
**3*(2048*a**4*b*c**5*d**3 - 768*a**3*b**3*c**4*d**3 - 192*a**2*b**5*c**3* 
d**3 + 112*a*b**7*c**2*d**3 - 12*b**9*c*d**3) + x**2*(512*a**5*c**5*d**3 + 
 896*a**4*b**2*c**4*d**3 - 736*a**3*b**4*c**3*d**3 + 136*a**2*b**6*c**2*d* 
*3 + 4*a*b**8*c*d**3 - 2*b**10*d**3) + x*(512*a**5*b*c**4*d**3 - 128*a**4* 
b**3*c**3*d**3 - 96*a**3*b**5*c**2*d**3 + 40*a**2*b**7*c*d**3 - 4*a*b**9*d 
**3))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 553 vs. \(2 (152) = 304\).

Time = 0.07 (sec) , antiderivative size = 553, normalized size of antiderivative = 3.59 \[ \int \frac {1}{(b d+2 c d x)^3 \left (a+b x+c x^2\right )^3} \, dx=\frac {96 \, c^{4} x^{4} + 192 \, b c^{3} x^{3} - b^{4} + 20 \, a b^{2} c + 32 \, a^{2} c^{2} + 36 \, {\left (3 \, b^{2} c^{2} + 4 \, a c^{3}\right )} x^{2} + 12 \, {\left (b^{3} c + 12 \, a b c^{2}\right )} x}{2 \, {\left (4 \, {\left (b^{6} c^{4} - 12 \, a b^{4} c^{5} + 48 \, a^{2} b^{2} c^{6} - 64 \, a^{3} c^{7}\right )} d^{3} x^{6} + 12 \, {\left (b^{7} c^{3} - 12 \, a b^{5} c^{4} + 48 \, a^{2} b^{3} c^{5} - 64 \, a^{3} b c^{6}\right )} d^{3} x^{5} + {\left (13 \, b^{8} c^{2} - 148 \, a b^{6} c^{3} + 528 \, a^{2} b^{4} c^{4} - 448 \, a^{3} b^{2} c^{5} - 512 \, a^{4} c^{6}\right )} d^{3} x^{4} + 2 \, {\left (3 \, b^{9} c - 28 \, a b^{7} c^{2} + 48 \, a^{2} b^{5} c^{3} + 192 \, a^{3} b^{3} c^{4} - 512 \, a^{4} b c^{5}\right )} d^{3} x^{3} + {\left (b^{10} - 2 \, a b^{8} c - 68 \, a^{2} b^{6} c^{2} + 368 \, a^{3} b^{4} c^{3} - 448 \, a^{4} b^{2} c^{4} - 256 \, a^{5} c^{5}\right )} d^{3} x^{2} + 2 \, {\left (a b^{9} - 10 \, a^{2} b^{7} c + 24 \, a^{3} b^{5} c^{2} + 32 \, a^{4} b^{3} c^{3} - 128 \, a^{5} b c^{4}\right )} d^{3} x + {\left (a^{2} b^{8} - 12 \, a^{3} b^{6} c + 48 \, a^{4} b^{4} c^{2} - 64 \, a^{5} b^{2} c^{3}\right )} d^{3}\right )}} + \frac {48 \, c^{2} \log \left (c x^{2} + b x + a\right )}{{\left (b^{8} - 16 \, a b^{6} c + 96 \, a^{2} b^{4} c^{2} - 256 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}\right )} d^{3}} - \frac {96 \, c^{2} \log \left (2 \, c x + b\right )}{{\left (b^{8} - 16 \, a b^{6} c + 96 \, a^{2} b^{4} c^{2} - 256 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}\right )} d^{3}} \] Input:

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

Output:

1/2*(96*c^4*x^4 + 192*b*c^3*x^3 - b^4 + 20*a*b^2*c + 32*a^2*c^2 + 36*(3*b^ 
2*c^2 + 4*a*c^3)*x^2 + 12*(b^3*c + 12*a*b*c^2)*x)/(4*(b^6*c^4 - 12*a*b^4*c 
^5 + 48*a^2*b^2*c^6 - 64*a^3*c^7)*d^3*x^6 + 12*(b^7*c^3 - 12*a*b^5*c^4 + 4 
8*a^2*b^3*c^5 - 64*a^3*b*c^6)*d^3*x^5 + (13*b^8*c^2 - 148*a*b^6*c^3 + 528* 
a^2*b^4*c^4 - 448*a^3*b^2*c^5 - 512*a^4*c^6)*d^3*x^4 + 2*(3*b^9*c - 28*a*b 
^7*c^2 + 48*a^2*b^5*c^3 + 192*a^3*b^3*c^4 - 512*a^4*b*c^5)*d^3*x^3 + (b^10 
 - 2*a*b^8*c - 68*a^2*b^6*c^2 + 368*a^3*b^4*c^3 - 448*a^4*b^2*c^4 - 256*a^ 
5*c^5)*d^3*x^2 + 2*(a*b^9 - 10*a^2*b^7*c + 24*a^3*b^5*c^2 + 32*a^4*b^3*c^3 
 - 128*a^5*b*c^4)*d^3*x + (a^2*b^8 - 12*a^3*b^6*c + 48*a^4*b^4*c^2 - 64*a^ 
5*b^2*c^3)*d^3) + 48*c^2*log(c*x^2 + b*x + a)/((b^8 - 16*a*b^6*c + 96*a^2* 
b^4*c^2 - 256*a^3*b^2*c^3 + 256*a^4*c^4)*d^3) - 96*c^2*log(2*c*x + b)/((b^ 
8 - 16*a*b^6*c + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 + 256*a^4*c^4)*d^3)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.96 \[ \int \frac {1}{(b d+2 c d x)^3 \left (a+b x+c x^2\right )^3} \, dx=-\frac {96 \, c^{3} \log \left ({\left | 2 \, c x + b \right |}\right )}{b^{8} c d^{3} - 16 \, a b^{6} c^{2} d^{3} + 96 \, a^{2} b^{4} c^{3} d^{3} - 256 \, a^{3} b^{2} c^{4} d^{3} + 256 \, a^{4} c^{5} d^{3}} + \frac {48 \, c^{2} \log \left (c x^{2} + b x + a\right )}{b^{8} d^{3} - 16 \, a b^{6} c d^{3} + 96 \, a^{2} b^{4} c^{2} d^{3} - 256 \, a^{3} b^{2} c^{3} d^{3} + 256 \, a^{4} c^{4} d^{3}} + \frac {96 \, c^{4} x^{4} + 192 \, b c^{3} x^{3} + 108 \, b^{2} c^{2} x^{2} + 144 \, a c^{3} x^{2} + 12 \, b^{3} c x + 144 \, a b c^{2} x - b^{4} + 20 \, a b^{2} c + 32 \, a^{2} c^{2}}{2 \, {\left (b^{6} d^{3} - 12 \, a b^{4} c d^{3} + 48 \, a^{2} b^{2} c^{2} d^{3} - 64 \, a^{3} c^{3} d^{3}\right )} {\left (2 \, c^{2} x^{3} + 3 \, b c x^{2} + b^{2} x + 2 \, a c x + a b\right )}^{2}} \] Input:

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

Output:

-96*c^3*log(abs(2*c*x + b))/(b^8*c*d^3 - 16*a*b^6*c^2*d^3 + 96*a^2*b^4*c^3 
*d^3 - 256*a^3*b^2*c^4*d^3 + 256*a^4*c^5*d^3) + 48*c^2*log(c*x^2 + b*x + a 
)/(b^8*d^3 - 16*a*b^6*c*d^3 + 96*a^2*b^4*c^2*d^3 - 256*a^3*b^2*c^3*d^3 + 2 
56*a^4*c^4*d^3) + 1/2*(96*c^4*x^4 + 192*b*c^3*x^3 + 108*b^2*c^2*x^2 + 144* 
a*c^3*x^2 + 12*b^3*c*x + 144*a*b*c^2*x - b^4 + 20*a*b^2*c + 32*a^2*c^2)/(( 
b^6*d^3 - 12*a*b^4*c*d^3 + 48*a^2*b^2*c^2*d^3 - 64*a^3*c^3*d^3)*(2*c^2*x^3 
 + 3*b*c*x^2 + b^2*x + 2*a*c*x + a*b)^2)
 

Mupad [B] (verification not implemented)

Time = 5.66 (sec) , antiderivative size = 522, normalized size of antiderivative = 3.39 \[ \int \frac {1}{(b d+2 c d x)^3 \left (a+b x+c x^2\right )^3} \, dx=\frac {\frac {32\,a^2\,c^2+20\,a\,b^2\,c-b^4}{2\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}+\frac {48\,c^4\,x^4}{-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}+\frac {96\,b\,c^3\,x^3}{-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}+\frac {18\,c\,x^2\,\left (3\,b^2\,c+4\,a\,c^2\right )}{-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}+\frac {6\,b\,c\,x\,\left (b^2+12\,a\,c\right )}{-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}}{x\,\left (4\,c\,a^2\,b\,d^3+2\,a\,b^3\,d^3\right )+x^4\,\left (13\,b^2\,c^2\,d^3+8\,a\,c^3\,d^3\right )+x^3\,\left (6\,b^3\,c\,d^3+16\,a\,b\,c^2\,d^3\right )+x^2\,\left (4\,a^2\,c^2\,d^3+10\,a\,b^2\,c\,d^3+b^4\,d^3\right )+a^2\,b^2\,d^3+4\,c^4\,d^3\,x^6+12\,b\,c^3\,d^3\,x^5}-\frac {96\,c^2\,\ln \left (b+2\,c\,x\right )}{256\,a^4\,c^4\,d^3-256\,a^3\,b^2\,c^3\,d^3+96\,a^2\,b^4\,c^2\,d^3-16\,a\,b^6\,c\,d^3+b^8\,d^3}+\frac {48\,c^2\,\ln \left (c\,x^2+b\,x+a\right )}{256\,a^4\,c^4\,d^3-256\,a^3\,b^2\,c^3\,d^3+96\,a^2\,b^4\,c^2\,d^3-16\,a\,b^6\,c\,d^3+b^8\,d^3} \] Input:

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

Output:

((32*a^2*c^2 - b^4 + 20*a*b^2*c)/(2*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 1 
2*a*b^4*c)) + (48*c^4*x^4)/(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c 
) + (96*b*c^3*x^3)/(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c) + (18* 
c*x^2*(4*a*c^2 + 3*b^2*c))/(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c 
) + (6*b*c*x*(12*a*c + b^2))/(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4 
*c))/(x*(2*a*b^3*d^3 + 4*a^2*b*c*d^3) + x^4*(8*a*c^3*d^3 + 13*b^2*c^2*d^3) 
 + x^3*(6*b^3*c*d^3 + 16*a*b*c^2*d^3) + x^2*(b^4*d^3 + 4*a^2*c^2*d^3 + 10* 
a*b^2*c*d^3) + a^2*b^2*d^3 + 4*c^4*d^3*x^6 + 12*b*c^3*d^3*x^5) - (96*c^2*l 
og(b + 2*c*x))/(b^8*d^3 + 256*a^4*c^4*d^3 + 96*a^2*b^4*c^2*d^3 - 256*a^3*b 
^2*c^3*d^3 - 16*a*b^6*c*d^3) + (48*c^2*log(a + b*x + c*x^2))/(b^8*d^3 + 25 
6*a^4*c^4*d^3 + 96*a^2*b^4*c^2*d^3 - 256*a^3*b^2*c^3*d^3 - 16*a*b^6*c*d^3)
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 1062, normalized size of antiderivative = 6.90 \[ \int \frac {1}{(b d+2 c d x)^3 \left (a+b x+c x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(96*log(a + b*x + c*x**2)*a**2*b**2*c**2 + 384*log(a + b*x + c*x**2)*a**2* 
b*c**3*x + 384*log(a + b*x + c*x**2)*a**2*c**4*x**2 + 192*log(a + b*x + c* 
x**2)*a*b**3*c**2*x + 960*log(a + b*x + c*x**2)*a*b**2*c**3*x**2 + 1536*lo 
g(a + b*x + c*x**2)*a*b*c**4*x**3 + 768*log(a + b*x + c*x**2)*a*c**5*x**4 
+ 96*log(a + b*x + c*x**2)*b**4*c**2*x**2 + 576*log(a + b*x + c*x**2)*b**3 
*c**3*x**3 + 1248*log(a + b*x + c*x**2)*b**2*c**4*x**4 + 1152*log(a + b*x 
+ c*x**2)*b*c**5*x**5 + 384*log(a + b*x + c*x**2)*c**6*x**6 - 192*log(b + 
2*c*x)*a**2*b**2*c**2 - 768*log(b + 2*c*x)*a**2*b*c**3*x - 768*log(b + 2*c 
*x)*a**2*c**4*x**2 - 384*log(b + 2*c*x)*a*b**3*c**2*x - 1920*log(b + 2*c*x 
)*a*b**2*c**3*x**2 - 3072*log(b + 2*c*x)*a*b*c**4*x**3 - 1536*log(b + 2*c* 
x)*a*c**5*x**4 - 192*log(b + 2*c*x)*b**4*c**2*x**2 - 1152*log(b + 2*c*x)*b 
**3*c**3*x**3 - 2496*log(b + 2*c*x)*b**2*c**4*x**4 - 2304*log(b + 2*c*x)*b 
*c**5*x**5 - 768*log(b + 2*c*x)*c**6*x**6 - 128*a**3*c**3 - 48*a**2*b**2*c 
**2 - 576*a**2*b*c**3*x - 576*a**2*c**4*x**2 + 24*a*b**4*c + 96*a*b**3*c** 
2*x - 288*a*b**2*c**3*x**2 - 768*a*b*c**4*x**3 - 384*a*c**5*x**4 - b**6 + 
12*b**5*c*x + 108*b**4*c**2*x**2 + 192*b**3*c**3*x**3 + 96*b**2*c**4*x**4) 
/(2*d**3*(256*a**6*b**2*c**4 + 1024*a**6*b*c**5*x + 1024*a**6*c**6*x**2 - 
256*a**5*b**4*c**3 - 512*a**5*b**3*c**4*x + 1536*a**5*b**2*c**5*x**2 + 409 
6*a**5*b*c**6*x**3 + 2048*a**5*c**7*x**4 + 96*a**4*b**6*c**2 - 128*a**4*b* 
*5*c**3*x - 1920*a**4*b**4*c**4*x**2 - 2560*a**4*b**3*c**5*x**3 + 1280*...