\(\int \frac {1}{(b d+2 c d x)^3 (a+b x+c x^2)^3} \, dx\) [1189]

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {701, 707, 695, 31, 642} \[ \int \frac {1}{(b d+2 c d x)^3 \left (a+b x+c x^2\right )^3} \, dx=\frac {48 c^2 \log \left (a+b x+c x^2\right )}{d^3 \left (b^2-4 a c\right )^4}+\frac {48 c^2}{d^3 \left (b^2-4 a c\right )^3 (b+2 c x)^2}-\frac {96 c^2 \log (b+2 c x)}{d^3 \left (b^2-4 a c\right )^4}+\frac {6 c}{d^3 \left (b^2-4 a c\right )^2 (b+2 c x)^2 \left (a+b x+c x^2\right )}-\frac {1}{2 d^3 \left (b^2-4 a c\right ) (b+2 c x)^2 \left (a+b x+c x^2\right )^2} \]

[In]

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

[Out]

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[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 695

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

Rule 701

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> 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] - Dist[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] && NeQ[b^2 - 4*a*c, 0]
&& EqQ[2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] && LtQ[p, -1] &&  !GtQ[m, 1] && RationalQ[m] && IntegerQ[2*p]

Rule 707

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> 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] + Dist[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] && NeQ[b^2 - 4*a*
c, 0] && EqQ[2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] && LtQ[m, -1] && (IntegerQ[2*p] || (IntegerQ[m] && Rationa
lQ[p]) || IntegerQ[(m + 2*p + 3)/2])

Rubi steps \begin{align*} \text {integral}& = -\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) \int \frac {1}{(b d+2 c d x)^3 \left (a+b x+c x^2\right )^2} \, dx}{b^2-4 a c} \\ & = -\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 {\left (48 c^2\right ) \int \frac {1}{(b d+2 c d x)^3 \left (a+b x+c x^2\right )} \, dx}{\left (b^2-4 a c\right )^2} \\ & = \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 {\left (48 c^2\right ) \int \frac {1}{(b d+2 c d x) \left (a+b x+c x^2\right )} \, dx}{\left (b^2-4 a c\right )^3 d^2} \\ & = \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 {\left (48 c^2\right ) \int \frac {b d+2 c d x}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right )^4 d^4}-\frac {\left (192 c^3\right ) \int \frac {1}{b+2 c x} \, dx}{\left (b^2-4 a c\right )^4 d^3} \\ & = \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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.72 \[ \int \frac {1}{(b d+2 c d x)^3 \left (a+b x+c x^2\right )^3} \, dx=\frac {\frac {32 c^2 \left (b^2-4 a c\right )}{(b+2 c x)^2}-\frac {\left (b^2-4 a c\right )^2}{(a+x (b+c x))^2}+\frac {16 c \left (b^2-4 a c\right )}{a+x (b+c x)}-192 c^2 \log (b+2 c x)+96 c^2 \log (a+x (b+c x))}{2 \left (b^2-4 a c\right )^4 d^3} \]

[In]

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

[Out]

((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)

Maple [A] (verified)

Time = 2.34 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.99

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

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

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

Time = 0.63 (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=-\frac {b^{6} - 24 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} + 128 \, a^{3} c^{3} - 96 \, {\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x^{4} - 192 \, {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} x^{3} - 36 \, {\left (3 \, b^{4} c^{2} - 8 \, a b^{2} c^{3} - 16 \, a^{2} c^{4}\right )} x^{2} - 12 \, {\left (b^{5} c + 8 \, a b^{3} c^{2} - 48 \, a^{2} b c^{3}\right )} x - 96 \, {\left (4 \, c^{6} x^{6} + 12 \, b c^{5} x^{5} + a^{2} b^{2} c^{2} + {\left (13 \, b^{2} c^{4} + 8 \, a c^{5}\right )} x^{4} + 2 \, {\left (3 \, b^{3} c^{3} + 8 \, a b c^{4}\right )} x^{3} + {\left (b^{4} c^{2} + 10 \, a b^{2} c^{3} + 4 \, a^{2} c^{4}\right )} x^{2} + 2 \, {\left (a b^{3} c^{2} + 2 \, a^{2} b c^{3}\right )} x\right )} \log \left (c x^{2} + b x + a\right ) + 192 \, {\left (4 \, c^{6} x^{6} + 12 \, b c^{5} x^{5} + a^{2} b^{2} c^{2} + {\left (13 \, b^{2} c^{4} + 8 \, a c^{5}\right )} x^{4} + 2 \, {\left (3 \, b^{3} c^{3} + 8 \, a b c^{4}\right )} x^{3} + {\left (b^{4} c^{2} + 10 \, a b^{2} c^{3} + 4 \, a^{2} c^{4}\right )} x^{2} + 2 \, {\left (a b^{3} c^{2} + 2 \, a^{2} b c^{3}\right )} x\right )} \log \left (2 \, c x + b\right )}{2 \, {\left (4 \, {\left (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}\right )} d^{3} x^{6} + 12 \, {\left (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}\right )} d^{3} x^{5} + {\left (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}\right )} d^{3} x^{4} + 2 \, {\left (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}\right )} d^{3} x^{3} + {\left (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}\right )} d^{3} x^{2} + 2 \, {\left (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}\right )} d^{3} x + {\left (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}\right )} d^{3}\right )}} \]

[In]

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

[Out]

-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 - 16*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 = 3.04 (sec) , antiderivative size = 600, normalized size of antiderivative = 3.90 \[ \int \frac {1}{(b d+2 c d x)^3 \left (a+b x+c x^2\right )^3} \, dx=- \frac {96 c^{2} \log {\left (\frac {b}{2 c} + x \right )}}{d^{3} \left (4 a c - b^{2}\right )^{4}} + \frac {48 c^{2} \log {\left (\frac {a}{c} + \frac {b x}{c} + x^{2} \right )}}{d^{3} \left (4 a c - b^{2}\right )^{4}} + \frac {- 32 a^{2} c^{2} - 20 a b^{2} c + b^{4} - 192 b c^{3} x^{3} - 96 c^{4} x^{4} + x^{2} \left (- 144 a c^{3} - 108 b^{2} c^{2}\right ) + x \left (- 144 a b c^{2} - 12 b^{3} c\right )}{128 a^{5} b^{2} c^{3} d^{3} - 96 a^{4} b^{4} c^{2} d^{3} + 24 a^{3} b^{6} c d^{3} - 2 a^{2} b^{8} d^{3} + x^{6} \cdot \left (512 a^{3} c^{7} d^{3} - 384 a^{2} b^{2} c^{6} d^{3} + 96 a b^{4} c^{5} d^{3} - 8 b^{6} c^{4} d^{3}\right ) + x^{5} \cdot \left (1536 a^{3} b c^{6} d^{3} - 1152 a^{2} b^{3} c^{5} d^{3} + 288 a b^{5} c^{4} d^{3} - 24 b^{7} c^{3} d^{3}\right ) + x^{4} \cdot \left (1024 a^{4} c^{6} d^{3} + 896 a^{3} b^{2} c^{5} d^{3} - 1056 a^{2} b^{4} c^{4} d^{3} + 296 a b^{6} c^{3} d^{3} - 26 b^{8} c^{2} d^{3}\right ) + x^{3} \cdot \left (2048 a^{4} b c^{5} d^{3} - 768 a^{3} b^{3} c^{4} d^{3} - 192 a^{2} b^{5} c^{3} d^{3} + 112 a b^{7} c^{2} d^{3} - 12 b^{9} c d^{3}\right ) + x^{2} \cdot \left (512 a^{5} c^{5} d^{3} + 896 a^{4} b^{2} c^{4} d^{3} - 736 a^{3} b^{4} c^{3} d^{3} + 136 a^{2} b^{6} c^{2} d^{3} + 4 a b^{8} c d^{3} - 2 b^{10} d^{3}\right ) + x \left (512 a^{5} b c^{4} d^{3} - 128 a^{4} b^{3} c^{3} d^{3} - 96 a^{3} b^{5} c^{2} d^{3} + 40 a^{2} b^{7} c d^{3} - 4 a b^{9} d^{3}\right )} \]

[In]

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

[Out]

-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 = 0.24 (sec) , antiderivative size = 553, normalized size of antiderivative = 3.59 \[ \int \frac {1}{(b d+2 c d x)^3 \left (a+b x+c x^2\right )^3} \, dx=\frac {96 \, c^{4} x^{4} + 192 \, b c^{3} x^{3} - b^{4} + 20 \, a b^{2} c + 32 \, a^{2} c^{2} + 36 \, {\left (3 \, b^{2} c^{2} + 4 \, a c^{3}\right )} x^{2} + 12 \, {\left (b^{3} c + 12 \, a b c^{2}\right )} x}{2 \, {\left (4 \, {\left (b^{6} c^{4} - 12 \, a b^{4} c^{5} + 48 \, a^{2} b^{2} c^{6} - 64 \, a^{3} c^{7}\right )} d^{3} x^{6} + 12 \, {\left (b^{7} c^{3} - 12 \, a b^{5} c^{4} + 48 \, a^{2} b^{3} c^{5} - 64 \, a^{3} b c^{6}\right )} d^{3} x^{5} + {\left (13 \, b^{8} c^{2} - 148 \, a b^{6} c^{3} + 528 \, a^{2} b^{4} c^{4} - 448 \, a^{3} b^{2} c^{5} - 512 \, a^{4} c^{6}\right )} d^{3} x^{4} + 2 \, {\left (3 \, b^{9} c - 28 \, a b^{7} c^{2} + 48 \, a^{2} b^{5} c^{3} + 192 \, a^{3} b^{3} c^{4} - 512 \, a^{4} b c^{5}\right )} d^{3} x^{3} + {\left (b^{10} - 2 \, a b^{8} c - 68 \, a^{2} b^{6} c^{2} + 368 \, a^{3} b^{4} c^{3} - 448 \, a^{4} b^{2} c^{4} - 256 \, a^{5} c^{5}\right )} d^{3} x^{2} + 2 \, {\left (a b^{9} - 10 \, a^{2} b^{7} c + 24 \, a^{3} b^{5} c^{2} + 32 \, a^{4} b^{3} c^{3} - 128 \, a^{5} b c^{4}\right )} d^{3} x + {\left (a^{2} b^{8} - 12 \, a^{3} b^{6} c + 48 \, a^{4} b^{4} c^{2} - 64 \, a^{5} b^{2} c^{3}\right )} d^{3}\right )}} + \frac {48 \, c^{2} \log \left (c x^{2} + b x + a\right )}{{\left (b^{8} - 16 \, a b^{6} c + 96 \, a^{2} b^{4} c^{2} - 256 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}\right )} d^{3}} - \frac {96 \, c^{2} \log \left (2 \, c x + b\right )}{{\left (b^{8} - 16 \, a b^{6} c + 96 \, a^{2} b^{4} c^{2} - 256 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}\right )} d^{3}} \]

[In]

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

[Out]

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 + 1
2*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 +
48*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 - 51
2*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 + 25
6*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)

none

Time = 0.29 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.96 \[ \int \frac {1}{(b d+2 c d x)^3 \left (a+b x+c x^2\right )^3} \, dx=-\frac {96 \, c^{3} \log \left ({\left | 2 \, c x + b \right |}\right )}{b^{8} c d^{3} - 16 \, a b^{6} c^{2} d^{3} + 96 \, a^{2} b^{4} c^{3} d^{3} - 256 \, a^{3} b^{2} c^{4} d^{3} + 256 \, a^{4} c^{5} d^{3}} + \frac {48 \, c^{2} \log \left (c x^{2} + b x + a\right )}{b^{8} d^{3} - 16 \, a b^{6} c d^{3} + 96 \, a^{2} b^{4} c^{2} d^{3} - 256 \, a^{3} b^{2} c^{3} d^{3} + 256 \, a^{4} c^{4} d^{3}} + \frac {96 \, c^{4} x^{4} + 192 \, b c^{3} x^{3} + 108 \, b^{2} c^{2} x^{2} + 144 \, a c^{3} x^{2} + 12 \, b^{3} c x + 144 \, a b c^{2} x - b^{4} + 20 \, a b^{2} c + 32 \, a^{2} c^{2}}{2 \, {\left (b^{6} d^{3} - 12 \, a b^{4} c d^{3} + 48 \, a^{2} b^{2} c^{2} d^{3} - 64 \, a^{3} c^{3} d^{3}\right )} {\left (2 \, c^{2} x^{3} + 3 \, b c x^{2} + b^{2} x + 2 \, a c x + a b\right )}^{2}} \]

[In]

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

[Out]

-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 +
256*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 = 10.18 (sec) , antiderivative size = 522, normalized size of antiderivative = 3.39 \[ \int \frac {1}{(b d+2 c d x)^3 \left (a+b x+c x^2\right )^3} \, dx=\frac {\frac {32\,a^2\,c^2+20\,a\,b^2\,c-b^4}{2\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}+\frac {48\,c^4\,x^4}{-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}+\frac {96\,b\,c^3\,x^3}{-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}+\frac {18\,c\,x^2\,\left (3\,b^2\,c+4\,a\,c^2\right )}{-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}+\frac {6\,b\,c\,x\,\left (b^2+12\,a\,c\right )}{-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}}{x\,\left (4\,c\,a^2\,b\,d^3+2\,a\,b^3\,d^3\right )+x^4\,\left (13\,b^2\,c^2\,d^3+8\,a\,c^3\,d^3\right )+x^3\,\left (6\,b^3\,c\,d^3+16\,a\,b\,c^2\,d^3\right )+x^2\,\left (4\,a^2\,c^2\,d^3+10\,a\,b^2\,c\,d^3+b^4\,d^3\right )+a^2\,b^2\,d^3+4\,c^4\,d^3\,x^6+12\,b\,c^3\,d^3\,x^5}-\frac {96\,c^2\,\ln \left (b+2\,c\,x\right )}{256\,a^4\,c^4\,d^3-256\,a^3\,b^2\,c^3\,d^3+96\,a^2\,b^4\,c^2\,d^3-16\,a\,b^6\,c\,d^3+b^8\,d^3}+\frac {48\,c^2\,\ln \left (c\,x^2+b\,x+a\right )}{256\,a^4\,c^4\,d^3-256\,a^3\,b^2\,c^3\,d^3+96\,a^2\,b^4\,c^2\,d^3-16\,a\,b^6\,c\,d^3+b^8\,d^3} \]

[In]

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

[Out]

((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 - 12*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) + (1
8*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*log(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 + 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)