\(\int \frac {1}{(a c+(b c+a d) x+b d x^2)^4} \, dx\) [22]

Optimal result

Integrand size = 21, antiderivative size = 193 \[ \int \frac {1}{\left (a c+(b c+a d) x+b d x^2\right )^4} \, dx=-\frac {b^3}{3 (b c-a d)^4 (a+b x)^3}+\frac {2 b^3 d}{(b c-a d)^5 (a+b x)^2}-\frac {10 b^3 d^2}{(b c-a d)^6 (a+b x)}-\frac {d^3}{3 (b c-a d)^4 (c+d x)^3}-\frac {2 b d^3}{(b c-a d)^5 (c+d x)^2}-\frac {10 b^2 d^3}{(b c-a d)^6 (c+d x)}-\frac {20 b^3 d^3 \log (a+b x)}{(b c-a d)^7}+\frac {20 b^3 d^3 \log (c+d x)}{(b c-a d)^7} \] Output:

-1/3*b^3/(-a*d+b*c)^4/(b*x+a)^3+2*b^3*d/(-a*d+b*c)^5/(b*x+a)^2-10*b^3*d^2/ 
(-a*d+b*c)^6/(b*x+a)-1/3*d^3/(-a*d+b*c)^4/(d*x+c)^3-2*b*d^3/(-a*d+b*c)^5/( 
d*x+c)^2-10*b^2*d^3/(-a*d+b*c)^6/(d*x+c)-20*b^3*d^3*ln(b*x+a)/(-a*d+b*c)^7 
+20*b^3*d^3*ln(d*x+c)/(-a*d+b*c)^7
 

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.92 \[ \int \frac {1}{\left (a c+(b c+a d) x+b d x^2\right )^4} \, dx=-\frac {\frac {b^3 (b c-a d)^3}{(a+b x)^3}-\frac {6 b^3 d (b c-a d)^2}{(a+b x)^2}+\frac {30 b^3 d^2 (b c-a d)}{a+b x}-\frac {d^3 (-b c+a d)^3}{(c+d x)^3}+\frac {6 b d^3 (b c-a d)^2}{(c+d x)^2}+\frac {30 b^2 d^3 (b c-a d)}{c+d x}+60 b^3 d^3 \log (a+b x)-60 b^3 d^3 \log (c+d x)}{3 (b c-a d)^7} \] Input:

Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)^(-4),x]
 

Output:

-1/3*((b^3*(b*c - a*d)^3)/(a + b*x)^3 - (6*b^3*d*(b*c - a*d)^2)/(a + b*x)^ 
2 + (30*b^3*d^2*(b*c - a*d))/(a + b*x) - (d^3*(-(b*c) + a*d)^3)/(c + d*x)^ 
3 + (6*b*d^3*(b*c - a*d)^2)/(c + d*x)^2 + (30*b^2*d^3*(b*c - a*d))/(c + d* 
x) + 60*b^3*d^3*Log[a + b*x] - 60*b^3*d^3*Log[c + d*x])/(b*c - a*d)^7
 

Rubi [A] (verified)

Time = 0.82 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.09, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {1084, 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.

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

\(\Big \downarrow \) 1084

\(\displaystyle b^4 d^4 \int \left (\frac {20}{b (b c-a d)^7 (c+d x)}+\frac {10}{b^2 (b c-a d)^6 (c+d x)^2}+\frac {4}{b^3 (b c-a d)^5 (c+d x)^3}+\frac {1}{b^4 (b c-a d)^4 (c+d x)^4}-\frac {20}{d (b c-a d)^7 (a+b x)}+\frac {10}{d^2 (b c-a d)^6 (a+b x)^2}-\frac {4}{d^3 (b c-a d)^5 (a+b x)^3}+\frac {1}{d^4 (b c-a d)^4 (a+b x)^4}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle b^4 d^4 \left (-\frac {1}{3 b^4 d (c+d x)^3 (b c-a d)^4}-\frac {2}{b^3 d (c+d x)^2 (b c-a d)^5}-\frac {10}{b^2 d (c+d x) (b c-a d)^6}-\frac {1}{3 b d^4 (a+b x)^3 (b c-a d)^4}+\frac {2}{b d^3 (a+b x)^2 (b c-a d)^5}-\frac {10}{b d^2 (a+b x) (b c-a d)^6}-\frac {20 \log (a+b x)}{b d (b c-a d)^7}+\frac {20 \log (c+d x)}{b d (b c-a d)^7}\right )\)

Input:

Int[(a*c + (b*c + a*d)*x + b*d*x^2)^(-4),x]
 

Output:

b^4*d^4*(-1/3*1/(b*d^4*(b*c - a*d)^4*(a + b*x)^3) + 2/(b*d^3*(b*c - a*d)^5 
*(a + b*x)^2) - 10/(b*d^2*(b*c - a*d)^6*(a + b*x)) - 1/(3*b^4*d*(b*c - a*d 
)^4*(c + d*x)^3) - 2/(b^3*d*(b*c - a*d)^5*(c + d*x)^2) - 10/(b^2*d*(b*c - 
a*d)^6*(c + d*x)) - (20*Log[a + b*x])/(b*d*(b*c - a*d)^7) + (20*Log[c + d* 
x])/(b*d*(b*c - a*d)^7))
 

Defintions of rubi rules used

rule 1084

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Rt[b^2 
- 4*a*c, 2]}, Simp[1/c^p   Int[ExpandIntegrand[(b/2 - q/2 + c*x)^p*(b/2 + q 
/2 + c*x)^p, x], x], x] /;  !FractionalPowerFactorQ[q]] /; FreeQ[{a, b, c}, 
 x] && IntegerQ[p] && NiceSqrtQ[b^2 - 4*a*c]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 0.84 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.98

method	result	size
default	\(-\frac {b^{3}}{3 \left (a d -b c \right )^{4} \left (b x +a \right )^{3}}+\frac {20 b^{3} d^{3} \ln \left (b x +a \right )}{\left (a d -b c \right )^{7}}-\frac {10 b^{3} d^{2}}{\left (a d -b c \right )^{6} \left (b x +a \right )}-\frac {2 b^{3} d}{\left (a d -b c \right )^{5} \left (b x +a \right )^{2}}-\frac {d^{3}}{3 \left (a d -b c \right )^{4} \left (d x +c \right )^{3}}-\frac {20 b^{3} d^{3} \ln \left (d x +c \right )}{\left (a d -b c \right )^{7}}-\frac {10 d^{3} b^{2}}{\left (a d -b c \right )^{6} \left (d x +c \right )}+\frac {2 d^{3} b}{\left (a d -b c \right )^{5} \left (d x +c \right )^{2}}\)	\(190\)
risch	\(\frac {-\frac {20 b^{5} d^{5} x^{5}}{a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +b^{6} c^{6}}-\frac {50 b^{4} d^{4} \left (a d +b c \right ) x^{4}}{a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +b^{6} c^{6}}-\frac {10 \left (11 a^{2} d^{2}+38 a b c d +11 b^{2} c^{2}\right ) b^{3} d^{3} x^{3}}{3 \left (a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +b^{6} c^{6}\right )}-\frac {5 b^{2} d^{2} \left (a^{3} d^{3}+19 a^{2} b c \,d^{2}+19 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) x^{2}}{a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +b^{6} c^{6}}+\frac {\left (a^{4} d^{4}-14 a^{3} b c \,d^{3}-74 a^{2} b^{2} c^{2} d^{2}-14 a \,b^{3} c^{3} d +c^{4} b^{4}\right ) b d x}{a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +b^{6} c^{6}}-\frac {a^{5} d^{5}-8 a^{4} b c \,d^{4}+37 a^{3} b^{2} c^{2} d^{3}+37 a^{2} b^{3} c^{3} d^{2}-8 a \,b^{4} c^{4} d +b^{5} c^{5}}{3 \left (a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +b^{6} c^{6}\right )}}{\left (b d \,x^{2}+a d x +c b x +a c \right )^{3}}-\frac {20 b^{3} d^{3} \ln \left (d x +c \right )}{a^{7} d^{7}-7 a^{6} b c \,d^{6}+21 a^{5} b^{2} c^{2} d^{5}-35 a^{4} b^{3} c^{3} d^{4}+35 a^{3} b^{4} c^{4} d^{3}-21 a^{2} b^{5} c^{5} d^{2}+7 a \,b^{6} c^{6} d -b^{7} c^{7}}+\frac {20 b^{3} d^{3} \ln \left (-b x -a \right )}{a^{7} d^{7}-7 a^{6} b c \,d^{6}+21 a^{5} b^{2} c^{2} d^{5}-35 a^{4} b^{3} c^{3} d^{4}+35 a^{3} b^{4} c^{4} d^{3}-21 a^{2} b^{5} c^{5} d^{2}+7 a \,b^{6} c^{6} d -b^{7} c^{7}}\)	\(944\)
norman	\(\frac {-\frac {20 b^{5} d^{5} x^{5}}{a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +b^{6} c^{6}}+\frac {\left (-50 a \,b^{6} d^{7}-50 b^{7} c \,d^{6}\right ) x^{4}}{d^{2} b^{2} \left (a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +b^{6} c^{6}\right )}+\frac {\left (-5 a^{3} b^{5} d^{8}-95 a^{2} b^{6} c \,d^{7}-95 a \,b^{7} c^{2} d^{6}-5 b^{8} c^{3} d^{5}\right ) x^{2}}{d^{3} b^{3} \left (a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +b^{6} c^{6}\right )}+\frac {\left (a^{4} b^{4} d^{8}-14 a^{3} b^{5} c \,d^{7}-74 a^{2} b^{6} c^{2} d^{6}-14 a \,b^{7} c^{3} d^{5}+b^{8} c^{4} d^{4}\right ) x}{d^{3} b^{3} \left (a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +b^{6} c^{6}\right )}+\frac {-a^{5} b^{3} d^{8}+8 a^{4} b^{4} c \,d^{7}-37 a^{3} b^{5} c^{2} d^{6}-37 a^{2} b^{6} c^{3} d^{5}+8 a \,b^{7} c^{4} d^{4}-b^{8} c^{5} d^{3}}{3 d^{3} b^{3} \left (a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +b^{6} c^{6}\right )}+\frac {\left (-110 a^{2} b^{6} d^{8}-380 a \,b^{7} c \,d^{7}-110 b^{8} c^{2} d^{6}\right ) x^{3}}{3 d^{3} b^{3} \left (a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +b^{6} c^{6}\right )}}{\left (b x +a \right )^{3} \left (d x +c \right )^{3}}+\frac {20 b^{3} d^{3} \ln \left (b x +a \right )}{a^{7} d^{7}-7 a^{6} b c \,d^{6}+21 a^{5} b^{2} c^{2} d^{5}-35 a^{4} b^{3} c^{3} d^{4}+35 a^{3} b^{4} c^{4} d^{3}-21 a^{2} b^{5} c^{5} d^{2}+7 a \,b^{6} c^{6} d -b^{7} c^{7}}-\frac {20 b^{3} d^{3} \ln \left (d x +c \right )}{a^{7} d^{7}-7 a^{6} b c \,d^{6}+21 a^{5} b^{2} c^{2} d^{5}-35 a^{4} b^{3} c^{3} d^{4}+35 a^{3} b^{4} c^{4} d^{3}-21 a^{2} b^{5} c^{5} d^{2}+7 a \,b^{6} c^{6} d -b^{7} c^{7}}\)	\(999\)
parallelrisch	\(\text {Expression too large to display}\)	\(1087\)

Input:

int(1/(a*c+(a*d+b*c)*x+b*d*x^2)^4,x,method=_RETURNVERBOSE)
 

Output:

-1/3*b^3/(a*d-b*c)^4/(b*x+a)^3+20*b^3/(a*d-b*c)^7*d^3*ln(b*x+a)-10*b^3/(a* 
d-b*c)^6*d^2/(b*x+a)-2*b^3/(a*d-b*c)^5*d/(b*x+a)^2-1/3*d^3/(a*d-b*c)^4/(d* 
x+c)^3-20*b^3/(a*d-b*c)^7*d^3*ln(d*x+c)-10*d^3/(a*d-b*c)^6*b^2/(d*x+c)+2*d 
^3/(a*d-b*c)^5*b/(d*x+c)^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1520 vs. \(2 (189) = 378\).

Time = 0.13 (sec) , antiderivative size = 1520, normalized size of antiderivative = 7.88 \[ \int \frac {1}{\left (a c+(b c+a d) x+b d x^2\right )^4} \, dx=\text {Too large to display} \] Input:

integrate(1/(a*c+(a*d+b*c)*x+b*d*x^2)^4,x, algorithm="fricas")
 

Output:

-1/3*(b^6*c^6 - 9*a*b^5*c^5*d + 45*a^2*b^4*c^4*d^2 - 45*a^4*b^2*c^2*d^4 + 
9*a^5*b*c*d^5 - a^6*d^6 + 60*(b^6*c*d^5 - a*b^5*d^6)*x^5 + 150*(b^6*c^2*d^ 
4 - a^2*b^4*d^6)*x^4 + 10*(11*b^6*c^3*d^3 + 27*a*b^5*c^2*d^4 - 27*a^2*b^4* 
c*d^5 - 11*a^3*b^3*d^6)*x^3 + 15*(b^6*c^4*d^2 + 18*a*b^5*c^3*d^3 - 18*a^3* 
b^3*c*d^5 - a^4*b^2*d^6)*x^2 - 3*(b^6*c^5*d - 15*a*b^5*c^4*d^2 - 60*a^2*b^ 
4*c^3*d^3 + 60*a^3*b^3*c^2*d^4 + 15*a^4*b^2*c*d^5 - a^5*b*d^6)*x + 60*(b^6 
*d^6*x^6 + a^3*b^3*c^3*d^3 + 3*(b^6*c*d^5 + a*b^5*d^6)*x^5 + 3*(b^6*c^2*d^ 
4 + 3*a*b^5*c*d^5 + a^2*b^4*d^6)*x^4 + (b^6*c^3*d^3 + 9*a*b^5*c^2*d^4 + 9* 
a^2*b^4*c*d^5 + a^3*b^3*d^6)*x^3 + 3*(a*b^5*c^3*d^3 + 3*a^2*b^4*c^2*d^4 + 
a^3*b^3*c*d^5)*x^2 + 3*(a^2*b^4*c^3*d^3 + a^3*b^3*c^2*d^4)*x)*log(b*x + a) 
 - 60*(b^6*d^6*x^6 + a^3*b^3*c^3*d^3 + 3*(b^6*c*d^5 + a*b^5*d^6)*x^5 + 3*( 
b^6*c^2*d^4 + 3*a*b^5*c*d^5 + a^2*b^4*d^6)*x^4 + (b^6*c^3*d^3 + 9*a*b^5*c^ 
2*d^4 + 9*a^2*b^4*c*d^5 + a^3*b^3*d^6)*x^3 + 3*(a*b^5*c^3*d^3 + 3*a^2*b^4* 
c^2*d^4 + a^3*b^3*c*d^5)*x^2 + 3*(a^2*b^4*c^3*d^3 + a^3*b^3*c^2*d^4)*x)*lo 
g(d*x + c))/(a^3*b^7*c^10 - 7*a^4*b^6*c^9*d + 21*a^5*b^5*c^8*d^2 - 35*a^6* 
b^4*c^7*d^3 + 35*a^7*b^3*c^6*d^4 - 21*a^8*b^2*c^5*d^5 + 7*a^9*b*c^4*d^6 - 
a^10*c^3*d^7 + (b^10*c^7*d^3 - 7*a*b^9*c^6*d^4 + 21*a^2*b^8*c^5*d^5 - 35*a 
^3*b^7*c^4*d^6 + 35*a^4*b^6*c^3*d^7 - 21*a^5*b^5*c^2*d^8 + 7*a^6*b^4*c*d^9 
 - a^7*b^3*d^10)*x^6 + 3*(b^10*c^8*d^2 - 6*a*b^9*c^7*d^3 + 14*a^2*b^8*c^6* 
d^4 - 14*a^3*b^7*c^5*d^5 + 14*a^5*b^5*c^3*d^7 - 14*a^6*b^4*c^2*d^8 + 6*...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1590 vs. \(2 (175) = 350\).

Time = 2.65 (sec) , antiderivative size = 1590, normalized size of antiderivative = 8.24 \[ \int \frac {1}{\left (a c+(b c+a d) x+b d x^2\right )^4} \, dx=\text {Too large to display} \] Input:

integrate(1/(a*c+(a*d+b*c)*x+b*d*x**2)**4,x)
 

Output:

-20*b**3*d**3*log(x + (-20*a**8*b**3*d**11/(a*d - b*c)**7 + 160*a**7*b**4* 
c*d**10/(a*d - b*c)**7 - 560*a**6*b**5*c**2*d**9/(a*d - b*c)**7 + 1120*a** 
5*b**6*c**3*d**8/(a*d - b*c)**7 - 1400*a**4*b**7*c**4*d**7/(a*d - b*c)**7 
+ 1120*a**3*b**8*c**5*d**6/(a*d - b*c)**7 - 560*a**2*b**9*c**6*d**5/(a*d - 
 b*c)**7 + 160*a*b**10*c**7*d**4/(a*d - b*c)**7 + 20*a*b**3*d**4 - 20*b**1 
1*c**8*d**3/(a*d - b*c)**7 + 20*b**4*c*d**3)/(40*b**4*d**4))/(a*d - b*c)** 
7 + 20*b**3*d**3*log(x + (20*a**8*b**3*d**11/(a*d - b*c)**7 - 160*a**7*b** 
4*c*d**10/(a*d - b*c)**7 + 560*a**6*b**5*c**2*d**9/(a*d - b*c)**7 - 1120*a 
**5*b**6*c**3*d**8/(a*d - b*c)**7 + 1400*a**4*b**7*c**4*d**7/(a*d - b*c)** 
7 - 1120*a**3*b**8*c**5*d**6/(a*d - b*c)**7 + 560*a**2*b**9*c**6*d**5/(a*d 
 - b*c)**7 - 160*a*b**10*c**7*d**4/(a*d - b*c)**7 + 20*a*b**3*d**4 + 20*b* 
*11*c**8*d**3/(a*d - b*c)**7 + 20*b**4*c*d**3)/(40*b**4*d**4))/(a*d - b*c) 
**7 + (-a**5*d**5 + 8*a**4*b*c*d**4 - 37*a**3*b**2*c**2*d**3 - 37*a**2*b** 
3*c**3*d**2 + 8*a*b**4*c**4*d - b**5*c**5 - 60*b**5*d**5*x**5 + x**4*(-150 
*a*b**4*d**5 - 150*b**5*c*d**4) + x**3*(-110*a**2*b**3*d**5 - 380*a*b**4*c 
*d**4 - 110*b**5*c**2*d**3) + x**2*(-15*a**3*b**2*d**5 - 285*a**2*b**3*c*d 
**4 - 285*a*b**4*c**2*d**3 - 15*b**5*c**3*d**2) + x*(3*a**4*b*d**5 - 42*a* 
*3*b**2*c*d**4 - 222*a**2*b**3*c**2*d**3 - 42*a*b**4*c**3*d**2 + 3*b**5*c* 
*4*d))/(3*a**9*c**3*d**6 - 18*a**8*b*c**4*d**5 + 45*a**7*b**2*c**5*d**4 - 
60*a**6*b**3*c**6*d**3 + 45*a**5*b**4*c**7*d**2 - 18*a**4*b**5*c**8*d +...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1206 vs. \(2 (189) = 378\).

Time = 0.07 (sec) , antiderivative size = 1206, normalized size of antiderivative = 6.25 \[ \int \frac {1}{\left (a c+(b c+a d) x+b d x^2\right )^4} \, dx =\text {Too large to display} \] Input:

integrate(1/(a*c+(a*d+b*c)*x+b*d*x^2)^4,x, algorithm="maxima")
 

Output:

-20*b^3*d^3*log(b*x + a)/(b^7*c^7 - 7*a*b^6*c^6*d + 21*a^2*b^5*c^5*d^2 - 3 
5*a^3*b^4*c^4*d^3 + 35*a^4*b^3*c^3*d^4 - 21*a^5*b^2*c^2*d^5 + 7*a^6*b*c*d^ 
6 - a^7*d^7) + 20*b^3*d^3*log(d*x + c)/(b^7*c^7 - 7*a*b^6*c^6*d + 21*a^2*b 
^5*c^5*d^2 - 35*a^3*b^4*c^4*d^3 + 35*a^4*b^3*c^3*d^4 - 21*a^5*b^2*c^2*d^5 
+ 7*a^6*b*c*d^6 - a^7*d^7) - 1/3*(60*b^5*d^5*x^5 + b^5*c^5 - 8*a*b^4*c^4*d 
 + 37*a^2*b^3*c^3*d^2 + 37*a^3*b^2*c^2*d^3 - 8*a^4*b*c*d^4 + a^5*d^5 + 150 
*(b^5*c*d^4 + a*b^4*d^5)*x^4 + 10*(11*b^5*c^2*d^3 + 38*a*b^4*c*d^4 + 11*a^ 
2*b^3*d^5)*x^3 + 15*(b^5*c^3*d^2 + 19*a*b^4*c^2*d^3 + 19*a^2*b^3*c*d^4 + a 
^3*b^2*d^5)*x^2 - 3*(b^5*c^4*d - 14*a*b^4*c^3*d^2 - 74*a^2*b^3*c^2*d^3 - 1 
4*a^3*b^2*c*d^4 + a^4*b*d^5)*x)/(a^3*b^6*c^9 - 6*a^4*b^5*c^8*d + 15*a^5*b^ 
4*c^7*d^2 - 20*a^6*b^3*c^6*d^3 + 15*a^7*b^2*c^5*d^4 - 6*a^8*b*c^4*d^5 + a^ 
9*c^3*d^6 + (b^9*c^6*d^3 - 6*a*b^8*c^5*d^4 + 15*a^2*b^7*c^4*d^5 - 20*a^3*b 
^6*c^3*d^6 + 15*a^4*b^5*c^2*d^7 - 6*a^5*b^4*c*d^8 + a^6*b^3*d^9)*x^6 + 3*( 
b^9*c^7*d^2 - 5*a*b^8*c^6*d^3 + 9*a^2*b^7*c^5*d^4 - 5*a^3*b^6*c^4*d^5 - 5* 
a^4*b^5*c^3*d^6 + 9*a^5*b^4*c^2*d^7 - 5*a^6*b^3*c*d^8 + a^7*b^2*d^9)*x^5 + 
 3*(b^9*c^8*d - 3*a*b^8*c^7*d^2 - 2*a^2*b^7*c^6*d^3 + 19*a^3*b^6*c^5*d^4 - 
 30*a^4*b^5*c^4*d^5 + 19*a^5*b^4*c^3*d^6 - 2*a^6*b^3*c^2*d^7 - 3*a^7*b^2*c 
*d^8 + a^8*b*d^9)*x^4 + (b^9*c^9 + 3*a*b^8*c^8*d - 30*a^2*b^7*c^7*d^2 + 62 
*a^3*b^6*c^6*d^3 - 36*a^4*b^5*c^5*d^4 - 36*a^5*b^4*c^4*d^5 + 62*a^6*b^3*c^ 
3*d^6 - 30*a^7*b^2*c^2*d^7 + 3*a^8*b*c*d^8 + a^9*d^9)*x^3 + 3*(a*b^8*c^...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 584 vs. \(2 (189) = 378\).

Time = 0.17 (sec) , antiderivative size = 584, normalized size of antiderivative = 3.03 \[ \int \frac {1}{\left (a c+(b c+a d) x+b d x^2\right )^4} \, dx=-\frac {20 \, b^{4} d^{3} \log \left ({\left | b x + a \right |}\right )}{b^{8} c^{7} - 7 \, a b^{7} c^{6} d + 21 \, a^{2} b^{6} c^{5} d^{2} - 35 \, a^{3} b^{5} c^{4} d^{3} + 35 \, a^{4} b^{4} c^{3} d^{4} - 21 \, a^{5} b^{3} c^{2} d^{5} + 7 \, a^{6} b^{2} c d^{6} - a^{7} b d^{7}} + \frac {20 \, b^{3} d^{4} \log \left ({\left | d x + c \right |}\right )}{b^{7} c^{7} d - 7 \, a b^{6} c^{6} d^{2} + 21 \, a^{2} b^{5} c^{5} d^{3} - 35 \, a^{3} b^{4} c^{4} d^{4} + 35 \, a^{4} b^{3} c^{3} d^{5} - 21 \, a^{5} b^{2} c^{2} d^{6} + 7 \, a^{6} b c d^{7} - a^{7} d^{8}} - \frac {60 \, b^{5} d^{5} x^{5} + 150 \, b^{5} c d^{4} x^{4} + 150 \, a b^{4} d^{5} x^{4} + 110 \, b^{5} c^{2} d^{3} x^{3} + 380 \, a b^{4} c d^{4} x^{3} + 110 \, a^{2} b^{3} d^{5} x^{3} + 15 \, b^{5} c^{3} d^{2} x^{2} + 285 \, a b^{4} c^{2} d^{3} x^{2} + 285 \, a^{2} b^{3} c d^{4} x^{2} + 15 \, a^{3} b^{2} d^{5} x^{2} - 3 \, b^{5} c^{4} d x + 42 \, a b^{4} c^{3} d^{2} x + 222 \, a^{2} b^{3} c^{2} d^{3} x + 42 \, a^{3} b^{2} c d^{4} x - 3 \, a^{4} b d^{5} x + b^{5} c^{5} - 8 \, a b^{4} c^{4} d + 37 \, a^{2} b^{3} c^{3} d^{2} + 37 \, a^{3} b^{2} c^{2} d^{3} - 8 \, a^{4} b c d^{4} + a^{5} d^{5}}{3 \, {\left (b^{6} c^{6} - 6 \, a b^{5} c^{5} d + 15 \, a^{2} b^{4} c^{4} d^{2} - 20 \, a^{3} b^{3} c^{3} d^{3} + 15 \, a^{4} b^{2} c^{2} d^{4} - 6 \, a^{5} b c d^{5} + a^{6} d^{6}\right )} {\left (b d x^{2} + b c x + a d x + a c\right )}^{3}} \] Input:

integrate(1/(a*c+(a*d+b*c)*x+b*d*x^2)^4,x, algorithm="giac")
 

Output:

-20*b^4*d^3*log(abs(b*x + a))/(b^8*c^7 - 7*a*b^7*c^6*d + 21*a^2*b^6*c^5*d^ 
2 - 35*a^3*b^5*c^4*d^3 + 35*a^4*b^4*c^3*d^4 - 21*a^5*b^3*c^2*d^5 + 7*a^6*b 
^2*c*d^6 - a^7*b*d^7) + 20*b^3*d^4*log(abs(d*x + c))/(b^7*c^7*d - 7*a*b^6* 
c^6*d^2 + 21*a^2*b^5*c^5*d^3 - 35*a^3*b^4*c^4*d^4 + 35*a^4*b^3*c^3*d^5 - 2 
1*a^5*b^2*c^2*d^6 + 7*a^6*b*c*d^7 - a^7*d^8) - 1/3*(60*b^5*d^5*x^5 + 150*b 
^5*c*d^4*x^4 + 150*a*b^4*d^5*x^4 + 110*b^5*c^2*d^3*x^3 + 380*a*b^4*c*d^4*x 
^3 + 110*a^2*b^3*d^5*x^3 + 15*b^5*c^3*d^2*x^2 + 285*a*b^4*c^2*d^3*x^2 + 28 
5*a^2*b^3*c*d^4*x^2 + 15*a^3*b^2*d^5*x^2 - 3*b^5*c^4*d*x + 42*a*b^4*c^3*d^ 
2*x + 222*a^2*b^3*c^2*d^3*x + 42*a^3*b^2*c*d^4*x - 3*a^4*b*d^5*x + b^5*c^5 
 - 8*a*b^4*c^4*d + 37*a^2*b^3*c^3*d^2 + 37*a^3*b^2*c^2*d^3 - 8*a^4*b*c*d^4 
 + a^5*d^5)/((b^6*c^6 - 6*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^ 
3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 + a^6*d^6)*(b*d*x^2 + b*c*x + a 
*d*x + a*c)^3)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a c+(b c+a d) x+b d x^2\right )^4} \, dx=\left \{\begin {array}{cl} -\frac {20\,\left (\frac {a\,d}{2}+\frac {b\,c}{2}+b\,d\,x\right )\,\left (\frac {b\,d}{30\,\left ({\left (a\,d+b\,c\right )}^2-4\,a\,b\,c\,d\right )\,{\left (b\,d\,x^2+\left (a\,d+b\,c\right )\,x+a\,c\right )}^3}-\frac {b^2\,d^2}{6\,{\left ({\left (a\,d+b\,c\right )}^2-4\,a\,b\,c\,d\right )}^2\,{\left (b\,d\,x^2+\left (a\,d+b\,c\right )\,x+a\,c\right )}^2}+\frac {b^3\,d^3}{{\left ({\left (a\,d+b\,c\right )}^2-4\,a\,b\,c\,d\right )}^3\,\left (b\,d\,x^2+\left (a\,d+b\,c\right )\,x+a\,c\right )}\right )}{b\,d}-\frac {20\,b^3\,d^3\,\ln \left (\frac {\frac {a\,d}{2}-\sqrt {\frac {{\left (a\,d+b\,c\right )}^2}{4}-a\,b\,c\,d}+\frac {b\,c}{2}+b\,d\,x}{\sqrt {\frac {{\left (a\,d+b\,c\right )}^2}{4}-a\,b\,c\,d}+\frac {a\,d}{2}+\frac {b\,c}{2}+b\,d\,x}\right )}{{\left ({\left (a\,d+b\,c\right )}^2-4\,a\,b\,c\,d\right )}^{7/2}} & \text {\ if\ \ }0<{\left (a\,d+b\,c\right )}^2-4\,a\,b\,c\,d\\ -\frac {20\,\left (\frac {a\,d}{2}+\frac {b\,c}{2}+b\,d\,x\right )\,\left (\frac {b\,d}{30\,\left ({\left (a\,d+b\,c\right )}^2-4\,a\,b\,c\,d\right )\,{\left (b\,d\,x^2+\left (a\,d+b\,c\right )\,x+a\,c\right )}^3}-\frac {b^2\,d^2}{6\,{\left ({\left (a\,d+b\,c\right )}^2-4\,a\,b\,c\,d\right )}^2\,{\left (b\,d\,x^2+\left (a\,d+b\,c\right )\,x+a\,c\right )}^2}+\frac {b^3\,d^3}{{\left ({\left (a\,d+b\,c\right )}^2-4\,a\,b\,c\,d\right )}^3\,\left (b\,d\,x^2+\left (a\,d+b\,c\right )\,x+a\,c\right )}\right )}{b\,d}-\frac {20\,b^3\,d^3\,\mathrm {atan}\left (\frac {\frac {a\,d}{2}+\frac {b\,c}{2}+b\,d\,x}{\sqrt {a\,b\,c\,d-\frac {{\left (a\,d+b\,c\right )}^2}{4}}}\right )}{{\left ({\left (a\,d+b\,c\right )}^2-4\,a\,b\,c\,d\right )}^3\,\sqrt {a\,b\,c\,d-\frac {{\left (a\,d+b\,c\right )}^2}{4}}} & \text {\ if\ \ }{\left (a\,d+b\,c\right )}^2-4\,a\,b\,c\,d<0\\ \int \frac {1}{{\left (b\,d\,x^2+\left (a\,d+b\,c\right )\,x+a\,c\right )}^4} \,d x & \text {\ if\ \ }{\left (a\,d+b\,c\right )}^2-4\,a\,b\,c\,d\notin \mathbb {R}\vee {\left (a\,d+b\,c\right )}^2=4\,a\,b\,c\,d \end {array}\right . \] Input:

int(1/(a*c + x*(a*d + b*c) + b*d*x^2)^4,x)
 

Output:

piecewise(0 < (a*d + b*c)^2 - 4*a*b*c*d, - (20*((a*d)/2 + (b*c)/2 + b*d*x) 
*((b*d)/(30*((a*d + b*c)^2 - 4*a*b*c*d)*(a*c + x*(a*d + b*c) + b*d*x^2)^3) 
 - (b^2*d^2)/(6*((a*d + b*c)^2 - 4*a*b*c*d)^2*(a*c + x*(a*d + b*c) + b*d*x 
^2)^2) + (b^3*d^3)/(((a*d + b*c)^2 - 4*a*b*c*d)^3*(a*c + x*(a*d + b*c) + b 
*d*x^2))))/(b*d) - (20*b^3*d^3*log((- ((a*d + b*c)^2/4 - a*b*c*d)^(1/2) + 
(a*d)/2 + (b*c)/2 + b*d*x)/(((a*d + b*c)^2/4 - a*b*c*d)^(1/2) + (a*d)/2 + 
(b*c)/2 + b*d*x)))/((a*d + b*c)^2 - 4*a*b*c*d)^(7/2), (a*d + b*c)^2 - 4*a* 
b*c*d < 0, - (20*((a*d)/2 + (b*c)/2 + b*d*x)*((b*d)/(30*((a*d + b*c)^2 - 4 
*a*b*c*d)*(a*c + x*(a*d + b*c) + b*d*x^2)^3) - (b^2*d^2)/(6*((a*d + b*c)^2 
 - 4*a*b*c*d)^2*(a*c + x*(a*d + b*c) + b*d*x^2)^2) + (b^3*d^3)/(((a*d + b* 
c)^2 - 4*a*b*c*d)^3*(a*c + x*(a*d + b*c) + b*d*x^2))))/(b*d) - (20*b^3*d^3 
*atan(((a*d)/2 + (b*c)/2 + b*d*x)/(- (a*d + b*c)^2/4 + a*b*c*d)^(1/2)))/(( 
(a*d + b*c)^2 - 4*a*b*c*d)^3*(- (a*d + b*c)^2/4 + a*b*c*d)^(1/2)), ~in((a* 
d + b*c)^2 - 4*a*b*c*d, 'real') | (a*d + b*c)^2 == 4*a*b*c*d, int(1/(a*c + 
 x*(a*d + b*c) + b*d*x^2)^4, x))
 

Reduce [B] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 2310, normalized size of antiderivative = 11.97 \[ \int \frac {1}{\left (a c+(b c+a d) x+b d x^2\right )^4} \, dx =\text {Too large to display} \] Input:

int(1/(a*c+(a*d+b*c)*x+b*d*x^2)^4,x)
 

Output:

(60*log(a + b*x)*a**4*b**3*c**3*d**4 + 180*log(a + b*x)*a**4*b**3*c**2*d** 
5*x + 180*log(a + b*x)*a**4*b**3*c*d**6*x**2 + 60*log(a + b*x)*a**4*b**3*d 
**7*x**3 + 60*log(a + b*x)*a**3*b**4*c**4*d**3 + 360*log(a + b*x)*a**3*b** 
4*c**3*d**4*x + 720*log(a + b*x)*a**3*b**4*c**2*d**5*x**2 + 600*log(a + b* 
x)*a**3*b**4*c*d**6*x**3 + 180*log(a + b*x)*a**3*b**4*d**7*x**4 + 180*log( 
a + b*x)*a**2*b**5*c**4*d**3*x + 720*log(a + b*x)*a**2*b**5*c**3*d**4*x**2 
 + 1080*log(a + b*x)*a**2*b**5*c**2*d**5*x**3 + 720*log(a + b*x)*a**2*b**5 
*c*d**6*x**4 + 180*log(a + b*x)*a**2*b**5*d**7*x**5 + 180*log(a + b*x)*a*b 
**6*c**4*d**3*x**2 + 600*log(a + b*x)*a*b**6*c**3*d**4*x**3 + 720*log(a + 
b*x)*a*b**6*c**2*d**5*x**4 + 360*log(a + b*x)*a*b**6*c*d**6*x**5 + 60*log( 
a + b*x)*a*b**6*d**7*x**6 + 60*log(a + b*x)*b**7*c**4*d**3*x**3 + 180*log( 
a + b*x)*b**7*c**3*d**4*x**4 + 180*log(a + b*x)*b**7*c**2*d**5*x**5 + 60*l 
og(a + b*x)*b**7*c*d**6*x**6 - 60*log(c + d*x)*a**4*b**3*c**3*d**4 - 180*l 
og(c + d*x)*a**4*b**3*c**2*d**5*x - 180*log(c + d*x)*a**4*b**3*c*d**6*x**2 
 - 60*log(c + d*x)*a**4*b**3*d**7*x**3 - 60*log(c + d*x)*a**3*b**4*c**4*d* 
*3 - 360*log(c + d*x)*a**3*b**4*c**3*d**4*x - 720*log(c + d*x)*a**3*b**4*c 
**2*d**5*x**2 - 600*log(c + d*x)*a**3*b**4*c*d**6*x**3 - 180*log(c + d*x)* 
a**3*b**4*d**7*x**4 - 180*log(c + d*x)*a**2*b**5*c**4*d**3*x - 720*log(c + 
 d*x)*a**2*b**5*c**3*d**4*x**2 - 1080*log(c + d*x)*a**2*b**5*c**2*d**5*x** 
3 - 720*log(c + d*x)*a**2*b**5*c*d**6*x**4 - 180*log(c + d*x)*a**2*b**5...