\(\int \frac {A+B x+C x^2+D x^3}{(a+b x) (c+d x)^4} \, dx\) [52]

Optimal result

Integrand size = 30, antiderivative size = 274 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x) (c+d x)^4} \, dx=\frac {c^2 C d-B c d^2+A d^3-c^3 D}{3 d^3 (b c-a d) (c+d x)^3}+\frac {a d \left (2 c C d-B d^2-3 c^2 D\right )-b \left (c^2 C d-A d^3-2 c^3 D\right )}{2 d^3 (b c-a d)^2 (c+d x)^2}+\frac {A b^2 d^3-b^2 c^3 D+a^2 d^2 (C d-3 c D)-a b \left (B d^3-3 c^2 d D\right )}{d^3 (b c-a d)^3 (c+d x)}+\frac {\left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) \log (a+b x)}{(b c-a d)^4}-\frac {\left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) \log (c+d x)}{(b c-a d)^4} \] Output:

1/3*(A*d^3-B*c*d^2+C*c^2*d-D*c^3)/d^3/(-a*d+b*c)/(d*x+c)^3+1/2*(a*d*(-B*d^ 
2+2*C*c*d-3*D*c^2)-b*(-A*d^3+C*c^2*d-2*D*c^3))/d^3/(-a*d+b*c)^2/(d*x+c)^2+ 
(A*b^2*d^3-b^2*c^3*D+a^2*d^2*(C*d-3*D*c)-a*b*(B*d^3-3*D*c^2*d))/d^3/(-a*d+ 
b*c)^3/(d*x+c)+(A*b^3-a*(B*b^2-C*a*b+D*a^2))*ln(b*x+a)/(-a*d+b*c)^4-(A*b^3 
-a*(B*b^2-C*a*b+D*a^2))*ln(d*x+c)/(-a*d+b*c)^4
 

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.01 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x) (c+d x)^4} \, dx=\frac {-c^2 C d+B c d^2-A d^3+c^3 D}{3 d^3 (-b c+a d) (c+d x)^3}+\frac {-b c^2 C d+2 a c C d^2+A b d^3-a B d^3+2 b c^3 D-3 a c^2 d D}{2 d^3 (-b c+a d)^2 (c+d x)^2}+\frac {-A b^2 d^3+a b B d^3-a^2 C d^3+b^2 c^3 D-3 a b c^2 d D+3 a^2 c d^2 D}{d^3 (-b c+a d)^3 (c+d x)}+\frac {\left (A b^3-a b^2 B+a^2 b C-a^3 D\right ) \log (a+b x)}{(b c-a d)^4}+\frac {\left (-A b^3+a b^2 B-a^2 b C+a^3 D\right ) \log (c+d x)}{(b c-a d)^4} \] Input:

Integrate[(A + B*x + C*x^2 + D*x^3)/((a + b*x)*(c + d*x)^4),x]
 

Output:

(-(c^2*C*d) + B*c*d^2 - A*d^3 + c^3*D)/(3*d^3*(-(b*c) + a*d)*(c + d*x)^3) 
+ (-(b*c^2*C*d) + 2*a*c*C*d^2 + A*b*d^3 - a*B*d^3 + 2*b*c^3*D - 3*a*c^2*d* 
D)/(2*d^3*(-(b*c) + a*d)^2*(c + d*x)^2) + (-(A*b^2*d^3) + a*b*B*d^3 - a^2* 
C*d^3 + b^2*c^3*D - 3*a*b*c^2*d*D + 3*a^2*c*d^2*D)/(d^3*(-(b*c) + a*d)^3*( 
c + d*x)) + ((A*b^3 - a*b^2*B + a^2*b*C - a^3*D)*Log[a + b*x])/(b*c - a*d) 
^4 + ((-(A*b^3) + a*b^2*B - a^2*b*C + a^3*D)*Log[c + d*x])/(b*c - a*d)^4
 

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2123, 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.

\(\displaystyle \int \frac {A+B x+C x^2+D x^3}{(a+b x) (c+d x)^4} \, dx\)

\(\Big \downarrow \) 2123

\(\displaystyle \int \left (\frac {-a^2 d^2 (C d-3 c D)+a b \left (B d^3-3 c^2 d D\right )-A b^2 d^3+b^2 c^3 D}{d^2 (c+d x)^2 (b c-a d)^3}+\frac {b \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{(a+b x) (b c-a d)^4}+\frac {d \left (a \left (a^2 D-a b C+b^2 B\right )-A b^3\right )}{(c+d x) (b c-a d)^4}+\frac {A d^3-B c d^2+c^3 (-D)+c^2 C d}{d^2 (c+d x)^4 (a d-b c)}+\frac {b \left (-A d^3-2 c^3 D+c^2 C d\right )-a d \left (-B d^2-3 c^2 D+2 c C d\right )}{d^2 (c+d x)^3 (b c-a d)^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {a^2 d^2 (C d-3 c D)-a b \left (B d^3-3 c^2 d D\right )+A b^2 d^3-b^2 c^3 D}{d^3 (c+d x) (b c-a d)^3}+\frac {\log (a+b x) \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{(b c-a d)^4}-\frac {\log (c+d x) \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{(b c-a d)^4}+\frac {A d^3-B c d^2+c^3 (-D)+c^2 C d}{3 d^3 (c+d x)^3 (b c-a d)}+\frac {a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (-A d^3-2 c^3 D+c^2 C d\right )}{2 d^3 (c+d x)^2 (b c-a d)^2}\)

Input:

Int[(A + B*x + C*x^2 + D*x^3)/((a + b*x)*(c + d*x)^4),x]
 

Output:

(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)/(3*d^3*(b*c - a*d)*(c + d*x)^3) + (a*d 
*(2*c*C*d - B*d^2 - 3*c^2*D) - b*(c^2*C*d - A*d^3 - 2*c^3*D))/(2*d^3*(b*c 
- a*d)^2*(c + d*x)^2) + (A*b^2*d^3 - b^2*c^3*D + a^2*d^2*(C*d - 3*c*D) - a 
*b*(B*d^3 - 3*c^2*d*D))/(d^3*(b*c - a*d)^3*(c + d*x)) + ((A*b^3 - a*(b^2*B 
 - a*b*C + a^2*D))*Log[a + b*x])/(b*c - a*d)^4 - ((A*b^3 - a*(b^2*B - a*b* 
C + a^2*D))*Log[c + d*x])/(b*c - a*d)^4
 

Defintions of rubi rules used

rule 2009

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

rule 2123

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] 
:> Int[ExpandIntegrand[Px*(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c 
, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2])
 

Maple [A] (verified)

Time = 0.50 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.00

method	result	size
default	\(-\frac {A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}}{3 d^{3} \left (a d -b c \right ) \left (x d +c \right )^{3}}-\frac {-A b \,d^{3}+B a \,d^{3}-2 C a c \,d^{2}+C b \,c^{2} d +3 D a \,c^{2} d -2 D b \,c^{3}}{2 d^{3} \left (a d -b c \right )^{2} \left (x d +c \right )^{2}}-\frac {A \,b^{2} d^{3}-a b B \,d^{3}+C \,a^{2} d^{3}-3 D a^{2} c \,d^{2}+3 D a b \,c^{2} d -b^{2} c^{3} D}{\left (a d -b c \right )^{3} d^{3} \left (x d +c \right )}-\frac {\left (b^{3} A -a \,b^{2} B +a^{2} b C -a^{3} D\right ) \ln \left (x d +c \right )}{\left (a d -b c \right )^{4}}+\frac {\left (b^{3} A -a \,b^{2} B +a^{2} b C -a^{3} D\right ) \ln \left (b x +a \right )}{\left (a d -b c \right )^{4}}\)	\(275\)
norman	\(\frac {-\frac {2 A \,a^{2} d^{5}-7 A a b c \,d^{4}+11 A \,b^{2} c^{2} d^{3}+B \,a^{2} c \,d^{4}-5 B a b \,c^{2} d^{3}-2 B \,b^{2} c^{3} d^{2}+2 C \,a^{2} c^{2} d^{3}+5 C a b \,c^{3} d^{2}-C \,b^{2} c^{4} d -11 D a^{2} c^{3} d^{2}+7 D a b \,c^{4} d -2 D b^{2} c^{5}}{6 d^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {\left (A \,b^{2} d^{3}-a b B \,d^{3}+C \,a^{2} d^{3}-3 D a^{2} c \,d^{2}+3 D a b \,c^{2} d -b^{2} c^{3} D\right ) x^{2}}{d \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {\left (A a b \,d^{4}-5 A \,b^{2} c \,d^{3}-B \,a^{2} d^{4}+5 B a b c \,d^{3}-2 C \,a^{2} c \,d^{3}-3 C a b \,c^{2} d^{2}+C \,b^{2} c^{3} d +9 D a^{2} c^{2} d^{2}-7 D a b \,c^{3} d +2 D b^{2} c^{4}\right ) x}{2 d^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}}{\left (x d +c \right )^{3}}+\frac {\left (b^{3} A -a \,b^{2} B +a^{2} b C -a^{3} D\right ) \ln \left (b x +a \right )}{d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}}-\frac {\left (b^{3} A -a \,b^{2} B +a^{2} b C -a^{3} D\right ) \ln \left (x d +c \right )}{d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}}\)	\(586\)
parallelrisch	\(\text {Expression too large to display}\)	\(1174\)

Input:

int((D*x^3+C*x^2+B*x+A)/(b*x+a)/(d*x+c)^4,x,method=_RETURNVERBOSE)
 

Output:

-1/3/d^3*(A*d^3-B*c*d^2+C*c^2*d-D*c^3)/(a*d-b*c)/(d*x+c)^3-1/2*(-A*b*d^3+B 
*a*d^3-2*C*a*c*d^2+C*b*c^2*d+3*D*a*c^2*d-2*D*b*c^3)/d^3/(a*d-b*c)^2/(d*x+c 
)^2-(A*b^2*d^3-B*a*b*d^3+C*a^2*d^3-3*D*a^2*c*d^2+3*D*a*b*c^2*d-D*b^2*c^3)/ 
(a*d-b*c)^3/d^3/(d*x+c)-(A*b^3-B*a*b^2+C*a^2*b-D*a^3)/(a*d-b*c)^4*ln(d*x+c 
)+(A*b^3-B*a*b^2+C*a^2*b-D*a^3)/(a*d-b*c)^4*ln(b*x+a)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 909 vs. \(2 (272) = 544\).

Time = 0.10 (sec) , antiderivative size = 909, normalized size of antiderivative = 3.32 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x) (c+d x)^4} \, dx =\text {Too large to display} \] Input:

integrate((D*x^3+C*x^2+B*x+A)/(b*x+a)/(d*x+c)^4,x, algorithm="fricas")
 

Output:

-1/6*(2*D*b^3*c^6 + 2*A*a^3*d^6 - (9*D*a*b^2 - C*b^3)*c^5*d + 2*(9*D*a^2*b 
 - 3*C*a*b^2 + B*b^3)*c^4*d^2 - (11*D*a^3 - 3*C*a^2*b - 3*B*a*b^2 + 11*A*b 
^3)*c^3*d^3 + 2*(C*a^3 - 3*B*a^2*b + 9*A*a*b^2)*c^2*d^4 + (B*a^3 - 9*A*a^2 
*b)*c*d^5 + 6*(D*b^3*c^4*d^2 - 4*D*a*b^2*c^3*d^3 + 6*D*a^2*b*c^2*d^4 - (3* 
D*a^3 + C*a^2*b - B*a*b^2 + A*b^3)*c*d^5 + (C*a^3 - B*a^2*b + A*a*b^2)*d^6 
)*x^2 + 3*(2*D*b^3*c^5*d - (9*D*a*b^2 - C*b^3)*c^4*d^2 + 4*(4*D*a^2*b - C* 
a*b^2)*c^3*d^3 - (9*D*a^3 - C*a^2*b - 5*B*a*b^2 + 5*A*b^3)*c^2*d^4 + 2*(C* 
a^3 - 3*B*a^2*b + 3*A*a*b^2)*c*d^5 + (B*a^3 - A*a^2*b)*d^6)*x + 6*((D*a^3 
- C*a^2*b + B*a*b^2 - A*b^3)*d^6*x^3 + 3*(D*a^3 - C*a^2*b + B*a*b^2 - A*b^ 
3)*c*d^5*x^2 + 3*(D*a^3 - C*a^2*b + B*a*b^2 - A*b^3)*c^2*d^4*x + (D*a^3 - 
C*a^2*b + B*a*b^2 - A*b^3)*c^3*d^3)*log(b*x + a) - 6*((D*a^3 - C*a^2*b + B 
*a*b^2 - A*b^3)*d^6*x^3 + 3*(D*a^3 - C*a^2*b + B*a*b^2 - A*b^3)*c*d^5*x^2 
+ 3*(D*a^3 - C*a^2*b + B*a*b^2 - A*b^3)*c^2*d^4*x + (D*a^3 - C*a^2*b + B*a 
*b^2 - A*b^3)*c^3*d^3)*log(d*x + c))/(b^4*c^7*d^3 - 4*a*b^3*c^6*d^4 + 6*a^ 
2*b^2*c^5*d^5 - 4*a^3*b*c^4*d^6 + a^4*c^3*d^7 + (b^4*c^4*d^6 - 4*a*b^3*c^3 
*d^7 + 6*a^2*b^2*c^2*d^8 - 4*a^3*b*c*d^9 + a^4*d^10)*x^3 + 3*(b^4*c^5*d^5 
- 4*a*b^3*c^4*d^6 + 6*a^2*b^2*c^3*d^7 - 4*a^3*b*c^2*d^8 + a^4*c*d^9)*x^2 + 
 3*(b^4*c^6*d^4 - 4*a*b^3*c^5*d^5 + 6*a^2*b^2*c^4*d^6 - 4*a^3*b*c^3*d^7 + 
a^4*c^2*d^8)*x)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1360 vs. \(2 (252) = 504\).

Time = 28.12 (sec) , antiderivative size = 1360, normalized size of antiderivative = 4.96 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x) (c+d x)^4} \, dx=\text {Too large to display} \] Input:

integrate((D*x**3+C*x**2+B*x+A)/(b*x+a)/(d*x+c)**4,x)
 

Output:

(-2*A*a**2*d**5 + 7*A*a*b*c*d**4 - 11*A*b**2*c**2*d**3 - B*a**2*c*d**4 + 5 
*B*a*b*c**2*d**3 + 2*B*b**2*c**3*d**2 - 2*C*a**2*c**2*d**3 - 5*C*a*b*c**3* 
d**2 + C*b**2*c**4*d + 11*D*a**2*c**3*d**2 - 7*D*a*b*c**4*d + 2*D*b**2*c** 
5 + x**2*(-6*A*b**2*d**5 + 6*B*a*b*d**5 - 6*C*a**2*d**5 + 18*D*a**2*c*d**4 
 - 18*D*a*b*c**2*d**3 + 6*D*b**2*c**3*d**2) + x*(3*A*a*b*d**5 - 15*A*b**2* 
c*d**4 - 3*B*a**2*d**5 + 15*B*a*b*c*d**4 - 6*C*a**2*c*d**4 - 9*C*a*b*c**2* 
d**3 + 3*C*b**2*c**3*d**2 + 27*D*a**2*c**2*d**3 - 21*D*a*b*c**3*d**2 + 6*D 
*b**2*c**4*d))/(6*a**3*c**3*d**6 - 18*a**2*b*c**4*d**5 + 18*a*b**2*c**5*d* 
*4 - 6*b**3*c**6*d**3 + x**3*(6*a**3*d**9 - 18*a**2*b*c*d**8 + 18*a*b**2*c 
**2*d**7 - 6*b**3*c**3*d**6) + x**2*(18*a**3*c*d**8 - 54*a**2*b*c**2*d**7 
+ 54*a*b**2*c**3*d**6 - 18*b**3*c**4*d**5) + x*(18*a**3*c**2*d**7 - 54*a** 
2*b*c**3*d**6 + 54*a*b**2*c**4*d**5 - 18*b**3*c**5*d**4)) + (-A*b**3 + B*a 
*b**2 - C*a**2*b + D*a**3)*log(x + (-A*a*b**3*d - A*b**4*c + B*a**2*b**2*d 
 + B*a*b**3*c - C*a**3*b*d - C*a**2*b**2*c + D*a**4*d + D*a**3*b*c - a**5* 
d**5*(-A*b**3 + B*a*b**2 - C*a**2*b + D*a**3)/(a*d - b*c)**4 + 5*a**4*b*c* 
d**4*(-A*b**3 + B*a*b**2 - C*a**2*b + D*a**3)/(a*d - b*c)**4 - 10*a**3*b** 
2*c**2*d**3*(-A*b**3 + B*a*b**2 - C*a**2*b + D*a**3)/(a*d - b*c)**4 + 10*a 
**2*b**3*c**3*d**2*(-A*b**3 + B*a*b**2 - C*a**2*b + D*a**3)/(a*d - b*c)**4 
 - 5*a*b**4*c**4*d*(-A*b**3 + B*a*b**2 - C*a**2*b + D*a**3)/(a*d - b*c)**4 
 + b**5*c**5*(-A*b**3 + B*a*b**2 - C*a**2*b + D*a**3)/(a*d - b*c)**4)/(...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 619 vs. \(2 (272) = 544\).

Time = 0.07 (sec) , antiderivative size = 619, normalized size of antiderivative = 2.26 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x) (c+d x)^4} \, dx=-\frac {{\left (D a^{3} - C a^{2} b + B a b^{2} - A b^{3}\right )} \log \left (b x + a\right )}{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}} + \frac {{\left (D a^{3} - C a^{2} b + B a b^{2} - A b^{3}\right )} \log \left (d x + c\right )}{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}} - \frac {2 \, D b^{2} c^{5} - 2 \, A a^{2} d^{5} - {\left (7 \, D a b - C b^{2}\right )} c^{4} d + {\left (11 \, D a^{2} - 5 \, C a b + 2 \, B b^{2}\right )} c^{3} d^{2} - {\left (2 \, C a^{2} - 5 \, B a b + 11 \, A b^{2}\right )} c^{2} d^{3} - {\left (B a^{2} - 7 \, A a b\right )} c d^{4} + 6 \, {\left (D b^{2} c^{3} d^{2} - 3 \, D a b c^{2} d^{3} + 3 \, D a^{2} c d^{4} - {\left (C a^{2} - B a b + A b^{2}\right )} d^{5}\right )} x^{2} + 3 \, {\left (2 \, D b^{2} c^{4} d - {\left (7 \, D a b - C b^{2}\right )} c^{3} d^{2} + 3 \, {\left (3 \, D a^{2} - C a b\right )} c^{2} d^{3} - {\left (2 \, C a^{2} - 5 \, B a b + 5 \, A b^{2}\right )} c d^{4} - {\left (B a^{2} - A a b\right )} d^{5}\right )} x}{6 \, {\left (b^{3} c^{6} d^{3} - 3 \, a b^{2} c^{5} d^{4} + 3 \, a^{2} b c^{4} d^{5} - a^{3} c^{3} d^{6} + {\left (b^{3} c^{3} d^{6} - 3 \, a b^{2} c^{2} d^{7} + 3 \, a^{2} b c d^{8} - a^{3} d^{9}\right )} x^{3} + 3 \, {\left (b^{3} c^{4} d^{5} - 3 \, a b^{2} c^{3} d^{6} + 3 \, a^{2} b c^{2} d^{7} - a^{3} c d^{8}\right )} x^{2} + 3 \, {\left (b^{3} c^{5} d^{4} - 3 \, a b^{2} c^{4} d^{5} + 3 \, a^{2} b c^{3} d^{6} - a^{3} c^{2} d^{7}\right )} x\right )}} \] Input:

integrate((D*x^3+C*x^2+B*x+A)/(b*x+a)/(d*x+c)^4,x, algorithm="maxima")
 

Output:

-(D*a^3 - C*a^2*b + B*a*b^2 - A*b^3)*log(b*x + a)/(b^4*c^4 - 4*a*b^3*c^3*d 
 + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4) + (D*a^3 - C*a^2*b + B*a*b 
^2 - A*b^3)*log(d*x + c)/(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4* 
a^3*b*c*d^3 + a^4*d^4) - 1/6*(2*D*b^2*c^5 - 2*A*a^2*d^5 - (7*D*a*b - C*b^2 
)*c^4*d + (11*D*a^2 - 5*C*a*b + 2*B*b^2)*c^3*d^2 - (2*C*a^2 - 5*B*a*b + 11 
*A*b^2)*c^2*d^3 - (B*a^2 - 7*A*a*b)*c*d^4 + 6*(D*b^2*c^3*d^2 - 3*D*a*b*c^2 
*d^3 + 3*D*a^2*c*d^4 - (C*a^2 - B*a*b + A*b^2)*d^5)*x^2 + 3*(2*D*b^2*c^4*d 
 - (7*D*a*b - C*b^2)*c^3*d^2 + 3*(3*D*a^2 - C*a*b)*c^2*d^3 - (2*C*a^2 - 5* 
B*a*b + 5*A*b^2)*c*d^4 - (B*a^2 - A*a*b)*d^5)*x)/(b^3*c^6*d^3 - 3*a*b^2*c^ 
5*d^4 + 3*a^2*b*c^4*d^5 - a^3*c^3*d^6 + (b^3*c^3*d^6 - 3*a*b^2*c^2*d^7 + 3 
*a^2*b*c*d^8 - a^3*d^9)*x^3 + 3*(b^3*c^4*d^5 - 3*a*b^2*c^3*d^6 + 3*a^2*b*c 
^2*d^7 - a^3*c*d^8)*x^2 + 3*(b^3*c^5*d^4 - 3*a*b^2*c^4*d^5 + 3*a^2*b*c^3*d 
^6 - a^3*c^2*d^7)*x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 664 vs. \(2 (272) = 544\).

Time = 0.13 (sec) , antiderivative size = 664, normalized size of antiderivative = 2.42 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x) (c+d x)^4} \, dx=-\frac {{\left (D a^{3} b - C a^{2} b^{2} + B a b^{3} - A b^{4}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}} + \frac {{\left (D a^{3} d - C a^{2} b d + B a b^{2} d - A b^{3} d\right )} \log \left ({\left | d x + c \right |}\right )}{b^{4} c^{4} d - 4 \, a b^{3} c^{3} d^{2} + 6 \, a^{2} b^{2} c^{2} d^{3} - 4 \, a^{3} b c d^{4} + a^{4} d^{5}} - \frac {2 \, D b^{3} c^{6} - 9 \, D a b^{2} c^{5} d + C b^{3} c^{5} d + 18 \, D a^{2} b c^{4} d^{2} - 6 \, C a b^{2} c^{4} d^{2} + 2 \, B b^{3} c^{4} d^{2} - 11 \, D a^{3} c^{3} d^{3} + 3 \, C a^{2} b c^{3} d^{3} + 3 \, B a b^{2} c^{3} d^{3} - 11 \, A b^{3} c^{3} d^{3} + 2 \, C a^{3} c^{2} d^{4} - 6 \, B a^{2} b c^{2} d^{4} + 18 \, A a b^{2} c^{2} d^{4} + B a^{3} c d^{5} - 9 \, A a^{2} b c d^{5} + 2 \, A a^{3} d^{6} + 6 \, {\left (D b^{3} c^{4} d^{2} - 4 \, D a b^{2} c^{3} d^{3} + 6 \, D a^{2} b c^{2} d^{4} - 3 \, D a^{3} c d^{5} - C a^{2} b c d^{5} + B a b^{2} c d^{5} - A b^{3} c d^{5} + C a^{3} d^{6} - B a^{2} b d^{6} + A a b^{2} d^{6}\right )} x^{2} + 3 \, {\left (2 \, D b^{3} c^{5} d - 9 \, D a b^{2} c^{4} d^{2} + C b^{3} c^{4} d^{2} + 16 \, D a^{2} b c^{3} d^{3} - 4 \, C a b^{2} c^{3} d^{3} - 9 \, D a^{3} c^{2} d^{4} + C a^{2} b c^{2} d^{4} + 5 \, B a b^{2} c^{2} d^{4} - 5 \, A b^{3} c^{2} d^{4} + 2 \, C a^{3} c d^{5} - 6 \, B a^{2} b c d^{5} + 6 \, A a b^{2} c d^{5} + B a^{3} d^{6} - A a^{2} b d^{6}\right )} x}{6 \, {\left (b c - a d\right )}^{4} {\left (d x + c\right )}^{3} d^{3}} \] Input:

integrate((D*x^3+C*x^2+B*x+A)/(b*x+a)/(d*x+c)^4,x, algorithm="giac")
 

Output:

-(D*a^3*b - C*a^2*b^2 + B*a*b^3 - A*b^4)*log(abs(b*x + a))/(b^5*c^4 - 4*a* 
b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4) + (D*a^3*d - 
C*a^2*b*d + B*a*b^2*d - A*b^3*d)*log(abs(d*x + c))/(b^4*c^4*d - 4*a*b^3*c^ 
3*d^2 + 6*a^2*b^2*c^2*d^3 - 4*a^3*b*c*d^4 + a^4*d^5) - 1/6*(2*D*b^3*c^6 - 
9*D*a*b^2*c^5*d + C*b^3*c^5*d + 18*D*a^2*b*c^4*d^2 - 6*C*a*b^2*c^4*d^2 + 2 
*B*b^3*c^4*d^2 - 11*D*a^3*c^3*d^3 + 3*C*a^2*b*c^3*d^3 + 3*B*a*b^2*c^3*d^3 
- 11*A*b^3*c^3*d^3 + 2*C*a^3*c^2*d^4 - 6*B*a^2*b*c^2*d^4 + 18*A*a*b^2*c^2* 
d^4 + B*a^3*c*d^5 - 9*A*a^2*b*c*d^5 + 2*A*a^3*d^6 + 6*(D*b^3*c^4*d^2 - 4*D 
*a*b^2*c^3*d^3 + 6*D*a^2*b*c^2*d^4 - 3*D*a^3*c*d^5 - C*a^2*b*c*d^5 + B*a*b 
^2*c*d^5 - A*b^3*c*d^5 + C*a^3*d^6 - B*a^2*b*d^6 + A*a*b^2*d^6)*x^2 + 3*(2 
*D*b^3*c^5*d - 9*D*a*b^2*c^4*d^2 + C*b^3*c^4*d^2 + 16*D*a^2*b*c^3*d^3 - 4* 
C*a*b^2*c^3*d^3 - 9*D*a^3*c^2*d^4 + C*a^2*b*c^2*d^4 + 5*B*a*b^2*c^2*d^4 - 
5*A*b^3*c^2*d^4 + 2*C*a^3*c*d^5 - 6*B*a^2*b*c*d^5 + 6*A*a*b^2*c*d^5 + B*a^ 
3*d^6 - A*a^2*b*d^6)*x)/((b*c - a*d)^4*(d*x + c)^3*d^3)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x) (c+d x)^4} \, dx=\int \frac {A+B\,x+C\,x^2+x^3\,D}{\left (a+b\,x\right )\,{\left (c+d\,x\right )}^4} \,d x \] Input:

int((A + B*x + C*x^2 + x^3*D)/((a + b*x)*(c + d*x)^4),x)
 

Output:

int((A + B*x + C*x^2 + x^3*D)/((a + b*x)*(c + d*x)^4), x)
 

Reduce [B] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 479, normalized size of antiderivative = 1.75 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x) (c+d x)^4} \, dx=\frac {-6 \,\mathrm {log}\left (b x +a \right ) a^{2} c^{3} d^{2}-18 \,\mathrm {log}\left (b x +a \right ) a^{2} c^{2} d^{3} x -18 \,\mathrm {log}\left (b x +a \right ) a^{2} c \,d^{4} x^{2}-6 \,\mathrm {log}\left (b x +a \right ) a^{2} d^{5} x^{3}+6 \,\mathrm {log}\left (d x +c \right ) a^{2} c^{3} d^{2}+18 \,\mathrm {log}\left (d x +c \right ) a^{2} c^{2} d^{3} x +18 \,\mathrm {log}\left (d x +c \right ) a^{2} c \,d^{4} x^{2}+6 \,\mathrm {log}\left (d x +c \right ) a^{2} d^{5} x^{3}-2 a^{3} d^{4}+6 a^{2} b c \,d^{3}+5 a^{2} c^{3} d^{2}+9 a^{2} c^{2} d^{3} x -4 a^{2} d^{5} x^{3}-6 a \,b^{2} c^{2} d^{2}-6 a b \,c^{4} d -12 a b \,c^{3} d^{2} x +6 a b c \,d^{4} x^{3}+2 b^{3} c^{3} d +b^{2} c^{5}+3 b^{2} c^{4} d x -2 b^{2} c^{2} d^{3} x^{3}}{6 d^{2} \left (a^{3} d^{6} x^{3}-3 a^{2} b c \,d^{5} x^{3}+3 a \,b^{2} c^{2} d^{4} x^{3}-b^{3} c^{3} d^{3} x^{3}+3 a^{3} c \,d^{5} x^{2}-9 a^{2} b \,c^{2} d^{4} x^{2}+9 a \,b^{2} c^{3} d^{3} x^{2}-3 b^{3} c^{4} d^{2} x^{2}+3 a^{3} c^{2} d^{4} x -9 a^{2} b \,c^{3} d^{3} x +9 a \,b^{2} c^{4} d^{2} x -3 b^{3} c^{5} d x +a^{3} c^{3} d^{3}-3 a^{2} b \,c^{4} d^{2}+3 a \,b^{2} c^{5} d -b^{3} c^{6}\right )} \] Input:

int((D*x^3+C*x^2+B*x+A)/(b*x+a)/(d*x+c)^4,x)
 

Output:

( - 6*log(a + b*x)*a**2*c**3*d**2 - 18*log(a + b*x)*a**2*c**2*d**3*x - 18* 
log(a + b*x)*a**2*c*d**4*x**2 - 6*log(a + b*x)*a**2*d**5*x**3 + 6*log(c + 
d*x)*a**2*c**3*d**2 + 18*log(c + d*x)*a**2*c**2*d**3*x + 18*log(c + d*x)*a 
**2*c*d**4*x**2 + 6*log(c + d*x)*a**2*d**5*x**3 - 2*a**3*d**4 + 6*a**2*b*c 
*d**3 + 5*a**2*c**3*d**2 + 9*a**2*c**2*d**3*x - 4*a**2*d**5*x**3 - 6*a*b** 
2*c**2*d**2 - 6*a*b*c**4*d - 12*a*b*c**3*d**2*x + 6*a*b*c*d**4*x**3 + 2*b* 
*3*c**3*d + b**2*c**5 + 3*b**2*c**4*d*x - 2*b**2*c**2*d**3*x**3)/(6*d**2*( 
a**3*c**3*d**3 + 3*a**3*c**2*d**4*x + 3*a**3*c*d**5*x**2 + a**3*d**6*x**3 
- 3*a**2*b*c**4*d**2 - 9*a**2*b*c**3*d**3*x - 9*a**2*b*c**2*d**4*x**2 - 3* 
a**2*b*c*d**5*x**3 + 3*a*b**2*c**5*d + 9*a*b**2*c**4*d**2*x + 9*a*b**2*c** 
3*d**3*x**2 + 3*a*b**2*c**2*d**4*x**3 - b**3*c**6 - 3*b**3*c**5*d*x - 3*b* 
*3*c**4*d**2*x**2 - b**3*c**3*d**3*x**3))