\(\int \frac {A+B x+C x^2+D x^3}{(a+b x)^2 (c+d x)^4} \, dx\) [53]

Optimal result

Integrand size = 30, antiderivative size = 352 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^2 (c+d x)^4} \, dx=-\frac {A b^3-a \left (b^2 B-a b C+a^2 D\right )}{(b c-a d)^4 (a+b x)}-\frac {c^2 C d-B c d^2+A d^3-c^3 D}{3 d^2 (b c-a d)^2 (c+d x)^3}-\frac {a d \left (2 c C d-B d^2-3 c^2 D\right )-b \left (B c d^2-2 A d^3-c^3 D\right )}{2 d^2 (b c-a d)^3 (c+d x)^2}+\frac {b^2 (B c-3 A d)-a b (2 c C-2 B d)-a^2 (C d-3 c D)}{(b c-a d)^4 (c+d x)}+\frac {\left (b^3 (B c-4 A d)-a b^2 (2 c C-3 B d)+a^3 d D-a^2 b (2 C d-3 c D)\right ) \log (a+b x)}{(b c-a d)^5}-\frac {\left (b^3 (B c-4 A d)-a b^2 (2 c C-3 B d)+a^3 d D-a^2 b (2 C d-3 c D)\right ) \log (c+d x)}{(b c-a d)^5} \] Output:

-(A*b^3-a*(B*b^2-C*a*b+D*a^2))/(-a*d+b*c)^4/(b*x+a)-1/3*(A*d^3-B*c*d^2+C*c 
^2*d-D*c^3)/d^2/(-a*d+b*c)^2/(d*x+c)^3-1/2*(a*d*(-B*d^2+2*C*c*d-3*D*c^2)-b 
*(-2*A*d^3+B*c*d^2-D*c^3))/d^2/(-a*d+b*c)^3/(d*x+c)^2+(b^2*(-3*A*d+B*c)-a* 
b*(-2*B*d+2*C*c)-a^2*(C*d-3*D*c))/(-a*d+b*c)^4/(d*x+c)+(b^3*(-4*A*d+B*c)-a 
*b^2*(-3*B*d+2*C*c)+a^3*d*D-a^2*b*(2*C*d-3*D*c))*ln(b*x+a)/(-a*d+b*c)^5-(b 
^3*(-4*A*d+B*c)-a*b^2*(-3*B*d+2*C*c)+a^3*d*D-a^2*b*(2*C*d-3*D*c))*ln(d*x+c 
)/(-a*d+b*c)^5
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 345, normalized size of antiderivative = 0.98 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^2 (c+d x)^4} \, dx=\frac {-A b^3+a \left (b^2 B-a b C+a^2 D\right )}{(b c-a d)^4 (a+b x)}+\frac {-c^2 C d+B c d^2-A d^3+c^3 D}{3 d^2 (b c-a d)^2 (c+d x)^3}+\frac {-a d \left (-2 c C d+B d^2+3 c^2 D\right )+b \left (-B c d^2+2 A d^3+c^3 D\right )}{2 d^2 (-b c+a d)^3 (c+d x)^2}+\frac {b^2 (B c-3 A d)+a b (-2 c C+2 B d)+a^2 (-C d+3 c D)}{(b c-a d)^4 (c+d x)}+\frac {\left (b^3 (B c-4 A d)+a b^2 (-2 c C+3 B d)+a^3 d D+a^2 b (-2 C d+3 c D)\right ) \log (a+b x)}{(b c-a d)^5}-\frac {\left (b^3 (B c-4 A d)+a b^2 (-2 c C+3 B d)+a^3 d D+a^2 b (-2 C d+3 c D)\right ) \log (c+d x)}{(b c-a d)^5} \] Input:

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

Output:

(-(A*b^3) + a*(b^2*B - a*b*C + a^2*D))/((b*c - a*d)^4*(a + b*x)) + (-(c^2* 
C*d) + B*c*d^2 - A*d^3 + c^3*D)/(3*d^2*(b*c - a*d)^2*(c + d*x)^3) + (-(a*d 
*(-2*c*C*d + B*d^2 + 3*c^2*D)) + b*(-(B*c*d^2) + 2*A*d^3 + c^3*D))/(2*d^2* 
(-(b*c) + a*d)^3*(c + d*x)^2) + (b^2*(B*c - 3*A*d) + a*b*(-2*c*C + 2*B*d) 
+ a^2*(-(C*d) + 3*c*D))/((b*c - a*d)^4*(c + d*x)) + ((b^3*(B*c - 4*A*d) + 
a*b^2*(-2*c*C + 3*B*d) + a^3*d*D + a^2*b*(-2*C*d + 3*c*D))*Log[a + b*x])/( 
b*c - a*d)^5 - ((b^3*(B*c - 4*A*d) + a*b^2*(-2*c*C + 3*B*d) + a^3*d*D + a^ 
2*b*(-2*C*d + 3*c*D))*Log[c + d*x])/(b*c - a*d)^5
 

Rubi [A] (verified)

Time = 0.85 (sec) , antiderivative size = 352, 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.

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

\(\Big \downarrow \) 2123

\(\displaystyle \int \left (\frac {d \left (a^2 (C d-3 c D)+2 a b (c C-B d)-b^2 (B c-3 A d)\right )}{(c+d x)^2 (b c-a d)^4}+\frac {b \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{(a+b x)^2 (b c-a d)^4}+\frac {b \left (a^3 d D-a^2 b (2 C d-3 c D)-a b^2 (2 c C-3 B d)+b^3 (B c-4 A d)\right )}{(a+b x) (b c-a d)^5}+\frac {d \left (a^3 (-d) D+a^2 b (2 C d-3 c D)+a b^2 (2 c C-3 B d)-b^3 (B c-4 A d)\right )}{(c+d x) (b c-a d)^5}+\frac {A d^3-B c d^2+c^3 (-D)+c^2 C d}{d (c+d x)^4 (a d-b c)^2}+\frac {a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (-2 A d^3+B c d^2+c^3 (-D)\right )}{d (c+d x)^3 (b c-a d)^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-\left (a^2 (C d-3 c D)\right )-a b (2 c C-2 B d)+b^2 (B c-3 A d)}{(c+d x) (b c-a d)^4}-\frac {A b^3-a \left (a^2 D-a b C+b^2 B\right )}{(a+b x) (b c-a d)^4}+\frac {\log (a+b x) \left (a^3 d D-a^2 b (2 C d-3 c D)-a b^2 (2 c C-3 B d)+b^3 (B c-4 A d)\right )}{(b c-a d)^5}-\frac {\log (c+d x) \left (a^3 d D-a^2 b (2 C d-3 c D)-a b^2 (2 c C-3 B d)+b^3 (B c-4 A d)\right )}{(b c-a d)^5}-\frac {A d^3-B c d^2+c^3 (-D)+c^2 C d}{3 d^2 (c+d x)^3 (b c-a d)^2}-\frac {a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (-2 A d^3+B c d^2+c^3 (-D)\right )}{2 d^2 (c+d x)^2 (b c-a d)^3}\)

Input:

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

Output:

-((A*b^3 - a*(b^2*B - a*b*C + a^2*D))/((b*c - a*d)^4*(a + b*x))) - (c^2*C* 
d - B*c*d^2 + A*d^3 - c^3*D)/(3*d^2*(b*c - a*d)^2*(c + d*x)^3) - (a*d*(2*c 
*C*d - B*d^2 - 3*c^2*D) - b*(B*c*d^2 - 2*A*d^3 - c^3*D))/(2*d^2*(b*c - a*d 
)^3*(c + d*x)^2) + (b^2*(B*c - 3*A*d) - a*b*(2*c*C - 2*B*d) - a^2*(C*d - 3 
*c*D))/((b*c - a*d)^4*(c + d*x)) + ((b^3*(B*c - 4*A*d) - a*b^2*(2*c*C - 3* 
B*d) + a^3*d*D - a^2*b*(2*C*d - 3*c*D))*Log[a + b*x])/(b*c - a*d)^5 - ((b^ 
3*(B*c - 4*A*d) - a*b^2*(2*c*C - 3*B*d) + a^3*d*D - a^2*b*(2*C*d - 3*c*D)) 
*Log[c + d*x])/(b*c - a*d)^5
 

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 = 0.49 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.03

method	result	size
default	\(-\frac {3 d \,b^{2} A -2 B a b d -B \,b^{2} c +C \,a^{2} d +2 C a b c -3 a^{2} c D}{\left (a d -b c \right )^{4} \left (x d +c \right )}-\frac {A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}}{3 d^{2} \left (a d -b c \right )^{2} \left (x d +c \right )^{3}}-\frac {-2 A b \,d^{3}+B a \,d^{3}+B b c \,d^{2}-2 C a c \,d^{2}+3 D a \,c^{2} d -D b \,c^{3}}{2 d^{2} \left (a d -b c \right )^{3} \left (x d +c \right )^{2}}-\frac {\left (4 b^{3} d A -3 B a \,b^{2} d -B \,b^{3} c +2 C \,a^{2} b d +2 a \,b^{2} c C -a^{3} d D-3 a^{2} b c D\right ) \ln \left (x d +c \right )}{\left (a d -b c \right )^{5}}+\frac {\left (4 b^{3} d A -3 B a \,b^{2} d -B \,b^{3} c +2 C \,a^{2} b d +2 a \,b^{2} c C -a^{3} d D-3 a^{2} b c D\right ) \ln \left (b x +a \right )}{\left (a d -b c \right )^{5}}-\frac {b^{3} A -a \,b^{2} B +a^{2} b C -a^{3} D}{\left (a d -b c \right )^{4} \left (b x +a \right )}\)	\(361\)
norman	\(\text {Expression too large to display}\)	\(1084\)
parallelrisch	\(\text {Expression too large to display}\)	\(2939\)

Input:

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

Output:

-(3*A*b^2*d-2*B*a*b*d-B*b^2*c+C*a^2*d+2*C*a*b*c-3*D*a^2*c)/(a*d-b*c)^4/(d* 
x+c)-1/3*(A*d^3-B*c*d^2+C*c^2*d-D*c^3)/d^2/(a*d-b*c)^2/(d*x+c)^3-1/2*(-2*A 
*b*d^3+B*a*d^3+B*b*c*d^2-2*C*a*c*d^2+3*D*a*c^2*d-D*b*c^3)/d^2/(a*d-b*c)^3/ 
(d*x+c)^2-(4*A*b^3*d-3*B*a*b^2*d-B*b^3*c+2*C*a^2*b*d+2*C*a*b^2*c-D*a^3*d-3 
*D*a^2*b*c)/(a*d-b*c)^5*ln(d*x+c)+(4*A*b^3*d-3*B*a*b^2*d-B*b^3*c+2*C*a^2*b 
*d+2*C*a*b^2*c-D*a^3*d-3*D*a^2*b*c)/(a*d-b*c)^5*ln(b*x+a)-(A*b^3-B*a*b^2+C 
*a^2*b-D*a^3)/(a*d-b*c)^4/(b*x+a)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1929 vs. \(2 (344) = 688\).

Time = 0.15 (sec) , antiderivative size = 1929, normalized size of antiderivative = 5.48 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^2 (c+d x)^4} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/6*(D*a*b^3*c^6 - 2*A*a^4*d^6 - (9*D*a^2*b^2 - 2*C*a*b^3)*c^5*d - (9*D*a 
^3*b - 18*C*a^2*b^2 + 17*B*a*b^3 - 6*A*b^4)*c^4*d^2 + (17*D*a^4 - 18*C*a^3 
*b + 9*B*a^2*b^2 + 20*A*a*b^3)*c^3*d^3 - (2*C*a^4 - 9*B*a^3*b + 36*A*a^2*b 
^2)*c^2*d^4 - (B*a^4 - 12*A*a^3*b)*c*d^5 - 6*((3*D*a^2*b^2 - 2*C*a*b^3 + B 
*b^4)*c^2*d^4 - 2*(D*a^3*b - B*a*b^3 + 2*A*b^4)*c*d^5 - (D*a^4 - 2*C*a^3*b 
 + 3*B*a^2*b^2 - 4*A*a*b^3)*d^6)*x^3 + 3*(D*b^4*c^5*d - 5*D*a*b^3*c^4*d^2 
- 5*(D*a^2*b^2 - 2*C*a*b^3 + B*b^4)*c^3*d^3 - (3*D*a^3*b - 2*C*a^2*b^2 + 1 
1*B*a*b^3 - 20*A*b^4)*c^2*d^4 + (12*D*a^4 - 10*C*a^3*b + 13*B*a^2*b^2 - 16 
*A*a*b^3)*c*d^5 - (2*C*a^4 - 3*B*a^3*b + 4*A*a^2*b^2)*d^6)*x^2 + (D*b^4*c^ 
6 - 2*(3*D*a*b^3 - C*b^4)*c^5*d - (18*D*a^2*b^2 - 12*C*a*b^3 + 11*B*b^4)*c 
^4*d^2 - 2*(11*D*a^3*b - 18*C*a^2*b^2 + 15*B*a*b^3 - 22*A*b^4)*c^3*d^3 + ( 
45*D*a^4 - 44*C*a^3*b + 18*B*a^2*b^2 - 12*A*a*b^3)*c^2*d^4 - 2*(3*C*a^4 - 
13*B*a^3*b + 18*A*a^2*b^2)*c*d^5 - (3*B*a^4 - 4*A*a^3*b)*d^6)*x - 6*((3*D* 
a^3*b - 2*C*a^2*b^2 + B*a*b^3)*c^4*d^2 + (D*a^4 - 2*C*a^3*b + 3*B*a^2*b^2 
- 4*A*a*b^3)*c^3*d^3 + ((3*D*a^2*b^2 - 2*C*a*b^3 + B*b^4)*c*d^5 + (D*a^3*b 
 - 2*C*a^2*b^2 + 3*B*a*b^3 - 4*A*b^4)*d^6)*x^4 + (3*(3*D*a^2*b^2 - 2*C*a*b 
^3 + B*b^4)*c^2*d^4 + 2*(3*D*a^3*b - 4*C*a^2*b^2 + 5*B*a*b^3 - 6*A*b^4)*c* 
d^5 + (D*a^4 - 2*C*a^3*b + 3*B*a^2*b^2 - 4*A*a*b^3)*d^6)*x^3 + 3*((3*D*a^2 
*b^2 - 2*C*a*b^3 + B*b^4)*c^3*d^3 + 4*(D*a^3*b - C*a^2*b^2 + B*a*b^3 - A*b 
^4)*c^2*d^4 + (D*a^4 - 2*C*a^3*b + 3*B*a^2*b^2 - 4*A*a*b^3)*c*d^5)*x^2 ...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2739 vs. \(2 (328) = 656\).

Time = 120.68 (sec) , antiderivative size = 2739, normalized size of antiderivative = 7.78 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^2 (c+d x)^4} \, dx=\text {Too large to display} \] Input:

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

Output:

(-2*A*a**3*d**5 + 10*A*a**2*b*c*d**4 - 26*A*a*b**2*c**2*d**3 - 6*A*b**3*c* 
*3*d**2 - B*a**3*c*d**4 + 8*B*a**2*b*c**2*d**3 + 17*B*a*b**2*c**3*d**2 - 2 
*C*a**3*c**2*d**3 - 20*C*a**2*b*c**3*d**2 - 2*C*a*b**2*c**4*d + 17*D*a**3* 
c**3*d**2 + 8*D*a**2*b*c**4*d - D*a*b**2*c**5 + x**3*(-24*A*b**3*d**5 + 18 
*B*a*b**2*d**5 + 6*B*b**3*c*d**4 - 12*C*a**2*b*d**5 - 12*C*a*b**2*c*d**4 + 
 6*D*a**3*d**5 + 18*D*a**2*b*c*d**4) + x**2*(-12*A*a*b**2*d**5 - 60*A*b**3 
*c*d**4 + 9*B*a**2*b*d**5 + 48*B*a*b**2*c*d**4 + 15*B*b**3*c**2*d**3 - 6*C 
*a**3*d**5 - 36*C*a**2*b*c*d**4 - 30*C*a*b**2*c**2*d**3 + 36*D*a**3*c*d**4 
 + 27*D*a**2*b*c**2*d**3 + 12*D*a*b**2*c**3*d**2 - 3*D*b**3*c**4*d) + x*(4 
*A*a**2*b*d**5 - 32*A*a*b**2*c*d**4 - 44*A*b**3*c**2*d**3 - 3*B*a**3*d**5 
+ 23*B*a**2*b*c*d**4 + 41*B*a*b**2*c**2*d**3 + 11*B*b**3*c**3*d**2 - 6*C*a 
**3*c*d**4 - 50*C*a**2*b*c**2*d**3 - 14*C*a*b**2*c**3*d**2 - 2*C*b**3*c**4 
*d + 45*D*a**3*c**2*d**3 + 23*D*a**2*b*c**3*d**2 + 5*D*a*b**2*c**4*d - D*b 
**3*c**5))/(6*a**5*c**3*d**6 - 24*a**4*b*c**4*d**5 + 36*a**3*b**2*c**5*d** 
4 - 24*a**2*b**3*c**6*d**3 + 6*a*b**4*c**7*d**2 + x**4*(6*a**4*b*d**9 - 24 
*a**3*b**2*c*d**8 + 36*a**2*b**3*c**2*d**7 - 24*a*b**4*c**3*d**6 + 6*b**5* 
c**4*d**5) + x**3*(6*a**5*d**9 - 6*a**4*b*c*d**8 - 36*a**3*b**2*c**2*d**7 
+ 84*a**2*b**3*c**3*d**6 - 66*a*b**4*c**4*d**5 + 18*b**5*c**5*d**4) + x**2 
*(18*a**5*c*d**8 - 54*a**4*b*c**2*d**7 + 36*a**3*b**2*c**3*d**6 + 36*a**2* 
b**3*c**4*d**5 - 54*a*b**4*c**5*d**4 + 18*b**5*c**6*d**3) + x*(18*a**5*...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1038 vs. \(2 (344) = 688\).

Time = 0.09 (sec) , antiderivative size = 1038, normalized size of antiderivative = 2.95 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^2 (c+d x)^4} \, dx =\text {Too large to display} \] Input:

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

Output:

((3*D*a^2*b - 2*C*a*b^2 + B*b^3)*c + (D*a^3 - 2*C*a^2*b + 3*B*a*b^2 - 4*A* 
b^3)*d)*log(b*x + a)/(b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10*a^ 
3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5) - ((3*D*a^2*b - 2*C*a*b^2 + B*b^3 
)*c + (D*a^3 - 2*C*a^2*b + 3*B*a*b^2 - 4*A*b^3)*d)*log(d*x + c)/(b^5*c^5 - 
 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - 
 a^5*d^5) - 1/6*(D*a*b^2*c^5 + 2*A*a^3*d^5 - 2*(4*D*a^2*b - C*a*b^2)*c^4*d 
 - (17*D*a^3 - 20*C*a^2*b + 17*B*a*b^2 - 6*A*b^3)*c^3*d^2 + 2*(C*a^3 - 4*B 
*a^2*b + 13*A*a*b^2)*c^2*d^3 + (B*a^3 - 10*A*a^2*b)*c*d^4 - 6*((3*D*a^2*b 
- 2*C*a*b^2 + B*b^3)*c*d^4 + (D*a^3 - 2*C*a^2*b + 3*B*a*b^2 - 4*A*b^3)*d^5 
)*x^3 + 3*(D*b^3*c^4*d - 4*D*a*b^2*c^3*d^2 - (9*D*a^2*b - 10*C*a*b^2 + 5*B 
*b^3)*c^2*d^3 - 4*(3*D*a^3 - 3*C*a^2*b + 4*B*a*b^2 - 5*A*b^3)*c*d^4 + (2*C 
*a^3 - 3*B*a^2*b + 4*A*a*b^2)*d^5)*x^2 + (D*b^3*c^5 - (5*D*a*b^2 - 2*C*b^3 
)*c^4*d - (23*D*a^2*b - 14*C*a*b^2 + 11*B*b^3)*c^3*d^2 - (45*D*a^3 - 50*C* 
a^2*b + 41*B*a*b^2 - 44*A*b^3)*c^2*d^3 + (6*C*a^3 - 23*B*a^2*b + 32*A*a*b^ 
2)*c*d^4 + (3*B*a^3 - 4*A*a^2*b)*d^5)*x)/(a*b^4*c^7*d^2 - 4*a^2*b^3*c^6*d^ 
3 + 6*a^3*b^2*c^5*d^4 - 4*a^4*b*c^4*d^5 + a^5*c^3*d^6 + (b^5*c^4*d^5 - 4*a 
*b^4*c^3*d^6 + 6*a^2*b^3*c^2*d^7 - 4*a^3*b^2*c*d^8 + a^4*b*d^9)*x^4 + (3*b 
^5*c^5*d^4 - 11*a*b^4*c^4*d^5 + 14*a^2*b^3*c^3*d^6 - 6*a^3*b^2*c^2*d^7 - a 
^4*b*c*d^8 + a^5*d^9)*x^3 + 3*(b^5*c^6*d^3 - 3*a*b^4*c^5*d^4 + 2*a^2*b^3*c 
^4*d^5 + 2*a^3*b^2*c^3*d^6 - 3*a^4*b*c^2*d^7 + a^5*c*d^8)*x^2 + (b^5*c^...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 772 vs. \(2 (344) = 688\).

Time = 0.14 (sec) , antiderivative size = 772, normalized size of antiderivative = 2.19 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^2 (c+d x)^4} \, dx=-\frac {{\left (3 \, D a^{2} b^{2} c - 2 \, C a b^{3} c + B b^{4} c + D a^{3} b d - 2 \, C a^{2} b^{2} d + 3 \, B a b^{3} d - 4 \, A b^{4} d\right )} \log \left ({\left | \frac {b c}{b x + a} - \frac {a d}{b x + a} + d \right |}\right )}{b^{6} c^{5} - 5 \, a b^{5} c^{4} d + 10 \, a^{2} b^{4} c^{3} d^{2} - 10 \, a^{3} b^{3} c^{2} d^{3} + 5 \, a^{4} b^{2} c d^{4} - a^{5} b d^{5}} + \frac {\frac {D a^{3} b^{4}}{b x + a} - \frac {C a^{2} b^{5}}{b x + a} + \frac {B a b^{6}}{b x + a} - \frac {A b^{7}}{b x + a}}{b^{8} c^{4} - 4 \, a b^{7} c^{3} d + 6 \, a^{2} b^{6} c^{2} d^{2} - 4 \, a^{3} b^{5} c d^{3} + a^{4} b^{4} d^{4}} + \frac {D b^{3} c^{3} d - 9 \, D a b^{2} c^{2} d^{2} + 2 \, C b^{3} c^{2} d^{2} - 18 \, D a^{2} b c d^{3} + 18 \, C a b^{2} c d^{3} - 11 \, B b^{3} c d^{3} + 6 \, C a^{2} b d^{4} - 15 \, B a b^{2} d^{4} + 26 \, A b^{3} d^{4} + \frac {3 \, {\left (D b^{5} c^{4} - 10 \, D a b^{4} c^{3} d + 2 \, C b^{5} c^{3} d - 3 \, D a^{2} b^{3} c^{2} d^{2} + 12 \, C a b^{4} c^{2} d^{2} - 9 \, B b^{5} c^{2} d^{2} + 12 \, D a^{3} b^{2} c d^{3} - 10 \, C a^{2} b^{3} c d^{3} - 2 \, B a b^{4} c d^{3} + 20 \, A b^{5} c d^{3} - 4 \, C a^{3} b^{2} d^{4} + 11 \, B a^{2} b^{3} d^{4} - 20 \, A a b^{4} d^{4}\right )}}{{\left (b x + a\right )} b} - \frac {6 \, {\left (3 \, D a b^{6} c^{4} - C b^{7} c^{4} - 3 \, D a^{2} b^{5} c^{3} d - 2 \, C a b^{6} c^{3} d + 3 \, B b^{7} c^{3} d - 3 \, D a^{3} b^{4} c^{2} d^{2} + 6 \, C a^{2} b^{5} c^{2} d^{2} - 3 \, B a b^{6} c^{2} d^{2} - 6 \, A b^{7} c^{2} d^{2} + 3 \, D a^{4} b^{3} c d^{3} - 2 \, C a^{3} b^{4} c d^{3} - 3 \, B a^{2} b^{5} c d^{3} + 12 \, A a b^{6} c d^{3} - C a^{4} b^{3} d^{4} + 3 \, B a^{3} b^{4} d^{4} - 6 \, A a^{2} b^{5} d^{4}\right )}}{{\left (b x + a\right )}^{2} b^{2}}}{6 \, {\left (b c - a d\right )}^{5} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )}^{3}} \] Input:

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

Output:

-(3*D*a^2*b^2*c - 2*C*a*b^3*c + B*b^4*c + D*a^3*b*d - 2*C*a^2*b^2*d + 3*B* 
a*b^3*d - 4*A*b^4*d)*log(abs(b*c/(b*x + a) - a*d/(b*x + a) + d))/(b^6*c^5 
- 5*a*b^5*c^4*d + 10*a^2*b^4*c^3*d^2 - 10*a^3*b^3*c^2*d^3 + 5*a^4*b^2*c*d^ 
4 - a^5*b*d^5) + (D*a^3*b^4/(b*x + a) - C*a^2*b^5/(b*x + a) + B*a*b^6/(b*x 
 + a) - A*b^7/(b*x + a))/(b^8*c^4 - 4*a*b^7*c^3*d + 6*a^2*b^6*c^2*d^2 - 4* 
a^3*b^5*c*d^3 + a^4*b^4*d^4) + 1/6*(D*b^3*c^3*d - 9*D*a*b^2*c^2*d^2 + 2*C* 
b^3*c^2*d^2 - 18*D*a^2*b*c*d^3 + 18*C*a*b^2*c*d^3 - 11*B*b^3*c*d^3 + 6*C*a 
^2*b*d^4 - 15*B*a*b^2*d^4 + 26*A*b^3*d^4 + 3*(D*b^5*c^4 - 10*D*a*b^4*c^3*d 
 + 2*C*b^5*c^3*d - 3*D*a^2*b^3*c^2*d^2 + 12*C*a*b^4*c^2*d^2 - 9*B*b^5*c^2* 
d^2 + 12*D*a^3*b^2*c*d^3 - 10*C*a^2*b^3*c*d^3 - 2*B*a*b^4*c*d^3 + 20*A*b^5 
*c*d^3 - 4*C*a^3*b^2*d^4 + 11*B*a^2*b^3*d^4 - 20*A*a*b^4*d^4)/((b*x + a)*b 
) - 6*(3*D*a*b^6*c^4 - C*b^7*c^4 - 3*D*a^2*b^5*c^3*d - 2*C*a*b^6*c^3*d + 3 
*B*b^7*c^3*d - 3*D*a^3*b^4*c^2*d^2 + 6*C*a^2*b^5*c^2*d^2 - 3*B*a*b^6*c^2*d 
^2 - 6*A*b^7*c^2*d^2 + 3*D*a^4*b^3*c*d^3 - 2*C*a^3*b^4*c*d^3 - 3*B*a^2*b^5 
*c*d^3 + 12*A*a*b^6*c*d^3 - C*a^4*b^3*d^4 + 3*B*a^3*b^4*d^4 - 6*A*a^2*b^5* 
d^4)/((b*x + a)^2*b^2))/((b*c - a*d)^5*(b*c/(b*x + a) - a*d/(b*x + a) + d) 
^3)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 2067, normalized size of antiderivative = 5.87 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^2 (c+d x)^4} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 6*log(a + b*x)*a**4*c**3*d**3 - 18*log(a + b*x)*a**4*c**2*d**4*x - 18* 
log(a + b*x)*a**4*c*d**5*x**2 - 6*log(a + b*x)*a**4*d**6*x**3 - 30*log(a + 
 b*x)*a**3*b*c**4*d**2 - 96*log(a + b*x)*a**3*b*c**3*d**3*x - 108*log(a + 
b*x)*a**3*b*c**2*d**4*x**2 - 48*log(a + b*x)*a**3*b*c*d**5*x**3 - 6*log(a 
+ b*x)*a**3*b*d**6*x**4 + 6*log(a + b*x)*a**2*b**3*c**3*d**2 + 18*log(a + 
b*x)*a**2*b**3*c**2*d**3*x + 18*log(a + b*x)*a**2*b**3*c*d**4*x**2 + 6*log 
(a + b*x)*a**2*b**3*d**5*x**3 - 36*log(a + b*x)*a**2*b**2*c**5*d - 138*log 
(a + b*x)*a**2*b**2*c**4*d**2*x - 198*log(a + b*x)*a**2*b**2*c**3*d**3*x** 
2 - 126*log(a + b*x)*a**2*b**2*c**2*d**4*x**3 - 30*log(a + b*x)*a**2*b**2* 
c*d**5*x**4 + 18*log(a + b*x)*a*b**4*c**4*d + 60*log(a + b*x)*a*b**4*c**3* 
d**2*x + 72*log(a + b*x)*a*b**4*c**2*d**3*x**2 + 36*log(a + b*x)*a*b**4*c* 
d**4*x**3 + 6*log(a + b*x)*a*b**4*d**5*x**4 - 36*log(a + b*x)*a*b**3*c**5* 
d*x - 108*log(a + b*x)*a*b**3*c**4*d**2*x**2 - 108*log(a + b*x)*a*b**3*c** 
3*d**3*x**3 - 36*log(a + b*x)*a*b**3*c**2*d**4*x**4 + 18*log(a + b*x)*b**5 
*c**4*d*x + 54*log(a + b*x)*b**5*c**3*d**2*x**2 + 54*log(a + b*x)*b**5*c** 
2*d**3*x**3 + 18*log(a + b*x)*b**5*c*d**4*x**4 + 6*log(c + d*x)*a**4*c**3* 
d**3 + 18*log(c + d*x)*a**4*c**2*d**4*x + 18*log(c + d*x)*a**4*c*d**5*x**2 
 + 6*log(c + d*x)*a**4*d**6*x**3 + 30*log(c + d*x)*a**3*b*c**4*d**2 + 96*l 
og(c + d*x)*a**3*b*c**3*d**3*x + 108*log(c + d*x)*a**3*b*c**2*d**4*x**2 + 
48*log(c + d*x)*a**3*b*c*d**5*x**3 + 6*log(c + d*x)*a**3*b*d**6*x**4 - ...