\(\int \frac {A+B x+C x^2+D x^3}{(a+b x)^3 (c+d x)^3} \, dx\) [47]

Optimal result

Integrand size = 30, antiderivative size = 362 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^3 (c+d x)^3} \, dx=\frac {-A b^3+a \left (b^2 B-a b C+a^2 D\right )}{2 b (b c-a d)^3 (a+b x)^2}-\frac {b^2 (B c-3 A d)-a b (2 c C-2 B d)-a^2 (C d-3 c D)}{(b c-a d)^4 (a+b x)}+\frac {c^2 C d-B c d^2+A d^3-c^3 D}{2 d (b c-a d)^3 (c+d x)^2}+\frac {b \left (c^2 C-2 B c d+3 A d^2\right )+a \left (2 c C d-B d^2-3 c^2 D\right )}{(b c-a d)^4 (c+d x)}+\frac {\left (b^2 \left (c^2 C-3 B c d+6 A d^2\right )+a^2 d (C d-3 c D)+a b \left (4 c C d-3 B d^2-3 c^2 D\right )\right ) \log (a+b x)}{(b c-a d)^5}-\frac {\left (b^2 \left (c^2 C-3 B c d+6 A d^2\right )+a^2 d (C d-3 c D)+a b \left (4 c C d-3 B d^2-3 c^2 D\right )\right ) \log (c+d x)}{(b c-a d)^5} \] Output:

1/2*(-A*b^3+a*(B*b^2-C*a*b+D*a^2))/b/(-a*d+b*c)^3/(b*x+a)^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/(b*x+a)+1/2*(A*d^3-B* 
c*d^2+C*c^2*d-D*c^3)/d/(-a*d+b*c)^3/(d*x+c)^2+(b*(3*A*d^2-2*B*c*d+C*c^2)+a 
*(-B*d^2+2*C*c*d-3*D*c^2))/(-a*d+b*c)^4/(d*x+c)+(b^2*(6*A*d^2-3*B*c*d+C*c^ 
2)+a^2*d*(C*d-3*D*c)+a*b*(-3*B*d^2+4*C*c*d-3*D*c^2))*ln(b*x+a)/(-a*d+b*c)^ 
5-(b^2*(6*A*d^2-3*B*c*d+C*c^2)+a^2*d*(C*d-3*D*c)+a*b*(-3*B*d^2+4*C*c*d-3*D 
*c^2))*ln(d*x+c)/(-a*d+b*c)^5
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 360, normalized size of antiderivative = 0.99 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^3 (c+d x)^3} \, dx=\frac {-A b^3+a \left (b^2 B-a b C+a^2 D\right )}{2 b (b c-a d)^3 (a+b x)^2}+\frac {b^2 (-B c+3 A d)+2 a b (c C-B d)+a^2 (C d-3 c D)}{(b c-a d)^4 (a+b x)}+\frac {-c^2 C d+B c d^2-A d^3+c^3 D}{2 d (-b c+a d)^3 (c+d x)^2}+\frac {b \left (c^2 C-2 B c d+3 A d^2\right )-a \left (-2 c C d+B d^2+3 c^2 D\right )}{(b c-a d)^4 (c+d x)}+\frac {\left (b^2 \left (c^2 C-3 B c d+6 A d^2\right )+a^2 d (C d-3 c D)+a b \left (4 c C d-3 B d^2-3 c^2 D\right )\right ) \log (a+b x)}{(b c-a d)^5}-\frac {\left (b^2 \left (c^2 C-3 B c d+6 A d^2\right )+a^2 d (C d-3 c D)+a b \left (4 c C d-3 B d^2-3 c^2 D\right )\right ) \log (c+d x)}{(b c-a d)^5} \] Input:

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

Output:

(-(A*b^3) + a*(b^2*B - a*b*C + a^2*D))/(2*b*(b*c - a*d)^3*(a + b*x)^2) + ( 
b^2*(-(B*c) + 3*A*d) + 2*a*b*(c*C - B*d) + a^2*(C*d - 3*c*D))/((b*c - a*d) 
^4*(a + b*x)) + (-(c^2*C*d) + B*c*d^2 - A*d^3 + c^3*D)/(2*d*(-(b*c) + a*d) 
^3*(c + d*x)^2) + (b*(c^2*C - 2*B*c*d + 3*A*d^2) - a*(-2*c*C*d + B*d^2 + 3 
*c^2*D))/((b*c - a*d)^4*(c + d*x)) + ((b^2*(c^2*C - 3*B*c*d + 6*A*d^2) + a 
^2*d*(C*d - 3*c*D) + a*b*(4*c*C*d - 3*B*d^2 - 3*c^2*D))*Log[a + b*x])/(b*c 
 - a*d)^5 - ((b^2*(c^2*C - 3*B*c*d + 6*A*d^2) + a^2*d*(C*d - 3*c*D) + a*b* 
(4*c*C*d - 3*B*d^2 - 3*c^2*D))*Log[c + d*x])/(b*c - a*d)^5
 

Rubi [A] (verified)

Time = 0.87 (sec) , antiderivative size = 362, 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.

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

\(\Big \downarrow \) 2123

\(\displaystyle \int \left (\frac {b \left (a^2 d (C d-3 c D)+a b \left (-3 B d^2-3 c^2 D+4 c C d\right )+b^2 \left (6 A d^2-3 B c d+c^2 C\right )\right )}{(a+b x) (b c-a d)^5}+\frac {d \left (a^2 (-d) (C d-3 c D)-a b \left (-3 B d^2-3 c^2 D+4 c C d\right )-b^2 \left (6 A d^2-3 B c d+c^2 C\right )\right )}{(c+d x) (b c-a d)^5}+\frac {b \left (-\left (a^2 (C d-3 c D)\right )-a b (2 c C-2 B d)+b^2 (B c-3 A d)\right )}{(a+b x)^2 (b c-a d)^4}+\frac {A b^3-a \left (a^2 D-a b C+b^2 B\right )}{(a+b x)^3 (b c-a d)^3}+\frac {d \left (-a \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (3 A d^2-2 B c d+c^2 C\right )\right )}{(c+d x)^2 (b c-a d)^4}+\frac {-A d^3+B c d^2+c^3 D-c^2 C d}{(c+d x)^3 (b c-a d)^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\log (a+b x) \left (a^2 d (C d-3 c D)+a b \left (-3 B d^2-3 c^2 D+4 c C d\right )+b^2 \left (6 A d^2-3 B c d+c^2 C\right )\right )}{(b c-a d)^5}-\frac {\log (c+d x) \left (a^2 d (C d-3 c D)+a b \left (-3 B d^2-3 c^2 D+4 c C d\right )+b^2 \left (6 A d^2-3 B c d+c^2 C\right )\right )}{(b c-a d)^5}-\frac {-\left (a^2 (C d-3 c D)\right )-a b (2 c C-2 B d)+b^2 (B c-3 A d)}{(a+b x) (b c-a d)^4}-\frac {A b^3-a \left (a^2 D-a b C+b^2 B\right )}{2 b (a+b x)^2 (b c-a d)^3}+\frac {a \left (-B d^2-3 c^2 D+2 c C d\right )+b \left (3 A d^2-2 B c d+c^2 C\right )}{(c+d x) (b c-a d)^4}+\frac {A d^3-B c d^2+c^3 (-D)+c^2 C d}{2 d (c+d x)^2 (b c-a d)^3}\)

Input:

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

Output:

-1/2*(A*b^3 - a*(b^2*B - a*b*C + a^2*D))/(b*(b*c - a*d)^3*(a + b*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*(a + b*x)) + (c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)/(2*d*(b*c - a*d)^3*(c + 
 d*x)^2) + (b*(c^2*C - 2*B*c*d + 3*A*d^2) + a*(2*c*C*d - B*d^2 - 3*c^2*D)) 
/((b*c - a*d)^4*(c + d*x)) + ((b^2*(c^2*C - 3*B*c*d + 6*A*d^2) + a^2*d*(C* 
d - 3*c*D) + a*b*(4*c*C*d - 3*B*d^2 - 3*c^2*D))*Log[a + b*x])/(b*c - a*d)^ 
5 - ((b^2*(c^2*C - 3*B*c*d + 6*A*d^2) + a^2*d*(C*d - 3*c*D) + a*b*(4*c*C*d 
 - 3*B*d^2 - 3*c^2*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 = 0.50 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.04

method	result	size
default	\(-\frac {A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}}{2 \left (a d -b c \right )^{3} d \left (x d +c \right )^{2}}+\frac {\left (6 A \,b^{2} d^{2}-3 B a b \,d^{2}-3 B \,b^{2} c d +C \,a^{2} d^{2}+4 C a b c d +C \,b^{2} c^{2}-3 D a^{2} c d -3 D a b \,c^{2}\right ) \ln \left (x d +c \right )}{\left (a d -b c \right )^{5}}+\frac {3 b \,d^{2} A -B a \,d^{2}-2 B b c d +2 C a c d +C b \,c^{2}-3 a \,c^{2} D}{\left (a d -b c \right )^{4} \left (x d +c \right )}-\frac {-b^{3} A +a \,b^{2} B -a^{2} b C +a^{3} D}{2 \left (a d -b c \right )^{3} b \left (b x +a \right )^{2}}+\frac {3 d \,b^{2} A -2 B a b d -B \,b^{2} c +C \,a^{2} d +2 C a b c -3 a^{2} c D}{\left (a d -b c \right )^{4} \left (b x +a \right )}-\frac {\left (6 A \,b^{2} d^{2}-3 B a b \,d^{2}-3 B \,b^{2} c d +C \,a^{2} d^{2}+4 C a b c d +C \,b^{2} c^{2}-3 D a^{2} c d -3 D a b \,c^{2}\right ) \ln \left (b x +a \right )}{\left (a d -b c \right )^{5}}\)	\(377\)
norman	\(\text {Expression too large to display}\)	\(1128\)
parallelrisch	\(\text {Expression too large to display}\)	\(3272\)

Input:

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

Output:

-1/2*(A*d^3-B*c*d^2+C*c^2*d-D*c^3)/(a*d-b*c)^3/d/(d*x+c)^2+(6*A*b^2*d^2-3* 
B*a*b*d^2-3*B*b^2*c*d+C*a^2*d^2+4*C*a*b*c*d+C*b^2*c^2-3*D*a^2*c*d-3*D*a*b* 
c^2)/(a*d-b*c)^5*ln(d*x+c)+(3*A*b*d^2-B*a*d^2-2*B*b*c*d+2*C*a*c*d+C*b*c^2- 
3*D*a*c^2)/(a*d-b*c)^4/(d*x+c)-1/2*(-A*b^3+B*a*b^2-C*a^2*b+D*a^3)/(a*d-b*c 
)^3/b/(b*x+a)^2+(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/(b*x+a)-(6*A*b^2*d^2-3*B*a*b*d^2-3*B*b^2*c*d+C*a^2*d^2+4*C*a*b 
*c*d+C*b^2*c^2-3*D*a^2*c*d-3*D*a*b*c^2)/(a*d-b*c)^5*ln(b*x+a)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2183 vs. \(2 (362) = 724\).

Time = 0.15 (sec) , antiderivative size = 2183, normalized size of antiderivative = 6.03 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^3 (c+d x)^3} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/2*(D*a^2*b^3*c^5 - A*a^4*b*d^5 + (9*D*a^3*b^2 - 6*C*a^2*b^3 + B*a*b^4 + 
 A*b^5)*c^4*d - (9*D*a^4*b - 9*B*a^2*b^3 + 8*A*a*b^4)*c^3*d^2 - (D*a^5 - 6 
*C*a^4*b + 9*B*a^3*b^2)*c^2*d^3 - (B*a^4*b - 8*A*a^3*b^2)*c*d^4 + 2*((3*D* 
a*b^4 - C*b^5)*c^3*d^2 - 3*(C*a*b^4 - B*b^5)*c^2*d^3 - 3*(D*a^3*b^2 - C*a^ 
2*b^3 + 2*A*b^5)*c*d^4 + (C*a^3*b^2 - 3*B*a^2*b^3 + 6*A*a*b^4)*d^5)*x^3 + 
(D*b^5*c^5 + (4*D*a*b^4 - 3*C*b^5)*c^4*d + (19*D*a^2*b^3 - 12*C*a*b^4 + 9* 
B*b^5)*c^3*d^2 - (19*D*a^3*b^2 - 9*B*a*b^4 + 18*A*b^5)*c^2*d^3 - (4*D*a^4* 
b - 12*C*a^3*b^2 + 9*B*a^2*b^3)*c*d^4 - (D*a^5 - 3*C*a^4*b + 9*B*a^3*b^2 - 
 18*A*a^2*b^3)*d^5)*x^2 + 2*(D*a*b^4*c^5 + (7*D*a^2*b^3 - 5*C*a*b^4 + B*b^ 
5)*c^4*d - (3*C*a^2*b^3 - 7*B*a*b^4 + 2*A*b^5)*c^3*d^2 - (7*D*a^4*b - 3*C* 
a^3*b^2 + 12*A*a*b^4)*c^2*d^3 - (D*a^5 - 5*C*a^4*b + 7*B*a^3*b^2 - 12*A*a^ 
2*b^3)*c*d^4 - (B*a^4*b - 2*A*a^3*b^2)*d^5)*x + 2*((3*D*a^3*b^2 - C*a^2*b^ 
3)*c^4*d + (3*D*a^4*b - 4*C*a^3*b^2 + 3*B*a^2*b^3)*c^3*d^2 - (C*a^4*b - 3* 
B*a^3*b^2 + 6*A*a^2*b^3)*c^2*d^3 + ((3*D*a*b^4 - C*b^5)*c^2*d^3 + (3*D*a^2 
*b^3 - 4*C*a*b^4 + 3*B*b^5)*c*d^4 - (C*a^2*b^3 - 3*B*a*b^4 + 6*A*b^5)*d^5) 
*x^4 + 2*((3*D*a*b^4 - C*b^5)*c^3*d^2 + (6*D*a^2*b^3 - 5*C*a*b^4 + 3*B*b^5 
)*c^2*d^3 + (3*D*a^3*b^2 - 5*C*a^2*b^3 + 6*B*a*b^4 - 6*A*b^5)*c*d^4 - (C*a 
^3*b^2 - 3*B*a^2*b^3 + 6*A*a*b^4)*d^5)*x^3 + ((3*D*a*b^4 - C*b^5)*c^4*d + 
(15*D*a^2*b^3 - 8*C*a*b^4 + 3*B*b^5)*c^3*d^2 + 3*(5*D*a^3*b^2 - 6*C*a^2*b^ 
3 + 5*B*a*b^4 - 2*A*b^5)*c^2*d^3 + (3*D*a^4*b - 8*C*a^3*b^2 + 15*B*a^2*...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3074 vs. \(2 (343) = 686\).

Time = 149.07 (sec) , antiderivative size = 3074, normalized size of antiderivative = 8.49 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^3 (c+d x)^3} \, dx=\text {Too large to display} \] Input:

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

Output:

(-A*a**3*b*d**4 + 7*A*a**2*b**2*c*d**3 + 7*A*a*b**3*c**2*d**2 - A*b**4*c** 
3*d - B*a**3*b*c*d**3 - 10*B*a**2*b**2*c**2*d**2 - B*a*b**3*c**3*d + 6*C*a 
**3*b*c**2*d**2 + 6*C*a**2*b**2*c**3*d - D*a**4*c**2*d**2 - 10*D*a**3*b*c* 
*3*d - D*a**2*b**2*c**4 + x**3*(12*A*b**4*d**4 - 6*B*a*b**3*d**4 - 6*B*b** 
4*c*d**3 + 2*C*a**2*b**2*d**4 + 8*C*a*b**3*c*d**3 + 2*C*b**4*c**2*d**2 - 6 
*D*a**2*b**2*c*d**3 - 6*D*a*b**3*c**2*d**2) + x**2*(18*A*a*b**3*d**4 + 18* 
A*b**4*c*d**3 - 9*B*a**2*b**2*d**4 - 18*B*a*b**3*c*d**3 - 9*B*b**4*c**2*d* 
*2 + 3*C*a**3*b*d**4 + 15*C*a**2*b**2*c*d**3 + 15*C*a*b**3*c**2*d**2 + 3*C 
*b**4*c**3*d - D*a**4*d**4 - 5*D*a**3*b*c*d**3 - 24*D*a**2*b**2*c**2*d**2 
- 5*D*a*b**3*c**3*d - D*b**4*c**4) + x*(4*A*a**2*b**2*d**4 + 28*A*a*b**3*c 
*d**3 + 4*A*b**4*c**2*d**2 - 2*B*a**3*b*d**4 - 16*B*a**2*b**2*c*d**3 - 16* 
B*a*b**3*c**2*d**2 - 2*B*b**4*c**3*d + 10*C*a**3*b*c*d**3 + 16*C*a**2*b**2 
*c**2*d**2 + 10*C*a*b**3*c**3*d - 2*D*a**4*c*d**3 - 16*D*a**3*b*c**2*d**2 
- 16*D*a**2*b**2*c**3*d - 2*D*a*b**3*c**4))/(2*a**6*b*c**2*d**5 - 8*a**5*b 
**2*c**3*d**4 + 12*a**4*b**3*c**4*d**3 - 8*a**3*b**4*c**5*d**2 + 2*a**2*b* 
*5*c**6*d + x**4*(2*a**4*b**3*d**7 - 8*a**3*b**4*c*d**6 + 12*a**2*b**5*c** 
2*d**5 - 8*a*b**6*c**3*d**4 + 2*b**7*c**4*d**3) + x**3*(4*a**5*b**2*d**7 - 
 12*a**4*b**3*c*d**6 + 8*a**3*b**4*c**2*d**5 + 8*a**2*b**5*c**3*d**4 - 12* 
a*b**6*c**4*d**3 + 4*b**7*c**5*d**2) + x**2*(2*a**6*b*d**7 - 18*a**4*b**3* 
c**2*d**5 + 32*a**3*b**4*c**3*d**4 - 18*a**2*b**5*c**4*d**3 + 2*b**7*c*...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1075 vs. \(2 (362) = 724\).

Time = 0.07 (sec) , antiderivative size = 1075, normalized size of antiderivative = 2.97 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^3 (c+d x)^3} \, dx =\text {Too large to display} \] Input:

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

Output:

-((3*D*a*b - C*b^2)*c^2 + (3*D*a^2 - 4*C*a*b + 3*B*b^2)*c*d - (C*a^2 - 3*B 
*a*b + 6*A*b^2)*d^2)*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*b - C*b^2) 
*c^2 + (3*D*a^2 - 4*C*a*b + 3*B*b^2)*c*d - (C*a^2 - 3*B*a*b + 6*A*b^2)*d^2 
)*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) - 1/2*(D*a^2*b^2*c^4 + A*a^3*b*d^4 + (1 
0*D*a^3*b - 6*C*a^2*b^2 + B*a*b^3 + A*b^4)*c^3*d + (D*a^4 - 6*C*a^3*b + 10 
*B*a^2*b^2 - 7*A*a*b^3)*c^2*d^2 + (B*a^3*b - 7*A*a^2*b^2)*c*d^3 + 2*((3*D* 
a*b^3 - C*b^4)*c^2*d^2 + (3*D*a^2*b^2 - 4*C*a*b^3 + 3*B*b^4)*c*d^3 - (C*a^ 
2*b^2 - 3*B*a*b^3 + 6*A*b^4)*d^4)*x^3 + (D*b^4*c^4 + (5*D*a*b^3 - 3*C*b^4) 
*c^3*d + 3*(8*D*a^2*b^2 - 5*C*a*b^3 + 3*B*b^4)*c^2*d^2 + (5*D*a^3*b - 15*C 
*a^2*b^2 + 18*B*a*b^3 - 18*A*b^4)*c*d^3 + (D*a^4 - 3*C*a^3*b + 9*B*a^2*b^2 
 - 18*A*a*b^3)*d^4)*x^2 + 2*(D*a*b^3*c^4 + (8*D*a^2*b^2 - 5*C*a*b^3 + B*b^ 
4)*c^3*d + 2*(4*D*a^3*b - 4*C*a^2*b^2 + 4*B*a*b^3 - A*b^4)*c^2*d^2 + (D*a^ 
4 - 5*C*a^3*b + 8*B*a^2*b^2 - 14*A*a*b^3)*c*d^3 + (B*a^3*b - 2*A*a^2*b^2)* 
d^4)*x)/(a^2*b^5*c^6*d - 4*a^3*b^4*c^5*d^2 + 6*a^4*b^3*c^4*d^3 - 4*a^5*b^2 
*c^3*d^4 + a^6*b*c^2*d^5 + (b^7*c^4*d^3 - 4*a*b^6*c^3*d^4 + 6*a^2*b^5*c^2* 
d^5 - 4*a^3*b^4*c*d^6 + a^4*b^3*d^7)*x^4 + 2*(b^7*c^5*d^2 - 3*a*b^6*c^4*d^ 
3 + 2*a^2*b^5*c^3*d^4 + 2*a^3*b^4*c^2*d^5 - 3*a^4*b^3*c*d^6 + a^5*b^2*d^7) 
*x^3 + (b^7*c^6*d - 9*a^2*b^5*c^4*d^3 + 16*a^3*b^4*c^3*d^4 - 9*a^4*b^3*...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1012 vs. \(2 (362) = 724\).

Time = 0.13 (sec) , antiderivative size = 1012, normalized size of antiderivative = 2.80 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^3 (c+d x)^3} \, dx =\text {Too large to display} \] Input:

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

Output:

-(3*D*a*b^2*c^2 - C*b^3*c^2 + 3*D*a^2*b*c*d - 4*C*a*b^2*c*d + 3*B*b^3*c*d 
- C*a^2*b*d^2 + 3*B*a*b^2*d^2 - 6*A*b^3*d^2)*log(abs(b*x + a))/(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) + (3*D*a*b*c^2*d - C*b^2*c^2*d + 3*D*a^2*c*d^2 - 4*C*a*b*c*d^ 
2 + 3*B*b^2*c*d^2 - C*a^2*d^3 + 3*B*a*b*d^3 - 6*A*b^2*d^3)*log(abs(d*x + c 
))/(b^5*c^5*d - 5*a*b^4*c^4*d^2 + 10*a^2*b^3*c^3*d^3 - 10*a^3*b^2*c^2*d^4 
+ 5*a^4*b*c*d^5 - a^5*d^6) - 1/2*(6*D*a*b^3*c^2*d^2*x^3 - 2*C*b^4*c^2*d^2* 
x^3 + 6*D*a^2*b^2*c*d^3*x^3 - 8*C*a*b^3*c*d^3*x^3 + 6*B*b^4*c*d^3*x^3 - 2* 
C*a^2*b^2*d^4*x^3 + 6*B*a*b^3*d^4*x^3 - 12*A*b^4*d^4*x^3 + D*b^4*c^4*x^2 + 
 5*D*a*b^3*c^3*d*x^2 - 3*C*b^4*c^3*d*x^2 + 24*D*a^2*b^2*c^2*d^2*x^2 - 15*C 
*a*b^3*c^2*d^2*x^2 + 9*B*b^4*c^2*d^2*x^2 + 5*D*a^3*b*c*d^3*x^2 - 15*C*a^2* 
b^2*c*d^3*x^2 + 18*B*a*b^3*c*d^3*x^2 - 18*A*b^4*c*d^3*x^2 + D*a^4*d^4*x^2 
- 3*C*a^3*b*d^4*x^2 + 9*B*a^2*b^2*d^4*x^2 - 18*A*a*b^3*d^4*x^2 + 2*D*a*b^3 
*c^4*x + 16*D*a^2*b^2*c^3*d*x - 10*C*a*b^3*c^3*d*x + 2*B*b^4*c^3*d*x + 16* 
D*a^3*b*c^2*d^2*x - 16*C*a^2*b^2*c^2*d^2*x + 16*B*a*b^3*c^2*d^2*x - 4*A*b^ 
4*c^2*d^2*x + 2*D*a^4*c*d^3*x - 10*C*a^3*b*c*d^3*x + 16*B*a^2*b^2*c*d^3*x 
- 28*A*a*b^3*c*d^3*x + 2*B*a^3*b*d^4*x - 4*A*a^2*b^2*d^4*x + D*a^2*b^2*c^4 
 + 10*D*a^3*b*c^3*d - 6*C*a^2*b^2*c^3*d + B*a*b^3*c^3*d + A*b^4*c^3*d + D* 
a^4*c^2*d^2 - 6*C*a^3*b*c^2*d^2 + 10*B*a^2*b^2*c^2*d^2 - 7*A*a*b^3*c^2*d^2 
 + B*a^3*b*c*d^3 - 7*A*a^2*b^2*c*d^3 + A*a^3*b*d^4)/((b^5*c^4*d - 4*a*b...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 2124, normalized size of antiderivative = 5.87 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^3 (c+d x)^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(4*log(a + b*x)*a**4*b*c**3*d**2 + 8*log(a + b*x)*a**4*b*c**2*d**3*x + 4*l 
og(a + b*x)*a**4*b*c*d**4*x**2 - 6*log(a + b*x)*a**3*b**3*c**2*d**2 - 12*l 
og(a + b*x)*a**3*b**3*c*d**3*x - 6*log(a + b*x)*a**3*b**3*d**4*x**2 + 6*lo 
g(a + b*x)*a**3*b**2*c**4*d + 20*log(a + b*x)*a**3*b**2*c**3*d**2*x + 22*l 
og(a + b*x)*a**3*b**2*c**2*d**3*x**2 + 8*log(a + b*x)*a**3*b**2*c*d**4*x** 
3 - 6*log(a + b*x)*a**2*b**4*c**3*d - 24*log(a + b*x)*a**2*b**4*c**2*d**2* 
x - 30*log(a + b*x)*a**2*b**4*c*d**3*x**2 - 12*log(a + b*x)*a**2*b**4*d**4 
*x**3 + 2*log(a + b*x)*a**2*b**3*c**5 + 16*log(a + b*x)*a**2*b**3*c**4*d*x 
 + 30*log(a + b*x)*a**2*b**3*c**3*d**2*x**2 + 20*log(a + b*x)*a**2*b**3*c* 
*2*d**3*x**3 + 4*log(a + b*x)*a**2*b**3*c*d**4*x**4 - 12*log(a + b*x)*a*b* 
*5*c**3*d*x - 30*log(a + b*x)*a*b**5*c**2*d**2*x**2 - 24*log(a + b*x)*a*b* 
*5*c*d**3*x**3 - 6*log(a + b*x)*a*b**5*d**4*x**4 + 4*log(a + b*x)*a*b**4*c 
**5*x + 14*log(a + b*x)*a*b**4*c**4*d*x**2 + 16*log(a + b*x)*a*b**4*c**3*d 
**2*x**3 + 6*log(a + b*x)*a*b**4*c**2*d**3*x**4 - 6*log(a + b*x)*b**6*c**3 
*d*x**2 - 12*log(a + b*x)*b**6*c**2*d**2*x**3 - 6*log(a + b*x)*b**6*c*d**3 
*x**4 + 2*log(a + b*x)*b**5*c**5*x**2 + 4*log(a + b*x)*b**5*c**4*d*x**3 + 
2*log(a + b*x)*b**5*c**3*d**2*x**4 - 4*log(c + d*x)*a**4*b*c**3*d**2 - 8*l 
og(c + d*x)*a**4*b*c**2*d**3*x - 4*log(c + d*x)*a**4*b*c*d**4*x**2 + 6*log 
(c + d*x)*a**3*b**3*c**2*d**2 + 12*log(c + d*x)*a**3*b**3*c*d**3*x + 6*log 
(c + d*x)*a**3*b**3*d**4*x**2 - 6*log(c + d*x)*a**3*b**2*c**4*d - 20*lo...