\(\int \frac {A+B x+C x^2+D x^3}{(a+b x)^2 (c+d x)^3} \, dx\) [46]

Optimal result

Integrand size = 30, antiderivative size = 276 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^2 (c+d x)^3} \, dx=-\frac {A b^3-a \left (b^2 B-a b C+a^2 D\right )}{b (b c-a d)^3 (a+b x)}-\frac {c^2 C d-B c d^2+A d^3-c^3 D}{2 d^2 (b c-a d)^2 (c+d x)^2}-\frac {a d \left (2 c C d-B d^2-3 c^2 D\right )-b \left (B c d^2-2 A d^3-c^3 D\right )}{d^2 (b c-a d)^3 (c+d x)}+\frac {\left (b^2 (B c-3 A d)-a b (2 c C-2 B d)-a^2 (C d-3 c D)\right ) \log (a+b x)}{(b c-a d)^4}-\frac {\left (b^2 (B c-3 A d)-a b (2 c C-2 B d)-a^2 (C d-3 c D)\right ) \log (c+d x)}{(b c-a d)^4} \] Output:

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 270, normalized size of antiderivative = 0.98 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^2 (c+d x)^3} \, dx=\frac {-A b^3+a \left (b^2 B-a b C+a^2 D\right )}{b (b c-a d)^3 (a+b x)}+\frac {-c^2 C d+B c d^2-A d^3+c^3 D}{2 d^2 (b c-a d)^2 (c+d x)^2}+\frac {-a d \left (-2 c C d+B d^2+3 c^2 D\right )+b \left (-B c d^2+2 A d^3+c^3 D\right )}{d^2 (-b c+a d)^3 (c+d x)}+\frac {\left (b^2 (B c-3 A d)+a b (-2 c C+2 B d)+a^2 (-C d+3 c D)\right ) \log (a+b x)}{(b c-a d)^4}+\frac {\left (b^2 (-B c+3 A d)+2 a b (c C-B d)+a^2 (C d-3 c D)\right ) \log (c+d x)}{(b c-a d)^4} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 276, 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.

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

\(\Big \downarrow \) 2123

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-((A*b^3 - a*(b^2*B - a*b*C + a^2*D))/(b*(b*c - a*d)^3*(a + b*x))) - (c^2* 
C*d - B*c*d^2 + A*d^3 - c^3*D)/(2*d^2*(b*c - a*d)^2*(c + d*x)^2) - (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))/(d^2*(b*c - a*d 
)^3*(c + d*x)) + ((b^2*(B*c - 3*A*d) - a*b*(2*c*C - 2*B*d) - a^2*(C*d - 3* 
c*D))*Log[a + b*x])/(b*c - a*d)^4 - ((b^2*(B*c - 3*A*d) - a*b*(2*c*C - 2*B 
*d) - a^2*(C*d - 3*c*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 = 0.51 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.00

method	result	size
default	\(\frac {\left (3 d \,b^{2} A -2 B a b d -B \,b^{2} c +C \,a^{2} d +2 C a b c -3 a^{2} c D\right ) \ln \left (x d +c \right )}{\left (a d -b c \right )^{4}}-\frac {A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}}{2 d^{2} \left (a d -b c \right )^{2} \left (x d +c \right )^{2}}-\frac {-2 A b \,d^{3}+B a \,d^{3}+B b c \,d^{2}-2 C a c \,d^{2}+3 D a \,c^{2} d -D b \,c^{3}}{\left (a d -b c \right )^{3} d^{2} \left (x d +c \right )}-\frac {-b^{3} A +a \,b^{2} B -a^{2} b C +a^{3} D}{\left (a d -b c \right )^{3} b \left (b x +a \right )}-\frac {\left (3 d \,b^{2} A -2 B a b d -B \,b^{2} c +C \,a^{2} d +2 C a b c -3 a^{2} c D\right ) \ln \left (b x +a \right )}{\left (a d -b c \right )^{4}}\)	\(277\)
norman	\(\frac {\frac {\left (3 A \,b^{3} d^{3}-2 B a \,b^{2} d^{3}-B \,b^{3} c \,d^{2}+a^{2} b C \,d^{3}+2 C a \,b^{2} c \,d^{2}-a^{3} d^{3} D-3 D a \,b^{2} c^{2} d +D b^{3} c^{3}\right ) x^{2}}{b d \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {A \,a^{2} b \,d^{4}-5 A a \,b^{2} c \,d^{3}-2 A \,b^{3} c^{2} d^{2}+B \,a^{2} b c \,d^{3}+5 B a \,b^{2} c^{2} d^{2}-5 C \,a^{2} b \,c^{2} d^{2}-C a \,b^{2} c^{3} d +2 D a^{3} c^{2} d^{2}+5 D a^{2} b \,c^{3} d -D a \,b^{2} c^{4}}{2 d^{2} b \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {\left (3 a \,b^{2} A \,d^{4}+9 A \,b^{3} c \,d^{3}-2 B \,a^{2} b \,d^{4}-7 B a \,b^{2} c \,d^{3}-3 B \,b^{3} c^{2} d^{2}+8 C \,a^{2} b c \,d^{3}+3 C a \,b^{2} c^{2} d^{2}+C \,b^{3} c^{3} d -4 D a^{3} c \,d^{3}-6 D a^{2} b \,c^{2} d^{2}-3 D a \,b^{2} c^{3} d +D b^{3} c^{4}\right ) x}{2 d^{2} b \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}}{\left (b x +a \right ) \left (x d +c \right )^{2}}+\frac {\left (3 d \,b^{2} A -2 B a b d -B \,b^{2} c +C \,a^{2} d +2 C a b c -3 a^{2} c D\right ) \ln \left (x d +c \right )}{d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}}-\frac {\left (3 d \,b^{2} A -2 B a b d -B \,b^{2} c +C \,a^{2} d +2 C a b c -3 a^{2} c D\right ) \ln \left (b x +a \right )}{d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}}\)	\(676\)
parallelrisch	\(\text {Expression too large to display}\)	\(1921\)

Input:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1351 vs. \(2 (270) = 540\).

Time = 0.11 (sec) , antiderivative size = 1351, normalized size of antiderivative = 4.89 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^2 (c+d x)^3} \, dx=\text {Too large to display} \] Input:

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

Output:

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1912 vs. \(2 (250) = 500\).

Time = 42.63 (sec) , antiderivative size = 1912, normalized size of antiderivative = 6.93 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^2 (c+d x)^3} \, dx=\text {Too large to display} \] Input:

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 715 vs. \(2 (270) = 540\).

Time = 0.07 (sec) , antiderivative size = 715, normalized size of antiderivative = 2.59 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^2 (c+d x)^3} \, dx=\frac {{\left ({\left (3 \, D a^{2} - 2 \, C a b + B b^{2}\right )} c - {\left (C a^{2} - 2 \, B a b + 3 \, A b^{2}\right )} d\right )} \log \left (b x + a\right )}{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}} - \frac {{\left ({\left (3 \, D a^{2} - 2 \, C a b + B b^{2}\right )} c - {\left (C a^{2} - 2 \, B a b + 3 \, A b^{2}\right )} d\right )} \log \left (d x + c\right )}{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}} - \frac {D a b^{2} c^{4} - A a^{2} b d^{4} - {\left (5 \, D a^{2} b - C a b^{2}\right )} c^{3} d - {\left (2 \, D a^{3} - 5 \, C a^{2} b + 5 \, B a b^{2} - 2 \, A b^{3}\right )} c^{2} d^{2} - {\left (B a^{2} b - 5 \, A a b^{2}\right )} c d^{3} + 2 \, {\left (D b^{3} c^{3} d - 3 \, D a b^{2} c^{2} d^{2} + {\left (2 \, C a b^{2} - B b^{3}\right )} c d^{3} - {\left (D a^{3} - C a^{2} b + 2 \, B a b^{2} - 3 \, A b^{3}\right )} d^{4}\right )} x^{2} + {\left (D b^{3} c^{4} - {\left (3 \, D a b^{2} - C b^{3}\right )} c^{3} d - 3 \, {\left (2 \, D a^{2} b - C a b^{2} + B b^{3}\right )} c^{2} d^{2} - {\left (4 \, D a^{3} - 8 \, C a^{2} b + 7 \, B a b^{2} - 9 \, A b^{3}\right )} c d^{3} - {\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} d^{4}\right )} x}{2 \, {\left (a b^{4} c^{5} d^{2} - 3 \, a^{2} b^{3} c^{4} d^{3} + 3 \, a^{3} b^{2} c^{3} d^{4} - a^{4} b c^{2} d^{5} + {\left (b^{5} c^{3} d^{4} - 3 \, a b^{4} c^{2} d^{5} + 3 \, a^{2} b^{3} c d^{6} - a^{3} b^{2} d^{7}\right )} x^{3} + {\left (2 \, b^{5} c^{4} d^{3} - 5 \, a b^{4} c^{3} d^{4} + 3 \, a^{2} b^{3} c^{2} d^{5} + a^{3} b^{2} c d^{6} - a^{4} b d^{7}\right )} x^{2} + {\left (b^{5} c^{5} d^{2} - a b^{4} c^{4} d^{3} - 3 \, a^{2} b^{3} c^{3} d^{4} + 5 \, a^{3} b^{2} c^{2} d^{5} - 2 \, a^{4} b c d^{6}\right )} x\right )}} \] Input:

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 452, normalized size of antiderivative = 1.64 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^2 (c+d x)^3} \, dx=-\frac {{\left (3 \, D a^{2} b c - 2 \, C a b^{2} c + B b^{3} c - C a^{2} b d + 2 \, B a b^{2} d - 3 \, A b^{3} d\right )} \log \left ({\left | \frac {b c}{b x + a} - \frac {a d}{b x + a} + d \right |}\right )}{b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}} + \frac {\frac {D a^{3} b^{2}}{b x + a} - \frac {C a^{2} b^{3}}{b x + a} + \frac {B a b^{4}}{b x + a} - \frac {A b^{5}}{b x + a}}{b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}} + \frac {D b^{2} c^{3} - 6 \, D a b c^{2} d + C b^{2} c^{2} d + 4 \, C a b c d^{2} - 3 \, B b^{2} c d^{2} - 2 \, B a b d^{3} + 5 \, A b^{2} d^{3} - \frac {2 \, {\left (3 \, D a b^{3} c^{3} - C b^{4} c^{3} - 3 \, D a^{2} b^{2} c^{2} d - C a b^{3} c^{2} d + 2 \, B b^{4} c^{2} d + 2 \, C a^{2} b^{2} c d^{2} - B a b^{3} c d^{2} - 3 \, A b^{4} c d^{2} - B a^{2} b^{2} d^{3} + 3 \, A a b^{3} d^{3}\right )}}{{\left (b x + a\right )} b}}{2 \, {\left (b c - a d\right )}^{4} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )}^{2}} \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 1252, normalized size of antiderivative = 4.54 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^2 (c+d x)^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(4*log(a + b*x)*a**3*c**3*d**2 + 8*log(a + b*x)*a**3*c**2*d**3*x + 4*log(a 
 + b*x)*a**3*c*d**4*x**2 - 2*log(a + b*x)*a**2*b**2*c**2*d**2 - 4*log(a + 
b*x)*a**2*b**2*c*d**3*x - 2*log(a + b*x)*a**2*b**2*d**4*x**2 + 8*log(a + b 
*x)*a**2*b*c**4*d + 20*log(a + b*x)*a**2*b*c**3*d**2*x + 16*log(a + b*x)*a 
**2*b*c**2*d**3*x**2 + 4*log(a + b*x)*a**2*b*c*d**4*x**3 - 4*log(a + b*x)* 
a*b**3*c**3*d - 10*log(a + b*x)*a*b**3*c**2*d**2*x - 8*log(a + b*x)*a*b**3 
*c*d**3*x**2 - 2*log(a + b*x)*a*b**3*d**4*x**3 + 8*log(a + b*x)*a*b**2*c** 
4*d*x + 16*log(a + b*x)*a*b**2*c**3*d**2*x**2 + 8*log(a + b*x)*a*b**2*c**2 
*d**3*x**3 - 4*log(a + b*x)*b**4*c**3*d*x - 8*log(a + b*x)*b**4*c**2*d**2* 
x**2 - 4*log(a + b*x)*b**4*c*d**3*x**3 - 4*log(c + d*x)*a**3*c**3*d**2 - 8 
*log(c + d*x)*a**3*c**2*d**3*x - 4*log(c + d*x)*a**3*c*d**4*x**2 + 2*log(c 
 + d*x)*a**2*b**2*c**2*d**2 + 4*log(c + d*x)*a**2*b**2*c*d**3*x + 2*log(c 
+ d*x)*a**2*b**2*d**4*x**2 - 8*log(c + d*x)*a**2*b*c**4*d - 20*log(c + d*x 
)*a**2*b*c**3*d**2*x - 16*log(c + d*x)*a**2*b*c**2*d**3*x**2 - 4*log(c + d 
*x)*a**2*b*c*d**4*x**3 + 4*log(c + d*x)*a*b**3*c**3*d + 10*log(c + d*x)*a* 
b**3*c**2*d**2*x + 8*log(c + d*x)*a*b**3*c*d**3*x**2 + 2*log(c + d*x)*a*b* 
*3*d**4*x**3 - 8*log(c + d*x)*a*b**2*c**4*d*x - 16*log(c + d*x)*a*b**2*c** 
3*d**2*x**2 - 8*log(c + d*x)*a*b**2*c**2*d**3*x**3 + 4*log(c + d*x)*b**4*c 
**3*d*x + 8*log(c + d*x)*b**4*c**2*d**2*x**2 + 4*log(c + d*x)*b**4*c*d**3* 
x**3 - a**4*d**4 + 2*a**3*b*c*d**3 + a**3*b*d**4*x - 6*a**3*c**3*d**2 -...