\(\int \frac {A+B x}{(d+e x)^2 (b x+c x^2)^3} \, dx\) [51]

Optimal result

Integrand size = 24, antiderivative size = 331 \[ \int \frac {A+B x}{(d+e x)^2 \left (b x+c x^2\right )^3} \, dx=-\frac {A}{2 b^3 d^2 x^2}-\frac {b B d-3 A c d-2 A b e}{b^4 d^3 x}-\frac {c^3 (b B-A c)}{2 b^3 (c d-b e)^2 (b+c x)^2}-\frac {c^3 \left (2 b B c d-3 A c^2 d-4 b^2 B e+5 A b c e\right )}{b^4 (c d-b e)^3 (b+c x)}+\frac {e^4 (B d-A e)}{d^3 (c d-b e)^3 (d+e x)}+\frac {\left (6 A c^2 d^2-b^2 e (2 B d-3 A e)-3 b c d (B d-2 A e)\right ) \log (x)}{b^5 d^4}-\frac {c^3 \left (6 A c^3 d^2-10 b^3 B e^2+5 b^2 c e (2 B d+3 A e)-3 b c^2 d (B d+6 A e)\right ) \log (b+c x)}{b^5 (c d-b e)^4}-\frac {e^4 (B d (5 c d-2 b e)-3 A e (2 c d-b e)) \log (d+e x)}{d^4 (c d-b e)^4} \] Output:

-1/2*A/b^3/d^2/x^2-(-2*A*b*e-3*A*c*d+B*b*d)/b^4/d^3/x-1/2*c^3*(-A*c+B*b)/b 
^3/(-b*e+c*d)^2/(c*x+b)^2-c^3*(5*A*b*c*e-3*A*c^2*d-4*B*b^2*e+2*B*b*c*d)/b^ 
4/(-b*e+c*d)^3/(c*x+b)+e^4*(-A*e+B*d)/d^3/(-b*e+c*d)^3/(e*x+d)+(6*A*c^2*d^ 
2-b^2*e*(-3*A*e+2*B*d)-3*b*c*d*(-2*A*e+B*d))*ln(x)/b^5/d^4-c^3*(6*A*c^3*d^ 
2-10*b^3*B*e^2+5*b^2*c*e*(3*A*e+2*B*d)-3*b*c^2*d*(6*A*e+B*d))*ln(c*x+b)/b^ 
5/(-b*e+c*d)^4-e^4*(B*d*(-2*b*e+5*c*d)-3*A*e*(-b*e+2*c*d))*ln(e*x+d)/d^4/( 
-b*e+c*d)^4
 

Mathematica [A] (verified)

Time = 0.70 (sec) , antiderivative size = 328, normalized size of antiderivative = 0.99 \[ \int \frac {A+B x}{(d+e x)^2 \left (b x+c x^2\right )^3} \, dx=-\frac {A}{2 b^3 d^2 x^2}+\frac {-b B d+3 A c d+2 A b e}{b^4 d^3 x}+\frac {c^3 (-b B+A c)}{2 b^3 (c d-b e)^2 (b+c x)^2}+\frac {c^3 \left (-3 A c^2 d-4 b^2 B e+b c (2 B d+5 A e)\right )}{b^4 (-c d+b e)^3 (b+c x)}+\frac {e^4 (B d-A e)}{d^3 (c d-b e)^3 (d+e x)}-\frac {\left (-6 A c^2 d^2+b^2 e (2 B d-3 A e)+3 b c d (B d-2 A e)\right ) \log (x)}{b^5 d^4}+\frac {c^3 \left (-6 A c^3 d^2+10 b^3 B e^2-5 b^2 c e (2 B d+3 A e)+3 b c^2 d (B d+6 A e)\right ) \log (b+c x)}{b^5 (c d-b e)^4}-\frac {e^4 (B d (5 c d-2 b e)+3 A e (-2 c d+b e)) \log (d+e x)}{d^4 (c d-b e)^4} \] Input:

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

Output:

-1/2*A/(b^3*d^2*x^2) + (-(b*B*d) + 3*A*c*d + 2*A*b*e)/(b^4*d^3*x) + (c^3*( 
-(b*B) + A*c))/(2*b^3*(c*d - b*e)^2*(b + c*x)^2) + (c^3*(-3*A*c^2*d - 4*b^ 
2*B*e + b*c*(2*B*d + 5*A*e)))/(b^4*(-(c*d) + b*e)^3*(b + c*x)) + (e^4*(B*d 
 - A*e))/(d^3*(c*d - b*e)^3*(d + e*x)) - ((-6*A*c^2*d^2 + b^2*e*(2*B*d - 3 
*A*e) + 3*b*c*d*(B*d - 2*A*e))*Log[x])/(b^5*d^4) + (c^3*(-6*A*c^3*d^2 + 10 
*b^3*B*e^2 - 5*b^2*c*e*(2*B*d + 3*A*e) + 3*b*c^2*d*(B*d + 6*A*e))*Log[b + 
c*x])/(b^5*(c*d - b*e)^4) - (e^4*(B*d*(5*c*d - 2*b*e) + 3*A*e*(-2*c*d + b* 
e))*Log[d + e*x])/(d^4*(c*d - b*e)^4)
 

Rubi [A] (verified)

Time = 1.45 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1206, 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.

\(\displaystyle \int \frac {A+B x}{\left (b x+c x^2\right )^3 (d+e x)^2} \, dx\)

\(\Big \downarrow \) 1206

\(\displaystyle \int \left (\frac {-2 A b e-3 A c d+b B d}{b^4 d^3 x^2}+\frac {c^4 (b B-A c)}{b^3 (b+c x)^3 (b e-c d)^2}+\frac {A}{b^3 d^2 x^3}+\frac {b^2 (-e) (2 B d-3 A e)-3 b c d (B d-2 A e)+6 A c^2 d^2}{b^5 d^4 x}+\frac {c^4 \left (-5 A b c e+3 A c^2 d+4 b^2 B e-2 b B c d\right )}{b^4 (b+c x)^2 (b e-c d)^3}+\frac {c^4 \left (-5 b^2 c e (3 A e+2 B d)+3 b c^2 d (6 A e+B d)-6 A c^3 d^2+10 b^3 B e^2\right )}{b^5 (b+c x) (c d-b e)^4}+\frac {e^5 (3 A e (2 c d-b e)-B d (5 c d-2 b e))}{d^4 (d+e x) (c d-b e)^4}-\frac {e^5 (B d-A e)}{d^3 (d+e x)^2 (c d-b e)^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {-2 A b e-3 A c d+b B d}{b^4 d^3 x}-\frac {c^3 (b B-A c)}{2 b^3 (b+c x)^2 (c d-b e)^2}-\frac {A}{2 b^3 d^2 x^2}+\frac {\log (x) \left (b^2 (-e) (2 B d-3 A e)-3 b c d (B d-2 A e)+6 A c^2 d^2\right )}{b^5 d^4}-\frac {c^3 \left (5 A b c e-3 A c^2 d-4 b^2 B e+2 b B c d\right )}{b^4 (b+c x) (c d-b e)^3}-\frac {c^3 \log (b+c x) \left (5 b^2 c e (3 A e+2 B d)-3 b c^2 d (6 A e+B d)+6 A c^3 d^2-10 b^3 B e^2\right )}{b^5 (c d-b e)^4}-\frac {e^4 \log (d+e x) (B d (5 c d-2 b e)-3 A e (2 c d-b e))}{d^4 (c d-b e)^4}+\frac {e^4 (B d-A e)}{d^3 (d+e x) (c d-b e)^3}\)

Input:

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

Output:

-1/2*A/(b^3*d^2*x^2) - (b*B*d - 3*A*c*d - 2*A*b*e)/(b^4*d^3*x) - (c^3*(b*B 
 - A*c))/(2*b^3*(c*d - b*e)^2*(b + c*x)^2) - (c^3*(2*b*B*c*d - 3*A*c^2*d - 
 4*b^2*B*e + 5*A*b*c*e))/(b^4*(c*d - b*e)^3*(b + c*x)) + (e^4*(B*d - A*e)) 
/(d^3*(c*d - b*e)^3*(d + e*x)) + ((6*A*c^2*d^2 - b^2*e*(2*B*d - 3*A*e) - 3 
*b*c*d*(B*d - 2*A*e))*Log[x])/(b^5*d^4) - (c^3*(6*A*c^3*d^2 - 10*b^3*B*e^2 
 + 5*b^2*c*e*(2*B*d + 3*A*e) - 3*b*c^2*d*(B*d + 6*A*e))*Log[b + c*x])/(b^5 
*(c*d - b*e)^4) - (e^4*(B*d*(5*c*d - 2*b*e) - 3*A*e*(2*c*d - b*e))*Log[d + 
 e*x])/(d^4*(c*d - b*e)^4)
 

Defintions of rubi rules used

rule 1206

Int[((d_) + (e_.)*(x_))^(m_.)*((f_) + (g_.)*(x_))^(n_.)*((b_.)*(x_) + (c_.) 
*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[x^p*(d + e*x)^m*(f + g*x)^n 
*(b + c*x)^p, x], x] /; FreeQ[{b, c, d, e, f, g}, x] && ILtQ[p, -1] && Inte 
gersQ[m, n]
 

rule 2009

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

Maple [A] (verified)

Time = 1.14 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.02

method	result	size
default	\(\frac {c^{3} \left (5 A b c e -3 A \,c^{2} d -4 b^{2} B e +2 B b c d \right )}{b^{4} \left (b e -c d \right )^{3} \left (c x +b \right )}-\frac {c^{3} \left (15 A \,b^{2} c \,e^{2}-18 A b \,c^{2} d e +6 A \,c^{3} d^{2}-10 b^{3} B \,e^{2}+10 B \,b^{2} c d e -3 B b \,c^{2} d^{2}\right ) \ln \left (c x +b \right )}{b^{5} \left (b e -c d \right )^{4}}+\frac {\left (A c -B b \right ) c^{3}}{2 b^{3} \left (b e -c d \right )^{2} \left (c x +b \right )^{2}}-\frac {e^{4} \left (3 A b \,e^{2}-6 A c d e -2 B b d e +5 B c \,d^{2}\right ) \ln \left (e x +d \right )}{\left (b e -c d \right )^{4} d^{4}}+\frac {\left (A e -B d \right ) e^{4}}{\left (b e -c d \right )^{3} d^{3} \left (e x +d \right )}-\frac {A}{2 b^{3} d^{2} x^{2}}-\frac {-2 A b e -3 A c d +B b d}{b^{4} d^{3} x}+\frac {\left (3 A \,b^{2} e^{2}+6 A b c d e +6 A \,c^{2} d^{2}-2 B \,b^{2} d e -3 B b c \,d^{2}\right ) \ln \left (x \right )}{b^{5} d^{4}}\)	\(339\)
norman	\(\frac {-\frac {A}{2 b d}+\frac {\left (3 A b e +4 A c d -2 B b d \right ) x}{2 b^{2} d^{2}}-\frac {\left (3 A \,b^{6} e^{6}-2 A \,b^{4} c^{2} d^{2} e^{4}-6 A \,b^{3} c^{3} d^{3} e^{3}+20 A b \,c^{5} d^{5} e -12 A \,c^{6} d^{6}-2 B \,b^{6} d \,e^{5}+B \,b^{5} c \,d^{2} e^{4}+4 B \,b^{3} c^{3} d^{4} e^{2}-12 B \,b^{2} c^{4} d^{5} e +6 B b \,c^{5} d^{6}\right ) x^{3}}{d^{4} b^{4} \left (b^{3} e^{3}-3 d \,e^{2} b^{2} c +3 d^{2} e b \,c^{2}-d^{3} c^{3}\right )}-\frac {c \left (12 A \,b^{6} e^{6}-20 A \,b^{4} c^{2} d^{2} e^{4}-9 A \,b^{3} c^{3} d^{3} e^{3}+39 A \,b^{2} c^{4} d^{4} e^{2}+8 A b \,c^{5} d^{5} e -18 A \,c^{6} d^{6}-8 B \,b^{6} d \,e^{5}+4 B \,b^{5} c \,d^{2} e^{4}+8 B \,b^{4} c^{2} d^{3} e^{3}-18 B \,b^{3} c^{3} d^{4} e^{2}-7 B \,b^{2} c^{4} d^{5} e +9 B b \,c^{5} d^{6}\right ) x^{4}}{2 d^{4} b^{5} \left (b^{3} e^{3}-3 d \,e^{2} b^{2} c +3 d^{2} e b \,c^{2}-d^{3} c^{3}\right )}-\frac {e \,c^{2} \left (6 A \,b^{5} e^{5}-13 A \,b^{3} c^{2} d^{2} e^{3}-A \,b^{2} c^{3} d^{3} e^{2}+32 A b \,c^{4} d^{4} e -18 A \,c^{5} d^{5}-4 B \,b^{5} d \,e^{4}+2 B \,b^{4} c \,d^{2} e^{3}+6 B \,b^{3} c^{2} d^{3} e^{2}-19 B \,b^{2} c^{3} d^{4} e +9 B b \,c^{4} d^{5}\right ) x^{5}}{2 d^{4} b^{5} \left (b^{3} e^{3}-3 d \,e^{2} b^{2} c +3 d^{2} e b \,c^{2}-d^{3} c^{3}\right )}}{\left (e x +d \right ) x^{2} \left (c x +b \right )^{2}}+\frac {\left (3 A \,b^{2} e^{2}+6 A b c d e +6 A \,c^{2} d^{2}-2 B \,b^{2} d e -3 B b c \,d^{2}\right ) \ln \left (x \right )}{b^{5} d^{4}}-\frac {c^{3} \left (15 A \,b^{2} c \,e^{2}-18 A b \,c^{2} d e +6 A \,c^{3} d^{2}-10 b^{3} B \,e^{2}+10 B \,b^{2} c d e -3 B b \,c^{2} d^{2}\right ) \ln \left (c x +b \right )}{\left (b^{4} e^{4}-4 d \,e^{3} b^{3} c +6 d^{2} e^{2} b^{2} c^{2}-4 d^{3} e b \,c^{3}+d^{4} c^{4}\right ) b^{5}}-\frac {e^{4} \left (3 A b \,e^{2}-6 A c d e -2 B b d e +5 B c \,d^{2}\right ) \ln \left (e x +d \right )}{d^{4} \left (b^{4} e^{4}-4 d \,e^{3} b^{3} c +6 d^{2} e^{2} b^{2} c^{2}-4 d^{3} e b \,c^{3}+d^{4} c^{4}\right )}\)	\(864\)
risch	\(\text {Expression too large to display}\)	\(1328\)
parallelrisch	\(\text {Expression too large to display}\)	\(2880\)

Input:

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

Output:

c^3*(5*A*b*c*e-3*A*c^2*d-4*B*b^2*e+2*B*b*c*d)/b^4/(b*e-c*d)^3/(c*x+b)-c^3* 
(15*A*b^2*c*e^2-18*A*b*c^2*d*e+6*A*c^3*d^2-10*B*b^3*e^2+10*B*b^2*c*d*e-3*B 
*b*c^2*d^2)/b^5/(b*e-c*d)^4*ln(c*x+b)+1/2*(A*c-B*b)*c^3/b^3/(b*e-c*d)^2/(c 
*x+b)^2-e^4*(3*A*b*e^2-6*A*c*d*e-2*B*b*d*e+5*B*c*d^2)/(b*e-c*d)^4/d^4*ln(e 
*x+d)+(A*e-B*d)*e^4/(b*e-c*d)^3/d^3/(e*x+d)-1/2*A/b^3/d^2/x^2-(-2*A*b*e-3* 
A*c*d+B*b*d)/b^4/d^3/x+(3*A*b^2*e^2+6*A*b*c*d*e+6*A*c^2*d^2-2*B*b^2*d*e-3* 
B*b*c*d^2)/b^5/d^4*ln(x)
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

integrate((B*x+A)/(e*x+d)**2/(c*x**2+b*x)**3,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1043 vs. \(2 (327) = 654\).

Time = 0.08 (sec) , antiderivative size = 1043, normalized size of antiderivative = 3.15 \[ \int \frac {A+B x}{(d+e x)^2 \left (b x+c x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(3*(B*b*c^5 - 2*A*c^6)*d^2 - 2*(5*B*b^2*c^4 - 9*A*b*c^5)*d*e + 5*(2*B*b^3* 
c^3 - 3*A*b^2*c^4)*e^2)*log(c*x + b)/(b^5*c^4*d^4 - 4*b^6*c^3*d^3*e + 6*b^ 
7*c^2*d^2*e^2 - 4*b^8*c*d*e^3 + b^9*e^4) - (5*B*c*d^2*e^4 + 3*A*b*e^6 - 2* 
(B*b + 3*A*c)*d*e^5)*log(e*x + d)/(c^4*d^8 - 4*b*c^3*d^7*e + 6*b^2*c^2*d^6 
*e^2 - 4*b^3*c*d^5*e^3 + b^4*d^4*e^4) - 1/2*(A*b^3*c^3*d^5 - 3*A*b^4*c^2*d 
^4*e + 3*A*b^5*c*d^3*e^2 - A*b^6*d^2*e^3 + 2*(3*A*b^4*c^2*e^5 + 3*(B*b*c^5 
 - 2*A*c^6)*d^4*e - (7*B*b^2*c^4 - 12*A*b*c^5)*d^3*e^2 + 3*(B*b^3*c^3 - A* 
b^2*c^4)*d^2*e^3 - (2*B*b^4*c^2 + 3*A*b^3*c^3)*d*e^4)*x^4 + (12*A*b^5*c*e^ 
5 + 6*(B*b*c^5 - 2*A*c^6)*d^5 - (5*B*b^2*c^4 - 6*A*b*c^5)*d^4*e - 15*(B*b^ 
3*c^3 - 2*A*b^2*c^4)*d^3*e^2 + 5*(2*B*b^4*c^2 - 3*A*b^3*c^3)*d^2*e^3 - (8* 
B*b^5*c + 9*A*b^4*c^2)*d*e^4)*x^3 - (4*B*b^6*d*e^4 - 6*A*b^6*e^5 - 9*(B*b^ 
2*c^4 - 2*A*b*c^5)*d^5 + (19*B*b^3*c^3 - 32*A*b^2*c^4)*d^4*e - (6*B*b^4*c^ 
2 - A*b^3*c^3)*d^3*e^2 - (2*B*b^5*c - 13*A*b^4*c^2)*d^2*e^3)*x^2 + (3*A*b^ 
6*d*e^4 + 2*(B*b^3*c^3 - 2*A*b^2*c^4)*d^5 - 3*(2*B*b^4*c^2 - 3*A*b^3*c^3)* 
d^4*e + 3*(2*B*b^5*c - A*b^4*c^2)*d^3*e^2 - (2*B*b^6 + 5*A*b^5*c)*d^2*e^3) 
*x)/((b^4*c^5*d^6*e - 3*b^5*c^4*d^5*e^2 + 3*b^6*c^3*d^4*e^3 - b^7*c^2*d^3* 
e^4)*x^5 + (b^4*c^5*d^7 - b^5*c^4*d^6*e - 3*b^6*c^3*d^5*e^2 + 5*b^7*c^2*d^ 
4*e^3 - 2*b^8*c*d^3*e^4)*x^4 + (2*b^5*c^4*d^7 - 5*b^6*c^3*d^6*e + 3*b^7*c^ 
2*d^5*e^2 + b^8*c*d^4*e^3 - b^9*d^3*e^4)*x^3 + (b^6*c^3*d^7 - 3*b^7*c^2*d^ 
6*e + 3*b^8*c*d^5*e^2 - b^9*d^4*e^3)*x^2) + (3*A*b^2*e^2 - 3*(B*b*c - 2...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1350 vs. \(2 (327) = 654\).

Time = 0.29 (sec) , antiderivative size = 1350, normalized size of antiderivative = 4.08 \[ \int \frac {A+B x}{(d+e x)^2 \left (b x+c x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

1/2*(5*B*c*d^2*e^4 - 2*B*b*d*e^5 - 6*A*c*d*e^5 + 3*A*b*e^6)*log(abs(c - 2* 
c*d/(e*x + d) + c*d^2/(e*x + d)^2 + b*e/(e*x + d) - b*d*e/(e*x + d)^2))/(c 
^4*d^8 - 4*b*c^3*d^7*e + 6*b^2*c^2*d^6*e^2 - 4*b^3*c*d^5*e^3 + b^4*d^4*e^4 
) + (B*d*e^10/(e*x + d) - A*e^11/(e*x + d))/(c^3*d^6*e^6 - 3*b*c^2*d^5*e^7 
 + 3*b^2*c*d^4*e^8 - b^3*d^3*e^9) - 1/2*(6*B*b*c^5*d^6*e^2 - 12*A*c^6*d^6* 
e^2 - 20*B*b^2*c^4*d^5*e^3 + 36*A*b*c^5*d^5*e^3 + 20*B*b^3*c^3*d^4*e^4 - 3 
0*A*b^2*c^4*d^4*e^4 - 5*B*b^5*c*d^2*e^6 + 2*B*b^6*d*e^7 + 6*A*b^5*c*d*e^7 
- 3*A*b^6*e^8)*log(abs(2*c*d*e - 2*c*d^2*e/(e*x + d) - b*e^2 + 2*b*d*e^2/( 
e*x + d) - e^2*abs(b))/abs(2*c*d*e - 2*c*d^2*e/(e*x + d) - b*e^2 + 2*b*d*e 
^2/(e*x + d) + e^2*abs(b)))/((b^4*c^4*d^8 - 4*b^5*c^3*d^7*e + 6*b^6*c^2*d^ 
6*e^2 - 4*b^7*c*d^5*e^3 + b^8*d^4*e^4)*e^2*abs(b)) - 1/2*(6*B*b*c^6*d^5*e 
- 12*A*c^7*d^5*e - 17*B*b^2*c^5*d^4*e^2 + 30*A*b*c^6*d^4*e^2 + 12*B*b^3*c^ 
4*d^3*e^3 - 16*A*b^2*c^5*d^3*e^3 - 8*B*b^4*c^3*d^2*e^4 - 6*A*b^3*c^4*d^2*e 
^4 + 2*B*b^5*c^2*d*e^5 + 14*A*b^4*c^3*d*e^5 - 5*A*b^5*c^2*e^6 - 2*(9*B*b*c 
^6*d^6*e^2 - 18*A*c^7*d^6*e^2 - 30*B*b^2*c^5*d^5*e^3 + 54*A*b*c^6*d^5*e^3 
+ 31*B*b^3*c^4*d^4*e^4 - 47*A*b^2*c^5*d^4*e^4 - 24*B*b^4*c^3*d^3*e^5 + 4*A 
*b^3*c^4*d^3*e^5 + 11*B*b^5*c^2*d^2*e^6 + 29*A*b^4*c^3*d^2*e^6 - 2*B*b^6*c 
*d*e^7 - 22*A*b^5*c^2*d*e^7 + 5*A*b^6*c*e^8)/((e*x + d)*e) + (18*B*b*c^6*d 
^7*e^3 - 36*A*c^7*d^7*e^3 - 69*B*b^2*c^5*d^6*e^4 + 126*A*b*c^6*d^6*e^4 + 9 
0*B*b^3*c^4*d^5*e^5 - 144*A*b^2*c^5*d^5*e^5 - 80*B*b^4*c^3*d^4*e^6 + 45...
 

Mupad [B] (verification not implemented)

Time = 12.94 (sec) , antiderivative size = 879, normalized size of antiderivative = 2.66 \[ \int \frac {A+B x}{(d+e x)^2 \left (b x+c x^2\right )^3} \, dx=\frac {\ln \left (x\right )\,\left (b^2\,\left (3\,A\,e^2-2\,B\,d\,e\right )-b\,\left (3\,B\,c\,d^2-6\,A\,c\,d\,e\right )+6\,A\,c^2\,d^2\right )}{b^5\,d^4}-\frac {\ln \left (b+c\,x\right )\,\left (d^2\,\left (6\,A\,c^6-3\,B\,b\,c^5\right )-d\,\left (18\,A\,b\,c^5\,e-10\,B\,b^2\,c^4\,e\right )+15\,A\,b^2\,c^4\,e^2-10\,B\,b^3\,c^3\,e^2\right )}{b^9\,e^4-4\,b^8\,c\,d\,e^3+6\,b^7\,c^2\,d^2\,e^2-4\,b^6\,c^3\,d^3\,e+b^5\,c^4\,d^4}-\frac {\ln \left (d+e\,x\right )\,\left (c\,\left (5\,B\,d^2\,e^4-6\,A\,d\,e^5\right )+b\,\left (3\,A\,e^6-2\,B\,d\,e^5\right )\right )}{b^4\,d^4\,e^4-4\,b^3\,c\,d^5\,e^3+6\,b^2\,c^2\,d^6\,e^2-4\,b\,c^3\,d^7\,e+c^4\,d^8}-\frac {\frac {A}{2\,b\,d}-\frac {x\,\left (3\,A\,b\,e+4\,A\,c\,d-2\,B\,b\,d\right )}{2\,b^2\,d^2}+\frac {x^3\,\left (8\,B\,b^5\,c\,d\,e^4-12\,A\,b^5\,c\,e^5-10\,B\,b^4\,c^2\,d^2\,e^3+9\,A\,b^4\,c^2\,d\,e^4+15\,B\,b^3\,c^3\,d^3\,e^2+15\,A\,b^3\,c^3\,d^2\,e^3+5\,B\,b^2\,c^4\,d^4\,e-30\,A\,b^2\,c^4\,d^3\,e^2-6\,B\,b\,c^5\,d^5-6\,A\,b\,c^5\,d^4\,e+12\,A\,c^6\,d^5\right )}{2\,b^4\,d^3\,\left (b^3\,e^3-3\,b^2\,c\,d\,e^2+3\,b\,c^2\,d^2\,e-c^3\,d^3\right )}-\frac {x^2\,\left (-4\,B\,b^5\,d\,e^4+6\,A\,b^5\,e^5+2\,B\,b^4\,c\,d^2\,e^3+6\,B\,b^3\,c^2\,d^3\,e^2-13\,A\,b^3\,c^2\,d^2\,e^3-19\,B\,b^2\,c^3\,d^4\,e-A\,b^2\,c^3\,d^3\,e^2+9\,B\,b\,c^4\,d^5+32\,A\,b\,c^4\,d^4\,e-18\,A\,c^5\,d^5\right )}{2\,b^3\,d^3\,\left (b^3\,e^3-3\,b^2\,c\,d\,e^2+3\,b\,c^2\,d^2\,e-c^3\,d^3\right )}+\frac {c^2\,e\,x^4\,\left (2\,B\,b^4\,d\,e^3-3\,A\,b^4\,e^4-3\,B\,b^3\,c\,d^2\,e^2+3\,A\,b^3\,c\,d\,e^3+7\,B\,b^2\,c^2\,d^3\,e+3\,A\,b^2\,c^2\,d^2\,e^2-3\,B\,b\,c^3\,d^4-12\,A\,b\,c^3\,d^3\,e+6\,A\,c^4\,d^4\right )}{b^4\,d^3\,\left (b^3\,e^3-3\,b^2\,c\,d\,e^2+3\,b\,c^2\,d^2\,e-c^3\,d^3\right )}}{x^3\,\left (e\,b^2+2\,c\,d\,b\right )+x^4\,\left (d\,c^2+2\,b\,e\,c\right )+b^2\,d\,x^2+c^2\,e\,x^5} \] Input:

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

Output:

(log(x)*(b^2*(3*A*e^2 - 2*B*d*e) - b*(3*B*c*d^2 - 6*A*c*d*e) + 6*A*c^2*d^2 
))/(b^5*d^4) - (log(b + c*x)*(d^2*(6*A*c^6 - 3*B*b*c^5) - d*(18*A*b*c^5*e 
- 10*B*b^2*c^4*e) + 15*A*b^2*c^4*e^2 - 10*B*b^3*c^3*e^2))/(b^9*e^4 + b^5*c 
^4*d^4 - 4*b^6*c^3*d^3*e + 6*b^7*c^2*d^2*e^2 - 4*b^8*c*d*e^3) - (log(d + e 
*x)*(c*(5*B*d^2*e^4 - 6*A*d*e^5) + b*(3*A*e^6 - 2*B*d*e^5)))/(c^4*d^8 + b^ 
4*d^4*e^4 - 4*b^3*c*d^5*e^3 + 6*b^2*c^2*d^6*e^2 - 4*b*c^3*d^7*e) - (A/(2*b 
*d) - (x*(3*A*b*e + 4*A*c*d - 2*B*b*d))/(2*b^2*d^2) + (x^3*(12*A*c^6*d^5 - 
 12*A*b^5*c*e^5 - 6*B*b*c^5*d^5 + 9*A*b^4*c^2*d*e^4 + 5*B*b^2*c^4*d^4*e - 
30*A*b^2*c^4*d^3*e^2 + 15*A*b^3*c^3*d^2*e^3 + 15*B*b^3*c^3*d^3*e^2 - 10*B* 
b^4*c^2*d^2*e^3 - 6*A*b*c^5*d^4*e + 8*B*b^5*c*d*e^4))/(2*b^4*d^3*(b^3*e^3 
- c^3*d^3 + 3*b*c^2*d^2*e - 3*b^2*c*d*e^2)) - (x^2*(6*A*b^5*e^5 - 18*A*c^5 
*d^5 + 9*B*b*c^4*d^5 - 4*B*b^5*d*e^4 - 19*B*b^2*c^3*d^4*e + 2*B*b^4*c*d^2* 
e^3 - A*b^2*c^3*d^3*e^2 - 13*A*b^3*c^2*d^2*e^3 + 6*B*b^3*c^2*d^3*e^2 + 32* 
A*b*c^4*d^4*e))/(2*b^3*d^3*(b^3*e^3 - c^3*d^3 + 3*b*c^2*d^2*e - 3*b^2*c*d* 
e^2)) + (c^2*e*x^4*(6*A*c^4*d^4 - 3*A*b^4*e^4 - 3*B*b*c^3*d^4 + 2*B*b^4*d* 
e^3 + 7*B*b^2*c^2*d^3*e - 3*B*b^3*c*d^2*e^2 + 3*A*b^2*c^2*d^2*e^2 - 12*A*b 
*c^3*d^3*e + 3*A*b^3*c*d*e^3))/(b^4*d^3*(b^3*e^3 - c^3*d^3 + 3*b*c^2*d^2*e 
 - 3*b^2*c*d*e^2)))/(x^3*(b^2*e + 2*b*c*d) + x^4*(c^2*d + 2*b*c*e) + b^2*d 
*x^2 + c^2*e*x^5)
 

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 3700, normalized size of antiderivative = 11.18 \[ \int \frac {A+B x}{(d+e x)^2 \left (b x+c x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 60*log(b + c*x)*a*b**5*c**4*d**5*e**3*x**2 - 60*log(b + c*x)*a*b**5*c* 
*4*d**4*e**4*x**3 + 42*log(b + c*x)*a*b**4*c**5*d**6*e**2*x**2 - 78*log(b 
+ c*x)*a*b**4*c**5*d**5*e**3*x**3 - 120*log(b + c*x)*a*b**4*c**5*d**4*e**4 
*x**4 + 12*log(b + c*x)*a*b**3*c**6*d**7*e*x**2 + 96*log(b + c*x)*a*b**3*c 
**6*d**6*e**2*x**3 + 24*log(b + c*x)*a*b**3*c**6*d**5*e**3*x**4 - 60*log(b 
 + c*x)*a*b**3*c**6*d**4*e**4*x**5 - 12*log(b + c*x)*a*b**2*c**7*d**8*x**2 
 + 12*log(b + c*x)*a*b**2*c**7*d**7*e*x**3 + 66*log(b + c*x)*a*b**2*c**7*d 
**6*e**2*x**4 + 42*log(b + c*x)*a*b**2*c**7*d**5*e**3*x**5 - 24*log(b + c* 
x)*a*b*c**8*d**8*x**3 - 12*log(b + c*x)*a*b*c**8*d**7*e*x**4 + 12*log(b + 
c*x)*a*b*c**8*d**6*e**2*x**5 - 12*log(b + c*x)*a*c**9*d**8*x**4 - 12*log(b 
 + c*x)*a*c**9*d**7*e*x**5 + 40*log(b + c*x)*b**7*c**3*d**5*e**3*x**2 + 40 
*log(b + c*x)*b**7*c**3*d**4*e**4*x**3 - 20*log(b + c*x)*b**6*c**4*d**6*e* 
*2*x**2 + 60*log(b + c*x)*b**6*c**4*d**5*e**3*x**3 + 80*log(b + c*x)*b**6* 
c**4*d**4*e**4*x**4 - 8*log(b + c*x)*b**5*c**5*d**7*e*x**2 - 48*log(b + c* 
x)*b**5*c**5*d**6*e**2*x**3 + 40*log(b + c*x)*b**5*c**5*d**4*e**4*x**5 + 6 
*log(b + c*x)*b**4*c**6*d**8*x**2 - 10*log(b + c*x)*b**4*c**6*d**7*e*x**3 
- 36*log(b + c*x)*b**4*c**6*d**6*e**2*x**4 - 20*log(b + c*x)*b**4*c**6*d** 
5*e**3*x**5 + 12*log(b + c*x)*b**3*c**7*d**8*x**3 + 4*log(b + c*x)*b**3*c* 
*7*d**7*e*x**4 - 8*log(b + c*x)*b**3*c**7*d**6*e**2*x**5 + 6*log(b + c*x)* 
b**2*c**8*d**8*x**4 + 6*log(b + c*x)*b**2*c**8*d**7*e*x**5 - 12*log(d +...