\(\int \frac {A+B x}{(d+e x) (b x+c x^2)^3} \, dx\) [50]

Optimal result

Integrand size = 24, antiderivative size = 279 \[ \int \frac {A+B x}{(d+e x) \left (b x+c x^2\right )^3} \, dx=-\frac {A}{2 b^3 d x^2}-\frac {b B d-3 A c d-A b e}{b^4 d^2 x}-\frac {c^2 (b B-A c)}{2 b^3 (c d-b e) (b+c x)^2}+\frac {c^2 \left (3 A c^2 d+3 b^2 B e-2 b c (B d+2 A e)\right )}{b^4 (c d-b e)^2 (b+c x)}+\frac {\left (6 A c^2 d^2-3 b c d (B d-A e)-b^2 e (B d-A e)\right ) \log (x)}{b^5 d^3}-\frac {c^2 \left (6 A c^3 d^2-6 b^3 B e^2-3 b c^2 d (B d+5 A e)+2 b^2 c e (4 B d+5 A e)\right ) \log (b+c x)}{b^5 (c d-b e)^3}-\frac {e^4 (B d-A e) \log (d+e x)}{d^3 (c d-b e)^3} \] Output:

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

Mathematica [A] (verified)

Time = 1.30 (sec) , antiderivative size = 276, normalized size of antiderivative = 0.99 \[ \int \frac {A+B x}{(d+e x) \left (b x+c x^2\right )^3} \, dx=-\frac {A}{2 b^3 d x^2}+\frac {-b B d+3 A c d+A b e}{b^4 d^2 x}+\frac {c^2 (b B-A c)}{2 b^3 (-c d+b e) (b+c x)^2}+\frac {c^2 \left (3 A c^2 d+3 b^2 B e-2 b c (B d+2 A e)\right )}{b^4 (c d-b e)^2 (b+c x)}-\frac {\left (-6 A c^2 d^2+3 b c d (B d-A e)+b^2 e (B d-A e)\right ) \log (x)}{b^5 d^3}+\frac {c^2 \left (6 A c^3 d^2-6 b^3 B e^2-3 b c^2 d (B d+5 A e)+2 b^2 c e (4 B d+5 A e)\right ) \log (b+c x)}{b^5 (-c d+b e)^3}+\frac {e^4 (-B d+A e) \log (d+e x)}{d^3 (c d-b e)^3} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.15 (sec) , antiderivative size = 279, 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.

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

\(\Big \downarrow \) 1206

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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 = 1.00 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.03

method	result	size
default	\(-\frac {c^{2} \left (4 A b c e -3 A \,c^{2} d -3 b^{2} B e +2 B b c d \right )}{b^{4} \left (b e -c d \right )^{2} \left (c x +b \right )}+\frac {c^{2} \left (10 A \,b^{2} c \,e^{2}-15 A b \,c^{2} d e +6 A \,c^{3} d^{2}-6 b^{3} B \,e^{2}+8 B \,b^{2} c d e -3 B b \,c^{2} d^{2}\right ) \ln \left (c x +b \right )}{b^{5} \left (b e -c d \right )^{3}}-\frac {\left (A c -B b \right ) c^{2}}{2 b^{3} \left (b e -c d \right ) \left (c x +b \right )^{2}}-\frac {\left (A e -B d \right ) e^{4} \ln \left (e x +d \right )}{\left (b e -c d \right )^{3} d^{3}}-\frac {A}{2 b^{3} d \,x^{2}}-\frac {-A b e -3 A c d +B b d}{b^{4} d^{2} x}+\frac {\left (A \,b^{2} e^{2}+3 A b c d e +6 A \,c^{2} d^{2}-B \,b^{2} d e -3 B b c \,d^{2}\right ) \ln \left (x \right )}{b^{5} d^{3}}\)	\(287\)
norman	\(\frac {\frac {\left (A b e +2 A c d -B b d \right ) x}{b^{2} d^{2}}+\frac {\left (A \,b^{3} c^{3} e^{3}+A \,b^{2} c^{4} d \,e^{2}-9 A b \,c^{5} d^{2} e +6 A \,c^{6} d^{3}-B \,b^{3} c^{3} d \,e^{2}+5 B \,b^{2} c^{4} d^{2} e -3 B b \,c^{5} d^{3}\right ) x^{3}}{c \,d^{2} \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right ) b^{4}}-\frac {A}{2 b d}+\frac {\left (4 A \,b^{3} c^{3} e^{3}+3 A \,b^{2} c^{4} d \,e^{2}-27 A b \,c^{5} d^{2} e +18 A \,c^{6} d^{3}-4 B \,b^{3} c^{3} d \,e^{2}+15 B \,b^{2} c^{4} d^{2} e -9 B b \,c^{5} d^{3}\right ) x^{2}}{2 d^{2} c^{2} \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right ) b^{3}}}{x^{2} \left (c x +b \right )^{2}}+\frac {\left (A \,b^{2} e^{2}+3 A b c d e +6 A \,c^{2} d^{2}-B \,b^{2} d e -3 B b c \,d^{2}\right ) \ln \left (x \right )}{b^{5} d^{3}}+\frac {c^{2} \left (10 A \,b^{2} c \,e^{2}-15 A b \,c^{2} d e +6 A \,c^{3} d^{2}-6 b^{3} B \,e^{2}+8 B \,b^{2} c d e -3 B b \,c^{2} d^{2}\right ) \ln \left (c x +b \right )}{\left (b^{3} e^{3}-3 d \,e^{2} b^{2} c +3 d^{2} e b \,c^{2}-d^{3} c^{3}\right ) b^{5}}-\frac {e^{4} \left (A e -B d \right ) \ln \left (e x +d \right )}{d^{3} \left (b^{3} e^{3}-3 d \,e^{2} b^{2} c +3 d^{2} e b \,c^{2}-d^{3} c^{3}\right )}\)	\(499\)
risch	\(\frac {\frac {\left (A \,b^{3} e^{3}+A \,b^{2} c d \,e^{2}-9 A b \,c^{2} d^{2} e +6 A \,c^{3} d^{3}-B \,b^{3} d \,e^{2}+5 b^{2} B c \,d^{2} e -3 B b \,c^{2} d^{3}\right ) c^{2} x^{3}}{b^{4} d^{2} \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right )}+\frac {c \left (4 A \,b^{3} e^{3}+3 A \,b^{2} c d \,e^{2}-27 A b \,c^{2} d^{2} e +18 A \,c^{3} d^{3}-4 B \,b^{3} d \,e^{2}+15 b^{2} B c \,d^{2} e -9 B b \,c^{2} d^{3}\right ) x^{2}}{2 b^{3} d^{2} \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right )}+\frac {\left (A b e +2 A c d -B b d \right ) x}{b^{2} d^{2}}-\frac {A}{2 b d}}{x^{2} \left (c x +b \right )^{2}}+\frac {\ln \left (-x \right ) A \,e^{2}}{b^{3} d^{3}}+\frac {3 \ln \left (-x \right ) A c e}{b^{4} d^{2}}+\frac {6 \ln \left (-x \right ) A \,c^{2}}{b^{5} d}-\frac {\ln \left (-x \right ) B e}{b^{3} d^{2}}-\frac {3 \ln \left (-x \right ) B c}{b^{4} d}-\frac {e^{5} \ln \left (-e x -d \right ) A}{d^{3} \left (b^{3} e^{3}-3 d \,e^{2} b^{2} c +3 d^{2} e b \,c^{2}-d^{3} c^{3}\right )}+\frac {e^{4} \ln \left (-e x -d \right ) B}{d^{2} \left (b^{3} e^{3}-3 d \,e^{2} b^{2} c +3 d^{2} e b \,c^{2}-d^{3} c^{3}\right )}+\frac {10 c^{3} \ln \left (c x +b \right ) A \,e^{2}}{\left (b^{3} e^{3}-3 d \,e^{2} b^{2} c +3 d^{2} e b \,c^{2}-d^{3} c^{3}\right ) b^{3}}-\frac {15 c^{4} \ln \left (c x +b \right ) A d e}{\left (b^{3} e^{3}-3 d \,e^{2} b^{2} c +3 d^{2} e b \,c^{2}-d^{3} c^{3}\right ) b^{4}}+\frac {6 c^{5} \ln \left (c x +b \right ) A \,d^{2}}{\left (b^{3} e^{3}-3 d \,e^{2} b^{2} c +3 d^{2} e b \,c^{2}-d^{3} c^{3}\right ) b^{5}}-\frac {6 c^{2} \ln \left (c x +b \right ) B \,e^{2}}{\left (b^{3} e^{3}-3 d \,e^{2} b^{2} c +3 d^{2} e b \,c^{2}-d^{3} c^{3}\right ) b^{2}}+\frac {8 c^{3} \ln \left (c x +b \right ) B d e}{\left (b^{3} e^{3}-3 d \,e^{2} b^{2} c +3 d^{2} e b \,c^{2}-d^{3} c^{3}\right ) b^{3}}-\frac {3 c^{4} \ln \left (c x +b \right ) B \,d^{2}}{\left (b^{3} e^{3}-3 d \,e^{2} b^{2} c +3 d^{2} e b \,c^{2}-d^{3} c^{3}\right ) b^{4}}\)	\(778\)
parallelrisch	\(\text {Expression too large to display}\)	\(1475\)

Input:

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

Output:

-c^2*(4*A*b*c*e-3*A*c^2*d-3*B*b^2*e+2*B*b*c*d)/b^4/(b*e-c*d)^2/(c*x+b)+c^2 
*(10*A*b^2*c*e^2-15*A*b*c^2*d*e+6*A*c^3*d^2-6*B*b^3*e^2+8*B*b^2*c*d*e-3*B* 
b*c^2*d^2)/b^5/(b*e-c*d)^3*ln(c*x+b)-1/2*(A*c-B*b)*c^2/b^3/(b*e-c*d)/(c*x+ 
b)^2-(A*e-B*d)*e^4/(b*e-c*d)^3/d^3*ln(e*x+d)-1/2*A/b^3/d/x^2-(-A*b*e-3*A*c 
*d+B*b*d)/b^4/d^2/x+(A*b^2*e^2+3*A*b*c*d*e+6*A*c^2*d^2-B*b^2*d*e-3*B*b*c*d 
^2)/b^5/d^3*ln(x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1161 vs. \(2 (275) = 550\).

Time = 173.86 (sec) , antiderivative size = 1161, normalized size of antiderivative = 4.16 \[ \int \frac {A+B x}{(d+e x) \left (b x+c x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/2*(A*b^4*c^3*d^5 - 3*A*b^5*c^2*d^4*e + 3*A*b^6*c*d^3*e^2 - A*b^7*d^2*e^ 
3 - 2*(B*b^5*c^2*d^2*e^3 - A*b^5*c^2*d*e^4 - 3*(B*b^2*c^5 - 2*A*b*c^6)*d^5 
 + (8*B*b^3*c^4 - 15*A*b^2*c^5)*d^4*e - 2*(3*B*b^4*c^3 - 5*A*b^3*c^4)*d^3* 
e^2)*x^3 + (4*A*b^6*c*d*e^4 + 9*(B*b^3*c^4 - 2*A*b^2*c^5)*d^5 - 3*(8*B*b^4 
*c^3 - 15*A*b^3*c^4)*d^4*e + (19*B*b^5*c^2 - 30*A*b^4*c^3)*d^3*e^2 - (4*B* 
b^6*c + A*b^5*c^2)*d^2*e^3)*x^2 + 2*(A*b^7*d*e^4 + (B*b^4*c^3 - 2*A*b^3*c^ 
4)*d^5 - (3*B*b^5*c^2 - 5*A*b^4*c^3)*d^4*e + 3*(B*b^6*c - A*b^5*c^2)*d^3*e 
^2 - (B*b^7 + A*b^6*c)*d^2*e^3)*x - 2*((3*(B*b*c^6 - 2*A*c^7)*d^5 - (8*B*b 
^2*c^5 - 15*A*b*c^6)*d^4*e + 2*(3*B*b^3*c^4 - 5*A*b^2*c^5)*d^3*e^2)*x^4 + 
2*(3*(B*b^2*c^5 - 2*A*b*c^6)*d^5 - (8*B*b^3*c^4 - 15*A*b^2*c^5)*d^4*e + 2* 
(3*B*b^4*c^3 - 5*A*b^3*c^4)*d^3*e^2)*x^3 + (3*(B*b^3*c^4 - 2*A*b^2*c^5)*d^ 
5 - (8*B*b^4*c^3 - 15*A*b^3*c^4)*d^4*e + 2*(3*B*b^5*c^2 - 5*A*b^4*c^3)*d^3 
*e^2)*x^2)*log(c*x + b) + 2*((B*b^5*c^2*d*e^4 - A*b^5*c^2*e^5)*x^4 + 2*(B* 
b^6*c*d*e^4 - A*b^6*c*e^5)*x^3 + (B*b^7*d*e^4 - A*b^7*e^5)*x^2)*log(e*x + 
d) - 2*((B*b^5*c^2*d*e^4 - A*b^5*c^2*e^5 - 3*(B*b*c^6 - 2*A*c^7)*d^5 + (8* 
B*b^2*c^5 - 15*A*b*c^6)*d^4*e - 2*(3*B*b^3*c^4 - 5*A*b^2*c^5)*d^3*e^2)*x^4 
 + 2*(B*b^6*c*d*e^4 - A*b^6*c*e^5 - 3*(B*b^2*c^5 - 2*A*b*c^6)*d^5 + (8*B*b 
^3*c^4 - 15*A*b^2*c^5)*d^4*e - 2*(3*B*b^4*c^3 - 5*A*b^3*c^4)*d^3*e^2)*x^3 
+ (B*b^7*d*e^4 - A*b^7*e^5 - 3*(B*b^3*c^4 - 2*A*b^2*c^5)*d^5 + (8*B*b^4*c^ 
3 - 15*A*b^3*c^4)*d^4*e - 2*(3*B*b^5*c^2 - 5*A*b^4*c^3)*d^3*e^2)*x^2)*l...
 

Sympy [F(-1)]

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

integrate((B*x+A)/(e*x+d)/(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. 612 vs. \(2 (275) = 550\).

Time = 0.07 (sec) , antiderivative size = 612, normalized size of antiderivative = 2.19 \[ \int \frac {A+B x}{(d+e x) \left (b x+c x^2\right )^3} \, dx=\frac {{\left (3 \, {\left (B b c^{4} - 2 \, A c^{5}\right )} d^{2} - {\left (8 \, B b^{2} c^{3} - 15 \, A b c^{4}\right )} d e + 2 \, {\left (3 \, B b^{3} c^{2} - 5 \, A b^{2} c^{3}\right )} e^{2}\right )} \log \left (c x + b\right )}{b^{5} c^{3} d^{3} - 3 \, b^{6} c^{2} d^{2} e + 3 \, b^{7} c d e^{2} - b^{8} e^{3}} - \frac {{\left (B d e^{4} - A e^{5}\right )} \log \left (e x + d\right )}{c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}} - \frac {A b^{3} c^{2} d^{3} - 2 \, A b^{4} c d^{2} e + A b^{5} d e^{2} - 2 \, {\left (A b^{3} c^{2} e^{3} - 3 \, {\left (B b c^{4} - 2 \, A c^{5}\right )} d^{3} + {\left (5 \, B b^{2} c^{3} - 9 \, A b c^{4}\right )} d^{2} e - {\left (B b^{3} c^{2} - A b^{2} c^{3}\right )} d e^{2}\right )} x^{3} - {\left (4 \, A b^{4} c e^{3} - 9 \, {\left (B b^{2} c^{3} - 2 \, A b c^{4}\right )} d^{3} + 3 \, {\left (5 \, B b^{3} c^{2} - 9 \, A b^{2} c^{3}\right )} d^{2} e - {\left (4 \, B b^{4} c - 3 \, A b^{3} c^{2}\right )} d e^{2}\right )} x^{2} + 2 \, {\left (B b^{5} d e^{2} - A b^{5} e^{3} + {\left (B b^{3} c^{2} - 2 \, A b^{2} c^{3}\right )} d^{3} - {\left (2 \, B b^{4} c - 3 \, A b^{3} c^{2}\right )} d^{2} e\right )} x}{2 \, {\left ({\left (b^{4} c^{4} d^{4} - 2 \, b^{5} c^{3} d^{3} e + b^{6} c^{2} d^{2} e^{2}\right )} x^{4} + 2 \, {\left (b^{5} c^{3} d^{4} - 2 \, b^{6} c^{2} d^{3} e + b^{7} c d^{2} e^{2}\right )} x^{3} + {\left (b^{6} c^{2} d^{4} - 2 \, b^{7} c d^{3} e + b^{8} d^{2} e^{2}\right )} x^{2}\right )}} + \frac {{\left (A b^{2} e^{2} - 3 \, {\left (B b c - 2 \, A c^{2}\right )} d^{2} - {\left (B b^{2} - 3 \, A b c\right )} d e\right )} \log \left (x\right )}{b^{5} d^{3}} \] Input:

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 660 vs. \(2 (275) = 550\).

Time = 0.27 (sec) , antiderivative size = 660, normalized size of antiderivative = 2.37 \[ \int \frac {A+B x}{(d+e x) \left (b x+c x^2\right )^3} \, dx=\frac {{\left (3 \, B b c^{5} d^{2} - 6 \, A c^{6} d^{2} - 8 \, B b^{2} c^{4} d e + 15 \, A b c^{5} d e + 6 \, B b^{3} c^{3} e^{2} - 10 \, A b^{2} c^{4} e^{2}\right )} \log \left ({\left | c x + b \right |}\right )}{b^{5} c^{4} d^{3} - 3 \, b^{6} c^{3} d^{2} e + 3 \, b^{7} c^{2} d e^{2} - b^{8} c e^{3}} - \frac {{\left (B d e^{5} - A e^{6}\right )} \log \left ({\left | e x + d \right |}\right )}{c^{3} d^{6} e - 3 \, b c^{2} d^{5} e^{2} + 3 \, b^{2} c d^{4} e^{3} - b^{3} d^{3} e^{4}} - \frac {{\left (3 \, B b c d^{2} - 6 \, A c^{2} d^{2} + B b^{2} d e - 3 \, A b c d e - A b^{2} e^{2}\right )} \log \left ({\left | x \right |}\right )}{b^{5} d^{3}} - \frac {A b^{3} c^{3} d^{5} - 3 \, A b^{4} c^{2} d^{4} e + 3 \, A b^{5} c d^{3} e^{2} - A b^{6} d^{2} e^{3} + 2 \, {\left (3 \, B b c^{5} d^{5} - 6 \, A c^{6} d^{5} - 8 \, B b^{2} c^{4} d^{4} e + 15 \, A b c^{5} d^{4} e + 6 \, B b^{3} c^{3} d^{3} e^{2} - 10 \, A b^{2} c^{4} d^{3} e^{2} - B b^{4} c^{2} d^{2} e^{3} + A b^{4} c^{2} d e^{4}\right )} x^{3} + {\left (9 \, B b^{2} c^{4} d^{5} - 18 \, A b c^{5} d^{5} - 24 \, B b^{3} c^{3} d^{4} e + 45 \, A b^{2} c^{4} d^{4} e + 19 \, B b^{4} c^{2} d^{3} e^{2} - 30 \, A b^{3} c^{3} d^{3} e^{2} - 4 \, B b^{5} c d^{2} e^{3} - A b^{4} c^{2} d^{2} e^{3} + 4 \, A b^{5} c d e^{4}\right )} x^{2} + 2 \, {\left (B b^{3} c^{3} d^{5} - 2 \, A b^{2} c^{4} d^{5} - 3 \, B b^{4} c^{2} d^{4} e + 5 \, A b^{3} c^{3} d^{4} e + 3 \, B b^{5} c d^{3} e^{2} - 3 \, A b^{4} c^{2} d^{3} e^{2} - B b^{6} d^{2} e^{3} - A b^{5} c d^{2} e^{3} + A b^{6} d e^{4}\right )} x}{2 \, {\left (c d - b e\right )}^{3} {\left (c x + b\right )}^{2} b^{4} d^{3} x^{2}} \] Input:

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

Output:

(3*B*b*c^5*d^2 - 6*A*c^6*d^2 - 8*B*b^2*c^4*d*e + 15*A*b*c^5*d*e + 6*B*b^3* 
c^3*e^2 - 10*A*b^2*c^4*e^2)*log(abs(c*x + b))/(b^5*c^4*d^3 - 3*b^6*c^3*d^2 
*e + 3*b^7*c^2*d*e^2 - b^8*c*e^3) - (B*d*e^5 - A*e^6)*log(abs(e*x + d))/(c 
^3*d^6*e - 3*b*c^2*d^5*e^2 + 3*b^2*c*d^4*e^3 - b^3*d^3*e^4) - (3*B*b*c*d^2 
 - 6*A*c^2*d^2 + B*b^2*d*e - 3*A*b*c*d*e - A*b^2*e^2)*log(abs(x))/(b^5*d^3 
) - 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*B*b*c^5*d^5 - 6*A*c^6*d^5 - 8*B*b^2*c^4*d^4*e + 15*A*b*c^5*d^4 
*e + 6*B*b^3*c^3*d^3*e^2 - 10*A*b^2*c^4*d^3*e^2 - B*b^4*c^2*d^2*e^3 + A*b^ 
4*c^2*d*e^4)*x^3 + (9*B*b^2*c^4*d^5 - 18*A*b*c^5*d^5 - 24*B*b^3*c^3*d^4*e 
+ 45*A*b^2*c^4*d^4*e + 19*B*b^4*c^2*d^3*e^2 - 30*A*b^3*c^3*d^3*e^2 - 4*B*b 
^5*c*d^2*e^3 - A*b^4*c^2*d^2*e^3 + 4*A*b^5*c*d*e^4)*x^2 + 2*(B*b^3*c^3*d^5 
 - 2*A*b^2*c^4*d^5 - 3*B*b^4*c^2*d^4*e + 5*A*b^3*c^3*d^4*e + 3*B*b^5*c*d^3 
*e^2 - 3*A*b^4*c^2*d^3*e^2 - B*b^6*d^2*e^3 - A*b^5*c*d^2*e^3 + A*b^6*d*e^4 
)*x)/((c*d - b*e)^3*(c*x + b)^2*b^4*d^3*x^2)
 

Mupad [B] (verification not implemented)

Time = 12.15 (sec) , antiderivative size = 512, normalized size of antiderivative = 1.84 \[ \int \frac {A+B x}{(d+e x) \left (b x+c x^2\right )^3} \, dx=\frac {\frac {x\,\left (A\,b\,e+2\,A\,c\,d-B\,b\,d\right )}{b^2\,d^2}-\frac {A}{2\,b\,d}+\frac {x^3\,\left (-B\,b^3\,c^2\,d\,e^2+A\,b^3\,c^2\,e^3+5\,B\,b^2\,c^3\,d^2\,e+A\,b^2\,c^3\,d\,e^2-3\,B\,b\,c^4\,d^3-9\,A\,b\,c^4\,d^2\,e+6\,A\,c^5\,d^3\right )}{b^4\,d^2\,\left (b^2\,e^2-2\,b\,c\,d\,e+c^2\,d^2\right )}+\frac {x^2\,\left (-4\,B\,b^3\,c\,d\,e^2+4\,A\,b^3\,c\,e^3+15\,B\,b^2\,c^2\,d^2\,e+3\,A\,b^2\,c^2\,d\,e^2-9\,B\,b\,c^3\,d^3-27\,A\,b\,c^3\,d^2\,e+18\,A\,c^4\,d^3\right )}{2\,b^3\,d^2\,\left (b^2\,e^2-2\,b\,c\,d\,e+c^2\,d^2\right )}}{b^2\,x^2+2\,b\,c\,x^3+c^2\,x^4}+\frac {\ln \left (d+e\,x\right )\,\left (A\,e^5-B\,d\,e^4\right )}{-b^3\,d^3\,e^3+3\,b^2\,c\,d^4\,e^2-3\,b\,c^2\,d^5\,e+c^3\,d^6}+\frac {\ln \left (b+c\,x\right )\,\left (d^2\,\left (6\,A\,c^5-3\,B\,b\,c^4\right )-d\,\left (15\,A\,b\,c^4\,e-8\,B\,b^2\,c^3\,e\right )+10\,A\,b^2\,c^3\,e^2-6\,B\,b^3\,c^2\,e^2\right )}{b^8\,e^3-3\,b^7\,c\,d\,e^2+3\,b^6\,c^2\,d^2\,e-b^5\,c^3\,d^3}+\frac {\ln \left (x\right )\,\left (d^2\,\left (6\,A\,c^2-3\,B\,b\,c\right )-d\,\left (B\,b^2\,e-3\,A\,b\,c\,e\right )+A\,b^2\,e^2\right )}{b^5\,d^3} \] Input:

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

Output:

((x*(A*b*e + 2*A*c*d - B*b*d))/(b^2*d^2) - A/(2*b*d) + (x^3*(6*A*c^5*d^3 - 
 3*B*b*c^4*d^3 + A*b^3*c^2*e^3 + A*b^2*c^3*d*e^2 + 5*B*b^2*c^3*d^2*e - B*b 
^3*c^2*d*e^2 - 9*A*b*c^4*d^2*e))/(b^4*d^2*(b^2*e^2 + c^2*d^2 - 2*b*c*d*e)) 
 + (x^2*(18*A*c^4*d^3 + 4*A*b^3*c*e^3 - 9*B*b*c^3*d^3 + 3*A*b^2*c^2*d*e^2 
+ 15*B*b^2*c^2*d^2*e - 27*A*b*c^3*d^2*e - 4*B*b^3*c*d*e^2))/(2*b^3*d^2*(b^ 
2*e^2 + c^2*d^2 - 2*b*c*d*e)))/(b^2*x^2 + c^2*x^4 + 2*b*c*x^3) + (log(d + 
e*x)*(A*e^5 - B*d*e^4))/(c^3*d^6 - b^3*d^3*e^3 + 3*b^2*c*d^4*e^2 - 3*b*c^2 
*d^5*e) + (log(b + c*x)*(d^2*(6*A*c^5 - 3*B*b*c^4) - d*(15*A*b*c^4*e - 8*B 
*b^2*c^3*e) + 10*A*b^2*c^3*e^2 - 6*B*b^3*c^2*e^2))/(b^8*e^3 - b^5*c^3*d^3 
+ 3*b^6*c^2*d^2*e - 3*b^7*c*d*e^2) + (log(x)*(d^2*(6*A*c^2 - 3*B*b*c) - d* 
(B*b^2*e - 3*A*b*c*e) + A*b^2*e^2))/(b^5*d^3)
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 1492, normalized size of antiderivative = 5.35 \[ \int \frac {A+B x}{(d+e x) \left (b x+c x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(20*log(b + c*x)*a*b**4*c**3*d**3*e**2*x**2 - 30*log(b + c*x)*a*b**3*c**4* 
d**4*e*x**2 + 40*log(b + c*x)*a*b**3*c**4*d**3*e**2*x**3 + 12*log(b + c*x) 
*a*b**2*c**5*d**5*x**2 - 60*log(b + c*x)*a*b**2*c**5*d**4*e*x**3 + 20*log( 
b + c*x)*a*b**2*c**5*d**3*e**2*x**4 + 24*log(b + c*x)*a*b*c**6*d**5*x**3 - 
 30*log(b + c*x)*a*b*c**6*d**4*e*x**4 + 12*log(b + c*x)*a*c**7*d**5*x**4 - 
 12*log(b + c*x)*b**6*c**2*d**3*e**2*x**2 + 16*log(b + c*x)*b**5*c**3*d**4 
*e*x**2 - 24*log(b + c*x)*b**5*c**3*d**3*e**2*x**3 - 6*log(b + c*x)*b**4*c 
**4*d**5*x**2 + 32*log(b + c*x)*b**4*c**4*d**4*e*x**3 - 12*log(b + c*x)*b* 
*4*c**4*d**3*e**2*x**4 - 12*log(b + c*x)*b**3*c**5*d**5*x**3 + 16*log(b + 
c*x)*b**3*c**5*d**4*e*x**4 - 6*log(b + c*x)*b**2*c**6*d**5*x**4 - 2*log(d 
+ e*x)*a*b**7*e**5*x**2 - 4*log(d + e*x)*a*b**6*c*e**5*x**3 - 2*log(d + e* 
x)*a*b**5*c**2*e**5*x**4 + 2*log(d + e*x)*b**8*d*e**4*x**2 + 4*log(d + e*x 
)*b**7*c*d*e**4*x**3 + 2*log(d + e*x)*b**6*c**2*d*e**4*x**4 + 2*log(x)*a*b 
**7*e**5*x**2 + 4*log(x)*a*b**6*c*e**5*x**3 + 2*log(x)*a*b**5*c**2*e**5*x* 
*4 - 20*log(x)*a*b**4*c**3*d**3*e**2*x**2 + 30*log(x)*a*b**3*c**4*d**4*e*x 
**2 - 40*log(x)*a*b**3*c**4*d**3*e**2*x**3 - 12*log(x)*a*b**2*c**5*d**5*x* 
*2 + 60*log(x)*a*b**2*c**5*d**4*e*x**3 - 20*log(x)*a*b**2*c**5*d**3*e**2*x 
**4 - 24*log(x)*a*b*c**6*d**5*x**3 + 30*log(x)*a*b*c**6*d**4*e*x**4 - 12*l 
og(x)*a*c**7*d**5*x**4 - 2*log(x)*b**8*d*e**4*x**2 - 4*log(x)*b**7*c*d*e** 
4*x**3 + 12*log(x)*b**6*c**2*d**3*e**2*x**2 - 2*log(x)*b**6*c**2*d*e**4...