\(\int \frac {A+B x}{(a+b x) (d+e x)^4} \, dx\) [109]

Optimal result

Integrand size = 20, antiderivative size = 146 \[ \int \frac {A+B x}{(a+b x) (d+e x)^4} \, dx=\frac {-B d+A e}{3 e (b d-a e) (d+e x)^3}+\frac {A b-a B}{2 (b d-a e)^2 (d+e x)^2}+\frac {b (A b-a B)}{(b d-a e)^3 (d+e x)}+\frac {b^2 (A b-a B) \log (a+b x)}{(b d-a e)^4}-\frac {b^2 (A b-a B) \log (d+e x)}{(b d-a e)^4} \] Output:

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

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.99 \[ \int \frac {A+B x}{(a+b x) (d+e x)^4} \, dx=\frac {B d-A e}{3 e (-b d+a e) (d+e x)^3}+\frac {A b-a B}{2 (b d-a e)^2 (d+e x)^2}+\frac {b (A b-a B)}{(b d-a e)^3 (d+e x)}+\frac {b^2 (A b-a B) \log (a+b x)}{(b d-a e)^4}+\frac {b^2 (-A b+a B) \log (d+e x)}{(b d-a e)^4} \] Input:

Integrate[(A + B*x)/((a + b*x)*(d + e*x)^4),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {86, 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.

\(\displaystyle \int \frac {A+B x}{(a+b x) (d+e x)^4} \, dx\)

\(\Big \downarrow \) 86

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {b^2 (A b-a B) \log (a+b x)}{(b d-a e)^4}-\frac {b^2 (A b-a B) \log (d+e x)}{(b d-a e)^4}+\frac {b (A b-a B)}{(d+e x) (b d-a e)^3}+\frac {A b-a B}{2 (d+e x)^2 (b d-a e)^2}-\frac {B d-A e}{3 e (d+e x)^3 (b d-a e)}\)

Input:

Int[(A + B*x)/((a + b*x)*(d + e*x)^4),x]
 

Output:

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

Defintions of rubi rules used

rule 86

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ 
.), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; 
 FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 
] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p 
+ 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
 

rule 2009

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

Maple [A] (verified)

Time = 0.33 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.99

method	result	size
default	\(\frac {\left (A b -B a \right ) b^{2} \ln \left (b x +a \right )}{\left (a e -d b \right )^{4}}-\frac {A e -B d}{3 \left (a e -d b \right ) e \left (e x +d \right )^{3}}-\frac {\left (A b -B a \right ) b}{\left (a e -d b \right )^{3} \left (e x +d \right )}+\frac {A b -B a}{2 \left (a e -d b \right )^{2} \left (e x +d \right )^{2}}-\frac {\left (A b -B a \right ) b^{2} \ln \left (e x +d \right )}{\left (a e -d b \right )^{4}}\)	\(144\)
norman	\(\frac {-\frac {2 A \,a^{2} e^{5}-7 A a b d \,e^{4}+11 A \,b^{2} d^{2} e^{3}+B \,a^{2} d \,e^{4}-5 B a b \,d^{2} e^{3}-2 B \,b^{2} d^{3} e^{2}}{6 e^{3} \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}-\frac {\left (A \,b^{2} e^{3}-B a b \,e^{3}\right ) x^{2}}{e \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}+\frac {\left (A a b \,e^{4}-5 A \,b^{2} d \,e^{3}-B \,a^{2} e^{4}+5 B a b d \,e^{3}\right ) x}{2 e^{2} \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}}{\left (e x +d \right )^{3}}+\frac {b^{2} \left (A b -B a \right ) \ln \left (b x +a \right )}{a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}}-\frac {b^{2} \left (A b -B a \right ) \ln \left (e x +d \right )}{a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}}\)	\(399\)
risch	\(\frac {-\frac {b \,e^{2} \left (A b -B a \right ) x^{2}}{a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}}+\frac {\left (a e -5 d b \right ) e \left (A b -B a \right ) x}{2 a^{3} e^{3}-6 a^{2} b d \,e^{2}+6 a \,b^{2} d^{2} e -2 b^{3} d^{3}}-\frac {2 a^{2} A \,e^{3}-7 A a b d \,e^{2}+11 A \,b^{2} d^{2} e +B \,a^{2} d \,e^{2}-5 B a b \,d^{2} e -2 b^{2} B \,d^{3}}{6 e \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}}{\left (e x +d \right )^{3}}-\frac {b^{3} \ln \left (e x +d \right ) A}{a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}}+\frac {b^{2} \ln \left (e x +d \right ) B a}{a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}}+\frac {b^{3} \ln \left (-b x -a \right ) A}{a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}}-\frac {b^{2} \ln \left (-b x -a \right ) B a}{a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}}\)	\(481\)
parallelrisch	\(\frac {-18 B \ln \left (b x +a \right ) x^{2} a \,b^{2} d \,e^{5}+18 B \ln \left (e x +d \right ) x^{2} a \,b^{2} d \,e^{5}-18 B \ln \left (b x +a \right ) x a \,b^{2} d^{2} e^{4}+18 B \ln \left (e x +d \right ) x a \,b^{2} d^{2} e^{4}-3 B x \,a^{3} e^{6}+6 B \,x^{2} a^{2} b \,e^{6}+18 A \ln \left (b x +a \right ) x \,b^{3} d^{2} e^{4}-18 A \ln \left (e x +d \right ) x \,b^{3} d^{2} e^{4}-6 B \ln \left (b x +a \right ) a \,b^{2} d^{3} e^{3}+6 B \ln \left (e x +d \right ) a \,b^{2} d^{3} e^{3}-2 A \,a^{3} e^{6}+3 A x \,a^{2} b \,e^{6}+15 A x \,b^{3} d^{2} e^{4}+6 A \ln \left (b x +a \right ) x^{3} b^{3} e^{6}-6 A \ln \left (e x +d \right ) x^{3} b^{3} e^{6}+6 A \ln \left (b x +a \right ) b^{3} d^{3} e^{3}-6 A \ln \left (e x +d \right ) b^{3} d^{3} e^{3}-6 A \,x^{2} a \,b^{2} e^{6}+6 A \,x^{2} b^{3} d \,e^{5}-6 B \,x^{2} a \,b^{2} d \,e^{5}-18 A x a \,b^{2} d \,e^{5}+18 B x \,a^{2} b d \,e^{5}-15 B x a \,b^{2} d^{2} e^{4}-6 B \ln \left (b x +a \right ) x^{3} a \,b^{2} e^{6}+6 B \ln \left (e x +d \right ) x^{3} a \,b^{2} e^{6}+18 A \ln \left (b x +a \right ) x^{2} b^{3} d \,e^{5}-18 A \ln \left (e x +d \right ) x^{2} b^{3} d \,e^{5}+9 A \,a^{2} b d \,e^{5}-18 A a \,b^{2} d^{2} e^{4}+6 B \,a^{2} b \,d^{2} e^{4}-3 B a \,b^{2} d^{3} e^{3}+11 A \,b^{3} d^{3} e^{3}-B \,a^{3} d \,e^{5}-2 B \,b^{3} d^{4} e^{2}}{6 \left (a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) \left (e x +d \right )^{3} e^{3}}\)	\(587\)

Input:

int((B*x+A)/(b*x+a)/(e*x+d)^4,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 608 vs. \(2 (143) = 286\).

Time = 0.08 (sec) , antiderivative size = 608, normalized size of antiderivative = 4.16 \[ \int \frac {A+B x}{(a+b x) (d+e x)^4} \, dx=-\frac {2 \, B b^{3} d^{4} + 2 \, A a^{3} e^{4} + {\left (3 \, B a b^{2} - 11 \, A b^{3}\right )} d^{3} e - 6 \, {\left (B a^{2} b - 3 \, A a b^{2}\right )} d^{2} e^{2} + {\left (B a^{3} - 9 \, A a^{2} b\right )} d e^{3} + 6 \, {\left ({\left (B a b^{2} - A b^{3}\right )} d e^{3} - {\left (B a^{2} b - A a b^{2}\right )} e^{4}\right )} x^{2} + 3 \, {\left (5 \, {\left (B a b^{2} - A b^{3}\right )} d^{2} e^{2} - 6 \, {\left (B a^{2} b - A a b^{2}\right )} d e^{3} + {\left (B a^{3} - A a^{2} b\right )} e^{4}\right )} x + 6 \, {\left ({\left (B a b^{2} - A b^{3}\right )} e^{4} x^{3} + 3 \, {\left (B a b^{2} - A b^{3}\right )} d e^{3} x^{2} + 3 \, {\left (B a b^{2} - A b^{3}\right )} d^{2} e^{2} x + {\left (B a b^{2} - A b^{3}\right )} d^{3} e\right )} \log \left (b x + a\right ) - 6 \, {\left ({\left (B a b^{2} - A b^{3}\right )} e^{4} x^{3} + 3 \, {\left (B a b^{2} - A b^{3}\right )} d e^{3} x^{2} + 3 \, {\left (B a b^{2} - A b^{3}\right )} d^{2} e^{2} x + {\left (B a b^{2} - A b^{3}\right )} d^{3} e\right )} \log \left (e x + d\right )}{6 \, {\left (b^{4} d^{7} e - 4 \, a b^{3} d^{6} e^{2} + 6 \, a^{2} b^{2} d^{5} e^{3} - 4 \, a^{3} b d^{4} e^{4} + a^{4} d^{3} e^{5} + {\left (b^{4} d^{4} e^{4} - 4 \, a b^{3} d^{3} e^{5} + 6 \, a^{2} b^{2} d^{2} e^{6} - 4 \, a^{3} b d e^{7} + a^{4} e^{8}\right )} x^{3} + 3 \, {\left (b^{4} d^{5} e^{3} - 4 \, a b^{3} d^{4} e^{4} + 6 \, a^{2} b^{2} d^{3} e^{5} - 4 \, a^{3} b d^{2} e^{6} + a^{4} d e^{7}\right )} x^{2} + 3 \, {\left (b^{4} d^{6} e^{2} - 4 \, a b^{3} d^{5} e^{3} + 6 \, a^{2} b^{2} d^{4} e^{4} - 4 \, a^{3} b d^{3} e^{5} + a^{4} d^{2} e^{6}\right )} x\right )}} \] Input:

integrate((B*x+A)/(b*x+a)/(e*x+d)^4,x, algorithm="fricas")
 

Output:

-1/6*(2*B*b^3*d^4 + 2*A*a^3*e^4 + (3*B*a*b^2 - 11*A*b^3)*d^3*e - 6*(B*a^2* 
b - 3*A*a*b^2)*d^2*e^2 + (B*a^3 - 9*A*a^2*b)*d*e^3 + 6*((B*a*b^2 - A*b^3)* 
d*e^3 - (B*a^2*b - A*a*b^2)*e^4)*x^2 + 3*(5*(B*a*b^2 - A*b^3)*d^2*e^2 - 6* 
(B*a^2*b - A*a*b^2)*d*e^3 + (B*a^3 - A*a^2*b)*e^4)*x + 6*((B*a*b^2 - A*b^3 
)*e^4*x^3 + 3*(B*a*b^2 - A*b^3)*d*e^3*x^2 + 3*(B*a*b^2 - A*b^3)*d^2*e^2*x 
+ (B*a*b^2 - A*b^3)*d^3*e)*log(b*x + a) - 6*((B*a*b^2 - A*b^3)*e^4*x^3 + 3 
*(B*a*b^2 - A*b^3)*d*e^3*x^2 + 3*(B*a*b^2 - A*b^3)*d^2*e^2*x + (B*a*b^2 - 
A*b^3)*d^3*e)*log(e*x + d))/(b^4*d^7*e - 4*a*b^3*d^6*e^2 + 6*a^2*b^2*d^5*e 
^3 - 4*a^3*b*d^4*e^4 + a^4*d^3*e^5 + (b^4*d^4*e^4 - 4*a*b^3*d^3*e^5 + 6*a^ 
2*b^2*d^2*e^6 - 4*a^3*b*d*e^7 + a^4*e^8)*x^3 + 3*(b^4*d^5*e^3 - 4*a*b^3*d^ 
4*e^4 + 6*a^2*b^2*d^3*e^5 - 4*a^3*b*d^2*e^6 + a^4*d*e^7)*x^2 + 3*(b^4*d^6* 
e^2 - 4*a*b^3*d^5*e^3 + 6*a^2*b^2*d^4*e^4 - 4*a^3*b*d^3*e^5 + a^4*d^2*e^6) 
*x)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 818 vs. \(2 (122) = 244\).

Time = 1.60 (sec) , antiderivative size = 818, normalized size of antiderivative = 5.60 \[ \int \frac {A+B x}{(a+b x) (d+e x)^4} \, dx =\text {Too large to display} \] Input:

integrate((B*x+A)/(b*x+a)/(e*x+d)**4,x)
 

Output:

b**2*(-A*b + B*a)*log(x + (-A*a*b**3*e - A*b**4*d + B*a**2*b**2*e + B*a*b* 
*3*d - a**5*b**2*e**5*(-A*b + B*a)/(a*e - b*d)**4 + 5*a**4*b**3*d*e**4*(-A 
*b + B*a)/(a*e - b*d)**4 - 10*a**3*b**4*d**2*e**3*(-A*b + B*a)/(a*e - b*d) 
**4 + 10*a**2*b**5*d**3*e**2*(-A*b + B*a)/(a*e - b*d)**4 - 5*a*b**6*d**4*e 
*(-A*b + B*a)/(a*e - b*d)**4 + b**7*d**5*(-A*b + B*a)/(a*e - b*d)**4)/(-2* 
A*b**4*e + 2*B*a*b**3*e))/(a*e - b*d)**4 - b**2*(-A*b + B*a)*log(x + (-A*a 
*b**3*e - A*b**4*d + B*a**2*b**2*e + B*a*b**3*d + a**5*b**2*e**5*(-A*b + B 
*a)/(a*e - b*d)**4 - 5*a**4*b**3*d*e**4*(-A*b + B*a)/(a*e - b*d)**4 + 10*a 
**3*b**4*d**2*e**3*(-A*b + B*a)/(a*e - b*d)**4 - 10*a**2*b**5*d**3*e**2*(- 
A*b + B*a)/(a*e - b*d)**4 + 5*a*b**6*d**4*e*(-A*b + B*a)/(a*e - b*d)**4 - 
b**7*d**5*(-A*b + B*a)/(a*e - b*d)**4)/(-2*A*b**4*e + 2*B*a*b**3*e))/(a*e 
- b*d)**4 + (-2*A*a**2*e**3 + 7*A*a*b*d*e**2 - 11*A*b**2*d**2*e - B*a**2*d 
*e**2 + 5*B*a*b*d**2*e + 2*B*b**2*d**3 + x**2*(-6*A*b**2*e**3 + 6*B*a*b*e* 
*3) + x*(3*A*a*b*e**3 - 15*A*b**2*d*e**2 - 3*B*a**2*e**3 + 15*B*a*b*d*e**2 
))/(6*a**3*d**3*e**4 - 18*a**2*b*d**4*e**3 + 18*a*b**2*d**5*e**2 - 6*b**3* 
d**6*e + x**3*(6*a**3*e**7 - 18*a**2*b*d*e**6 + 18*a*b**2*d**2*e**5 - 6*b* 
*3*d**3*e**4) + x**2*(18*a**3*d*e**6 - 54*a**2*b*d**2*e**5 + 54*a*b**2*d** 
3*e**4 - 18*b**3*d**4*e**3) + x*(18*a**3*d**2*e**5 - 54*a**2*b*d**3*e**4 + 
 54*a*b**2*d**4*e**3 - 18*b**3*d**5*e**2))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 444 vs. \(2 (143) = 286\).

Time = 0.05 (sec) , antiderivative size = 444, normalized size of antiderivative = 3.04 \[ \int \frac {A+B x}{(a+b x) (d+e x)^4} \, dx=-\frac {{\left (B a b^{2} - A b^{3}\right )} \log \left (b x + a\right )}{b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}} + \frac {{\left (B a b^{2} - A b^{3}\right )} \log \left (e x + d\right )}{b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}} - \frac {2 \, B b^{2} d^{3} - 2 \, A a^{2} e^{3} + 6 \, {\left (B a b - A b^{2}\right )} e^{3} x^{2} + {\left (5 \, B a b - 11 \, A b^{2}\right )} d^{2} e - {\left (B a^{2} - 7 \, A a b\right )} d e^{2} + 3 \, {\left (5 \, {\left (B a b - A b^{2}\right )} d e^{2} - {\left (B a^{2} - A a b\right )} e^{3}\right )} x}{6 \, {\left (b^{3} d^{6} e - 3 \, a b^{2} d^{5} e^{2} + 3 \, a^{2} b d^{4} e^{3} - a^{3} d^{3} e^{4} + {\left (b^{3} d^{3} e^{4} - 3 \, a b^{2} d^{2} e^{5} + 3 \, a^{2} b d e^{6} - a^{3} e^{7}\right )} x^{3} + 3 \, {\left (b^{3} d^{4} e^{3} - 3 \, a b^{2} d^{3} e^{4} + 3 \, a^{2} b d^{2} e^{5} - a^{3} d e^{6}\right )} x^{2} + 3 \, {\left (b^{3} d^{5} e^{2} - 3 \, a b^{2} d^{4} e^{3} + 3 \, a^{2} b d^{3} e^{4} - a^{3} d^{2} e^{5}\right )} x\right )}} \] Input:

integrate((B*x+A)/(b*x+a)/(e*x+d)^4,x, algorithm="maxima")
 

Output:

-(B*a*b^2 - A*b^3)*log(b*x + a)/(b^4*d^4 - 4*a*b^3*d^3*e + 6*a^2*b^2*d^2*e 
^2 - 4*a^3*b*d*e^3 + a^4*e^4) + (B*a*b^2 - A*b^3)*log(e*x + d)/(b^4*d^4 - 
4*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 - 4*a^3*b*d*e^3 + a^4*e^4) - 1/6*(2*B*b^ 
2*d^3 - 2*A*a^2*e^3 + 6*(B*a*b - A*b^2)*e^3*x^2 + (5*B*a*b - 11*A*b^2)*d^2 
*e - (B*a^2 - 7*A*a*b)*d*e^2 + 3*(5*(B*a*b - A*b^2)*d*e^2 - (B*a^2 - A*a*b 
)*e^3)*x)/(b^3*d^6*e - 3*a*b^2*d^5*e^2 + 3*a^2*b*d^4*e^3 - a^3*d^3*e^4 + ( 
b^3*d^3*e^4 - 3*a*b^2*d^2*e^5 + 3*a^2*b*d*e^6 - a^3*e^7)*x^3 + 3*(b^3*d^4* 
e^3 - 3*a*b^2*d^3*e^4 + 3*a^2*b*d^2*e^5 - a^3*d*e^6)*x^2 + 3*(b^3*d^5*e^2 
- 3*a*b^2*d^4*e^3 + 3*a^2*b*d^3*e^4 - a^3*d^2*e^5)*x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 376 vs. \(2 (143) = 286\).

Time = 0.13 (sec) , antiderivative size = 376, normalized size of antiderivative = 2.58 \[ \int \frac {A+B x}{(a+b x) (d+e x)^4} \, dx=-\frac {{\left (B a b^{3} - A b^{4}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{5} d^{4} - 4 \, a b^{4} d^{3} e + 6 \, a^{2} b^{3} d^{2} e^{2} - 4 \, a^{3} b^{2} d e^{3} + a^{4} b e^{4}} + \frac {{\left (B a b^{2} e - A b^{3} e\right )} \log \left ({\left | e x + d \right |}\right )}{b^{4} d^{4} e - 4 \, a b^{3} d^{3} e^{2} + 6 \, a^{2} b^{2} d^{2} e^{3} - 4 \, a^{3} b d e^{4} + a^{4} e^{5}} - \frac {2 \, B b^{3} d^{4} + 3 \, B a b^{2} d^{3} e - 11 \, A b^{3} d^{3} e - 6 \, B a^{2} b d^{2} e^{2} + 18 \, A a b^{2} d^{2} e^{2} + B a^{3} d e^{3} - 9 \, A a^{2} b d e^{3} + 2 \, A a^{3} e^{4} + 6 \, {\left (B a b^{2} d e^{3} - A b^{3} d e^{3} - B a^{2} b e^{4} + A a b^{2} e^{4}\right )} x^{2} + 3 \, {\left (5 \, B a b^{2} d^{2} e^{2} - 5 \, A b^{3} d^{2} e^{2} - 6 \, B a^{2} b d e^{3} + 6 \, A a b^{2} d e^{3} + B a^{3} e^{4} - A a^{2} b e^{4}\right )} x}{6 \, {\left (b d - a e\right )}^{4} {\left (e x + d\right )}^{3} e} \] Input:

integrate((B*x+A)/(b*x+a)/(e*x+d)^4,x, algorithm="giac")
 

Output:

-(B*a*b^3 - A*b^4)*log(abs(b*x + a))/(b^5*d^4 - 4*a*b^4*d^3*e + 6*a^2*b^3* 
d^2*e^2 - 4*a^3*b^2*d*e^3 + a^4*b*e^4) + (B*a*b^2*e - A*b^3*e)*log(abs(e*x 
 + d))/(b^4*d^4*e - 4*a*b^3*d^3*e^2 + 6*a^2*b^2*d^2*e^3 - 4*a^3*b*d*e^4 + 
a^4*e^5) - 1/6*(2*B*b^3*d^4 + 3*B*a*b^2*d^3*e - 11*A*b^3*d^3*e - 6*B*a^2*b 
*d^2*e^2 + 18*A*a*b^2*d^2*e^2 + B*a^3*d*e^3 - 9*A*a^2*b*d*e^3 + 2*A*a^3*e^ 
4 + 6*(B*a*b^2*d*e^3 - A*b^3*d*e^3 - B*a^2*b*e^4 + A*a*b^2*e^4)*x^2 + 3*(5 
*B*a*b^2*d^2*e^2 - 5*A*b^3*d^2*e^2 - 6*B*a^2*b*d*e^3 + 6*A*a*b^2*d*e^3 + B 
*a^3*e^4 - A*a^2*b*e^4)*x)/((b*d - a*e)^4*(e*x + d)^3*e)
 

Mupad [B] (verification not implemented)

Time = 1.17 (sec) , antiderivative size = 399, normalized size of antiderivative = 2.73 \[ \int \frac {A+B x}{(a+b x) (d+e x)^4} \, dx=\frac {2\,b^2\,\mathrm {atanh}\left (\frac {\left (\frac {a^4\,e^4-2\,a^3\,b\,d\,e^3+2\,a\,b^3\,d^3\,e-b^4\,d^4}{a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3}+2\,b\,e\,x\right )\,\left (a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3\right )}{{\left (a\,e-b\,d\right )}^4}\right )\,\left (A\,b-B\,a\right )}{{\left (a\,e-b\,d\right )}^4}-\frac {\frac {B\,a^2\,d\,e^2+2\,A\,a^2\,e^3-5\,B\,a\,b\,d^2\,e-7\,A\,a\,b\,d\,e^2-2\,B\,b^2\,d^3+11\,A\,b^2\,d^2\,e}{6\,e\,\left (a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3\right )}-\frac {x\,\left (A\,b-B\,a\right )\,\left (a\,e^2-5\,b\,d\,e\right )}{2\,\left (a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3\right )}+\frac {b\,e^2\,x^2\,\left (A\,b-B\,a\right )}{a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3}}{d^3+3\,d^2\,e\,x+3\,d\,e^2\,x^2+e^3\,x^3} \] Input:

int((A + B*x)/((a + b*x)*(d + e*x)^4),x)
 

Output:

(2*b^2*atanh((((a^4*e^4 - b^4*d^4 + 2*a*b^3*d^3*e - 2*a^3*b*d*e^3)/(a^3*e^ 
3 - b^3*d^3 + 3*a*b^2*d^2*e - 3*a^2*b*d*e^2) + 2*b*e*x)*(a^3*e^3 - b^3*d^3 
 + 3*a*b^2*d^2*e - 3*a^2*b*d*e^2))/(a*e - b*d)^4)*(A*b - B*a))/(a*e - b*d) 
^4 - ((2*A*a^2*e^3 - 2*B*b^2*d^3 + 11*A*b^2*d^2*e + B*a^2*d*e^2 - 7*A*a*b* 
d*e^2 - 5*B*a*b*d^2*e)/(6*e*(a^3*e^3 - b^3*d^3 + 3*a*b^2*d^2*e - 3*a^2*b*d 
*e^2)) - (x*(A*b - B*a)*(a*e^2 - 5*b*d*e))/(2*(a^3*e^3 - b^3*d^3 + 3*a*b^2 
*d^2*e - 3*a^2*b*d*e^2)) + (b*e^2*x^2*(A*b - B*a))/(a^3*e^3 - b^3*d^3 + 3* 
a*b^2*d^2*e - 3*a^2*b*d*e^2))/(d^3 + e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.23 \[ \int \frac {A+B x}{(a+b x) (d+e x)^4} \, dx=-\frac {1}{3 e \left (e^{3} x^{3}+3 d \,e^{2} x^{2}+3 d^{2} e x +d^{3}\right )} \] Input:

int((B*x+A)/(b*x+a)/(e*x+d)^4,x)
                                                                                    
                                                                                    
 

Output:

( - 1)/(3*e*(d**3 + 3*d**2*e*x + 3*d*e**2*x**2 + e**3*x**3))