\(\int \frac {A+B x}{(a+b x) (d+e x)^4} \, dx\) [1120]

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {78} \[ \int \frac {A+B x}{(a+b x) (d+e x)^4} \, dx=\frac {b^2 (A b-a B) \log (a+b x)}{(b d-a e)^4}-\frac {b^2 (A b-a B) \log (d+e x)}{(b d-a e)^4}+\frac {b (A b-a B)}{(d+e x) (b d-a e)^3}+\frac {A b-a B}{2 (d+e x)^2 (b d-a e)^2}-\frac {B d-A e}{3 e (d+e x)^3 (b d-a e)} \]

[In]

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

[Out]

-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

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((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]))))

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {b^3 (A b-a B)}{(b d-a e)^4 (a+b x)}+\frac {B d-A e}{(b d-a e) (d+e x)^4}+\frac {(-A b+a B) e}{(b d-a e)^2 (d+e x)^3}+\frac {b (A b-a B) e}{(-b d+a e)^3 (d+e x)^2}-\frac {b^2 (A b-a B) e}{(-b d+a e)^4 (d+e x)}\right ) \, 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} \\ \end{align*}

Mathematica [A] (verified)

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

[In]

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

[Out]

(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

Maple [A] (verified)

Time = 0.78 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.99

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

[In]

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

[Out]

(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 = 0.24 (sec) , antiderivative size = 608, normalized size of antiderivative = 4.16 \[ \int \frac {A+B x}{(a+b x) (d+e x)^4} \, dx=-\frac {2 \, B b^{3} d^{4} + 2 \, A a^{3} e^{4} + {\left (3 \, B a b^{2} - 11 \, A b^{3}\right )} d^{3} e - 6 \, {\left (B a^{2} b - 3 \, A a b^{2}\right )} d^{2} e^{2} + {\left (B a^{3} - 9 \, A a^{2} b\right )} d e^{3} + 6 \, {\left ({\left (B a b^{2} - A b^{3}\right )} d e^{3} - {\left (B a^{2} b - A a b^{2}\right )} e^{4}\right )} x^{2} + 3 \, {\left (5 \, {\left (B a b^{2} - A b^{3}\right )} d^{2} e^{2} - 6 \, {\left (B a^{2} b - A a b^{2}\right )} d e^{3} + {\left (B a^{3} - A a^{2} b\right )} e^{4}\right )} x + 6 \, {\left ({\left (B a b^{2} - A b^{3}\right )} e^{4} x^{3} + 3 \, {\left (B a b^{2} - A b^{3}\right )} d e^{3} x^{2} + 3 \, {\left (B a b^{2} - A b^{3}\right )} d^{2} e^{2} x + {\left (B a b^{2} - A b^{3}\right )} d^{3} e\right )} \log \left (b x + a\right ) - 6 \, {\left ({\left (B a b^{2} - A b^{3}\right )} e^{4} x^{3} + 3 \, {\left (B a b^{2} - A b^{3}\right )} d e^{3} x^{2} + 3 \, {\left (B a b^{2} - A b^{3}\right )} d^{2} e^{2} x + {\left (B a b^{2} - A b^{3}\right )} d^{3} e\right )} \log \left (e x + d\right )}{6 \, {\left (b^{4} d^{7} e - 4 \, a b^{3} d^{6} e^{2} + 6 \, a^{2} b^{2} d^{5} e^{3} - 4 \, a^{3} b d^{4} e^{4} + a^{4} d^{3} e^{5} + {\left (b^{4} d^{4} e^{4} - 4 \, a b^{3} d^{3} e^{5} + 6 \, a^{2} b^{2} d^{2} e^{6} - 4 \, a^{3} b d e^{7} + a^{4} e^{8}\right )} x^{3} + 3 \, {\left (b^{4} d^{5} e^{3} - 4 \, a b^{3} d^{4} e^{4} + 6 \, a^{2} b^{2} d^{3} e^{5} - 4 \, a^{3} b d^{2} e^{6} + a^{4} d e^{7}\right )} x^{2} + 3 \, {\left (b^{4} d^{6} e^{2} - 4 \, a b^{3} d^{5} e^{3} + 6 \, a^{2} b^{2} d^{4} e^{4} - 4 \, a^{3} b d^{3} e^{5} + a^{4} d^{2} e^{6}\right )} x\right )}} \]

[In]

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

[Out]

-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 = 1.65 (sec) , antiderivative size = 818, normalized size of antiderivative = 5.60 \[ \int \frac {A+B x}{(a+b x) (d+e x)^4} \, dx=\frac {b^{2} \left (- A b + B a\right ) \log {\left (x + \frac {- A a b^{3} e - A b^{4} d + B a^{2} b^{2} e + B a b^{3} d - \frac {a^{5} b^{2} e^{5} \left (- A b + B a\right )}{\left (a e - b d\right )^{4}} + \frac {5 a^{4} b^{3} d e^{4} \left (- A b + B a\right )}{\left (a e - b d\right )^{4}} - \frac {10 a^{3} b^{4} d^{2} e^{3} \left (- A b + B a\right )}{\left (a e - b d\right )^{4}} + \frac {10 a^{2} b^{5} d^{3} e^{2} \left (- A b + B a\right )}{\left (a e - b d\right )^{4}} - \frac {5 a b^{6} d^{4} e \left (- A b + B a\right )}{\left (a e - b d\right )^{4}} + \frac {b^{7} d^{5} \left (- A b + B a\right )}{\left (a e - b d\right )^{4}}}{- 2 A b^{4} e + 2 B a b^{3} e} \right )}}{\left (a e - b d\right )^{4}} - \frac {b^{2} \left (- A b + B a\right ) \log {\left (x + \frac {- A a b^{3} e - A b^{4} d + B a^{2} b^{2} e + B a b^{3} d + \frac {a^{5} b^{2} e^{5} \left (- A b + B a\right )}{\left (a e - b d\right )^{4}} - \frac {5 a^{4} b^{3} d e^{4} \left (- A b + B a\right )}{\left (a e - b d\right )^{4}} + \frac {10 a^{3} b^{4} d^{2} e^{3} \left (- A b + B a\right )}{\left (a e - b d\right )^{4}} - \frac {10 a^{2} b^{5} d^{3} e^{2} \left (- A b + B a\right )}{\left (a e - b d\right )^{4}} + \frac {5 a b^{6} d^{4} e \left (- A b + B a\right )}{\left (a e - b d\right )^{4}} - \frac {b^{7} d^{5} \left (- A b + B a\right )}{\left (a e - b d\right )^{4}}}{- 2 A b^{4} e + 2 B a b^{3} e} \right )}}{\left (a e - b d\right )^{4}} + \frac {- 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} \left (- 6 A b^{2} e^{3} + 6 B a b e^{3}\right ) + x \left (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}\right )}{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} \cdot \left (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}\right ) + x^{2} \cdot \left (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}\right ) + x \left (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}\right )} \]

[In]

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

[Out]

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 = 0.22 (sec) , antiderivative size = 444, normalized size of antiderivative = 3.04 \[ \int \frac {A+B x}{(a+b x) (d+e x)^4} \, dx=-\frac {{\left (B a b^{2} - A b^{3}\right )} \log \left (b x + a\right )}{b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}} + \frac {{\left (B a b^{2} - A b^{3}\right )} \log \left (e x + d\right )}{b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}} - \frac {2 \, B b^{2} d^{3} - 2 \, A a^{2} e^{3} + 6 \, {\left (B a b - A b^{2}\right )} e^{3} x^{2} + {\left (5 \, B a b - 11 \, A b^{2}\right )} d^{2} e - {\left (B a^{2} - 7 \, A a b\right )} d e^{2} + 3 \, {\left (5 \, {\left (B a b - A b^{2}\right )} d e^{2} - {\left (B a^{2} - A a b\right )} e^{3}\right )} x}{6 \, {\left (b^{3} d^{6} e - 3 \, a b^{2} d^{5} e^{2} + 3 \, a^{2} b d^{4} e^{3} - a^{3} d^{3} e^{4} + {\left (b^{3} d^{3} e^{4} - 3 \, a b^{2} d^{2} e^{5} + 3 \, a^{2} b d e^{6} - a^{3} e^{7}\right )} x^{3} + 3 \, {\left (b^{3} d^{4} e^{3} - 3 \, a b^{2} d^{3} e^{4} + 3 \, a^{2} b d^{2} e^{5} - a^{3} d e^{6}\right )} x^{2} + 3 \, {\left (b^{3} d^{5} e^{2} - 3 \, a b^{2} d^{4} e^{3} + 3 \, a^{2} b d^{3} e^{4} - a^{3} d^{2} e^{5}\right )} x\right )}} \]

[In]

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

[Out]

-(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 = 0.30 (sec) , antiderivative size = 376, normalized size of antiderivative = 2.58 \[ \int \frac {A+B x}{(a+b x) (d+e x)^4} \, dx=-\frac {{\left (B a b^{3} - A b^{4}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{5} d^{4} - 4 \, a b^{4} d^{3} e + 6 \, a^{2} b^{3} d^{2} e^{2} - 4 \, a^{3} b^{2} d e^{3} + a^{4} b e^{4}} + \frac {{\left (B a b^{2} e - A b^{3} e\right )} \log \left ({\left | e x + d \right |}\right )}{b^{4} d^{4} e - 4 \, a b^{3} d^{3} e^{2} + 6 \, a^{2} b^{2} d^{2} e^{3} - 4 \, a^{3} b d e^{4} + a^{4} e^{5}} - \frac {2 \, B b^{3} d^{4} + 3 \, B a b^{2} d^{3} e - 11 \, A b^{3} d^{3} e - 6 \, B a^{2} b d^{2} e^{2} + 18 \, A a b^{2} d^{2} e^{2} + B a^{3} d e^{3} - 9 \, A a^{2} b d e^{3} + 2 \, A a^{3} e^{4} + 6 \, {\left (B a b^{2} d e^{3} - A b^{3} d e^{3} - B a^{2} b e^{4} + A a b^{2} e^{4}\right )} x^{2} + 3 \, {\left (5 \, B a b^{2} d^{2} e^{2} - 5 \, A b^{3} d^{2} e^{2} - 6 \, B a^{2} b d e^{3} + 6 \, A a b^{2} d e^{3} + B a^{3} e^{4} - A a^{2} b e^{4}\right )} x}{6 \, {\left (b d - a e\right )}^{4} {\left (e x + d\right )}^{3} e} \]

[In]

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

[Out]

-(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 = 1.57 (sec) , antiderivative size = 399, normalized size of antiderivative = 2.73 \[ \int \frac {A+B x}{(a+b x) (d+e x)^4} \, dx=\frac {2\,b^2\,\mathrm {atanh}\left (\frac {\left (\frac {a^4\,e^4-2\,a^3\,b\,d\,e^3+2\,a\,b^3\,d^3\,e-b^4\,d^4}{a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3}+2\,b\,e\,x\right )\,\left (a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3\right )}{{\left (a\,e-b\,d\right )}^4}\right )\,\left (A\,b-B\,a\right )}{{\left (a\,e-b\,d\right )}^4}-\frac {\frac {B\,a^2\,d\,e^2+2\,A\,a^2\,e^3-5\,B\,a\,b\,d^2\,e-7\,A\,a\,b\,d\,e^2-2\,B\,b^2\,d^3+11\,A\,b^2\,d^2\,e}{6\,e\,\left (a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3\right )}-\frac {x\,\left (A\,b-B\,a\right )\,\left (a\,e^2-5\,b\,d\,e\right )}{2\,\left (a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3\right )}+\frac {b\,e^2\,x^2\,\left (A\,b-B\,a\right )}{a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3}}{d^3+3\,d^2\,e\,x+3\,d\,e^2\,x^2+e^3\,x^3} \]

[In]

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

[Out]

(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)