\(\int \frac {A+B x}{(a+b x) (d+e x)^5} \, dx\) [1121]

Optimal result

Integrand size = 20, antiderivative size = 178 \[ \int \frac {A+B x}{(a+b x) (d+e x)^5} \, dx=\frac {-B d+A e}{4 e (b d-a e) (d+e x)^4}+\frac {A b-a B}{3 (b d-a e)^2 (d+e x)^3}+\frac {b (A b-a B)}{2 (b d-a e)^3 (d+e x)^2}+\frac {b^2 (A b-a B)}{(b d-a e)^4 (d+e x)}+\frac {b^3 (A b-a B) \log (a+b x)}{(b d-a e)^5}-\frac {b^3 (A b-a B) \log (d+e x)}{(b d-a e)^5} \]

[Out]

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

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 178, 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} \[ \int \frac {A+B x}{(a+b x) (d+e x)^5} \, dx=\frac {b^3 (A b-a B) \log (a+b x)}{(b d-a e)^5}-\frac {b^3 (A b-a B) \log (d+e x)}{(b d-a e)^5}+\frac {b^2 (A b-a B)}{(d+e x) (b d-a e)^4}+\frac {b (A b-a B)}{2 (d+e x)^2 (b d-a e)^3}+\frac {A b-a B}{3 (d+e x)^3 (b d-a e)^2}-\frac {B d-A e}{4 e (d+e x)^4 (b d-a e)} \]

[In]

Int[(A + B*x)/((a + b*x)*(d + e*x)^5),x]

[Out]

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

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^4 (A b-a B)}{(b d-a e)^5 (a+b x)}+\frac {B d-A e}{(b d-a e) (d+e x)^5}+\frac {(-A b+a B) e}{(b d-a e)^2 (d+e x)^4}+\frac {b (A b-a B) e}{(-b d+a e)^3 (d+e x)^3}-\frac {b^2 (A b-a B) e}{(-b d+a e)^4 (d+e x)^2}+\frac {b^3 (A b-a B) e}{(-b d+a e)^5 (d+e x)}\right ) \, dx \\ & = -\frac {B d-A e}{4 e (b d-a e) (d+e x)^4}+\frac {A b-a B}{3 (b d-a e)^2 (d+e x)^3}+\frac {b (A b-a B)}{2 (b d-a e)^3 (d+e x)^2}+\frac {b^2 (A b-a B)}{(b d-a e)^4 (d+e x)}+\frac {b^3 (A b-a B) \log (a+b x)}{(b d-a e)^5}-\frac {b^3 (A b-a B) \log (d+e x)}{(b d-a e)^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.03 \[ \int \frac {A+B x}{(a+b x) (d+e x)^5} \, dx=\frac {-3 (b d-a e)^4 (B d-A e)+4 (A b-a B) e (b d-a e)^3 (d+e x)+6 b (A b-a B) e (b d-a e)^2 (d+e x)^2+12 b^2 (A b-a B) e (b d-a e) (d+e x)^3+12 b^3 (A b-a B) e (d+e x)^4 \log (a+b x)-12 b^3 (A b-a B) e (d+e x)^4 \log (d+e x)}{12 e (b d-a e)^5 (d+e x)^4} \]

[In]

Integrate[(A + B*x)/((a + b*x)*(d + e*x)^5),x]

[Out]

(-3*(b*d - a*e)^4*(B*d - A*e) + 4*(A*b - a*B)*e*(b*d - a*e)^3*(d + e*x) + 6*b*(A*b - a*B)*e*(b*d - a*e)^2*(d +
e*x)^2 + 12*b^2*(A*b - a*B)*e*(b*d - a*e)*(d + e*x)^3 + 12*b^3*(A*b - a*B)*e*(d + e*x)^4*Log[a + b*x] - 12*b^
3*(A*b - a*B)*e*(d + e*x)^4*Log[d + e*x])/(12*e*(b*d - a*e)^5*(d + e*x)^4)

Maple [A] (verified)

Time = 0.83 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.97

method	result	size
default	\(-\frac {\left (A b -B a \right ) b^{3} \ln \left (b x +a \right )}{\left (a e -b d \right )^{5}}-\frac {A e -B d}{4 \left (a e -b d \right ) e \left (e x +d \right )^{4}}-\frac {\left (A b -B a \right ) b}{2 \left (a e -b d \right )^{3} \left (e x +d \right )^{2}}+\frac {\left (A b -B a \right ) b^{3} \ln \left (e x +d \right )}{\left (a e -b d \right )^{5}}+\frac {A b -B a}{3 \left (a e -b d \right )^{2} \left (e x +d \right )^{3}}+\frac {\left (A b -B a \right ) b^{2}}{\left (a e -b d \right )^{4} \left (e x +d \right )}\)	\(173\)
norman	\(\frac {\frac {\left (A \,b^{3} e^{4}-B a \,b^{2} e^{4}\right ) x^{3}}{e \left (a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}-\frac {3 A \,a^{3} e^{7}-13 A \,a^{2} b d \,e^{6}+23 A a \,b^{2} d^{2} e^{5}-25 A \,b^{3} d^{3} e^{4}+B \,a^{3} d \,e^{6}-5 B \,a^{2} b \,d^{2} e^{5}+13 B a \,b^{2} d^{3} e^{4}+3 B \,b^{3} d^{4} e^{3}}{12 e^{4} \left (a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}-\frac {\left (A a \,b^{2} e^{5}-7 A \,b^{3} d \,e^{4}-B \,a^{2} b \,e^{5}+7 B a \,b^{2} d \,e^{4}\right ) x^{2}}{2 e^{2} \left (a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}+\frac {\left (A \,a^{2} b \,e^{6}-5 A a \,b^{2} d \,e^{5}+13 A \,b^{3} d^{2} e^{4}-B \,a^{3} e^{6}+5 B \,a^{2} b d \,e^{5}-13 B a \,b^{2} d^{2} e^{4}\right ) x}{3 e^{3} \left (a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}}{\left (e x +d \right )^{4}}+\frac {b^{3} \left (A b -B a \right ) \ln \left (e x +d \right )}{a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}}-\frac {b^{3} \left (A b -B a \right ) \ln \left (b x +a \right )}{a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}}\)	\(629\)
risch	\(\frac {\frac {e^{3} b^{2} \left (A b -B a \right ) x^{3}}{a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}}-\frac {\left (a e -7 b d \right ) e^{2} b \left (A b -B a \right ) x^{2}}{2 \left (a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}+\frac {e \left (A \,a^{2} b \,e^{2}-5 A a \,b^{2} d e +13 A \,b^{3} d^{2}-B \,a^{3} e^{2}+5 B \,a^{2} b d e -13 B a \,b^{2} d^{2}\right ) x}{3 a^{4} e^{4}-12 a^{3} b d \,e^{3}+18 a^{2} b^{2} d^{2} e^{2}-12 a \,b^{3} d^{3} e +3 b^{4} d^{4}}-\frac {3 a^{3} A \,e^{4}-13 A \,a^{2} b d \,e^{3}+23 A a \,b^{2} d^{2} e^{2}-25 A \,b^{3} d^{3} e +B \,a^{3} d \,e^{3}-5 B \,a^{2} b \,d^{2} e^{2}+13 B a \,b^{2} d^{3} e +3 b^{3} B \,d^{4}}{12 e \left (a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}}{\left (e x +d \right )^{4}}-\frac {b^{4} \ln \left (b x +a \right ) A}{a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}}+\frac {b^{3} \ln \left (b x +a \right ) B a}{a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}}+\frac {b^{4} \ln \left (-e x -d \right ) A}{a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}}-\frac {b^{3} \ln \left (-e x -d \right ) B a}{a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}}\)	\(727\)
parallelrisch	\(-\frac {-48 B \ln \left (b x +a \right ) x^{3} a \,b^{3} d \,e^{7}+48 B \ln \left (e x +d \right ) x^{3} a \,b^{3} d \,e^{7}-72 B \ln \left (b x +a \right ) x^{2} a \,b^{3} d^{2} e^{6}+72 B \ln \left (e x +d \right ) x^{2} a \,b^{3} d^{2} e^{6}-48 B \ln \left (b x +a \right ) x a \,b^{3} d^{3} e^{5}+48 B \ln \left (e x +d \right ) x a \,b^{3} d^{3} e^{5}+12 B \ln \left (e x +d \right ) a \,b^{3} d^{4} e^{4}-12 B \,x^{3} a \,b^{3} d \,e^{7}-48 A \,x^{2} a \,b^{3} d \,e^{7}+48 B \,x^{2} a^{2} b^{2} d \,e^{7}-42 B \,x^{2} a \,b^{3} d^{2} e^{6}+24 A x \,a^{2} b^{2} d \,e^{7}-72 A x a \,b^{3} d^{2} e^{6}-24 B x \,a^{3} b d \,e^{7}+36 A \,a^{2} b^{2} d^{2} e^{6}-48 A a \,b^{3} d^{3} e^{5}-6 B \,a^{3} b \,d^{2} e^{6}+18 B \,a^{2} b^{2} d^{3} e^{5}-10 B a \,b^{3} d^{4} e^{4}+25 A \,b^{4} d^{4} e^{4}+B \,a^{4} d \,e^{7}-3 B \,b^{4} d^{5} e^{3}+72 B x \,a^{2} b^{2} d^{2} e^{6}-52 B x a \,b^{3} d^{3} e^{5}-12 B \ln \left (b x +a \right ) x^{4} a \,b^{3} e^{8}+12 B \ln \left (e x +d \right ) x^{4} a \,b^{3} e^{8}+48 A \ln \left (b x +a \right ) x^{3} b^{4} d \,e^{7}-48 A \ln \left (e x +d \right ) x^{3} b^{4} d \,e^{7}+72 A \ln \left (b x +a \right ) x^{2} b^{4} d^{2} e^{6}-72 A \ln \left (e x +d \right ) x^{2} b^{4} d^{2} e^{6}+48 A \ln \left (b x +a \right ) x \,b^{4} d^{3} e^{5}-48 A \ln \left (e x +d \right ) x \,b^{4} d^{3} e^{5}-12 B \ln \left (b x +a \right ) a \,b^{3} d^{4} e^{4}-4 A x \,a^{3} b \,e^{8}+52 A x \,b^{4} d^{3} e^{5}+12 A \ln \left (b x +a \right ) x^{4} b^{4} e^{8}-12 A \ln \left (e x +d \right ) x^{4} b^{4} e^{8}+12 A \ln \left (b x +a \right ) b^{4} d^{4} e^{4}-12 A \ln \left (e x +d \right ) b^{4} d^{4} e^{4}-12 A \,x^{3} a \,b^{3} e^{8}+12 A \,x^{3} b^{4} d \,e^{7}+12 B \,x^{3} a^{2} b^{2} e^{8}+6 A \,x^{2} a^{2} b^{2} e^{8}+42 A \,x^{2} b^{4} d^{2} e^{6}-6 B \,x^{2} a^{3} b \,e^{8}-16 A \,a^{3} b d \,e^{7}+4 B x \,a^{4} e^{8}+3 A \,a^{4} e^{8}}{12 \left (a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right ) \left (e x +d \right )^{4} e^{4}}\)	\(840\)

[In]

int((B*x+A)/(b*x+a)/(e*x+d)^5,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 922 vs. \(2 (173) = 346\).

Time = 0.25 (sec) , antiderivative size = 922, normalized size of antiderivative = 5.18 \[ \int \frac {A+B x}{(a+b x) (d+e x)^5} \, dx=-\frac {3 \, B b^{4} d^{5} - 3 \, A a^{4} e^{5} + 5 \, {\left (2 \, B a b^{3} - 5 \, A b^{4}\right )} d^{4} e - 6 \, {\left (3 \, B a^{2} b^{2} - 8 \, A a b^{3}\right )} d^{3} e^{2} + 6 \, {\left (B a^{3} b - 6 \, A a^{2} b^{2}\right )} d^{2} e^{3} - {\left (B a^{4} - 16 \, A a^{3} b\right )} d e^{4} + 12 \, {\left ({\left (B a b^{3} - A b^{4}\right )} d e^{4} - {\left (B a^{2} b^{2} - A a b^{3}\right )} e^{5}\right )} x^{3} + 6 \, {\left (7 \, {\left (B a b^{3} - A b^{4}\right )} d^{2} e^{3} - 8 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} d e^{4} + {\left (B a^{3} b - A a^{2} b^{2}\right )} e^{5}\right )} x^{2} + 4 \, {\left (13 \, {\left (B a b^{3} - A b^{4}\right )} d^{3} e^{2} - 18 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} d^{2} e^{3} + 6 \, {\left (B a^{3} b - A a^{2} b^{2}\right )} d e^{4} - {\left (B a^{4} - A a^{3} b\right )} e^{5}\right )} x + 12 \, {\left ({\left (B a b^{3} - A b^{4}\right )} e^{5} x^{4} + 4 \, {\left (B a b^{3} - A b^{4}\right )} d e^{4} x^{3} + 6 \, {\left (B a b^{3} - A b^{4}\right )} d^{2} e^{3} x^{2} + 4 \, {\left (B a b^{3} - A b^{4}\right )} d^{3} e^{2} x + {\left (B a b^{3} - A b^{4}\right )} d^{4} e\right )} \log \left (b x + a\right ) - 12 \, {\left ({\left (B a b^{3} - A b^{4}\right )} e^{5} x^{4} + 4 \, {\left (B a b^{3} - A b^{4}\right )} d e^{4} x^{3} + 6 \, {\left (B a b^{3} - A b^{4}\right )} d^{2} e^{3} x^{2} + 4 \, {\left (B a b^{3} - A b^{4}\right )} d^{3} e^{2} x + {\left (B a b^{3} - A b^{4}\right )} d^{4} e\right )} \log \left (e x + d\right )}{12 \, {\left (b^{5} d^{9} e - 5 \, a b^{4} d^{8} e^{2} + 10 \, a^{2} b^{3} d^{7} e^{3} - 10 \, a^{3} b^{2} d^{6} e^{4} + 5 \, a^{4} b d^{5} e^{5} - a^{5} d^{4} e^{6} + {\left (b^{5} d^{5} e^{5} - 5 \, a b^{4} d^{4} e^{6} + 10 \, a^{2} b^{3} d^{3} e^{7} - 10 \, a^{3} b^{2} d^{2} e^{8} + 5 \, a^{4} b d e^{9} - a^{5} e^{10}\right )} x^{4} + 4 \, {\left (b^{5} d^{6} e^{4} - 5 \, a b^{4} d^{5} e^{5} + 10 \, a^{2} b^{3} d^{4} e^{6} - 10 \, a^{3} b^{2} d^{3} e^{7} + 5 \, a^{4} b d^{2} e^{8} - a^{5} d e^{9}\right )} x^{3} + 6 \, {\left (b^{5} d^{7} e^{3} - 5 \, a b^{4} d^{6} e^{4} + 10 \, a^{2} b^{3} d^{5} e^{5} - 10 \, a^{3} b^{2} d^{4} e^{6} + 5 \, a^{4} b d^{3} e^{7} - a^{5} d^{2} e^{8}\right )} x^{2} + 4 \, {\left (b^{5} d^{8} e^{2} - 5 \, a b^{4} d^{7} e^{3} + 10 \, a^{2} b^{3} d^{6} e^{4} - 10 \, a^{3} b^{2} d^{5} e^{5} + 5 \, a^{4} b d^{4} e^{6} - a^{5} d^{3} e^{7}\right )} x\right )}} \]

[In]

integrate((B*x+A)/(b*x+a)/(e*x+d)^5,x, algorithm="fricas")

[Out]

-1/12*(3*B*b^4*d^5 - 3*A*a^4*e^5 + 5*(2*B*a*b^3 - 5*A*b^4)*d^4*e - 6*(3*B*a^2*b^2 - 8*A*a*b^3)*d^3*e^2 + 6*(B*
a^3*b - 6*A*a^2*b^2)*d^2*e^3 - (B*a^4 - 16*A*a^3*b)*d*e^4 + 12*((B*a*b^3 - A*b^4)*d*e^4 - (B*a^2*b^2 - A*a*b^3
)*e^5)*x^3 + 6*(7*(B*a*b^3 - A*b^4)*d^2*e^3 - 8*(B*a^2*b^2 - A*a*b^3)*d*e^4 + (B*a^3*b - A*a^2*b^2)*e^5)*x^2 +
4*(13*(B*a*b^3 - A*b^4)*d^3*e^2 - 18*(B*a^2*b^2 - A*a*b^3)*d^2*e^3 + 6*(B*a^3*b - A*a^2*b^2)*d*e^4 - (B*a^4 -
A*a^3*b)*e^5)*x + 12*((B*a*b^3 - A*b^4)*e^5*x^4 + 4*(B*a*b^3 - A*b^4)*d*e^4*x^3 + 6*(B*a*b^3 - A*b^4)*d^2*e^3
*x^2 + 4*(B*a*b^3 - A*b^4)*d^3*e^2*x + (B*a*b^3 - A*b^4)*d^4*e)*log(b*x + a) - 12*((B*a*b^3 - A*b^4)*e^5*x^4 +
4*(B*a*b^3 - A*b^4)*d*e^4*x^3 + 6*(B*a*b^3 - A*b^4)*d^2*e^3*x^2 + 4*(B*a*b^3 - A*b^4)*d^3*e^2*x + (B*a*b^3 -
A*b^4)*d^4*e)*log(e*x + d))/(b^5*d^9*e - 5*a*b^4*d^8*e^2 + 10*a^2*b^3*d^7*e^3 - 10*a^3*b^2*d^6*e^4 + 5*a^4*b*d
^5*e^5 - a^5*d^4*e^6 + (b^5*d^5*e^5 - 5*a*b^4*d^4*e^6 + 10*a^2*b^3*d^3*e^7 - 10*a^3*b^2*d^2*e^8 + 5*a^4*b*d*e^
9 - a^5*e^10)*x^4 + 4*(b^5*d^6*e^4 - 5*a*b^4*d^5*e^5 + 10*a^2*b^3*d^4*e^6 - 10*a^3*b^2*d^3*e^7 + 5*a^4*b*d^2*e
^8 - a^5*d*e^9)*x^3 + 6*(b^5*d^7*e^3 - 5*a*b^4*d^6*e^4 + 10*a^2*b^3*d^5*e^5 - 10*a^3*b^2*d^4*e^6 + 5*a^4*b*d^3
*e^7 - a^5*d^2*e^8)*x^2 + 4*(b^5*d^8*e^2 - 5*a*b^4*d^7*e^3 + 10*a^2*b^3*d^6*e^4 - 10*a^3*b^2*d^5*e^5 + 5*a^4*b
*d^4*e^6 - a^5*d^3*e^7)*x)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1132 vs. \(2 (150) = 300\).

Time = 2.32 (sec) , antiderivative size = 1132, normalized size of antiderivative = 6.36 \[ \int \frac {A+B x}{(a+b x) (d+e x)^5} \, dx=- \frac {b^{3} \left (- A b + B a\right ) \log {\left (x + \frac {- A a b^{4} e - A b^{5} d + B a^{2} b^{3} e + B a b^{4} d - \frac {a^{6} b^{3} e^{6} \left (- A b + B a\right )}{\left (a e - b d\right )^{5}} + \frac {6 a^{5} b^{4} d e^{5} \left (- A b + B a\right )}{\left (a e - b d\right )^{5}} - \frac {15 a^{4} b^{5} d^{2} e^{4} \left (- A b + B a\right )}{\left (a e - b d\right )^{5}} + \frac {20 a^{3} b^{6} d^{3} e^{3} \left (- A b + B a\right )}{\left (a e - b d\right )^{5}} - \frac {15 a^{2} b^{7} d^{4} e^{2} \left (- A b + B a\right )}{\left (a e - b d\right )^{5}} + \frac {6 a b^{8} d^{5} e \left (- A b + B a\right )}{\left (a e - b d\right )^{5}} - \frac {b^{9} d^{6} \left (- A b + B a\right )}{\left (a e - b d\right )^{5}}}{- 2 A b^{5} e + 2 B a b^{4} e} \right )}}{\left (a e - b d\right )^{5}} + \frac {b^{3} \left (- A b + B a\right ) \log {\left (x + \frac {- A a b^{4} e - A b^{5} d + B a^{2} b^{3} e + B a b^{4} d + \frac {a^{6} b^{3} e^{6} \left (- A b + B a\right )}{\left (a e - b d\right )^{5}} - \frac {6 a^{5} b^{4} d e^{5} \left (- A b + B a\right )}{\left (a e - b d\right )^{5}} + \frac {15 a^{4} b^{5} d^{2} e^{4} \left (- A b + B a\right )}{\left (a e - b d\right )^{5}} - \frac {20 a^{3} b^{6} d^{3} e^{3} \left (- A b + B a\right )}{\left (a e - b d\right )^{5}} + \frac {15 a^{2} b^{7} d^{4} e^{2} \left (- A b + B a\right )}{\left (a e - b d\right )^{5}} - \frac {6 a b^{8} d^{5} e \left (- A b + B a\right )}{\left (a e - b d\right )^{5}} + \frac {b^{9} d^{6} \left (- A b + B a\right )}{\left (a e - b d\right )^{5}}}{- 2 A b^{5} e + 2 B a b^{4} e} \right )}}{\left (a e - b d\right )^{5}} + \frac {- 3 A a^{3} e^{4} + 13 A a^{2} b d e^{3} - 23 A a b^{2} d^{2} e^{2} + 25 A b^{3} d^{3} e - B a^{3} d e^{3} + 5 B a^{2} b d^{2} e^{2} - 13 B a b^{2} d^{3} e - 3 B b^{3} d^{4} + x^{3} \cdot \left (12 A b^{3} e^{4} - 12 B a b^{2} e^{4}\right ) + x^{2} \left (- 6 A a b^{2} e^{4} + 42 A b^{3} d e^{3} + 6 B a^{2} b e^{4} - 42 B a b^{2} d e^{3}\right ) + x \left (4 A a^{2} b e^{4} - 20 A a b^{2} d e^{3} + 52 A b^{3} d^{2} e^{2} - 4 B a^{3} e^{4} + 20 B a^{2} b d e^{3} - 52 B a b^{2} d^{2} e^{2}\right )}{12 a^{4} d^{4} e^{5} - 48 a^{3} b d^{5} e^{4} + 72 a^{2} b^{2} d^{6} e^{3} - 48 a b^{3} d^{7} e^{2} + 12 b^{4} d^{8} e + x^{4} \cdot \left (12 a^{4} e^{9} - 48 a^{3} b d e^{8} + 72 a^{2} b^{2} d^{2} e^{7} - 48 a b^{3} d^{3} e^{6} + 12 b^{4} d^{4} e^{5}\right ) + x^{3} \cdot \left (48 a^{4} d e^{8} - 192 a^{3} b d^{2} e^{7} + 288 a^{2} b^{2} d^{3} e^{6} - 192 a b^{3} d^{4} e^{5} + 48 b^{4} d^{5} e^{4}\right ) + x^{2} \cdot \left (72 a^{4} d^{2} e^{7} - 288 a^{3} b d^{3} e^{6} + 432 a^{2} b^{2} d^{4} e^{5} - 288 a b^{3} d^{5} e^{4} + 72 b^{4} d^{6} e^{3}\right ) + x \left (48 a^{4} d^{3} e^{6} - 192 a^{3} b d^{4} e^{5} + 288 a^{2} b^{2} d^{5} e^{4} - 192 a b^{3} d^{6} e^{3} + 48 b^{4} d^{7} e^{2}\right )} \]

[In]

integrate((B*x+A)/(b*x+a)/(e*x+d)**5,x)

[Out]

-b**3*(-A*b + B*a)*log(x + (-A*a*b**4*e - A*b**5*d + B*a**2*b**3*e + B*a*b**4*d - a**6*b**3*e**6*(-A*b + B*a)/
(a*e - b*d)**5 + 6*a**5*b**4*d*e**5*(-A*b + B*a)/(a*e - b*d)**5 - 15*a**4*b**5*d**2*e**4*(-A*b + B*a)/(a*e - b
*d)**5 + 20*a**3*b**6*d**3*e**3*(-A*b + B*a)/(a*e - b*d)**5 - 15*a**2*b**7*d**4*e**2*(-A*b + B*a)/(a*e - b*d)*
*5 + 6*a*b**8*d**5*e*(-A*b + B*a)/(a*e - b*d)**5 - b**9*d**6*(-A*b + B*a)/(a*e - b*d)**5)/(-2*A*b**5*e + 2*B*a
*b**4*e))/(a*e - b*d)**5 + b**3*(-A*b + B*a)*log(x + (-A*a*b**4*e - A*b**5*d + B*a**2*b**3*e + B*a*b**4*d + a*
*6*b**3*e**6*(-A*b + B*a)/(a*e - b*d)**5 - 6*a**5*b**4*d*e**5*(-A*b + B*a)/(a*e - b*d)**5 + 15*a**4*b**5*d**2*
e**4*(-A*b + B*a)/(a*e - b*d)**5 - 20*a**3*b**6*d**3*e**3*(-A*b + B*a)/(a*e - b*d)**5 + 15*a**2*b**7*d**4*e**2
*(-A*b + B*a)/(a*e - b*d)**5 - 6*a*b**8*d**5*e*(-A*b + B*a)/(a*e - b*d)**5 + b**9*d**6*(-A*b + B*a)/(a*e - b*d
)**5)/(-2*A*b**5*e + 2*B*a*b**4*e))/(a*e - b*d)**5 + (-3*A*a**3*e**4 + 13*A*a**2*b*d*e**3 - 23*A*a*b**2*d**2*e
**2 + 25*A*b**3*d**3*e - B*a**3*d*e**3 + 5*B*a**2*b*d**2*e**2 - 13*B*a*b**2*d**3*e - 3*B*b**3*d**4 + x**3*(12*
A*b**3*e**4 - 12*B*a*b**2*e**4) + x**2*(-6*A*a*b**2*e**4 + 42*A*b**3*d*e**3 + 6*B*a**2*b*e**4 - 42*B*a*b**2*d*
e**3) + x*(4*A*a**2*b*e**4 - 20*A*a*b**2*d*e**3 + 52*A*b**3*d**2*e**2 - 4*B*a**3*e**4 + 20*B*a**2*b*d*e**3 - 5
2*B*a*b**2*d**2*e**2))/(12*a**4*d**4*e**5 - 48*a**3*b*d**5*e**4 + 72*a**2*b**2*d**6*e**3 - 48*a*b**3*d**7*e**2
+ 12*b**4*d**8*e + x**4*(12*a**4*e**9 - 48*a**3*b*d*e**8 + 72*a**2*b**2*d**2*e**7 - 48*a*b**3*d**3*e**6 + 12*
b**4*d**4*e**5) + x**3*(48*a**4*d*e**8 - 192*a**3*b*d**2*e**7 + 288*a**2*b**2*d**3*e**6 - 192*a*b**3*d**4*e**5
+ 48*b**4*d**5*e**4) + x**2*(72*a**4*d**2*e**7 - 288*a**3*b*d**3*e**6 + 432*a**2*b**2*d**4*e**5 - 288*a*b**3*
d**5*e**4 + 72*b**4*d**6*e**3) + x*(48*a**4*d**3*e**6 - 192*a**3*b*d**4*e**5 + 288*a**2*b**2*d**5*e**4 - 192*a
*b**3*d**6*e**3 + 48*b**4*d**7*e**2))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 687 vs. \(2 (173) = 346\).

Time = 0.26 (sec) , antiderivative size = 687, normalized size of antiderivative = 3.86 \[ \int \frac {A+B x}{(a+b x) (d+e x)^5} \, dx=-\frac {{\left (B a b^{3} - A b^{4}\right )} \log \left (b x + a\right )}{b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}} + \frac {{\left (B a b^{3} - A b^{4}\right )} \log \left (e x + d\right )}{b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}} - \frac {3 \, B b^{3} d^{4} + 3 \, A a^{3} e^{4} + 12 \, {\left (B a b^{2} - A b^{3}\right )} e^{4} x^{3} + {\left (13 \, B a b^{2} - 25 \, A b^{3}\right )} d^{3} e - {\left (5 \, B a^{2} b - 23 \, A a b^{2}\right )} d^{2} e^{2} + {\left (B a^{3} - 13 \, A a^{2} b\right )} d e^{3} + 6 \, {\left (7 \, {\left (B a b^{2} - A b^{3}\right )} d e^{3} - {\left (B a^{2} b - A a b^{2}\right )} e^{4}\right )} x^{2} + 4 \, {\left (13 \, {\left (B a b^{2} - A b^{3}\right )} d^{2} e^{2} - 5 \, {\left (B a^{2} b - A a b^{2}\right )} d e^{3} + {\left (B a^{3} - A a^{2} b\right )} e^{4}\right )} x}{12 \, {\left (b^{4} d^{8} e - 4 \, a b^{3} d^{7} e^{2} + 6 \, a^{2} b^{2} d^{6} e^{3} - 4 \, a^{3} b d^{5} e^{4} + a^{4} d^{4} e^{5} + {\left (b^{4} d^{4} e^{5} - 4 \, a b^{3} d^{3} e^{6} + 6 \, a^{2} b^{2} d^{2} e^{7} - 4 \, a^{3} b d e^{8} + a^{4} e^{9}\right )} x^{4} + 4 \, {\left (b^{4} d^{5} e^{4} - 4 \, a b^{3} d^{4} e^{5} + 6 \, a^{2} b^{2} d^{3} e^{6} - 4 \, a^{3} b d^{2} e^{7} + a^{4} d e^{8}\right )} x^{3} + 6 \, {\left (b^{4} d^{6} e^{3} - 4 \, a b^{3} d^{5} e^{4} + 6 \, a^{2} b^{2} d^{4} e^{5} - 4 \, a^{3} b d^{3} e^{6} + a^{4} d^{2} e^{7}\right )} x^{2} + 4 \, {\left (b^{4} d^{7} e^{2} - 4 \, a b^{3} d^{6} e^{3} + 6 \, a^{2} b^{2} d^{5} e^{4} - 4 \, a^{3} b d^{4} e^{5} + a^{4} d^{3} e^{6}\right )} x\right )}} \]

[In]

integrate((B*x+A)/(b*x+a)/(e*x+d)^5,x, algorithm="maxima")

[Out]

-(B*a*b^3 - A*b^4)*log(b*x + a)/(b^5*d^5 - 5*a*b^4*d^4*e + 10*a^2*b^3*d^3*e^2 - 10*a^3*b^2*d^2*e^3 + 5*a^4*b*d
*e^4 - a^5*e^5) + (B*a*b^3 - A*b^4)*log(e*x + d)/(b^5*d^5 - 5*a*b^4*d^4*e + 10*a^2*b^3*d^3*e^2 - 10*a^3*b^2*d^
2*e^3 + 5*a^4*b*d*e^4 - a^5*e^5) - 1/12*(3*B*b^3*d^4 + 3*A*a^3*e^4 + 12*(B*a*b^2 - A*b^3)*e^4*x^3 + (13*B*a*b^
2 - 25*A*b^3)*d^3*e - (5*B*a^2*b - 23*A*a*b^2)*d^2*e^2 + (B*a^3 - 13*A*a^2*b)*d*e^3 + 6*(7*(B*a*b^2 - A*b^3)*d
*e^3 - (B*a^2*b - A*a*b^2)*e^4)*x^2 + 4*(13*(B*a*b^2 - A*b^3)*d^2*e^2 - 5*(B*a^2*b - A*a*b^2)*d*e^3 + (B*a^3 -
A*a^2*b)*e^4)*x)/(b^4*d^8*e - 4*a*b^3*d^7*e^2 + 6*a^2*b^2*d^6*e^3 - 4*a^3*b*d^5*e^4 + a^4*d^4*e^5 + (b^4*d^4*
e^5 - 4*a*b^3*d^3*e^6 + 6*a^2*b^2*d^2*e^7 - 4*a^3*b*d*e^8 + a^4*e^9)*x^4 + 4*(b^4*d^5*e^4 - 4*a*b^3*d^4*e^5 +
6*a^2*b^2*d^3*e^6 - 4*a^3*b*d^2*e^7 + a^4*d*e^8)*x^3 + 6*(b^4*d^6*e^3 - 4*a*b^3*d^5*e^4 + 6*a^2*b^2*d^4*e^5 -
4*a^3*b*d^3*e^6 + a^4*d^2*e^7)*x^2 + 4*(b^4*d^7*e^2 - 4*a*b^3*d^6*e^3 + 6*a^2*b^2*d^5*e^4 - 4*a^3*b*d^4*e^5 +
a^4*d^3*e^6)*x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 530 vs. \(2 (173) = 346\).

Time = 0.30 (sec) , antiderivative size = 530, normalized size of antiderivative = 2.98 \[ \int \frac {A+B x}{(a+b x) (d+e x)^5} \, dx=-\frac {{\left (B a b^{3} e - A b^{4} e\right )} \log \left ({\left | b - \frac {b d}{e x + d} + \frac {a e}{e x + d} \right |}\right )}{b^{5} d^{5} e - 5 \, a b^{4} d^{4} e^{2} + 10 \, a^{2} b^{3} d^{3} e^{3} - 10 \, a^{3} b^{2} d^{2} e^{4} + 5 \, a^{4} b d e^{5} - a^{5} e^{6}} - \frac {\frac {3 \, B b^{3} d^{4} e^{3}}{{\left (e x + d\right )}^{4}} + \frac {12 \, B a b^{2} e^{4}}{e x + d} - \frac {12 \, A b^{3} e^{4}}{e x + d} + \frac {6 \, B a b^{2} d e^{4}}{{\left (e x + d\right )}^{2}} - \frac {6 \, A b^{3} d e^{4}}{{\left (e x + d\right )}^{2}} + \frac {4 \, B a b^{2} d^{2} e^{4}}{{\left (e x + d\right )}^{3}} - \frac {4 \, A b^{3} d^{2} e^{4}}{{\left (e x + d\right )}^{3}} - \frac {9 \, B a b^{2} d^{3} e^{4}}{{\left (e x + d\right )}^{4}} - \frac {3 \, A b^{3} d^{3} e^{4}}{{\left (e x + d\right )}^{4}} - \frac {6 \, B a^{2} b e^{5}}{{\left (e x + d\right )}^{2}} + \frac {6 \, A a b^{2} e^{5}}{{\left (e x + d\right )}^{2}} - \frac {8 \, B a^{2} b d e^{5}}{{\left (e x + d\right )}^{3}} + \frac {8 \, A a b^{2} d e^{5}}{{\left (e x + d\right )}^{3}} + \frac {9 \, B a^{2} b d^{2} e^{5}}{{\left (e x + d\right )}^{4}} + \frac {9 \, A a b^{2} d^{2} e^{5}}{{\left (e x + d\right )}^{4}} + \frac {4 \, B a^{3} e^{6}}{{\left (e x + d\right )}^{3}} - \frac {4 \, A a^{2} b e^{6}}{{\left (e x + d\right )}^{3}} - \frac {3 \, B a^{3} d e^{6}}{{\left (e x + d\right )}^{4}} - \frac {9 \, A a^{2} b d e^{6}}{{\left (e x + d\right )}^{4}} + \frac {3 \, A a^{3} e^{7}}{{\left (e x + d\right )}^{4}}}{12 \, {\left (b^{4} d^{4} e^{4} - 4 \, a b^{3} d^{3} e^{5} + 6 \, a^{2} b^{2} d^{2} e^{6} - 4 \, a^{3} b d e^{7} + a^{4} e^{8}\right )}} \]

[In]

integrate((B*x+A)/(b*x+a)/(e*x+d)^5,x, algorithm="giac")

[Out]

-(B*a*b^3*e - A*b^4*e)*log(abs(b - b*d/(e*x + d) + a*e/(e*x + d)))/(b^5*d^5*e - 5*a*b^4*d^4*e^2 + 10*a^2*b^3*d
^3*e^3 - 10*a^3*b^2*d^2*e^4 + 5*a^4*b*d*e^5 - a^5*e^6) - 1/12*(3*B*b^3*d^4*e^3/(e*x + d)^4 + 12*B*a*b^2*e^4/(e
*x + d) - 12*A*b^3*e^4/(e*x + d) + 6*B*a*b^2*d*e^4/(e*x + d)^2 - 6*A*b^3*d*e^4/(e*x + d)^2 + 4*B*a*b^2*d^2*e^4
/(e*x + d)^3 - 4*A*b^3*d^2*e^4/(e*x + d)^3 - 9*B*a*b^2*d^3*e^4/(e*x + d)^4 - 3*A*b^3*d^3*e^4/(e*x + d)^4 - 6*B
*a^2*b*e^5/(e*x + d)^2 + 6*A*a*b^2*e^5/(e*x + d)^2 - 8*B*a^2*b*d*e^5/(e*x + d)^3 + 8*A*a*b^2*d*e^5/(e*x + d)^3
+ 9*B*a^2*b*d^2*e^5/(e*x + d)^4 + 9*A*a*b^2*d^2*e^5/(e*x + d)^4 + 4*B*a^3*e^6/(e*x + d)^3 - 4*A*a^2*b*e^6/(e*
x + d)^3 - 3*B*a^3*d*e^6/(e*x + d)^4 - 9*A*a^2*b*d*e^6/(e*x + d)^4 + 3*A*a^3*e^7/(e*x + d)^4)/(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)

Mupad [B] (verification not implemented)

Time = 1.71 (sec) , antiderivative size = 629, normalized size of antiderivative = 3.53 \[ \int \frac {A+B x}{(a+b x) (d+e x)^5} \, dx=-\frac {\frac {B\,a^3\,d\,e^3+3\,A\,a^3\,e^4-5\,B\,a^2\,b\,d^2\,e^2-13\,A\,a^2\,b\,d\,e^3+13\,B\,a\,b^2\,d^3\,e+23\,A\,a\,b^2\,d^2\,e^2+3\,B\,b^3\,d^4-25\,A\,b^3\,d^3\,e}{12\,e\,\left (a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4\right )}-\frac {x\,\left (A\,b-B\,a\right )\,\left (a^2\,e^3-5\,a\,b\,d\,e^2+13\,b^2\,d^2\,e\right )}{3\,\left (a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4\right )}+\frac {b\,x^2\,\left (A\,b-B\,a\right )\,\left (a\,e^3-7\,b\,d\,e^2\right )}{2\,\left (a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4\right )}-\frac {b^2\,e^3\,x^3\,\left (A\,b-B\,a\right )}{a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4}}{d^4+4\,d^3\,e\,x+6\,d^2\,e^2\,x^2+4\,d\,e^3\,x^3+e^4\,x^4}-\frac {2\,b^3\,\mathrm {atanh}\left (\frac {\left (\frac {a^5\,e^5-3\,a^4\,b\,d\,e^4+2\,a^3\,b^2\,d^2\,e^3+2\,a^2\,b^3\,d^3\,e^2-3\,a\,b^4\,d^4\,e+b^5\,d^5}{a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4}+2\,b\,e\,x\right )\,\left (a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4\right )}{{\left (a\,e-b\,d\right )}^5}\right )\,\left (A\,b-B\,a\right )}{{\left (a\,e-b\,d\right )}^5} \]

[In]

int((A + B*x)/((a + b*x)*(d + e*x)^5),x)

[Out]

- ((3*A*a^3*e^4 + 3*B*b^3*d^4 - 25*A*b^3*d^3*e + B*a^3*d*e^3 + 23*A*a*b^2*d^2*e^2 - 5*B*a^2*b*d^2*e^2 - 13*A*a
^2*b*d*e^3 + 13*B*a*b^2*d^3*e)/(12*e*(a^4*e^4 + b^4*d^4 + 6*a^2*b^2*d^2*e^2 - 4*a*b^3*d^3*e - 4*a^3*b*d*e^3))
- (x*(A*b - B*a)*(a^2*e^3 + 13*b^2*d^2*e - 5*a*b*d*e^2))/(3*(a^4*e^4 + b^4*d^4 + 6*a^2*b^2*d^2*e^2 - 4*a*b^3*d
^3*e - 4*a^3*b*d*e^3)) + (b*x^2*(A*b - B*a)*(a*e^3 - 7*b*d*e^2))/(2*(a^4*e^4 + b^4*d^4 + 6*a^2*b^2*d^2*e^2 - 4
*a*b^3*d^3*e - 4*a^3*b*d*e^3)) - (b^2*e^3*x^3*(A*b - B*a))/(a^4*e^4 + b^4*d^4 + 6*a^2*b^2*d^2*e^2 - 4*a*b^3*d^
3*e - 4*a^3*b*d*e^3))/(d^4 + e^4*x^4 + 4*d*e^3*x^3 + 6*d^2*e^2*x^2 + 4*d^3*e*x) - (2*b^3*atanh((((a^5*e^5 + b^
5*d^5 + 2*a^2*b^3*d^3*e^2 + 2*a^3*b^2*d^2*e^3 - 3*a*b^4*d^4*e - 3*a^4*b*d*e^4)/(a^4*e^4 + b^4*d^4 + 6*a^2*b^2*
d^2*e^2 - 4*a*b^3*d^3*e - 4*a^3*b*d*e^3) + 2*b*e*x)*(a^4*e^4 + b^4*d^4 + 6*a^2*b^2*d^2*e^2 - 4*a*b^3*d^3*e - 4
*a^3*b*d*e^3))/(a*e - b*d)^5)*(A*b - B*a))/(a*e - b*d)^5