\(\int \frac {(A+B x) (d+e x)^5}{a+b x} \, dx\) [1111]

Optimal result

Integrand size = 20, antiderivative size = 187 \[ \int \frac {(A+B x) (d+e x)^5}{a+b x} \, dx=\frac {(A b-a B) e (b d-a e)^4 x}{b^6}+\frac {(A b-a B) (b d-a e)^3 (d+e x)^2}{2 b^5}+\frac {(A b-a B) (b d-a e)^2 (d+e x)^3}{3 b^4}+\frac {(A b-a B) (b d-a e) (d+e x)^4}{4 b^3}+\frac {(A b-a B) (d+e x)^5}{5 b^2}+\frac {B (d+e x)^6}{6 b e}+\frac {(A b-a B) (b d-a e)^5 \log (a+b x)}{b^7} \]

[Out]

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

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 187, 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} \[ \int \frac {(A+B x) (d+e x)^5}{a+b x} \, dx=\frac {(A b-a B) (b d-a e)^5 \log (a+b x)}{b^7}+\frac {e x (A b-a B) (b d-a e)^4}{b^6}+\frac {(d+e x)^2 (A b-a B) (b d-a e)^3}{2 b^5}+\frac {(d+e x)^3 (A b-a B) (b d-a e)^2}{3 b^4}+\frac {(d+e x)^4 (A b-a B) (b d-a e)}{4 b^3}+\frac {(d+e x)^5 (A b-a B)}{5 b^2}+\frac {B (d+e x)^6}{6 b e} \]

[In]

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

[Out]

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

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 {(A b-a B) e (b d-a e)^4}{b^6}+\frac {(A b-a B) (b d-a e)^5}{b^6 (a+b x)}+\frac {(A b-a B) e (b d-a e)^3 (d+e x)}{b^5}+\frac {(A b-a B) e (b d-a e)^2 (d+e x)^2}{b^4}+\frac {(A b-a B) e (b d-a e) (d+e x)^3}{b^3}+\frac {(A b-a B) e (d+e x)^4}{b^2}+\frac {B (d+e x)^5}{b}\right ) \, dx \\ & = \frac {(A b-a B) e (b d-a e)^4 x}{b^6}+\frac {(A b-a B) (b d-a e)^3 (d+e x)^2}{2 b^5}+\frac {(A b-a B) (b d-a e)^2 (d+e x)^3}{3 b^4}+\frac {(A b-a B) (b d-a e) (d+e x)^4}{4 b^3}+\frac {(A b-a B) (d+e x)^5}{5 b^2}+\frac {B (d+e x)^6}{6 b e}+\frac {(A b-a B) (b d-a e)^5 \log (a+b x)}{b^7} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.97 \[ \int \frac {(A+B x) (d+e x)^5}{a+b x} \, dx=\frac {b x \left (-60 a^5 B e^5+30 a^4 b e^4 (10 B d+2 A e+B e x)-10 a^3 b^2 e^3 \left (3 A e (10 d+e x)+B \left (60 d^2+15 d e x+2 e^2 x^2\right )\right )+5 a^2 b^3 e^2 \left (2 A e \left (60 d^2+15 d e x+2 e^2 x^2\right )+B \left (120 d^3+60 d^2 e x+20 d e^2 x^2+3 e^3 x^3\right )\right )-a b^4 e \left (5 A e \left (120 d^3+60 d^2 e x+20 d e^2 x^2+3 e^3 x^3\right )+B \left (300 d^4+300 d^3 e x+200 d^2 e^2 x^2+75 d e^3 x^3+12 e^4 x^4\right )\right )+b^5 \left (A e \left (300 d^4+300 d^3 e x+200 d^2 e^2 x^2+75 d e^3 x^3+12 e^4 x^4\right )+10 B \left (6 d^5+15 d^4 e x+20 d^3 e^2 x^2+15 d^2 e^3 x^3+6 d e^4 x^4+e^5 x^5\right )\right )\right )+60 (A b-a B) (b d-a e)^5 \log (a+b x)}{60 b^7} \]

[In]

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

[Out]

(b*x*(-60*a^5*B*e^5 + 30*a^4*b*e^4*(10*B*d + 2*A*e + B*e*x) - 10*a^3*b^2*e^3*(3*A*e*(10*d + e*x) + B*(60*d^2 +
15*d*e*x + 2*e^2*x^2)) + 5*a^2*b^3*e^2*(2*A*e*(60*d^2 + 15*d*e*x + 2*e^2*x^2) + B*(120*d^3 + 60*d^2*e*x + 20*
d*e^2*x^2 + 3*e^3*x^3)) - a*b^4*e*(5*A*e*(120*d^3 + 60*d^2*e*x + 20*d*e^2*x^2 + 3*e^3*x^3) + B*(300*d^4 + 300*
d^3*e*x + 200*d^2*e^2*x^2 + 75*d*e^3*x^3 + 12*e^4*x^4)) + b^5*(A*e*(300*d^4 + 300*d^3*e*x + 200*d^2*e^2*x^2 +
75*d*e^3*x^3 + 12*e^4*x^4) + 10*B*(6*d^5 + 15*d^4*e*x + 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 + 6*d*e^4*x^4 + e^5*x^
5))) + 60*(A*b - a*B)*(b*d - a*e)^5*Log[a + b*x])/(60*b^7)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(574\) vs. \(2(177)=354\).

Time = 2.13 (sec) , antiderivative size = 575, normalized size of antiderivative = 3.07

method	result	size
norman	\(\frac {\left (A \,a^{4} b \,e^{5}-5 A \,a^{3} b^{2} d \,e^{4}+10 A \,a^{2} b^{3} d^{2} e^{3}-10 A a \,b^{4} d^{3} e^{2}+5 A \,b^{5} d^{4} e -B \,a^{5} e^{5}+5 B \,a^{4} b d \,e^{4}-10 B \,a^{3} b^{2} d^{2} e^{3}+10 B \,a^{2} b^{3} d^{3} e^{2}-5 B a \,b^{4} d^{4} e +B \,b^{5} d^{5}\right ) x}{b^{6}}+\frac {B \,e^{5} x^{6}}{6 b}-\frac {e \left (A \,a^{3} b \,e^{4}-5 A \,a^{2} b^{2} d \,e^{3}+10 A a \,b^{3} d^{2} e^{2}-10 A \,b^{4} d^{3} e -B \,a^{4} e^{4}+5 B \,a^{3} b d \,e^{3}-10 B \,a^{2} b^{2} d^{2} e^{2}+10 B a \,b^{3} d^{3} e -5 B \,b^{4} d^{4}\right ) x^{2}}{2 b^{5}}+\frac {e^{2} \left (A \,a^{2} b \,e^{3}-5 A a \,b^{2} d \,e^{2}+10 A \,b^{3} d^{2} e -B \,a^{3} e^{3}+5 B \,a^{2} b d \,e^{2}-10 B a \,b^{2} d^{2} e +10 b^{3} B \,d^{3}\right ) x^{3}}{3 b^{4}}-\frac {e^{3} \left (A a b \,e^{2}-5 A \,b^{2} d e -B \,a^{2} e^{2}+5 B a b d e -10 b^{2} B \,d^{2}\right ) x^{4}}{4 b^{3}}+\frac {e^{4} \left (A b e -B a e +5 B b d \right ) x^{5}}{5 b^{2}}-\frac {\left (A \,a^{5} b \,e^{5}-5 A \,a^{4} b^{2} d \,e^{4}+10 A \,a^{3} b^{3} d^{2} e^{3}-10 A \,a^{2} b^{4} d^{3} e^{2}+5 A a \,b^{5} d^{4} e -A \,b^{6} d^{5}-B \,a^{6} e^{5}+5 B \,a^{5} b d \,e^{4}-10 B \,a^{4} b^{2} d^{2} e^{3}+10 B \,a^{3} b^{3} d^{3} e^{2}-5 B \,a^{2} b^{4} d^{4} e +B a \,b^{5} d^{5}\right ) \ln \left (b x +a \right )}{b^{7}}\)	\(575\)
default	\(\frac {-10 B \,a^{3} b^{2} d^{2} e^{3} x +10 B \,a^{2} b^{3} d^{3} e^{2} x -5 B a \,b^{4} d^{4} e x +5 B \,a^{2} b^{3} d^{2} e^{3} x^{2}-5 B a \,b^{4} d^{3} e^{2} x^{2}-5 A \,a^{3} b^{2} d \,e^{4} x +10 A \,a^{2} b^{3} d^{2} e^{3} x -10 A a \,b^{4} d^{3} e^{2} x +5 B \,a^{4} b d \,e^{4} x +\frac {1}{4} B \,a^{2} b^{3} e^{5} x^{4}-B \,a^{5} e^{5} x +B \,b^{5} d^{5} x -\frac {1}{5} B a \,b^{4} e^{5} x^{5}+B \,b^{5} d \,e^{4} x^{5}+\frac {5}{2} B \,b^{5} d^{2} e^{3} x^{4}+\frac {1}{3} A \,a^{2} b^{3} e^{5} x^{3}-\frac {1}{3} B \,a^{3} b^{2} e^{5} x^{3}+\frac {10}{3} B \,b^{5} d^{3} e^{2} x^{3}+\frac {5}{4} A \,b^{5} d \,e^{4} x^{4}+5 A \,b^{5} d^{3} e^{2} x^{2}+\frac {1}{2} B \,a^{4} b \,e^{5} x^{2}+\frac {5}{2} B \,b^{5} d^{4} e \,x^{2}+A \,a^{4} b \,e^{5} x +5 A \,b^{5} d^{4} e x -\frac {1}{2} A \,a^{3} b^{2} e^{5} x^{2}-\frac {1}{4} A a \,b^{4} e^{5} x^{4}+\frac {5}{3} B \,a^{2} b^{3} d \,e^{4} x^{3}-\frac {10}{3} B a \,b^{4} d^{2} e^{3} x^{3}+\frac {5}{2} A \,a^{2} b^{3} d \,e^{4} x^{2}-5 A a \,b^{4} d^{2} e^{3} x^{2}-\frac {5}{2} B \,a^{3} b^{2} d \,e^{4} x^{2}+\frac {10}{3} A \,b^{5} d^{2} e^{3} x^{3}+\frac {1}{5} A \,b^{5} e^{5} x^{5}+\frac {1}{6} b^{5} B \,x^{6} e^{5}-\frac {5}{4} B a \,b^{4} d \,e^{4} x^{4}-\frac {5}{3} A a \,b^{4} d \,e^{4} x^{3}}{b^{6}}+\frac {\left (-A \,a^{5} b \,e^{5}+5 A \,a^{4} b^{2} d \,e^{4}-10 A \,a^{3} b^{3} d^{2} e^{3}+10 A \,a^{2} b^{4} d^{3} e^{2}-5 A a \,b^{5} d^{4} e +A \,b^{6} d^{5}+B \,a^{6} e^{5}-5 B \,a^{5} b d \,e^{4}+10 B \,a^{4} b^{2} d^{2} e^{3}-10 B \,a^{3} b^{3} d^{3} e^{2}+5 B \,a^{2} b^{4} d^{4} e -B a \,b^{5} d^{5}\right ) \ln \left (b x +a \right )}{b^{7}}\)	\(665\)
risch	\(\frac {5 B \,a^{2} d \,e^{4} x^{3}}{3 b^{3}}-\frac {10 B a \,d^{2} e^{3} x^{3}}{3 b^{2}}+\frac {5 A \,a^{2} d \,e^{4} x^{2}}{2 b^{3}}-\frac {5 A a \,d^{2} e^{3} x^{2}}{b^{2}}+\frac {B \,d^{5} x}{b}+\frac {A \,e^{5} x^{5}}{5 b}-\frac {5 B \,a^{3} d \,e^{4} x^{2}}{2 b^{4}}-\frac {5 B a d \,e^{4} x^{4}}{4 b^{2}}-\frac {5 A a d \,e^{4} x^{3}}{3 b^{2}}+\frac {5 \ln \left (b x +a \right ) A \,a^{4} d \,e^{4}}{b^{5}}-\frac {10 \ln \left (b x +a \right ) A \,a^{3} d^{2} e^{3}}{b^{4}}+\frac {10 \ln \left (b x +a \right ) A \,a^{2} d^{3} e^{2}}{b^{3}}-\frac {5 \ln \left (b x +a \right ) A a \,d^{4} e}{b^{2}}-\frac {5 \ln \left (b x +a \right ) B \,a^{5} d \,e^{4}}{b^{6}}+\frac {10 \ln \left (b x +a \right ) B \,a^{4} d^{2} e^{3}}{b^{5}}-\frac {10 \ln \left (b x +a \right ) B \,a^{3} d^{3} e^{2}}{b^{4}}+\frac {5 \ln \left (b x +a \right ) B \,a^{2} d^{4} e}{b^{3}}-\frac {5 A \,a^{3} d \,e^{4} x}{b^{4}}+\frac {10 A \,a^{2} d^{2} e^{3} x}{b^{3}}-\frac {10 A a \,d^{3} e^{2} x}{b^{2}}+\frac {5 B \,a^{4} d \,e^{4} x}{b^{5}}+\frac {B \,a^{2} e^{5} x^{4}}{4 b^{3}}-\frac {B \,a^{5} e^{5} x}{b^{6}}-\frac {10 B \,a^{3} d^{2} e^{3} x}{b^{4}}+\frac {10 B \,a^{2} d^{3} e^{2} x}{b^{3}}-\frac {5 B a \,d^{4} e x}{b^{2}}+\frac {5 B \,a^{2} d^{2} e^{3} x^{2}}{b^{3}}-\frac {5 B a \,d^{3} e^{2} x^{2}}{b^{2}}+\frac {\ln \left (b x +a \right ) A \,d^{5}}{b}-\frac {\ln \left (b x +a \right ) A \,a^{5} e^{5}}{b^{6}}+\frac {\ln \left (b x +a \right ) B \,a^{6} e^{5}}{b^{7}}-\frac {\ln \left (b x +a \right ) B a \,d^{5}}{b^{2}}+\frac {B \,e^{5} x^{6}}{6 b}-\frac {B a \,e^{5} x^{5}}{5 b^{2}}+\frac {B d \,e^{4} x^{5}}{b}+\frac {5 B \,d^{2} e^{3} x^{4}}{2 b}+\frac {A \,a^{2} e^{5} x^{3}}{3 b^{3}}-\frac {B \,a^{3} e^{5} x^{3}}{3 b^{4}}+\frac {10 B \,d^{3} e^{2} x^{3}}{3 b}+\frac {5 A d \,e^{4} x^{4}}{4 b}+\frac {5 A \,d^{3} e^{2} x^{2}}{b}+\frac {B \,a^{4} e^{5} x^{2}}{2 b^{5}}+\frac {5 B \,d^{4} e \,x^{2}}{2 b}+\frac {A \,a^{4} e^{5} x}{b^{5}}+\frac {5 A \,d^{4} e x}{b}-\frac {A \,a^{3} e^{5} x^{2}}{2 b^{4}}-\frac {A a \,e^{5} x^{4}}{4 b^{2}}+\frac {10 A \,d^{2} e^{3} x^{3}}{3 b}\)	\(737\)
parallelrisch	\(-\frac {-600 A \ln \left (b x +a \right ) a^{2} b^{4} d^{3} e^{2}+300 A \ln \left (b x +a \right ) a \,b^{5} d^{4} e +300 B \ln \left (b x +a \right ) a^{5} b d \,e^{4}-600 B \ln \left (b x +a \right ) a^{4} b^{2} d^{2} e^{3}+600 B \ln \left (b x +a \right ) a^{3} b^{3} d^{3} e^{2}-300 B \ln \left (b x +a \right ) a^{2} b^{4} d^{4} e -300 B x \,a^{4} b^{2} d \,e^{4}+600 B x \,a^{3} b^{3} d^{2} e^{3}-600 B x \,a^{2} b^{4} d^{3} e^{2}+300 B x a \,b^{5} d^{4} e +100 A \,x^{3} a \,b^{5} d \,e^{4}-100 B \,x^{3} a^{2} b^{4} d \,e^{4}+200 B \,x^{3} a \,b^{5} d^{2} e^{3}-150 A \,x^{2} a^{2} b^{4} d \,e^{4}+300 A \,x^{2} a \,b^{5} d^{2} e^{3}+150 B \,x^{2} a^{3} b^{3} d \,e^{4}-300 B \,x^{2} a^{2} b^{4} d^{2} e^{3}+300 B \,x^{2} a \,b^{5} d^{3} e^{2}+300 A x \,a^{3} b^{3} d \,e^{4}-600 A x \,a^{2} b^{4} d^{2} e^{3}+600 A x a \,b^{5} d^{3} e^{2}+75 B \,x^{4} a \,b^{5} d \,e^{4}-15 B \,x^{4} a^{2} b^{4} e^{5}-150 B \,x^{4} b^{6} d^{2} e^{3}-20 A \,x^{3} a^{2} b^{4} e^{5}-200 A \,x^{3} b^{6} d^{2} e^{3}+20 B \,x^{3} a^{3} b^{3} e^{5}-200 B \,x^{3} b^{6} d^{3} e^{2}+30 A \,x^{2} a^{3} b^{3} e^{5}-300 A \,x^{2} b^{6} d^{3} e^{2}-30 B \,x^{2} a^{4} b^{2} e^{5}+60 B \ln \left (b x +a \right ) a \,b^{5} d^{5}+60 A \ln \left (b x +a \right ) a^{5} b \,e^{5}-150 B \,x^{2} b^{6} d^{4} e -60 A x \,a^{4} b^{2} e^{5}-300 A x \,b^{6} d^{4} e +60 B x \,a^{5} b \,e^{5}+12 B \,x^{5} a \,b^{5} e^{5}-60 B \,x^{5} b^{6} d \,e^{4}+15 A \,x^{4} a \,b^{5} e^{5}-75 A \,x^{4} b^{6} d \,e^{4}-60 A \ln \left (b x +a \right ) b^{6} d^{5}-60 B \ln \left (b x +a \right ) a^{6} e^{5}-10 B \,x^{6} e^{5} b^{6}-12 A \,x^{5} b^{6} e^{5}-60 B x \,b^{6} d^{5}-300 A \ln \left (b x +a \right ) a^{4} b^{2} d \,e^{4}+600 A \ln \left (b x +a \right ) a^{3} b^{3} d^{2} e^{3}}{60 b^{7}}\)	\(738\)

[In]

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

[Out]

(A*a^4*b*e^5-5*A*a^3*b^2*d*e^4+10*A*a^2*b^3*d^2*e^3-10*A*a*b^4*d^3*e^2+5*A*b^5*d^4*e-B*a^5*e^5+5*B*a^4*b*d*e^4
-10*B*a^3*b^2*d^2*e^3+10*B*a^2*b^3*d^3*e^2-5*B*a*b^4*d^4*e+B*b^5*d^5)/b^6*x+1/6*B/b*e^5*x^6-1/2/b^5*e*(A*a^3*b
*e^4-5*A*a^2*b^2*d*e^3+10*A*a*b^3*d^2*e^2-10*A*b^4*d^3*e-B*a^4*e^4+5*B*a^3*b*d*e^3-10*B*a^2*b^2*d^2*e^2+10*B*a
*b^3*d^3*e-5*B*b^4*d^4)*x^2+1/3/b^4*e^2*(A*a^2*b*e^3-5*A*a*b^2*d*e^2+10*A*b^3*d^2*e-B*a^3*e^3+5*B*a^2*b*d*e^2-
10*B*a*b^2*d^2*e+10*B*b^3*d^3)*x^3-1/4/b^3*e^3*(A*a*b*e^2-5*A*b^2*d*e-B*a^2*e^2+5*B*a*b*d*e-10*B*b^2*d^2)*x^4+
1/5/b^2*e^4*(A*b*e-B*a*e+5*B*b*d)*x^5-(A*a^5*b*e^5-5*A*a^4*b^2*d*e^4+10*A*a^3*b^3*d^2*e^3-10*A*a^2*b^4*d^3*e^2
+5*A*a*b^5*d^4*e-A*b^6*d^5-B*a^6*e^5+5*B*a^5*b*d*e^4-10*B*a^4*b^2*d^2*e^3+10*B*a^3*b^3*d^3*e^2-5*B*a^2*b^4*d^4
*e+B*a*b^5*d^5)/b^7*ln(b*x+a)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 566 vs. \(2 (179) = 358\).

Time = 0.23 (sec) , antiderivative size = 566, normalized size of antiderivative = 3.03 \[ \int \frac {(A+B x) (d+e x)^5}{a+b x} \, dx=\frac {10 \, B b^{6} e^{5} x^{6} + 12 \, {\left (5 \, B b^{6} d e^{4} - {\left (B a b^{5} - A b^{6}\right )} e^{5}\right )} x^{5} + 15 \, {\left (10 \, B b^{6} d^{2} e^{3} - 5 \, {\left (B a b^{5} - A b^{6}\right )} d e^{4} + {\left (B a^{2} b^{4} - A a b^{5}\right )} e^{5}\right )} x^{4} + 20 \, {\left (10 \, B b^{6} d^{3} e^{2} - 10 \, {\left (B a b^{5} - A b^{6}\right )} d^{2} e^{3} + 5 \, {\left (B a^{2} b^{4} - A a b^{5}\right )} d e^{4} - {\left (B a^{3} b^{3} - A a^{2} b^{4}\right )} e^{5}\right )} x^{3} + 30 \, {\left (5 \, B b^{6} d^{4} e - 10 \, {\left (B a b^{5} - A b^{6}\right )} d^{3} e^{2} + 10 \, {\left (B a^{2} b^{4} - A a b^{5}\right )} d^{2} e^{3} - 5 \, {\left (B a^{3} b^{3} - A a^{2} b^{4}\right )} d e^{4} + {\left (B a^{4} b^{2} - A a^{3} b^{3}\right )} e^{5}\right )} x^{2} + 60 \, {\left (B b^{6} d^{5} - 5 \, {\left (B a b^{5} - A b^{6}\right )} d^{4} e + 10 \, {\left (B a^{2} b^{4} - A a b^{5}\right )} d^{3} e^{2} - 10 \, {\left (B a^{3} b^{3} - A a^{2} b^{4}\right )} d^{2} e^{3} + 5 \, {\left (B a^{4} b^{2} - A a^{3} b^{3}\right )} d e^{4} - {\left (B a^{5} b - A a^{4} b^{2}\right )} e^{5}\right )} x - 60 \, {\left ({\left (B a b^{5} - A b^{6}\right )} d^{5} - 5 \, {\left (B a^{2} b^{4} - A a b^{5}\right )} d^{4} e + 10 \, {\left (B a^{3} b^{3} - A a^{2} b^{4}\right )} d^{3} e^{2} - 10 \, {\left (B a^{4} b^{2} - A a^{3} b^{3}\right )} d^{2} e^{3} + 5 \, {\left (B a^{5} b - A a^{4} b^{2}\right )} d e^{4} - {\left (B a^{6} - A a^{5} b\right )} e^{5}\right )} \log \left (b x + a\right )}{60 \, b^{7}} \]

[In]

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

[Out]

1/60*(10*B*b^6*e^5*x^6 + 12*(5*B*b^6*d*e^4 - (B*a*b^5 - A*b^6)*e^5)*x^5 + 15*(10*B*b^6*d^2*e^3 - 5*(B*a*b^5 -
A*b^6)*d*e^4 + (B*a^2*b^4 - A*a*b^5)*e^5)*x^4 + 20*(10*B*b^6*d^3*e^2 - 10*(B*a*b^5 - A*b^6)*d^2*e^3 + 5*(B*a^2
*b^4 - A*a*b^5)*d*e^4 - (B*a^3*b^3 - A*a^2*b^4)*e^5)*x^3 + 30*(5*B*b^6*d^4*e - 10*(B*a*b^5 - A*b^6)*d^3*e^2 +
10*(B*a^2*b^4 - A*a*b^5)*d^2*e^3 - 5*(B*a^3*b^3 - A*a^2*b^4)*d*e^4 + (B*a^4*b^2 - A*a^3*b^3)*e^5)*x^2 + 60*(B*
b^6*d^5 - 5*(B*a*b^5 - A*b^6)*d^4*e + 10*(B*a^2*b^4 - A*a*b^5)*d^3*e^2 - 10*(B*a^3*b^3 - A*a^2*b^4)*d^2*e^3 +
5*(B*a^4*b^2 - A*a^3*b^3)*d*e^4 - (B*a^5*b - A*a^4*b^2)*e^5)*x - 60*((B*a*b^5 - A*b^6)*d^5 - 5*(B*a^2*b^4 - A*
a*b^5)*d^4*e + 10*(B*a^3*b^3 - A*a^2*b^4)*d^3*e^2 - 10*(B*a^4*b^2 - A*a^3*b^3)*d^2*e^3 + 5*(B*a^5*b - A*a^4*b^
2)*d*e^4 - (B*a^6 - A*a^5*b)*e^5)*log(b*x + a))/b^7

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 534 vs. \(2 (165) = 330\).

Time = 0.72 (sec) , antiderivative size = 534, normalized size of antiderivative = 2.86 \[ \int \frac {(A+B x) (d+e x)^5}{a+b x} \, dx=\frac {B e^{5} x^{6}}{6 b} + x^{5} \left (\frac {A e^{5}}{5 b} - \frac {B a e^{5}}{5 b^{2}} + \frac {B d e^{4}}{b}\right ) + x^{4} \left (- \frac {A a e^{5}}{4 b^{2}} + \frac {5 A d e^{4}}{4 b} + \frac {B a^{2} e^{5}}{4 b^{3}} - \frac {5 B a d e^{4}}{4 b^{2}} + \frac {5 B d^{2} e^{3}}{2 b}\right ) + x^{3} \left (\frac {A a^{2} e^{5}}{3 b^{3}} - \frac {5 A a d e^{4}}{3 b^{2}} + \frac {10 A d^{2} e^{3}}{3 b} - \frac {B a^{3} e^{5}}{3 b^{4}} + \frac {5 B a^{2} d e^{4}}{3 b^{3}} - \frac {10 B a d^{2} e^{3}}{3 b^{2}} + \frac {10 B d^{3} e^{2}}{3 b}\right ) + x^{2} \left (- \frac {A a^{3} e^{5}}{2 b^{4}} + \frac {5 A a^{2} d e^{4}}{2 b^{3}} - \frac {5 A a d^{2} e^{3}}{b^{2}} + \frac {5 A d^{3} e^{2}}{b} + \frac {B a^{4} e^{5}}{2 b^{5}} - \frac {5 B a^{3} d e^{4}}{2 b^{4}} + \frac {5 B a^{2} d^{2} e^{3}}{b^{3}} - \frac {5 B a d^{3} e^{2}}{b^{2}} + \frac {5 B d^{4} e}{2 b}\right ) + x \left (\frac {A a^{4} e^{5}}{b^{5}} - \frac {5 A a^{3} d e^{4}}{b^{4}} + \frac {10 A a^{2} d^{2} e^{3}}{b^{3}} - \frac {10 A a d^{3} e^{2}}{b^{2}} + \frac {5 A d^{4} e}{b} - \frac {B a^{5} e^{5}}{b^{6}} + \frac {5 B a^{4} d e^{4}}{b^{5}} - \frac {10 B a^{3} d^{2} e^{3}}{b^{4}} + \frac {10 B a^{2} d^{3} e^{2}}{b^{3}} - \frac {5 B a d^{4} e}{b^{2}} + \frac {B d^{5}}{b}\right ) + \frac {\left (- A b + B a\right ) \left (a e - b d\right )^{5} \log {\left (a + b x \right )}}{b^{7}} \]

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 563 vs. \(2 (179) = 358\).

Time = 0.21 (sec) , antiderivative size = 563, normalized size of antiderivative = 3.01 \[ \int \frac {(A+B x) (d+e x)^5}{a+b x} \, dx=\frac {10 \, B b^{5} e^{5} x^{6} + 12 \, {\left (5 \, B b^{5} d e^{4} - {\left (B a b^{4} - A b^{5}\right )} e^{5}\right )} x^{5} + 15 \, {\left (10 \, B b^{5} d^{2} e^{3} - 5 \, {\left (B a b^{4} - A b^{5}\right )} d e^{4} + {\left (B a^{2} b^{3} - A a b^{4}\right )} e^{5}\right )} x^{4} + 20 \, {\left (10 \, B b^{5} d^{3} e^{2} - 10 \, {\left (B a b^{4} - A b^{5}\right )} d^{2} e^{3} + 5 \, {\left (B a^{2} b^{3} - A a b^{4}\right )} d e^{4} - {\left (B a^{3} b^{2} - A a^{2} b^{3}\right )} e^{5}\right )} x^{3} + 30 \, {\left (5 \, B b^{5} d^{4} e - 10 \, {\left (B a b^{4} - A b^{5}\right )} d^{3} e^{2} + 10 \, {\left (B a^{2} b^{3} - A a b^{4}\right )} d^{2} e^{3} - 5 \, {\left (B a^{3} b^{2} - A a^{2} b^{3}\right )} d e^{4} + {\left (B a^{4} b - A a^{3} b^{2}\right )} e^{5}\right )} x^{2} + 60 \, {\left (B b^{5} d^{5} - 5 \, {\left (B a b^{4} - A b^{5}\right )} d^{4} e + 10 \, {\left (B a^{2} b^{3} - A a b^{4}\right )} d^{3} e^{2} - 10 \, {\left (B a^{3} b^{2} - A a^{2} b^{3}\right )} d^{2} e^{3} + 5 \, {\left (B a^{4} b - A a^{3} b^{2}\right )} d e^{4} - {\left (B a^{5} - A a^{4} b\right )} e^{5}\right )} x}{60 \, b^{6}} - \frac {{\left ({\left (B a b^{5} - A b^{6}\right )} d^{5} - 5 \, {\left (B a^{2} b^{4} - A a b^{5}\right )} d^{4} e + 10 \, {\left (B a^{3} b^{3} - A a^{2} b^{4}\right )} d^{3} e^{2} - 10 \, {\left (B a^{4} b^{2} - A a^{3} b^{3}\right )} d^{2} e^{3} + 5 \, {\left (B a^{5} b - A a^{4} b^{2}\right )} d e^{4} - {\left (B a^{6} - A a^{5} b\right )} e^{5}\right )} \log \left (b x + a\right )}{b^{7}} \]

[In]

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

[Out]

1/60*(10*B*b^5*e^5*x^6 + 12*(5*B*b^5*d*e^4 - (B*a*b^4 - A*b^5)*e^5)*x^5 + 15*(10*B*b^5*d^2*e^3 - 5*(B*a*b^4 -
A*b^5)*d*e^4 + (B*a^2*b^3 - A*a*b^4)*e^5)*x^4 + 20*(10*B*b^5*d^3*e^2 - 10*(B*a*b^4 - A*b^5)*d^2*e^3 + 5*(B*a^2
*b^3 - A*a*b^4)*d*e^4 - (B*a^3*b^2 - A*a^2*b^3)*e^5)*x^3 + 30*(5*B*b^5*d^4*e - 10*(B*a*b^4 - A*b^5)*d^3*e^2 +
10*(B*a^2*b^3 - A*a*b^4)*d^2*e^3 - 5*(B*a^3*b^2 - A*a^2*b^3)*d*e^4 + (B*a^4*b - A*a^3*b^2)*e^5)*x^2 + 60*(B*b^
5*d^5 - 5*(B*a*b^4 - A*b^5)*d^4*e + 10*(B*a^2*b^3 - A*a*b^4)*d^3*e^2 - 10*(B*a^3*b^2 - A*a^2*b^3)*d^2*e^3 + 5*
(B*a^4*b - A*a^3*b^2)*d*e^4 - (B*a^5 - A*a^4*b)*e^5)*x)/b^6 - ((B*a*b^5 - A*b^6)*d^5 - 5*(B*a^2*b^4 - A*a*b^5)
*d^4*e + 10*(B*a^3*b^3 - A*a^2*b^4)*d^3*e^2 - 10*(B*a^4*b^2 - A*a^3*b^3)*d^2*e^3 + 5*(B*a^5*b - A*a^4*b^2)*d*e
^4 - (B*a^6 - A*a^5*b)*e^5)*log(b*x + a)/b^7

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 670 vs. \(2 (179) = 358\).

Time = 0.29 (sec) , antiderivative size = 670, normalized size of antiderivative = 3.58 \[ \int \frac {(A+B x) (d+e x)^5}{a+b x} \, dx=\frac {10 \, B b^{5} e^{5} x^{6} + 60 \, B b^{5} d e^{4} x^{5} - 12 \, B a b^{4} e^{5} x^{5} + 12 \, A b^{5} e^{5} x^{5} + 150 \, B b^{5} d^{2} e^{3} x^{4} - 75 \, B a b^{4} d e^{4} x^{4} + 75 \, A b^{5} d e^{4} x^{4} + 15 \, B a^{2} b^{3} e^{5} x^{4} - 15 \, A a b^{4} e^{5} x^{4} + 200 \, B b^{5} d^{3} e^{2} x^{3} - 200 \, B a b^{4} d^{2} e^{3} x^{3} + 200 \, A b^{5} d^{2} e^{3} x^{3} + 100 \, B a^{2} b^{3} d e^{4} x^{3} - 100 \, A a b^{4} d e^{4} x^{3} - 20 \, B a^{3} b^{2} e^{5} x^{3} + 20 \, A a^{2} b^{3} e^{5} x^{3} + 150 \, B b^{5} d^{4} e x^{2} - 300 \, B a b^{4} d^{3} e^{2} x^{2} + 300 \, A b^{5} d^{3} e^{2} x^{2} + 300 \, B a^{2} b^{3} d^{2} e^{3} x^{2} - 300 \, A a b^{4} d^{2} e^{3} x^{2} - 150 \, B a^{3} b^{2} d e^{4} x^{2} + 150 \, A a^{2} b^{3} d e^{4} x^{2} + 30 \, B a^{4} b e^{5} x^{2} - 30 \, A a^{3} b^{2} e^{5} x^{2} + 60 \, B b^{5} d^{5} x - 300 \, B a b^{4} d^{4} e x + 300 \, A b^{5} d^{4} e x + 600 \, B a^{2} b^{3} d^{3} e^{2} x - 600 \, A a b^{4} d^{3} e^{2} x - 600 \, B a^{3} b^{2} d^{2} e^{3} x + 600 \, A a^{2} b^{3} d^{2} e^{3} x + 300 \, B a^{4} b d e^{4} x - 300 \, A a^{3} b^{2} d e^{4} x - 60 \, B a^{5} e^{5} x + 60 \, A a^{4} b e^{5} x}{60 \, b^{6}} - \frac {{\left (B a b^{5} d^{5} - A b^{6} d^{5} - 5 \, B a^{2} b^{4} d^{4} e + 5 \, A a b^{5} d^{4} e + 10 \, B a^{3} b^{3} d^{3} e^{2} - 10 \, A a^{2} b^{4} d^{3} e^{2} - 10 \, B a^{4} b^{2} d^{2} e^{3} + 10 \, A a^{3} b^{3} d^{2} e^{3} + 5 \, B a^{5} b d e^{4} - 5 \, A a^{4} b^{2} d e^{4} - B a^{6} e^{5} + A a^{5} b e^{5}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{7}} \]

[In]

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

[Out]

1/60*(10*B*b^5*e^5*x^6 + 60*B*b^5*d*e^4*x^5 - 12*B*a*b^4*e^5*x^5 + 12*A*b^5*e^5*x^5 + 150*B*b^5*d^2*e^3*x^4 -
75*B*a*b^4*d*e^4*x^4 + 75*A*b^5*d*e^4*x^4 + 15*B*a^2*b^3*e^5*x^4 - 15*A*a*b^4*e^5*x^4 + 200*B*b^5*d^3*e^2*x^3
- 200*B*a*b^4*d^2*e^3*x^3 + 200*A*b^5*d^2*e^3*x^3 + 100*B*a^2*b^3*d*e^4*x^3 - 100*A*a*b^4*d*e^4*x^3 - 20*B*a^3
*b^2*e^5*x^3 + 20*A*a^2*b^3*e^5*x^3 + 150*B*b^5*d^4*e*x^2 - 300*B*a*b^4*d^3*e^2*x^2 + 300*A*b^5*d^3*e^2*x^2 +
300*B*a^2*b^3*d^2*e^3*x^2 - 300*A*a*b^4*d^2*e^3*x^2 - 150*B*a^3*b^2*d*e^4*x^2 + 150*A*a^2*b^3*d*e^4*x^2 + 30*B
*a^4*b*e^5*x^2 - 30*A*a^3*b^2*e^5*x^2 + 60*B*b^5*d^5*x - 300*B*a*b^4*d^4*e*x + 300*A*b^5*d^4*e*x + 600*B*a^2*b
^3*d^3*e^2*x - 600*A*a*b^4*d^3*e^2*x - 600*B*a^3*b^2*d^2*e^3*x + 600*A*a^2*b^3*d^2*e^3*x + 300*B*a^4*b*d*e^4*x
- 300*A*a^3*b^2*d*e^4*x - 60*B*a^5*e^5*x + 60*A*a^4*b*e^5*x)/b^6 - (B*a*b^5*d^5 - A*b^6*d^5 - 5*B*a^2*b^4*d^4
*e + 5*A*a*b^5*d^4*e + 10*B*a^3*b^3*d^3*e^2 - 10*A*a^2*b^4*d^3*e^2 - 10*B*a^4*b^2*d^2*e^3 + 10*A*a^3*b^3*d^2*e
^3 + 5*B*a^5*b*d*e^4 - 5*A*a^4*b^2*d*e^4 - B*a^6*e^5 + A*a^5*b*e^5)*log(abs(b*x + a))/b^7

Mupad [B] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 565, normalized size of antiderivative = 3.02 \[ \int \frac {(A+B x) (d+e x)^5}{a+b x} \, dx=x\,\left (\frac {B\,d^5+5\,A\,e\,d^4}{b}+\frac {a\,\left (\frac {a\,\left (\frac {a\,\left (\frac {a\,\left (\frac {A\,e^5+5\,B\,d\,e^4}{b}-\frac {B\,a\,e^5}{b^2}\right )}{b}-\frac {5\,d\,e^3\,\left (A\,e+2\,B\,d\right )}{b}\right )}{b}+\frac {10\,d^2\,e^2\,\left (A\,e+B\,d\right )}{b}\right )}{b}-\frac {5\,d^3\,e\,\left (2\,A\,e+B\,d\right )}{b}\right )}{b}\right )+x^3\,\left (\frac {a\,\left (\frac {a\,\left (\frac {A\,e^5+5\,B\,d\,e^4}{b}-\frac {B\,a\,e^5}{b^2}\right )}{b}-\frac {5\,d\,e^3\,\left (A\,e+2\,B\,d\right )}{b}\right )}{3\,b}+\frac {10\,d^2\,e^2\,\left (A\,e+B\,d\right )}{3\,b}\right )-x^4\,\left (\frac {a\,\left (\frac {A\,e^5+5\,B\,d\,e^4}{b}-\frac {B\,a\,e^5}{b^2}\right )}{4\,b}-\frac {5\,d\,e^3\,\left (A\,e+2\,B\,d\right )}{4\,b}\right )+x^5\,\left (\frac {A\,e^5+5\,B\,d\,e^4}{5\,b}-\frac {B\,a\,e^5}{5\,b^2}\right )-x^2\,\left (\frac {a\,\left (\frac {a\,\left (\frac {a\,\left (\frac {A\,e^5+5\,B\,d\,e^4}{b}-\frac {B\,a\,e^5}{b^2}\right )}{b}-\frac {5\,d\,e^3\,\left (A\,e+2\,B\,d\right )}{b}\right )}{b}+\frac {10\,d^2\,e^2\,\left (A\,e+B\,d\right )}{b}\right )}{2\,b}-\frac {5\,d^3\,e\,\left (2\,A\,e+B\,d\right )}{2\,b}\right )+\frac {\ln \left (a+b\,x\right )\,\left (B\,a^6\,e^5-5\,B\,a^5\,b\,d\,e^4-A\,a^5\,b\,e^5+10\,B\,a^4\,b^2\,d^2\,e^3+5\,A\,a^4\,b^2\,d\,e^4-10\,B\,a^3\,b^3\,d^3\,e^2-10\,A\,a^3\,b^3\,d^2\,e^3+5\,B\,a^2\,b^4\,d^4\,e+10\,A\,a^2\,b^4\,d^3\,e^2-B\,a\,b^5\,d^5-5\,A\,a\,b^5\,d^4\,e+A\,b^6\,d^5\right )}{b^7}+\frac {B\,e^5\,x^6}{6\,b} \]

[In]

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

[Out]

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