\(\int \frac {(a+b x)^6 (A+B x)}{d+e x} \, dx\) [1060]

Optimal result

Integrand size = 20, antiderivative size = 220 \[ \int \frac {(a+b x)^6 (A+B x)}{d+e x} \, dx=\frac {b (b d-a e)^5 (B d-A e) x}{e^7}-\frac {(b d-a e)^4 (B d-A e) (a+b x)^2}{2 e^6}+\frac {(b d-a e)^3 (B d-A e) (a+b x)^3}{3 e^5}-\frac {(b d-a e)^2 (B d-A e) (a+b x)^4}{4 e^4}+\frac {(b d-a e) (B d-A e) (a+b x)^5}{5 e^3}-\frac {(B d-A e) (a+b x)^6}{6 e^2}+\frac {B (a+b x)^7}{7 b e}-\frac {(b d-a e)^6 (B d-A e) \log (d+e x)}{e^8} \]

[Out]

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

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 220, 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} \[ \int \frac {(a+b x)^6 (A+B x)}{d+e x} \, dx=-\frac {(b d-a e)^6 (B d-A e) \log (d+e x)}{e^8}+\frac {b x (b d-a e)^5 (B d-A e)}{e^7}-\frac {(a+b x)^2 (b d-a e)^4 (B d-A e)}{2 e^6}+\frac {(a+b x)^3 (b d-a e)^3 (B d-A e)}{3 e^5}-\frac {(a+b x)^4 (b d-a e)^2 (B d-A e)}{4 e^4}+\frac {(a+b x)^5 (b d-a e) (B d-A e)}{5 e^3}-\frac {(a+b x)^6 (B d-A e)}{6 e^2}+\frac {B (a+b x)^7}{7 b e} \]

[In]

Int[((a + b*x)^6*(A + B*x))/(d + e*x),x]

[Out]

(b*(b*d - a*e)^5*(B*d - A*e)*x)/e^7 - ((b*d - a*e)^4*(B*d - A*e)*(a + b*x)^2)/(2*e^6) + ((b*d - a*e)^3*(B*d -
A*e)*(a + b*x)^3)/(3*e^5) - ((b*d - a*e)^2*(B*d - A*e)*(a + b*x)^4)/(4*e^4) + ((b*d - a*e)*(B*d - A*e)*(a + b*
x)^5)/(5*e^3) - ((B*d - A*e)*(a + b*x)^6)/(6*e^2) + (B*(a + b*x)^7)/(7*b*e) - ((b*d - a*e)^6*(B*d - A*e)*Log[d
+ e*x])/e^8

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 (b d-a e)^5 (-B d+A e)}{e^7}+\frac {b (b d-a e)^4 (-B d+A e) (a+b x)}{e^6}-\frac {b (b d-a e)^3 (-B d+A e) (a+b x)^2}{e^5}+\frac {b (b d-a e)^2 (-B d+A e) (a+b x)^3}{e^4}-\frac {b (b d-a e) (-B d+A e) (a+b x)^4}{e^3}+\frac {b (-B d+A e) (a+b x)^5}{e^2}+\frac {B (a+b x)^6}{e}+\frac {(-b d+a e)^6 (-B d+A e)}{e^7 (d+e x)}\right ) \, dx \\ & = \frac {b (b d-a e)^5 (B d-A e) x}{e^7}-\frac {(b d-a e)^4 (B d-A e) (a+b x)^2}{2 e^6}+\frac {(b d-a e)^3 (B d-A e) (a+b x)^3}{3 e^5}-\frac {(b d-a e)^2 (B d-A e) (a+b x)^4}{4 e^4}+\frac {(b d-a e) (B d-A e) (a+b x)^5}{5 e^3}-\frac {(B d-A e) (a+b x)^6}{6 e^2}+\frac {B (a+b x)^7}{7 b e}-\frac {(b d-a e)^6 (B d-A e) \log (d+e x)}{e^8} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(501\) vs. \(2(220)=440\).

Time = 0.17 (sec) , antiderivative size = 501, normalized size of antiderivative = 2.28 \[ \int \frac {(a+b x)^6 (A+B x)}{d+e x} \, dx=\frac {e x \left (420 a^6 B e^6+1260 a^5 b e^5 (-2 B d+2 A e+B e x)+1050 a^4 b^2 e^4 \left (3 A e (-2 d+e x)+B \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )+700 a^3 b^3 e^3 \left (2 A e \left (6 d^2-3 d e x+2 e^2 x^2\right )+B \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )\right )+105 a^2 b^4 e^2 \left (5 A e \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )+B \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )\right )+42 a b^5 e \left (A e \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )+B \left (-60 d^5+30 d^4 e x-20 d^3 e^2 x^2+15 d^2 e^3 x^3-12 d e^4 x^4+10 e^5 x^5\right )\right )+b^6 \left (7 A e \left (-60 d^5+30 d^4 e x-20 d^3 e^2 x^2+15 d^2 e^3 x^3-12 d e^4 x^4+10 e^5 x^5\right )+B \left (420 d^6-210 d^5 e x+140 d^4 e^2 x^2-105 d^3 e^3 x^3+84 d^2 e^4 x^4-70 d e^5 x^5+60 e^6 x^6\right )\right )\right )-420 (b d-a e)^6 (B d-A e) \log (d+e x)}{420 e^8} \]

[In]

Integrate[((a + b*x)^6*(A + B*x))/(d + e*x),x]

[Out]

(e*x*(420*a^6*B*e^6 + 1260*a^5*b*e^5*(-2*B*d + 2*A*e + B*e*x) + 1050*a^4*b^2*e^4*(3*A*e*(-2*d + e*x) + B*(6*d^
2 - 3*d*e*x + 2*e^2*x^2)) + 700*a^3*b^3*e^3*(2*A*e*(6*d^2 - 3*d*e*x + 2*e^2*x^2) + B*(-12*d^3 + 6*d^2*e*x - 4*
d*e^2*x^2 + 3*e^3*x^3)) + 105*a^2*b^4*e^2*(5*A*e*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3) + B*(60*d^4 -
30*d^3*e*x + 20*d^2*e^2*x^2 - 15*d*e^3*x^3 + 12*e^4*x^4)) + 42*a*b^5*e*(A*e*(60*d^4 - 30*d^3*e*x + 20*d^2*e^2
*x^2 - 15*d*e^3*x^3 + 12*e^4*x^4) + B*(-60*d^5 + 30*d^4*e*x - 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 - 12*d*e^4*x^4 +
10*e^5*x^5)) + b^6*(7*A*e*(-60*d^5 + 30*d^4*e*x - 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 - 12*d*e^4*x^4 + 10*e^5*x^5
) + B*(420*d^6 - 210*d^5*e*x + 140*d^4*e^2*x^2 - 105*d^3*e^3*x^3 + 84*d^2*e^4*x^4 - 70*d*e^5*x^5 + 60*e^6*x^6)
)) - 420*(b*d - a*e)^6*(B*d - A*e)*Log[d + e*x])/(420*e^8)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(775\) vs. \(2(208)=416\).

Time = 0.70 (sec) , antiderivative size = 776, normalized size of antiderivative = 3.53

method	result	size
norman	\(\frac {\left (6 A \,a^{5} b \,e^{6}-15 A \,a^{4} b^{2} d \,e^{5}+20 A \,a^{3} b^{3} d^{2} e^{4}-15 A \,a^{2} b^{4} d^{3} e^{3}+6 A a \,b^{5} d^{4} e^{2}-A \,b^{6} d^{5} e +B \,a^{6} e^{6}-6 B \,a^{5} b d \,e^{5}+15 B \,a^{4} b^{2} d^{2} e^{4}-20 B \,a^{3} b^{3} d^{3} e^{3}+15 B \,a^{2} b^{4} d^{4} e^{2}-6 B a \,b^{5} d^{5} e +b^{6} B \,d^{6}\right ) x}{e^{7}}+\frac {b \left (15 A \,a^{4} b \,e^{5}-20 A \,a^{3} b^{2} d \,e^{4}+15 A \,a^{2} b^{3} d^{2} e^{3}-6 A a \,b^{4} d^{3} e^{2}+A \,b^{5} d^{4} e +6 B \,a^{5} e^{5}-15 B \,a^{4} b d \,e^{4}+20 B \,a^{3} b^{2} d^{2} e^{3}-15 B \,a^{2} b^{3} d^{3} e^{2}+6 B a \,b^{4} d^{4} e -B \,b^{5} d^{5}\right ) x^{2}}{2 e^{6}}+\frac {b^{2} \left (20 A \,a^{3} b \,e^{4}-15 A \,a^{2} b^{2} d \,e^{3}+6 A a \,b^{3} d^{2} e^{2}-A \,b^{4} d^{3} e +15 B \,a^{4} e^{4}-20 B \,a^{3} b d \,e^{3}+15 B \,a^{2} b^{2} d^{2} e^{2}-6 B a \,b^{3} d^{3} e +B \,b^{4} d^{4}\right ) x^{3}}{3 e^{5}}+\frac {b^{3} \left (15 A \,a^{2} b \,e^{3}-6 A a \,b^{2} d \,e^{2}+A \,b^{3} d^{2} e +20 B \,a^{3} e^{3}-15 B \,a^{2} b d \,e^{2}+6 B a \,b^{2} d^{2} e -b^{3} B \,d^{3}\right ) x^{4}}{4 e^{4}}+\frac {b^{4} \left (6 A a b \,e^{2}-A \,b^{2} d e +15 B \,a^{2} e^{2}-6 B a b d e +b^{2} B \,d^{2}\right ) x^{5}}{5 e^{3}}+\frac {b^{5} \left (A b e +6 B a e -B b d \right ) x^{6}}{6 e^{2}}+\frac {b^{6} B \,x^{7}}{7 e}+\frac {\left (A \,a^{6} e^{7}-6 A \,a^{5} b d \,e^{6}+15 A \,a^{4} b^{2} d^{2} e^{5}-20 A \,a^{3} b^{3} d^{3} e^{4}+15 A \,a^{2} b^{4} d^{4} e^{3}-6 A a \,b^{5} d^{5} e^{2}+A \,b^{6} d^{6} e -B \,a^{6} d \,e^{6}+6 B \,a^{5} b \,d^{2} e^{5}-15 B \,a^{4} b^{2} d^{3} e^{4}+20 B \,a^{3} b^{3} d^{4} e^{3}-15 B \,a^{2} b^{4} d^{5} e^{2}+6 B a \,b^{5} d^{6} e -b^{6} B \,d^{7}\right ) \ln \left (e x +d \right )}{e^{8}}\)	\(776\)
default	\(\frac {-3 A a \,b^{5} d^{3} e^{3} x^{2}-\frac {15}{2} B \,a^{4} b^{2} d \,e^{5} x^{2}+10 B \,a^{3} b^{3} d^{2} e^{4} x^{2}-\frac {15}{2} B \,a^{2} b^{4} d^{3} e^{3} x^{2}+3 B a \,b^{5} d^{4} e^{2} x^{2}+2 A a \,b^{5} d^{2} e^{4} x^{3}+B \,a^{6} e^{6} x +b^{6} B \,d^{6} x -\frac {1}{2} B \,b^{6} d^{5} e \,x^{2}+6 A \,a^{5} b \,e^{6} x -A \,b^{6} d^{5} e x +\frac {1}{3} B \,b^{6} d^{4} e^{2} x^{3}+\frac {15}{2} A \,a^{4} b^{2} e^{6} x^{2}+\frac {1}{2} A \,b^{6} d^{4} e^{2} x^{2}+3 B \,a^{5} b \,e^{6} x^{2}+\frac {1}{4} A \,b^{6} d^{2} e^{4} x^{4}+5 B \,a^{3} b^{3} e^{6} x^{4}-\frac {1}{4} B \,b^{6} d^{3} e^{3} x^{4}+\frac {20}{3} A \,a^{3} b^{3} e^{6} x^{3}-\frac {1}{3} A \,b^{6} d^{3} e^{3} x^{3}+5 B \,a^{4} b^{2} e^{6} x^{3}+\frac {6}{5} A a \,b^{5} e^{6} x^{5}-\frac {1}{5} A \,b^{6} d \,e^{5} x^{5}+3 B \,a^{2} b^{4} e^{6} x^{5}+\frac {1}{5} B \,b^{6} d^{2} e^{4} x^{5}+\frac {15}{4} A \,a^{2} b^{4} e^{6} x^{4}-\frac {1}{6} B \,b^{6} d \,e^{5} x^{6}+B a \,b^{5} e^{6} x^{6}-\frac {20}{3} B \,a^{3} b^{3} d \,e^{5} x^{3}+\frac {1}{6} A \,b^{6} e^{6} x^{6}+\frac {1}{7} b^{6} B \,x^{7} e^{6}+5 B \,a^{2} b^{4} d^{2} e^{4} x^{3}-2 B a \,b^{5} d^{3} e^{3} x^{3}-10 A \,a^{3} b^{3} d \,e^{5} x^{2}+\frac {15}{2} A \,a^{2} b^{4} d^{2} e^{4} x^{2}-5 A \,a^{2} b^{4} d \,e^{5} x^{3}-15 A \,a^{4} b^{2} d \,e^{5} x +20 A \,a^{3} b^{3} d^{2} e^{4} x -15 A \,a^{2} b^{4} d^{3} e^{3} x +6 A a \,b^{5} d^{4} e^{2} x -6 B \,a^{5} b d \,e^{5} x +15 B \,a^{4} b^{2} d^{2} e^{4} x -20 B \,a^{3} b^{3} d^{3} e^{3} x +15 B \,a^{2} b^{4} d^{4} e^{2} x -6 B a \,b^{5} d^{5} e x -\frac {6}{5} B a \,b^{5} d \,e^{5} x^{5}-\frac {3}{2} A a \,b^{5} d \,e^{5} x^{4}-\frac {15}{4} B \,a^{2} b^{4} d \,e^{5} x^{4}+\frac {3}{2} B a \,b^{5} d^{2} e^{4} x^{4}}{e^{7}}+\frac {\left (A \,a^{6} e^{7}-6 A \,a^{5} b d \,e^{6}+15 A \,a^{4} b^{2} d^{2} e^{5}-20 A \,a^{3} b^{3} d^{3} e^{4}+15 A \,a^{2} b^{4} d^{4} e^{3}-6 A a \,b^{5} d^{5} e^{2}+A \,b^{6} d^{6} e -B \,a^{6} d \,e^{6}+6 B \,a^{5} b \,d^{2} e^{5}-15 B \,a^{4} b^{2} d^{3} e^{4}+20 B \,a^{3} b^{3} d^{4} e^{3}-15 B \,a^{2} b^{4} d^{5} e^{2}+6 B a \,b^{5} d^{6} e -b^{6} B \,d^{7}\right ) \ln \left (e x +d \right )}{e^{8}}\)	\(905\)
risch	\(\frac {15 \ln \left (e x +d \right ) A \,a^{2} b^{4} d^{4}}{e^{5}}-\frac {6 \ln \left (e x +d \right ) A a \,b^{5} d^{5}}{e^{6}}+\frac {b^{6} B \,d^{6} x}{e^{7}}-\frac {B \,b^{6} d^{5} x^{2}}{2 e^{6}}+\frac {20 A \,a^{3} b^{3} d^{2} x}{e^{3}}-\frac {3 A a \,b^{5} d^{3} x^{2}}{e^{4}}-\frac {15 B \,a^{4} b^{2} d \,x^{2}}{2 e^{2}}+\frac {10 B \,a^{3} b^{3} d^{2} x^{2}}{e^{3}}-\frac {15 B \,a^{2} b^{4} d^{3} x^{2}}{2 e^{4}}+\frac {3 B a \,b^{5} d^{4} x^{2}}{e^{5}}+\frac {2 A a \,b^{5} d^{2} x^{3}}{e^{3}}-\frac {6 \ln \left (e x +d \right ) A \,a^{5} b d}{e^{2}}+\frac {15 \ln \left (e x +d \right ) A \,a^{4} b^{2} d^{2}}{e^{3}}-\frac {20 \ln \left (e x +d \right ) A \,a^{3} b^{3} d^{3}}{e^{4}}-\frac {5 A \,a^{2} b^{4} d \,x^{3}}{e^{2}}-\frac {15 A \,a^{4} b^{2} d x}{e^{2}}+\frac {6 \ln \left (e x +d \right ) B \,a^{5} b \,d^{2}}{e^{3}}-\frac {2 B a \,b^{5} d^{3} x^{3}}{e^{4}}-\frac {\ln \left (e x +d \right ) b^{6} B \,d^{7}}{e^{8}}+\frac {20 A \,a^{3} b^{3} x^{3}}{3 e}-\frac {A \,b^{6} d^{3} x^{3}}{3 e^{4}}+\frac {5 B \,a^{4} b^{2} x^{3}}{e}+\frac {6 A a \,b^{5} x^{5}}{5 e}-\frac {A \,b^{6} d \,x^{5}}{5 e^{2}}+\frac {3 B \,a^{2} b^{4} x^{5}}{e}+\frac {B \,b^{6} d^{2} x^{5}}{5 e^{3}}+\frac {15 A \,a^{2} b^{4} x^{4}}{4 e}-\frac {B \,b^{6} d \,x^{6}}{6 e^{2}}+\frac {B a \,b^{5} x^{6}}{e}-\frac {10 A \,a^{3} b^{3} d \,x^{2}}{e^{2}}+\frac {b^{6} B \,x^{7}}{7 e}-\frac {\ln \left (e x +d \right ) B \,a^{6} d}{e^{2}}+\frac {B \,a^{6} x}{e}+\frac {A \,b^{6} x^{6}}{6 e}+\frac {\ln \left (e x +d \right ) A \,a^{6}}{e}+\frac {5 B \,a^{2} b^{4} d^{2} x^{3}}{e^{3}}+\frac {6 A \,a^{5} b x}{e}-\frac {A \,b^{6} d^{5} x}{e^{6}}+\frac {B \,b^{6} d^{4} x^{3}}{3 e^{5}}+\frac {15 A \,a^{4} b^{2} x^{2}}{2 e}+\frac {A \,b^{6} d^{4} x^{2}}{2 e^{5}}+\frac {3 B \,a^{5} b \,x^{2}}{e}+\frac {A \,b^{6} d^{2} x^{4}}{4 e^{3}}+\frac {5 B \,a^{3} b^{3} x^{4}}{e}-\frac {B \,b^{6} d^{3} x^{4}}{4 e^{4}}+\frac {15 A \,a^{2} b^{4} d^{2} x^{2}}{2 e^{3}}+\frac {6 \ln \left (e x +d \right ) B a \,b^{5} d^{6}}{e^{7}}+\frac {20 \ln \left (e x +d \right ) B \,a^{3} b^{3} d^{4}}{e^{5}}-\frac {15 \ln \left (e x +d \right ) B \,a^{2} b^{4} d^{5}}{e^{6}}-\frac {20 B \,a^{3} b^{3} d \,x^{3}}{3 e^{2}}+\frac {\ln \left (e x +d \right ) A \,b^{6} d^{6}}{e^{7}}-\frac {15 A \,a^{2} b^{4} d^{3} x}{e^{4}}+\frac {6 A a \,b^{5} d^{4} x}{e^{5}}-\frac {6 B \,a^{5} b d x}{e^{2}}+\frac {3 B a \,b^{5} d^{2} x^{4}}{2 e^{3}}+\frac {15 B \,a^{4} b^{2} d^{2} x}{e^{3}}-\frac {20 B \,a^{3} b^{3} d^{3} x}{e^{4}}+\frac {15 B \,a^{2} b^{4} d^{4} x}{e^{5}}-\frac {6 B a \,b^{5} d^{5} x}{e^{6}}-\frac {6 B a \,b^{5} d \,x^{5}}{5 e^{2}}-\frac {3 A a \,b^{5} d \,x^{4}}{2 e^{2}}-\frac {15 B \,a^{2} b^{4} d \,x^{4}}{4 e^{2}}-\frac {15 \ln \left (e x +d \right ) B \,a^{4} b^{2} d^{3}}{e^{4}}\)	\(989\)
parallelrisch	\(\frac {6300 B x \,a^{4} b^{2} d^{2} e^{5}+1260 B \,x^{2} a \,b^{5} d^{4} e^{3}+6300 B x \,a^{2} b^{4} d^{4} e^{3}-2520 B x a \,b^{5} d^{5} e^{2}-4200 A \,x^{2} a^{3} b^{3} d \,e^{6}-6300 A x \,a^{2} b^{4} d^{3} e^{4}+2520 A x a \,b^{5} d^{4} e^{3}-2520 A \ln \left (e x +d \right ) a^{5} b d \,e^{6}+6300 A \ln \left (e x +d \right ) a^{4} b^{2} d^{2} e^{5}-8400 A \ln \left (e x +d \right ) a^{3} b^{3} d^{3} e^{4}+2520 A x \,a^{5} b \,e^{7}-420 A x \,b^{6} d^{5} e^{2}+420 B x \,b^{6} d^{6} e +3150 A \,x^{2} a^{4} b^{2} e^{7}+210 A \,x^{2} b^{6} d^{4} e^{3}+1260 B \,x^{2} a^{5} b \,e^{7}-210 B \,x^{2} b^{6} d^{5} e^{2}+2800 A \,x^{3} a^{3} b^{3} e^{7}-140 A \,x^{3} b^{6} d^{3} e^{4}+2100 B \,x^{3} a^{4} b^{2} e^{7}+140 B \,x^{3} b^{6} d^{4} e^{3}+1575 A \,x^{4} a^{2} b^{4} e^{7}+105 A \,x^{4} b^{6} d^{2} e^{5}+2100 B \,x^{4} a^{3} b^{3} e^{7}-105 B \,x^{4} b^{6} d^{3} e^{4}+6300 A \ln \left (e x +d \right ) a^{2} b^{4} d^{4} e^{3}-2520 A \ln \left (e x +d \right ) a \,b^{5} d^{5} e^{2}+2520 B \ln \left (e x +d \right ) a^{5} b \,d^{2} e^{5}-6300 B \ln \left (e x +d \right ) a^{4} b^{2} d^{3} e^{4}+8400 B \ln \left (e x +d \right ) a^{3} b^{3} d^{4} e^{3}-2100 A \,x^{3} a^{2} b^{4} d \,e^{6}+4200 B \,x^{2} a^{3} b^{3} d^{2} e^{5}-3150 B \,x^{2} a^{2} b^{4} d^{3} e^{4}-6300 A x \,a^{4} b^{2} d \,e^{6}-8400 B x \,a^{3} b^{3} d^{3} e^{4}+8400 A x \,a^{3} b^{3} d^{2} e^{5}+3150 A \,x^{2} a^{2} b^{4} d^{2} e^{5}-504 B \,x^{5} a \,b^{5} d \,e^{6}-1260 A \,x^{2} a \,b^{5} d^{3} e^{4}-3150 B \,x^{2} a^{4} b^{2} d \,e^{6}-6300 B \ln \left (e x +d \right ) a^{2} b^{4} d^{5} e^{2}+2520 B \ln \left (e x +d \right ) a \,b^{5} d^{6} e +840 A \,x^{3} a \,b^{5} d^{2} e^{5}-2800 B \,x^{3} a^{3} b^{3} d \,e^{6}+2100 B \,x^{3} a^{2} b^{4} d^{2} e^{5}-840 B \,x^{3} a \,b^{5} d^{3} e^{4}-630 A \,x^{4} a \,b^{5} d \,e^{6}-1575 B \,x^{4} a^{2} b^{4} d \,e^{6}+630 B \,x^{4} a \,b^{5} d^{2} e^{5}-2520 B x \,a^{5} b d \,e^{6}+504 A \,x^{5} a \,b^{5} e^{7}-84 A \,x^{5} b^{6} d \,e^{6}+1260 B \,x^{5} a^{2} b^{4} e^{7}+84 B \,x^{5} b^{6} d^{2} e^{5}+420 B \,x^{6} a \,b^{5} e^{7}-70 B \,x^{6} b^{6} d \,e^{6}+420 A \ln \left (e x +d \right ) b^{6} d^{6} e -420 B \ln \left (e x +d \right ) a^{6} d \,e^{6}+60 B \,x^{7} b^{6} e^{7}+420 B x \,a^{6} e^{7}+70 A \,x^{6} b^{6} e^{7}+420 A \ln \left (e x +d \right ) a^{6} e^{7}-420 B \ln \left (e x +d \right ) b^{6} d^{7}}{420 e^{8}}\)	\(990\)

[In]

int((b*x+a)^6*(B*x+A)/(e*x+d),x,method=_RETURNVERBOSE)

[Out]

(6*A*a^5*b*e^6-15*A*a^4*b^2*d*e^5+20*A*a^3*b^3*d^2*e^4-15*A*a^2*b^4*d^3*e^3+6*A*a*b^5*d^4*e^2-A*b^6*d^5*e+B*a^
6*e^6-6*B*a^5*b*d*e^5+15*B*a^4*b^2*d^2*e^4-20*B*a^3*b^3*d^3*e^3+15*B*a^2*b^4*d^4*e^2-6*B*a*b^5*d^5*e+B*b^6*d^6
)/e^7*x+1/2*b/e^6*(15*A*a^4*b*e^5-20*A*a^3*b^2*d*e^4+15*A*a^2*b^3*d^2*e^3-6*A*a*b^4*d^3*e^2+A*b^5*d^4*e+6*B*a^
5*e^5-15*B*a^4*b*d*e^4+20*B*a^3*b^2*d^2*e^3-15*B*a^2*b^3*d^3*e^2+6*B*a*b^4*d^4*e-B*b^5*d^5)*x^2+1/3*b^2/e^5*(2
0*A*a^3*b*e^4-15*A*a^2*b^2*d*e^3+6*A*a*b^3*d^2*e^2-A*b^4*d^3*e+15*B*a^4*e^4-20*B*a^3*b*d*e^3+15*B*a^2*b^2*d^2*
e^2-6*B*a*b^3*d^3*e+B*b^4*d^4)*x^3+1/4*b^3/e^4*(15*A*a^2*b*e^3-6*A*a*b^2*d*e^2+A*b^3*d^2*e+20*B*a^3*e^3-15*B*a
^2*b*d*e^2+6*B*a*b^2*d^2*e-B*b^3*d^3)*x^4+1/5*b^4/e^3*(6*A*a*b*e^2-A*b^2*d*e+15*B*a^2*e^2-6*B*a*b*d*e+B*b^2*d^
2)*x^5+1/6*b^5/e^2*(A*b*e+6*B*a*e-B*b*d)*x^6+1/7*b^6*B/e*x^7+(A*a^6*e^7-6*A*a^5*b*d*e^6+15*A*a^4*b^2*d^2*e^5-2
0*A*a^3*b^3*d^3*e^4+15*A*a^2*b^4*d^4*e^3-6*A*a*b^5*d^5*e^2+A*b^6*d^6*e-B*a^6*d*e^6+6*B*a^5*b*d^2*e^5-15*B*a^4*
b^2*d^3*e^4+20*B*a^3*b^3*d^4*e^3-15*B*a^2*b^4*d^5*e^2+6*B*a*b^5*d^6*e-B*b^6*d^7)/e^8*ln(e*x+d)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 763 vs. \(2 (208) = 416\).

Time = 0.22 (sec) , antiderivative size = 763, normalized size of antiderivative = 3.47 \[ \int \frac {(a+b x)^6 (A+B x)}{d+e x} \, dx=\frac {60 \, B b^{6} e^{7} x^{7} - 70 \, {\left (B b^{6} d e^{6} - {\left (6 \, B a b^{5} + A b^{6}\right )} e^{7}\right )} x^{6} + 84 \, {\left (B b^{6} d^{2} e^{5} - {\left (6 \, B a b^{5} + A b^{6}\right )} d e^{6} + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} e^{7}\right )} x^{5} - 105 \, {\left (B b^{6} d^{3} e^{4} - {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} e^{5} + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d e^{6} - 5 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} e^{7}\right )} x^{4} + 140 \, {\left (B b^{6} d^{4} e^{3} - {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} e^{4} + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} e^{5} - 5 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d e^{6} + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} e^{7}\right )} x^{3} - 210 \, {\left (B b^{6} d^{5} e^{2} - {\left (6 \, B a b^{5} + A b^{6}\right )} d^{4} e^{3} + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{3} e^{4} - 5 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} e^{5} + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d e^{6} - 3 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} e^{7}\right )} x^{2} + 420 \, {\left (B b^{6} d^{6} e - {\left (6 \, B a b^{5} + A b^{6}\right )} d^{5} e^{2} + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{4} e^{3} - 5 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{3} e^{4} + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} e^{5} - 3 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d e^{6} + {\left (B a^{6} + 6 \, A a^{5} b\right )} e^{7}\right )} x - 420 \, {\left (B b^{6} d^{7} - A a^{6} e^{7} - {\left (6 \, B a b^{5} + A b^{6}\right )} d^{6} e + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{5} e^{2} - 5 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{4} e^{3} + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{3} e^{4} - 3 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2} e^{5} + {\left (B a^{6} + 6 \, A a^{5} b\right )} d e^{6}\right )} \log \left (e x + d\right )}{420 \, e^{8}} \]

[In]

integrate((b*x+a)^6*(B*x+A)/(e*x+d),x, algorithm="fricas")

[Out]

1/420*(60*B*b^6*e^7*x^7 - 70*(B*b^6*d*e^6 - (6*B*a*b^5 + A*b^6)*e^7)*x^6 + 84*(B*b^6*d^2*e^5 - (6*B*a*b^5 + A*
b^6)*d*e^6 + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*e^7)*x^5 - 105*(B*b^6*d^3*e^4 - (6*B*a*b^5 + A*b^6)*d^2*e^5 + 3*(5*B*
a^2*b^4 + 2*A*a*b^5)*d*e^6 - 5*(4*B*a^3*b^3 + 3*A*a^2*b^4)*e^7)*x^4 + 140*(B*b^6*d^4*e^3 - (6*B*a*b^5 + A*b^6)
*d^3*e^4 + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^5 - 5*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d*e^6 + 5*(3*B*a^4*b^2 + 4*A*a^
3*b^3)*e^7)*x^3 - 210*(B*b^6*d^5*e^2 - (6*B*a*b^5 + A*b^6)*d^4*e^3 + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^4 - 5*(
4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^5 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d*e^6 - 3*(2*B*a^5*b + 5*A*a^4*b^2)*e^7)*x^
2 + 420*(B*b^6*d^6*e - (6*B*a*b^5 + A*b^6)*d^5*e^2 + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^3 - 5*(4*B*a^3*b^3 + 3*
A*a^2*b^4)*d^3*e^4 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*e^5 - 3*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^6 + (B*a^6 + 6*A*
a^5*b)*e^7)*x - 420*(B*b^6*d^7 - A*a^6*e^7 - (6*B*a*b^5 + A*b^6)*d^6*e + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^2 -
5*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^3 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 - 3*(2*B*a^5*b + 5*A*a^4*b^2)*d
^2*e^5 + (B*a^6 + 6*A*a^5*b)*d*e^6)*log(e*x + d))/e^8

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 736 vs. \(2 (189) = 378\).

Time = 0.91 (sec) , antiderivative size = 736, normalized size of antiderivative = 3.35 \[ \int \frac {(a+b x)^6 (A+B x)}{d+e x} \, dx=\frac {B b^{6} x^{7}}{7 e} + x^{6} \left (\frac {A b^{6}}{6 e} + \frac {B a b^{5}}{e} - \frac {B b^{6} d}{6 e^{2}}\right ) + x^{5} \cdot \left (\frac {6 A a b^{5}}{5 e} - \frac {A b^{6} d}{5 e^{2}} + \frac {3 B a^{2} b^{4}}{e} - \frac {6 B a b^{5} d}{5 e^{2}} + \frac {B b^{6} d^{2}}{5 e^{3}}\right ) + x^{4} \cdot \left (\frac {15 A a^{2} b^{4}}{4 e} - \frac {3 A a b^{5} d}{2 e^{2}} + \frac {A b^{6} d^{2}}{4 e^{3}} + \frac {5 B a^{3} b^{3}}{e} - \frac {15 B a^{2} b^{4} d}{4 e^{2}} + \frac {3 B a b^{5} d^{2}}{2 e^{3}} - \frac {B b^{6} d^{3}}{4 e^{4}}\right ) + x^{3} \cdot \left (\frac {20 A a^{3} b^{3}}{3 e} - \frac {5 A a^{2} b^{4} d}{e^{2}} + \frac {2 A a b^{5} d^{2}}{e^{3}} - \frac {A b^{6} d^{3}}{3 e^{4}} + \frac {5 B a^{4} b^{2}}{e} - \frac {20 B a^{3} b^{3} d}{3 e^{2}} + \frac {5 B a^{2} b^{4} d^{2}}{e^{3}} - \frac {2 B a b^{5} d^{3}}{e^{4}} + \frac {B b^{6} d^{4}}{3 e^{5}}\right ) + x^{2} \cdot \left (\frac {15 A a^{4} b^{2}}{2 e} - \frac {10 A a^{3} b^{3} d}{e^{2}} + \frac {15 A a^{2} b^{4} d^{2}}{2 e^{3}} - \frac {3 A a b^{5} d^{3}}{e^{4}} + \frac {A b^{6} d^{4}}{2 e^{5}} + \frac {3 B a^{5} b}{e} - \frac {15 B a^{4} b^{2} d}{2 e^{2}} + \frac {10 B a^{3} b^{3} d^{2}}{e^{3}} - \frac {15 B a^{2} b^{4} d^{3}}{2 e^{4}} + \frac {3 B a b^{5} d^{4}}{e^{5}} - \frac {B b^{6} d^{5}}{2 e^{6}}\right ) + x \left (\frac {6 A a^{5} b}{e} - \frac {15 A a^{4} b^{2} d}{e^{2}} + \frac {20 A a^{3} b^{3} d^{2}}{e^{3}} - \frac {15 A a^{2} b^{4} d^{3}}{e^{4}} + \frac {6 A a b^{5} d^{4}}{e^{5}} - \frac {A b^{6} d^{5}}{e^{6}} + \frac {B a^{6}}{e} - \frac {6 B a^{5} b d}{e^{2}} + \frac {15 B a^{4} b^{2} d^{2}}{e^{3}} - \frac {20 B a^{3} b^{3} d^{3}}{e^{4}} + \frac {15 B a^{2} b^{4} d^{4}}{e^{5}} - \frac {6 B a b^{5} d^{5}}{e^{6}} + \frac {B b^{6} d^{6}}{e^{7}}\right ) - \frac {\left (- A e + B d\right ) \left (a e - b d\right )^{6} \log {\left (d + e x \right )}}{e^{8}} \]

[In]

integrate((b*x+a)**6*(B*x+A)/(e*x+d),x)

[Out]

B*b**6*x**7/(7*e) + x**6*(A*b**6/(6*e) + B*a*b**5/e - B*b**6*d/(6*e**2)) + x**5*(6*A*a*b**5/(5*e) - A*b**6*d/(
5*e**2) + 3*B*a**2*b**4/e - 6*B*a*b**5*d/(5*e**2) + B*b**6*d**2/(5*e**3)) + x**4*(15*A*a**2*b**4/(4*e) - 3*A*a
*b**5*d/(2*e**2) + A*b**6*d**2/(4*e**3) + 5*B*a**3*b**3/e - 15*B*a**2*b**4*d/(4*e**2) + 3*B*a*b**5*d**2/(2*e**
3) - B*b**6*d**3/(4*e**4)) + x**3*(20*A*a**3*b**3/(3*e) - 5*A*a**2*b**4*d/e**2 + 2*A*a*b**5*d**2/e**3 - A*b**6
*d**3/(3*e**4) + 5*B*a**4*b**2/e - 20*B*a**3*b**3*d/(3*e**2) + 5*B*a**2*b**4*d**2/e**3 - 2*B*a*b**5*d**3/e**4
+ B*b**6*d**4/(3*e**5)) + x**2*(15*A*a**4*b**2/(2*e) - 10*A*a**3*b**3*d/e**2 + 15*A*a**2*b**4*d**2/(2*e**3) -
3*A*a*b**5*d**3/e**4 + A*b**6*d**4/(2*e**5) + 3*B*a**5*b/e - 15*B*a**4*b**2*d/(2*e**2) + 10*B*a**3*b**3*d**2/e
**3 - 15*B*a**2*b**4*d**3/(2*e**4) + 3*B*a*b**5*d**4/e**5 - B*b**6*d**5/(2*e**6)) + x*(6*A*a**5*b/e - 15*A*a**
4*b**2*d/e**2 + 20*A*a**3*b**3*d**2/e**3 - 15*A*a**2*b**4*d**3/e**4 + 6*A*a*b**5*d**4/e**5 - A*b**6*d**5/e**6
+ B*a**6/e - 6*B*a**5*b*d/e**2 + 15*B*a**4*b**2*d**2/e**3 - 20*B*a**3*b**3*d**3/e**4 + 15*B*a**2*b**4*d**4/e**
5 - 6*B*a*b**5*d**5/e**6 + B*b**6*d**6/e**7) - (-A*e + B*d)*(a*e - b*d)**6*log(d + e*x)/e**8

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 762 vs. \(2 (208) = 416\).

Time = 0.21 (sec) , antiderivative size = 762, normalized size of antiderivative = 3.46 \[ \int \frac {(a+b x)^6 (A+B x)}{d+e x} \, dx=\frac {60 \, B b^{6} e^{6} x^{7} - 70 \, {\left (B b^{6} d e^{5} - {\left (6 \, B a b^{5} + A b^{6}\right )} e^{6}\right )} x^{6} + 84 \, {\left (B b^{6} d^{2} e^{4} - {\left (6 \, B a b^{5} + A b^{6}\right )} d e^{5} + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} e^{6}\right )} x^{5} - 105 \, {\left (B b^{6} d^{3} e^{3} - {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} e^{4} + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d e^{5} - 5 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} e^{6}\right )} x^{4} + 140 \, {\left (B b^{6} d^{4} e^{2} - {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} e^{3} + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} e^{4} - 5 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d e^{5} + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} e^{6}\right )} x^{3} - 210 \, {\left (B b^{6} d^{5} e - {\left (6 \, B a b^{5} + A b^{6}\right )} d^{4} e^{2} + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{3} e^{3} - 5 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} e^{4} + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d e^{5} - 3 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} e^{6}\right )} x^{2} + 420 \, {\left (B b^{6} d^{6} - {\left (6 \, B a b^{5} + A b^{6}\right )} d^{5} e + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{4} e^{2} - 5 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{3} e^{3} + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} e^{4} - 3 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d e^{5} + {\left (B a^{6} + 6 \, A a^{5} b\right )} e^{6}\right )} x}{420 \, e^{7}} - \frac {{\left (B b^{6} d^{7} - A a^{6} e^{7} - {\left (6 \, B a b^{5} + A b^{6}\right )} d^{6} e + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{5} e^{2} - 5 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{4} e^{3} + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{3} e^{4} - 3 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2} e^{5} + {\left (B a^{6} + 6 \, A a^{5} b\right )} d e^{6}\right )} \log \left (e x + d\right )}{e^{8}} \]

[In]

integrate((b*x+a)^6*(B*x+A)/(e*x+d),x, algorithm="maxima")

[Out]

1/420*(60*B*b^6*e^6*x^7 - 70*(B*b^6*d*e^5 - (6*B*a*b^5 + A*b^6)*e^6)*x^6 + 84*(B*b^6*d^2*e^4 - (6*B*a*b^5 + A*
b^6)*d*e^5 + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*e^6)*x^5 - 105*(B*b^6*d^3*e^3 - (6*B*a*b^5 + A*b^6)*d^2*e^4 + 3*(5*B*
a^2*b^4 + 2*A*a*b^5)*d*e^5 - 5*(4*B*a^3*b^3 + 3*A*a^2*b^4)*e^6)*x^4 + 140*(B*b^6*d^4*e^2 - (6*B*a*b^5 + A*b^6)
*d^3*e^3 + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^4 - 5*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d*e^5 + 5*(3*B*a^4*b^2 + 4*A*a^
3*b^3)*e^6)*x^3 - 210*(B*b^6*d^5*e - (6*B*a*b^5 + A*b^6)*d^4*e^2 + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^3 - 5*(4*
B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^4 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d*e^5 - 3*(2*B*a^5*b + 5*A*a^4*b^2)*e^6)*x^2
+ 420*(B*b^6*d^6 - (6*B*a*b^5 + A*b^6)*d^5*e + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^2 - 5*(4*B*a^3*b^3 + 3*A*a^2*
b^4)*d^3*e^3 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*e^4 - 3*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^5 + (B*a^6 + 6*A*a^5*b)
*e^6)*x)/e^7 - (B*b^6*d^7 - A*a^6*e^7 - (6*B*a*b^5 + A*b^6)*d^6*e + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^2 - 5*(4
*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^3 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 - 3*(2*B*a^5*b + 5*A*a^4*b^2)*d^2*e^
5 + (B*a^6 + 6*A*a^5*b)*d*e^6)*log(e*x + d)/e^8

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 910 vs. \(2 (208) = 416\).

Time = 0.30 (sec) , antiderivative size = 910, normalized size of antiderivative = 4.14 \[ \int \frac {(a+b x)^6 (A+B x)}{d+e x} \, dx=\frac {60 \, B b^{6} e^{6} x^{7} - 70 \, B b^{6} d e^{5} x^{6} + 420 \, B a b^{5} e^{6} x^{6} + 70 \, A b^{6} e^{6} x^{6} + 84 \, B b^{6} d^{2} e^{4} x^{5} - 504 \, B a b^{5} d e^{5} x^{5} - 84 \, A b^{6} d e^{5} x^{5} + 1260 \, B a^{2} b^{4} e^{6} x^{5} + 504 \, A a b^{5} e^{6} x^{5} - 105 \, B b^{6} d^{3} e^{3} x^{4} + 630 \, B a b^{5} d^{2} e^{4} x^{4} + 105 \, A b^{6} d^{2} e^{4} x^{4} - 1575 \, B a^{2} b^{4} d e^{5} x^{4} - 630 \, A a b^{5} d e^{5} x^{4} + 2100 \, B a^{3} b^{3} e^{6} x^{4} + 1575 \, A a^{2} b^{4} e^{6} x^{4} + 140 \, B b^{6} d^{4} e^{2} x^{3} - 840 \, B a b^{5} d^{3} e^{3} x^{3} - 140 \, A b^{6} d^{3} e^{3} x^{3} + 2100 \, B a^{2} b^{4} d^{2} e^{4} x^{3} + 840 \, A a b^{5} d^{2} e^{4} x^{3} - 2800 \, B a^{3} b^{3} d e^{5} x^{3} - 2100 \, A a^{2} b^{4} d e^{5} x^{3} + 2100 \, B a^{4} b^{2} e^{6} x^{3} + 2800 \, A a^{3} b^{3} e^{6} x^{3} - 210 \, B b^{6} d^{5} e x^{2} + 1260 \, B a b^{5} d^{4} e^{2} x^{2} + 210 \, A b^{6} d^{4} e^{2} x^{2} - 3150 \, B a^{2} b^{4} d^{3} e^{3} x^{2} - 1260 \, A a b^{5} d^{3} e^{3} x^{2} + 4200 \, B a^{3} b^{3} d^{2} e^{4} x^{2} + 3150 \, A a^{2} b^{4} d^{2} e^{4} x^{2} - 3150 \, B a^{4} b^{2} d e^{5} x^{2} - 4200 \, A a^{3} b^{3} d e^{5} x^{2} + 1260 \, B a^{5} b e^{6} x^{2} + 3150 \, A a^{4} b^{2} e^{6} x^{2} + 420 \, B b^{6} d^{6} x - 2520 \, B a b^{5} d^{5} e x - 420 \, A b^{6} d^{5} e x + 6300 \, B a^{2} b^{4} d^{4} e^{2} x + 2520 \, A a b^{5} d^{4} e^{2} x - 8400 \, B a^{3} b^{3} d^{3} e^{3} x - 6300 \, A a^{2} b^{4} d^{3} e^{3} x + 6300 \, B a^{4} b^{2} d^{2} e^{4} x + 8400 \, A a^{3} b^{3} d^{2} e^{4} x - 2520 \, B a^{5} b d e^{5} x - 6300 \, A a^{4} b^{2} d e^{5} x + 420 \, B a^{6} e^{6} x + 2520 \, A a^{5} b e^{6} x}{420 \, e^{7}} - \frac {{\left (B b^{6} d^{7} - 6 \, B a b^{5} d^{6} e - A b^{6} d^{6} e + 15 \, B a^{2} b^{4} d^{5} e^{2} + 6 \, A a b^{5} d^{5} e^{2} - 20 \, B a^{3} b^{3} d^{4} e^{3} - 15 \, A a^{2} b^{4} d^{4} e^{3} + 15 \, B a^{4} b^{2} d^{3} e^{4} + 20 \, A a^{3} b^{3} d^{3} e^{4} - 6 \, B a^{5} b d^{2} e^{5} - 15 \, A a^{4} b^{2} d^{2} e^{5} + B a^{6} d e^{6} + 6 \, A a^{5} b d e^{6} - A a^{6} e^{7}\right )} \log \left ({\left | e x + d \right |}\right )}{e^{8}} \]

[In]

integrate((b*x+a)^6*(B*x+A)/(e*x+d),x, algorithm="giac")

[Out]

1/420*(60*B*b^6*e^6*x^7 - 70*B*b^6*d*e^5*x^6 + 420*B*a*b^5*e^6*x^6 + 70*A*b^6*e^6*x^6 + 84*B*b^6*d^2*e^4*x^5 -
504*B*a*b^5*d*e^5*x^5 - 84*A*b^6*d*e^5*x^5 + 1260*B*a^2*b^4*e^6*x^5 + 504*A*a*b^5*e^6*x^5 - 105*B*b^6*d^3*e^3
*x^4 + 630*B*a*b^5*d^2*e^4*x^4 + 105*A*b^6*d^2*e^4*x^4 - 1575*B*a^2*b^4*d*e^5*x^4 - 630*A*a*b^5*d*e^5*x^4 + 21
00*B*a^3*b^3*e^6*x^4 + 1575*A*a^2*b^4*e^6*x^4 + 140*B*b^6*d^4*e^2*x^3 - 840*B*a*b^5*d^3*e^3*x^3 - 140*A*b^6*d^
3*e^3*x^3 + 2100*B*a^2*b^4*d^2*e^4*x^3 + 840*A*a*b^5*d^2*e^4*x^3 - 2800*B*a^3*b^3*d*e^5*x^3 - 2100*A*a^2*b^4*d
*e^5*x^3 + 2100*B*a^4*b^2*e^6*x^3 + 2800*A*a^3*b^3*e^6*x^3 - 210*B*b^6*d^5*e*x^2 + 1260*B*a*b^5*d^4*e^2*x^2 +
210*A*b^6*d^4*e^2*x^2 - 3150*B*a^2*b^4*d^3*e^3*x^2 - 1260*A*a*b^5*d^3*e^3*x^2 + 4200*B*a^3*b^3*d^2*e^4*x^2 + 3
150*A*a^2*b^4*d^2*e^4*x^2 - 3150*B*a^4*b^2*d*e^5*x^2 - 4200*A*a^3*b^3*d*e^5*x^2 + 1260*B*a^5*b*e^6*x^2 + 3150*
A*a^4*b^2*e^6*x^2 + 420*B*b^6*d^6*x - 2520*B*a*b^5*d^5*e*x - 420*A*b^6*d^5*e*x + 6300*B*a^2*b^4*d^4*e^2*x + 25
20*A*a*b^5*d^4*e^2*x - 8400*B*a^3*b^3*d^3*e^3*x - 6300*A*a^2*b^4*d^3*e^3*x + 6300*B*a^4*b^2*d^2*e^4*x + 8400*A
*a^3*b^3*d^2*e^4*x - 2520*B*a^5*b*d*e^5*x - 6300*A*a^4*b^2*d*e^5*x + 420*B*a^6*e^6*x + 2520*A*a^5*b*e^6*x)/e^7
- (B*b^6*d^7 - 6*B*a*b^5*d^6*e - A*b^6*d^6*e + 15*B*a^2*b^4*d^5*e^2 + 6*A*a*b^5*d^5*e^2 - 20*B*a^3*b^3*d^4*e^
3 - 15*A*a^2*b^4*d^4*e^3 + 15*B*a^4*b^2*d^3*e^4 + 20*A*a^3*b^3*d^3*e^4 - 6*B*a^5*b*d^2*e^5 - 15*A*a^4*b^2*d^2*
e^5 + B*a^6*d*e^6 + 6*A*a^5*b*d*e^6 - A*a^6*e^7)*log(abs(e*x + d))/e^8

Mupad [B] (verification not implemented)

Time = 1.42 (sec) , antiderivative size = 769, normalized size of antiderivative = 3.50 \[ \int \frac {(a+b x)^6 (A+B x)}{d+e x} \, dx=x\,\left (\frac {B\,a^6+6\,A\,b\,a^5}{e}-\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {A\,b^6+6\,B\,a\,b^5}{e}-\frac {B\,b^6\,d}{e^2}\right )}{e}-\frac {3\,a\,b^4\,\left (2\,A\,b+5\,B\,a\right )}{e}\right )}{e}+\frac {5\,a^2\,b^3\,\left (3\,A\,b+4\,B\,a\right )}{e}\right )}{e}-\frac {5\,a^3\,b^2\,\left (4\,A\,b+3\,B\,a\right )}{e}\right )}{e}+\frac {3\,a^4\,b\,\left (5\,A\,b+2\,B\,a\right )}{e}\right )}{e}\right )-x^3\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {A\,b^6+6\,B\,a\,b^5}{e}-\frac {B\,b^6\,d}{e^2}\right )}{e}-\frac {3\,a\,b^4\,\left (2\,A\,b+5\,B\,a\right )}{e}\right )}{e}+\frac {5\,a^2\,b^3\,\left (3\,A\,b+4\,B\,a\right )}{e}\right )}{3\,e}-\frac {5\,a^3\,b^2\,\left (4\,A\,b+3\,B\,a\right )}{3\,e}\right )+x^4\,\left (\frac {d\,\left (\frac {d\,\left (\frac {A\,b^6+6\,B\,a\,b^5}{e}-\frac {B\,b^6\,d}{e^2}\right )}{e}-\frac {3\,a\,b^4\,\left (2\,A\,b+5\,B\,a\right )}{e}\right )}{4\,e}+\frac {5\,a^2\,b^3\,\left (3\,A\,b+4\,B\,a\right )}{4\,e}\right )-x^5\,\left (\frac {d\,\left (\frac {A\,b^6+6\,B\,a\,b^5}{e}-\frac {B\,b^6\,d}{e^2}\right )}{5\,e}-\frac {3\,a\,b^4\,\left (2\,A\,b+5\,B\,a\right )}{5\,e}\right )+x^6\,\left (\frac {A\,b^6+6\,B\,a\,b^5}{6\,e}-\frac {B\,b^6\,d}{6\,e^2}\right )+x^2\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {A\,b^6+6\,B\,a\,b^5}{e}-\frac {B\,b^6\,d}{e^2}\right )}{e}-\frac {3\,a\,b^4\,\left (2\,A\,b+5\,B\,a\right )}{e}\right )}{e}+\frac {5\,a^2\,b^3\,\left (3\,A\,b+4\,B\,a\right )}{e}\right )}{e}-\frac {5\,a^3\,b^2\,\left (4\,A\,b+3\,B\,a\right )}{e}\right )}{2\,e}+\frac {3\,a^4\,b\,\left (5\,A\,b+2\,B\,a\right )}{2\,e}\right )+\frac {\ln \left (d+e\,x\right )\,\left (-B\,a^6\,d\,e^6+A\,a^6\,e^7+6\,B\,a^5\,b\,d^2\,e^5-6\,A\,a^5\,b\,d\,e^6-15\,B\,a^4\,b^2\,d^3\,e^4+15\,A\,a^4\,b^2\,d^2\,e^5+20\,B\,a^3\,b^3\,d^4\,e^3-20\,A\,a^3\,b^3\,d^3\,e^4-15\,B\,a^2\,b^4\,d^5\,e^2+15\,A\,a^2\,b^4\,d^4\,e^3+6\,B\,a\,b^5\,d^6\,e-6\,A\,a\,b^5\,d^5\,e^2-B\,b^6\,d^7+A\,b^6\,d^6\,e\right )}{e^8}+\frac {B\,b^6\,x^7}{7\,e} \]

[In]

int(((A + B*x)*(a + b*x)^6)/(d + e*x),x)

[Out]

x*((B*a^6 + 6*A*a^5*b)/e - (d*((d*((d*((d*((d*((A*b^6 + 6*B*a*b^5)/e - (B*b^6*d)/e^2))/e - (3*a*b^4*(2*A*b + 5
*B*a))/e))/e + (5*a^2*b^3*(3*A*b + 4*B*a))/e))/e - (5*a^3*b^2*(4*A*b + 3*B*a))/e))/e + (3*a^4*b*(5*A*b + 2*B*a
))/e))/e) - x^3*((d*((d*((d*((A*b^6 + 6*B*a*b^5)/e - (B*b^6*d)/e^2))/e - (3*a*b^4*(2*A*b + 5*B*a))/e))/e + (5*
a^2*b^3*(3*A*b + 4*B*a))/e))/(3*e) - (5*a^3*b^2*(4*A*b + 3*B*a))/(3*e)) + x^4*((d*((d*((A*b^6 + 6*B*a*b^5)/e -
(B*b^6*d)/e^2))/e - (3*a*b^4*(2*A*b + 5*B*a))/e))/(4*e) + (5*a^2*b^3*(3*A*b + 4*B*a))/(4*e)) - x^5*((d*((A*b^
6 + 6*B*a*b^5)/e - (B*b^6*d)/e^2))/(5*e) - (3*a*b^4*(2*A*b + 5*B*a))/(5*e)) + x^6*((A*b^6 + 6*B*a*b^5)/(6*e) -
(B*b^6*d)/(6*e^2)) + x^2*((d*((d*((d*((d*((A*b^6 + 6*B*a*b^5)/e - (B*b^6*d)/e^2))/e - (3*a*b^4*(2*A*b + 5*B*a
))/e))/e + (5*a^2*b^3*(3*A*b + 4*B*a))/e))/e - (5*a^3*b^2*(4*A*b + 3*B*a))/e))/(2*e) + (3*a^4*b*(5*A*b + 2*B*a
))/(2*e)) + (log(d + e*x)*(A*a^6*e^7 - B*b^6*d^7 + A*b^6*d^6*e - B*a^6*d*e^6 - 6*A*a*b^5*d^5*e^2 + 6*B*a^5*b*d
^2*e^5 + 15*A*a^2*b^4*d^4*e^3 - 20*A*a^3*b^3*d^3*e^4 + 15*A*a^4*b^2*d^2*e^5 - 15*B*a^2*b^4*d^5*e^2 + 20*B*a^3*
b^3*d^4*e^3 - 15*B*a^4*b^2*d^3*e^4 - 6*A*a^5*b*d*e^6 + 6*B*a*b^5*d^6*e))/e^8 + (B*b^6*x^7)/(7*e)