\(\int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{10}} \, dx\) [1069]

Optimal result

Integrand size = 20, antiderivative size = 135 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{10}} \, dx=-\frac {(B d-A e) (a+b x)^7}{9 e (b d-a e) (d+e x)^9}+\frac {(7 b B d+2 A b e-9 a B e) (a+b x)^7}{72 e (b d-a e)^2 (d+e x)^8}+\frac {b (7 b B d+2 A b e-9 a B e) (a+b x)^7}{504 e (b d-a e)^3 (d+e x)^7} \]

[Out]

-1/9*(-A*e+B*d)*(b*x+a)^7/e/(-a*e+b*d)/(e*x+d)^9+1/72*(2*A*b*e-9*B*a*e+7*B*b*d)*(b*x+a)^7/e/(-a*e+b*d)^2/(e*x+
d)^8+1/504*b*(2*A*b*e-9*B*a*e+7*B*b*d)*(b*x+a)^7/e/(-a*e+b*d)^3/(e*x+d)^7

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {79, 47, 37} \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{10}} \, dx=\frac {b (a+b x)^7 (-9 a B e+2 A b e+7 b B d)}{504 e (d+e x)^7 (b d-a e)^3}+\frac {(a+b x)^7 (-9 a B e+2 A b e+7 b B d)}{72 e (d+e x)^8 (b d-a e)^2}-\frac {(a+b x)^7 (B d-A e)}{9 e (d+e x)^9 (b d-a e)} \]

[In]

Int[((a + b*x)^6*(A + B*x))/(d + e*x)^10,x]

[Out]

-1/9*((B*d - A*e)*(a + b*x)^7)/(e*(b*d - a*e)*(d + e*x)^9) + ((7*b*B*d + 2*A*b*e - 9*a*B*e)*(a + b*x)^7)/(72*e
*(b*d - a*e)^2*(d + e*x)^8) + (b*(7*b*B*d + 2*A*b*e - 9*a*B*e)*(a + b*x)^7)/(504*e*(b*d - a*e)^3*(d + e*x)^7)

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*(Simplify[m + n + 2]/((b*c - a*d)*(m + 1))), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1
) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

Rubi steps \begin{align*} \text {integral}& = -\frac {(B d-A e) (a+b x)^7}{9 e (b d-a e) (d+e x)^9}+\frac {(7 b B d+2 A b e-9 a B e) \int \frac {(a+b x)^6}{(d+e x)^9} \, dx}{9 e (b d-a e)} \\ & = -\frac {(B d-A e) (a+b x)^7}{9 e (b d-a e) (d+e x)^9}+\frac {(7 b B d+2 A b e-9 a B e) (a+b x)^7}{72 e (b d-a e)^2 (d+e x)^8}+\frac {(b (7 b B d+2 A b e-9 a B e)) \int \frac {(a+b x)^6}{(d+e x)^8} \, dx}{72 e (b d-a e)^2} \\ & = -\frac {(B d-A e) (a+b x)^7}{9 e (b d-a e) (d+e x)^9}+\frac {(7 b B d+2 A b e-9 a B e) (a+b x)^7}{72 e (b d-a e)^2 (d+e x)^8}+\frac {b (7 b B d+2 A b e-9 a B e) (a+b x)^7}{504 e (b d-a e)^3 (d+e x)^7} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(603\) vs. \(2(135)=270\).

Time = 0.18 (sec) , antiderivative size = 603, normalized size of antiderivative = 4.47 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{10}} \, dx=-\frac {7 a^6 e^6 (8 A e+B (d+9 e x))+6 a^5 b e^5 \left (7 A e (d+9 e x)+2 B \left (d^2+9 d e x+36 e^2 x^2\right )\right )+15 a^4 b^2 e^4 \left (2 A e \left (d^2+9 d e x+36 e^2 x^2\right )+B \left (d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right )\right )+4 a^3 b^3 e^3 \left (5 A e \left (d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right )+4 B \left (d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right )\right )+3 a^2 b^4 e^2 \left (4 A e \left (d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right )+5 B \left (d^5+9 d^4 e x+36 d^3 e^2 x^2+84 d^2 e^3 x^3+126 d e^4 x^4+126 e^5 x^5\right )\right )+6 a b^5 e \left (A e \left (d^5+9 d^4 e x+36 d^3 e^2 x^2+84 d^2 e^3 x^3+126 d e^4 x^4+126 e^5 x^5\right )+2 B \left (d^6+9 d^5 e x+36 d^4 e^2 x^2+84 d^3 e^3 x^3+126 d^2 e^4 x^4+126 d e^5 x^5+84 e^6 x^6\right )\right )+b^6 \left (2 A e \left (d^6+9 d^5 e x+36 d^4 e^2 x^2+84 d^3 e^3 x^3+126 d^2 e^4 x^4+126 d e^5 x^5+84 e^6 x^6\right )+7 B \left (d^7+9 d^6 e x+36 d^5 e^2 x^2+84 d^4 e^3 x^3+126 d^3 e^4 x^4+126 d^2 e^5 x^5+84 d e^6 x^6+36 e^7 x^7\right )\right )}{504 e^8 (d+e x)^9} \]

[In]

Integrate[((a + b*x)^6*(A + B*x))/(d + e*x)^10,x]

[Out]

-1/504*(7*a^6*e^6*(8*A*e + B*(d + 9*e*x)) + 6*a^5*b*e^5*(7*A*e*(d + 9*e*x) + 2*B*(d^2 + 9*d*e*x + 36*e^2*x^2))
+ 15*a^4*b^2*e^4*(2*A*e*(d^2 + 9*d*e*x + 36*e^2*x^2) + B*(d^3 + 9*d^2*e*x + 36*d*e^2*x^2 + 84*e^3*x^3)) + 4*a
^3*b^3*e^3*(5*A*e*(d^3 + 9*d^2*e*x + 36*d*e^2*x^2 + 84*e^3*x^3) + 4*B*(d^4 + 9*d^3*e*x + 36*d^2*e^2*x^2 + 84*d
*e^3*x^3 + 126*e^4*x^4)) + 3*a^2*b^4*e^2*(4*A*e*(d^4 + 9*d^3*e*x + 36*d^2*e^2*x^2 + 84*d*e^3*x^3 + 126*e^4*x^4
) + 5*B*(d^5 + 9*d^4*e*x + 36*d^3*e^2*x^2 + 84*d^2*e^3*x^3 + 126*d*e^4*x^4 + 126*e^5*x^5)) + 6*a*b^5*e*(A*e*(d
^5 + 9*d^4*e*x + 36*d^3*e^2*x^2 + 84*d^2*e^3*x^3 + 126*d*e^4*x^4 + 126*e^5*x^5) + 2*B*(d^6 + 9*d^5*e*x + 36*d^
4*e^2*x^2 + 84*d^3*e^3*x^3 + 126*d^2*e^4*x^4 + 126*d*e^5*x^5 + 84*e^6*x^6)) + b^6*(2*A*e*(d^6 + 9*d^5*e*x + 36
*d^4*e^2*x^2 + 84*d^3*e^3*x^3 + 126*d^2*e^4*x^4 + 126*d*e^5*x^5 + 84*e^6*x^6) + 7*B*(d^7 + 9*d^6*e*x + 36*d^5*
e^2*x^2 + 84*d^4*e^3*x^3 + 126*d^3*e^4*x^4 + 126*d^2*e^5*x^5 + 84*d*e^6*x^6 + 36*e^7*x^7)))/(e^8*(d + e*x)^9)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(788\) vs. \(2(129)=258\).

Time = 0.70 (sec) , antiderivative size = 789, normalized size of antiderivative = 5.84

method	result	size
risch	\(\frac {-\frac {b^{6} B \,x^{7}}{2 e}-\frac {b^{5} \left (2 A b e +12 B a e +7 B b d \right ) x^{6}}{6 e^{2}}-\frac {b^{4} \left (6 A a b \,e^{2}+2 A \,b^{2} d e +15 B \,a^{2} e^{2}+12 B a b d e +7 b^{2} B \,d^{2}\right ) x^{5}}{4 e^{3}}-\frac {b^{3} \left (12 A \,a^{2} b \,e^{3}+6 A a \,b^{2} d \,e^{2}+2 A \,b^{3} d^{2} e +16 B \,a^{3} e^{3}+15 B \,a^{2} b d \,e^{2}+12 B a \,b^{2} d^{2} e +7 b^{3} B \,d^{3}\right ) x^{4}}{4 e^{4}}-\frac {b^{2} \left (20 A \,a^{3} b \,e^{4}+12 A \,a^{2} b^{2} d \,e^{3}+6 A a \,b^{3} d^{2} e^{2}+2 A \,b^{4} d^{3} e +15 B \,a^{4} e^{4}+16 B \,a^{3} b d \,e^{3}+15 B \,a^{2} b^{2} d^{2} e^{2}+12 B a \,b^{3} d^{3} e +7 B \,b^{4} d^{4}\right ) x^{3}}{6 e^{5}}-\frac {b \left (30 A \,a^{4} b \,e^{5}+20 A \,a^{3} b^{2} d \,e^{4}+12 A \,a^{2} b^{3} d^{2} e^{3}+6 A a \,b^{4} d^{3} e^{2}+2 A \,b^{5} d^{4} e +12 B \,a^{5} e^{5}+15 B \,a^{4} b d \,e^{4}+16 B \,a^{3} b^{2} d^{2} e^{3}+15 B \,a^{2} b^{3} d^{3} e^{2}+12 B a \,b^{4} d^{4} e +7 B \,b^{5} d^{5}\right ) x^{2}}{14 e^{6}}-\frac {\left (42 A \,a^{5} b \,e^{6}+30 A \,a^{4} b^{2} d \,e^{5}+20 A \,a^{3} b^{3} d^{2} e^{4}+12 A \,a^{2} b^{4} d^{3} e^{3}+6 A a \,b^{5} d^{4} e^{2}+2 A \,b^{6} d^{5} e +7 B \,a^{6} e^{6}+12 B \,a^{5} b d \,e^{5}+15 B \,a^{4} b^{2} d^{2} e^{4}+16 B \,a^{3} b^{3} d^{3} e^{3}+15 B \,a^{2} b^{4} d^{4} e^{2}+12 B a \,b^{5} d^{5} e +7 b^{6} B \,d^{6}\right ) x}{56 e^{7}}-\frac {56 A \,a^{6} e^{7}+42 A \,a^{5} b d \,e^{6}+30 A \,a^{4} b^{2} d^{2} e^{5}+20 A \,a^{3} b^{3} d^{3} e^{4}+12 A \,a^{2} b^{4} d^{4} e^{3}+6 A a \,b^{5} d^{5} e^{2}+2 A \,b^{6} d^{6} e +7 B \,a^{6} d \,e^{6}+12 B \,a^{5} b \,d^{2} e^{5}+15 B \,a^{4} b^{2} d^{3} e^{4}+16 B \,a^{3} b^{3} d^{4} e^{3}+15 B \,a^{2} b^{4} d^{5} e^{2}+12 B a \,b^{5} d^{6} e +7 b^{6} B \,d^{7}}{504 e^{8}}}{\left (e x +d \right )^{9}}\)	\(789\)
default	\(-\frac {6 A \,a^{5} b \,e^{6}-30 A \,a^{4} b^{2} d \,e^{5}+60 A \,a^{3} b^{3} d^{2} e^{4}-60 A \,a^{2} b^{4} d^{3} e^{3}+30 A a \,b^{5} d^{4} e^{2}-6 A \,b^{6} d^{5} e +B \,a^{6} e^{6}-12 B \,a^{5} b d \,e^{5}+45 B \,a^{4} b^{2} d^{2} e^{4}-80 B \,a^{3} b^{3} d^{3} e^{3}+75 B \,a^{2} b^{4} d^{4} e^{2}-36 B a \,b^{5} d^{5} e +7 b^{6} B \,d^{6}}{8 e^{8} \left (e x +d \right )^{8}}-\frac {3 b \left (5 A \,a^{4} b \,e^{5}-20 A \,a^{3} b^{2} d \,e^{4}+30 A \,a^{2} b^{3} d^{2} e^{3}-20 A a \,b^{4} d^{3} e^{2}+5 A \,b^{5} d^{4} e +2 B \,a^{5} e^{5}-15 B \,a^{4} b d \,e^{4}+40 B \,a^{3} b^{2} d^{2} e^{3}-50 B \,a^{2} b^{3} d^{3} e^{2}+30 B a \,b^{4} d^{4} e -7 B \,b^{5} d^{5}\right )}{7 e^{8} \left (e x +d \right )^{7}}-\frac {b^{3} \left (3 A \,a^{2} b \,e^{3}-6 A a \,b^{2} d \,e^{2}+3 A \,b^{3} d^{2} e +4 B \,a^{3} e^{3}-15 B \,a^{2} b d \,e^{2}+18 B a \,b^{2} d^{2} e -7 b^{3} B \,d^{3}\right )}{e^{8} \left (e x +d \right )^{5}}-\frac {A \,a^{6} e^{7}-6 A \,a^{5} b d \,e^{6}+15 A \,a^{4} b^{2} d^{2} e^{5}-20 A \,a^{3} b^{3} d^{3} e^{4}+15 A \,a^{2} b^{4} d^{4} e^{3}-6 A a \,b^{5} d^{5} e^{2}+A \,b^{6} d^{6} e -B \,a^{6} d \,e^{6}+6 B \,a^{5} b \,d^{2} e^{5}-15 B \,a^{4} b^{2} d^{3} e^{4}+20 B \,a^{3} b^{3} d^{4} e^{3}-15 B \,a^{2} b^{4} d^{5} e^{2}+6 B a \,b^{5} d^{6} e -b^{6} B \,d^{7}}{9 e^{8} \left (e x +d \right )^{9}}-\frac {b^{5} \left (A b e +6 B a e -7 B b d \right )}{3 e^{8} \left (e x +d \right )^{3}}-\frac {b^{6} B}{2 e^{8} \left (e x +d \right )^{2}}-\frac {5 b^{2} \left (4 A \,a^{3} b \,e^{4}-12 A \,a^{2} b^{2} d \,e^{3}+12 A a \,b^{3} d^{2} e^{2}-4 A \,b^{4} d^{3} e +3 B \,a^{4} e^{4}-16 B \,a^{3} b d \,e^{3}+30 B \,a^{2} b^{2} d^{2} e^{2}-24 B a \,b^{3} d^{3} e +7 B \,b^{4} d^{4}\right )}{6 e^{8} \left (e x +d \right )^{6}}-\frac {3 b^{4} \left (2 A a b \,e^{2}-2 A \,b^{2} d e +5 B \,a^{2} e^{2}-12 B a b d e +7 b^{2} B \,d^{2}\right )}{4 e^{8} \left (e x +d \right )^{4}}\)	\(814\)
norman	\(\frac {-\frac {b^{6} B \,x^{7}}{2 e}-\frac {\left (2 A \,b^{6} e^{2}+12 B a \,b^{5} e^{2}+7 b^{6} B d e \right ) x^{6}}{6 e^{3}}-\frac {\left (6 A a \,b^{5} e^{3}+2 A \,b^{6} d \,e^{2}+15 B \,a^{2} b^{4} e^{3}+12 B a \,b^{5} d \,e^{2}+7 b^{6} B \,d^{2} e \right ) x^{5}}{4 e^{4}}-\frac {\left (12 A \,a^{2} b^{4} e^{4}+6 A a \,b^{5} d \,e^{3}+2 A \,b^{6} d^{2} e^{2}+16 B \,a^{3} b^{3} e^{4}+15 B \,a^{2} b^{4} d \,e^{3}+12 B a \,b^{5} d^{2} e^{2}+7 b^{6} B \,d^{3} e \right ) x^{4}}{4 e^{5}}-\frac {\left (20 A \,a^{3} b^{3} e^{5}+12 A \,a^{2} b^{4} d \,e^{4}+6 A a \,b^{5} d^{2} e^{3}+2 A \,b^{6} d^{3} e^{2}+15 B \,a^{4} b^{2} e^{5}+16 B \,a^{3} b^{3} d \,e^{4}+15 B \,a^{2} b^{4} d^{2} e^{3}+12 B a \,b^{5} d^{3} e^{2}+7 b^{6} B \,d^{4} e \right ) x^{3}}{6 e^{6}}-\frac {\left (30 A \,a^{4} b^{2} e^{6}+20 A \,a^{3} b^{3} d \,e^{5}+12 A \,a^{2} b^{4} d^{2} e^{4}+6 A a \,b^{5} d^{3} e^{3}+2 A \,b^{6} d^{4} e^{2}+12 B \,a^{5} b \,e^{6}+15 B \,a^{4} b^{2} d \,e^{5}+16 B \,a^{3} b^{3} d^{2} e^{4}+15 B \,a^{2} b^{4} d^{3} e^{3}+12 B a \,b^{5} d^{4} e^{2}+7 b^{6} B \,d^{5} e \right ) x^{2}}{14 e^{7}}-\frac {\left (42 A \,a^{5} b \,e^{7}+30 A \,a^{4} b^{2} d \,e^{6}+20 A \,a^{3} b^{3} d^{2} e^{5}+12 A \,a^{2} b^{4} d^{3} e^{4}+6 A a \,b^{5} d^{4} e^{3}+2 A \,b^{6} d^{5} e^{2}+7 B \,a^{6} e^{7}+12 B \,a^{5} b d \,e^{6}+15 B \,a^{4} b^{2} d^{2} e^{5}+16 B \,a^{3} b^{3} d^{3} e^{4}+15 B \,a^{2} b^{4} d^{4} e^{3}+12 B a \,b^{5} d^{5} e^{2}+7 b^{6} B \,d^{6} e \right ) x}{56 e^{8}}-\frac {56 A \,a^{6} e^{8}+42 A \,a^{5} b d \,e^{7}+30 A \,a^{4} b^{2} d^{2} e^{6}+20 A \,a^{3} b^{3} d^{3} e^{5}+12 A \,a^{2} b^{4} d^{4} e^{4}+6 A a \,b^{5} d^{5} e^{3}+2 A \,b^{6} d^{6} e^{2}+7 B \,a^{6} d \,e^{7}+12 B \,a^{5} b \,d^{2} e^{6}+15 B \,a^{4} b^{2} d^{3} e^{5}+16 B \,a^{3} b^{3} d^{4} e^{4}+15 B \,a^{2} b^{4} d^{5} e^{3}+12 B a \,b^{5} d^{6} e^{2}+7 b^{6} B \,d^{7} e}{504 e^{9}}}{\left (e x +d \right )^{9}}\)	\(844\)
gosper	\(-\frac {252 B \,x^{7} b^{6} e^{7}+168 A \,x^{6} b^{6} e^{7}+1008 B \,x^{6} a \,b^{5} e^{7}+588 B \,x^{6} b^{6} d \,e^{6}+756 A \,x^{5} a \,b^{5} e^{7}+252 A \,x^{5} b^{6} d \,e^{6}+1890 B \,x^{5} a^{2} b^{4} e^{7}+1512 B \,x^{5} a \,b^{5} d \,e^{6}+882 B \,x^{5} b^{6} d^{2} e^{5}+1512 A \,x^{4} a^{2} b^{4} e^{7}+756 A \,x^{4} a \,b^{5} d \,e^{6}+252 A \,x^{4} b^{6} d^{2} e^{5}+2016 B \,x^{4} a^{3} b^{3} e^{7}+1890 B \,x^{4} a^{2} b^{4} d \,e^{6}+1512 B \,x^{4} a \,b^{5} d^{2} e^{5}+882 B \,x^{4} b^{6} d^{3} e^{4}+1680 A \,x^{3} a^{3} b^{3} e^{7}+1008 A \,x^{3} a^{2} b^{4} d \,e^{6}+504 A \,x^{3} a \,b^{5} d^{2} e^{5}+168 A \,x^{3} b^{6} d^{3} e^{4}+1260 B \,x^{3} a^{4} b^{2} e^{7}+1344 B \,x^{3} a^{3} b^{3} d \,e^{6}+1260 B \,x^{3} a^{2} b^{4} d^{2} e^{5}+1008 B \,x^{3} a \,b^{5} d^{3} e^{4}+588 B \,x^{3} b^{6} d^{4} e^{3}+1080 A \,x^{2} a^{4} b^{2} e^{7}+720 A \,x^{2} a^{3} b^{3} d \,e^{6}+432 A \,x^{2} a^{2} b^{4} d^{2} e^{5}+216 A \,x^{2} a \,b^{5} d^{3} e^{4}+72 A \,x^{2} b^{6} d^{4} e^{3}+432 B \,x^{2} a^{5} b \,e^{7}+540 B \,x^{2} a^{4} b^{2} d \,e^{6}+576 B \,x^{2} a^{3} b^{3} d^{2} e^{5}+540 B \,x^{2} a^{2} b^{4} d^{3} e^{4}+432 B \,x^{2} a \,b^{5} d^{4} e^{3}+252 B \,x^{2} b^{6} d^{5} e^{2}+378 A x \,a^{5} b \,e^{7}+270 A x \,a^{4} b^{2} d \,e^{6}+180 A x \,a^{3} b^{3} d^{2} e^{5}+108 A x \,a^{2} b^{4} d^{3} e^{4}+54 A x a \,b^{5} d^{4} e^{3}+18 A x \,b^{6} d^{5} e^{2}+63 B x \,a^{6} e^{7}+108 B x \,a^{5} b d \,e^{6}+135 B x \,a^{4} b^{2} d^{2} e^{5}+144 B x \,a^{3} b^{3} d^{3} e^{4}+135 B x \,a^{2} b^{4} d^{4} e^{3}+108 B x a \,b^{5} d^{5} e^{2}+63 B x \,b^{6} d^{6} e +56 A \,a^{6} e^{7}+42 A \,a^{5} b d \,e^{6}+30 A \,a^{4} b^{2} d^{2} e^{5}+20 A \,a^{3} b^{3} d^{3} e^{4}+12 A \,a^{2} b^{4} d^{4} e^{3}+6 A a \,b^{5} d^{5} e^{2}+2 A \,b^{6} d^{6} e +7 B \,a^{6} d \,e^{6}+12 B \,a^{5} b \,d^{2} e^{5}+15 B \,a^{4} b^{2} d^{3} e^{4}+16 B \,a^{3} b^{3} d^{4} e^{3}+15 B \,a^{2} b^{4} d^{5} e^{2}+12 B a \,b^{5} d^{6} e +7 b^{6} B \,d^{7}}{504 e^{8} \left (e x +d \right )^{9}}\)	\(913\)
parallelrisch	\(-\frac {252 B \,b^{6} x^{7} e^{8}+168 A \,b^{6} e^{8} x^{6}+1008 B a \,b^{5} e^{8} x^{6}+588 B \,b^{6} d \,e^{7} x^{6}+756 A a \,b^{5} e^{8} x^{5}+252 A \,b^{6} d \,e^{7} x^{5}+1890 B \,a^{2} b^{4} e^{8} x^{5}+1512 B a \,b^{5} d \,e^{7} x^{5}+882 B \,b^{6} d^{2} e^{6} x^{5}+1512 A \,a^{2} b^{4} e^{8} x^{4}+756 A a \,b^{5} d \,e^{7} x^{4}+252 A \,b^{6} d^{2} e^{6} x^{4}+2016 B \,a^{3} b^{3} e^{8} x^{4}+1890 B \,a^{2} b^{4} d \,e^{7} x^{4}+1512 B a \,b^{5} d^{2} e^{6} x^{4}+882 B \,b^{6} d^{3} e^{5} x^{4}+1680 A \,a^{3} b^{3} e^{8} x^{3}+1008 A \,a^{2} b^{4} d \,e^{7} x^{3}+504 A a \,b^{5} d^{2} e^{6} x^{3}+168 A \,b^{6} d^{3} e^{5} x^{3}+1260 B \,a^{4} b^{2} e^{8} x^{3}+1344 B \,a^{3} b^{3} d \,e^{7} x^{3}+1260 B \,a^{2} b^{4} d^{2} e^{6} x^{3}+1008 B a \,b^{5} d^{3} e^{5} x^{3}+588 B \,b^{6} d^{4} e^{4} x^{3}+1080 A \,a^{4} b^{2} e^{8} x^{2}+720 A \,a^{3} b^{3} d \,e^{7} x^{2}+432 A \,a^{2} b^{4} d^{2} e^{6} x^{2}+216 A a \,b^{5} d^{3} e^{5} x^{2}+72 A \,b^{6} d^{4} e^{4} x^{2}+432 B \,a^{5} b \,e^{8} x^{2}+540 B \,a^{4} b^{2} d \,e^{7} x^{2}+576 B \,a^{3} b^{3} d^{2} e^{6} x^{2}+540 B \,a^{2} b^{4} d^{3} e^{5} x^{2}+432 B a \,b^{5} d^{4} e^{4} x^{2}+252 B \,b^{6} d^{5} e^{3} x^{2}+378 A \,a^{5} b \,e^{8} x +270 A \,a^{4} b^{2} d \,e^{7} x +180 A \,a^{3} b^{3} d^{2} e^{6} x +108 A \,a^{2} b^{4} d^{3} e^{5} x +54 A a \,b^{5} d^{4} e^{4} x +18 A \,b^{6} d^{5} e^{3} x +63 B \,a^{6} e^{8} x +108 B \,a^{5} b d \,e^{7} x +135 B \,a^{4} b^{2} d^{2} e^{6} x +144 B \,a^{3} b^{3} d^{3} e^{5} x +135 B \,a^{2} b^{4} d^{4} e^{4} x +108 B a \,b^{5} d^{5} e^{3} x +63 B \,b^{6} d^{6} e^{2} x +56 A \,a^{6} e^{8}+42 A \,a^{5} b d \,e^{7}+30 A \,a^{4} b^{2} d^{2} e^{6}+20 A \,a^{3} b^{3} d^{3} e^{5}+12 A \,a^{2} b^{4} d^{4} e^{4}+6 A a \,b^{5} d^{5} e^{3}+2 A \,b^{6} d^{6} e^{2}+7 B \,a^{6} d \,e^{7}+12 B \,a^{5} b \,d^{2} e^{6}+15 B \,a^{4} b^{2} d^{3} e^{5}+16 B \,a^{3} b^{3} d^{4} e^{4}+15 B \,a^{2} b^{4} d^{5} e^{3}+12 B a \,b^{5} d^{6} e^{2}+7 b^{6} B \,d^{7} e}{504 e^{9} \left (e x +d \right )^{9}}\)	\(920\)

[In]

int((b*x+a)^6*(B*x+A)/(e*x+d)^10,x,method=_RETURNVERBOSE)

[Out]

(-1/2*b^6*B/e*x^7-1/6*b^5/e^2*(2*A*b*e+12*B*a*e+7*B*b*d)*x^6-1/4*b^4/e^3*(6*A*a*b*e^2+2*A*b^2*d*e+15*B*a^2*e^2
+12*B*a*b*d*e+7*B*b^2*d^2)*x^5-1/4*b^3/e^4*(12*A*a^2*b*e^3+6*A*a*b^2*d*e^2+2*A*b^3*d^2*e+16*B*a^3*e^3+15*B*a^2
*b*d*e^2+12*B*a*b^2*d^2*e+7*B*b^3*d^3)*x^4-1/6*b^2/e^5*(20*A*a^3*b*e^4+12*A*a^2*b^2*d*e^3+6*A*a*b^3*d^2*e^2+2*
A*b^4*d^3*e+15*B*a^4*e^4+16*B*a^3*b*d*e^3+15*B*a^2*b^2*d^2*e^2+12*B*a*b^3*d^3*e+7*B*b^4*d^4)*x^3-1/14*b/e^6*(3
0*A*a^4*b*e^5+20*A*a^3*b^2*d*e^4+12*A*a^2*b^3*d^2*e^3+6*A*a*b^4*d^3*e^2+2*A*b^5*d^4*e+12*B*a^5*e^5+15*B*a^4*b*
d*e^4+16*B*a^3*b^2*d^2*e^3+15*B*a^2*b^3*d^3*e^2+12*B*a*b^4*d^4*e+7*B*b^5*d^5)*x^2-1/56/e^7*(42*A*a^5*b*e^6+30*
A*a^4*b^2*d*e^5+20*A*a^3*b^3*d^2*e^4+12*A*a^2*b^4*d^3*e^3+6*A*a*b^5*d^4*e^2+2*A*b^6*d^5*e+7*B*a^6*e^6+12*B*a^5
*b*d*e^5+15*B*a^4*b^2*d^2*e^4+16*B*a^3*b^3*d^3*e^3+15*B*a^2*b^4*d^4*e^2+12*B*a*b^5*d^5*e+7*B*b^6*d^6)*x-1/504/
e^8*(56*A*a^6*e^7+42*A*a^5*b*d*e^6+30*A*a^4*b^2*d^2*e^5+20*A*a^3*b^3*d^3*e^4+12*A*a^2*b^4*d^4*e^3+6*A*a*b^5*d^
5*e^2+2*A*b^6*d^6*e+7*B*a^6*d*e^6+12*B*a^5*b*d^2*e^5+15*B*a^4*b^2*d^3*e^4+16*B*a^3*b^3*d^4*e^3+15*B*a^2*b^4*d^
5*e^2+12*B*a*b^5*d^6*e+7*B*b^6*d^7))/(e*x+d)^9

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 861 vs. \(2 (129) = 258\).

Time = 0.24 (sec) , antiderivative size = 861, normalized size of antiderivative = 6.38 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{10}} \, dx=-\frac {252 \, B b^{6} e^{7} x^{7} + 7 \, B b^{6} d^{7} + 56 \, A a^{6} e^{7} + 2 \, {\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} + 4 \, {\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} + 6 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2} e^{5} + 7 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d e^{6} + 84 \, {\left (7 \, B b^{6} d e^{6} + 2 \, {\left (6 \, B a b^{5} + A b^{6}\right )} e^{7}\right )} x^{6} + 126 \, {\left (7 \, B b^{6} d^{2} e^{5} + 2 \, {\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} + 126 \, {\left (7 \, B b^{6} d^{3} e^{4} + 2 \, {\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} + 4 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} e^{7}\right )} x^{4} + 84 \, {\left (7 \, B b^{6} d^{4} e^{3} + 2 \, {\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} + 4 \, {\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} + 36 \, {\left (7 \, B b^{6} d^{5} e^{2} + 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} + 4 \, {\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} + 6 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} e^{7}\right )} x^{2} + 9 \, {\left (7 \, B b^{6} d^{6} e + 2 \, {\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} + 4 \, {\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} + 6 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d e^{6} + 7 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} e^{7}\right )} x}{504 \, {\left (e^{17} x^{9} + 9 \, d e^{16} x^{8} + 36 \, d^{2} e^{15} x^{7} + 84 \, d^{3} e^{14} x^{6} + 126 \, d^{4} e^{13} x^{5} + 126 \, d^{5} e^{12} x^{4} + 84 \, d^{6} e^{11} x^{3} + 36 \, d^{7} e^{10} x^{2} + 9 \, d^{8} e^{9} x + d^{9} e^{8}\right )}} \]

[In]

integrate((b*x+a)^6*(B*x+A)/(e*x+d)^10,x, algorithm="fricas")

[Out]

-1/504*(252*B*b^6*e^7*x^7 + 7*B*b^6*d^7 + 56*A*a^6*e^7 + 2*(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 + 4*(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 + 6*(2*B*a^5*b +
5*A*a^4*b^2)*d^2*e^5 + 7*(B*a^6 + 6*A*a^5*b)*d*e^6 + 84*(7*B*b^6*d*e^6 + 2*(6*B*a*b^5 + A*b^6)*e^7)*x^6 + 126*
(7*B*b^6*d^2*e^5 + 2*(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 + 126*(7*B*b^6*d^3*e^4 +
2*(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 + 4*(4*B*a^3*b^3 + 3*A*a^2*b^4)*e^7)*x^4 +
84*(7*B*b^6*d^4*e^3 + 2*(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 + 4*(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 + 36*(7*B*b^6*d^5*e^2 + 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 + 4*(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 + 6*(2*B*a^5*b + 5*A*a^4*b^2)*e^7)*x^2 + 9*(7*B*b^6*d^6*e + 2*(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 + 4*(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 + 6
*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^6 + 7*(B*a^6 + 6*A*a^5*b)*e^7)*x)/(e^17*x^9 + 9*d*e^16*x^8 + 36*d^2*e^15*x^7 +
84*d^3*e^14*x^6 + 126*d^4*e^13*x^5 + 126*d^5*e^12*x^4 + 84*d^6*e^11*x^3 + 36*d^7*e^10*x^2 + 9*d^8*e^9*x + d^9*
e^8)

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{10}} \, dx=\text {Timed out} \]

[In]

integrate((b*x+a)**6*(B*x+A)/(e*x+d)**10,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 861 vs. \(2 (129) = 258\).

Time = 0.24 (sec) , antiderivative size = 861, normalized size of antiderivative = 6.38 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{10}} \, dx=-\frac {252 \, B b^{6} e^{7} x^{7} + 7 \, B b^{6} d^{7} + 56 \, A a^{6} e^{7} + 2 \, {\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} + 4 \, {\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} + 6 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2} e^{5} + 7 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d e^{6} + 84 \, {\left (7 \, B b^{6} d e^{6} + 2 \, {\left (6 \, B a b^{5} + A b^{6}\right )} e^{7}\right )} x^{6} + 126 \, {\left (7 \, B b^{6} d^{2} e^{5} + 2 \, {\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} + 126 \, {\left (7 \, B b^{6} d^{3} e^{4} + 2 \, {\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} + 4 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} e^{7}\right )} x^{4} + 84 \, {\left (7 \, B b^{6} d^{4} e^{3} + 2 \, {\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} + 4 \, {\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} + 36 \, {\left (7 \, B b^{6} d^{5} e^{2} + 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} + 4 \, {\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} + 6 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} e^{7}\right )} x^{2} + 9 \, {\left (7 \, B b^{6} d^{6} e + 2 \, {\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} + 4 \, {\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} + 6 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d e^{6} + 7 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} e^{7}\right )} x}{504 \, {\left (e^{17} x^{9} + 9 \, d e^{16} x^{8} + 36 \, d^{2} e^{15} x^{7} + 84 \, d^{3} e^{14} x^{6} + 126 \, d^{4} e^{13} x^{5} + 126 \, d^{5} e^{12} x^{4} + 84 \, d^{6} e^{11} x^{3} + 36 \, d^{7} e^{10} x^{2} + 9 \, d^{8} e^{9} x + d^{9} e^{8}\right )}} \]

[In]

integrate((b*x+a)^6*(B*x+A)/(e*x+d)^10,x, algorithm="maxima")

[Out]

-1/504*(252*B*b^6*e^7*x^7 + 7*B*b^6*d^7 + 56*A*a^6*e^7 + 2*(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 + 4*(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 + 6*(2*B*a^5*b +
5*A*a^4*b^2)*d^2*e^5 + 7*(B*a^6 + 6*A*a^5*b)*d*e^6 + 84*(7*B*b^6*d*e^6 + 2*(6*B*a*b^5 + A*b^6)*e^7)*x^6 + 126*
(7*B*b^6*d^2*e^5 + 2*(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 + 126*(7*B*b^6*d^3*e^4 +
2*(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 + 4*(4*B*a^3*b^3 + 3*A*a^2*b^4)*e^7)*x^4 +
84*(7*B*b^6*d^4*e^3 + 2*(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 + 4*(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 + 36*(7*B*b^6*d^5*e^2 + 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 + 4*(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 + 6*(2*B*a^5*b + 5*A*a^4*b^2)*e^7)*x^2 + 9*(7*B*b^6*d^6*e + 2*(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 + 4*(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 + 6
*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^6 + 7*(B*a^6 + 6*A*a^5*b)*e^7)*x)/(e^17*x^9 + 9*d*e^16*x^8 + 36*d^2*e^15*x^7 +
84*d^3*e^14*x^6 + 126*d^4*e^13*x^5 + 126*d^5*e^12*x^4 + 84*d^6*e^11*x^3 + 36*d^7*e^10*x^2 + 9*d^8*e^9*x + d^9*
e^8)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 912 vs. \(2 (129) = 258\).

Time = 0.30 (sec) , antiderivative size = 912, normalized size of antiderivative = 6.76 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{10}} \, dx=-\frac {252 \, B b^{6} e^{7} x^{7} + 588 \, B b^{6} d e^{6} x^{6} + 1008 \, B a b^{5} e^{7} x^{6} + 168 \, A b^{6} e^{7} x^{6} + 882 \, B b^{6} d^{2} e^{5} x^{5} + 1512 \, B a b^{5} d e^{6} x^{5} + 252 \, A b^{6} d e^{6} x^{5} + 1890 \, B a^{2} b^{4} e^{7} x^{5} + 756 \, A a b^{5} e^{7} x^{5} + 882 \, B b^{6} d^{3} e^{4} x^{4} + 1512 \, B a b^{5} d^{2} e^{5} x^{4} + 252 \, A b^{6} d^{2} e^{5} x^{4} + 1890 \, B a^{2} b^{4} d e^{6} x^{4} + 756 \, A a b^{5} d e^{6} x^{4} + 2016 \, B a^{3} b^{3} e^{7} x^{4} + 1512 \, A a^{2} b^{4} e^{7} x^{4} + 588 \, B b^{6} d^{4} e^{3} x^{3} + 1008 \, B a b^{5} d^{3} e^{4} x^{3} + 168 \, A b^{6} d^{3} e^{4} x^{3} + 1260 \, B a^{2} b^{4} d^{2} e^{5} x^{3} + 504 \, A a b^{5} d^{2} e^{5} x^{3} + 1344 \, B a^{3} b^{3} d e^{6} x^{3} + 1008 \, A a^{2} b^{4} d e^{6} x^{3} + 1260 \, B a^{4} b^{2} e^{7} x^{3} + 1680 \, A a^{3} b^{3} e^{7} x^{3} + 252 \, B b^{6} d^{5} e^{2} x^{2} + 432 \, B a b^{5} d^{4} e^{3} x^{2} + 72 \, A b^{6} d^{4} e^{3} x^{2} + 540 \, B a^{2} b^{4} d^{3} e^{4} x^{2} + 216 \, A a b^{5} d^{3} e^{4} x^{2} + 576 \, B a^{3} b^{3} d^{2} e^{5} x^{2} + 432 \, A a^{2} b^{4} d^{2} e^{5} x^{2} + 540 \, B a^{4} b^{2} d e^{6} x^{2} + 720 \, A a^{3} b^{3} d e^{6} x^{2} + 432 \, B a^{5} b e^{7} x^{2} + 1080 \, A a^{4} b^{2} e^{7} x^{2} + 63 \, B b^{6} d^{6} e x + 108 \, B a b^{5} d^{5} e^{2} x + 18 \, A b^{6} d^{5} e^{2} x + 135 \, B a^{2} b^{4} d^{4} e^{3} x + 54 \, A a b^{5} d^{4} e^{3} x + 144 \, B a^{3} b^{3} d^{3} e^{4} x + 108 \, A a^{2} b^{4} d^{3} e^{4} x + 135 \, B a^{4} b^{2} d^{2} e^{5} x + 180 \, A a^{3} b^{3} d^{2} e^{5} x + 108 \, B a^{5} b d e^{6} x + 270 \, A a^{4} b^{2} d e^{6} x + 63 \, B a^{6} e^{7} x + 378 \, A a^{5} b e^{7} x + 7 \, B b^{6} d^{7} + 12 \, B a b^{5} d^{6} e + 2 \, 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} + 16 \, B a^{3} b^{3} d^{4} e^{3} + 12 \, 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} + 12 \, B a^{5} b d^{2} e^{5} + 30 \, A a^{4} b^{2} d^{2} e^{5} + 7 \, B a^{6} d e^{6} + 42 \, A a^{5} b d e^{6} + 56 \, A a^{6} e^{7}}{504 \, {\left (e x + d\right )}^{9} e^{8}} \]

[In]

integrate((b*x+a)^6*(B*x+A)/(e*x+d)^10,x, algorithm="giac")

[Out]

-1/504*(252*B*b^6*e^7*x^7 + 588*B*b^6*d*e^6*x^6 + 1008*B*a*b^5*e^7*x^6 + 168*A*b^6*e^7*x^6 + 882*B*b^6*d^2*e^5
*x^5 + 1512*B*a*b^5*d*e^6*x^5 + 252*A*b^6*d*e^6*x^5 + 1890*B*a^2*b^4*e^7*x^5 + 756*A*a*b^5*e^7*x^5 + 882*B*b^6
*d^3*e^4*x^4 + 1512*B*a*b^5*d^2*e^5*x^4 + 252*A*b^6*d^2*e^5*x^4 + 1890*B*a^2*b^4*d*e^6*x^4 + 756*A*a*b^5*d*e^6
*x^4 + 2016*B*a^3*b^3*e^7*x^4 + 1512*A*a^2*b^4*e^7*x^4 + 588*B*b^6*d^4*e^3*x^3 + 1008*B*a*b^5*d^3*e^4*x^3 + 16
8*A*b^6*d^3*e^4*x^3 + 1260*B*a^2*b^4*d^2*e^5*x^3 + 504*A*a*b^5*d^2*e^5*x^3 + 1344*B*a^3*b^3*d*e^6*x^3 + 1008*A
*a^2*b^4*d*e^6*x^3 + 1260*B*a^4*b^2*e^7*x^3 + 1680*A*a^3*b^3*e^7*x^3 + 252*B*b^6*d^5*e^2*x^2 + 432*B*a*b^5*d^4
*e^3*x^2 + 72*A*b^6*d^4*e^3*x^2 + 540*B*a^2*b^4*d^3*e^4*x^2 + 216*A*a*b^5*d^3*e^4*x^2 + 576*B*a^3*b^3*d^2*e^5*
x^2 + 432*A*a^2*b^4*d^2*e^5*x^2 + 540*B*a^4*b^2*d*e^6*x^2 + 720*A*a^3*b^3*d*e^6*x^2 + 432*B*a^5*b*e^7*x^2 + 10
80*A*a^4*b^2*e^7*x^2 + 63*B*b^6*d^6*e*x + 108*B*a*b^5*d^5*e^2*x + 18*A*b^6*d^5*e^2*x + 135*B*a^2*b^4*d^4*e^3*x
+ 54*A*a*b^5*d^4*e^3*x + 144*B*a^3*b^3*d^3*e^4*x + 108*A*a^2*b^4*d^3*e^4*x + 135*B*a^4*b^2*d^2*e^5*x + 180*A*
a^3*b^3*d^2*e^5*x + 108*B*a^5*b*d*e^6*x + 270*A*a^4*b^2*d*e^6*x + 63*B*a^6*e^7*x + 378*A*a^5*b*e^7*x + 7*B*b^6
*d^7 + 12*B*a*b^5*d^6*e + 2*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 + 16*B*a^3*b^3*d^4*e^3 + 12
*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 + 12*B*a^5*b*d^2*e^5 + 30*A*a^4*b^2*d^2*e^5 +
7*B*a^6*d*e^6 + 42*A*a^5*b*d*e^6 + 56*A*a^6*e^7)/((e*x + d)^9*e^8)

Mupad [B] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 877, normalized size of antiderivative = 6.50 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{10}} \, dx=-\frac {\frac {7\,B\,a^6\,d\,e^6+56\,A\,a^6\,e^7+12\,B\,a^5\,b\,d^2\,e^5+42\,A\,a^5\,b\,d\,e^6+15\,B\,a^4\,b^2\,d^3\,e^4+30\,A\,a^4\,b^2\,d^2\,e^5+16\,B\,a^3\,b^3\,d^4\,e^3+20\,A\,a^3\,b^3\,d^3\,e^4+15\,B\,a^2\,b^4\,d^5\,e^2+12\,A\,a^2\,b^4\,d^4\,e^3+12\,B\,a\,b^5\,d^6\,e+6\,A\,a\,b^5\,d^5\,e^2+7\,B\,b^6\,d^7+2\,A\,b^6\,d^6\,e}{504\,e^8}+\frac {x\,\left (7\,B\,a^6\,e^6+12\,B\,a^5\,b\,d\,e^5+42\,A\,a^5\,b\,e^6+15\,B\,a^4\,b^2\,d^2\,e^4+30\,A\,a^4\,b^2\,d\,e^5+16\,B\,a^3\,b^3\,d^3\,e^3+20\,A\,a^3\,b^3\,d^2\,e^4+15\,B\,a^2\,b^4\,d^4\,e^2+12\,A\,a^2\,b^4\,d^3\,e^3+12\,B\,a\,b^5\,d^5\,e+6\,A\,a\,b^5\,d^4\,e^2+7\,B\,b^6\,d^6+2\,A\,b^6\,d^5\,e\right )}{56\,e^7}+\frac {b^3\,x^4\,\left (16\,B\,a^3\,e^3+15\,B\,a^2\,b\,d\,e^2+12\,A\,a^2\,b\,e^3+12\,B\,a\,b^2\,d^2\,e+6\,A\,a\,b^2\,d\,e^2+7\,B\,b^3\,d^3+2\,A\,b^3\,d^2\,e\right )}{4\,e^4}+\frac {b^5\,x^6\,\left (2\,A\,b\,e+12\,B\,a\,e+7\,B\,b\,d\right )}{6\,e^2}+\frac {b\,x^2\,\left (12\,B\,a^5\,e^5+15\,B\,a^4\,b\,d\,e^4+30\,A\,a^4\,b\,e^5+16\,B\,a^3\,b^2\,d^2\,e^3+20\,A\,a^3\,b^2\,d\,e^4+15\,B\,a^2\,b^3\,d^3\,e^2+12\,A\,a^2\,b^3\,d^2\,e^3+12\,B\,a\,b^4\,d^4\,e+6\,A\,a\,b^4\,d^3\,e^2+7\,B\,b^5\,d^5+2\,A\,b^5\,d^4\,e\right )}{14\,e^6}+\frac {b^2\,x^3\,\left (15\,B\,a^4\,e^4+16\,B\,a^3\,b\,d\,e^3+20\,A\,a^3\,b\,e^4+15\,B\,a^2\,b^2\,d^2\,e^2+12\,A\,a^2\,b^2\,d\,e^3+12\,B\,a\,b^3\,d^3\,e+6\,A\,a\,b^3\,d^2\,e^2+7\,B\,b^4\,d^4+2\,A\,b^4\,d^3\,e\right )}{6\,e^5}+\frac {b^4\,x^5\,\left (15\,B\,a^2\,e^2+12\,B\,a\,b\,d\,e+6\,A\,a\,b\,e^2+7\,B\,b^2\,d^2+2\,A\,b^2\,d\,e\right )}{4\,e^3}+\frac {B\,b^6\,x^7}{2\,e}}{d^9+9\,d^8\,e\,x+36\,d^7\,e^2\,x^2+84\,d^6\,e^3\,x^3+126\,d^5\,e^4\,x^4+126\,d^4\,e^5\,x^5+84\,d^3\,e^6\,x^6+36\,d^2\,e^7\,x^7+9\,d\,e^8\,x^8+e^9\,x^9} \]

[In]

int(((A + B*x)*(a + b*x)^6)/(d + e*x)^10,x)

[Out]

-((56*A*a^6*e^7 + 7*B*b^6*d^7 + 2*A*b^6*d^6*e + 7*B*a^6*d*e^6 + 6*A*a*b^5*d^5*e^2 + 12*B*a^5*b*d^2*e^5 + 12*A*
a^2*b^4*d^4*e^3 + 20*A*a^3*b^3*d^3*e^4 + 30*A*a^4*b^2*d^2*e^5 + 15*B*a^2*b^4*d^5*e^2 + 16*B*a^3*b^3*d^4*e^3 +
15*B*a^4*b^2*d^3*e^4 + 42*A*a^5*b*d*e^6 + 12*B*a*b^5*d^6*e)/(504*e^8) + (x*(7*B*a^6*e^6 + 7*B*b^6*d^6 + 42*A*a
^5*b*e^6 + 2*A*b^6*d^5*e + 6*A*a*b^5*d^4*e^2 + 30*A*a^4*b^2*d*e^5 + 12*A*a^2*b^4*d^3*e^3 + 20*A*a^3*b^3*d^2*e^
4 + 15*B*a^2*b^4*d^4*e^2 + 16*B*a^3*b^3*d^3*e^3 + 15*B*a^4*b^2*d^2*e^4 + 12*B*a*b^5*d^5*e + 12*B*a^5*b*d*e^5))
/(56*e^7) + (b^3*x^4*(16*B*a^3*e^3 + 7*B*b^3*d^3 + 12*A*a^2*b*e^3 + 2*A*b^3*d^2*e + 6*A*a*b^2*d*e^2 + 12*B*a*b
^2*d^2*e + 15*B*a^2*b*d*e^2))/(4*e^4) + (b^5*x^6*(2*A*b*e + 12*B*a*e + 7*B*b*d))/(6*e^2) + (b*x^2*(12*B*a^5*e^
5 + 7*B*b^5*d^5 + 30*A*a^4*b*e^5 + 2*A*b^5*d^4*e + 6*A*a*b^4*d^3*e^2 + 20*A*a^3*b^2*d*e^4 + 12*A*a^2*b^3*d^2*e
^3 + 15*B*a^2*b^3*d^3*e^2 + 16*B*a^3*b^2*d^2*e^3 + 12*B*a*b^4*d^4*e + 15*B*a^4*b*d*e^4))/(14*e^6) + (b^2*x^3*(
15*B*a^4*e^4 + 7*B*b^4*d^4 + 20*A*a^3*b*e^4 + 2*A*b^4*d^3*e + 6*A*a*b^3*d^2*e^2 + 12*A*a^2*b^2*d*e^3 + 15*B*a^
2*b^2*d^2*e^2 + 12*B*a*b^3*d^3*e + 16*B*a^3*b*d*e^3))/(6*e^5) + (b^4*x^5*(15*B*a^2*e^2 + 7*B*b^2*d^2 + 6*A*a*b
*e^2 + 2*A*b^2*d*e + 12*B*a*b*d*e))/(4*e^3) + (B*b^6*x^7)/(2*e))/(d^9 + e^9*x^9 + 9*d*e^8*x^8 + 36*d^7*e^2*x^2
+ 84*d^6*e^3*x^3 + 126*d^5*e^4*x^4 + 126*d^4*e^5*x^5 + 84*d^3*e^6*x^6 + 36*d^2*e^7*x^7 + 9*d^8*e*x)