\(\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{17}} \, dx\) [1105]

Optimal result

Integrand size = 20, antiderivative size = 285 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{17}} \, dx=-\frac {(B d-A e) (a+b x)^{11}}{16 e (b d-a e) (d+e x)^{16}}+\frac {(11 b B d+5 A b e-16 a B e) (a+b x)^{11}}{240 e (b d-a e)^2 (d+e x)^{15}}+\frac {b (11 b B d+5 A b e-16 a B e) (a+b x)^{11}}{840 e (b d-a e)^3 (d+e x)^{14}}+\frac {b^2 (11 b B d+5 A b e-16 a B e) (a+b x)^{11}}{3640 e (b d-a e)^4 (d+e x)^{13}}+\frac {b^3 (11 b B d+5 A b e-16 a B e) (a+b x)^{11}}{21840 e (b d-a e)^5 (d+e x)^{12}}+\frac {b^4 (11 b B d+5 A b e-16 a B e) (a+b x)^{11}}{240240 e (b d-a e)^6 (d+e x)^{11}} \]

[Out]

-1/16*(-A*e+B*d)*(b*x+a)^11/e/(-a*e+b*d)/(e*x+d)^16+1/240*(5*A*b*e-16*B*a*e+11*B*b*d)*(b*x+a)^11/e/(-a*e+b*d)^
2/(e*x+d)^15+1/840*b*(5*A*b*e-16*B*a*e+11*B*b*d)*(b*x+a)^11/e/(-a*e+b*d)^3/(e*x+d)^14+1/3640*b^2*(5*A*b*e-16*B
*a*e+11*B*b*d)*(b*x+a)^11/e/(-a*e+b*d)^4/(e*x+d)^13+1/21840*b^3*(5*A*b*e-16*B*a*e+11*B*b*d)*(b*x+a)^11/e/(-a*e
+b*d)^5/(e*x+d)^12+1/240240*b^4*(5*A*b*e-16*B*a*e+11*B*b*d)*(b*x+a)^11/e/(-a*e+b*d)^6/(e*x+d)^11

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {79, 47, 37} \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{17}} \, dx=\frac {b^4 (a+b x)^{11} (-16 a B e+5 A b e+11 b B d)}{240240 e (d+e x)^{11} (b d-a e)^6}+\frac {b^3 (a+b x)^{11} (-16 a B e+5 A b e+11 b B d)}{21840 e (d+e x)^{12} (b d-a e)^5}+\frac {b^2 (a+b x)^{11} (-16 a B e+5 A b e+11 b B d)}{3640 e (d+e x)^{13} (b d-a e)^4}+\frac {b (a+b x)^{11} (-16 a B e+5 A b e+11 b B d)}{840 e (d+e x)^{14} (b d-a e)^3}+\frac {(a+b x)^{11} (-16 a B e+5 A b e+11 b B d)}{240 e (d+e x)^{15} (b d-a e)^2}-\frac {(a+b x)^{11} (B d-A e)}{16 e (d+e x)^{16} (b d-a e)} \]

[In]

Int[((a + b*x)^10*(A + B*x))/(d + e*x)^17,x]

[Out]

-1/16*((B*d - A*e)*(a + b*x)^11)/(e*(b*d - a*e)*(d + e*x)^16) + ((11*b*B*d + 5*A*b*e - 16*a*B*e)*(a + b*x)^11)
/(240*e*(b*d - a*e)^2*(d + e*x)^15) + (b*(11*b*B*d + 5*A*b*e - 16*a*B*e)*(a + b*x)^11)/(840*e*(b*d - a*e)^3*(d
+ e*x)^14) + (b^2*(11*b*B*d + 5*A*b*e - 16*a*B*e)*(a + b*x)^11)/(3640*e*(b*d - a*e)^4*(d + e*x)^13) + (b^3*(1
1*b*B*d + 5*A*b*e - 16*a*B*e)*(a + b*x)^11)/(21840*e*(b*d - a*e)^5*(d + e*x)^12) + (b^4*(11*b*B*d + 5*A*b*e -
16*a*B*e)*(a + b*x)^11)/(240240*e*(b*d - a*e)^6*(d + e*x)^11)

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)^{11}}{16 e (b d-a e) (d+e x)^{16}}+\frac {(11 b B d+5 A b e-16 a B e) \int \frac {(a+b x)^{10}}{(d+e x)^{16}} \, dx}{16 e (b d-a e)} \\ & = -\frac {(B d-A e) (a+b x)^{11}}{16 e (b d-a e) (d+e x)^{16}}+\frac {(11 b B d+5 A b e-16 a B e) (a+b x)^{11}}{240 e (b d-a e)^2 (d+e x)^{15}}+\frac {(b (11 b B d+5 A b e-16 a B e)) \int \frac {(a+b x)^{10}}{(d+e x)^{15}} \, dx}{60 e (b d-a e)^2} \\ & = -\frac {(B d-A e) (a+b x)^{11}}{16 e (b d-a e) (d+e x)^{16}}+\frac {(11 b B d+5 A b e-16 a B e) (a+b x)^{11}}{240 e (b d-a e)^2 (d+e x)^{15}}+\frac {b (11 b B d+5 A b e-16 a B e) (a+b x)^{11}}{840 e (b d-a e)^3 (d+e x)^{14}}+\frac {\left (b^2 (11 b B d+5 A b e-16 a B e)\right ) \int \frac {(a+b x)^{10}}{(d+e x)^{14}} \, dx}{280 e (b d-a e)^3} \\ & = -\frac {(B d-A e) (a+b x)^{11}}{16 e (b d-a e) (d+e x)^{16}}+\frac {(11 b B d+5 A b e-16 a B e) (a+b x)^{11}}{240 e (b d-a e)^2 (d+e x)^{15}}+\frac {b (11 b B d+5 A b e-16 a B e) (a+b x)^{11}}{840 e (b d-a e)^3 (d+e x)^{14}}+\frac {b^2 (11 b B d+5 A b e-16 a B e) (a+b x)^{11}}{3640 e (b d-a e)^4 (d+e x)^{13}}+\frac {\left (b^3 (11 b B d+5 A b e-16 a B e)\right ) \int \frac {(a+b x)^{10}}{(d+e x)^{13}} \, dx}{1820 e (b d-a e)^4} \\ & = -\frac {(B d-A e) (a+b x)^{11}}{16 e (b d-a e) (d+e x)^{16}}+\frac {(11 b B d+5 A b e-16 a B e) (a+b x)^{11}}{240 e (b d-a e)^2 (d+e x)^{15}}+\frac {b (11 b B d+5 A b e-16 a B e) (a+b x)^{11}}{840 e (b d-a e)^3 (d+e x)^{14}}+\frac {b^2 (11 b B d+5 A b e-16 a B e) (a+b x)^{11}}{3640 e (b d-a e)^4 (d+e x)^{13}}+\frac {b^3 (11 b B d+5 A b e-16 a B e) (a+b x)^{11}}{21840 e (b d-a e)^5 (d+e x)^{12}}+\frac {\left (b^4 (11 b B d+5 A b e-16 a B e)\right ) \int \frac {(a+b x)^{10}}{(d+e x)^{12}} \, dx}{21840 e (b d-a e)^5} \\ & = -\frac {(B d-A e) (a+b x)^{11}}{16 e (b d-a e) (d+e x)^{16}}+\frac {(11 b B d+5 A b e-16 a B e) (a+b x)^{11}}{240 e (b d-a e)^2 (d+e x)^{15}}+\frac {b (11 b B d+5 A b e-16 a B e) (a+b x)^{11}}{840 e (b d-a e)^3 (d+e x)^{14}}+\frac {b^2 (11 b B d+5 A b e-16 a B e) (a+b x)^{11}}{3640 e (b d-a e)^4 (d+e x)^{13}}+\frac {b^3 (11 b B d+5 A b e-16 a B e) (a+b x)^{11}}{21840 e (b d-a e)^5 (d+e x)^{12}}+\frac {b^4 (11 b B d+5 A b e-16 a B e) (a+b x)^{11}}{240240 e (b d-a e)^6 (d+e x)^{11}} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1429\) vs. \(2(285)=570\).

Time = 0.47 (sec) , antiderivative size = 1429, normalized size of antiderivative = 5.01 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{17}} \, dx=-\frac {1001 a^{10} e^{10} (15 A e+B (d+16 e x))+1430 a^9 b e^9 \left (7 A e (d+16 e x)+B \left (d^2+16 d e x+120 e^2 x^2\right )\right )+495 a^8 b^2 e^8 \left (13 A e \left (d^2+16 d e x+120 e^2 x^2\right )+3 B \left (d^3+16 d^2 e x+120 d e^2 x^2+560 e^3 x^3\right )\right )+1320 a^7 b^3 e^7 \left (3 A e \left (d^3+16 d^2 e x+120 d e^2 x^2+560 e^3 x^3\right )+B \left (d^4+16 d^3 e x+120 d^2 e^2 x^2+560 d e^3 x^3+1820 e^4 x^4\right )\right )+210 a^6 b^4 e^6 \left (11 A e \left (d^4+16 d^3 e x+120 d^2 e^2 x^2+560 d e^3 x^3+1820 e^4 x^4\right )+5 B \left (d^5+16 d^4 e x+120 d^3 e^2 x^2+560 d^2 e^3 x^3+1820 d e^4 x^4+4368 e^5 x^5\right )\right )+252 a^5 b^5 e^5 \left (5 A e \left (d^5+16 d^4 e x+120 d^3 e^2 x^2+560 d^2 e^3 x^3+1820 d e^4 x^4+4368 e^5 x^5\right )+3 B \left (d^6+16 d^5 e x+120 d^4 e^2 x^2+560 d^3 e^3 x^3+1820 d^2 e^4 x^4+4368 d e^5 x^5+8008 e^6 x^6\right )\right )+70 a^4 b^6 e^4 \left (9 A e \left (d^6+16 d^5 e x+120 d^4 e^2 x^2+560 d^3 e^3 x^3+1820 d^2 e^4 x^4+4368 d e^5 x^5+8008 e^6 x^6\right )+7 B \left (d^7+16 d^6 e x+120 d^5 e^2 x^2+560 d^4 e^3 x^3+1820 d^3 e^4 x^4+4368 d^2 e^5 x^5+8008 d e^6 x^6+11440 e^7 x^7\right )\right )+280 a^3 b^7 e^3 \left (A e \left (d^7+16 d^6 e x+120 d^5 e^2 x^2+560 d^4 e^3 x^3+1820 d^3 e^4 x^4+4368 d^2 e^5 x^5+8008 d e^6 x^6+11440 e^7 x^7\right )+B \left (d^8+16 d^7 e x+120 d^6 e^2 x^2+560 d^5 e^3 x^3+1820 d^4 e^4 x^4+4368 d^3 e^5 x^5+8008 d^2 e^6 x^6+11440 d e^7 x^7+12870 e^8 x^8\right )\right )+15 a^2 b^8 e^2 \left (7 A e \left (d^8+16 d^7 e x+120 d^6 e^2 x^2+560 d^5 e^3 x^3+1820 d^4 e^4 x^4+4368 d^3 e^5 x^5+8008 d^2 e^6 x^6+11440 d e^7 x^7+12870 e^8 x^8\right )+9 B \left (d^9+16 d^8 e x+120 d^7 e^2 x^2+560 d^6 e^3 x^3+1820 d^5 e^4 x^4+4368 d^4 e^5 x^5+8008 d^3 e^6 x^6+11440 d^2 e^7 x^7+12870 d e^8 x^8+11440 e^9 x^9\right )\right )+10 a b^9 e \left (3 A e \left (d^9+16 d^8 e x+120 d^7 e^2 x^2+560 d^6 e^3 x^3+1820 d^5 e^4 x^4+4368 d^4 e^5 x^5+8008 d^3 e^6 x^6+11440 d^2 e^7 x^7+12870 d e^8 x^8+11440 e^9 x^9\right )+5 B \left (d^{10}+16 d^9 e x+120 d^8 e^2 x^2+560 d^7 e^3 x^3+1820 d^6 e^4 x^4+4368 d^5 e^5 x^5+8008 d^4 e^6 x^6+11440 d^3 e^7 x^7+12870 d^2 e^8 x^8+11440 d e^9 x^9+8008 e^{10} x^{10}\right )\right )+b^{10} \left (5 A e \left (d^{10}+16 d^9 e x+120 d^8 e^2 x^2+560 d^7 e^3 x^3+1820 d^6 e^4 x^4+4368 d^5 e^5 x^5+8008 d^4 e^6 x^6+11440 d^3 e^7 x^7+12870 d^2 e^8 x^8+11440 d e^9 x^9+8008 e^{10} x^{10}\right )+11 B \left (d^{11}+16 d^{10} e x+120 d^9 e^2 x^2+560 d^8 e^3 x^3+1820 d^7 e^4 x^4+4368 d^6 e^5 x^5+8008 d^5 e^6 x^6+11440 d^4 e^7 x^7+12870 d^3 e^8 x^8+11440 d^2 e^9 x^9+8008 d e^{10} x^{10}+4368 e^{11} x^{11}\right )\right )}{240240 e^{12} (d+e x)^{16}} \]

[In]

Integrate[((a + b*x)^10*(A + B*x))/(d + e*x)^17,x]

[Out]

-1/240240*(1001*a^10*e^10*(15*A*e + B*(d + 16*e*x)) + 1430*a^9*b*e^9*(7*A*e*(d + 16*e*x) + B*(d^2 + 16*d*e*x +
120*e^2*x^2)) + 495*a^8*b^2*e^8*(13*A*e*(d^2 + 16*d*e*x + 120*e^2*x^2) + 3*B*(d^3 + 16*d^2*e*x + 120*d*e^2*x^
2 + 560*e^3*x^3)) + 1320*a^7*b^3*e^7*(3*A*e*(d^3 + 16*d^2*e*x + 120*d*e^2*x^2 + 560*e^3*x^3) + B*(d^4 + 16*d^3
*e*x + 120*d^2*e^2*x^2 + 560*d*e^3*x^3 + 1820*e^4*x^4)) + 210*a^6*b^4*e^6*(11*A*e*(d^4 + 16*d^3*e*x + 120*d^2*
e^2*x^2 + 560*d*e^3*x^3 + 1820*e^4*x^4) + 5*B*(d^5 + 16*d^4*e*x + 120*d^3*e^2*x^2 + 560*d^2*e^3*x^3 + 1820*d*e
^4*x^4 + 4368*e^5*x^5)) + 252*a^5*b^5*e^5*(5*A*e*(d^5 + 16*d^4*e*x + 120*d^3*e^2*x^2 + 560*d^2*e^3*x^3 + 1820*
d*e^4*x^4 + 4368*e^5*x^5) + 3*B*(d^6 + 16*d^5*e*x + 120*d^4*e^2*x^2 + 560*d^3*e^3*x^3 + 1820*d^2*e^4*x^4 + 436
8*d*e^5*x^5 + 8008*e^6*x^6)) + 70*a^4*b^6*e^4*(9*A*e*(d^6 + 16*d^5*e*x + 120*d^4*e^2*x^2 + 560*d^3*e^3*x^3 + 1
820*d^2*e^4*x^4 + 4368*d*e^5*x^5 + 8008*e^6*x^6) + 7*B*(d^7 + 16*d^6*e*x + 120*d^5*e^2*x^2 + 560*d^4*e^3*x^3 +
1820*d^3*e^4*x^4 + 4368*d^2*e^5*x^5 + 8008*d*e^6*x^6 + 11440*e^7*x^7)) + 280*a^3*b^7*e^3*(A*e*(d^7 + 16*d^6*e
*x + 120*d^5*e^2*x^2 + 560*d^4*e^3*x^3 + 1820*d^3*e^4*x^4 + 4368*d^2*e^5*x^5 + 8008*d*e^6*x^6 + 11440*e^7*x^7)
+ B*(d^8 + 16*d^7*e*x + 120*d^6*e^2*x^2 + 560*d^5*e^3*x^3 + 1820*d^4*e^4*x^4 + 4368*d^3*e^5*x^5 + 8008*d^2*e^
6*x^6 + 11440*d*e^7*x^7 + 12870*e^8*x^8)) + 15*a^2*b^8*e^2*(7*A*e*(d^8 + 16*d^7*e*x + 120*d^6*e^2*x^2 + 560*d^
5*e^3*x^3 + 1820*d^4*e^4*x^4 + 4368*d^3*e^5*x^5 + 8008*d^2*e^6*x^6 + 11440*d*e^7*x^7 + 12870*e^8*x^8) + 9*B*(d
^9 + 16*d^8*e*x + 120*d^7*e^2*x^2 + 560*d^6*e^3*x^3 + 1820*d^5*e^4*x^4 + 4368*d^4*e^5*x^5 + 8008*d^3*e^6*x^6 +
11440*d^2*e^7*x^7 + 12870*d*e^8*x^8 + 11440*e^9*x^9)) + 10*a*b^9*e*(3*A*e*(d^9 + 16*d^8*e*x + 120*d^7*e^2*x^2
+ 560*d^6*e^3*x^3 + 1820*d^5*e^4*x^4 + 4368*d^4*e^5*x^5 + 8008*d^3*e^6*x^6 + 11440*d^2*e^7*x^7 + 12870*d*e^8*
x^8 + 11440*e^9*x^9) + 5*B*(d^10 + 16*d^9*e*x + 120*d^8*e^2*x^2 + 560*d^7*e^3*x^3 + 1820*d^6*e^4*x^4 + 4368*d^
5*e^5*x^5 + 8008*d^4*e^6*x^6 + 11440*d^3*e^7*x^7 + 12870*d^2*e^8*x^8 + 11440*d*e^9*x^9 + 8008*e^10*x^10)) + b^
10*(5*A*e*(d^10 + 16*d^9*e*x + 120*d^8*e^2*x^2 + 560*d^7*e^3*x^3 + 1820*d^6*e^4*x^4 + 4368*d^5*e^5*x^5 + 8008*
d^4*e^6*x^6 + 11440*d^3*e^7*x^7 + 12870*d^2*e^8*x^8 + 11440*d*e^9*x^9 + 8008*e^10*x^10) + 11*B*(d^11 + 16*d^10
*e*x + 120*d^9*e^2*x^2 + 560*d^8*e^3*x^3 + 1820*d^7*e^4*x^4 + 4368*d^6*e^5*x^5 + 8008*d^5*e^6*x^6 + 11440*d^4*
e^7*x^7 + 12870*d^3*e^8*x^8 + 11440*d^2*e^9*x^9 + 8008*d*e^10*x^10 + 4368*e^11*x^11)))/(e^12*(d + e*x)^16)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1900\) vs. \(2(273)=546\).

Time = 2.13 (sec) , antiderivative size = 1901, normalized size of antiderivative = 6.67

method	result	size
risch	\(\text {Expression too large to display}\)	\(1901\)
default	\(\text {Expression too large to display}\)	\(1942\)
norman	\(\text {Expression too large to display}\)	\(2014\)
gosper	\(\text {Expression too large to display}\)	\(2233\)
parallelrisch	\(\text {Expression too large to display}\)	\(2242\)

[In]

int((b*x+a)^10*(B*x+A)/(e*x+d)^17,x,method=_RETURNVERBOSE)

[Out]

(-1/240240/e^12*(15015*A*a^10*e^11+10010*A*a^9*b*d*e^10+6435*A*a^8*b^2*d^2*e^9+3960*A*a^7*b^3*d^3*e^8+2310*A*a
^6*b^4*d^4*e^7+1260*A*a^5*b^5*d^5*e^6+630*A*a^4*b^6*d^6*e^5+280*A*a^3*b^7*d^7*e^4+105*A*a^2*b^8*d^8*e^3+30*A*a
*b^9*d^9*e^2+5*A*b^10*d^10*e+1001*B*a^10*d*e^10+1430*B*a^9*b*d^2*e^9+1485*B*a^8*b^2*d^3*e^8+1320*B*a^7*b^3*d^4
*e^7+1050*B*a^6*b^4*d^5*e^6+756*B*a^5*b^5*d^6*e^5+490*B*a^4*b^6*d^7*e^4+280*B*a^3*b^7*d^8*e^3+135*B*a^2*b^8*d^
9*e^2+50*B*a*b^9*d^10*e+11*B*b^10*d^11)-1/15015/e^11*(10010*A*a^9*b*e^10+6435*A*a^8*b^2*d*e^9+3960*A*a^7*b^3*d
^2*e^8+2310*A*a^6*b^4*d^3*e^7+1260*A*a^5*b^5*d^4*e^6+630*A*a^4*b^6*d^5*e^5+280*A*a^3*b^7*d^6*e^4+105*A*a^2*b^8
*d^7*e^3+30*A*a*b^9*d^8*e^2+5*A*b^10*d^9*e+1001*B*a^10*e^10+1430*B*a^9*b*d*e^9+1485*B*a^8*b^2*d^2*e^8+1320*B*a
^7*b^3*d^3*e^7+1050*B*a^6*b^4*d^4*e^6+756*B*a^5*b^5*d^5*e^5+490*B*a^4*b^6*d^6*e^4+280*B*a^3*b^7*d^7*e^3+135*B*
a^2*b^8*d^8*e^2+50*B*a*b^9*d^9*e+11*B*b^10*d^10)*x-1/2002*b/e^10*(6435*A*a^8*b*e^9+3960*A*a^7*b^2*d*e^8+2310*A
*a^6*b^3*d^2*e^7+1260*A*a^5*b^4*d^3*e^6+630*A*a^4*b^5*d^4*e^5+280*A*a^3*b^6*d^5*e^4+105*A*a^2*b^7*d^6*e^3+30*A
*a*b^8*d^7*e^2+5*A*b^9*d^8*e+1430*B*a^9*e^9+1485*B*a^8*b*d*e^8+1320*B*a^7*b^2*d^2*e^7+1050*B*a^6*b^3*d^3*e^6+7
56*B*a^5*b^4*d^4*e^5+490*B*a^4*b^5*d^5*e^4+280*B*a^3*b^6*d^6*e^3+135*B*a^2*b^7*d^7*e^2+50*B*a*b^8*d^8*e+11*B*b
^9*d^9)*x^2-1/429*b^2/e^9*(3960*A*a^7*b*e^8+2310*A*a^6*b^2*d*e^7+1260*A*a^5*b^3*d^2*e^6+630*A*a^4*b^4*d^3*e^5+
280*A*a^3*b^5*d^4*e^4+105*A*a^2*b^6*d^5*e^3+30*A*a*b^7*d^6*e^2+5*A*b^8*d^7*e+1485*B*a^8*e^8+1320*B*a^7*b*d*e^7
+1050*B*a^6*b^2*d^2*e^6+756*B*a^5*b^3*d^3*e^5+490*B*a^4*b^4*d^4*e^4+280*B*a^3*b^5*d^5*e^3+135*B*a^2*b^6*d^6*e^
2+50*B*a*b^7*d^7*e+11*B*b^8*d^8)*x^3-1/132*b^3/e^8*(2310*A*a^6*b*e^7+1260*A*a^5*b^2*d*e^6+630*A*a^4*b^3*d^2*e^
5+280*A*a^3*b^4*d^3*e^4+105*A*a^2*b^5*d^4*e^3+30*A*a*b^6*d^5*e^2+5*A*b^7*d^6*e+1320*B*a^7*e^7+1050*B*a^6*b*d*e
^6+756*B*a^5*b^2*d^2*e^5+490*B*a^4*b^3*d^3*e^4+280*B*a^3*b^4*d^4*e^3+135*B*a^2*b^5*d^5*e^2+50*B*a*b^6*d^6*e+11
*B*b^7*d^7)*x^4-1/55*b^4/e^7*(1260*A*a^5*b*e^6+630*A*a^4*b^2*d*e^5+280*A*a^3*b^3*d^2*e^4+105*A*a^2*b^4*d^3*e^3
+30*A*a*b^5*d^4*e^2+5*A*b^6*d^5*e+1050*B*a^6*e^6+756*B*a^5*b*d*e^5+490*B*a^4*b^2*d^2*e^4+280*B*a^3*b^3*d^3*e^3
+135*B*a^2*b^4*d^4*e^2+50*B*a*b^5*d^5*e+11*B*b^6*d^6)*x^5-1/30*b^5/e^6*(630*A*a^4*b*e^5+280*A*a^3*b^2*d*e^4+10
5*A*a^2*b^3*d^2*e^3+30*A*a*b^4*d^3*e^2+5*A*b^5*d^4*e+756*B*a^5*e^5+490*B*a^4*b*d*e^4+280*B*a^3*b^2*d^2*e^3+135
*B*a^2*b^3*d^3*e^2+50*B*a*b^4*d^4*e+11*B*b^5*d^5)*x^6-1/21*b^6/e^5*(280*A*a^3*b*e^4+105*A*a^2*b^2*d*e^3+30*A*a
*b^3*d^2*e^2+5*A*b^4*d^3*e+490*B*a^4*e^4+280*B*a^3*b*d*e^3+135*B*a^2*b^2*d^2*e^2+50*B*a*b^3*d^3*e+11*B*b^4*d^4
)*x^7-3/56*b^7/e^4*(105*A*a^2*b*e^3+30*A*a*b^2*d*e^2+5*A*b^3*d^2*e+280*B*a^3*e^3+135*B*a^2*b*d*e^2+50*B*a*b^2*
d^2*e+11*B*b^3*d^3)*x^8-1/21*b^8/e^3*(30*A*a*b*e^2+5*A*b^2*d*e+135*B*a^2*e^2+50*B*a*b*d*e+11*B*b^2*d^2)*x^9-1/
30*b^9/e^2*(5*A*b*e+50*B*a*e+11*B*b*d)*x^10-1/5*b^10*B/e*x^11)/(e*x+d)^16

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1984 vs. \(2 (273) = 546\).

Time = 0.31 (sec) , antiderivative size = 1984, normalized size of antiderivative = 6.96 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{17}} \, dx=\text {Too large to display} \]

[In]

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^17,x, algorithm="fricas")

[Out]

-1/240240*(48048*B*b^10*e^11*x^11 + 11*B*b^10*d^11 + 15015*A*a^10*e^11 + 5*(10*B*a*b^9 + A*b^10)*d^10*e + 15*(
9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 + 35*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 70*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7
*e^4 + 126*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 210*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 + 330*(4*B*a^7*b^3 +
7*A*a^6*b^4)*d^4*e^7 + 495*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^8 + 715*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9 + 1001*
(B*a^10 + 10*A*a^9*b)*d*e^10 + 8008*(11*B*b^10*d*e^10 + 5*(10*B*a*b^9 + A*b^10)*e^11)*x^10 + 11440*(11*B*b^10*
d^2*e^9 + 5*(10*B*a*b^9 + A*b^10)*d*e^10 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9 + 12870*(11*B*b^10*d^3*e^8 +
5*(10*B*a*b^9 + A*b^10)*d^2*e^9 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^10 + 35*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^11)*
x^8 + 11440*(11*B*b^10*d^4*e^7 + 5*(10*B*a*b^9 + A*b^10)*d^3*e^8 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^9 + 35*(
8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^10 + 70*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 + 8008*(11*B*b^10*d^5*e^6 + 5*(10
*B*a*b^9 + A*b^10)*d^4*e^7 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^8 + 35*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^9 + 7
0*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^10 + 126*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 4368*(11*B*b^10*d^6*e^5 + 5
*(10*B*a*b^9 + A*b^10)*d^5*e^6 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 + 35*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8
+ 70*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^9 + 126*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 + 210*(5*B*a^6*b^4 + 6*A*a^
5*b^5)*e^11)*x^5 + 1820*(11*B*b^10*d^7*e^4 + 5*(10*B*a*b^9 + A*b^10)*d^6*e^5 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d^
5*e^6 + 35*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 + 70*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^8 + 126*(6*B*a^5*b^5 + 5
*A*a^4*b^6)*d^2*e^9 + 210*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^10 + 330*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^11)*x^4 + 560
*(11*B*b^10*d^8*e^3 + 5*(10*B*a*b^9 + A*b^10)*d^7*e^4 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5 + 35*(8*B*a^3*b^7
+ 3*A*a^2*b^8)*d^5*e^6 + 70*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^7 + 126*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^8 + 2
10*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*e^9 + 330*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*e^10 + 495*(3*B*a^8*b^2 + 8*A*a^7*b
^3)*e^11)*x^3 + 120*(11*B*b^10*d^9*e^2 + 5*(10*B*a*b^9 + A*b^10)*d^8*e^3 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^
4 + 35*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^5 + 70*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*e^6 + 126*(6*B*a^5*b^5 + 5*A*a
^4*b^6)*d^4*e^7 + 210*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*e^8 + 330*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*e^9 + 495*(3*B
*a^8*b^2 + 8*A*a^7*b^3)*d*e^10 + 715*(2*B*a^9*b + 9*A*a^8*b^2)*e^11)*x^2 + 16*(11*B*b^10*d^10*e + 5*(10*B*a*b^
9 + A*b^10)*d^9*e^2 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^3 + 35*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4 + 70*(7*B*
a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5 + 126*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 + 210*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4
*e^7 + 330*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*e^8 + 495*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e^9 + 715*(2*B*a^9*b + 9*
A*a^8*b^2)*d*e^10 + 1001*(B*a^10 + 10*A*a^9*b)*e^11)*x)/(e^28*x^16 + 16*d*e^27*x^15 + 120*d^2*e^26*x^14 + 560*
d^3*e^25*x^13 + 1820*d^4*e^24*x^12 + 4368*d^5*e^23*x^11 + 8008*d^6*e^22*x^10 + 11440*d^7*e^21*x^9 + 12870*d^8*
e^20*x^8 + 11440*d^9*e^19*x^7 + 8008*d^10*e^18*x^6 + 4368*d^11*e^17*x^5 + 1820*d^12*e^16*x^4 + 560*d^13*e^15*x
^3 + 120*d^14*e^14*x^2 + 16*d^15*e^13*x + d^16*e^12)

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{17}} \, dx=\text {Timed out} \]

[In]

integrate((b*x+a)**10*(B*x+A)/(e*x+d)**17,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1984 vs. \(2 (273) = 546\).

Time = 0.31 (sec) , antiderivative size = 1984, normalized size of antiderivative = 6.96 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{17}} \, dx=\text {Too large to display} \]

[In]

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^17,x, algorithm="maxima")

[Out]

-1/240240*(48048*B*b^10*e^11*x^11 + 11*B*b^10*d^11 + 15015*A*a^10*e^11 + 5*(10*B*a*b^9 + A*b^10)*d^10*e + 15*(
9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 + 35*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 70*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7
*e^4 + 126*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 210*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 + 330*(4*B*a^7*b^3 +
7*A*a^6*b^4)*d^4*e^7 + 495*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^8 + 715*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9 + 1001*
(B*a^10 + 10*A*a^9*b)*d*e^10 + 8008*(11*B*b^10*d*e^10 + 5*(10*B*a*b^9 + A*b^10)*e^11)*x^10 + 11440*(11*B*b^10*
d^2*e^9 + 5*(10*B*a*b^9 + A*b^10)*d*e^10 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9 + 12870*(11*B*b^10*d^3*e^8 +
5*(10*B*a*b^9 + A*b^10)*d^2*e^9 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^10 + 35*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^11)*
x^8 + 11440*(11*B*b^10*d^4*e^7 + 5*(10*B*a*b^9 + A*b^10)*d^3*e^8 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^9 + 35*(
8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^10 + 70*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 + 8008*(11*B*b^10*d^5*e^6 + 5*(10
*B*a*b^9 + A*b^10)*d^4*e^7 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^8 + 35*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^9 + 7
0*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^10 + 126*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 4368*(11*B*b^10*d^6*e^5 + 5
*(10*B*a*b^9 + A*b^10)*d^5*e^6 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 + 35*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8
+ 70*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^9 + 126*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 + 210*(5*B*a^6*b^4 + 6*A*a^
5*b^5)*e^11)*x^5 + 1820*(11*B*b^10*d^7*e^4 + 5*(10*B*a*b^9 + A*b^10)*d^6*e^5 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d^
5*e^6 + 35*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 + 70*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^8 + 126*(6*B*a^5*b^5 + 5
*A*a^4*b^6)*d^2*e^9 + 210*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^10 + 330*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^11)*x^4 + 560
*(11*B*b^10*d^8*e^3 + 5*(10*B*a*b^9 + A*b^10)*d^7*e^4 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5 + 35*(8*B*a^3*b^7
+ 3*A*a^2*b^8)*d^5*e^6 + 70*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^7 + 126*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^8 + 2
10*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*e^9 + 330*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*e^10 + 495*(3*B*a^8*b^2 + 8*A*a^7*b
^3)*e^11)*x^3 + 120*(11*B*b^10*d^9*e^2 + 5*(10*B*a*b^9 + A*b^10)*d^8*e^3 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^
4 + 35*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^5 + 70*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*e^6 + 126*(6*B*a^5*b^5 + 5*A*a
^4*b^6)*d^4*e^7 + 210*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*e^8 + 330*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*e^9 + 495*(3*B
*a^8*b^2 + 8*A*a^7*b^3)*d*e^10 + 715*(2*B*a^9*b + 9*A*a^8*b^2)*e^11)*x^2 + 16*(11*B*b^10*d^10*e + 5*(10*B*a*b^
9 + A*b^10)*d^9*e^2 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^3 + 35*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4 + 70*(7*B*
a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5 + 126*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 + 210*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4
*e^7 + 330*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*e^8 + 495*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e^9 + 715*(2*B*a^9*b + 9*
A*a^8*b^2)*d*e^10 + 1001*(B*a^10 + 10*A*a^9*b)*e^11)*x)/(e^28*x^16 + 16*d*e^27*x^15 + 120*d^2*e^26*x^14 + 560*
d^3*e^25*x^13 + 1820*d^4*e^24*x^12 + 4368*d^5*e^23*x^11 + 8008*d^6*e^22*x^10 + 11440*d^7*e^21*x^9 + 12870*d^8*
e^20*x^8 + 11440*d^9*e^19*x^7 + 8008*d^10*e^18*x^6 + 4368*d^11*e^17*x^5 + 1820*d^12*e^16*x^4 + 560*d^13*e^15*x
^3 + 120*d^14*e^14*x^2 + 16*d^15*e^13*x + d^16*e^12)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2232 vs. \(2 (273) = 546\).

Time = 0.29 (sec) , antiderivative size = 2232, normalized size of antiderivative = 7.83 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{17}} \, dx=\text {Too large to display} \]

[In]

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^17,x, algorithm="giac")

[Out]

-1/240240*(48048*B*b^10*e^11*x^11 + 88088*B*b^10*d*e^10*x^10 + 400400*B*a*b^9*e^11*x^10 + 40040*A*b^10*e^11*x^
10 + 125840*B*b^10*d^2*e^9*x^9 + 572000*B*a*b^9*d*e^10*x^9 + 57200*A*b^10*d*e^10*x^9 + 1544400*B*a^2*b^8*e^11*
x^9 + 343200*A*a*b^9*e^11*x^9 + 141570*B*b^10*d^3*e^8*x^8 + 643500*B*a*b^9*d^2*e^9*x^8 + 64350*A*b^10*d^2*e^9*
x^8 + 1737450*B*a^2*b^8*d*e^10*x^8 + 386100*A*a*b^9*d*e^10*x^8 + 3603600*B*a^3*b^7*e^11*x^8 + 1351350*A*a^2*b^
8*e^11*x^8 + 125840*B*b^10*d^4*e^7*x^7 + 572000*B*a*b^9*d^3*e^8*x^7 + 57200*A*b^10*d^3*e^8*x^7 + 1544400*B*a^2
*b^8*d^2*e^9*x^7 + 343200*A*a*b^9*d^2*e^9*x^7 + 3203200*B*a^3*b^7*d*e^10*x^7 + 1201200*A*a^2*b^8*d*e^10*x^7 +
5605600*B*a^4*b^6*e^11*x^7 + 3203200*A*a^3*b^7*e^11*x^7 + 88088*B*b^10*d^5*e^6*x^6 + 400400*B*a*b^9*d^4*e^7*x^
6 + 40040*A*b^10*d^4*e^7*x^6 + 1081080*B*a^2*b^8*d^3*e^8*x^6 + 240240*A*a*b^9*d^3*e^8*x^6 + 2242240*B*a^3*b^7*
d^2*e^9*x^6 + 840840*A*a^2*b^8*d^2*e^9*x^6 + 3923920*B*a^4*b^6*d*e^10*x^6 + 2242240*A*a^3*b^7*d*e^10*x^6 + 605
4048*B*a^5*b^5*e^11*x^6 + 5045040*A*a^4*b^6*e^11*x^6 + 48048*B*b^10*d^6*e^5*x^5 + 218400*B*a*b^9*d^5*e^6*x^5 +
21840*A*b^10*d^5*e^6*x^5 + 589680*B*a^2*b^8*d^4*e^7*x^5 + 131040*A*a*b^9*d^4*e^7*x^5 + 1223040*B*a^3*b^7*d^3*
e^8*x^5 + 458640*A*a^2*b^8*d^3*e^8*x^5 + 2140320*B*a^4*b^6*d^2*e^9*x^5 + 1223040*A*a^3*b^7*d^2*e^9*x^5 + 33022
08*B*a^5*b^5*d*e^10*x^5 + 2751840*A*a^4*b^6*d*e^10*x^5 + 4586400*B*a^6*b^4*e^11*x^5 + 5503680*A*a^5*b^5*e^11*x
^5 + 20020*B*b^10*d^7*e^4*x^4 + 91000*B*a*b^9*d^6*e^5*x^4 + 9100*A*b^10*d^6*e^5*x^4 + 245700*B*a^2*b^8*d^5*e^6
*x^4 + 54600*A*a*b^9*d^5*e^6*x^4 + 509600*B*a^3*b^7*d^4*e^7*x^4 + 191100*A*a^2*b^8*d^4*e^7*x^4 + 891800*B*a^4*
b^6*d^3*e^8*x^4 + 509600*A*a^3*b^7*d^3*e^8*x^4 + 1375920*B*a^5*b^5*d^2*e^9*x^4 + 1146600*A*a^4*b^6*d^2*e^9*x^4
+ 1911000*B*a^6*b^4*d*e^10*x^4 + 2293200*A*a^5*b^5*d*e^10*x^4 + 2402400*B*a^7*b^3*e^11*x^4 + 4204200*A*a^6*b^
4*e^11*x^4 + 6160*B*b^10*d^8*e^3*x^3 + 28000*B*a*b^9*d^7*e^4*x^3 + 2800*A*b^10*d^7*e^4*x^3 + 75600*B*a^2*b^8*d
^6*e^5*x^3 + 16800*A*a*b^9*d^6*e^5*x^3 + 156800*B*a^3*b^7*d^5*e^6*x^3 + 58800*A*a^2*b^8*d^5*e^6*x^3 + 274400*B
*a^4*b^6*d^4*e^7*x^3 + 156800*A*a^3*b^7*d^4*e^7*x^3 + 423360*B*a^5*b^5*d^3*e^8*x^3 + 352800*A*a^4*b^6*d^3*e^8*
x^3 + 588000*B*a^6*b^4*d^2*e^9*x^3 + 705600*A*a^5*b^5*d^2*e^9*x^3 + 739200*B*a^7*b^3*d*e^10*x^3 + 1293600*A*a^
6*b^4*d*e^10*x^3 + 831600*B*a^8*b^2*e^11*x^3 + 2217600*A*a^7*b^3*e^11*x^3 + 1320*B*b^10*d^9*e^2*x^2 + 6000*B*a
*b^9*d^8*e^3*x^2 + 600*A*b^10*d^8*e^3*x^2 + 16200*B*a^2*b^8*d^7*e^4*x^2 + 3600*A*a*b^9*d^7*e^4*x^2 + 33600*B*a
^3*b^7*d^6*e^5*x^2 + 12600*A*a^2*b^8*d^6*e^5*x^2 + 58800*B*a^4*b^6*d^5*e^6*x^2 + 33600*A*a^3*b^7*d^5*e^6*x^2 +
90720*B*a^5*b^5*d^4*e^7*x^2 + 75600*A*a^4*b^6*d^4*e^7*x^2 + 126000*B*a^6*b^4*d^3*e^8*x^2 + 151200*A*a^5*b^5*d
^3*e^8*x^2 + 158400*B*a^7*b^3*d^2*e^9*x^2 + 277200*A*a^6*b^4*d^2*e^9*x^2 + 178200*B*a^8*b^2*d*e^10*x^2 + 47520
0*A*a^7*b^3*d*e^10*x^2 + 171600*B*a^9*b*e^11*x^2 + 772200*A*a^8*b^2*e^11*x^2 + 176*B*b^10*d^10*e*x + 800*B*a*b
^9*d^9*e^2*x + 80*A*b^10*d^9*e^2*x + 2160*B*a^2*b^8*d^8*e^3*x + 480*A*a*b^9*d^8*e^3*x + 4480*B*a^3*b^7*d^7*e^4
*x + 1680*A*a^2*b^8*d^7*e^4*x + 7840*B*a^4*b^6*d^6*e^5*x + 4480*A*a^3*b^7*d^6*e^5*x + 12096*B*a^5*b^5*d^5*e^6*
x + 10080*A*a^4*b^6*d^5*e^6*x + 16800*B*a^6*b^4*d^4*e^7*x + 20160*A*a^5*b^5*d^4*e^7*x + 21120*B*a^7*b^3*d^3*e^
8*x + 36960*A*a^6*b^4*d^3*e^8*x + 23760*B*a^8*b^2*d^2*e^9*x + 63360*A*a^7*b^3*d^2*e^9*x + 22880*B*a^9*b*d*e^10
*x + 102960*A*a^8*b^2*d*e^10*x + 16016*B*a^10*e^11*x + 160160*A*a^9*b*e^11*x + 11*B*b^10*d^11 + 50*B*a*b^9*d^1
0*e + 5*A*b^10*d^10*e + 135*B*a^2*b^8*d^9*e^2 + 30*A*a*b^9*d^9*e^2 + 280*B*a^3*b^7*d^8*e^3 + 105*A*a^2*b^8*d^8
*e^3 + 490*B*a^4*b^6*d^7*e^4 + 280*A*a^3*b^7*d^7*e^4 + 756*B*a^5*b^5*d^6*e^5 + 630*A*a^4*b^6*d^6*e^5 + 1050*B*
a^6*b^4*d^5*e^6 + 1260*A*a^5*b^5*d^5*e^6 + 1320*B*a^7*b^3*d^4*e^7 + 2310*A*a^6*b^4*d^4*e^7 + 1485*B*a^8*b^2*d^
3*e^8 + 3960*A*a^7*b^3*d^3*e^8 + 1430*B*a^9*b*d^2*e^9 + 6435*A*a^8*b^2*d^2*e^9 + 1001*B*a^10*d*e^10 + 10010*A*
a^9*b*d*e^10 + 15015*A*a^10*e^11)/((e*x + d)^16*e^12)

Mupad [B] (verification not implemented)

Time = 0.65 (sec) , antiderivative size = 2066, normalized size of antiderivative = 7.25 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{17}} \, dx=\text {Too large to display} \]

[In]

int(((A + B*x)*(a + b*x)^10)/(d + e*x)^17,x)

[Out]

-((15015*A*a^10*e^11 + 11*B*b^10*d^11 + 5*A*b^10*d^10*e + 1001*B*a^10*d*e^10 + 30*A*a*b^9*d^9*e^2 + 1430*B*a^9
*b*d^2*e^9 + 105*A*a^2*b^8*d^8*e^3 + 280*A*a^3*b^7*d^7*e^4 + 630*A*a^4*b^6*d^6*e^5 + 1260*A*a^5*b^5*d^5*e^6 +
2310*A*a^6*b^4*d^4*e^7 + 3960*A*a^7*b^3*d^3*e^8 + 6435*A*a^8*b^2*d^2*e^9 + 135*B*a^2*b^8*d^9*e^2 + 280*B*a^3*b
^7*d^8*e^3 + 490*B*a^4*b^6*d^7*e^4 + 756*B*a^5*b^5*d^6*e^5 + 1050*B*a^6*b^4*d^5*e^6 + 1320*B*a^7*b^3*d^4*e^7 +
1485*B*a^8*b^2*d^3*e^8 + 10010*A*a^9*b*d*e^10 + 50*B*a*b^9*d^10*e)/(240240*e^12) + (x*(1001*B*a^10*e^10 + 11*
B*b^10*d^10 + 10010*A*a^9*b*e^10 + 5*A*b^10*d^9*e + 30*A*a*b^9*d^8*e^2 + 6435*A*a^8*b^2*d*e^9 + 105*A*a^2*b^8*
d^7*e^3 + 280*A*a^3*b^7*d^6*e^4 + 630*A*a^4*b^6*d^5*e^5 + 1260*A*a^5*b^5*d^4*e^6 + 2310*A*a^6*b^4*d^3*e^7 + 39
60*A*a^7*b^3*d^2*e^8 + 135*B*a^2*b^8*d^8*e^2 + 280*B*a^3*b^7*d^7*e^3 + 490*B*a^4*b^6*d^6*e^4 + 756*B*a^5*b^5*d
^5*e^5 + 1050*B*a^6*b^4*d^4*e^6 + 1320*B*a^7*b^3*d^3*e^7 + 1485*B*a^8*b^2*d^2*e^8 + 50*B*a*b^9*d^9*e + 1430*B*
a^9*b*d*e^9))/(15015*e^11) + (3*b^7*x^8*(280*B*a^3*e^3 + 11*B*b^3*d^3 + 105*A*a^2*b*e^3 + 5*A*b^3*d^2*e + 30*A
*a*b^2*d*e^2 + 50*B*a*b^2*d^2*e + 135*B*a^2*b*d*e^2))/(56*e^4) + (b^4*x^5*(1050*B*a^6*e^6 + 11*B*b^6*d^6 + 126
0*A*a^5*b*e^6 + 5*A*b^6*d^5*e + 30*A*a*b^5*d^4*e^2 + 630*A*a^4*b^2*d*e^5 + 105*A*a^2*b^4*d^3*e^3 + 280*A*a^3*b
^3*d^2*e^4 + 135*B*a^2*b^4*d^4*e^2 + 280*B*a^3*b^3*d^3*e^3 + 490*B*a^4*b^2*d^2*e^4 + 50*B*a*b^5*d^5*e + 756*B*
a^5*b*d*e^5))/(55*e^7) + (b^9*x^10*(5*A*b*e + 50*B*a*e + 11*B*b*d))/(30*e^2) + (b^6*x^7*(490*B*a^4*e^4 + 11*B*
b^4*d^4 + 280*A*a^3*b*e^4 + 5*A*b^4*d^3*e + 30*A*a*b^3*d^2*e^2 + 105*A*a^2*b^2*d*e^3 + 135*B*a^2*b^2*d^2*e^2 +
50*B*a*b^3*d^3*e + 280*B*a^3*b*d*e^3))/(21*e^5) + (b^3*x^4*(1320*B*a^7*e^7 + 11*B*b^7*d^7 + 2310*A*a^6*b*e^7
+ 5*A*b^7*d^6*e + 30*A*a*b^6*d^5*e^2 + 1260*A*a^5*b^2*d*e^6 + 105*A*a^2*b^5*d^4*e^3 + 280*A*a^3*b^4*d^3*e^4 +
630*A*a^4*b^3*d^2*e^5 + 135*B*a^2*b^5*d^5*e^2 + 280*B*a^3*b^4*d^4*e^3 + 490*B*a^4*b^3*d^3*e^4 + 756*B*a^5*b^2*
d^2*e^5 + 50*B*a*b^6*d^6*e + 1050*B*a^6*b*d*e^6))/(132*e^8) + (b*x^2*(1430*B*a^9*e^9 + 11*B*b^9*d^9 + 6435*A*a
^8*b*e^9 + 5*A*b^9*d^8*e + 30*A*a*b^8*d^7*e^2 + 3960*A*a^7*b^2*d*e^8 + 105*A*a^2*b^7*d^6*e^3 + 280*A*a^3*b^6*d
^5*e^4 + 630*A*a^4*b^5*d^4*e^5 + 1260*A*a^5*b^4*d^3*e^6 + 2310*A*a^6*b^3*d^2*e^7 + 135*B*a^2*b^7*d^7*e^2 + 280
*B*a^3*b^6*d^6*e^3 + 490*B*a^4*b^5*d^5*e^4 + 756*B*a^5*b^4*d^4*e^5 + 1050*B*a^6*b^3*d^3*e^6 + 1320*B*a^7*b^2*d
^2*e^7 + 50*B*a*b^8*d^8*e + 1485*B*a^8*b*d*e^8))/(2002*e^10) + (b^8*x^9*(135*B*a^2*e^2 + 11*B*b^2*d^2 + 30*A*a
*b*e^2 + 5*A*b^2*d*e + 50*B*a*b*d*e))/(21*e^3) + (b^5*x^6*(756*B*a^5*e^5 + 11*B*b^5*d^5 + 630*A*a^4*b*e^5 + 5*
A*b^5*d^4*e + 30*A*a*b^4*d^3*e^2 + 280*A*a^3*b^2*d*e^4 + 105*A*a^2*b^3*d^2*e^3 + 135*B*a^2*b^3*d^3*e^2 + 280*B
*a^3*b^2*d^2*e^3 + 50*B*a*b^4*d^4*e + 490*B*a^4*b*d*e^4))/(30*e^6) + (b^2*x^3*(1485*B*a^8*e^8 + 11*B*b^8*d^8 +
3960*A*a^7*b*e^8 + 5*A*b^8*d^7*e + 30*A*a*b^7*d^6*e^2 + 2310*A*a^6*b^2*d*e^7 + 105*A*a^2*b^6*d^5*e^3 + 280*A*
a^3*b^5*d^4*e^4 + 630*A*a^4*b^4*d^3*e^5 + 1260*A*a^5*b^3*d^2*e^6 + 135*B*a^2*b^6*d^6*e^2 + 280*B*a^3*b^5*d^5*e
^3 + 490*B*a^4*b^4*d^4*e^4 + 756*B*a^5*b^3*d^3*e^5 + 1050*B*a^6*b^2*d^2*e^6 + 50*B*a*b^7*d^7*e + 1320*B*a^7*b*
d*e^7))/(429*e^9) + (B*b^10*x^11)/(5*e))/(d^16 + e^16*x^16 + 16*d*e^15*x^15 + 120*d^14*e^2*x^2 + 560*d^13*e^3*
x^3 + 1820*d^12*e^4*x^4 + 4368*d^11*e^5*x^5 + 8008*d^10*e^6*x^6 + 11440*d^9*e^7*x^7 + 12870*d^8*e^8*x^8 + 1144
0*d^7*e^9*x^9 + 8008*d^6*e^10*x^10 + 4368*d^5*e^11*x^11 + 1820*d^4*e^12*x^12 + 560*d^3*e^13*x^13 + 120*d^2*e^1
4*x^14 + 16*d^15*e*x)