\(\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{18}} \, dx\) [1106]

Optimal result

Integrand size = 20, antiderivative size = 335 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{18}} \, dx=-\frac {(B d-A e) (a+b x)^{11}}{17 e (b d-a e) (d+e x)^{17}}+\frac {(11 b B d+6 A b e-17 a B e) (a+b x)^{11}}{272 e (b d-a e)^2 (d+e x)^{16}}+\frac {b (11 b B d+6 A b e-17 a B e) (a+b x)^{11}}{816 e (b d-a e)^3 (d+e x)^{15}}+\frac {b^2 (11 b B d+6 A b e-17 a B e) (a+b x)^{11}}{2856 e (b d-a e)^4 (d+e x)^{14}}+\frac {b^3 (11 b B d+6 A b e-17 a B e) (a+b x)^{11}}{12376 e (b d-a e)^5 (d+e x)^{13}}+\frac {b^4 (11 b B d+6 A b e-17 a B e) (a+b x)^{11}}{74256 e (b d-a e)^6 (d+e x)^{12}}+\frac {b^5 (11 b B d+6 A b e-17 a B e) (a+b x)^{11}}{816816 e (b d-a e)^7 (d+e x)^{11}} \]

[Out]

-1/17*(-A*e+B*d)*(b*x+a)^11/e/(-a*e+b*d)/(e*x+d)^17+1/272*(6*A*b*e-17*B*a*e+11*B*b*d)*(b*x+a)^11/e/(-a*e+b*d)^
2/(e*x+d)^16+1/816*b*(6*A*b*e-17*B*a*e+11*B*b*d)*(b*x+a)^11/e/(-a*e+b*d)^3/(e*x+d)^15+1/2856*b^2*(6*A*b*e-17*B
*a*e+11*B*b*d)*(b*x+a)^11/e/(-a*e+b*d)^4/(e*x+d)^14+1/12376*b^3*(6*A*b*e-17*B*a*e+11*B*b*d)*(b*x+a)^11/e/(-a*e
+b*d)^5/(e*x+d)^13+1/74256*b^4*(6*A*b*e-17*B*a*e+11*B*b*d)*(b*x+a)^11/e/(-a*e+b*d)^6/(e*x+d)^12+1/816816*b^5*(
6*A*b*e-17*B*a*e+11*B*b*d)*(b*x+a)^11/e/(-a*e+b*d)^7/(e*x+d)^11

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {79, 47, 37} \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{18}} \, dx=\frac {b^5 (a+b x)^{11} (-17 a B e+6 A b e+11 b B d)}{816816 e (d+e x)^{11} (b d-a e)^7}+\frac {b^4 (a+b x)^{11} (-17 a B e+6 A b e+11 b B d)}{74256 e (d+e x)^{12} (b d-a e)^6}+\frac {b^3 (a+b x)^{11} (-17 a B e+6 A b e+11 b B d)}{12376 e (d+e x)^{13} (b d-a e)^5}+\frac {b^2 (a+b x)^{11} (-17 a B e+6 A b e+11 b B d)}{2856 e (d+e x)^{14} (b d-a e)^4}+\frac {b (a+b x)^{11} (-17 a B e+6 A b e+11 b B d)}{816 e (d+e x)^{15} (b d-a e)^3}+\frac {(a+b x)^{11} (-17 a B e+6 A b e+11 b B d)}{272 e (d+e x)^{16} (b d-a e)^2}-\frac {(a+b x)^{11} (B d-A e)}{17 e (d+e x)^{17} (b d-a e)} \]

[In]

Int[((a + b*x)^10*(A + B*x))/(d + e*x)^18,x]

[Out]

-1/17*((B*d - A*e)*(a + b*x)^11)/(e*(b*d - a*e)*(d + e*x)^17) + ((11*b*B*d + 6*A*b*e - 17*a*B*e)*(a + b*x)^11)
/(272*e*(b*d - a*e)^2*(d + e*x)^16) + (b*(11*b*B*d + 6*A*b*e - 17*a*B*e)*(a + b*x)^11)/(816*e*(b*d - a*e)^3*(d
+ e*x)^15) + (b^2*(11*b*B*d + 6*A*b*e - 17*a*B*e)*(a + b*x)^11)/(2856*e*(b*d - a*e)^4*(d + e*x)^14) + (b^3*(1
1*b*B*d + 6*A*b*e - 17*a*B*e)*(a + b*x)^11)/(12376*e*(b*d - a*e)^5*(d + e*x)^13) + (b^4*(11*b*B*d + 6*A*b*e -
17*a*B*e)*(a + b*x)^11)/(74256*e*(b*d - a*e)^6*(d + e*x)^12) + (b^5*(11*b*B*d + 6*A*b*e - 17*a*B*e)*(a + b*x)^
11)/(816816*e*(b*d - a*e)^7*(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}}{17 e (b d-a e) (d+e x)^{17}}+\frac {(11 b B d+6 A b e-17 a B e) \int \frac {(a+b x)^{10}}{(d+e x)^{17}} \, dx}{17 e (b d-a e)} \\ & = -\frac {(B d-A e) (a+b x)^{11}}{17 e (b d-a e) (d+e x)^{17}}+\frac {(11 b B d+6 A b e-17 a B e) (a+b x)^{11}}{272 e (b d-a e)^2 (d+e x)^{16}}+\frac {(5 b (11 b B d+6 A b e-17 a B e)) \int \frac {(a+b x)^{10}}{(d+e x)^{16}} \, dx}{272 e (b d-a e)^2} \\ & = -\frac {(B d-A e) (a+b x)^{11}}{17 e (b d-a e) (d+e x)^{17}}+\frac {(11 b B d+6 A b e-17 a B e) (a+b x)^{11}}{272 e (b d-a e)^2 (d+e x)^{16}}+\frac {b (11 b B d+6 A b e-17 a B e) (a+b x)^{11}}{816 e (b d-a e)^3 (d+e x)^{15}}+\frac {\left (b^2 (11 b B d+6 A b e-17 a B e)\right ) \int \frac {(a+b x)^{10}}{(d+e x)^{15}} \, dx}{204 e (b d-a e)^3} \\ & = -\frac {(B d-A e) (a+b x)^{11}}{17 e (b d-a e) (d+e x)^{17}}+\frac {(11 b B d+6 A b e-17 a B e) (a+b x)^{11}}{272 e (b d-a e)^2 (d+e x)^{16}}+\frac {b (11 b B d+6 A b e-17 a B e) (a+b x)^{11}}{816 e (b d-a e)^3 (d+e x)^{15}}+\frac {b^2 (11 b B d+6 A b e-17 a B e) (a+b x)^{11}}{2856 e (b d-a e)^4 (d+e x)^{14}}+\frac {\left (b^3 (11 b B d+6 A b e-17 a B e)\right ) \int \frac {(a+b x)^{10}}{(d+e x)^{14}} \, dx}{952 e (b d-a e)^4} \\ & = -\frac {(B d-A e) (a+b x)^{11}}{17 e (b d-a e) (d+e x)^{17}}+\frac {(11 b B d+6 A b e-17 a B e) (a+b x)^{11}}{272 e (b d-a e)^2 (d+e x)^{16}}+\frac {b (11 b B d+6 A b e-17 a B e) (a+b x)^{11}}{816 e (b d-a e)^3 (d+e x)^{15}}+\frac {b^2 (11 b B d+6 A b e-17 a B e) (a+b x)^{11}}{2856 e (b d-a e)^4 (d+e x)^{14}}+\frac {b^3 (11 b B d+6 A b e-17 a B e) (a+b x)^{11}}{12376 e (b d-a e)^5 (d+e x)^{13}}+\frac {\left (b^4 (11 b B d+6 A b e-17 a B e)\right ) \int \frac {(a+b x)^{10}}{(d+e x)^{13}} \, dx}{6188 e (b d-a e)^5} \\ & = -\frac {(B d-A e) (a+b x)^{11}}{17 e (b d-a e) (d+e x)^{17}}+\frac {(11 b B d+6 A b e-17 a B e) (a+b x)^{11}}{272 e (b d-a e)^2 (d+e x)^{16}}+\frac {b (11 b B d+6 A b e-17 a B e) (a+b x)^{11}}{816 e (b d-a e)^3 (d+e x)^{15}}+\frac {b^2 (11 b B d+6 A b e-17 a B e) (a+b x)^{11}}{2856 e (b d-a e)^4 (d+e x)^{14}}+\frac {b^3 (11 b B d+6 A b e-17 a B e) (a+b x)^{11}}{12376 e (b d-a e)^5 (d+e x)^{13}}+\frac {b^4 (11 b B d+6 A b e-17 a B e) (a+b x)^{11}}{74256 e (b d-a e)^6 (d+e x)^{12}}+\frac {\left (b^5 (11 b B d+6 A b e-17 a B e)\right ) \int \frac {(a+b x)^{10}}{(d+e x)^{12}} \, dx}{74256 e (b d-a e)^6} \\ & = -\frac {(B d-A e) (a+b x)^{11}}{17 e (b d-a e) (d+e x)^{17}}+\frac {(11 b B d+6 A b e-17 a B e) (a+b x)^{11}}{272 e (b d-a e)^2 (d+e x)^{16}}+\frac {b (11 b B d+6 A b e-17 a B e) (a+b x)^{11}}{816 e (b d-a e)^3 (d+e x)^{15}}+\frac {b^2 (11 b B d+6 A b e-17 a B e) (a+b x)^{11}}{2856 e (b d-a e)^4 (d+e x)^{14}}+\frac {b^3 (11 b B d+6 A b e-17 a B e) (a+b x)^{11}}{12376 e (b d-a e)^5 (d+e x)^{13}}+\frac {b^4 (11 b B d+6 A b e-17 a B e) (a+b x)^{11}}{74256 e (b d-a e)^6 (d+e x)^{12}}+\frac {b^5 (11 b B d+6 A b e-17 a B e) (a+b x)^{11}}{816816 e (b d-a e)^7 (d+e x)^{11}} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1433\) vs. \(2(335)=670\).

Time = 0.55 (sec) , antiderivative size = 1433, normalized size of antiderivative = 4.28 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{18}} \, dx=-\frac {3003 a^{10} e^{10} (16 A e+B (d+17 e x))+2002 a^9 b e^9 \left (15 A e (d+17 e x)+2 B \left (d^2+17 d e x+136 e^2 x^2\right )\right )+1287 a^8 b^2 e^8 \left (14 A e \left (d^2+17 d e x+136 e^2 x^2\right )+3 B \left (d^3+17 d^2 e x+136 d e^2 x^2+680 e^3 x^3\right )\right )+792 a^7 b^3 e^7 \left (13 A e \left (d^3+17 d^2 e x+136 d e^2 x^2+680 e^3 x^3\right )+4 B \left (d^4+17 d^3 e x+136 d^2 e^2 x^2+680 d e^3 x^3+2380 e^4 x^4\right )\right )+462 a^6 b^4 e^6 \left (12 A e \left (d^4+17 d^3 e x+136 d^2 e^2 x^2+680 d e^3 x^3+2380 e^4 x^4\right )+5 B \left (d^5+17 d^4 e x+136 d^3 e^2 x^2+680 d^2 e^3 x^3+2380 d e^4 x^4+6188 e^5 x^5\right )\right )+252 a^5 b^5 e^5 \left (11 A e \left (d^5+17 d^4 e x+136 d^3 e^2 x^2+680 d^2 e^3 x^3+2380 d e^4 x^4+6188 e^5 x^5\right )+6 B \left (d^6+17 d^5 e x+136 d^4 e^2 x^2+680 d^3 e^3 x^3+2380 d^2 e^4 x^4+6188 d e^5 x^5+12376 e^6 x^6\right )\right )+126 a^4 b^6 e^4 \left (10 A e \left (d^6+17 d^5 e x+136 d^4 e^2 x^2+680 d^3 e^3 x^3+2380 d^2 e^4 x^4+6188 d e^5 x^5+12376 e^6 x^6\right )+7 B \left (d^7+17 d^6 e x+136 d^5 e^2 x^2+680 d^4 e^3 x^3+2380 d^3 e^4 x^4+6188 d^2 e^5 x^5+12376 d e^6 x^6+19448 e^7 x^7\right )\right )+56 a^3 b^7 e^3 \left (9 A e \left (d^7+17 d^6 e x+136 d^5 e^2 x^2+680 d^4 e^3 x^3+2380 d^3 e^4 x^4+6188 d^2 e^5 x^5+12376 d e^6 x^6+19448 e^7 x^7\right )+8 B \left (d^8+17 d^7 e x+136 d^6 e^2 x^2+680 d^5 e^3 x^3+2380 d^4 e^4 x^4+6188 d^3 e^5 x^5+12376 d^2 e^6 x^6+19448 d e^7 x^7+24310 e^8 x^8\right )\right )+21 a^2 b^8 e^2 \left (8 A e \left (d^8+17 d^7 e x+136 d^6 e^2 x^2+680 d^5 e^3 x^3+2380 d^4 e^4 x^4+6188 d^3 e^5 x^5+12376 d^2 e^6 x^6+19448 d e^7 x^7+24310 e^8 x^8\right )+9 B \left (d^9+17 d^8 e x+136 d^7 e^2 x^2+680 d^6 e^3 x^3+2380 d^5 e^4 x^4+6188 d^4 e^5 x^5+12376 d^3 e^6 x^6+19448 d^2 e^7 x^7+24310 d e^8 x^8+24310 e^9 x^9\right )\right )+6 a b^9 e \left (7 A e \left (d^9+17 d^8 e x+136 d^7 e^2 x^2+680 d^6 e^3 x^3+2380 d^5 e^4 x^4+6188 d^4 e^5 x^5+12376 d^3 e^6 x^6+19448 d^2 e^7 x^7+24310 d e^8 x^8+24310 e^9 x^9\right )+10 B \left (d^{10}+17 d^9 e x+136 d^8 e^2 x^2+680 d^7 e^3 x^3+2380 d^6 e^4 x^4+6188 d^5 e^5 x^5+12376 d^4 e^6 x^6+19448 d^3 e^7 x^7+24310 d^2 e^8 x^8+24310 d e^9 x^9+19448 e^{10} x^{10}\right )\right )+b^{10} \left (6 A e \left (d^{10}+17 d^9 e x+136 d^8 e^2 x^2+680 d^7 e^3 x^3+2380 d^6 e^4 x^4+6188 d^5 e^5 x^5+12376 d^4 e^6 x^6+19448 d^3 e^7 x^7+24310 d^2 e^8 x^8+24310 d e^9 x^9+19448 e^{10} x^{10}\right )+11 B \left (d^{11}+17 d^{10} e x+136 d^9 e^2 x^2+680 d^8 e^3 x^3+2380 d^7 e^4 x^4+6188 d^6 e^5 x^5+12376 d^5 e^6 x^6+19448 d^4 e^7 x^7+24310 d^3 e^8 x^8+24310 d^2 e^9 x^9+19448 d e^{10} x^{10}+12376 e^{11} x^{11}\right )\right )}{816816 e^{12} (d+e x)^{17}} \]

[In]

Integrate[((a + b*x)^10*(A + B*x))/(d + e*x)^18,x]

[Out]

-1/816816*(3003*a^10*e^10*(16*A*e + B*(d + 17*e*x)) + 2002*a^9*b*e^9*(15*A*e*(d + 17*e*x) + 2*B*(d^2 + 17*d*e*
x + 136*e^2*x^2)) + 1287*a^8*b^2*e^8*(14*A*e*(d^2 + 17*d*e*x + 136*e^2*x^2) + 3*B*(d^3 + 17*d^2*e*x + 136*d*e^
2*x^2 + 680*e^3*x^3)) + 792*a^7*b^3*e^7*(13*A*e*(d^3 + 17*d^2*e*x + 136*d*e^2*x^2 + 680*e^3*x^3) + 4*B*(d^4 +
17*d^3*e*x + 136*d^2*e^2*x^2 + 680*d*e^3*x^3 + 2380*e^4*x^4)) + 462*a^6*b^4*e^6*(12*A*e*(d^4 + 17*d^3*e*x + 13
6*d^2*e^2*x^2 + 680*d*e^3*x^3 + 2380*e^4*x^4) + 5*B*(d^5 + 17*d^4*e*x + 136*d^3*e^2*x^2 + 680*d^2*e^3*x^3 + 23
80*d*e^4*x^4 + 6188*e^5*x^5)) + 252*a^5*b^5*e^5*(11*A*e*(d^5 + 17*d^4*e*x + 136*d^3*e^2*x^2 + 680*d^2*e^3*x^3
+ 2380*d*e^4*x^4 + 6188*e^5*x^5) + 6*B*(d^6 + 17*d^5*e*x + 136*d^4*e^2*x^2 + 680*d^3*e^3*x^3 + 2380*d^2*e^4*x^
4 + 6188*d*e^5*x^5 + 12376*e^6*x^6)) + 126*a^4*b^6*e^4*(10*A*e*(d^6 + 17*d^5*e*x + 136*d^4*e^2*x^2 + 680*d^3*e
^3*x^3 + 2380*d^2*e^4*x^4 + 6188*d*e^5*x^5 + 12376*e^6*x^6) + 7*B*(d^7 + 17*d^6*e*x + 136*d^5*e^2*x^2 + 680*d^
4*e^3*x^3 + 2380*d^3*e^4*x^4 + 6188*d^2*e^5*x^5 + 12376*d*e^6*x^6 + 19448*e^7*x^7)) + 56*a^3*b^7*e^3*(9*A*e*(d
^7 + 17*d^6*e*x + 136*d^5*e^2*x^2 + 680*d^4*e^3*x^3 + 2380*d^3*e^4*x^4 + 6188*d^2*e^5*x^5 + 12376*d*e^6*x^6 +
19448*e^7*x^7) + 8*B*(d^8 + 17*d^7*e*x + 136*d^6*e^2*x^2 + 680*d^5*e^3*x^3 + 2380*d^4*e^4*x^4 + 6188*d^3*e^5*x
^5 + 12376*d^2*e^6*x^6 + 19448*d*e^7*x^7 + 24310*e^8*x^8)) + 21*a^2*b^8*e^2*(8*A*e*(d^8 + 17*d^7*e*x + 136*d^6
*e^2*x^2 + 680*d^5*e^3*x^3 + 2380*d^4*e^4*x^4 + 6188*d^3*e^5*x^5 + 12376*d^2*e^6*x^6 + 19448*d*e^7*x^7 + 24310
*e^8*x^8) + 9*B*(d^9 + 17*d^8*e*x + 136*d^7*e^2*x^2 + 680*d^6*e^3*x^3 + 2380*d^5*e^4*x^4 + 6188*d^4*e^5*x^5 +
12376*d^3*e^6*x^6 + 19448*d^2*e^7*x^7 + 24310*d*e^8*x^8 + 24310*e^9*x^9)) + 6*a*b^9*e*(7*A*e*(d^9 + 17*d^8*e*x
+ 136*d^7*e^2*x^2 + 680*d^6*e^3*x^3 + 2380*d^5*e^4*x^4 + 6188*d^4*e^5*x^5 + 12376*d^3*e^6*x^6 + 19448*d^2*e^7
*x^7 + 24310*d*e^8*x^8 + 24310*e^9*x^9) + 10*B*(d^10 + 17*d^9*e*x + 136*d^8*e^2*x^2 + 680*d^7*e^3*x^3 + 2380*d
^6*e^4*x^4 + 6188*d^5*e^5*x^5 + 12376*d^4*e^6*x^6 + 19448*d^3*e^7*x^7 + 24310*d^2*e^8*x^8 + 24310*d*e^9*x^9 +
19448*e^10*x^10)) + b^10*(6*A*e*(d^10 + 17*d^9*e*x + 136*d^8*e^2*x^2 + 680*d^7*e^3*x^3 + 2380*d^6*e^4*x^4 + 61
88*d^5*e^5*x^5 + 12376*d^4*e^6*x^6 + 19448*d^3*e^7*x^7 + 24310*d^2*e^8*x^8 + 24310*d*e^9*x^9 + 19448*e^10*x^10
) + 11*B*(d^11 + 17*d^10*e*x + 136*d^9*e^2*x^2 + 680*d^8*e^3*x^3 + 2380*d^7*e^4*x^4 + 6188*d^6*e^5*x^5 + 12376
*d^5*e^6*x^6 + 19448*d^4*e^7*x^7 + 24310*d^3*e^8*x^8 + 24310*d^2*e^9*x^9 + 19448*d*e^10*x^10 + 12376*e^11*x^11
)))/(e^12*(d + e*x)^17)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1900\) vs. \(2(321)=642\).

Time = 2.13 (sec) , antiderivative size = 1901, normalized size of antiderivative = 5.67

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)^18,x,method=_RETURNVERBOSE)

[Out]

(-1/816816/e^12*(48048*A*a^10*e^11+30030*A*a^9*b*d*e^10+18018*A*a^8*b^2*d^2*e^9+10296*A*a^7*b^3*d^3*e^8+5544*A
*a^6*b^4*d^4*e^7+2772*A*a^5*b^5*d^5*e^6+1260*A*a^4*b^6*d^6*e^5+504*A*a^3*b^7*d^7*e^4+168*A*a^2*b^8*d^8*e^3+42*
A*a*b^9*d^9*e^2+6*A*b^10*d^10*e+3003*B*a^10*d*e^10+4004*B*a^9*b*d^2*e^9+3861*B*a^8*b^2*d^3*e^8+3168*B*a^7*b^3*
d^4*e^7+2310*B*a^6*b^4*d^5*e^6+1512*B*a^5*b^5*d^6*e^5+882*B*a^4*b^6*d^7*e^4+448*B*a^3*b^7*d^8*e^3+189*B*a^2*b^
8*d^9*e^2+60*B*a*b^9*d^10*e+11*B*b^10*d^11)-1/48048/e^11*(30030*A*a^9*b*e^10+18018*A*a^8*b^2*d*e^9+10296*A*a^7
*b^3*d^2*e^8+5544*A*a^6*b^4*d^3*e^7+2772*A*a^5*b^5*d^4*e^6+1260*A*a^4*b^6*d^5*e^5+504*A*a^3*b^7*d^6*e^4+168*A*
a^2*b^8*d^7*e^3+42*A*a*b^9*d^8*e^2+6*A*b^10*d^9*e+3003*B*a^10*e^10+4004*B*a^9*b*d*e^9+3861*B*a^8*b^2*d^2*e^8+3
168*B*a^7*b^3*d^3*e^7+2310*B*a^6*b^4*d^4*e^6+1512*B*a^5*b^5*d^5*e^5+882*B*a^4*b^6*d^6*e^4+448*B*a^3*b^7*d^7*e^
3+189*B*a^2*b^8*d^8*e^2+60*B*a*b^9*d^9*e+11*B*b^10*d^10)*x-1/6006*b/e^10*(18018*A*a^8*b*e^9+10296*A*a^7*b^2*d*
e^8+5544*A*a^6*b^3*d^2*e^7+2772*A*a^5*b^4*d^3*e^6+1260*A*a^4*b^5*d^4*e^5+504*A*a^3*b^6*d^5*e^4+168*A*a^2*b^7*d
^6*e^3+42*A*a*b^8*d^7*e^2+6*A*b^9*d^8*e+4004*B*a^9*e^9+3861*B*a^8*b*d*e^8+3168*B*a^7*b^2*d^2*e^7+2310*B*a^6*b^
3*d^3*e^6+1512*B*a^5*b^4*d^4*e^5+882*B*a^4*b^5*d^5*e^4+448*B*a^3*b^6*d^6*e^3+189*B*a^2*b^7*d^7*e^2+60*B*a*b^8*
d^8*e+11*B*b^9*d^9)*x^2-5/6006*b^2/e^9*(10296*A*a^7*b*e^8+5544*A*a^6*b^2*d*e^7+2772*A*a^5*b^3*d^2*e^6+1260*A*a
^4*b^4*d^3*e^5+504*A*a^3*b^5*d^4*e^4+168*A*a^2*b^6*d^5*e^3+42*A*a*b^7*d^6*e^2+6*A*b^8*d^7*e+3861*B*a^8*e^8+316
8*B*a^7*b*d*e^7+2310*B*a^6*b^2*d^2*e^6+1512*B*a^5*b^3*d^3*e^5+882*B*a^4*b^4*d^4*e^4+448*B*a^3*b^5*d^5*e^3+189*
B*a^2*b^6*d^6*e^2+60*B*a*b^7*d^7*e+11*B*b^8*d^8)*x^3-5/1716*b^3/e^8*(5544*A*a^6*b*e^7+2772*A*a^5*b^2*d*e^6+126
0*A*a^4*b^3*d^2*e^5+504*A*a^3*b^4*d^3*e^4+168*A*a^2*b^5*d^4*e^3+42*A*a*b^6*d^5*e^2+6*A*b^7*d^6*e+3168*B*a^7*e^
7+2310*B*a^6*b*d*e^6+1512*B*a^5*b^2*d^2*e^5+882*B*a^4*b^3*d^3*e^4+448*B*a^3*b^4*d^4*e^3+189*B*a^2*b^5*d^5*e^2+
60*B*a*b^6*d^6*e+11*B*b^7*d^7)*x^4-1/132*b^4/e^7*(2772*A*a^5*b*e^6+1260*A*a^4*b^2*d*e^5+504*A*a^3*b^3*d^2*e^4+
168*A*a^2*b^4*d^3*e^3+42*A*a*b^5*d^4*e^2+6*A*b^6*d^5*e+2310*B*a^6*e^6+1512*B*a^5*b*d*e^5+882*B*a^4*b^2*d^2*e^4
+448*B*a^3*b^3*d^3*e^3+189*B*a^2*b^4*d^4*e^2+60*B*a*b^5*d^5*e+11*B*b^6*d^6)*x^5-1/66*b^5/e^6*(1260*A*a^4*b*e^5
+504*A*a^3*b^2*d*e^4+168*A*a^2*b^3*d^2*e^3+42*A*a*b^4*d^3*e^2+6*A*b^5*d^4*e+1512*B*a^5*e^5+882*B*a^4*b*d*e^4+4
48*B*a^3*b^2*d^2*e^3+189*B*a^2*b^3*d^3*e^2+60*B*a*b^4*d^4*e+11*B*b^5*d^5)*x^6-1/42*b^6/e^5*(504*A*a^3*b*e^4+16
8*A*a^2*b^2*d*e^3+42*A*a*b^3*d^2*e^2+6*A*b^4*d^3*e+882*B*a^4*e^4+448*B*a^3*b*d*e^3+189*B*a^2*b^2*d^2*e^2+60*B*
a*b^3*d^3*e+11*B*b^4*d^4)*x^7-5/168*b^7/e^4*(168*A*a^2*b*e^3+42*A*a*b^2*d*e^2+6*A*b^3*d^2*e+448*B*a^3*e^3+189*
B*a^2*b*d*e^2+60*B*a*b^2*d^2*e+11*B*b^3*d^3)*x^8-5/168*b^8/e^3*(42*A*a*b*e^2+6*A*b^2*d*e+189*B*a^2*e^2+60*B*a*
b*d*e+11*B*b^2*d^2)*x^9-1/42*b^9/e^2*(6*A*b*e+60*B*a*e+11*B*b*d)*x^10-1/6*b^10*B/e*x^11)/(e*x+d)^17

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1995 vs. \(2 (321) = 642\).

Time = 0.33 (sec) , antiderivative size = 1995, normalized size of antiderivative = 5.96 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{18}} \, dx=\text {Too large to display} \]

[In]

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^18,x, algorithm="fricas")

[Out]

-1/816816*(136136*B*b^10*e^11*x^11 + 11*B*b^10*d^11 + 48048*A*a^10*e^11 + 6*(10*B*a*b^9 + A*b^10)*d^10*e + 21*
(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 + 56*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 126*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d
^7*e^4 + 252*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 462*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 + 792*(4*B*a^7*b^3
+ 7*A*a^6*b^4)*d^4*e^7 + 1287*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^8 + 2002*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9 + 3
003*(B*a^10 + 10*A*a^9*b)*d*e^10 + 19448*(11*B*b^10*d*e^10 + 6*(10*B*a*b^9 + A*b^10)*e^11)*x^10 + 24310*(11*B*
b^10*d^2*e^9 + 6*(10*B*a*b^9 + A*b^10)*d*e^10 + 21*(9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9 + 24310*(11*B*b^10*d^3*
e^8 + 6*(10*B*a*b^9 + A*b^10)*d^2*e^9 + 21*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^10 + 56*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e
^11)*x^8 + 19448*(11*B*b^10*d^4*e^7 + 6*(10*B*a*b^9 + A*b^10)*d^3*e^8 + 21*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^9 +
56*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^10 + 126*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 + 12376*(11*B*b^10*d^5*e^6
+ 6*(10*B*a*b^9 + A*b^10)*d^4*e^7 + 21*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^8 + 56*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*
e^9 + 126*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^10 + 252*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 6188*(11*B*b^10*d^6
*e^5 + 6*(10*B*a*b^9 + A*b^10)*d^5*e^6 + 21*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 + 56*(8*B*a^3*b^7 + 3*A*a^2*b^8)
*d^3*e^8 + 126*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^9 + 252*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 + 462*(5*B*a^6*b^4
+ 6*A*a^5*b^5)*e^11)*x^5 + 2380*(11*B*b^10*d^7*e^4 + 6*(10*B*a*b^9 + A*b^10)*d^6*e^5 + 21*(9*B*a^2*b^8 + 2*A*
a*b^9)*d^5*e^6 + 56*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 + 126*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^8 + 252*(6*B*a
^5*b^5 + 5*A*a^4*b^6)*d^2*e^9 + 462*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^10 + 792*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^11)
*x^4 + 680*(11*B*b^10*d^8*e^3 + 6*(10*B*a*b^9 + A*b^10)*d^7*e^4 + 21*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5 + 56*(8
*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^6 + 126*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^7 + 252*(6*B*a^5*b^5 + 5*A*a^4*b^6)*
d^3*e^8 + 462*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*e^9 + 792*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*e^10 + 1287*(3*B*a^8*b^2
+ 8*A*a^7*b^3)*e^11)*x^3 + 136*(11*B*b^10*d^9*e^2 + 6*(10*B*a*b^9 + A*b^10)*d^8*e^3 + 21*(9*B*a^2*b^8 + 2*A*a
*b^9)*d^7*e^4 + 56*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^5 + 126*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*e^6 + 252*(6*B*a^
5*b^5 + 5*A*a^4*b^6)*d^4*e^7 + 462*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*e^8 + 792*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*e
^9 + 1287*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d*e^10 + 2002*(2*B*a^9*b + 9*A*a^8*b^2)*e^11)*x^2 + 17*(11*B*b^10*d^10*e
+ 6*(10*B*a*b^9 + A*b^10)*d^9*e^2 + 21*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^3 + 56*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7
*e^4 + 126*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5 + 252*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 + 462*(5*B*a^6*b^4 +
6*A*a^5*b^5)*d^4*e^7 + 792*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*e^8 + 1287*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e^9 + 20
02*(2*B*a^9*b + 9*A*a^8*b^2)*d*e^10 + 3003*(B*a^10 + 10*A*a^9*b)*e^11)*x)/(e^29*x^17 + 17*d*e^28*x^16 + 136*d^
2*e^27*x^15 + 680*d^3*e^26*x^14 + 2380*d^4*e^25*x^13 + 6188*d^5*e^24*x^12 + 12376*d^6*e^23*x^11 + 19448*d^7*e^
22*x^10 + 24310*d^8*e^21*x^9 + 24310*d^9*e^20*x^8 + 19448*d^10*e^19*x^7 + 12376*d^11*e^18*x^6 + 6188*d^12*e^17
*x^5 + 2380*d^13*e^16*x^4 + 680*d^14*e^15*x^3 + 136*d^15*e^14*x^2 + 17*d^16*e^13*x + d^17*e^12)

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{18}} \, dx=\text {Timed out} \]

[In]

integrate((b*x+a)**10*(B*x+A)/(e*x+d)**18,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1995 vs. \(2 (321) = 642\).

Time = 0.31 (sec) , antiderivative size = 1995, normalized size of antiderivative = 5.96 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{18}} \, dx=\text {Too large to display} \]

[In]

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^18,x, algorithm="maxima")

[Out]

-1/816816*(136136*B*b^10*e^11*x^11 + 11*B*b^10*d^11 + 48048*A*a^10*e^11 + 6*(10*B*a*b^9 + A*b^10)*d^10*e + 21*
(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 + 56*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 126*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d
^7*e^4 + 252*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 462*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 + 792*(4*B*a^7*b^3
+ 7*A*a^6*b^4)*d^4*e^7 + 1287*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^8 + 2002*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9 + 3
003*(B*a^10 + 10*A*a^9*b)*d*e^10 + 19448*(11*B*b^10*d*e^10 + 6*(10*B*a*b^9 + A*b^10)*e^11)*x^10 + 24310*(11*B*
b^10*d^2*e^9 + 6*(10*B*a*b^9 + A*b^10)*d*e^10 + 21*(9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9 + 24310*(11*B*b^10*d^3*
e^8 + 6*(10*B*a*b^9 + A*b^10)*d^2*e^9 + 21*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^10 + 56*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e
^11)*x^8 + 19448*(11*B*b^10*d^4*e^7 + 6*(10*B*a*b^9 + A*b^10)*d^3*e^8 + 21*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^9 +
56*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^10 + 126*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 + 12376*(11*B*b^10*d^5*e^6
+ 6*(10*B*a*b^9 + A*b^10)*d^4*e^7 + 21*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^8 + 56*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*
e^9 + 126*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^10 + 252*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 6188*(11*B*b^10*d^6
*e^5 + 6*(10*B*a*b^9 + A*b^10)*d^5*e^6 + 21*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 + 56*(8*B*a^3*b^7 + 3*A*a^2*b^8)
*d^3*e^8 + 126*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^9 + 252*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 + 462*(5*B*a^6*b^4
+ 6*A*a^5*b^5)*e^11)*x^5 + 2380*(11*B*b^10*d^7*e^4 + 6*(10*B*a*b^9 + A*b^10)*d^6*e^5 + 21*(9*B*a^2*b^8 + 2*A*
a*b^9)*d^5*e^6 + 56*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 + 126*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^8 + 252*(6*B*a
^5*b^5 + 5*A*a^4*b^6)*d^2*e^9 + 462*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^10 + 792*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^11)
*x^4 + 680*(11*B*b^10*d^8*e^3 + 6*(10*B*a*b^9 + A*b^10)*d^7*e^4 + 21*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5 + 56*(8
*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^6 + 126*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^7 + 252*(6*B*a^5*b^5 + 5*A*a^4*b^6)*
d^3*e^8 + 462*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*e^9 + 792*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*e^10 + 1287*(3*B*a^8*b^2
+ 8*A*a^7*b^3)*e^11)*x^3 + 136*(11*B*b^10*d^9*e^2 + 6*(10*B*a*b^9 + A*b^10)*d^8*e^3 + 21*(9*B*a^2*b^8 + 2*A*a
*b^9)*d^7*e^4 + 56*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^5 + 126*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*e^6 + 252*(6*B*a^
5*b^5 + 5*A*a^4*b^6)*d^4*e^7 + 462*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*e^8 + 792*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*e
^9 + 1287*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d*e^10 + 2002*(2*B*a^9*b + 9*A*a^8*b^2)*e^11)*x^2 + 17*(11*B*b^10*d^10*e
+ 6*(10*B*a*b^9 + A*b^10)*d^9*e^2 + 21*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^3 + 56*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7
*e^4 + 126*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5 + 252*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 + 462*(5*B*a^6*b^4 +
6*A*a^5*b^5)*d^4*e^7 + 792*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*e^8 + 1287*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e^9 + 20
02*(2*B*a^9*b + 9*A*a^8*b^2)*d*e^10 + 3003*(B*a^10 + 10*A*a^9*b)*e^11)*x)/(e^29*x^17 + 17*d*e^28*x^16 + 136*d^
2*e^27*x^15 + 680*d^3*e^26*x^14 + 2380*d^4*e^25*x^13 + 6188*d^5*e^24*x^12 + 12376*d^6*e^23*x^11 + 19448*d^7*e^
22*x^10 + 24310*d^8*e^21*x^9 + 24310*d^9*e^20*x^8 + 19448*d^10*e^19*x^7 + 12376*d^11*e^18*x^6 + 6188*d^12*e^17
*x^5 + 2380*d^13*e^16*x^4 + 680*d^14*e^15*x^3 + 136*d^15*e^14*x^2 + 17*d^16*e^13*x + d^17*e^12)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2232 vs. \(2 (321) = 642\).

Time = 0.31 (sec) , antiderivative size = 2232, normalized size of antiderivative = 6.66 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{18}} \, dx=\text {Too large to display} \]

[In]

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^18,x, algorithm="giac")

[Out]

-1/816816*(136136*B*b^10*e^11*x^11 + 213928*B*b^10*d*e^10*x^10 + 1166880*B*a*b^9*e^11*x^10 + 116688*A*b^10*e^1
1*x^10 + 267410*B*b^10*d^2*e^9*x^9 + 1458600*B*a*b^9*d*e^10*x^9 + 145860*A*b^10*d*e^10*x^9 + 4594590*B*a^2*b^8
*e^11*x^9 + 1021020*A*a*b^9*e^11*x^9 + 267410*B*b^10*d^3*e^8*x^8 + 1458600*B*a*b^9*d^2*e^9*x^8 + 145860*A*b^10
*d^2*e^9*x^8 + 4594590*B*a^2*b^8*d*e^10*x^8 + 1021020*A*a*b^9*d*e^10*x^8 + 10890880*B*a^3*b^7*e^11*x^8 + 40840
80*A*a^2*b^8*e^11*x^8 + 213928*B*b^10*d^4*e^7*x^7 + 1166880*B*a*b^9*d^3*e^8*x^7 + 116688*A*b^10*d^3*e^8*x^7 +
3675672*B*a^2*b^8*d^2*e^9*x^7 + 816816*A*a*b^9*d^2*e^9*x^7 + 8712704*B*a^3*b^7*d*e^10*x^7 + 3267264*A*a^2*b^8*
d*e^10*x^7 + 17153136*B*a^4*b^6*e^11*x^7 + 9801792*A*a^3*b^7*e^11*x^7 + 136136*B*b^10*d^5*e^6*x^6 + 742560*B*a
*b^9*d^4*e^7*x^6 + 74256*A*b^10*d^4*e^7*x^6 + 2339064*B*a^2*b^8*d^3*e^8*x^6 + 519792*A*a*b^9*d^3*e^8*x^6 + 554
4448*B*a^3*b^7*d^2*e^9*x^6 + 2079168*A*a^2*b^8*d^2*e^9*x^6 + 10915632*B*a^4*b^6*d*e^10*x^6 + 6237504*A*a^3*b^7
*d*e^10*x^6 + 18712512*B*a^5*b^5*e^11*x^6 + 15593760*A*a^4*b^6*e^11*x^6 + 68068*B*b^10*d^6*e^5*x^5 + 371280*B*
a*b^9*d^5*e^6*x^5 + 37128*A*b^10*d^5*e^6*x^5 + 1169532*B*a^2*b^8*d^4*e^7*x^5 + 259896*A*a*b^9*d^4*e^7*x^5 + 27
72224*B*a^3*b^7*d^3*e^8*x^5 + 1039584*A*a^2*b^8*d^3*e^8*x^5 + 5457816*B*a^4*b^6*d^2*e^9*x^5 + 3118752*A*a^3*b^
7*d^2*e^9*x^5 + 9356256*B*a^5*b^5*d*e^10*x^5 + 7796880*A*a^4*b^6*d*e^10*x^5 + 14294280*B*a^6*b^4*e^11*x^5 + 17
153136*A*a^5*b^5*e^11*x^5 + 26180*B*b^10*d^7*e^4*x^4 + 142800*B*a*b^9*d^6*e^5*x^4 + 14280*A*b^10*d^6*e^5*x^4 +
449820*B*a^2*b^8*d^5*e^6*x^4 + 99960*A*a*b^9*d^5*e^6*x^4 + 1066240*B*a^3*b^7*d^4*e^7*x^4 + 399840*A*a^2*b^8*d
^4*e^7*x^4 + 2099160*B*a^4*b^6*d^3*e^8*x^4 + 1199520*A*a^3*b^7*d^3*e^8*x^4 + 3598560*B*a^5*b^5*d^2*e^9*x^4 + 2
998800*A*a^4*b^6*d^2*e^9*x^4 + 5497800*B*a^6*b^4*d*e^10*x^4 + 6597360*A*a^5*b^5*d*e^10*x^4 + 7539840*B*a^7*b^3
*e^11*x^4 + 13194720*A*a^6*b^4*e^11*x^4 + 7480*B*b^10*d^8*e^3*x^3 + 40800*B*a*b^9*d^7*e^4*x^3 + 4080*A*b^10*d^
7*e^4*x^3 + 128520*B*a^2*b^8*d^6*e^5*x^3 + 28560*A*a*b^9*d^6*e^5*x^3 + 304640*B*a^3*b^7*d^5*e^6*x^3 + 114240*A
*a^2*b^8*d^5*e^6*x^3 + 599760*B*a^4*b^6*d^4*e^7*x^3 + 342720*A*a^3*b^7*d^4*e^7*x^3 + 1028160*B*a^5*b^5*d^3*e^8
*x^3 + 856800*A*a^4*b^6*d^3*e^8*x^3 + 1570800*B*a^6*b^4*d^2*e^9*x^3 + 1884960*A*a^5*b^5*d^2*e^9*x^3 + 2154240*
B*a^7*b^3*d*e^10*x^3 + 3769920*A*a^6*b^4*d*e^10*x^3 + 2625480*B*a^8*b^2*e^11*x^3 + 7001280*A*a^7*b^3*e^11*x^3
+ 1496*B*b^10*d^9*e^2*x^2 + 8160*B*a*b^9*d^8*e^3*x^2 + 816*A*b^10*d^8*e^3*x^2 + 25704*B*a^2*b^8*d^7*e^4*x^2 +
5712*A*a*b^9*d^7*e^4*x^2 + 60928*B*a^3*b^7*d^6*e^5*x^2 + 22848*A*a^2*b^8*d^6*e^5*x^2 + 119952*B*a^4*b^6*d^5*e^
6*x^2 + 68544*A*a^3*b^7*d^5*e^6*x^2 + 205632*B*a^5*b^5*d^4*e^7*x^2 + 171360*A*a^4*b^6*d^4*e^7*x^2 + 314160*B*a
^6*b^4*d^3*e^8*x^2 + 376992*A*a^5*b^5*d^3*e^8*x^2 + 430848*B*a^7*b^3*d^2*e^9*x^2 + 753984*A*a^6*b^4*d^2*e^9*x^
2 + 525096*B*a^8*b^2*d*e^10*x^2 + 1400256*A*a^7*b^3*d*e^10*x^2 + 544544*B*a^9*b*e^11*x^2 + 2450448*A*a^8*b^2*e
^11*x^2 + 187*B*b^10*d^10*e*x + 1020*B*a*b^9*d^9*e^2*x + 102*A*b^10*d^9*e^2*x + 3213*B*a^2*b^8*d^8*e^3*x + 714
*A*a*b^9*d^8*e^3*x + 7616*B*a^3*b^7*d^7*e^4*x + 2856*A*a^2*b^8*d^7*e^4*x + 14994*B*a^4*b^6*d^6*e^5*x + 8568*A*
a^3*b^7*d^6*e^5*x + 25704*B*a^5*b^5*d^5*e^6*x + 21420*A*a^4*b^6*d^5*e^6*x + 39270*B*a^6*b^4*d^4*e^7*x + 47124*
A*a^5*b^5*d^4*e^7*x + 53856*B*a^7*b^3*d^3*e^8*x + 94248*A*a^6*b^4*d^3*e^8*x + 65637*B*a^8*b^2*d^2*e^9*x + 1750
32*A*a^7*b^3*d^2*e^9*x + 68068*B*a^9*b*d*e^10*x + 306306*A*a^8*b^2*d*e^10*x + 51051*B*a^10*e^11*x + 510510*A*a
^9*b*e^11*x + 11*B*b^10*d^11 + 60*B*a*b^9*d^10*e + 6*A*b^10*d^10*e + 189*B*a^2*b^8*d^9*e^2 + 42*A*a*b^9*d^9*e^
2 + 448*B*a^3*b^7*d^8*e^3 + 168*A*a^2*b^8*d^8*e^3 + 882*B*a^4*b^6*d^7*e^4 + 504*A*a^3*b^7*d^7*e^4 + 1512*B*a^5
*b^5*d^6*e^5 + 1260*A*a^4*b^6*d^6*e^5 + 2310*B*a^6*b^4*d^5*e^6 + 2772*A*a^5*b^5*d^5*e^6 + 3168*B*a^7*b^3*d^4*e
^7 + 5544*A*a^6*b^4*d^4*e^7 + 3861*B*a^8*b^2*d^3*e^8 + 10296*A*a^7*b^3*d^3*e^8 + 4004*B*a^9*b*d^2*e^9 + 18018*
A*a^8*b^2*d^2*e^9 + 3003*B*a^10*d*e^10 + 30030*A*a^9*b*d*e^10 + 48048*A*a^10*e^11)/((e*x + d)^17*e^12)

Mupad [B] (verification not implemented)

Time = 0.74 (sec) , antiderivative size = 2077, normalized size of antiderivative = 6.20 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{18}} \, dx=\text {Too large to display} \]

[In]

int(((A + B*x)*(a + b*x)^10)/(d + e*x)^18,x)

[Out]

-((48048*A*a^10*e^11 + 11*B*b^10*d^11 + 6*A*b^10*d^10*e + 3003*B*a^10*d*e^10 + 42*A*a*b^9*d^9*e^2 + 4004*B*a^9
*b*d^2*e^9 + 168*A*a^2*b^8*d^8*e^3 + 504*A*a^3*b^7*d^7*e^4 + 1260*A*a^4*b^6*d^6*e^5 + 2772*A*a^5*b^5*d^5*e^6 +
5544*A*a^6*b^4*d^4*e^7 + 10296*A*a^7*b^3*d^3*e^8 + 18018*A*a^8*b^2*d^2*e^9 + 189*B*a^2*b^8*d^9*e^2 + 448*B*a^
3*b^7*d^8*e^3 + 882*B*a^4*b^6*d^7*e^4 + 1512*B*a^5*b^5*d^6*e^5 + 2310*B*a^6*b^4*d^5*e^6 + 3168*B*a^7*b^3*d^4*e
^7 + 3861*B*a^8*b^2*d^3*e^8 + 30030*A*a^9*b*d*e^10 + 60*B*a*b^9*d^10*e)/(816816*e^12) + (x*(3003*B*a^10*e^10 +
11*B*b^10*d^10 + 30030*A*a^9*b*e^10 + 6*A*b^10*d^9*e + 42*A*a*b^9*d^8*e^2 + 18018*A*a^8*b^2*d*e^9 + 168*A*a^2
*b^8*d^7*e^3 + 504*A*a^3*b^7*d^6*e^4 + 1260*A*a^4*b^6*d^5*e^5 + 2772*A*a^5*b^5*d^4*e^6 + 5544*A*a^6*b^4*d^3*e^
7 + 10296*A*a^7*b^3*d^2*e^8 + 189*B*a^2*b^8*d^8*e^2 + 448*B*a^3*b^7*d^7*e^3 + 882*B*a^4*b^6*d^6*e^4 + 1512*B*a
^5*b^5*d^5*e^5 + 2310*B*a^6*b^4*d^4*e^6 + 3168*B*a^7*b^3*d^3*e^7 + 3861*B*a^8*b^2*d^2*e^8 + 60*B*a*b^9*d^9*e +
4004*B*a^9*b*d*e^9))/(48048*e^11) + (5*b^7*x^8*(448*B*a^3*e^3 + 11*B*b^3*d^3 + 168*A*a^2*b*e^3 + 6*A*b^3*d^2*
e + 42*A*a*b^2*d*e^2 + 60*B*a*b^2*d^2*e + 189*B*a^2*b*d*e^2))/(168*e^4) + (b^4*x^5*(2310*B*a^6*e^6 + 11*B*b^6*
d^6 + 2772*A*a^5*b*e^6 + 6*A*b^6*d^5*e + 42*A*a*b^5*d^4*e^2 + 1260*A*a^4*b^2*d*e^5 + 168*A*a^2*b^4*d^3*e^3 + 5
04*A*a^3*b^3*d^2*e^4 + 189*B*a^2*b^4*d^4*e^2 + 448*B*a^3*b^3*d^3*e^3 + 882*B*a^4*b^2*d^2*e^4 + 60*B*a*b^5*d^5*
e + 1512*B*a^5*b*d*e^5))/(132*e^7) + (b^9*x^10*(6*A*b*e + 60*B*a*e + 11*B*b*d))/(42*e^2) + (b^6*x^7*(882*B*a^4
*e^4 + 11*B*b^4*d^4 + 504*A*a^3*b*e^4 + 6*A*b^4*d^3*e + 42*A*a*b^3*d^2*e^2 + 168*A*a^2*b^2*d*e^3 + 189*B*a^2*b
^2*d^2*e^2 + 60*B*a*b^3*d^3*e + 448*B*a^3*b*d*e^3))/(42*e^5) + (5*b^3*x^4*(3168*B*a^7*e^7 + 11*B*b^7*d^7 + 554
4*A*a^6*b*e^7 + 6*A*b^7*d^6*e + 42*A*a*b^6*d^5*e^2 + 2772*A*a^5*b^2*d*e^6 + 168*A*a^2*b^5*d^4*e^3 + 504*A*a^3*
b^4*d^3*e^4 + 1260*A*a^4*b^3*d^2*e^5 + 189*B*a^2*b^5*d^5*e^2 + 448*B*a^3*b^4*d^4*e^3 + 882*B*a^4*b^3*d^3*e^4 +
1512*B*a^5*b^2*d^2*e^5 + 60*B*a*b^6*d^6*e + 2310*B*a^6*b*d*e^6))/(1716*e^8) + (b*x^2*(4004*B*a^9*e^9 + 11*B*b
^9*d^9 + 18018*A*a^8*b*e^9 + 6*A*b^9*d^8*e + 42*A*a*b^8*d^7*e^2 + 10296*A*a^7*b^2*d*e^8 + 168*A*a^2*b^7*d^6*e^
3 + 504*A*a^3*b^6*d^5*e^4 + 1260*A*a^4*b^5*d^4*e^5 + 2772*A*a^5*b^4*d^3*e^6 + 5544*A*a^6*b^3*d^2*e^7 + 189*B*a
^2*b^7*d^7*e^2 + 448*B*a^3*b^6*d^6*e^3 + 882*B*a^4*b^5*d^5*e^4 + 1512*B*a^5*b^4*d^4*e^5 + 2310*B*a^6*b^3*d^3*e
^6 + 3168*B*a^7*b^2*d^2*e^7 + 60*B*a*b^8*d^8*e + 3861*B*a^8*b*d*e^8))/(6006*e^10) + (5*b^8*x^9*(189*B*a^2*e^2
+ 11*B*b^2*d^2 + 42*A*a*b*e^2 + 6*A*b^2*d*e + 60*B*a*b*d*e))/(168*e^3) + (b^5*x^6*(1512*B*a^5*e^5 + 11*B*b^5*d
^5 + 1260*A*a^4*b*e^5 + 6*A*b^5*d^4*e + 42*A*a*b^4*d^3*e^2 + 504*A*a^3*b^2*d*e^4 + 168*A*a^2*b^3*d^2*e^3 + 189
*B*a^2*b^3*d^3*e^2 + 448*B*a^3*b^2*d^2*e^3 + 60*B*a*b^4*d^4*e + 882*B*a^4*b*d*e^4))/(66*e^6) + (5*b^2*x^3*(386
1*B*a^8*e^8 + 11*B*b^8*d^8 + 10296*A*a^7*b*e^8 + 6*A*b^8*d^7*e + 42*A*a*b^7*d^6*e^2 + 5544*A*a^6*b^2*d*e^7 + 1
68*A*a^2*b^6*d^5*e^3 + 504*A*a^3*b^5*d^4*e^4 + 1260*A*a^4*b^4*d^3*e^5 + 2772*A*a^5*b^3*d^2*e^6 + 189*B*a^2*b^6
*d^6*e^2 + 448*B*a^3*b^5*d^5*e^3 + 882*B*a^4*b^4*d^4*e^4 + 1512*B*a^5*b^3*d^3*e^5 + 2310*B*a^6*b^2*d^2*e^6 + 6
0*B*a*b^7*d^7*e + 3168*B*a^7*b*d*e^7))/(6006*e^9) + (B*b^10*x^11)/(6*e))/(d^17 + e^17*x^17 + 17*d*e^16*x^16 +
136*d^15*e^2*x^2 + 680*d^14*e^3*x^3 + 2380*d^13*e^4*x^4 + 6188*d^12*e^5*x^5 + 12376*d^11*e^6*x^6 + 19448*d^10*
e^7*x^7 + 24310*d^9*e^8*x^8 + 24310*d^8*e^9*x^9 + 19448*d^7*e^10*x^10 + 12376*d^6*e^11*x^11 + 6188*d^5*e^12*x^
12 + 2380*d^4*e^13*x^13 + 680*d^3*e^14*x^14 + 136*d^2*e^15*x^15 + 17*d^16*e*x)