∫ (a+bx)10(A+Bx)____________ (d+ex)18 dx [1106]

Optimal. Leaf size=335 \[ -\frac {(B d-A e) (a+b x)^{11}}{17 e (b d-a e) (d+e x)^{17}}+\frac {(11 b B d+6 A b e-17 a B e) (a+b x)^{11}}{272 e (b d-a e)^2 (d+e x)^{16}}+\frac {b (11 b B d+6 A b e-17 a B e) (a+b x)^{11}}{816 e (b d-a e)^3 (d+e x)^{15}}+\frac {b^2 (11 b B d+6 A b e-17 a B e) (a+b x)^{11}}{2856 e (b d-a e)^4 (d+e x)^{14}}+\frac {b^3 (11 b B d+6 A b e-17 a B e) (a+b x)^{11}}{12376 e (b d-a e)^5 (d+e x)^{13}}+\frac {b^4 (11 b B d+6 A b e-17 a B e) (a+b x)^{11}}{74256 e (b d-a e)^6 (d+e x)^{12}}+\frac {b^5 (11 b B d+6 A b e-17 a B e) (a+b x)^{11}}{816816 e (b d-a e)^7 (d+e x)^{11}} \]

-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

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Rubi [A]
time = 0.12, antiderivative size = 335, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {79, 47, 37} \begin {gather*} \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)} \end {gather*}

-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)^

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

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]))) && (

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

\begin {align*} \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) \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*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1433\) vs. \(2(335)=670\).
time = 0.57, size = 1433, normalized size = 4.28 \begin {gather*} -\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}} \end {gather*}

-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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1941\) vs. \(2(321)=642\).
time = 0.13, size = 1942, normalized size = 5.80


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	\(\text {Expression too large to display}\)	\(2233\)

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

-30/13*b^3/e^12*(7*A*a^6*b*e^7-42*A*a^5*b^2*d*e^6+105*A*a^4*b^3*d^2*e^5-140*A*a^3*b^4*d^3*e^4+105*A*a^2*b^5*d^

4*e^3-42*A*a*b^6*d^5*e^2+7*A*b^7*d^6*e+4*B*a^7*e^7-35*B*a^6*b*d*e^6+126*B*a^5*b^2*d^2*e^5-245*B*a^4*b^3*d^3*e^

4+280*B*a^3*b^4*d^4*e^3-189*B*a^2*b^5*d^5*e^2+70*B*a*b^6*d^6*e-11*B*b^7*d^7)/(e*x+d)^13-1/16/e^12*(10*A*a^9*b*

e^10-90*A*a^8*b^2*d*e^9+360*A*a^7*b^3*d^2*e^8-840*A*a^6*b^4*d^3*e^7+1260*A*a^5*b^5*d^4*e^6-1260*A*a^4*b^6*d^5*

e^5+840*A*a^3*b^7*d^6*e^4-360*A*a^2*b^8*d^7*e^3+90*A*a*b^9*d^8*e^2-10*A*b^10*d^9*e+B*a^10*e^10-20*B*a^9*b*d*e^

9+135*B*a^8*b^2*d^2*e^8-480*B*a^7*b^3*d^3*e^7+1050*B*a^6*b^4*d^4*e^6-1512*B*a^5*b^5*d^5*e^5+1470*B*a^4*b^6*d^6

*e^4-960*B*a^3*b^7*d^7*e^3+405*B*a^2*b^8*d^8*e^2-100*B*a*b^9*d^9*e+11*B*b^10*d^10)/(e*x+d)^16-5/8*b^8/e^12*(2*

A*a*b*e^2-2*A*b^2*d*e+9*B*a^2*e^2-20*B*a*b*d*e+11*B*b^2*d^2)/(e*x+d)^8-5/3*b^7/e^12*(3*A*a^2*b*e^3-6*A*a*b^2*d

*e^2+3*A*b^3*d^2*e+8*B*a^3*e^3-27*B*a^2*b*d*e^2+30*B*a*b^2*d^2*e-11*B*b^3*d^3)/(e*x+d)^9-1/7*b^9/e^12*(A*b*e+1

0*B*a*e-11*B*b*d)/(e*x+d)^7-1/6*b^10*B/e^12/(e*x+d)^6-7/2*b^4/e^12*(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+5*B*a^6*e^6-36*B*a^5*b*d*e^5+105*B*a^4*b^2*d

^2*e^4-160*B*a^3*b^3*d^3*e^3+135*B*a^2*b^4*d^4*e^2-60*B*a*b^5*d^5*e+11*B*b^6*d^6)/(e*x+d)^12-3*b^6/e^12*(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+7*B*a^4*e^4-32*B*a^3*b*d*e^3+54*B*a^2*b^2*d^2*e^2

-40*B*a*b^3*d^3*e+11*B*b^4*d^4)/(e*x+d)^10-15/14*b^2/e^12*(8*A*a^7*b*e^8-56*A*a^6*b^2*d*e^7+168*A*a^5*b^3*d^2*

e^6-280*A*a^4*b^4*d^3*e^5+280*A*a^3*b^5*d^4*e^4-168*A*a^2*b^6*d^5*e^3+56*A*a*b^7*d^6*e^2-8*A*b^8*d^7*e+3*B*a^8

*e^8-32*B*a^7*b*d*e^7+140*B*a^6*b^2*d^2*e^6-336*B*a^5*b^3*d^3*e^5+490*B*a^4*b^4*d^4*e^4-448*B*a^3*b^5*d^5*e^3+

252*B*a^2*b^6*d^6*e^2-80*B*a*b^7*d^7*e+11*B*b^8*d^8)/(e*x+d)^14-42/11*b^5/e^12*(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+6*B*a^5*e^5-35*B*a^4*b*d*e^4+80*B*a^3*b^2*d^2*e^3-90*

B*a^2*b^3*d^3*e^2+50*B*a*b^4*d^4*e-11*B*b^5*d^5)/(e*x+d)^11-1/17*(A*a^10*e^11-10*A*a^9*b*d*e^10+45*A*a^8*b^2*d

^2*e^9-120*A*a^7*b^3*d^3*e^8+210*A*a^6*b^4*d^4*e^7-252*A*a^5*b^5*d^5*e^6+210*A*a^4*b^6*d^6*e^5-120*A*a^3*b^7*d

^7*e^4+45*A*a^2*b^8*d^8*e^3-10*A*a*b^9*d^9*e^2+A*b^10*d^10*e-B*a^10*d*e^10+10*B*a^9*b*d^2*e^9-45*B*a^8*b^2*d^3

*e^8+120*B*a^7*b^3*d^4*e^7-210*B*a^6*b^4*d^5*e^6+252*B*a^5*b^5*d^6*e^5-210*B*a^4*b^6*d^7*e^4+120*B*a^3*b^7*d^8

*e^3-45*B*a^2*b^8*d^9*e^2+10*B*a*b^9*d^10*e-B*b^10*d^11)/e^12/(e*x+d)^17-1/3*b/e^12*(9*A*a^8*b*e^9-72*A*a^7*b^

2*d*e^8+252*A*a^6*b^3*d^2*e^7-504*A*a^5*b^4*d^3*e^6+630*A*a^4*b^5*d^4*e^5-504*A*a^3*b^6*d^5*e^4+252*A*a^2*b^7*

d^6*e^3-72*A*a*b^8*d^7*e^2+9*A*b^9*d^8*e+2*B*a^9*e^9-27*B*a^8*b*d*e^8+144*B*a^7*b^2*d^2*e^7-420*B*a^6*b^3*d^3*

e^6+756*B*a^5*b^4*d^4*e^5-882*B*a^4*b^5*d^5*e^4+672*B*a^3*b^6*d^6*e^3-324*B*a^2*b^7*d^7*e^2+90*B*a*b^8*d^8*e-1

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 2006 vs. \(2 (341) = 682\).
time = 0.53, size = 2006, normalized size = 5.99 \begin {gather*} \text {Too large to display} \end {gather*}

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

-1/816816*(136136*B*b^10*x^11*e^11 + 11*B*b^10*d^11 + 48048*A*a^10*e^11 + 6*(10*B*a*b^9*e + A*b^10*e)*d^10 + 1

9448*(11*B*b^10*d*e^10 + 60*B*a*b^9*e^11 + 6*A*b^10*e^11)*x^10 + 21*(9*B*a^2*b^8*e^2 + 2*A*a*b^9*e^2)*d^9 + 24

310*(11*B*b^10*d^2*e^9 + 189*B*a^2*b^8*e^11 + 42*A*a*b^9*e^11 + 6*(10*B*a*b^9*e^10 + A*b^10*e^10)*d)*x^9 + 56*

(8*B*a^3*b^7*e^3 + 3*A*a^2*b^8*e^3)*d^8 + 24310*(11*B*b^10*d^3*e^8 + 448*B*a^3*b^7*e^11 + 168*A*a^2*b^8*e^11 +

6*(10*B*a*b^9*e^9 + A*b^10*e^9)*d^2 + 21*(9*B*a^2*b^8*e^10 + 2*A*a*b^9*e^10)*d)*x^8 + 126*(7*B*a^4*b^6*e^4 +

4*A*a^3*b^7*e^4)*d^7 + 19448*(11*B*b^10*d^4*e^7 + 882*B*a^4*b^6*e^11 + 504*A*a^3*b^7*e^11 + 6*(10*B*a*b^9*e^8

+ A*b^10*e^8)*d^3 + 21*(9*B*a^2*b^8*e^9 + 2*A*a*b^9*e^9)*d^2 + 56*(8*B*a^3*b^7*e^10 + 3*A*a^2*b^8*e^10)*d)*x^7

+ 252*(6*B*a^5*b^5*e^5 + 5*A*a^4*b^6*e^5)*d^6 + 12376*(11*B*b^10*d^5*e^6 + 1512*B*a^5*b^5*e^11 + 1260*A*a^4*b

^6*e^11 + 6*(10*B*a*b^9*e^7 + A*b^10*e^7)*d^4 + 21*(9*B*a^2*b^8*e^8 + 2*A*a*b^9*e^8)*d^3 + 56*(8*B*a^3*b^7*e^9

+ 3*A*a^2*b^8*e^9)*d^2 + 126*(7*B*a^4*b^6*e^10 + 4*A*a^3*b^7*e^10)*d)*x^6 + 462*(5*B*a^6*b^4*e^6 + 6*A*a^5*b^

5*e^6)*d^5 + 6188*(11*B*b^10*d^6*e^5 + 2310*B*a^6*b^4*e^11 + 2772*A*a^5*b^5*e^11 + 6*(10*B*a*b^9*e^6 + A*b^10*

e^6)*d^5 + 21*(9*B*a^2*b^8*e^7 + 2*A*a*b^9*e^7)*d^4 + 56*(8*B*a^3*b^7*e^8 + 3*A*a^2*b^8*e^8)*d^3 + 126*(7*B*a^

4*b^6*e^9 + 4*A*a^3*b^7*e^9)*d^2 + 252*(6*B*a^5*b^5*e^10 + 5*A*a^4*b^6*e^10)*d)*x^5 + 792*(4*B*a^7*b^3*e^7 + 7

*A*a^6*b^4*e^7)*d^4 + 2380*(11*B*b^10*d^7*e^4 + 3168*B*a^7*b^3*e^11 + 5544*A*a^6*b^4*e^11 + 6*(10*B*a*b^9*e^5

+ A*b^10*e^5)*d^6 + 21*(9*B*a^2*b^8*e^6 + 2*A*a*b^9*e^6)*d^5 + 56*(8*B*a^3*b^7*e^7 + 3*A*a^2*b^8*e^7)*d^4 + 12

6*(7*B*a^4*b^6*e^8 + 4*A*a^3*b^7*e^8)*d^3 + 252*(6*B*a^5*b^5*e^9 + 5*A*a^4*b^6*e^9)*d^2 + 462*(5*B*a^6*b^4*e^1

0 + 6*A*a^5*b^5*e^10)*d)*x^4 + 1287*(3*B*a^8*b^2*e^8 + 8*A*a^7*b^3*e^8)*d^3 + 680*(11*B*b^10*d^8*e^3 + 3861*B*

a^8*b^2*e^11 + 10296*A*a^7*b^3*e^11 + 6*(10*B*a*b^9*e^4 + A*b^10*e^4)*d^7 + 21*(9*B*a^2*b^8*e^5 + 2*A*a*b^9*e^

5)*d^6 + 56*(8*B*a^3*b^7*e^6 + 3*A*a^2*b^8*e^6)*d^5 + 126*(7*B*a^4*b^6*e^7 + 4*A*a^3*b^7*e^7)*d^4 + 252*(6*B*a

^5*b^5*e^8 + 5*A*a^4*b^6*e^8)*d^3 + 462*(5*B*a^6*b^4*e^9 + 6*A*a^5*b^5*e^9)*d^2 + 792*(4*B*a^7*b^3*e^10 + 7*A*

a^6*b^4*e^10)*d)*x^3 + 2002*(2*B*a^9*b*e^9 + 9*A*a^8*b^2*e^9)*d^2 + 136*(11*B*b^10*d^9*e^2 + 4004*B*a^9*b*e^11

+ 18018*A*a^8*b^2*e^11 + 6*(10*B*a*b^9*e^3 + A*b^10*e^3)*d^8 + 21*(9*B*a^2*b^8*e^4 + 2*A*a*b^9*e^4)*d^7 + 56*

(8*B*a^3*b^7*e^5 + 3*A*a^2*b^8*e^5)*d^6 + 126*(7*B*a^4*b^6*e^6 + 4*A*a^3*b^7*e^6)*d^5 + 252*(6*B*a^5*b^5*e^7 +

5*A*a^4*b^6*e^7)*d^4 + 462*(5*B*a^6*b^4*e^8 + 6*A*a^5*b^5*e^8)*d^3 + 792*(4*B*a^7*b^3*e^9 + 7*A*a^6*b^4*e^9)*

d^2 + 1287*(3*B*a^8*b^2*e^10 + 8*A*a^7*b^3*e^10)*d)*x^2 + 3003*(B*a^10*e^10 + 10*A*a^9*b*e^10)*d + 17*(11*B*b^

10*d^10*e + 3003*B*a^10*e^11 + 30030*A*a^9*b*e^11 + 6*(10*B*a*b^9*e^2 + A*b^10*e^2)*d^9 + 21*(9*B*a^2*b^8*e^3

+ 2*A*a*b^9*e^3)*d^8 + 56*(8*B*a^3*b^7*e^4 + 3*A*a^2*b^8*e^4)*d^7 + 126*(7*B*a^4*b^6*e^5 + 4*A*a^3*b^7*e^5)*d^

6 + 252*(6*B*a^5*b^5*e^6 + 5*A*a^4*b^6*e^6)*d^5 + 462*(5*B*a^6*b^4*e^7 + 6*A*a^5*b^5*e^7)*d^4 + 792*(4*B*a^7*b

^3*e^8 + 7*A*a^6*b^4*e^8)*d^3 + 1287*(3*B*a^8*b^2*e^9 + 8*A*a^7*b^3*e^9)*d^2 + 2002*(2*B*a^9*b*e^10 + 9*A*a^8*

b^2*e^10)*d)*x)/(x^17*e^29 + 17*d*x^16*e^28 + 136*d^2*x^15*e^27 + 680*d^3*x^14*e^26 + 2380*d^4*x^13*e^25 + 618

8*d^5*x^12*e^24 + 12376*d^6*x^11*e^23 + 19448*d^7*x^10*e^22 + 24310*d^8*x^9*e^21 + 24310*d^9*x^8*e^20 + 19448*

d^10*x^7*e^19 + 12376*d^11*x^6*e^18 + 6188*d^12*x^5*e^17 + 2380*d^13*x^4*e^16 + 680*d^14*x^3*e^15 + 136*d^15*x

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1913 vs. \(2 (341) = 682\).
time = 1.16, size = 1913, normalized size = 5.71 \begin {gather*} \text {Too large to display} \end {gather*}

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

-1/816816*(11*B*b^10*d^11 + (136136*B*b^10*x^11 + 48048*A*a^10 + 116688*(10*B*a*b^9 + A*b^10)*x^10 + 510510*(9

*B*a^2*b^8 + 2*A*a*b^9)*x^9 + 1361360*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 2450448*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^

7 + 3118752*(6*B*a^5*b^5 + 5*A*a^4*b^6)*x^6 + 2858856*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 1884960*(4*B*a^7*b^3 +

7*A*a^6*b^4)*x^4 + 875160*(3*B*a^8*b^2 + 8*A*a^7*b^3)*x^3 + 272272*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 51051*(B*a

^10 + 10*A*a^9*b)*x)*e^11 + (213928*B*b^10*d*x^10 + 145860*(10*B*a*b^9 + A*b^10)*d*x^9 + 510510*(9*B*a^2*b^8 +

2*A*a*b^9)*d*x^8 + 1089088*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*x^7 + 1559376*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*x^6 + 15

59376*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*x^5 + 1099560*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*x^4 + 538560*(4*B*a^7*b^3 + 7*

A*a^6*b^4)*d*x^3 + 175032*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d*x^2 + 34034*(2*B*a^9*b + 9*A*a^8*b^2)*d*x + 3003*(B*a^

10 + 10*A*a^9*b)*d)*e^10 + (267410*B*b^10*d^2*x^9 + 145860*(10*B*a*b^9 + A*b^10)*d^2*x^8 + 408408*(9*B*a^2*b^8

+ 2*A*a*b^9)*d^2*x^7 + 693056*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*x^6 + 779688*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*x^

5 + 599760*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*x^4 + 314160*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*x^3 + 107712*(4*B*a^7*

b^3 + 7*A*a^6*b^4)*d^2*x^2 + 21879*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*x + 2002*(2*B*a^9*b + 9*A*a^8*b^2)*d^2)*e^9

+ (267410*B*b^10*d^3*x^8 + 116688*(10*B*a*b^9 + A*b^10)*d^3*x^7 + 259896*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*x^6 +

346528*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*x^5 + 299880*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*x^4 + 171360*(6*B*a^5*b^5

+ 5*A*a^4*b^6)*d^3*x^3 + 62832*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*x^2 + 13464*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*x +

1287*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3)*e^8 + 2*(106964*B*b^10*d^4*x^7 + 37128*(10*B*a*b^9 + A*b^10)*d^4*x^6 +

64974*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*x^5 + 66640*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*x^4 + 42840*(7*B*a^4*b^6 + 4*A

*a^3*b^7)*d^4*x^3 + 17136*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*x^2 + 3927*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*x + 396*(

4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4)*e^7 + 14*(9724*B*b^10*d^5*x^6 + 2652*(10*B*a*b^9 + A*b^10)*d^5*x^5 + 3570*(9*B

*a^2*b^8 + 2*A*a*b^9)*d^5*x^4 + 2720*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*x^3 + 1224*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^

5*x^2 + 306*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*x + 33*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5)*e^6 + 14*(4862*B*b^10*d^6*

x^5 + 1020*(10*B*a*b^9 + A*b^10)*d^6*x^4 + 1020*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*x^3 + 544*(8*B*a^3*b^7 + 3*A*a^2

*b^8)*d^6*x^2 + 153*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*x + 18*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6)*e^5 + 2*(13090*B*b

^10*d^7*x^4 + 2040*(10*B*a*b^9 + A*b^10)*d^7*x^3 + 1428*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*x^2 + 476*(8*B*a^3*b^7 +

3*A*a^2*b^8)*d^7*x + 63*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7)*e^4 + (7480*B*b^10*d^8*x^3 + 816*(10*B*a*b^9 + A*b^1

0)*d^8*x^2 + 357*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*x + 56*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8)*e^3 + (1496*B*b^10*d^9*

x^2 + 102*(10*B*a*b^9 + A*b^10)*d^9*x + 21*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9)*e^2 + (187*B*b^10*d^10*x + 6*(10*B*a

*b^9 + A*b^10)*d^10)*e)/(x^17*e^29 + 17*d*x^16*e^28 + 136*d^2*x^15*e^27 + 680*d^3*x^14*e^26 + 2380*d^4*x^13*e^

25 + 6188*d^5*x^12*e^24 + 12376*d^6*x^11*e^23 + 19448*d^7*x^10*e^22 + 24310*d^8*x^9*e^21 + 24310*d^9*x^8*e^20

+ 19448*d^10*x^7*e^19 + 12376*d^11*x^6*e^18 + 6188*d^12*x^5*e^17 + 2380*d^13*x^4*e^16 + 680*d^14*x^3*e^15 + 13

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2096 vs. \(2 (341) = 682\).
time = 1.63, size = 2096, normalized size = 6.26 \begin {gather*} \text {Too large to display} \end {gather*}

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

-1/816816*(136136*B*b^10*x^11*e^11 + 213928*B*b^10*d*x^10*e^10 + 267410*B*b^10*d^2*x^9*e^9 + 267410*B*b^10*d^3

*x^8*e^8 + 213928*B*b^10*d^4*x^7*e^7 + 136136*B*b^10*d^5*x^6*e^6 + 68068*B*b^10*d^6*x^5*e^5 + 26180*B*b^10*d^7

*x^4*e^4 + 7480*B*b^10*d^8*x^3*e^3 + 1496*B*b^10*d^9*x^2*e^2 + 187*B*b^10*d^10*x*e + 11*B*b^10*d^11 + 1166880*

B*a*b^9*x^10*e^11 + 116688*A*b^10*x^10*e^11 + 1458600*B*a*b^9*d*x^9*e^10 + 145860*A*b^10*d*x^9*e^10 + 1458600*

B*a*b^9*d^2*x^8*e^9 + 145860*A*b^10*d^2*x^8*e^9 + 1166880*B*a*b^9*d^3*x^7*e^8 + 116688*A*b^10*d^3*x^7*e^8 + 74

2560*B*a*b^9*d^4*x^6*e^7 + 74256*A*b^10*d^4*x^6*e^7 + 371280*B*a*b^9*d^5*x^5*e^6 + 37128*A*b^10*d^5*x^5*e^6 +

142800*B*a*b^9*d^6*x^4*e^5 + 14280*A*b^10*d^6*x^4*e^5 + 40800*B*a*b^9*d^7*x^3*e^4 + 4080*A*b^10*d^7*x^3*e^4 +

8160*B*a*b^9*d^8*x^2*e^3 + 816*A*b^10*d^8*x^2*e^3 + 1020*B*a*b^9*d^9*x*e^2 + 102*A*b^10*d^9*x*e^2 + 60*B*a*b^9

*d^10*e + 6*A*b^10*d^10*e + 4594590*B*a^2*b^8*x^9*e^11 + 1021020*A*a*b^9*x^9*e^11 + 4594590*B*a^2*b^8*d*x^8*e^

10 + 1021020*A*a*b^9*d*x^8*e^10 + 3675672*B*a^2*b^8*d^2*x^7*e^9 + 816816*A*a*b^9*d^2*x^7*e^9 + 2339064*B*a^2*b

^8*d^3*x^6*e^8 + 519792*A*a*b^9*d^3*x^6*e^8 + 1169532*B*a^2*b^8*d^4*x^5*e^7 + 259896*A*a*b^9*d^4*x^5*e^7 + 449

820*B*a^2*b^8*d^5*x^4*e^6 + 99960*A*a*b^9*d^5*x^4*e^6 + 128520*B*a^2*b^8*d^6*x^3*e^5 + 28560*A*a*b^9*d^6*x^3*e

^5 + 25704*B*a^2*b^8*d^7*x^2*e^4 + 5712*A*a*b^9*d^7*x^2*e^4 + 3213*B*a^2*b^8*d^8*x*e^3 + 714*A*a*b^9*d^8*x*e^3

+ 189*B*a^2*b^8*d^9*e^2 + 42*A*a*b^9*d^9*e^2 + 10890880*B*a^3*b^7*x^8*e^11 + 4084080*A*a^2*b^8*x^8*e^11 + 871

2704*B*a^3*b^7*d*x^7*e^10 + 3267264*A*a^2*b^8*d*x^7*e^10 + 5544448*B*a^3*b^7*d^2*x^6*e^9 + 2079168*A*a^2*b^8*d

^2*x^6*e^9 + 2772224*B*a^3*b^7*d^3*x^5*e^8 + 1039584*A*a^2*b^8*d^3*x^5*e^8 + 1066240*B*a^3*b^7*d^4*x^4*e^7 + 3

99840*A*a^2*b^8*d^4*x^4*e^7 + 304640*B*a^3*b^7*d^5*x^3*e^6 + 114240*A*a^2*b^8*d^5*x^3*e^6 + 60928*B*a^3*b^7*d^

6*x^2*e^5 + 22848*A*a^2*b^8*d^6*x^2*e^5 + 7616*B*a^3*b^7*d^7*x*e^4 + 2856*A*a^2*b^8*d^7*x*e^4 + 448*B*a^3*b^7*

d^8*e^3 + 168*A*a^2*b^8*d^8*e^3 + 17153136*B*a^4*b^6*x^7*e^11 + 9801792*A*a^3*b^7*x^7*e^11 + 10915632*B*a^4*b^

6*d*x^6*e^10 + 6237504*A*a^3*b^7*d*x^6*e^10 + 5457816*B*a^4*b^6*d^2*x^5*e^9 + 3118752*A*a^3*b^7*d^2*x^5*e^9 +

2099160*B*a^4*b^6*d^3*x^4*e^8 + 1199520*A*a^3*b^7*d^3*x^4*e^8 + 599760*B*a^4*b^6*d^4*x^3*e^7 + 342720*A*a^3*b^

7*d^4*x^3*e^7 + 119952*B*a^4*b^6*d^5*x^2*e^6 + 68544*A*a^3*b^7*d^5*x^2*e^6 + 14994*B*a^4*b^6*d^6*x*e^5 + 8568*

A*a^3*b^7*d^6*x*e^5 + 882*B*a^4*b^6*d^7*e^4 + 504*A*a^3*b^7*d^7*e^4 + 18712512*B*a^5*b^5*x^6*e^11 + 15593760*A

*a^4*b^6*x^6*e^11 + 9356256*B*a^5*b^5*d*x^5*e^10 + 7796880*A*a^4*b^6*d*x^5*e^10 + 3598560*B*a^5*b^5*d^2*x^4*e^

9 + 2998800*A*a^4*b^6*d^2*x^4*e^9 + 1028160*B*a^5*b^5*d^3*x^3*e^8 + 856800*A*a^4*b^6*d^3*x^3*e^8 + 205632*B*a^

5*b^5*d^4*x^2*e^7 + 171360*A*a^4*b^6*d^4*x^2*e^7 + 25704*B*a^5*b^5*d^5*x*e^6 + 21420*A*a^4*b^6*d^5*x*e^6 + 151

2*B*a^5*b^5*d^6*e^5 + 1260*A*a^4*b^6*d^6*e^5 + 14294280*B*a^6*b^4*x^5*e^11 + 17153136*A*a^5*b^5*x^5*e^11 + 549

7800*B*a^6*b^4*d*x^4*e^10 + 6597360*A*a^5*b^5*d*x^4*e^10 + 1570800*B*a^6*b^4*d^2*x^3*e^9 + 1884960*A*a^5*b^5*d

^2*x^3*e^9 + 314160*B*a^6*b^4*d^3*x^2*e^8 + 376992*A*a^5*b^5*d^3*x^2*e^8 + 39270*B*a^6*b^4*d^4*x*e^7 + 47124*A

*a^5*b^5*d^4*x*e^7 + 2310*B*a^6*b^4*d^5*e^6 + 2772*A*a^5*b^5*d^5*e^6 + 7539840*B*a^7*b^3*x^4*e^11 + 13194720*A

*a^6*b^4*x^4*e^11 + 2154240*B*a^7*b^3*d*x^3*e^10 + 3769920*A*a^6*b^4*d*x^3*e^10 + 430848*B*a^7*b^3*d^2*x^2*e^9

+ 753984*A*a^6*b^4*d^2*x^2*e^9 + 53856*B*a^7*b^3*d^3*x*e^8 + 94248*A*a^6*b^4*d^3*x*e^8 + 3168*B*a^7*b^3*d^4*e

^7 + 5544*A*a^6*b^4*d^4*e^7 + 2625480*B*a^8*b^2*x^3*e^11 + 7001280*A*a^7*b^3*x^3*e^11 + 525096*B*a^8*b^2*d*x^2

*e^10 + 1400256*A*a^7*b^3*d*x^2*e^10 + 65637*B*a^8*b^2*d^2*x*e^9 + 175032*A*a^7*b^3*d^2*x*e^9 + 3861*B*a^8*b^2

*d^3*e^8 + 10296*A*a^7*b^3*d^3*e^8 + 544544*B*a^9*b*x^2*e^11 + 2450448*A*a^8*b^2*x^2*e^11 + 68068*B*a^9*b*d*x*

e^10 + 306306*A*a^8*b^2*d*x*e^10 + 4004*B*a^9*b*d^2*e^9 + 18018*A*a^8*b^2*d^2*e^9 + 51051*B*a^10*x*e^11 + 5105

10*A*a^9*b*x*e^11 + 3003*B*a^10*d*e^10 + 30030*A*a^9*b*d*e^10 + 48048*A*a^10*e^11)*e^(-12)/(x*e + d)^17

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Mupad [B]
time = 0.74, size = 2077, normalized size = 6.20 \begin {gather*} -\frac {\frac {3003\,B\,a^{10}\,d\,e^{10}+48048\,A\,a^{10}\,e^{11}+4004\,B\,a^9\,b\,d^2\,e^9+30030\,A\,a^9\,b\,d\,e^{10}+3861\,B\,a^8\,b^2\,d^3\,e^8+18018\,A\,a^8\,b^2\,d^2\,e^9+3168\,B\,a^7\,b^3\,d^4\,e^7+10296\,A\,a^7\,b^3\,d^3\,e^8+2310\,B\,a^6\,b^4\,d^5\,e^6+5544\,A\,a^6\,b^4\,d^4\,e^7+1512\,B\,a^5\,b^5\,d^6\,e^5+2772\,A\,a^5\,b^5\,d^5\,e^6+882\,B\,a^4\,b^6\,d^7\,e^4+1260\,A\,a^4\,b^6\,d^6\,e^5+448\,B\,a^3\,b^7\,d^8\,e^3+504\,A\,a^3\,b^7\,d^7\,e^4+189\,B\,a^2\,b^8\,d^9\,e^2+168\,A\,a^2\,b^8\,d^8\,e^3+60\,B\,a\,b^9\,d^{10}\,e+42\,A\,a\,b^9\,d^9\,e^2+11\,B\,b^{10}\,d^{11}+6\,A\,b^{10}\,d^{10}\,e}{816816\,e^{12}}+\frac {x\,\left (3003\,B\,a^{10}\,e^{10}+4004\,B\,a^9\,b\,d\,e^9+30030\,A\,a^9\,b\,e^{10}+3861\,B\,a^8\,b^2\,d^2\,e^8+18018\,A\,a^8\,b^2\,d\,e^9+3168\,B\,a^7\,b^3\,d^3\,e^7+10296\,A\,a^7\,b^3\,d^2\,e^8+2310\,B\,a^6\,b^4\,d^4\,e^6+5544\,A\,a^6\,b^4\,d^3\,e^7+1512\,B\,a^5\,b^5\,d^5\,e^5+2772\,A\,a^5\,b^5\,d^4\,e^6+882\,B\,a^4\,b^6\,d^6\,e^4+1260\,A\,a^4\,b^6\,d^5\,e^5+448\,B\,a^3\,b^7\,d^7\,e^3+504\,A\,a^3\,b^7\,d^6\,e^4+189\,B\,a^2\,b^8\,d^8\,e^2+168\,A\,a^2\,b^8\,d^7\,e^3+60\,B\,a\,b^9\,d^9\,e+42\,A\,a\,b^9\,d^8\,e^2+11\,B\,b^{10}\,d^{10}+6\,A\,b^{10}\,d^9\,e\right )}{48048\,e^{11}}+\frac {5\,b^7\,x^8\,\left (448\,B\,a^3\,e^3+189\,B\,a^2\,b\,d\,e^2+168\,A\,a^2\,b\,e^3+60\,B\,a\,b^2\,d^2\,e+42\,A\,a\,b^2\,d\,e^2+11\,B\,b^3\,d^3+6\,A\,b^3\,d^2\,e\right )}{168\,e^4}+\frac {b^4\,x^5\,\left (2310\,B\,a^6\,e^6+1512\,B\,a^5\,b\,d\,e^5+2772\,A\,a^5\,b\,e^6+882\,B\,a^4\,b^2\,d^2\,e^4+1260\,A\,a^4\,b^2\,d\,e^5+448\,B\,a^3\,b^3\,d^3\,e^3+504\,A\,a^3\,b^3\,d^2\,e^4+189\,B\,a^2\,b^4\,d^4\,e^2+168\,A\,a^2\,b^4\,d^3\,e^3+60\,B\,a\,b^5\,d^5\,e+42\,A\,a\,b^5\,d^4\,e^2+11\,B\,b^6\,d^6+6\,A\,b^6\,d^5\,e\right )}{132\,e^7}+\frac {b^9\,x^{10}\,\left (6\,A\,b\,e+60\,B\,a\,e+11\,B\,b\,d\right )}{42\,e^2}+\frac {b^6\,x^7\,\left (882\,B\,a^4\,e^4+448\,B\,a^3\,b\,d\,e^3+504\,A\,a^3\,b\,e^4+189\,B\,a^2\,b^2\,d^2\,e^2+168\,A\,a^2\,b^2\,d\,e^3+60\,B\,a\,b^3\,d^3\,e+42\,A\,a\,b^3\,d^2\,e^2+11\,B\,b^4\,d^4+6\,A\,b^4\,d^3\,e\right )}{42\,e^5}+\frac {5\,b^3\,x^4\,\left (3168\,B\,a^7\,e^7+2310\,B\,a^6\,b\,d\,e^6+5544\,A\,a^6\,b\,e^7+1512\,B\,a^5\,b^2\,d^2\,e^5+2772\,A\,a^5\,b^2\,d\,e^6+882\,B\,a^4\,b^3\,d^3\,e^4+1260\,A\,a^4\,b^3\,d^2\,e^5+448\,B\,a^3\,b^4\,d^4\,e^3+504\,A\,a^3\,b^4\,d^3\,e^4+189\,B\,a^2\,b^5\,d^5\,e^2+168\,A\,a^2\,b^5\,d^4\,e^3+60\,B\,a\,b^6\,d^6\,e+42\,A\,a\,b^6\,d^5\,e^2+11\,B\,b^7\,d^7+6\,A\,b^7\,d^6\,e\right )}{1716\,e^8}+\frac {b\,x^2\,\left (4004\,B\,a^9\,e^9+3861\,B\,a^8\,b\,d\,e^8+18018\,A\,a^8\,b\,e^9+3168\,B\,a^7\,b^2\,d^2\,e^7+10296\,A\,a^7\,b^2\,d\,e^8+2310\,B\,a^6\,b^3\,d^3\,e^6+5544\,A\,a^6\,b^3\,d^2\,e^7+1512\,B\,a^5\,b^4\,d^4\,e^5+2772\,A\,a^5\,b^4\,d^3\,e^6+882\,B\,a^4\,b^5\,d^5\,e^4+1260\,A\,a^4\,b^5\,d^4\,e^5+448\,B\,a^3\,b^6\,d^6\,e^3+504\,A\,a^3\,b^6\,d^5\,e^4+189\,B\,a^2\,b^7\,d^7\,e^2+168\,A\,a^2\,b^7\,d^6\,e^3+60\,B\,a\,b^8\,d^8\,e+42\,A\,a\,b^8\,d^7\,e^2+11\,B\,b^9\,d^9+6\,A\,b^9\,d^8\,e\right )}{6006\,e^{10}}+\frac {5\,b^8\,x^9\,\left (189\,B\,a^2\,e^2+60\,B\,a\,b\,d\,e+42\,A\,a\,b\,e^2+11\,B\,b^2\,d^2+6\,A\,b^2\,d\,e\right )}{168\,e^3}+\frac {b^5\,x^6\,\left (1512\,B\,a^5\,e^5+882\,B\,a^4\,b\,d\,e^4+1260\,A\,a^4\,b\,e^5+448\,B\,a^3\,b^2\,d^2\,e^3+504\,A\,a^3\,b^2\,d\,e^4+189\,B\,a^2\,b^3\,d^3\,e^2+168\,A\,a^2\,b^3\,d^2\,e^3+60\,B\,a\,b^4\,d^4\,e+42\,A\,a\,b^4\,d^3\,e^2+11\,B\,b^5\,d^5+6\,A\,b^5\,d^4\,e\right )}{66\,e^6}+\frac {5\,b^2\,x^3\,\left (3861\,B\,a^8\,e^8+3168\,B\,a^7\,b\,d\,e^7+10296\,A\,a^7\,b\,e^8+2310\,B\,a^6\,b^2\,d^2\,e^6+5544\,A\,a^6\,b^2\,d\,e^7+1512\,B\,a^5\,b^3\,d^3\,e^5+2772\,A\,a^5\,b^3\,d^2\,e^6+882\,B\,a^4\,b^4\,d^4\,e^4+1260\,A\,a^4\,b^4\,d^3\,e^5+448\,B\,a^3\,b^5\,d^5\,e^3+504\,A\,a^3\,b^5\,d^4\,e^4+189\,B\,a^2\,b^6\,d^6\,e^2+168\,A\,a^2\,b^6\,d^5\,e^3+60\,B\,a\,b^7\,d^7\,e+42\,A\,a\,b^7\,d^6\,e^2+11\,B\,b^8\,d^8+6\,A\,b^8\,d^7\,e\right )}{6006\,e^9}+\frac {B\,b^{10}\,x^{11}}{6\,e}}{d^{17}+17\,d^{16}\,e\,x+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\,e^{16}\,x^{16}+e^{17}\,x^{17}} \end {gather*}

-((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)

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