∫ (a+bx)10(A+Bx)____________ (d+ex)17 dx [1105]

Optimal. Leaf size=285 \[ -\frac {(B d-A e) (a+b x)^{11}}{16 e (b d-a e) (d+e x)^{16}}+\frac {(11 b B d+5 A b e-16 a B e) (a+b x)^{11}}{240 e (b d-a e)^2 (d+e x)^{15}}+\frac {b (11 b B d+5 A b e-16 a B e) (a+b x)^{11}}{840 e (b d-a e)^3 (d+e x)^{14}}+\frac {b^2 (11 b B d+5 A b e-16 a B e) (a+b x)^{11}}{3640 e (b d-a e)^4 (d+e x)^{13}}+\frac {b^3 (11 b B d+5 A b e-16 a B e) (a+b x)^{11}}{21840 e (b d-a e)^5 (d+e x)^{12}}+\frac {b^4 (11 b B d+5 A b e-16 a B e) (a+b x)^{11}}{240240 e (b d-a e)^6 (d+e x)^{11}} \]

-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

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Rubi [A]
time = 0.10, antiderivative size = 285, normalized size of antiderivative = 1.00, number of steps used = 6, 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^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)} \end {gather*}

-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)

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)^{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) \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*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1429\) vs. \(2(285)=570\).
time = 0.55, size = 1429, normalized size = 5.01 \begin {gather*} -\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}} \end {gather*}

-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)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1941\) vs. \(2(273)=546\).
time = 0.13, size = 1942, normalized size = 6.81


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)^17,x,method=_RETURNVERBOSE)

-15/13*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)^13-1/16*(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)^16-15/8*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)^8-10/3*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)^9-5/7*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)^7-1/6*b^9/e^12*(A*b

*e+10*B*a*e-11*B*b*d)/(e*x+d)^6-5/2*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)^12-1/5*b^10*B/e^12/(e*x+d)^5-21/5*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)^10-5/14*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-11*B*b^9*d^9)/(e*x+d)^14-42/11*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)^11-1/15/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+

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1996 vs. \(2 (290) = 580\).
time = 0.48, size = 1996, normalized size = 7.00 \begin {gather*} \text {Too large to display} \end {gather*}

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

-1/240240*(48048*B*b^10*x^11*e^11 + 11*B*b^10*d^11 + 15015*A*a^10*e^11 + 5*(10*B*a*b^9*e + A*b^10*e)*d^10 + 80

08*(11*B*b^10*d*e^10 + 50*B*a*b^9*e^11 + 5*A*b^10*e^11)*x^10 + 15*(9*B*a^2*b^8*e^2 + 2*A*a*b^9*e^2)*d^9 + 1144

0*(11*B*b^10*d^2*e^9 + 135*B*a^2*b^8*e^11 + 30*A*a*b^9*e^11 + 5*(10*B*a*b^9*e^10 + A*b^10*e^10)*d)*x^9 + 35*(8

*B*a^3*b^7*e^3 + 3*A*a^2*b^8*e^3)*d^8 + 12870*(11*B*b^10*d^3*e^8 + 280*B*a^3*b^7*e^11 + 105*A*a^2*b^8*e^11 + 5

*(10*B*a*b^9*e^9 + A*b^10*e^9)*d^2 + 15*(9*B*a^2*b^8*e^10 + 2*A*a*b^9*e^10)*d)*x^8 + 70*(7*B*a^4*b^6*e^4 + 4*A

*a^3*b^7*e^4)*d^7 + 11440*(11*B*b^10*d^4*e^7 + 490*B*a^4*b^6*e^11 + 280*A*a^3*b^7*e^11 + 5*(10*B*a*b^9*e^8 + A

*b^10*e^8)*d^3 + 15*(9*B*a^2*b^8*e^9 + 2*A*a*b^9*e^9)*d^2 + 35*(8*B*a^3*b^7*e^10 + 3*A*a^2*b^8*e^10)*d)*x^7 +

126*(6*B*a^5*b^5*e^5 + 5*A*a^4*b^6*e^5)*d^6 + 8008*(11*B*b^10*d^5*e^6 + 756*B*a^5*b^5*e^11 + 630*A*a^4*b^6*e^1

1 + 5*(10*B*a*b^9*e^7 + A*b^10*e^7)*d^4 + 15*(9*B*a^2*b^8*e^8 + 2*A*a*b^9*e^8)*d^3 + 35*(8*B*a^3*b^7*e^9 + 3*A

*a^2*b^8*e^9)*d^2 + 70*(7*B*a^4*b^6*e^10 + 4*A*a^3*b^7*e^10)*d)*x^6 + 210*(5*B*a^6*b^4*e^6 + 6*A*a^5*b^5*e^6)*

d^5 + 4368*(11*B*b^10*d^6*e^5 + 1050*B*a^6*b^4*e^11 + 1260*A*a^5*b^5*e^11 + 5*(10*B*a*b^9*e^6 + A*b^10*e^6)*d^

5 + 15*(9*B*a^2*b^8*e^7 + 2*A*a*b^9*e^7)*d^4 + 35*(8*B*a^3*b^7*e^8 + 3*A*a^2*b^8*e^8)*d^3 + 70*(7*B*a^4*b^6*e^

9 + 4*A*a^3*b^7*e^9)*d^2 + 126*(6*B*a^5*b^5*e^10 + 5*A*a^4*b^6*e^10)*d)*x^5 + 330*(4*B*a^7*b^3*e^7 + 7*A*a^6*b

^4*e^7)*d^4 + 1820*(11*B*b^10*d^7*e^4 + 1320*B*a^7*b^3*e^11 + 2310*A*a^6*b^4*e^11 + 5*(10*B*a*b^9*e^5 + A*b^10

*e^5)*d^6 + 15*(9*B*a^2*b^8*e^6 + 2*A*a*b^9*e^6)*d^5 + 35*(8*B*a^3*b^7*e^7 + 3*A*a^2*b^8*e^7)*d^4 + 70*(7*B*a^

4*b^6*e^8 + 4*A*a^3*b^7*e^8)*d^3 + 126*(6*B*a^5*b^5*e^9 + 5*A*a^4*b^6*e^9)*d^2 + 210*(5*B*a^6*b^4*e^10 + 6*A*a

^5*b^5*e^10)*d)*x^4 + 495*(3*B*a^8*b^2*e^8 + 8*A*a^7*b^3*e^8)*d^3 + 560*(11*B*b^10*d^8*e^3 + 1485*B*a^8*b^2*e^

11 + 3960*A*a^7*b^3*e^11 + 5*(10*B*a*b^9*e^4 + A*b^10*e^4)*d^7 + 15*(9*B*a^2*b^8*e^5 + 2*A*a*b^9*e^5)*d^6 + 35

*(8*B*a^3*b^7*e^6 + 3*A*a^2*b^8*e^6)*d^5 + 70*(7*B*a^4*b^6*e^7 + 4*A*a^3*b^7*e^7)*d^4 + 126*(6*B*a^5*b^5*e^8 +

5*A*a^4*b^6*e^8)*d^3 + 210*(5*B*a^6*b^4*e^9 + 6*A*a^5*b^5*e^9)*d^2 + 330*(4*B*a^7*b^3*e^10 + 7*A*a^6*b^4*e^10

)*d)*x^3 + 715*(2*B*a^9*b*e^9 + 9*A*a^8*b^2*e^9)*d^2 + 120*(11*B*b^10*d^9*e^2 + 1430*B*a^9*b*e^11 + 6435*A*a^8

*b^2*e^11 + 5*(10*B*a*b^9*e^3 + A*b^10*e^3)*d^8 + 15*(9*B*a^2*b^8*e^4 + 2*A*a*b^9*e^4)*d^7 + 35*(8*B*a^3*b^7*e

^5 + 3*A*a^2*b^8*e^5)*d^6 + 70*(7*B*a^4*b^6*e^6 + 4*A*a^3*b^7*e^6)*d^5 + 126*(6*B*a^5*b^5*e^7 + 5*A*a^4*b^6*e^

7)*d^4 + 210*(5*B*a^6*b^4*e^8 + 6*A*a^5*b^5*e^8)*d^3 + 330*(4*B*a^7*b^3*e^9 + 7*A*a^6*b^4*e^9)*d^2 + 495*(3*B*

a^8*b^2*e^10 + 8*A*a^7*b^3*e^10)*d)*x^2 + 1001*(B*a^10*e^10 + 10*A*a^9*b*e^10)*d + 16*(11*B*b^10*d^10*e + 1001

*B*a^10*e^11 + 10010*A*a^9*b*e^11 + 5*(10*B*a*b^9*e^2 + A*b^10*e^2)*d^9 + 15*(9*B*a^2*b^8*e^3 + 2*A*a*b^9*e^3)

*d^8 + 35*(8*B*a^3*b^7*e^4 + 3*A*a^2*b^8*e^4)*d^7 + 70*(7*B*a^4*b^6*e^5 + 4*A*a^3*b^7*e^5)*d^6 + 126*(6*B*a^5*

b^5*e^6 + 5*A*a^4*b^6*e^6)*d^5 + 210*(5*B*a^6*b^4*e^7 + 6*A*a^5*b^5*e^7)*d^4 + 330*(4*B*a^7*b^3*e^8 + 7*A*a^6*

b^4*e^8)*d^3 + 495*(3*B*a^8*b^2*e^9 + 8*A*a^7*b^3*e^9)*d^2 + 715*(2*B*a^9*b*e^10 + 9*A*a^8*b^2*e^10)*d)*x)/(x^

16*e^28 + 16*d*x^15*e^27 + 120*d^2*x^14*e^26 + 560*d^3*x^13*e^25 + 1820*d^4*x^12*e^24 + 4368*d^5*x^11*e^23 + 8

008*d^6*x^10*e^22 + 11440*d^7*x^9*e^21 + 12870*d^8*x^8*e^20 + 11440*d^9*x^7*e^19 + 8008*d^10*x^6*e^18 + 4368*d

^11*x^5*e^17 + 1820*d^12*x^4*e^16 + 560*d^13*x^3*e^15 + 120*d^14*x^2*e^14 + 16*d^15*x*e^13 + d^16*e^12)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1907 vs. \(2 (290) = 580\).
time = 1.38, size = 1907, normalized size = 6.69 \begin {gather*} \text {Too large to display} \end {gather*}

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

-1/240240*(11*B*b^10*d^11 + (48048*B*b^10*x^11 + 15015*A*a^10 + 40040*(10*B*a*b^9 + A*b^10)*x^10 + 171600*(9*B

*a^2*b^8 + 2*A*a*b^9)*x^9 + 450450*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 800800*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 +

1009008*(6*B*a^5*b^5 + 5*A*a^4*b^6)*x^6 + 917280*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 600600*(4*B*a^7*b^3 + 7*A*a

^6*b^4)*x^4 + 277200*(3*B*a^8*b^2 + 8*A*a^7*b^3)*x^3 + 85800*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 16016*(B*a^10 + 1

0*A*a^9*b)*x)*e^11 + (88088*B*b^10*d*x^10 + 57200*(10*B*a*b^9 + A*b^10)*d*x^9 + 193050*(9*B*a^2*b^8 + 2*A*a*b^

9)*d*x^8 + 400400*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*x^7 + 560560*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*x^6 + 550368*(6*B*a

^5*b^5 + 5*A*a^4*b^6)*d*x^5 + 382200*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*x^4 + 184800*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*

x^3 + 59400*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d*x^2 + 11440*(2*B*a^9*b + 9*A*a^8*b^2)*d*x + 1001*(B*a^10 + 10*A*a^9*

b)*d)*e^10 + 5*(25168*B*b^10*d^2*x^9 + 12870*(10*B*a*b^9 + A*b^10)*d^2*x^8 + 34320*(9*B*a^2*b^8 + 2*A*a*b^9)*d

^2*x^7 + 56056*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*x^6 + 61152*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*x^5 + 45864*(6*B*a^

5*b^5 + 5*A*a^4*b^6)*d^2*x^4 + 23520*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*x^3 + 7920*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^

2*x^2 + 1584*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*x + 143*(2*B*a^9*b + 9*A*a^8*b^2)*d^2)*e^9 + 5*(28314*B*b^10*d^3*

x^8 + 11440*(10*B*a*b^9 + A*b^10)*d^3*x^7 + 24024*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*x^6 + 30576*(8*B*a^3*b^7 + 3*A

*a^2*b^8)*d^3*x^5 + 25480*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*x^4 + 14112*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*x^3 + 50

40*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*x^2 + 1056*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*x + 99*(3*B*a^8*b^2 + 8*A*a^7*b^

3)*d^3)*e^8 + 10*(12584*B*b^10*d^4*x^7 + 4004*(10*B*a*b^9 + A*b^10)*d^4*x^6 + 6552*(9*B*a^2*b^8 + 2*A*a*b^9)*d

^4*x^5 + 6370*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*x^4 + 3920*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*x^3 + 1512*(6*B*a^5*b

^5 + 5*A*a^4*b^6)*d^4*x^2 + 336*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*x + 33*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4)*e^7 +

14*(6292*B*b^10*d^5*x^6 + 1560*(10*B*a*b^9 + A*b^10)*d^5*x^5 + 1950*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*x^4 + 1400*(

8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*x^3 + 600*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*x^2 + 144*(6*B*a^5*b^5 + 5*A*a^4*b^6)

*d^5*x + 15*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5)*e^6 + 14*(3432*B*b^10*d^6*x^5 + 650*(10*B*a*b^9 + A*b^10)*d^6*x^4

+ 600*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*x^3 + 300*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*x^2 + 80*(7*B*a^4*b^6 + 4*A*a^3

*b^7)*d^6*x + 9*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6)*e^5 + 10*(2002*B*b^10*d^7*x^4 + 280*(10*B*a*b^9 + A*b^10)*d^7

*x^3 + 180*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*x^2 + 56*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*x + 7*(7*B*a^4*b^6 + 4*A*a^3

*b^7)*d^7)*e^4 + 5*(1232*B*b^10*d^8*x^3 + 120*(10*B*a*b^9 + A*b^10)*d^8*x^2 + 48*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8

*x + 7*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8)*e^3 + 5*(264*B*b^10*d^9*x^2 + 16*(10*B*a*b^9 + A*b^10)*d^9*x + 3*(9*B*

a^2*b^8 + 2*A*a*b^9)*d^9)*e^2 + (176*B*b^10*d^10*x + 5*(10*B*a*b^9 + A*b^10)*d^10)*e)/(x^16*e^28 + 16*d*x^15*e

^27 + 120*d^2*x^14*e^26 + 560*d^3*x^13*e^25 + 1820*d^4*x^12*e^24 + 4368*d^5*x^11*e^23 + 8008*d^6*x^10*e^22 + 1

1440*d^7*x^9*e^21 + 12870*d^8*x^8*e^20 + 11440*d^9*x^7*e^19 + 8008*d^10*x^6*e^18 + 4368*d^11*x^5*e^17 + 1820*d

^12*x^4*e^16 + 560*d^13*x^3*e^15 + 120*d^14*x^2*e^14 + 16*d^15*x*e^13 + d^16*e^12)

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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. \(2 (290) = 580\).
time = 1.12, size = 2096, normalized size = 7.35 \begin {gather*} \text {Too large to display} \end {gather*}

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

-1/240240*(48048*B*b^10*x^11*e^11 + 88088*B*b^10*d*x^10*e^10 + 125840*B*b^10*d^2*x^9*e^9 + 141570*B*b^10*d^3*x

^8*e^8 + 125840*B*b^10*d^4*x^7*e^7 + 88088*B*b^10*d^5*x^6*e^6 + 48048*B*b^10*d^6*x^5*e^5 + 20020*B*b^10*d^7*x^

4*e^4 + 6160*B*b^10*d^8*x^3*e^3 + 1320*B*b^10*d^9*x^2*e^2 + 176*B*b^10*d^10*x*e + 11*B*b^10*d^11 + 400400*B*a*

b^9*x^10*e^11 + 40040*A*b^10*x^10*e^11 + 572000*B*a*b^9*d*x^9*e^10 + 57200*A*b^10*d*x^9*e^10 + 643500*B*a*b^9*

d^2*x^8*e^9 + 64350*A*b^10*d^2*x^8*e^9 + 572000*B*a*b^9*d^3*x^7*e^8 + 57200*A*b^10*d^3*x^7*e^8 + 400400*B*a*b^

9*d^4*x^6*e^7 + 40040*A*b^10*d^4*x^6*e^7 + 218400*B*a*b^9*d^5*x^5*e^6 + 21840*A*b^10*d^5*x^5*e^6 + 91000*B*a*b

^9*d^6*x^4*e^5 + 9100*A*b^10*d^6*x^4*e^5 + 28000*B*a*b^9*d^7*x^3*e^4 + 2800*A*b^10*d^7*x^3*e^4 + 6000*B*a*b^9*

d^8*x^2*e^3 + 600*A*b^10*d^8*x^2*e^3 + 800*B*a*b^9*d^9*x*e^2 + 80*A*b^10*d^9*x*e^2 + 50*B*a*b^9*d^10*e + 5*A*b

^10*d^10*e + 1544400*B*a^2*b^8*x^9*e^11 + 343200*A*a*b^9*x^9*e^11 + 1737450*B*a^2*b^8*d*x^8*e^10 + 386100*A*a*

b^9*d*x^8*e^10 + 1544400*B*a^2*b^8*d^2*x^7*e^9 + 343200*A*a*b^9*d^2*x^7*e^9 + 1081080*B*a^2*b^8*d^3*x^6*e^8 +

240240*A*a*b^9*d^3*x^6*e^8 + 589680*B*a^2*b^8*d^4*x^5*e^7 + 131040*A*a*b^9*d^4*x^5*e^7 + 245700*B*a^2*b^8*d^5*

x^4*e^6 + 54600*A*a*b^9*d^5*x^4*e^6 + 75600*B*a^2*b^8*d^6*x^3*e^5 + 16800*A*a*b^9*d^6*x^3*e^5 + 16200*B*a^2*b^

8*d^7*x^2*e^4 + 3600*A*a*b^9*d^7*x^2*e^4 + 2160*B*a^2*b^8*d^8*x*e^3 + 480*A*a*b^9*d^8*x*e^3 + 135*B*a^2*b^8*d^

9*e^2 + 30*A*a*b^9*d^9*e^2 + 3603600*B*a^3*b^7*x^8*e^11 + 1351350*A*a^2*b^8*x^8*e^11 + 3203200*B*a^3*b^7*d*x^7

*e^10 + 1201200*A*a^2*b^8*d*x^7*e^10 + 2242240*B*a^3*b^7*d^2*x^6*e^9 + 840840*A*a^2*b^8*d^2*x^6*e^9 + 1223040*

B*a^3*b^7*d^3*x^5*e^8 + 458640*A*a^2*b^8*d^3*x^5*e^8 + 509600*B*a^3*b^7*d^4*x^4*e^7 + 191100*A*a^2*b^8*d^4*x^4

*e^7 + 156800*B*a^3*b^7*d^5*x^3*e^6 + 58800*A*a^2*b^8*d^5*x^3*e^6 + 33600*B*a^3*b^7*d^6*x^2*e^5 + 12600*A*a^2*

b^8*d^6*x^2*e^5 + 4480*B*a^3*b^7*d^7*x*e^4 + 1680*A*a^2*b^8*d^7*x*e^4 + 280*B*a^3*b^7*d^8*e^3 + 105*A*a^2*b^8*

d^8*e^3 + 5605600*B*a^4*b^6*x^7*e^11 + 3203200*A*a^3*b^7*x^7*e^11 + 3923920*B*a^4*b^6*d*x^6*e^10 + 2242240*A*a

^3*b^7*d*x^6*e^10 + 2140320*B*a^4*b^6*d^2*x^5*e^9 + 1223040*A*a^3*b^7*d^2*x^5*e^9 + 891800*B*a^4*b^6*d^3*x^4*e

^8 + 509600*A*a^3*b^7*d^3*x^4*e^8 + 274400*B*a^4*b^6*d^4*x^3*e^7 + 156800*A*a^3*b^7*d^4*x^3*e^7 + 58800*B*a^4*

b^6*d^5*x^2*e^6 + 33600*A*a^3*b^7*d^5*x^2*e^6 + 7840*B*a^4*b^6*d^6*x*e^5 + 4480*A*a^3*b^7*d^6*x*e^5 + 490*B*a^

4*b^6*d^7*e^4 + 280*A*a^3*b^7*d^7*e^4 + 6054048*B*a^5*b^5*x^6*e^11 + 5045040*A*a^4*b^6*x^6*e^11 + 3302208*B*a^

5*b^5*d*x^5*e^10 + 2751840*A*a^4*b^6*d*x^5*e^10 + 1375920*B*a^5*b^5*d^2*x^4*e^9 + 1146600*A*a^4*b^6*d^2*x^4*e^

9 + 423360*B*a^5*b^5*d^3*x^3*e^8 + 352800*A*a^4*b^6*d^3*x^3*e^8 + 90720*B*a^5*b^5*d^4*x^2*e^7 + 75600*A*a^4*b^

6*d^4*x^2*e^7 + 12096*B*a^5*b^5*d^5*x*e^6 + 10080*A*a^4*b^6*d^5*x*e^6 + 756*B*a^5*b^5*d^6*e^5 + 630*A*a^4*b^6*

d^6*e^5 + 4586400*B*a^6*b^4*x^5*e^11 + 5503680*A*a^5*b^5*x^5*e^11 + 1911000*B*a^6*b^4*d*x^4*e^10 + 2293200*A*a

^5*b^5*d*x^4*e^10 + 588000*B*a^6*b^4*d^2*x^3*e^9 + 705600*A*a^5*b^5*d^2*x^3*e^9 + 126000*B*a^6*b^4*d^3*x^2*e^8

+ 151200*A*a^5*b^5*d^3*x^2*e^8 + 16800*B*a^6*b^4*d^4*x*e^7 + 20160*A*a^5*b^5*d^4*x*e^7 + 1050*B*a^6*b^4*d^5*e

^6 + 1260*A*a^5*b^5*d^5*e^6 + 2402400*B*a^7*b^3*x^4*e^11 + 4204200*A*a^6*b^4*x^4*e^11 + 739200*B*a^7*b^3*d*x^3

*e^10 + 1293600*A*a^6*b^4*d*x^3*e^10 + 158400*B*a^7*b^3*d^2*x^2*e^9 + 277200*A*a^6*b^4*d^2*x^2*e^9 + 21120*B*a

^7*b^3*d^3*x*e^8 + 36960*A*a^6*b^4*d^3*x*e^8 + 1320*B*a^7*b^3*d^4*e^7 + 2310*A*a^6*b^4*d^4*e^7 + 831600*B*a^8*

b^2*x^3*e^11 + 2217600*A*a^7*b^3*x^3*e^11 + 178200*B*a^8*b^2*d*x^2*e^10 + 475200*A*a^7*b^3*d*x^2*e^10 + 23760*

B*a^8*b^2*d^2*x*e^9 + 63360*A*a^7*b^3*d^2*x*e^9 + 1485*B*a^8*b^2*d^3*e^8 + 3960*A*a^7*b^3*d^3*e^8 + 171600*B*a

^9*b*x^2*e^11 + 772200*A*a^8*b^2*x^2*e^11 + 22880*B*a^9*b*d*x*e^10 + 102960*A*a^8*b^2*d*x*e^10 + 1430*B*a^9*b*

d^2*e^9 + 6435*A*a^8*b^2*d^2*e^9 + 16016*B*a^10*x*e^11 + 160160*A*a^9*b*x*e^11 + 1001*B*a^10*d*e^10 + 10010*A*

a^9*b*d*e^10 + 15015*A*a^10*e^11)*e^(-12)/(x*e + d)^16

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Mupad [B]
time = 0.64, size = 2066, normalized size = 7.25 \begin {gather*} -\frac {\frac {1001\,B\,a^{10}\,d\,e^{10}+15015\,A\,a^{10}\,e^{11}+1430\,B\,a^9\,b\,d^2\,e^9+10010\,A\,a^9\,b\,d\,e^{10}+1485\,B\,a^8\,b^2\,d^3\,e^8+6435\,A\,a^8\,b^2\,d^2\,e^9+1320\,B\,a^7\,b^3\,d^4\,e^7+3960\,A\,a^7\,b^3\,d^3\,e^8+1050\,B\,a^6\,b^4\,d^5\,e^6+2310\,A\,a^6\,b^4\,d^4\,e^7+756\,B\,a^5\,b^5\,d^6\,e^5+1260\,A\,a^5\,b^5\,d^5\,e^6+490\,B\,a^4\,b^6\,d^7\,e^4+630\,A\,a^4\,b^6\,d^6\,e^5+280\,B\,a^3\,b^7\,d^8\,e^3+280\,A\,a^3\,b^7\,d^7\,e^4+135\,B\,a^2\,b^8\,d^9\,e^2+105\,A\,a^2\,b^8\,d^8\,e^3+50\,B\,a\,b^9\,d^{10}\,e+30\,A\,a\,b^9\,d^9\,e^2+11\,B\,b^{10}\,d^{11}+5\,A\,b^{10}\,d^{10}\,e}{240240\,e^{12}}+\frac {x\,\left (1001\,B\,a^{10}\,e^{10}+1430\,B\,a^9\,b\,d\,e^9+10010\,A\,a^9\,b\,e^{10}+1485\,B\,a^8\,b^2\,d^2\,e^8+6435\,A\,a^8\,b^2\,d\,e^9+1320\,B\,a^7\,b^3\,d^3\,e^7+3960\,A\,a^7\,b^3\,d^2\,e^8+1050\,B\,a^6\,b^4\,d^4\,e^6+2310\,A\,a^6\,b^4\,d^3\,e^7+756\,B\,a^5\,b^5\,d^5\,e^5+1260\,A\,a^5\,b^5\,d^4\,e^6+490\,B\,a^4\,b^6\,d^6\,e^4+630\,A\,a^4\,b^6\,d^5\,e^5+280\,B\,a^3\,b^7\,d^7\,e^3+280\,A\,a^3\,b^7\,d^6\,e^4+135\,B\,a^2\,b^8\,d^8\,e^2+105\,A\,a^2\,b^8\,d^7\,e^3+50\,B\,a\,b^9\,d^9\,e+30\,A\,a\,b^9\,d^8\,e^2+11\,B\,b^{10}\,d^{10}+5\,A\,b^{10}\,d^9\,e\right )}{15015\,e^{11}}+\frac {3\,b^7\,x^8\,\left (280\,B\,a^3\,e^3+135\,B\,a^2\,b\,d\,e^2+105\,A\,a^2\,b\,e^3+50\,B\,a\,b^2\,d^2\,e+30\,A\,a\,b^2\,d\,e^2+11\,B\,b^3\,d^3+5\,A\,b^3\,d^2\,e\right )}{56\,e^4}+\frac {b^4\,x^5\,\left (1050\,B\,a^6\,e^6+756\,B\,a^5\,b\,d\,e^5+1260\,A\,a^5\,b\,e^6+490\,B\,a^4\,b^2\,d^2\,e^4+630\,A\,a^4\,b^2\,d\,e^5+280\,B\,a^3\,b^3\,d^3\,e^3+280\,A\,a^3\,b^3\,d^2\,e^4+135\,B\,a^2\,b^4\,d^4\,e^2+105\,A\,a^2\,b^4\,d^3\,e^3+50\,B\,a\,b^5\,d^5\,e+30\,A\,a\,b^5\,d^4\,e^2+11\,B\,b^6\,d^6+5\,A\,b^6\,d^5\,e\right )}{55\,e^7}+\frac {b^9\,x^{10}\,\left (5\,A\,b\,e+50\,B\,a\,e+11\,B\,b\,d\right )}{30\,e^2}+\frac {b^6\,x^7\,\left (490\,B\,a^4\,e^4+280\,B\,a^3\,b\,d\,e^3+280\,A\,a^3\,b\,e^4+135\,B\,a^2\,b^2\,d^2\,e^2+105\,A\,a^2\,b^2\,d\,e^3+50\,B\,a\,b^3\,d^3\,e+30\,A\,a\,b^3\,d^2\,e^2+11\,B\,b^4\,d^4+5\,A\,b^4\,d^3\,e\right )}{21\,e^5}+\frac {b^3\,x^4\,\left (1320\,B\,a^7\,e^7+1050\,B\,a^6\,b\,d\,e^6+2310\,A\,a^6\,b\,e^7+756\,B\,a^5\,b^2\,d^2\,e^5+1260\,A\,a^5\,b^2\,d\,e^6+490\,B\,a^4\,b^3\,d^3\,e^4+630\,A\,a^4\,b^3\,d^2\,e^5+280\,B\,a^3\,b^4\,d^4\,e^3+280\,A\,a^3\,b^4\,d^3\,e^4+135\,B\,a^2\,b^5\,d^5\,e^2+105\,A\,a^2\,b^5\,d^4\,e^3+50\,B\,a\,b^6\,d^6\,e+30\,A\,a\,b^6\,d^5\,e^2+11\,B\,b^7\,d^7+5\,A\,b^7\,d^6\,e\right )}{132\,e^8}+\frac {b\,x^2\,\left (1430\,B\,a^9\,e^9+1485\,B\,a^8\,b\,d\,e^8+6435\,A\,a^8\,b\,e^9+1320\,B\,a^7\,b^2\,d^2\,e^7+3960\,A\,a^7\,b^2\,d\,e^8+1050\,B\,a^6\,b^3\,d^3\,e^6+2310\,A\,a^6\,b^3\,d^2\,e^7+756\,B\,a^5\,b^4\,d^4\,e^5+1260\,A\,a^5\,b^4\,d^3\,e^6+490\,B\,a^4\,b^5\,d^5\,e^4+630\,A\,a^4\,b^5\,d^4\,e^5+280\,B\,a^3\,b^6\,d^6\,e^3+280\,A\,a^3\,b^6\,d^5\,e^4+135\,B\,a^2\,b^7\,d^7\,e^2+105\,A\,a^2\,b^7\,d^6\,e^3+50\,B\,a\,b^8\,d^8\,e+30\,A\,a\,b^8\,d^7\,e^2+11\,B\,b^9\,d^9+5\,A\,b^9\,d^8\,e\right )}{2002\,e^{10}}+\frac {b^8\,x^9\,\left (135\,B\,a^2\,e^2+50\,B\,a\,b\,d\,e+30\,A\,a\,b\,e^2+11\,B\,b^2\,d^2+5\,A\,b^2\,d\,e\right )}{21\,e^3}+\frac {b^5\,x^6\,\left (756\,B\,a^5\,e^5+490\,B\,a^4\,b\,d\,e^4+630\,A\,a^4\,b\,e^5+280\,B\,a^3\,b^2\,d^2\,e^3+280\,A\,a^3\,b^2\,d\,e^4+135\,B\,a^2\,b^3\,d^3\,e^2+105\,A\,a^2\,b^3\,d^2\,e^3+50\,B\,a\,b^4\,d^4\,e+30\,A\,a\,b^4\,d^3\,e^2+11\,B\,b^5\,d^5+5\,A\,b^5\,d^4\,e\right )}{30\,e^6}+\frac {b^2\,x^3\,\left (1485\,B\,a^8\,e^8+1320\,B\,a^7\,b\,d\,e^7+3960\,A\,a^7\,b\,e^8+1050\,B\,a^6\,b^2\,d^2\,e^6+2310\,A\,a^6\,b^2\,d\,e^7+756\,B\,a^5\,b^3\,d^3\,e^5+1260\,A\,a^5\,b^3\,d^2\,e^6+490\,B\,a^4\,b^4\,d^4\,e^4+630\,A\,a^4\,b^4\,d^3\,e^5+280\,B\,a^3\,b^5\,d^5\,e^3+280\,A\,a^3\,b^5\,d^4\,e^4+135\,B\,a^2\,b^6\,d^6\,e^2+105\,A\,a^2\,b^6\,d^5\,e^3+50\,B\,a\,b^7\,d^7\,e+30\,A\,a\,b^7\,d^6\,e^2+11\,B\,b^8\,d^8+5\,A\,b^8\,d^7\,e\right )}{429\,e^9}+\frac {B\,b^{10}\,x^{11}}{5\,e}}{d^{16}+16\,d^{15}\,e\,x+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+11440\,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^{14}\,x^{14}+16\,d\,e^{15}\,x^{15}+e^{16}\,x^{16}} \end {gather*}

-((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

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