3.12.0 ∫ (a+bx)10(A+Bx) (d+ex)16 dx [1104]

Integrand size = 20, antiderivative size = 235 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{16}} \, dx=-\frac {(B d-A e) (a+b x)^{11}}{15 e (b d-a e) (d+e x)^{15}}+\frac {(11 b B d+4 A b e-15 a B e) (a+b x)^{11}}{210 e (b d-a e)^2 (d+e x)^{14}}+\frac {b (11 b B d+4 A b e-15 a B e) (a+b x)^{11}}{910 e (b d-a e)^3 (d+e x)^{13}}+\frac {b^2 (11 b B d+4 A b e-15 a B e) (a+b x)^{11}}{5460 e (b d-a e)^4 (d+e x)^{12}}+\frac {b^3 (11 b B d+4 A b e-15 a B e) (a+b x)^{11}}{60060 e (b d-a e)^5 (d+e x)^{11}} \]

-1/15*(-A*e+B*d)*(b*x+a)^11/e/(-a*e+b*d)/(e*x+d)^15+1/210*(4*A*b*e-15*B*a*e+11*B*b*d)*(b*x+a)^11/e/(-a*e+b*d)^

2/(e*x+d)^14+1/910*b*(4*A*b*e-15*B*a*e+11*B*b*d)*(b*x+a)^11/e/(-a*e+b*d)^3/(e*x+d)^13+1/5460*b^2*(4*A*b*e-15*B

*a*e+11*B*b*d)*(b*x+a)^11/e/(-a*e+b*d)^4/(e*x+d)^12+1/60060*b^3*(4*A*b*e-15*B*a*e+11*B*b*d)*(b*x+a)^11/e/(-a*e

Rubi [A] (veriﬁed)

Time = 0.08 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {79, 47, 37} \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{16}} \, dx=\frac {b^3 (a+b x)^{11} (-15 a B e+4 A b e+11 b B d)}{60060 e (d+e x)^{11} (b d-a e)^5}+\frac {b^2 (a+b x)^{11} (-15 a B e+4 A b e+11 b B d)}{5460 e (d+e x)^{12} (b d-a e)^4}+\frac {b (a+b x)^{11} (-15 a B e+4 A b e+11 b B d)}{910 e (d+e x)^{13} (b d-a e)^3}+\frac {(a+b x)^{11} (-15 a B e+4 A b e+11 b B d)}{210 e (d+e x)^{14} (b d-a e)^2}-\frac {(a+b x)^{11} (B d-A e)}{15 e (d+e x)^{15} (b d-a e)} \]

-1/15*((B*d - A*e)*(a + b*x)^11)/(e*(b*d - a*e)*(d + e*x)^15) + ((11*b*B*d + 4*A*b*e - 15*a*B*e)*(a + b*x)^11)

/(210*e*(b*d - a*e)^2*(d + e*x)^14) + (b*(11*b*B*d + 4*A*b*e - 15*a*B*e)*(a + b*x)^11)/(910*e*(b*d - a*e)^3*(d

+ e*x)^13) + (b^2*(11*b*B*d + 4*A*b*e - 15*a*B*e)*(a + b*x)^11)/(5460*e*(b*d - a*e)^4*(d + e*x)^12) + (b^3*(1

1*b*B*d + 4*A*b*e - 15*a*B*e)*(a + b*x)^11)/(60060*e*(b*d - a*e)^5*(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

Rubi steps \begin{align*} \text {integral}& = -\frac {(B d-A e) (a+b x)^{11}}{15 e (b d-a e) (d+e x)^{15}}+\frac {(11 b B d+4 A b e-15 a B e) \int \frac {(a+b x)^{10}}{(d+e x)^{15}} \, dx}{15 e (b d-a e)} \\ & = -\frac {(B d-A e) (a+b x)^{11}}{15 e (b d-a e) (d+e x)^{15}}+\frac {(11 b B d+4 A b e-15 a B e) (a+b x)^{11}}{210 e (b d-a e)^2 (d+e x)^{14}}+\frac {(b (11 b B d+4 A b e-15 a B e)) \int \frac {(a+b x)^{10}}{(d+e x)^{14}} \, dx}{70 e (b d-a e)^2} \\ & = -\frac {(B d-A e) (a+b x)^{11}}{15 e (b d-a e) (d+e x)^{15}}+\frac {(11 b B d+4 A b e-15 a B e) (a+b x)^{11}}{210 e (b d-a e)^2 (d+e x)^{14}}+\frac {b (11 b B d+4 A b e-15 a B e) (a+b x)^{11}}{910 e (b d-a e)^3 (d+e x)^{13}}+\frac {\left (b^2 (11 b B d+4 A b e-15 a B e)\right ) \int \frac {(a+b x)^{10}}{(d+e x)^{13}} \, dx}{455 e (b d-a e)^3} \\ & = -\frac {(B d-A e) (a+b x)^{11}}{15 e (b d-a e) (d+e x)^{15}}+\frac {(11 b B d+4 A b e-15 a B e) (a+b x)^{11}}{210 e (b d-a e)^2 (d+e x)^{14}}+\frac {b (11 b B d+4 A b e-15 a B e) (a+b x)^{11}}{910 e (b d-a e)^3 (d+e x)^{13}}+\frac {b^2 (11 b B d+4 A b e-15 a B e) (a+b x)^{11}}{5460 e (b d-a e)^4 (d+e x)^{12}}+\frac {\left (b^3 (11 b B d+4 A b e-15 a B e)\right ) \int \frac {(a+b x)^{10}}{(d+e x)^{12}} \, dx}{5460 e (b d-a e)^4} \\ & = -\frac {(B d-A e) (a+b x)^{11}}{15 e (b d-a e) (d+e x)^{15}}+\frac {(11 b B d+4 A b e-15 a B e) (a+b x)^{11}}{210 e (b d-a e)^2 (d+e x)^{14}}+\frac {b (11 b B d+4 A b e-15 a B e) (a+b x)^{11}}{910 e (b d-a e)^3 (d+e x)^{13}}+\frac {b^2 (11 b B d+4 A b e-15 a B e) (a+b x)^{11}}{5460 e (b d-a e)^4 (d+e x)^{12}}+\frac {b^3 (11 b B d+4 A b e-15 a B e) (a+b x)^{11}}{60060 e (b d-a e)^5 (d+e x)^{11}} \\ \end{align*}

Mathematica [B] (veriﬁed)

Leaf count is larger than twice the leaf count of optimal. \(1430\) vs. \(2(235)=470\).

Time = 0.50 (sec) , antiderivative size = 1430, normalized size of antiderivative = 6.09 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{16}} \, dx=-\frac {286 a^{10} e^{10} (14 A e+B (d+15 e x))+220 a^9 b e^9 \left (13 A e (d+15 e x)+2 B \left (d^2+15 d e x+105 e^2 x^2\right )\right )+495 a^8 b^2 e^8 \left (4 A e \left (d^2+15 d e x+105 e^2 x^2\right )+B \left (d^3+15 d^2 e x+105 d e^2 x^2+455 e^3 x^3\right )\right )+120 a^7 b^3 e^7 \left (11 A e \left (d^3+15 d^2 e x+105 d e^2 x^2+455 e^3 x^3\right )+4 B \left (d^4+15 d^3 e x+105 d^2 e^2 x^2+455 d e^3 x^3+1365 e^4 x^4\right )\right )+420 a^6 b^4 e^6 \left (2 A e \left (d^4+15 d^3 e x+105 d^2 e^2 x^2+455 d e^3 x^3+1365 e^4 x^4\right )+B \left (d^5+15 d^4 e x+105 d^3 e^2 x^2+455 d^2 e^3 x^3+1365 d e^4 x^4+3003 e^5 x^5\right )\right )+168 a^5 b^5 e^5 \left (3 A e \left (d^5+15 d^4 e x+105 d^3 e^2 x^2+455 d^2 e^3 x^3+1365 d e^4 x^4+3003 e^5 x^5\right )+2 B \left (d^6+15 d^5 e x+105 d^4 e^2 x^2+455 d^3 e^3 x^3+1365 d^2 e^4 x^4+3003 d e^5 x^5+5005 e^6 x^6\right )\right )+35 a^4 b^6 e^4 \left (8 A e \left (d^6+15 d^5 e x+105 d^4 e^2 x^2+455 d^3 e^3 x^3+1365 d^2 e^4 x^4+3003 d e^5 x^5+5005 e^6 x^6\right )+7 B \left (d^7+15 d^6 e x+105 d^5 e^2 x^2+455 d^4 e^3 x^3+1365 d^3 e^4 x^4+3003 d^2 e^5 x^5+5005 d e^6 x^6+6435 e^7 x^7\right )\right )+20 a^3 b^7 e^3 \left (7 A e \left (d^7+15 d^6 e x+105 d^5 e^2 x^2+455 d^4 e^3 x^3+1365 d^3 e^4 x^4+3003 d^2 e^5 x^5+5005 d e^6 x^6+6435 e^7 x^7\right )+8 B \left (d^8+15 d^7 e x+105 d^6 e^2 x^2+455 d^5 e^3 x^3+1365 d^4 e^4 x^4+3003 d^3 e^5 x^5+5005 d^2 e^6 x^6+6435 d e^7 x^7+6435 e^8 x^8\right )\right )+30 a^2 b^8 e^2 \left (2 A e \left (d^8+15 d^7 e x+105 d^6 e^2 x^2+455 d^5 e^3 x^3+1365 d^4 e^4 x^4+3003 d^3 e^5 x^5+5005 d^2 e^6 x^6+6435 d e^7 x^7+6435 e^8 x^8\right )+3 B \left (d^9+15 d^8 e x+105 d^7 e^2 x^2+455 d^6 e^3 x^3+1365 d^5 e^4 x^4+3003 d^4 e^5 x^5+5005 d^3 e^6 x^6+6435 d^2 e^7 x^7+6435 d e^8 x^8+5005 e^9 x^9\right )\right )+20 a b^9 e \left (A e \left (d^9+15 d^8 e x+105 d^7 e^2 x^2+455 d^6 e^3 x^3+1365 d^5 e^4 x^4+3003 d^4 e^5 x^5+5005 d^3 e^6 x^6+6435 d^2 e^7 x^7+6435 d e^8 x^8+5005 e^9 x^9\right )+2 B \left (d^{10}+15 d^9 e x+105 d^8 e^2 x^2+455 d^7 e^3 x^3+1365 d^6 e^4 x^4+3003 d^5 e^5 x^5+5005 d^4 e^6 x^6+6435 d^3 e^7 x^7+6435 d^2 e^8 x^8+5005 d e^9 x^9+3003 e^{10} x^{10}\right )\right )+b^{10} \left (4 A e \left (d^{10}+15 d^9 e x+105 d^8 e^2 x^2+455 d^7 e^3 x^3+1365 d^6 e^4 x^4+3003 d^5 e^5 x^5+5005 d^4 e^6 x^6+6435 d^3 e^7 x^7+6435 d^2 e^8 x^8+5005 d e^9 x^9+3003 e^{10} x^{10}\right )+11 B \left (d^{11}+15 d^{10} e x+105 d^9 e^2 x^2+455 d^8 e^3 x^3+1365 d^7 e^4 x^4+3003 d^6 e^5 x^5+5005 d^5 e^6 x^6+6435 d^4 e^7 x^7+6435 d^3 e^8 x^8+5005 d^2 e^9 x^9+3003 d e^{10} x^{10}+1365 e^{11} x^{11}\right )\right )}{60060 e^{12} (d+e x)^{15}} \]

-1/60060*(286*a^10*e^10*(14*A*e + B*(d + 15*e*x)) + 220*a^9*b*e^9*(13*A*e*(d + 15*e*x) + 2*B*(d^2 + 15*d*e*x +

105*e^2*x^2)) + 495*a^8*b^2*e^8*(4*A*e*(d^2 + 15*d*e*x + 105*e^2*x^2) + B*(d^3 + 15*d^2*e*x + 105*d*e^2*x^2 +

455*e^3*x^3)) + 120*a^7*b^3*e^7*(11*A*e*(d^3 + 15*d^2*e*x + 105*d*e^2*x^2 + 455*e^3*x^3) + 4*B*(d^4 + 15*d^3*

e*x + 105*d^2*e^2*x^2 + 455*d*e^3*x^3 + 1365*e^4*x^4)) + 420*a^6*b^4*e^6*(2*A*e*(d^4 + 15*d^3*e*x + 105*d^2*e^

2*x^2 + 455*d*e^3*x^3 + 1365*e^4*x^4) + B*(d^5 + 15*d^4*e*x + 105*d^3*e^2*x^2 + 455*d^2*e^3*x^3 + 1365*d*e^4*x

^4 + 3003*e^5*x^5)) + 168*a^5*b^5*e^5*(3*A*e*(d^5 + 15*d^4*e*x + 105*d^3*e^2*x^2 + 455*d^2*e^3*x^3 + 1365*d*e^

4*x^4 + 3003*e^5*x^5) + 2*B*(d^6 + 15*d^5*e*x + 105*d^4*e^2*x^2 + 455*d^3*e^3*x^3 + 1365*d^2*e^4*x^4 + 3003*d*

e^5*x^5 + 5005*e^6*x^6)) + 35*a^4*b^6*e^4*(8*A*e*(d^6 + 15*d^5*e*x + 105*d^4*e^2*x^2 + 455*d^3*e^3*x^3 + 1365*

d^2*e^4*x^4 + 3003*d*e^5*x^5 + 5005*e^6*x^6) + 7*B*(d^7 + 15*d^6*e*x + 105*d^5*e^2*x^2 + 455*d^4*e^3*x^3 + 136

5*d^3*e^4*x^4 + 3003*d^2*e^5*x^5 + 5005*d*e^6*x^6 + 6435*e^7*x^7)) + 20*a^3*b^7*e^3*(7*A*e*(d^7 + 15*d^6*e*x +

105*d^5*e^2*x^2 + 455*d^4*e^3*x^3 + 1365*d^3*e^4*x^4 + 3003*d^2*e^5*x^5 + 5005*d*e^6*x^6 + 6435*e^7*x^7) + 8*

B*(d^8 + 15*d^7*e*x + 105*d^6*e^2*x^2 + 455*d^5*e^3*x^3 + 1365*d^4*e^4*x^4 + 3003*d^3*e^5*x^5 + 5005*d^2*e^6*x

^6 + 6435*d*e^7*x^7 + 6435*e^8*x^8)) + 30*a^2*b^8*e^2*(2*A*e*(d^8 + 15*d^7*e*x + 105*d^6*e^2*x^2 + 455*d^5*e^3

*x^3 + 1365*d^4*e^4*x^4 + 3003*d^3*e^5*x^5 + 5005*d^2*e^6*x^6 + 6435*d*e^7*x^7 + 6435*e^8*x^8) + 3*B*(d^9 + 15

*d^8*e*x + 105*d^7*e^2*x^2 + 455*d^6*e^3*x^3 + 1365*d^5*e^4*x^4 + 3003*d^4*e^5*x^5 + 5005*d^3*e^6*x^6 + 6435*d

^2*e^7*x^7 + 6435*d*e^8*x^8 + 5005*e^9*x^9)) + 20*a*b^9*e*(A*e*(d^9 + 15*d^8*e*x + 105*d^7*e^2*x^2 + 455*d^6*e

^3*x^3 + 1365*d^5*e^4*x^4 + 3003*d^4*e^5*x^5 + 5005*d^3*e^6*x^6 + 6435*d^2*e^7*x^7 + 6435*d*e^8*x^8 + 5005*e^9

*x^9) + 2*B*(d^10 + 15*d^9*e*x + 105*d^8*e^2*x^2 + 455*d^7*e^3*x^3 + 1365*d^6*e^4*x^4 + 3003*d^5*e^5*x^5 + 500

5*d^4*e^6*x^6 + 6435*d^3*e^7*x^7 + 6435*d^2*e^8*x^8 + 5005*d*e^9*x^9 + 3003*e^10*x^10)) + b^10*(4*A*e*(d^10 +

15*d^9*e*x + 105*d^8*e^2*x^2 + 455*d^7*e^3*x^3 + 1365*d^6*e^4*x^4 + 3003*d^5*e^5*x^5 + 5005*d^4*e^6*x^6 + 6435

*d^3*e^7*x^7 + 6435*d^2*e^8*x^8 + 5005*d*e^9*x^9 + 3003*e^10*x^10) + 11*B*(d^11 + 15*d^10*e*x + 105*d^9*e^2*x^

2 + 455*d^8*e^3*x^3 + 1365*d^7*e^4*x^4 + 3003*d^6*e^5*x^5 + 5005*d^5*e^6*x^6 + 6435*d^4*e^7*x^7 + 6435*d^3*e^8

*x^8 + 5005*d^2*e^9*x^9 + 3003*d*e^10*x^10 + 1365*e^11*x^11)))/(e^12*(d + e*x)^15)

Maple [B] (veriﬁed)

Leaf count of result is larger than twice the leaf count of optimal. \(1900\) vs. \(2(225)=450\).

Time = 2.13 (sec) , antiderivative size = 1901, normalized size of antiderivative = 8.09


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	\(\text {Expression too large to display}\)	\(2242\)

(-1/60060/e^12*(4004*A*a^10*e^11+2860*A*a^9*b*d*e^10+1980*A*a^8*b^2*d^2*e^9+1320*A*a^7*b^3*d^3*e^8+840*A*a^6*b

^4*d^4*e^7+504*A*a^5*b^5*d^5*e^6+280*A*a^4*b^6*d^6*e^5+140*A*a^3*b^7*d^7*e^4+60*A*a^2*b^8*d^8*e^3+20*A*a*b^9*d

^9*e^2+4*A*b^10*d^10*e+286*B*a^10*d*e^10+440*B*a^9*b*d^2*e^9+495*B*a^8*b^2*d^3*e^8+480*B*a^7*b^3*d^4*e^7+420*B

*a^6*b^4*d^5*e^6+336*B*a^5*b^5*d^6*e^5+245*B*a^4*b^6*d^7*e^4+160*B*a^3*b^7*d^8*e^3+90*B*a^2*b^8*d^9*e^2+40*B*a

*b^9*d^10*e+11*B*b^10*d^11)-1/4004/e^11*(2860*A*a^9*b*e^10+1980*A*a^8*b^2*d*e^9+1320*A*a^7*b^3*d^2*e^8+840*A*a

^6*b^4*d^3*e^7+504*A*a^5*b^5*d^4*e^6+280*A*a^4*b^6*d^5*e^5+140*A*a^3*b^7*d^6*e^4+60*A*a^2*b^8*d^7*e^3+20*A*a*b

^9*d^8*e^2+4*A*b^10*d^9*e+286*B*a^10*e^10+440*B*a^9*b*d*e^9+495*B*a^8*b^2*d^2*e^8+480*B*a^7*b^3*d^3*e^7+420*B*

a^6*b^4*d^4*e^6+336*B*a^5*b^5*d^5*e^5+245*B*a^4*b^6*d^6*e^4+160*B*a^3*b^7*d^7*e^3+90*B*a^2*b^8*d^8*e^2+40*B*a*

b^9*d^9*e+11*B*b^10*d^10)*x-1/572*b/e^10*(1980*A*a^8*b*e^9+1320*A*a^7*b^2*d*e^8+840*A*a^6*b^3*d^2*e^7+504*A*a^

5*b^4*d^3*e^6+280*A*a^4*b^5*d^4*e^5+140*A*a^3*b^6*d^5*e^4+60*A*a^2*b^7*d^6*e^3+20*A*a*b^8*d^7*e^2+4*A*b^9*d^8*

e+440*B*a^9*e^9+495*B*a^8*b*d*e^8+480*B*a^7*b^2*d^2*e^7+420*B*a^6*b^3*d^3*e^6+336*B*a^5*b^4*d^4*e^5+245*B*a^4*

b^5*d^5*e^4+160*B*a^3*b^6*d^6*e^3+90*B*a^2*b^7*d^7*e^2+40*B*a*b^8*d^8*e+11*B*b^9*d^9)*x^2-1/132*b^2/e^9*(1320*

A*a^7*b*e^8+840*A*a^6*b^2*d*e^7+504*A*a^5*b^3*d^2*e^6+280*A*a^4*b^4*d^3*e^5+140*A*a^3*b^5*d^4*e^4+60*A*a^2*b^6

*d^5*e^3+20*A*a*b^7*d^6*e^2+4*A*b^8*d^7*e+495*B*a^8*e^8+480*B*a^7*b*d*e^7+420*B*a^6*b^2*d^2*e^6+336*B*a^5*b^3*

d^3*e^5+245*B*a^4*b^4*d^4*e^4+160*B*a^3*b^5*d^5*e^3+90*B*a^2*b^6*d^6*e^2+40*B*a*b^7*d^7*e+11*B*b^8*d^8)*x^3-1/

44*b^3/e^8*(840*A*a^6*b*e^7+504*A*a^5*b^2*d*e^6+280*A*a^4*b^3*d^2*e^5+140*A*a^3*b^4*d^3*e^4+60*A*a^2*b^5*d^4*e

^3+20*A*a*b^6*d^5*e^2+4*A*b^7*d^6*e+480*B*a^7*e^7+420*B*a^6*b*d*e^6+336*B*a^5*b^2*d^2*e^5+245*B*a^4*b^3*d^3*e^

4+160*B*a^3*b^4*d^4*e^3+90*B*a^2*b^5*d^5*e^2+40*B*a*b^6*d^6*e+11*B*b^7*d^7)*x^4-1/20*b^4/e^7*(504*A*a^5*b*e^6+

280*A*a^4*b^2*d*e^5+140*A*a^3*b^3*d^2*e^4+60*A*a^2*b^4*d^3*e^3+20*A*a*b^5*d^4*e^2+4*A*b^6*d^5*e+420*B*a^6*e^6+

336*B*a^5*b*d*e^5+245*B*a^4*b^2*d^2*e^4+160*B*a^3*b^3*d^3*e^3+90*B*a^2*b^4*d^4*e^2+40*B*a*b^5*d^5*e+11*B*b^6*d

^6)*x^5-1/12*b^5/e^6*(280*A*a^4*b*e^5+140*A*a^3*b^2*d*e^4+60*A*a^2*b^3*d^2*e^3+20*A*a*b^4*d^3*e^2+4*A*b^5*d^4*

e+336*B*a^5*e^5+245*B*a^4*b*d*e^4+160*B*a^3*b^2*d^2*e^3+90*B*a^2*b^3*d^3*e^2+40*B*a*b^4*d^4*e+11*B*b^5*d^5)*x^

6-3/28*b^6/e^5*(140*A*a^3*b*e^4+60*A*a^2*b^2*d*e^3+20*A*a*b^3*d^2*e^2+4*A*b^4*d^3*e+245*B*a^4*e^4+160*B*a^3*b*

d*e^3+90*B*a^2*b^2*d^2*e^2+40*B*a*b^3*d^3*e+11*B*b^4*d^4)*x^7-3/28*b^7/e^4*(60*A*a^2*b*e^3+20*A*a*b^2*d*e^2+4*

A*b^3*d^2*e+160*B*a^3*e^3+90*B*a^2*b*d*e^2+40*B*a*b^2*d^2*e+11*B*b^3*d^3)*x^8-1/12*b^8/e^3*(20*A*a*b*e^2+4*A*b

^2*d*e+90*B*a^2*e^2+40*B*a*b*d*e+11*B*b^2*d^2)*x^9-1/20*b^9/e^2*(4*A*b*e+40*B*a*e+11*B*b*d)*x^10-1/4*b^10*B/e*

Fricas [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1973 vs. \(2 (225) = 450\).

Time = 0.29 (sec) , antiderivative size = 1973, normalized size of antiderivative = 8.40 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{16}} \, dx=\text {Too large to display} \]

-1/60060*(15015*B*b^10*e^11*x^11 + 11*B*b^10*d^11 + 4004*A*a^10*e^11 + 4*(10*B*a*b^9 + A*b^10)*d^10*e + 10*(9*

B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 + 20*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 35*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e

^4 + 56*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 84*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 + 120*(4*B*a^7*b^3 + 7*A*

a^6*b^4)*d^4*e^7 + 165*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^8 + 220*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9 + 286*(B*a^

10 + 10*A*a^9*b)*d*e^10 + 3003*(11*B*b^10*d*e^10 + 4*(10*B*a*b^9 + A*b^10)*e^11)*x^10 + 5005*(11*B*b^10*d^2*e^

9 + 4*(10*B*a*b^9 + A*b^10)*d*e^10 + 10*(9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9 + 6435*(11*B*b^10*d^3*e^8 + 4*(10*

B*a*b^9 + A*b^10)*d^2*e^9 + 10*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^10 + 20*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^11)*x^8 + 6

435*(11*B*b^10*d^4*e^7 + 4*(10*B*a*b^9 + A*b^10)*d^3*e^8 + 10*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^9 + 20*(8*B*a^3*

b^7 + 3*A*a^2*b^8)*d*e^10 + 35*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 + 5005*(11*B*b^10*d^5*e^6 + 4*(10*B*a*b^9

+ A*b^10)*d^4*e^7 + 10*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^8 + 20*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^9 + 35*(7*B*a

^4*b^6 + 4*A*a^3*b^7)*d*e^10 + 56*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 3003*(11*B*b^10*d^6*e^5 + 4*(10*B*a*

b^9 + A*b^10)*d^5*e^6 + 10*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 + 20*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8 + 35*(7*

B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^9 + 56*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 + 84*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^11

)*x^5 + 1365*(11*B*b^10*d^7*e^4 + 4*(10*B*a*b^9 + A*b^10)*d^6*e^5 + 10*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^6 + 20*

(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 + 35*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^8 + 56*(6*B*a^5*b^5 + 5*A*a^4*b^6)*

d^2*e^9 + 84*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^10 + 120*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^11)*x^4 + 455*(11*B*b^10*d

^8*e^3 + 4*(10*B*a*b^9 + A*b^10)*d^7*e^4 + 10*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5 + 20*(8*B*a^3*b^7 + 3*A*a^2*b^

8)*d^5*e^6 + 35*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^7 + 56*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^8 + 84*(5*B*a^6*b^4

+ 6*A*a^5*b^5)*d^2*e^9 + 120*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*e^10 + 165*(3*B*a^8*b^2 + 8*A*a^7*b^3)*e^11)*x^3 +

105*(11*B*b^10*d^9*e^2 + 4*(10*B*a*b^9 + A*b^10)*d^8*e^3 + 10*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^4 + 20*(8*B*a^3

*b^7 + 3*A*a^2*b^8)*d^6*e^5 + 35*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*e^6 + 56*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*e^7

+ 84*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*e^8 + 120*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*e^9 + 165*(3*B*a^8*b^2 + 8*A*a^

7*b^3)*d*e^10 + 220*(2*B*a^9*b + 9*A*a^8*b^2)*e^11)*x^2 + 15*(11*B*b^10*d^10*e + 4*(10*B*a*b^9 + A*b^10)*d^9*e

^2 + 10*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^3 + 20*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4 + 35*(7*B*a^4*b^6 + 4*A*a^3

*b^7)*d^6*e^5 + 56*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 + 84*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*e^7 + 120*(4*B*a^7

*b^3 + 7*A*a^6*b^4)*d^3*e^8 + 165*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e^9 + 220*(2*B*a^9*b + 9*A*a^8*b^2)*d*e^10 +

286*(B*a^10 + 10*A*a^9*b)*e^11)*x)/(e^27*x^15 + 15*d*e^26*x^14 + 105*d^2*e^25*x^13 + 455*d^3*e^24*x^12 + 1365

*d^4*e^23*x^11 + 3003*d^5*e^22*x^10 + 5005*d^6*e^21*x^9 + 6435*d^7*e^20*x^8 + 6435*d^8*e^19*x^7 + 5005*d^9*e^1

8*x^6 + 3003*d^10*e^17*x^5 + 1365*d^11*e^16*x^4 + 455*d^12*e^15*x^3 + 105*d^13*e^14*x^2 + 15*d^14*e^13*x + d^1

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{16}} \, dx=\text {Timed out} \]

Maxima [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1973 vs. \(2 (225) = 450\).

Time = 0.30 (sec) , antiderivative size = 1973, normalized size of antiderivative = 8.40 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{16}} \, dx=\text {Too large to display} \]