∫ (a+bx)10(A+Bx)____________ (d+ex)22 dx [1110]

Optimal. Leaf size=464 \[ \frac {(b d-a e)^{10} (B d-A e)}{21 e^{12} (d+e x)^{21}}-\frac {(b d-a e)^9 (11 b B d-10 A b e-a B e)}{20 e^{12} (d+e x)^{20}}+\frac {5 b (b d-a e)^8 (11 b B d-9 A b e-2 a B e)}{19 e^{12} (d+e x)^{19}}-\frac {5 b^2 (b d-a e)^7 (11 b B d-8 A b e-3 a B e)}{6 e^{12} (d+e x)^{18}}+\frac {30 b^3 (b d-a e)^6 (11 b B d-7 A b e-4 a B e)}{17 e^{12} (d+e x)^{17}}-\frac {21 b^4 (b d-a e)^5 (11 b B d-6 A b e-5 a B e)}{8 e^{12} (d+e x)^{16}}+\frac {14 b^5 (b d-a e)^4 (11 b B d-5 A b e-6 a B e)}{5 e^{12} (d+e x)^{15}}-\frac {15 b^6 (b d-a e)^3 (11 b B d-4 A b e-7 a B e)}{7 e^{12} (d+e x)^{14}}+\frac {15 b^7 (b d-a e)^2 (11 b B d-3 A b e-8 a B e)}{13 e^{12} (d+e x)^{13}}-\frac {5 b^8 (b d-a e) (11 b B d-2 A b e-9 a B e)}{12 e^{12} (d+e x)^{12}}+\frac {b^9 (11 b B d-A b e-10 a B e)}{11 e^{12} (d+e x)^{11}}-\frac {b^{10} B}{10 e^{12} (d+e x)^{10}} \]

1/21*(-a*e+b*d)^10*(-A*e+B*d)/e^12/(e*x+d)^21-1/20*(-a*e+b*d)^9*(-10*A*b*e-B*a*e+11*B*b*d)/e^12/(e*x+d)^20+5/1

9*b*(-a*e+b*d)^8*(-9*A*b*e-2*B*a*e+11*B*b*d)/e^12/(e*x+d)^19-5/6*b^2*(-a*e+b*d)^7*(-8*A*b*e-3*B*a*e+11*B*b*d)/

e^12/(e*x+d)^18+30/17*b^3*(-a*e+b*d)^6*(-7*A*b*e-4*B*a*e+11*B*b*d)/e^12/(e*x+d)^17-21/8*b^4*(-a*e+b*d)^5*(-6*A

*b*e-5*B*a*e+11*B*b*d)/e^12/(e*x+d)^16+14/5*b^5*(-a*e+b*d)^4*(-5*A*b*e-6*B*a*e+11*B*b*d)/e^12/(e*x+d)^15-15/7*

b^6*(-a*e+b*d)^3*(-4*A*b*e-7*B*a*e+11*B*b*d)/e^12/(e*x+d)^14+15/13*b^7*(-a*e+b*d)^2*(-3*A*b*e-8*B*a*e+11*B*b*d

)/e^12/(e*x+d)^13-5/12*b^8*(-a*e+b*d)*(-2*A*b*e-9*B*a*e+11*B*b*d)/e^12/(e*x+d)^12+1/11*b^9*(-A*b*e-10*B*a*e+11

*B*b*d)/e^12/(e*x+d)^11-1/10*b^10*B/e^12/(e*x+d)^10

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Rubi [A]
time = 0.42, antiderivative size = 464, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {78} \begin {gather*} \frac {b^9 (-10 a B e-A b e+11 b B d)}{11 e^{12} (d+e x)^{11}}-\frac {5 b^8 (b d-a e) (-9 a B e-2 A b e+11 b B d)}{12 e^{12} (d+e x)^{12}}+\frac {15 b^7 (b d-a e)^2 (-8 a B e-3 A b e+11 b B d)}{13 e^{12} (d+e x)^{13}}-\frac {15 b^6 (b d-a e)^3 (-7 a B e-4 A b e+11 b B d)}{7 e^{12} (d+e x)^{14}}+\frac {14 b^5 (b d-a e)^4 (-6 a B e-5 A b e+11 b B d)}{5 e^{12} (d+e x)^{15}}-\frac {21 b^4 (b d-a e)^5 (-5 a B e-6 A b e+11 b B d)}{8 e^{12} (d+e x)^{16}}+\frac {30 b^3 (b d-a e)^6 (-4 a B e-7 A b e+11 b B d)}{17 e^{12} (d+e x)^{17}}-\frac {5 b^2 (b d-a e)^7 (-3 a B e-8 A b e+11 b B d)}{6 e^{12} (d+e x)^{18}}+\frac {5 b (b d-a e)^8 (-2 a B e-9 A b e+11 b B d)}{19 e^{12} (d+e x)^{19}}-\frac {(b d-a e)^9 (-a B e-10 A b e+11 b B d)}{20 e^{12} (d+e x)^{20}}+\frac {(b d-a e)^{10} (B d-A e)}{21 e^{12} (d+e x)^{21}}-\frac {b^{10} B}{10 e^{12} (d+e x)^{10}} \end {gather*}

((b*d - a*e)^10*(B*d - A*e))/(21*e^12*(d + e*x)^21) - ((b*d - a*e)^9*(11*b*B*d - 10*A*b*e - a*B*e))/(20*e^12*(

d + e*x)^20) + (5*b*(b*d - a*e)^8*(11*b*B*d - 9*A*b*e - 2*a*B*e))/(19*e^12*(d + e*x)^19) - (5*b^2*(b*d - a*e)^

7*(11*b*B*d - 8*A*b*e - 3*a*B*e))/(6*e^12*(d + e*x)^18) + (30*b^3*(b*d - a*e)^6*(11*b*B*d - 7*A*b*e - 4*a*B*e)

)/(17*e^12*(d + e*x)^17) - (21*b^4*(b*d - a*e)^5*(11*b*B*d - 6*A*b*e - 5*a*B*e))/(8*e^12*(d + e*x)^16) + (14*b

^5*(b*d - a*e)^4*(11*b*B*d - 5*A*b*e - 6*a*B*e))/(5*e^12*(d + e*x)^15) - (15*b^6*(b*d - a*e)^3*(11*b*B*d - 4*A

*b*e - 7*a*B*e))/(7*e^12*(d + e*x)^14) + (15*b^7*(b*d - a*e)^2*(11*b*B*d - 3*A*b*e - 8*a*B*e))/(13*e^12*(d + e

*x)^13) - (5*b^8*(b*d - a*e)*(11*b*B*d - 2*A*b*e - 9*a*B*e))/(12*e^12*(d + e*x)^12) + (b^9*(11*b*B*d - A*b*e -

10*a*B*e))/(11*e^12*(d + e*x)^11) - (b^10*B)/(10*e^12*(d + e*x)^10)

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran

d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[

n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1

, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

\begin {align*} \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{22}} \, dx &=\int \left (\frac {(-b d+a e)^{10} (-B d+A e)}{e^{11} (d+e x)^{22}}+\frac {(-b d+a e)^9 (-11 b B d+10 A b e+a B e)}{e^{11} (d+e x)^{21}}+\frac {5 b (b d-a e)^8 (-11 b B d+9 A b e+2 a B e)}{e^{11} (d+e x)^{20}}-\frac {15 b^2 (b d-a e)^7 (-11 b B d+8 A b e+3 a B e)}{e^{11} (d+e x)^{19}}+\frac {30 b^3 (b d-a e)^6 (-11 b B d+7 A b e+4 a B e)}{e^{11} (d+e x)^{18}}-\frac {42 b^4 (b d-a e)^5 (-11 b B d+6 A b e+5 a B e)}{e^{11} (d+e x)^{17}}+\frac {42 b^5 (b d-a e)^4 (-11 b B d+5 A b e+6 a B e)}{e^{11} (d+e x)^{16}}-\frac {30 b^6 (b d-a e)^3 (-11 b B d+4 A b e+7 a B e)}{e^{11} (d+e x)^{15}}+\frac {15 b^7 (b d-a e)^2 (-11 b B d+3 A b e+8 a B e)}{e^{11} (d+e x)^{14}}-\frac {5 b^8 (b d-a e) (-11 b B d+2 A b e+9 a B e)}{e^{11} (d+e x)^{13}}+\frac {b^9 (-11 b B d+A b e+10 a B e)}{e^{11} (d+e x)^{12}}+\frac {b^{10} B}{e^{11} (d+e x)^{11}}\right ) \, dx\\ &=\frac {(b d-a e)^{10} (B d-A e)}{21 e^{12} (d+e x)^{21}}-\frac {(b d-a e)^9 (11 b B d-10 A b e-a B e)}{20 e^{12} (d+e x)^{20}}+\frac {5 b (b d-a e)^8 (11 b B d-9 A b e-2 a B e)}{19 e^{12} (d+e x)^{19}}-\frac {5 b^2 (b d-a e)^7 (11 b B d-8 A b e-3 a B e)}{6 e^{12} (d+e x)^{18}}+\frac {30 b^3 (b d-a e)^6 (11 b B d-7 A b e-4 a B e)}{17 e^{12} (d+e x)^{17}}-\frac {21 b^4 (b d-a e)^5 (11 b B d-6 A b e-5 a B e)}{8 e^{12} (d+e x)^{16}}+\frac {14 b^5 (b d-a e)^4 (11 b B d-5 A b e-6 a B e)}{5 e^{12} (d+e x)^{15}}-\frac {15 b^6 (b d-a e)^3 (11 b B d-4 A b e-7 a B e)}{7 e^{12} (d+e x)^{14}}+\frac {15 b^7 (b d-a e)^2 (11 b B d-3 A b e-8 a B e)}{13 e^{12} (d+e x)^{13}}-\frac {5 b^8 (b d-a e) (11 b B d-2 A b e-9 a B e)}{12 e^{12} (d+e x)^{12}}+\frac {b^9 (11 b B d-A b e-10 a B e)}{11 e^{12} (d+e x)^{11}}-\frac {b^{10} B}{10 e^{12} (d+e x)^{10}}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1431\) vs. \(2(464)=928\).
time = 0.59, size = 1431, normalized size = 3.08 \begin {gather*} -\frac {92378 a^{10} e^{10} (20 A e+B (d+21 e x))+48620 a^9 b e^9 \left (19 A e (d+21 e x)+2 B \left (d^2+21 d e x+210 e^2 x^2\right )\right )+72930 a^8 b^2 e^8 \left (6 A e \left (d^2+21 d e x+210 e^2 x^2\right )+B \left (d^3+21 d^2 e x+210 d e^2 x^2+1330 e^3 x^3\right )\right )+11440 a^7 b^3 e^7 \left (17 A e \left (d^3+21 d^2 e x+210 d e^2 x^2+1330 e^3 x^3\right )+4 B \left (d^4+21 d^3 e x+210 d^2 e^2 x^2+1330 d e^3 x^3+5985 e^4 x^4\right )\right )+5005 a^6 b^4 e^6 \left (16 A e \left (d^4+21 d^3 e x+210 d^2 e^2 x^2+1330 d e^3 x^3+5985 e^4 x^4\right )+5 B \left (d^5+21 d^4 e x+210 d^3 e^2 x^2+1330 d^2 e^3 x^3+5985 d e^4 x^4+20349 e^5 x^5\right )\right )+6006 a^5 b^5 e^5 \left (5 A e \left (d^5+21 d^4 e x+210 d^3 e^2 x^2+1330 d^2 e^3 x^3+5985 d e^4 x^4+20349 e^5 x^5\right )+2 B \left (d^6+21 d^5 e x+210 d^4 e^2 x^2+1330 d^3 e^3 x^3+5985 d^2 e^4 x^4+20349 d e^5 x^5+54264 e^6 x^6\right )\right )+5005 a^4 b^6 e^4 \left (2 A e \left (d^6+21 d^5 e x+210 d^4 e^2 x^2+1330 d^3 e^3 x^3+5985 d^2 e^4 x^4+20349 d e^5 x^5+54264 e^6 x^6\right )+B \left (d^7+21 d^6 e x+210 d^5 e^2 x^2+1330 d^4 e^3 x^3+5985 d^3 e^4 x^4+20349 d^2 e^5 x^5+54264 d e^6 x^6+116280 e^7 x^7\right )\right )+220 a^3 b^7 e^3 \left (13 A e \left (d^7+21 d^6 e x+210 d^5 e^2 x^2+1330 d^4 e^3 x^3+5985 d^3 e^4 x^4+20349 d^2 e^5 x^5+54264 d e^6 x^6+116280 e^7 x^7\right )+8 B \left (d^8+21 d^7 e x+210 d^6 e^2 x^2+1330 d^5 e^3 x^3+5985 d^4 e^4 x^4+20349 d^3 e^5 x^5+54264 d^2 e^6 x^6+116280 d e^7 x^7+203490 e^8 x^8\right )\right )+165 a^2 b^8 e^2 \left (4 A e \left (d^8+21 d^7 e x+210 d^6 e^2 x^2+1330 d^5 e^3 x^3+5985 d^4 e^4 x^4+20349 d^3 e^5 x^5+54264 d^2 e^6 x^6+116280 d e^7 x^7+203490 e^8 x^8\right )+3 B \left (d^9+21 d^8 e x+210 d^7 e^2 x^2+1330 d^6 e^3 x^3+5985 d^5 e^4 x^4+20349 d^4 e^5 x^5+54264 d^3 e^6 x^6+116280 d^2 e^7 x^7+203490 d e^8 x^8+293930 e^9 x^9\right )\right )+10 a b^9 e \left (11 A e \left (d^9+21 d^8 e x+210 d^7 e^2 x^2+1330 d^6 e^3 x^3+5985 d^5 e^4 x^4+20349 d^4 e^5 x^5+54264 d^3 e^6 x^6+116280 d^2 e^7 x^7+203490 d e^8 x^8+293930 e^9 x^9\right )+10 B \left (d^{10}+21 d^9 e x+210 d^8 e^2 x^2+1330 d^7 e^3 x^3+5985 d^6 e^4 x^4+20349 d^5 e^5 x^5+54264 d^4 e^6 x^6+116280 d^3 e^7 x^7+203490 d^2 e^8 x^8+293930 d e^9 x^9+352716 e^{10} x^{10}\right )\right )+b^{10} \left (10 A e \left (d^{10}+21 d^9 e x+210 d^8 e^2 x^2+1330 d^7 e^3 x^3+5985 d^6 e^4 x^4+20349 d^5 e^5 x^5+54264 d^4 e^6 x^6+116280 d^3 e^7 x^7+203490 d^2 e^8 x^8+293930 d e^9 x^9+352716 e^{10} x^{10}\right )+11 B \left (d^{11}+21 d^{10} e x+210 d^9 e^2 x^2+1330 d^8 e^3 x^3+5985 d^7 e^4 x^4+20349 d^6 e^5 x^5+54264 d^5 e^6 x^6+116280 d^4 e^7 x^7+203490 d^3 e^8 x^8+293930 d^2 e^9 x^9+352716 d e^{10} x^{10}+352716 e^{11} x^{11}\right )\right )}{38798760 e^{12} (d+e x)^{21}} \end {gather*}

-1/38798760*(92378*a^10*e^10*(20*A*e + B*(d + 21*e*x)) + 48620*a^9*b*e^9*(19*A*e*(d + 21*e*x) + 2*B*(d^2 + 21*

d*e*x + 210*e^2*x^2)) + 72930*a^8*b^2*e^8*(6*A*e*(d^2 + 21*d*e*x + 210*e^2*x^2) + B*(d^3 + 21*d^2*e*x + 210*d*

e^2*x^2 + 1330*e^3*x^3)) + 11440*a^7*b^3*e^7*(17*A*e*(d^3 + 21*d^2*e*x + 210*d*e^2*x^2 + 1330*e^3*x^3) + 4*B*(

d^4 + 21*d^3*e*x + 210*d^2*e^2*x^2 + 1330*d*e^3*x^3 + 5985*e^4*x^4)) + 5005*a^6*b^4*e^6*(16*A*e*(d^4 + 21*d^3*

e*x + 210*d^2*e^2*x^2 + 1330*d*e^3*x^3 + 5985*e^4*x^4) + 5*B*(d^5 + 21*d^4*e*x + 210*d^3*e^2*x^2 + 1330*d^2*e^

3*x^3 + 5985*d*e^4*x^4 + 20349*e^5*x^5)) + 6006*a^5*b^5*e^5*(5*A*e*(d^5 + 21*d^4*e*x + 210*d^3*e^2*x^2 + 1330*

d^2*e^3*x^3 + 5985*d*e^4*x^4 + 20349*e^5*x^5) + 2*B*(d^6 + 21*d^5*e*x + 210*d^4*e^2*x^2 + 1330*d^3*e^3*x^3 + 5

985*d^2*e^4*x^4 + 20349*d*e^5*x^5 + 54264*e^6*x^6)) + 5005*a^4*b^6*e^4*(2*A*e*(d^6 + 21*d^5*e*x + 210*d^4*e^2*

x^2 + 1330*d^3*e^3*x^3 + 5985*d^2*e^4*x^4 + 20349*d*e^5*x^5 + 54264*e^6*x^6) + B*(d^7 + 21*d^6*e*x + 210*d^5*e

^2*x^2 + 1330*d^4*e^3*x^3 + 5985*d^3*e^4*x^4 + 20349*d^2*e^5*x^5 + 54264*d*e^6*x^6 + 116280*e^7*x^7)) + 220*a^

3*b^7*e^3*(13*A*e*(d^7 + 21*d^6*e*x + 210*d^5*e^2*x^2 + 1330*d^4*e^3*x^3 + 5985*d^3*e^4*x^4 + 20349*d^2*e^5*x^

5 + 54264*d*e^6*x^6 + 116280*e^7*x^7) + 8*B*(d^8 + 21*d^7*e*x + 210*d^6*e^2*x^2 + 1330*d^5*e^3*x^3 + 5985*d^4*

e^4*x^4 + 20349*d^3*e^5*x^5 + 54264*d^2*e^6*x^6 + 116280*d*e^7*x^7 + 203490*e^8*x^8)) + 165*a^2*b^8*e^2*(4*A*e

*(d^8 + 21*d^7*e*x + 210*d^6*e^2*x^2 + 1330*d^5*e^3*x^3 + 5985*d^4*e^4*x^4 + 20349*d^3*e^5*x^5 + 54264*d^2*e^6

*x^6 + 116280*d*e^7*x^7 + 203490*e^8*x^8) + 3*B*(d^9 + 21*d^8*e*x + 210*d^7*e^2*x^2 + 1330*d^6*e^3*x^3 + 5985*

d^5*e^4*x^4 + 20349*d^4*e^5*x^5 + 54264*d^3*e^6*x^6 + 116280*d^2*e^7*x^7 + 203490*d*e^8*x^8 + 293930*e^9*x^9))

+ 10*a*b^9*e*(11*A*e*(d^9 + 21*d^8*e*x + 210*d^7*e^2*x^2 + 1330*d^6*e^3*x^3 + 5985*d^5*e^4*x^4 + 20349*d^4*e^

5*x^5 + 54264*d^3*e^6*x^6 + 116280*d^2*e^7*x^7 + 203490*d*e^8*x^8 + 293930*e^9*x^9) + 10*B*(d^10 + 21*d^9*e*x

+ 210*d^8*e^2*x^2 + 1330*d^7*e^3*x^3 + 5985*d^6*e^4*x^4 + 20349*d^5*e^5*x^5 + 54264*d^4*e^6*x^6 + 116280*d^3*e

^7*x^7 + 203490*d^2*e^8*x^8 + 293930*d*e^9*x^9 + 352716*e^10*x^10)) + b^10*(10*A*e*(d^10 + 21*d^9*e*x + 210*d^

8*e^2*x^2 + 1330*d^7*e^3*x^3 + 5985*d^6*e^4*x^4 + 20349*d^5*e^5*x^5 + 54264*d^4*e^6*x^6 + 116280*d^3*e^7*x^7 +

203490*d^2*e^8*x^8 + 293930*d*e^9*x^9 + 352716*e^10*x^10) + 11*B*(d^11 + 21*d^10*e*x + 210*d^9*e^2*x^2 + 1330

*d^8*e^3*x^3 + 5985*d^7*e^4*x^4 + 20349*d^6*e^5*x^5 + 54264*d^5*e^6*x^6 + 116280*d^4*e^7*x^7 + 203490*d^3*e^8*

x^8 + 293930*d^2*e^9*x^9 + 352716*d*e^10*x^10 + 352716*e^11*x^11)))/(e^12*(d + e*x)^21)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1941\) vs. \(2(440)=880\).
time = 0.18, size = 1942, normalized size = 4.19


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)^22,x,method=_RETURNVERBOSE)

-15/13*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)^13-1/20/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)^20-1/21*(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)^21-21/8*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)^16-5/12*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)^12-5/19*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)^19-5/6*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)^18-1/10*b^10*B/e^12/(e*x+d)^10-15/7*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)^14-1/11*b^9/e^12*(A*b*e+10*B*a*e-11*B*b*d)/(e*x+d)^11-30/17*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)^17-14/5*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^3-2

0*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*

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 2046 vs. \(2 (471) = 942\).
time = 0.55, size = 2046, normalized size = 4.41 \begin {gather*} \text {Too large to display} \end {gather*}

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^22,x, algorithm="maxima")

-1/38798760*(3879876*B*b^10*x^11*e^11 + 11*B*b^10*d^11 + 1847560*A*a^10*e^11 + 10*(10*B*a*b^9*e + A*b^10*e)*d^

10 + 352716*(11*B*b^10*d*e^10 + 100*B*a*b^9*e^11 + 10*A*b^10*e^11)*x^10 + 55*(9*B*a^2*b^8*e^2 + 2*A*a*b^9*e^2)

*d^9 + 293930*(11*B*b^10*d^2*e^9 + 495*B*a^2*b^8*e^11 + 110*A*a*b^9*e^11 + 10*(10*B*a*b^9*e^10 + A*b^10*e^10)*

d)*x^9 + 220*(8*B*a^3*b^7*e^3 + 3*A*a^2*b^8*e^3)*d^8 + 203490*(11*B*b^10*d^3*e^8 + 1760*B*a^3*b^7*e^11 + 660*A

*a^2*b^8*e^11 + 10*(10*B*a*b^9*e^9 + A*b^10*e^9)*d^2 + 55*(9*B*a^2*b^8*e^10 + 2*A*a*b^9*e^10)*d)*x^8 + 715*(7*

B*a^4*b^6*e^4 + 4*A*a^3*b^7*e^4)*d^7 + 116280*(11*B*b^10*d^4*e^7 + 5005*B*a^4*b^6*e^11 + 2860*A*a^3*b^7*e^11 +

10*(10*B*a*b^9*e^8 + A*b^10*e^8)*d^3 + 55*(9*B*a^2*b^8*e^9 + 2*A*a*b^9*e^9)*d^2 + 220*(8*B*a^3*b^7*e^10 + 3*A

*a^2*b^8*e^10)*d)*x^7 + 2002*(6*B*a^5*b^5*e^5 + 5*A*a^4*b^6*e^5)*d^6 + 54264*(11*B*b^10*d^5*e^6 + 12012*B*a^5*

b^5*e^11 + 10010*A*a^4*b^6*e^11 + 10*(10*B*a*b^9*e^7 + A*b^10*e^7)*d^4 + 55*(9*B*a^2*b^8*e^8 + 2*A*a*b^9*e^8)*

d^3 + 220*(8*B*a^3*b^7*e^9 + 3*A*a^2*b^8*e^9)*d^2 + 715*(7*B*a^4*b^6*e^10 + 4*A*a^3*b^7*e^10)*d)*x^6 + 5005*(5

*B*a^6*b^4*e^6 + 6*A*a^5*b^5*e^6)*d^5 + 20349*(11*B*b^10*d^6*e^5 + 25025*B*a^6*b^4*e^11 + 30030*A*a^5*b^5*e^11

+ 10*(10*B*a*b^9*e^6 + A*b^10*e^6)*d^5 + 55*(9*B*a^2*b^8*e^7 + 2*A*a*b^9*e^7)*d^4 + 220*(8*B*a^3*b^7*e^8 + 3*

A*a^2*b^8*e^8)*d^3 + 715*(7*B*a^4*b^6*e^9 + 4*A*a^3*b^7*e^9)*d^2 + 2002*(6*B*a^5*b^5*e^10 + 5*A*a^4*b^6*e^10)*

d)*x^5 + 11440*(4*B*a^7*b^3*e^7 + 7*A*a^6*b^4*e^7)*d^4 + 5985*(11*B*b^10*d^7*e^4 + 45760*B*a^7*b^3*e^11 + 8008

0*A*a^6*b^4*e^11 + 10*(10*B*a*b^9*e^5 + A*b^10*e^5)*d^6 + 55*(9*B*a^2*b^8*e^6 + 2*A*a*b^9*e^6)*d^5 + 220*(8*B*

a^3*b^7*e^7 + 3*A*a^2*b^8*e^7)*d^4 + 715*(7*B*a^4*b^6*e^8 + 4*A*a^3*b^7*e^8)*d^3 + 2002*(6*B*a^5*b^5*e^9 + 5*A

*a^4*b^6*e^9)*d^2 + 5005*(5*B*a^6*b^4*e^10 + 6*A*a^5*b^5*e^10)*d)*x^4 + 24310*(3*B*a^8*b^2*e^8 + 8*A*a^7*b^3*e

^8)*d^3 + 1330*(11*B*b^10*d^8*e^3 + 72930*B*a^8*b^2*e^11 + 194480*A*a^7*b^3*e^11 + 10*(10*B*a*b^9*e^4 + A*b^10

*e^4)*d^7 + 55*(9*B*a^2*b^8*e^5 + 2*A*a*b^9*e^5)*d^6 + 220*(8*B*a^3*b^7*e^6 + 3*A*a^2*b^8*e^6)*d^5 + 715*(7*B*

a^4*b^6*e^7 + 4*A*a^3*b^7*e^7)*d^4 + 2002*(6*B*a^5*b^5*e^8 + 5*A*a^4*b^6*e^8)*d^3 + 5005*(5*B*a^6*b^4*e^9 + 6*

A*a^5*b^5*e^9)*d^2 + 11440*(4*B*a^7*b^3*e^10 + 7*A*a^6*b^4*e^10)*d)*x^3 + 48620*(2*B*a^9*b*e^9 + 9*A*a^8*b^2*e

^9)*d^2 + 210*(11*B*b^10*d^9*e^2 + 97240*B*a^9*b*e^11 + 437580*A*a^8*b^2*e^11 + 10*(10*B*a*b^9*e^3 + A*b^10*e^

3)*d^8 + 55*(9*B*a^2*b^8*e^4 + 2*A*a*b^9*e^4)*d^7 + 220*(8*B*a^3*b^7*e^5 + 3*A*a^2*b^8*e^5)*d^6 + 715*(7*B*a^4

*b^6*e^6 + 4*A*a^3*b^7*e^6)*d^5 + 2002*(6*B*a^5*b^5*e^7 + 5*A*a^4*b^6*e^7)*d^4 + 5005*(5*B*a^6*b^4*e^8 + 6*A*a

^5*b^5*e^8)*d^3 + 11440*(4*B*a^7*b^3*e^9 + 7*A*a^6*b^4*e^9)*d^2 + 24310*(3*B*a^8*b^2*e^10 + 8*A*a^7*b^3*e^10)*

d)*x^2 + 92378*(B*a^10*e^10 + 10*A*a^9*b*e^10)*d + 21*(11*B*b^10*d^10*e + 92378*B*a^10*e^11 + 923780*A*a^9*b*e

^11 + 10*(10*B*a*b^9*e^2 + A*b^10*e^2)*d^9 + 55*(9*B*a^2*b^8*e^3 + 2*A*a*b^9*e^3)*d^8 + 220*(8*B*a^3*b^7*e^4 +

3*A*a^2*b^8*e^4)*d^7 + 715*(7*B*a^4*b^6*e^5 + 4*A*a^3*b^7*e^5)*d^6 + 2002*(6*B*a^5*b^5*e^6 + 5*A*a^4*b^6*e^6)

*d^5 + 5005*(5*B*a^6*b^4*e^7 + 6*A*a^5*b^5*e^7)*d^4 + 11440*(4*B*a^7*b^3*e^8 + 7*A*a^6*b^4*e^8)*d^3 + 24310*(3

*B*a^8*b^2*e^9 + 8*A*a^7*b^3*e^9)*d^2 + 48620*(2*B*a^9*b*e^10 + 9*A*a^8*b^2*e^10)*d)*x)/(x^21*e^33 + 21*d*x^20

*e^32 + 210*d^2*x^19*e^31 + 1330*d^3*x^18*e^30 + 5985*d^4*x^17*e^29 + 20349*d^5*x^16*e^28 + 54264*d^6*x^15*e^2

7 + 116280*d^7*x^14*e^26 + 203490*d^8*x^13*e^25 + 293930*d^9*x^12*e^24 + 352716*d^10*x^11*e^23 + 352716*d^11*x

^10*e^22 + 293930*d^12*x^9*e^21 + 203490*d^13*x^8*e^20 + 116280*d^14*x^7*e^19 + 54264*d^15*x^6*e^18 + 20349*d^

16*x^5*e^17 + 5985*d^17*x^4*e^16 + 1330*d^18*x^3*e^15 + 210*d^19*x^2*e^14 + 21*d^20*x*e^13 + d^21*e^12)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1957 vs. \(2 (471) = 942\).
time = 0.92, size = 1957, normalized size = 4.22 \begin {gather*} \text {Too large to display} \end {gather*}

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^22,x, algorithm="fricas")

-1/38798760*(11*B*b^10*d^11 + (3879876*B*b^10*x^11 + 1847560*A*a^10 + 3527160*(10*B*a*b^9 + A*b^10)*x^10 + 161

66150*(9*B*a^2*b^8 + 2*A*a*b^9)*x^9 + 44767800*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 83140200*(7*B*a^4*b^6 + 4*A*a

^3*b^7)*x^7 + 108636528*(6*B*a^5*b^5 + 5*A*a^4*b^6)*x^6 + 101846745*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 68468400

*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 + 32332300*(3*B*a^8*b^2 + 8*A*a^7*b^3)*x^3 + 10210200*(2*B*a^9*b + 9*A*a^8*b^

2)*x^2 + 1939938*(B*a^10 + 10*A*a^9*b)*x)*e^11 + (3879876*B*b^10*d*x^10 + 2939300*(10*B*a*b^9 + A*b^10)*d*x^9

+ 11191950*(9*B*a^2*b^8 + 2*A*a*b^9)*d*x^8 + 25581600*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*x^7 + 38798760*(7*B*a^4*b^

6 + 4*A*a^3*b^7)*d*x^6 + 40738698*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*x^5 + 29954925*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*x

^4 + 15215200*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*x^3 + 5105100*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d*x^2 + 1021020*(2*B*a^9

*b + 9*A*a^8*b^2)*d*x + 92378*(B*a^10 + 10*A*a^9*b)*d)*e^10 + 5*(646646*B*b^10*d^2*x^9 + 406980*(10*B*a*b^9 +

A*b^10)*d^2*x^8 + 1279080*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*x^7 + 2387616*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*x^6 + 29

09907*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*x^5 + 2396394*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*x^4 + 1331330*(5*B*a^6*b^4

+ 6*A*a^5*b^5)*d^2*x^3 + 480480*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*x^2 + 102102*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*

x + 9724*(2*B*a^9*b + 9*A*a^8*b^2)*d^2)*e^9 + 5*(447678*B*b^10*d^3*x^8 + 232560*(10*B*a*b^9 + A*b^10)*d^3*x^7

+ 596904*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*x^6 + 895356*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*x^5 + 855855*(7*B*a^4*b^6

+ 4*A*a^3*b^7)*d^3*x^4 + 532532*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*x^3 + 210210*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*x

^2 + 48048*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*x + 4862*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3)*e^8 + 5*(255816*B*b^10*d^

4*x^7 + 108528*(10*B*a*b^9 + A*b^10)*d^4*x^6 + 223839*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*x^5 + 263340*(8*B*a^3*b^7

+ 3*A*a^2*b^8)*d^4*x^4 + 190190*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*x^3 + 84084*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*x^

2 + 21021*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*x + 2288*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4)*e^7 + 7*(85272*B*b^10*d^5*

x^6 + 29070*(10*B*a*b^9 + A*b^10)*d^5*x^5 + 47025*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*x^4 + 41800*(8*B*a^3*b^7 + 3*A

*a^2*b^8)*d^5*x^3 + 21450*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*x^2 + 6006*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*x + 715*(

5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5)*e^6 + 7*(31977*B*b^10*d^6*x^5 + 8550*(10*B*a*b^9 + A*b^10)*d^6*x^4 + 10450*(9*

B*a^2*b^8 + 2*A*a*b^9)*d^6*x^3 + 6600*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*x^2 + 2145*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d

^6*x + 286*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6)*e^5 + 5*(13167*B*b^10*d^7*x^4 + 2660*(10*B*a*b^9 + A*b^10)*d^7*x^3

+ 2310*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*x^2 + 924*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*x + 143*(7*B*a^4*b^6 + 4*A*a^3

*b^7)*d^7)*e^4 + 5*(2926*B*b^10*d^8*x^3 + 420*(10*B*a*b^9 + A*b^10)*d^8*x^2 + 231*(9*B*a^2*b^8 + 2*A*a*b^9)*d^

8*x + 44*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8)*e^3 + 5*(462*B*b^10*d^9*x^2 + 42*(10*B*a*b^9 + A*b^10)*d^9*x + 11*(9

*B*a^2*b^8 + 2*A*a*b^9)*d^9)*e^2 + (231*B*b^10*d^10*x + 10*(10*B*a*b^9 + A*b^10)*d^10)*e)/(x^21*e^33 + 21*d*x^

20*e^32 + 210*d^2*x^19*e^31 + 1330*d^3*x^18*e^30 + 5985*d^4*x^17*e^29 + 20349*d^5*x^16*e^28 + 54264*d^6*x^15*e

^27 + 116280*d^7*x^14*e^26 + 203490*d^8*x^13*e^25 + 293930*d^9*x^12*e^24 + 352716*d^10*x^11*e^23 + 352716*d^11

*x^10*e^22 + 293930*d^12*x^9*e^21 + 203490*d^13*x^8*e^20 + 116280*d^14*x^7*e^19 + 54264*d^15*x^6*e^18 + 20349*

d^16*x^5*e^17 + 5985*d^17*x^4*e^16 + 1330*d^18*x^3*e^15 + 210*d^19*x^2*e^14 + 21*d^20*x*e^13 + d^21*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. \(2 (471) = 942\).
time = 2.15, size = 2096, normalized size = 4.52 \begin {gather*} \text {Too large to display} \end {gather*}

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^22,x, algorithm="giac")

-1/38798760*(3879876*B*b^10*x^11*e^11 + 3879876*B*b^10*d*x^10*e^10 + 3233230*B*b^10*d^2*x^9*e^9 + 2238390*B*b^

10*d^3*x^8*e^8 + 1279080*B*b^10*d^4*x^7*e^7 + 596904*B*b^10*d^5*x^6*e^6 + 223839*B*b^10*d^6*x^5*e^5 + 65835*B*

b^10*d^7*x^4*e^4 + 14630*B*b^10*d^8*x^3*e^3 + 2310*B*b^10*d^9*x^2*e^2 + 231*B*b^10*d^10*x*e + 11*B*b^10*d^11 +

35271600*B*a*b^9*x^10*e^11 + 3527160*A*b^10*x^10*e^11 + 29393000*B*a*b^9*d*x^9*e^10 + 2939300*A*b^10*d*x^9*e^

10 + 20349000*B*a*b^9*d^2*x^8*e^9 + 2034900*A*b^10*d^2*x^8*e^9 + 11628000*B*a*b^9*d^3*x^7*e^8 + 1162800*A*b^10

*d^3*x^7*e^8 + 5426400*B*a*b^9*d^4*x^6*e^7 + 542640*A*b^10*d^4*x^6*e^7 + 2034900*B*a*b^9*d^5*x^5*e^6 + 203490*

A*b^10*d^5*x^5*e^6 + 598500*B*a*b^9*d^6*x^4*e^5 + 59850*A*b^10*d^6*x^4*e^5 + 133000*B*a*b^9*d^7*x^3*e^4 + 1330

0*A*b^10*d^7*x^3*e^4 + 21000*B*a*b^9*d^8*x^2*e^3 + 2100*A*b^10*d^8*x^2*e^3 + 2100*B*a*b^9*d^9*x*e^2 + 210*A*b^

10*d^9*x*e^2 + 100*B*a*b^9*d^10*e + 10*A*b^10*d^10*e + 145495350*B*a^2*b^8*x^9*e^11 + 32332300*A*a*b^9*x^9*e^1

1 + 100727550*B*a^2*b^8*d*x^8*e^10 + 22383900*A*a*b^9*d*x^8*e^10 + 57558600*B*a^2*b^8*d^2*x^7*e^9 + 12790800*A

*a*b^9*d^2*x^7*e^9 + 26860680*B*a^2*b^8*d^3*x^6*e^8 + 5969040*A*a*b^9*d^3*x^6*e^8 + 10072755*B*a^2*b^8*d^4*x^5

*e^7 + 2238390*A*a*b^9*d^4*x^5*e^7 + 2962575*B*a^2*b^8*d^5*x^4*e^6 + 658350*A*a*b^9*d^5*x^4*e^6 + 658350*B*a^2

*b^8*d^6*x^3*e^5 + 146300*A*a*b^9*d^6*x^3*e^5 + 103950*B*a^2*b^8*d^7*x^2*e^4 + 23100*A*a*b^9*d^7*x^2*e^4 + 103

95*B*a^2*b^8*d^8*x*e^3 + 2310*A*a*b^9*d^8*x*e^3 + 495*B*a^2*b^8*d^9*e^2 + 110*A*a*b^9*d^9*e^2 + 358142400*B*a^

3*b^7*x^8*e^11 + 134303400*A*a^2*b^8*x^8*e^11 + 204652800*B*a^3*b^7*d*x^7*e^10 + 76744800*A*a^2*b^8*d*x^7*e^10

+ 95504640*B*a^3*b^7*d^2*x^6*e^9 + 35814240*A*a^2*b^8*d^2*x^6*e^9 + 35814240*B*a^3*b^7*d^3*x^5*e^8 + 13430340

*A*a^2*b^8*d^3*x^5*e^8 + 10533600*B*a^3*b^7*d^4*x^4*e^7 + 3950100*A*a^2*b^8*d^4*x^4*e^7 + 2340800*B*a^3*b^7*d^

5*x^3*e^6 + 877800*A*a^2*b^8*d^5*x^3*e^6 + 369600*B*a^3*b^7*d^6*x^2*e^5 + 138600*A*a^2*b^8*d^6*x^2*e^5 + 36960

*B*a^3*b^7*d^7*x*e^4 + 13860*A*a^2*b^8*d^7*x*e^4 + 1760*B*a^3*b^7*d^8*e^3 + 660*A*a^2*b^8*d^8*e^3 + 581981400*

B*a^4*b^6*x^7*e^11 + 332560800*A*a^3*b^7*x^7*e^11 + 271591320*B*a^4*b^6*d*x^6*e^10 + 155195040*A*a^3*b^7*d*x^6

*e^10 + 101846745*B*a^4*b^6*d^2*x^5*e^9 + 58198140*A*a^3*b^7*d^2*x^5*e^9 + 29954925*B*a^4*b^6*d^3*x^4*e^8 + 17

117100*A*a^3*b^7*d^3*x^4*e^8 + 6656650*B*a^4*b^6*d^4*x^3*e^7 + 3803800*A*a^3*b^7*d^4*x^3*e^7 + 1051050*B*a^4*b

^6*d^5*x^2*e^6 + 600600*A*a^3*b^7*d^5*x^2*e^6 + 105105*B*a^4*b^6*d^6*x*e^5 + 60060*A*a^3*b^7*d^6*x*e^5 + 5005*

B*a^4*b^6*d^7*e^4 + 2860*A*a^3*b^7*d^7*e^4 + 651819168*B*a^5*b^5*x^6*e^11 + 543182640*A*a^4*b^6*x^6*e^11 + 244

432188*B*a^5*b^5*d*x^5*e^10 + 203693490*A*a^4*b^6*d*x^5*e^10 + 71891820*B*a^5*b^5*d^2*x^4*e^9 + 59909850*A*a^4

*b^6*d^2*x^4*e^9 + 15975960*B*a^5*b^5*d^3*x^3*e^8 + 13313300*A*a^4*b^6*d^3*x^3*e^8 + 2522520*B*a^5*b^5*d^4*x^2

*e^7 + 2102100*A*a^4*b^6*d^4*x^2*e^7 + 252252*B*a^5*b^5*d^5*x*e^6 + 210210*A*a^4*b^6*d^5*x*e^6 + 12012*B*a^5*b

^5*d^6*e^5 + 10010*A*a^4*b^6*d^6*e^5 + 509233725*B*a^6*b^4*x^5*e^11 + 611080470*A*a^5*b^5*x^5*e^11 + 149774625

*B*a^6*b^4*d*x^4*e^10 + 179729550*A*a^5*b^5*d*x^4*e^10 + 33283250*B*a^6*b^4*d^2*x^3*e^9 + 39939900*A*a^5*b^5*d

^2*x^3*e^9 + 5255250*B*a^6*b^4*d^3*x^2*e^8 + 6306300*A*a^5*b^5*d^3*x^2*e^8 + 525525*B*a^6*b^4*d^4*x*e^7 + 6306

30*A*a^5*b^5*d^4*x*e^7 + 25025*B*a^6*b^4*d^5*e^6 + 30030*A*a^5*b^5*d^5*e^6 + 273873600*B*a^7*b^3*x^4*e^11 + 47

9278800*A*a^6*b^4*x^4*e^11 + 60860800*B*a^7*b^3*d*x^3*e^10 + 106506400*A*a^6*b^4*d*x^3*e^10 + 9609600*B*a^7*b^

3*d^2*x^2*e^9 + 16816800*A*a^6*b^4*d^2*x^2*e^9 + 960960*B*a^7*b^3*d^3*x*e^8 + 1681680*A*a^6*b^4*d^3*x*e^8 + 45

760*B*a^7*b^3*d^4*e^7 + 80080*A*a^6*b^4*d^4*e^7 + 96996900*B*a^8*b^2*x^3*e^11 + 258658400*A*a^7*b^3*x^3*e^11 +

15315300*B*a^8*b^2*d*x^2*e^10 + 40840800*A*a^7*b^3*d*x^2*e^10 + 1531530*B*a^8*b^2*d^2*x*e^9 + 4084080*A*a^7*b

^3*d^2*x*e^9 + 72930*B*a^8*b^2*d^3*e^8 + 194480*A*a^7*b^3*d^3*e^8 + 20420400*B*a^9*b*x^2*e^11 + 91891800*A*a^8

*b^2*x^2*e^11 + 2042040*B*a^9*b*d*x*e^10 + 9189180*A*a^8*b^2*d*x*e^10 + 97240*B*a^9*b*d^2*e^9 + 437580*A*a^8*b

^2*d^2*e^9 + 1939938*B*a^10*x*e^11 + 19399380*A*a^9*b*x*e^11 + 92378*B*a^10*d*e^10 + 923780*A*a^9*b*d*e^10 + 1

________________________________________________________________________________________

Mupad [B]
time = 2.12, size = 2121, normalized size = 4.57 \begin {gather*} -\frac {\frac {92378\,B\,a^{10}\,d\,e^{10}+1847560\,A\,a^{10}\,e^{11}+97240\,B\,a^9\,b\,d^2\,e^9+923780\,A\,a^9\,b\,d\,e^{10}+72930\,B\,a^8\,b^2\,d^3\,e^8+437580\,A\,a^8\,b^2\,d^2\,e^9+45760\,B\,a^7\,b^3\,d^4\,e^7+194480\,A\,a^7\,b^3\,d^3\,e^8+25025\,B\,a^6\,b^4\,d^5\,e^6+80080\,A\,a^6\,b^4\,d^4\,e^7+12012\,B\,a^5\,b^5\,d^6\,e^5+30030\,A\,a^5\,b^5\,d^5\,e^6+5005\,B\,a^4\,b^6\,d^7\,e^4+10010\,A\,a^4\,b^6\,d^6\,e^5+1760\,B\,a^3\,b^7\,d^8\,e^3+2860\,A\,a^3\,b^7\,d^7\,e^4+495\,B\,a^2\,b^8\,d^9\,e^2+660\,A\,a^2\,b^8\,d^8\,e^3+100\,B\,a\,b^9\,d^{10}\,e+110\,A\,a\,b^9\,d^9\,e^2+11\,B\,b^{10}\,d^{11}+10\,A\,b^{10}\,d^{10}\,e}{38798760\,e^{12}}+\frac {x\,\left (92378\,B\,a^{10}\,e^{10}+97240\,B\,a^9\,b\,d\,e^9+923780\,A\,a^9\,b\,e^{10}+72930\,B\,a^8\,b^2\,d^2\,e^8+437580\,A\,a^8\,b^2\,d\,e^9+45760\,B\,a^7\,b^3\,d^3\,e^7+194480\,A\,a^7\,b^3\,d^2\,e^8+25025\,B\,a^6\,b^4\,d^4\,e^6+80080\,A\,a^6\,b^4\,d^3\,e^7+12012\,B\,a^5\,b^5\,d^5\,e^5+30030\,A\,a^5\,b^5\,d^4\,e^6+5005\,B\,a^4\,b^6\,d^6\,e^4+10010\,A\,a^4\,b^6\,d^5\,e^5+1760\,B\,a^3\,b^7\,d^7\,e^3+2860\,A\,a^3\,b^7\,d^6\,e^4+495\,B\,a^2\,b^8\,d^8\,e^2+660\,A\,a^2\,b^8\,d^7\,e^3+100\,B\,a\,b^9\,d^9\,e+110\,A\,a\,b^9\,d^8\,e^2+11\,B\,b^{10}\,d^{10}+10\,A\,b^{10}\,d^9\,e\right )}{1847560\,e^{11}}+\frac {3\,b^7\,x^8\,\left (1760\,B\,a^3\,e^3+495\,B\,a^2\,b\,d\,e^2+660\,A\,a^2\,b\,e^3+100\,B\,a\,b^2\,d^2\,e+110\,A\,a\,b^2\,d\,e^2+11\,B\,b^3\,d^3+10\,A\,b^3\,d^2\,e\right )}{572\,e^4}+\frac {3\,b^4\,x^5\,\left (25025\,B\,a^6\,e^6+12012\,B\,a^5\,b\,d\,e^5+30030\,A\,a^5\,b\,e^6+5005\,B\,a^4\,b^2\,d^2\,e^4+10010\,A\,a^4\,b^2\,d\,e^5+1760\,B\,a^3\,b^3\,d^3\,e^3+2860\,A\,a^3\,b^3\,d^2\,e^4+495\,B\,a^2\,b^4\,d^4\,e^2+660\,A\,a^2\,b^4\,d^3\,e^3+100\,B\,a\,b^5\,d^5\,e+110\,A\,a\,b^5\,d^4\,e^2+11\,B\,b^6\,d^6+10\,A\,b^6\,d^5\,e\right )}{5720\,e^7}+\frac {b^9\,x^{10}\,\left (10\,A\,b\,e+100\,B\,a\,e+11\,B\,b\,d\right )}{110\,e^2}+\frac {3\,b^6\,x^7\,\left (5005\,B\,a^4\,e^4+1760\,B\,a^3\,b\,d\,e^3+2860\,A\,a^3\,b\,e^4+495\,B\,a^2\,b^2\,d^2\,e^2+660\,A\,a^2\,b^2\,d\,e^3+100\,B\,a\,b^3\,d^3\,e+110\,A\,a\,b^3\,d^2\,e^2+11\,B\,b^4\,d^4+10\,A\,b^4\,d^3\,e\right )}{1001\,e^5}+\frac {3\,b^3\,x^4\,\left (45760\,B\,a^7\,e^7+25025\,B\,a^6\,b\,d\,e^6+80080\,A\,a^6\,b\,e^7+12012\,B\,a^5\,b^2\,d^2\,e^5+30030\,A\,a^5\,b^2\,d\,e^6+5005\,B\,a^4\,b^3\,d^3\,e^4+10010\,A\,a^4\,b^3\,d^2\,e^5+1760\,B\,a^3\,b^4\,d^4\,e^3+2860\,A\,a^3\,b^4\,d^3\,e^4+495\,B\,a^2\,b^5\,d^5\,e^2+660\,A\,a^2\,b^5\,d^4\,e^3+100\,B\,a\,b^6\,d^6\,e+110\,A\,a\,b^6\,d^5\,e^2+11\,B\,b^7\,d^7+10\,A\,b^7\,d^6\,e\right )}{19448\,e^8}+\frac {b\,x^2\,\left (97240\,B\,a^9\,e^9+72930\,B\,a^8\,b\,d\,e^8+437580\,A\,a^8\,b\,e^9+45760\,B\,a^7\,b^2\,d^2\,e^7+194480\,A\,a^7\,b^2\,d\,e^8+25025\,B\,a^6\,b^3\,d^3\,e^6+80080\,A\,a^6\,b^3\,d^2\,e^7+12012\,B\,a^5\,b^4\,d^4\,e^5+30030\,A\,a^5\,b^4\,d^3\,e^6+5005\,B\,a^4\,b^5\,d^5\,e^4+10010\,A\,a^4\,b^5\,d^4\,e^5+1760\,B\,a^3\,b^6\,d^6\,e^3+2860\,A\,a^3\,b^6\,d^5\,e^4+495\,B\,a^2\,b^7\,d^7\,e^2+660\,A\,a^2\,b^7\,d^6\,e^3+100\,B\,a\,b^8\,d^8\,e+110\,A\,a\,b^8\,d^7\,e^2+11\,B\,b^9\,d^9+10\,A\,b^9\,d^8\,e\right )}{184756\,e^{10}}+\frac {b^8\,x^9\,\left (495\,B\,a^2\,e^2+100\,B\,a\,b\,d\,e+110\,A\,a\,b\,e^2+11\,B\,b^2\,d^2+10\,A\,b^2\,d\,e\right )}{132\,e^3}+\frac {b^5\,x^6\,\left (12012\,B\,a^5\,e^5+5005\,B\,a^4\,b\,d\,e^4+10010\,A\,a^4\,b\,e^5+1760\,B\,a^3\,b^2\,d^2\,e^3+2860\,A\,a^3\,b^2\,d\,e^4+495\,B\,a^2\,b^3\,d^3\,e^2+660\,A\,a^2\,b^3\,d^2\,e^3+100\,B\,a\,b^4\,d^4\,e+110\,A\,a\,b^4\,d^3\,e^2+11\,B\,b^5\,d^5+10\,A\,b^5\,d^4\,e\right )}{715\,e^6}+\frac {b^2\,x^3\,\left (72930\,B\,a^8\,e^8+45760\,B\,a^7\,b\,d\,e^7+194480\,A\,a^7\,b\,e^8+25025\,B\,a^6\,b^2\,d^2\,e^6+80080\,A\,a^6\,b^2\,d\,e^7+12012\,B\,a^5\,b^3\,d^3\,e^5+30030\,A\,a^5\,b^3\,d^2\,e^6+5005\,B\,a^4\,b^4\,d^4\,e^4+10010\,A\,a^4\,b^4\,d^3\,e^5+1760\,B\,a^3\,b^5\,d^5\,e^3+2860\,A\,a^3\,b^5\,d^4\,e^4+495\,B\,a^2\,b^6\,d^6\,e^2+660\,A\,a^2\,b^6\,d^5\,e^3+100\,B\,a\,b^7\,d^7\,e+110\,A\,a\,b^7\,d^6\,e^2+11\,B\,b^8\,d^8+10\,A\,b^8\,d^7\,e\right )}{29172\,e^9}+\frac {B\,b^{10}\,x^{11}}{10\,e}}{d^{21}+21\,d^{20}\,e\,x+210\,d^{19}\,e^2\,x^2+1330\,d^{18}\,e^3\,x^3+5985\,d^{17}\,e^4\,x^4+20349\,d^{16}\,e^5\,x^5+54264\,d^{15}\,e^6\,x^6+116280\,d^{14}\,e^7\,x^7+203490\,d^{13}\,e^8\,x^8+293930\,d^{12}\,e^9\,x^9+352716\,d^{11}\,e^{10}\,x^{10}+352716\,d^{10}\,e^{11}\,x^{11}+293930\,d^9\,e^{12}\,x^{12}+203490\,d^8\,e^{13}\,x^{13}+116280\,d^7\,e^{14}\,x^{14}+54264\,d^6\,e^{15}\,x^{15}+20349\,d^5\,e^{16}\,x^{16}+5985\,d^4\,e^{17}\,x^{17}+1330\,d^3\,e^{18}\,x^{18}+210\,d^2\,e^{19}\,x^{19}+21\,d\,e^{20}\,x^{20}+e^{21}\,x^{21}} \end {gather*}

-((1847560*A*a^10*e^11 + 11*B*b^10*d^11 + 10*A*b^10*d^10*e + 92378*B*a^10*d*e^10 + 110*A*a*b^9*d^9*e^2 + 97240

*B*a^9*b*d^2*e^9 + 660*A*a^2*b^8*d^8*e^3 + 2860*A*a^3*b^7*d^7*e^4 + 10010*A*a^4*b^6*d^6*e^5 + 30030*A*a^5*b^5*

d^5*e^6 + 80080*A*a^6*b^4*d^4*e^7 + 194480*A*a^7*b^3*d^3*e^8 + 437580*A*a^8*b^2*d^2*e^9 + 495*B*a^2*b^8*d^9*e^

2 + 1760*B*a^3*b^7*d^8*e^3 + 5005*B*a^4*b^6*d^7*e^4 + 12012*B*a^5*b^5*d^6*e^5 + 25025*B*a^6*b^4*d^5*e^6 + 4576

0*B*a^7*b^3*d^4*e^7 + 72930*B*a^8*b^2*d^3*e^8 + 923780*A*a^9*b*d*e^10 + 100*B*a*b^9*d^10*e)/(38798760*e^12) +

(x*(92378*B*a^10*e^10 + 11*B*b^10*d^10 + 923780*A*a^9*b*e^10 + 10*A*b^10*d^9*e + 110*A*a*b^9*d^8*e^2 + 437580*

A*a^8*b^2*d*e^9 + 660*A*a^2*b^8*d^7*e^3 + 2860*A*a^3*b^7*d^6*e^4 + 10010*A*a^4*b^6*d^5*e^5 + 30030*A*a^5*b^5*d

^4*e^6 + 80080*A*a^6*b^4*d^3*e^7 + 194480*A*a^7*b^3*d^2*e^8 + 495*B*a^2*b^8*d^8*e^2 + 1760*B*a^3*b^7*d^7*e^3 +

5005*B*a^4*b^6*d^6*e^4 + 12012*B*a^5*b^5*d^5*e^5 + 25025*B*a^6*b^4*d^4*e^6 + 45760*B*a^7*b^3*d^3*e^7 + 72930*

B*a^8*b^2*d^2*e^8 + 100*B*a*b^9*d^9*e + 97240*B*a^9*b*d*e^9))/(1847560*e^11) + (3*b^7*x^8*(1760*B*a^3*e^3 + 11

*B*b^3*d^3 + 660*A*a^2*b*e^3 + 10*A*b^3*d^2*e + 110*A*a*b^2*d*e^2 + 100*B*a*b^2*d^2*e + 495*B*a^2*b*d*e^2))/(5

72*e^4) + (3*b^4*x^5*(25025*B*a^6*e^6 + 11*B*b^6*d^6 + 30030*A*a^5*b*e^6 + 10*A*b^6*d^5*e + 110*A*a*b^5*d^4*e^

2 + 10010*A*a^4*b^2*d*e^5 + 660*A*a^2*b^4*d^3*e^3 + 2860*A*a^3*b^3*d^2*e^4 + 495*B*a^2*b^4*d^4*e^2 + 1760*B*a^

3*b^3*d^3*e^3 + 5005*B*a^4*b^2*d^2*e^4 + 100*B*a*b^5*d^5*e + 12012*B*a^5*b*d*e^5))/(5720*e^7) + (b^9*x^10*(10*

A*b*e + 100*B*a*e + 11*B*b*d))/(110*e^2) + (3*b^6*x^7*(5005*B*a^4*e^4 + 11*B*b^4*d^4 + 2860*A*a^3*b*e^4 + 10*A

*b^4*d^3*e + 110*A*a*b^3*d^2*e^2 + 660*A*a^2*b^2*d*e^3 + 495*B*a^2*b^2*d^2*e^2 + 100*B*a*b^3*d^3*e + 1760*B*a^

3*b*d*e^3))/(1001*e^5) + (3*b^3*x^4*(45760*B*a^7*e^7 + 11*B*b^7*d^7 + 80080*A*a^6*b*e^7 + 10*A*b^7*d^6*e + 110

*A*a*b^6*d^5*e^2 + 30030*A*a^5*b^2*d*e^6 + 660*A*a^2*b^5*d^4*e^3 + 2860*A*a^3*b^4*d^3*e^4 + 10010*A*a^4*b^3*d^

2*e^5 + 495*B*a^2*b^5*d^5*e^2 + 1760*B*a^3*b^4*d^4*e^3 + 5005*B*a^4*b^3*d^3*e^4 + 12012*B*a^5*b^2*d^2*e^5 + 10

0*B*a*b^6*d^6*e + 25025*B*a^6*b*d*e^6))/(19448*e^8) + (b*x^2*(97240*B*a^9*e^9 + 11*B*b^9*d^9 + 437580*A*a^8*b*

e^9 + 10*A*b^9*d^8*e + 110*A*a*b^8*d^7*e^2 + 194480*A*a^7*b^2*d*e^8 + 660*A*a^2*b^7*d^6*e^3 + 2860*A*a^3*b^6*d

^5*e^4 + 10010*A*a^4*b^5*d^4*e^5 + 30030*A*a^5*b^4*d^3*e^6 + 80080*A*a^6*b^3*d^2*e^7 + 495*B*a^2*b^7*d^7*e^2 +

1760*B*a^3*b^6*d^6*e^3 + 5005*B*a^4*b^5*d^5*e^4 + 12012*B*a^5*b^4*d^4*e^5 + 25025*B*a^6*b^3*d^3*e^6 + 45760*B

*a^7*b^2*d^2*e^7 + 100*B*a*b^8*d^8*e + 72930*B*a^8*b*d*e^8))/(184756*e^10) + (b^8*x^9*(495*B*a^2*e^2 + 11*B*b^

2*d^2 + 110*A*a*b*e^2 + 10*A*b^2*d*e + 100*B*a*b*d*e))/(132*e^3) + (b^5*x^6*(12012*B*a^5*e^5 + 11*B*b^5*d^5 +

10010*A*a^4*b*e^5 + 10*A*b^5*d^4*e + 110*A*a*b^4*d^3*e^2 + 2860*A*a^3*b^2*d*e^4 + 660*A*a^2*b^3*d^2*e^3 + 495*

B*a^2*b^3*d^3*e^2 + 1760*B*a^3*b^2*d^2*e^3 + 100*B*a*b^4*d^4*e + 5005*B*a^4*b*d*e^4))/(715*e^6) + (b^2*x^3*(72

930*B*a^8*e^8 + 11*B*b^8*d^8 + 194480*A*a^7*b*e^8 + 10*A*b^8*d^7*e + 110*A*a*b^7*d^6*e^2 + 80080*A*a^6*b^2*d*e

^7 + 660*A*a^2*b^6*d^5*e^3 + 2860*A*a^3*b^5*d^4*e^4 + 10010*A*a^4*b^4*d^3*e^5 + 30030*A*a^5*b^3*d^2*e^6 + 495*

B*a^2*b^6*d^6*e^2 + 1760*B*a^3*b^5*d^5*e^3 + 5005*B*a^4*b^4*d^4*e^4 + 12012*B*a^5*b^3*d^3*e^5 + 25025*B*a^6*b^

2*d^2*e^6 + 100*B*a*b^7*d^7*e + 45760*B*a^7*b*d*e^7))/(29172*e^9) + (B*b^10*x^11)/(10*e))/(d^21 + e^21*x^21 +

21*d*e^20*x^20 + 210*d^19*e^2*x^2 + 1330*d^18*e^3*x^3 + 5985*d^17*e^4*x^4 + 20349*d^16*e^5*x^5 + 54264*d^15*e^

6*x^6 + 116280*d^14*e^7*x^7 + 203490*d^13*e^8*x^8 + 293930*d^12*e^9*x^9 + 352716*d^11*e^10*x^10 + 352716*d^10*

e^11*x^11 + 293930*d^9*e^12*x^12 + 203490*d^8*e^13*x^13 + 116280*d^7*e^14*x^14 + 54264*d^6*e^15*x^15 + 20349*d

^5*e^16*x^16 + 5985*d^4*e^17*x^17 + 1330*d^3*e^18*x^18 + 210*d^2*e^19*x^19 + 21*d^20*e*x)

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3.12.10 \(\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{22}} \, dx\) [1110]