3.1110 \(\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{22}} \, dx\)

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.66, 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 = {77} \[ \frac {b^9 (-10 a B e-A b e+11 b B d)}{11 e^{12} (d+e x)^{11}}-\frac {5 b^8 (b d-a e) (-9 a B e-2 A b e+11 b B d)}{12 e^{12} (d+e x)^{12}}+\frac {15 b^7 (b d-a e)^2 (-8 a B e-3 A b e+11 b B d)}{13 e^{12} (d+e x)^{13}}-\frac {15 b^6 (b d-a e)^3 (-7 a B e-4 A b e+11 b B d)}{7 e^{12} (d+e x)^{14}}+\frac {14 b^5 (b d-a e)^4 (-6 a B e-5 A b e+11 b B d)}{5 e^{12} (d+e x)^{15}}-\frac {21 b^4 (b d-a e)^5 (-5 a B e-6 A b e+11 b B d)}{8 e^{12} (d+e x)^{16}}+\frac {30 b^3 (b d-a e)^6 (-4 a B e-7 A b e+11 b B d)}{17 e^{12} (d+e x)^{17}}-\frac {5 b^2 (b d-a e)^7 (-3 a B e-8 A b e+11 b B d)}{6 e^{12} (d+e x)^{18}}+\frac {5 b (b d-a e)^8 (-2 a B e-9 A b e+11 b B d)}{19 e^{12} (d+e x)^{19}}-\frac {(b d-a e)^9 (-a B e-10 A b e+11 b B d)}{20 e^{12} (d+e x)^{20}}+\frac {(b d-a e)^{10} (B d-A e)}{21 e^{12} (d+e x)^{21}}-\frac {b^{10} B}{10 e^{12} (d+e x)^{10}} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^10*(A + B*x))/(d + e*x)^22,x]

[Out]

((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)

Rule 77

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]))))

Rubi steps

\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]  time = 1.02, size = 1431, normalized size = 3.08 \[ -\frac {\left (10 A e \left (d^{10}+21 e x d^9+210 e^2 x^2 d^8+1330 e^3 x^3 d^7+5985 e^4 x^4 d^6+20349 e^5 x^5 d^5+54264 e^6 x^6 d^4+116280 e^7 x^7 d^3+203490 e^8 x^8 d^2+293930 e^9 x^9 d+352716 e^{10} x^{10}\right )+11 B \left (d^{11}+21 e x d^{10}+210 e^2 x^2 d^9+1330 e^3 x^3 d^8+5985 e^4 x^4 d^7+20349 e^5 x^5 d^6+54264 e^6 x^6 d^5+116280 e^7 x^7 d^4+203490 e^8 x^8 d^3+293930 e^9 x^9 d^2+352716 e^{10} x^{10} d+352716 e^{11} x^{11}\right )\right ) b^{10}+10 a e \left (11 A e \left (d^9+21 e x d^8+210 e^2 x^2 d^7+1330 e^3 x^3 d^6+5985 e^4 x^4 d^5+20349 e^5 x^5 d^4+54264 e^6 x^6 d^3+116280 e^7 x^7 d^2+203490 e^8 x^8 d+293930 e^9 x^9\right )+10 B \left (d^{10}+21 e x d^9+210 e^2 x^2 d^8+1330 e^3 x^3 d^7+5985 e^4 x^4 d^6+20349 e^5 x^5 d^5+54264 e^6 x^6 d^4+116280 e^7 x^7 d^3+203490 e^8 x^8 d^2+293930 e^9 x^9 d+352716 e^{10} x^{10}\right )\right ) b^9+165 a^2 e^2 \left (4 A e \left (d^8+21 e x d^7+210 e^2 x^2 d^6+1330 e^3 x^3 d^5+5985 e^4 x^4 d^4+20349 e^5 x^5 d^3+54264 e^6 x^6 d^2+116280 e^7 x^7 d+203490 e^8 x^8\right )+3 B \left (d^9+21 e x d^8+210 e^2 x^2 d^7+1330 e^3 x^3 d^6+5985 e^4 x^4 d^5+20349 e^5 x^5 d^4+54264 e^6 x^6 d^3+116280 e^7 x^7 d^2+203490 e^8 x^8 d+293930 e^9 x^9\right )\right ) b^8+220 a^3 e^3 \left (13 A e \left (d^7+21 e x d^6+210 e^2 x^2 d^5+1330 e^3 x^3 d^4+5985 e^4 x^4 d^3+20349 e^5 x^5 d^2+54264 e^6 x^6 d+116280 e^7 x^7\right )+8 B \left (d^8+21 e x d^7+210 e^2 x^2 d^6+1330 e^3 x^3 d^5+5985 e^4 x^4 d^4+20349 e^5 x^5 d^3+54264 e^6 x^6 d^2+116280 e^7 x^7 d+203490 e^8 x^8\right )\right ) b^7+5005 a^4 e^4 \left (2 A e \left (d^6+21 e x d^5+210 e^2 x^2 d^4+1330 e^3 x^3 d^3+5985 e^4 x^4 d^2+20349 e^5 x^5 d+54264 e^6 x^6\right )+B \left (d^7+21 e x d^6+210 e^2 x^2 d^5+1330 e^3 x^3 d^4+5985 e^4 x^4 d^3+20349 e^5 x^5 d^2+54264 e^6 x^6 d+116280 e^7 x^7\right )\right ) b^6+6006 a^5 e^5 \left (5 A e \left (d^5+21 e x d^4+210 e^2 x^2 d^3+1330 e^3 x^3 d^2+5985 e^4 x^4 d+20349 e^5 x^5\right )+2 B \left (d^6+21 e x d^5+210 e^2 x^2 d^4+1330 e^3 x^3 d^3+5985 e^4 x^4 d^2+20349 e^5 x^5 d+54264 e^6 x^6\right )\right ) b^5+5005 a^6 e^6 \left (16 A e \left (d^4+21 e x d^3+210 e^2 x^2 d^2+1330 e^3 x^3 d+5985 e^4 x^4\right )+5 B \left (d^5+21 e x d^4+210 e^2 x^2 d^3+1330 e^3 x^3 d^2+5985 e^4 x^4 d+20349 e^5 x^5\right )\right ) b^4+11440 a^7 e^7 \left (17 A e \left (d^3+21 e x d^2+210 e^2 x^2 d+1330 e^3 x^3\right )+4 B \left (d^4+21 e x d^3+210 e^2 x^2 d^2+1330 e^3 x^3 d+5985 e^4 x^4\right )\right ) b^3+72930 a^8 e^8 \left (6 A e \left (d^2+21 e x d+210 e^2 x^2\right )+B \left (d^3+21 e x d^2+210 e^2 x^2 d+1330 e^3 x^3\right )\right ) b^2+48620 a^9 e^9 \left (19 A e (d+21 e x)+2 B \left (d^2+21 e x d+210 e^2 x^2\right )\right ) b+92378 a^{10} e^{10} (20 A e+B (d+21 e x))}{38798760 e^{12} (d+e x)^{21}} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^10*(A + B*x))/(d + e*x)^22,x]

[Out]

-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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fricas [B]  time = 0.86, size = 2039, normalized size = 4.39 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/38798760*(3879876*B*b^10*e^11*x^11 + 11*B*b^10*d^11 + 1847560*A*a^10*e^11 + 10*(10*B*a*b^9 + A*b^10)*d^10*e
+ 55*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 + 220*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 715*(7*B*a^4*b^6 + 4*A*a^3
*b^7)*d^7*e^4 + 2002*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 5005*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 + 11440*(4
*B*a^7*b^3 + 7*A*a^6*b^4)*d^4*e^7 + 24310*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^8 + 48620*(2*B*a^9*b + 9*A*a^8*b^2
)*d^2*e^9 + 92378*(B*a^10 + 10*A*a^9*b)*d*e^10 + 352716*(11*B*b^10*d*e^10 + 10*(10*B*a*b^9 + A*b^10)*e^11)*x^1
0 + 293930*(11*B*b^10*d^2*e^9 + 10*(10*B*a*b^9 + A*b^10)*d*e^10 + 55*(9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9 + 203
490*(11*B*b^10*d^3*e^8 + 10*(10*B*a*b^9 + A*b^10)*d^2*e^9 + 55*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^10 + 220*(8*B*a^3
*b^7 + 3*A*a^2*b^8)*e^11)*x^8 + 116280*(11*B*b^10*d^4*e^7 + 10*(10*B*a*b^9 + A*b^10)*d^3*e^8 + 55*(9*B*a^2*b^8
+ 2*A*a*b^9)*d^2*e^9 + 220*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^10 + 715*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 + 5
4264*(11*B*b^10*d^5*e^6 + 10*(10*B*a*b^9 + A*b^10)*d^4*e^7 + 55*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^8 + 220*(8*B*a
^3*b^7 + 3*A*a^2*b^8)*d^2*e^9 + 715*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^10 + 2002*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11
)*x^6 + 20349*(11*B*b^10*d^6*e^5 + 10*(10*B*a*b^9 + A*b^10)*d^5*e^6 + 55*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 + 2
20*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8 + 715*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^9 + 2002*(6*B*a^5*b^5 + 5*A*a^4
*b^6)*d*e^10 + 5005*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^11)*x^5 + 5985*(11*B*b^10*d^7*e^4 + 10*(10*B*a*b^9 + A*b^10)
*d^6*e^5 + 55*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^6 + 220*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 + 715*(7*B*a^4*b^6 +
4*A*a^3*b^7)*d^3*e^8 + 2002*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^9 + 5005*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^10 + 1
1440*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^11)*x^4 + 1330*(11*B*b^10*d^8*e^3 + 10*(10*B*a*b^9 + A*b^10)*d^7*e^4 + 55*(
9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5 + 220*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^6 + 715*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d
^4*e^7 + 2002*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^8 + 5005*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*e^9 + 11440*(4*B*a^7*
b^3 + 7*A*a^6*b^4)*d*e^10 + 24310*(3*B*a^8*b^2 + 8*A*a^7*b^3)*e^11)*x^3 + 210*(11*B*b^10*d^9*e^2 + 10*(10*B*a*
b^9 + A*b^10)*d^8*e^3 + 55*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^4 + 220*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^5 + 715*(
7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*e^6 + 2002*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*e^7 + 5005*(5*B*a^6*b^4 + 6*A*a^5*b^
5)*d^3*e^8 + 11440*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*e^9 + 24310*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d*e^10 + 48620*(2*B
*a^9*b + 9*A*a^8*b^2)*e^11)*x^2 + 21*(11*B*b^10*d^10*e + 10*(10*B*a*b^9 + A*b^10)*d^9*e^2 + 55*(9*B*a^2*b^8 +
2*A*a*b^9)*d^8*e^3 + 220*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4 + 715*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5 + 2002*
(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 + 5005*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*e^7 + 11440*(4*B*a^7*b^3 + 7*A*a^6*
b^4)*d^3*e^8 + 24310*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e^9 + 48620*(2*B*a^9*b + 9*A*a^8*b^2)*d*e^10 + 92378*(B*a
^10 + 10*A*a^9*b)*e^11)*x)/(e^33*x^21 + 21*d*e^32*x^20 + 210*d^2*e^31*x^19 + 1330*d^3*e^30*x^18 + 5985*d^4*e^2
9*x^17 + 20349*d^5*e^28*x^16 + 54264*d^6*e^27*x^15 + 116280*d^7*e^26*x^14 + 203490*d^8*e^25*x^13 + 293930*d^9*
e^24*x^12 + 352716*d^10*e^23*x^11 + 352716*d^11*e^22*x^10 + 293930*d^12*e^21*x^9 + 203490*d^13*e^20*x^8 + 1162
80*d^14*e^19*x^7 + 54264*d^15*e^18*x^6 + 20349*d^16*e^17*x^5 + 5985*d^17*e^16*x^4 + 1330*d^18*e^15*x^3 + 210*d
^19*e^14*x^2 + 21*d^20*e^13*x + d^21*e^12)

________________________________________________________________________________________

giac [B]  time = 1.36, size = 2096, normalized size = 4.52 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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
847560*A*a^10*e^11)*e^(-12)/(x*e + d)^21

________________________________________________________________________________________

maple [B]  time = 0.01, size = 1942, normalized size = 4.19 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^10*(B*x+A)/(e*x+d)^22,x)

[Out]

-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)^21-30/17*b^3*(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^12/(e*x+d)^17-14/5*b^5*
(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^12/(e*x+d)^15-5/19*b*(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^12/(e*x+d)^19-15/13*b^7*(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^12/(e*x+d)^13-5/6*b^2*(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^12/(e*x+d)^18-15/7*b^6*(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^12/(e*x+d)^14-21/8*b^4*(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^12/(e*x+d)^16-5/12*b^8*(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^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)^10-1/20*(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^12/(e*x+d)^20

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maxima [B]  time = 1.47, size = 2039, normalized size = 4.39 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/38798760*(3879876*B*b^10*e^11*x^11 + 11*B*b^10*d^11 + 1847560*A*a^10*e^11 + 10*(10*B*a*b^9 + A*b^10)*d^10*e
+ 55*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 + 220*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 715*(7*B*a^4*b^6 + 4*A*a^3
*b^7)*d^7*e^4 + 2002*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 5005*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 + 11440*(4
*B*a^7*b^3 + 7*A*a^6*b^4)*d^4*e^7 + 24310*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^8 + 48620*(2*B*a^9*b + 9*A*a^8*b^2
)*d^2*e^9 + 92378*(B*a^10 + 10*A*a^9*b)*d*e^10 + 352716*(11*B*b^10*d*e^10 + 10*(10*B*a*b^9 + A*b^10)*e^11)*x^1
0 + 293930*(11*B*b^10*d^2*e^9 + 10*(10*B*a*b^9 + A*b^10)*d*e^10 + 55*(9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9 + 203
490*(11*B*b^10*d^3*e^8 + 10*(10*B*a*b^9 + A*b^10)*d^2*e^9 + 55*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^10 + 220*(8*B*a^3
*b^7 + 3*A*a^2*b^8)*e^11)*x^8 + 116280*(11*B*b^10*d^4*e^7 + 10*(10*B*a*b^9 + A*b^10)*d^3*e^8 + 55*(9*B*a^2*b^8
+ 2*A*a*b^9)*d^2*e^9 + 220*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^10 + 715*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 + 5
4264*(11*B*b^10*d^5*e^6 + 10*(10*B*a*b^9 + A*b^10)*d^4*e^7 + 55*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^8 + 220*(8*B*a
^3*b^7 + 3*A*a^2*b^8)*d^2*e^9 + 715*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^10 + 2002*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11
)*x^6 + 20349*(11*B*b^10*d^6*e^5 + 10*(10*B*a*b^9 + A*b^10)*d^5*e^6 + 55*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 + 2
20*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8 + 715*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^9 + 2002*(6*B*a^5*b^5 + 5*A*a^4
*b^6)*d*e^10 + 5005*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^11)*x^5 + 5985*(11*B*b^10*d^7*e^4 + 10*(10*B*a*b^9 + A*b^10)
*d^6*e^5 + 55*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^6 + 220*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 + 715*(7*B*a^4*b^6 +
4*A*a^3*b^7)*d^3*e^8 + 2002*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^9 + 5005*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^10 + 1
1440*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^11)*x^4 + 1330*(11*B*b^10*d^8*e^3 + 10*(10*B*a*b^9 + A*b^10)*d^7*e^4 + 55*(
9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5 + 220*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^6 + 715*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d
^4*e^7 + 2002*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^8 + 5005*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*e^9 + 11440*(4*B*a^7*
b^3 + 7*A*a^6*b^4)*d*e^10 + 24310*(3*B*a^8*b^2 + 8*A*a^7*b^3)*e^11)*x^3 + 210*(11*B*b^10*d^9*e^2 + 10*(10*B*a*
b^9 + A*b^10)*d^8*e^3 + 55*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^4 + 220*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^5 + 715*(
7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*e^6 + 2002*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*e^7 + 5005*(5*B*a^6*b^4 + 6*A*a^5*b^
5)*d^3*e^8 + 11440*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*e^9 + 24310*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d*e^10 + 48620*(2*B
*a^9*b + 9*A*a^8*b^2)*e^11)*x^2 + 21*(11*B*b^10*d^10*e + 10*(10*B*a*b^9 + A*b^10)*d^9*e^2 + 55*(9*B*a^2*b^8 +
2*A*a*b^9)*d^8*e^3 + 220*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4 + 715*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5 + 2002*
(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 + 5005*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*e^7 + 11440*(4*B*a^7*b^3 + 7*A*a^6*
b^4)*d^3*e^8 + 24310*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e^9 + 48620*(2*B*a^9*b + 9*A*a^8*b^2)*d*e^10 + 92378*(B*a
^10 + 10*A*a^9*b)*e^11)*x)/(e^33*x^21 + 21*d*e^32*x^20 + 210*d^2*e^31*x^19 + 1330*d^3*e^30*x^18 + 5985*d^4*e^2
9*x^17 + 20349*d^5*e^28*x^16 + 54264*d^6*e^27*x^15 + 116280*d^7*e^26*x^14 + 203490*d^8*e^25*x^13 + 293930*d^9*
e^24*x^12 + 352716*d^10*e^23*x^11 + 352716*d^11*e^22*x^10 + 293930*d^12*e^21*x^9 + 203490*d^13*e^20*x^8 + 1162
80*d^14*e^19*x^7 + 54264*d^15*e^18*x^6 + 20349*d^16*e^17*x^5 + 5985*d^17*e^16*x^4 + 1330*d^18*e^15*x^3 + 210*d
^19*e^14*x^2 + 21*d^20*e^13*x + d^21*e^12)

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mupad [B]  time = 2.12, size = 2121, normalized size = 4.57 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x)^10)/(d + e*x)^22,x)

[Out]

-((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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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**10*(B*x+A)/(e*x+d)**22,x)

[Out]

Timed out

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