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3.1090 \(\int \frac{(a+b x)^{10} (A+B x)}{(d+e x)^2} \, dx\)

Optimal. Leaf size=445 \[ \frac{15 b^2 (d+e x)^2 (b d-a e)^7 (-3 a B e-8 A b e+11 b B d)}{2 e^{12}}-\frac{10 b^3 (d+e x)^3 (b d-a e)^6 (-4 a B e-7 A b e+11 b B d)}{e^{12}}+\frac{21 b^4 (d+e x)^4 (b d-a e)^5 (-5 a B e-6 A b e+11 b B d)}{2 e^{12}}-\frac{42 b^5 (d+e x)^5 (b d-a e)^4 (-6 a B e-5 A b e+11 b B d)}{5 e^{12}}+\frac{5 b^6 (d+e x)^6 (b d-a e)^3 (-7 a B e-4 A b e+11 b B d)}{e^{12}}-\frac{15 b^7 (d+e x)^7 (b d-a e)^2 (-8 a B e-3 A b e+11 b B d)}{7 e^{12}}+\frac{5 b^8 (d+e x)^8 (b d-a e) (-9 a B e-2 A b e+11 b B d)}{8 e^{12}}-\frac{b^9 (d+e x)^9 (-10 a B e-A b e+11 b B d)}{9 e^{12}}+\frac{(b d-a e)^{10} (B d-A e)}{e^{12} (d+e x)}-\frac{5 b x (b d-a e)^8 (-2 a B e-9 A b e+11 b B d)}{e^{11}}+\frac{(b d-a e)^9 \log (d+e x) (-a B e-10 A b e+11 b B d)}{e^{12}}+\frac{b^{10} B (d+e x)^{10}}{10 e^{12}} \]

[Out]

(-5*b*(b*d - a*e)^8*(11*b*B*d - 9*A*b*e - 2*a*B*e)*x)/e^11 + ((b*d - a*e)^10*(B*d - A*e))/(e^12*(d + e*x)) + (
15*b^2*(b*d - a*e)^7*(11*b*B*d - 8*A*b*e - 3*a*B*e)*(d + e*x)^2)/(2*e^12) - (10*b^3*(b*d - a*e)^6*(11*b*B*d -
7*A*b*e - 4*a*B*e)*(d + e*x)^3)/e^12 + (21*b^4*(b*d - a*e)^5*(11*b*B*d - 6*A*b*e - 5*a*B*e)*(d + e*x)^4)/(2*e^
12) - (42*b^5*(b*d - a*e)^4*(11*b*B*d - 5*A*b*e - 6*a*B*e)*(d + e*x)^5)/(5*e^12) + (5*b^6*(b*d - a*e)^3*(11*b*
B*d - 4*A*b*e - 7*a*B*e)*(d + e*x)^6)/e^12 - (15*b^7*(b*d - a*e)^2*(11*b*B*d - 3*A*b*e - 8*a*B*e)*(d + e*x)^7)
/(7*e^12) + (5*b^8*(b*d - a*e)*(11*b*B*d - 2*A*b*e - 9*a*B*e)*(d + e*x)^8)/(8*e^12) - (b^9*(11*b*B*d - A*b*e -
 10*a*B*e)*(d + e*x)^9)/(9*e^12) + (b^10*B*(d + e*x)^10)/(10*e^12) + ((b*d - a*e)^9*(11*b*B*d - 10*A*b*e - a*B
*e)*Log[d + e*x])/e^12

________________________________________________________________________________________

Rubi [A]  time = 1.6825, antiderivative size = 445, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {77} \[ \frac{15 b^2 (d+e x)^2 (b d-a e)^7 (-3 a B e-8 A b e+11 b B d)}{2 e^{12}}-\frac{10 b^3 (d+e x)^3 (b d-a e)^6 (-4 a B e-7 A b e+11 b B d)}{e^{12}}+\frac{21 b^4 (d+e x)^4 (b d-a e)^5 (-5 a B e-6 A b e+11 b B d)}{2 e^{12}}-\frac{42 b^5 (d+e x)^5 (b d-a e)^4 (-6 a B e-5 A b e+11 b B d)}{5 e^{12}}+\frac{5 b^6 (d+e x)^6 (b d-a e)^3 (-7 a B e-4 A b e+11 b B d)}{e^{12}}-\frac{15 b^7 (d+e x)^7 (b d-a e)^2 (-8 a B e-3 A b e+11 b B d)}{7 e^{12}}+\frac{5 b^8 (d+e x)^8 (b d-a e) (-9 a B e-2 A b e+11 b B d)}{8 e^{12}}-\frac{b^9 (d+e x)^9 (-10 a B e-A b e+11 b B d)}{9 e^{12}}+\frac{(b d-a e)^{10} (B d-A e)}{e^{12} (d+e x)}-\frac{5 b x (b d-a e)^8 (-2 a B e-9 A b e+11 b B d)}{e^{11}}+\frac{(b d-a e)^9 \log (d+e x) (-a B e-10 A b e+11 b B d)}{e^{12}}+\frac{b^{10} B (d+e x)^{10}}{10 e^{12}} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^10*(A + B*x))/(d + e*x)^2,x]

[Out]

(-5*b*(b*d - a*e)^8*(11*b*B*d - 9*A*b*e - 2*a*B*e)*x)/e^11 + ((b*d - a*e)^10*(B*d - A*e))/(e^12*(d + e*x)) + (
15*b^2*(b*d - a*e)^7*(11*b*B*d - 8*A*b*e - 3*a*B*e)*(d + e*x)^2)/(2*e^12) - (10*b^3*(b*d - a*e)^6*(11*b*B*d -
7*A*b*e - 4*a*B*e)*(d + e*x)^3)/e^12 + (21*b^4*(b*d - a*e)^5*(11*b*B*d - 6*A*b*e - 5*a*B*e)*(d + e*x)^4)/(2*e^
12) - (42*b^5*(b*d - a*e)^4*(11*b*B*d - 5*A*b*e - 6*a*B*e)*(d + e*x)^5)/(5*e^12) + (5*b^6*(b*d - a*e)^3*(11*b*
B*d - 4*A*b*e - 7*a*B*e)*(d + e*x)^6)/e^12 - (15*b^7*(b*d - a*e)^2*(11*b*B*d - 3*A*b*e - 8*a*B*e)*(d + e*x)^7)
/(7*e^12) + (5*b^8*(b*d - a*e)*(11*b*B*d - 2*A*b*e - 9*a*B*e)*(d + e*x)^8)/(8*e^12) - (b^9*(11*b*B*d - A*b*e -
 10*a*B*e)*(d + e*x)^9)/(9*e^12) + (b^10*B*(d + e*x)^10)/(10*e^12) + ((b*d - a*e)^9*(11*b*B*d - 10*A*b*e - a*B
*e)*Log[d + e*x])/e^12

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

Mathematica [B]  time = 0.682045, size = 1486, normalized size = 3.34 \[ \frac{\left (10 A e \left (-252 d^{10}+2268 e x d^9+1260 e^2 x^2 d^8-420 e^3 x^3 d^7+210 e^4 x^4 d^6-126 e^5 x^5 d^5+84 e^6 x^6 d^4-60 e^7 x^7 d^3+45 e^8 x^8 d^2-35 e^9 x^9 d+28 e^{10} x^{10}\right )+B \left (2520 d^{11}-25200 e x d^{10}-13860 e^2 x^2 d^9+4620 e^3 x^3 d^8-2310 e^4 x^4 d^7+1386 e^5 x^5 d^6-924 e^6 x^6 d^5+660 e^7 x^7 d^4-495 e^8 x^8 d^3+385 e^9 x^9 d^2-308 e^{10} x^{10} d+252 e^{11} x^{11}\right )\right ) b^{10}+10 a e \left (9 A e \left (280 d^9-2240 e x d^8-1260 e^2 x^2 d^7+420 e^3 x^3 d^6-210 e^4 x^4 d^5+126 e^5 x^5 d^4-84 e^6 x^6 d^3+60 e^7 x^7 d^2-45 e^8 x^8 d+35 e^9 x^9\right )-10 B \left (252 d^{10}-2268 e x d^9-1260 e^2 x^2 d^8+420 e^3 x^3 d^7-210 e^4 x^4 d^6+126 e^5 x^5 d^5-84 e^6 x^6 d^4+60 e^7 x^7 d^3-45 e^8 x^8 d^2+35 e^9 x^9 d-28 e^{10} x^{10}\right )\right ) b^9+135 a^2 e^2 \left (8 A e \left (-105 d^8+735 e x d^7+420 e^2 x^2 d^6-140 e^3 x^3 d^5+70 e^4 x^4 d^4-42 e^5 x^5 d^3+28 e^6 x^6 d^2-20 e^7 x^7 d+15 e^8 x^8\right )+3 B \left (280 d^9-2240 e x d^8-1260 e^2 x^2 d^7+420 e^3 x^3 d^6-210 e^4 x^4 d^5+126 e^5 x^5 d^4-84 e^6 x^6 d^3+60 e^7 x^7 d^2-45 e^8 x^8 d+35 e^9 x^9\right )\right ) b^8+720 a^3 e^3 \left (7 A e \left (60 d^7-360 e x d^6-210 e^2 x^2 d^5+70 e^3 x^3 d^4-35 e^4 x^4 d^3+21 e^5 x^5 d^2-14 e^6 x^6 d+10 e^7 x^7\right )-4 B \left (105 d^8-735 e x d^7-420 e^2 x^2 d^6+140 e^3 x^3 d^5-70 e^4 x^4 d^4+42 e^5 x^5 d^3-28 e^6 x^6 d^2+20 e^7 x^7 d-15 e^8 x^8\right )\right ) b^7+8820 a^4 e^4 \left (6 A e \left (-10 d^6+50 e x d^5+30 e^2 x^2 d^4-10 e^3 x^3 d^3+5 e^4 x^4 d^2-3 e^5 x^5 d+2 e^6 x^6\right )+B \left (60 d^7-360 e x d^6-210 e^2 x^2 d^5+70 e^3 x^3 d^4-35 e^4 x^4 d^3+21 e^5 x^5 d^2-14 e^6 x^6 d+10 e^7 x^7\right )\right ) b^6+10584 a^5 e^5 \left (5 A e \left (12 d^5-48 e x d^4-30 e^2 x^2 d^3+10 e^3 x^3 d^2-5 e^4 x^4 d+3 e^5 x^5\right )-6 B \left (10 d^6-50 e x d^5-30 e^2 x^2 d^4+10 e^3 x^3 d^3-5 e^4 x^4 d^2+3 e^5 x^5 d-2 e^6 x^6\right )\right ) b^5+44100 a^6 e^6 \left (4 A e \left (-3 d^4+9 e x d^3+6 e^2 x^2 d^2-2 e^3 x^3 d+e^4 x^4\right )+B \left (12 d^5-48 e x d^4-30 e^2 x^2 d^3+10 e^3 x^3 d^2-5 e^4 x^4 d+3 e^5 x^5\right )\right ) b^4+50400 a^7 e^7 \left (3 A e \left (2 d^3-4 e x d^2-3 e^2 x^2 d+e^3 x^3\right )+2 B \left (-3 d^4+9 e x d^3+6 e^2 x^2 d^2-2 e^3 x^3 d+e^4 x^4\right )\right ) b^3+56700 a^8 e^8 \left (2 A e \left (-d^2+e x d+e^2 x^2\right )+B \left (2 d^3-4 e x d^2-3 e^2 x^2 d+e^3 x^3\right )\right ) b^2+25200 a^9 e^9 \left (A d e+B \left (-d^2+e x d+e^2 x^2\right )\right ) b-2520 a^{10} e^{10} (A e-B d)+2520 (b d-a e)^9 (11 b B d-10 A b e-a B e) (d+e x) \log (d+e x)}{2520 e^{12} (d+e x)} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^10*(A + B*x))/(d + e*x)^2,x]

[Out]

(-2520*a^10*e^10*(-(B*d) + A*e) + 25200*a^9*b*e^9*(A*d*e + B*(-d^2 + d*e*x + e^2*x^2)) + 56700*a^8*b^2*e^8*(2*
A*e*(-d^2 + d*e*x + e^2*x^2) + B*(2*d^3 - 4*d^2*e*x - 3*d*e^2*x^2 + e^3*x^3)) + 50400*a^7*b^3*e^7*(3*A*e*(2*d^
3 - 4*d^2*e*x - 3*d*e^2*x^2 + e^3*x^3) + 2*B*(-3*d^4 + 9*d^3*e*x + 6*d^2*e^2*x^2 - 2*d*e^3*x^3 + e^4*x^4)) + 4
4100*a^6*b^4*e^6*(4*A*e*(-3*d^4 + 9*d^3*e*x + 6*d^2*e^2*x^2 - 2*d*e^3*x^3 + e^4*x^4) + B*(12*d^5 - 48*d^4*e*x
- 30*d^3*e^2*x^2 + 10*d^2*e^3*x^3 - 5*d*e^4*x^4 + 3*e^5*x^5)) + 10584*a^5*b^5*e^5*(5*A*e*(12*d^5 - 48*d^4*e*x
- 30*d^3*e^2*x^2 + 10*d^2*e^3*x^3 - 5*d*e^4*x^4 + 3*e^5*x^5) - 6*B*(10*d^6 - 50*d^5*e*x - 30*d^4*e^2*x^2 + 10*
d^3*e^3*x^3 - 5*d^2*e^4*x^4 + 3*d*e^5*x^5 - 2*e^6*x^6)) + 8820*a^4*b^6*e^4*(6*A*e*(-10*d^6 + 50*d^5*e*x + 30*d
^4*e^2*x^2 - 10*d^3*e^3*x^3 + 5*d^2*e^4*x^4 - 3*d*e^5*x^5 + 2*e^6*x^6) + B*(60*d^7 - 360*d^6*e*x - 210*d^5*e^2
*x^2 + 70*d^4*e^3*x^3 - 35*d^3*e^4*x^4 + 21*d^2*e^5*x^5 - 14*d*e^6*x^6 + 10*e^7*x^7)) + 720*a^3*b^7*e^3*(7*A*e
*(60*d^7 - 360*d^6*e*x - 210*d^5*e^2*x^2 + 70*d^4*e^3*x^3 - 35*d^3*e^4*x^4 + 21*d^2*e^5*x^5 - 14*d*e^6*x^6 + 1
0*e^7*x^7) - 4*B*(105*d^8 - 735*d^7*e*x - 420*d^6*e^2*x^2 + 140*d^5*e^3*x^3 - 70*d^4*e^4*x^4 + 42*d^3*e^5*x^5
- 28*d^2*e^6*x^6 + 20*d*e^7*x^7 - 15*e^8*x^8)) + 135*a^2*b^8*e^2*(8*A*e*(-105*d^8 + 735*d^7*e*x + 420*d^6*e^2*
x^2 - 140*d^5*e^3*x^3 + 70*d^4*e^4*x^4 - 42*d^3*e^5*x^5 + 28*d^2*e^6*x^6 - 20*d*e^7*x^7 + 15*e^8*x^8) + 3*B*(2
80*d^9 - 2240*d^8*e*x - 1260*d^7*e^2*x^2 + 420*d^6*e^3*x^3 - 210*d^5*e^4*x^4 + 126*d^4*e^5*x^5 - 84*d^3*e^6*x^
6 + 60*d^2*e^7*x^7 - 45*d*e^8*x^8 + 35*e^9*x^9)) + 10*a*b^9*e*(9*A*e*(280*d^9 - 2240*d^8*e*x - 1260*d^7*e^2*x^
2 + 420*d^6*e^3*x^3 - 210*d^5*e^4*x^4 + 126*d^4*e^5*x^5 - 84*d^3*e^6*x^6 + 60*d^2*e^7*x^7 - 45*d*e^8*x^8 + 35*
e^9*x^9) - 10*B*(252*d^10 - 2268*d^9*e*x - 1260*d^8*e^2*x^2 + 420*d^7*e^3*x^3 - 210*d^6*e^4*x^4 + 126*d^5*e^5*
x^5 - 84*d^4*e^6*x^6 + 60*d^3*e^7*x^7 - 45*d^2*e^8*x^8 + 35*d*e^9*x^9 - 28*e^10*x^10)) + b^10*(10*A*e*(-252*d^
10 + 2268*d^9*e*x + 1260*d^8*e^2*x^2 - 420*d^7*e^3*x^3 + 210*d^6*e^4*x^4 - 126*d^5*e^5*x^5 + 84*d^4*e^6*x^6 -
60*d^3*e^7*x^7 + 45*d^2*e^8*x^8 - 35*d*e^9*x^9 + 28*e^10*x^10) + B*(2520*d^11 - 25200*d^10*e*x - 13860*d^9*e^2
*x^2 + 4620*d^8*e^3*x^3 - 2310*d^7*e^4*x^4 + 1386*d^6*e^5*x^5 - 924*d^5*e^6*x^6 + 660*d^4*e^7*x^7 - 495*d^3*e^
8*x^8 + 385*d^2*e^9*x^9 - 308*d*e^10*x^10 + 252*e^11*x^11)) + 2520*(b*d - a*e)^9*(11*b*B*d - 10*A*b*e - a*B*e)
*(d + e*x)*Log[d + e*x])/(2520*e^12*(d + e*x))

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Maple [B]  time = 0.022, size = 2447, normalized size = 5.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^10*(B*x+A)/(e*x+d)^2,x)

[Out]

9/2*b^10/e^10*B*x^2*d^8+45*b^2/e^2*A*a^8*x-90/e^3*ln(e*x+d)*A*a^8*b^2*d+360/e^4*ln(e*x+d)*A*a^7*b^3*d^2-840/e^
5*ln(e*x+d)*A*a^6*b^4*d^3+1260/e^6*ln(e*x+d)*A*a^5*b^5*d^4-1260/e^7*ln(e*x+d)*A*a^4*b^6*d^5+840/e^8*ln(e*x+d)*
A*a^3*b^7*d^6-360/e^9*ln(e*x+d)*A*a^2*b^8*d^7+90/e^10*ln(e*x+d)*A*a*b^9*d^8-20/e^3*ln(e*x+d)*B*a^9*b*d+135/e^4
*ln(e*x+d)*B*a^8*b^2*d^2-480/e^5*ln(e*x+d)*B*a^7*b^3*d^3+1050/e^6*ln(e*x+d)*B*a^6*b^4*d^4-1512/e^7*ln(e*x+d)*B
*a^5*b^5*d^5+1470/e^8*ln(e*x+d)*B*a^4*b^6*d^6-960/e^9*ln(e*x+d)*B*a^3*b^7*d^7+405/e^10*ln(e*x+d)*B*a^2*b^8*d^8
-100/e^11*ln(e*x+d)*B*a*b^9*d^9+30/7*b^9/e^4*B*x^7*a*d^2-15*b^8/e^3*A*x^6*a^2*d+5*b^9/e^4*A*x^6*a*d^2-40*b^7/e
^3*B*x^6*a^3*d+45/2*b^8/e^4*B*x^6*a^2*d^2-20/3*b^9/e^5*B*x^6*a*d^3-48*b^7/e^3*A*x^5*a^3*d+27*b^8/e^4*A*x^5*a^2
*d^2-8*b^9/e^5*A*x^5*a*d^3+1050*b^6/e^6*A*a^4*d^4*x-720*b^7/e^7*A*a^3*d^5*x+315*b^8/e^8*A*a^2*d^6*x-80*b^9/e^9
*A*a*d^7*x-90*b^2/e^3*B*a^8*d*x+360*b^3/e^4*B*a^7*d^2*x-840*b^4/e^5*B*a^6*d^3*x+1260*b^5/e^6*B*a^5*d^4*x-1260*
b^6/e^7*B*a^4*d^5*x+840*b^7/e^8*B*a^3*d^6*x-84*b^6/e^3*B*x^5*a^4*d+72*b^7/e^4*B*x^5*a^3*d^2-36*b^8/e^5*B*x^5*a
^2*d^3+10*b^9/e^6*B*x^5*a*d^4-105*b^6/e^3*A*x^4*a^4*d+90*b^7/e^4*A*x^4*a^3*d^2-45*b^8/e^5*A*x^4*a^2*d^3+25/2*b
^9/e^6*A*x^4*a*d^4-90/7*b^8/e^3*B*x^7*a^2*d-504*b^5/e^5*B*x^2*a^5*d^3+525*b^6/e^6*B*x^2*a^4*d^4-360*b^7/e^7*B*
x^2*a^3*d^5+315/2*b^8/e^8*B*x^2*a^2*d^6-40*b^9/e^9*B*x^2*a*d^7-240*b^3/e^3*A*a^7*d*x+630*b^4/e^4*A*a^6*d^2*x-1
008*b^5/e^5*A*a^5*d^3*x+225/4*b^8/e^6*B*x^4*a^2*d^4-15*b^9/e^7*B*x^4*a*d^5-168*b^5/e^3*A*x^3*a^5*d+210*b^6/e^4
*A*x^3*a^4*d^2+10/e^2/(e*x+d)*A*d*a^9*b-45/e^3/(e*x+d)*A*a^8*b^2*d^2+120/e^4/(e*x+d)*A*a^7*b^3*d^3-210/e^5/(e*
x+d)*A*a^6*b^4*d^4+252/e^6/(e*x+d)*A*a^5*b^5*d^5-210/e^7/(e*x+d)*A*a^4*b^6*d^6+120/e^8/(e*x+d)*A*a^3*b^7*d^7-4
5/e^9/(e*x+d)*A*a^2*b^8*d^8+10/e^10/(e*x+d)*A*a*b^9*d^9-10/e^3/(e*x+d)*B*a^9*b*d^2-2/9*b^10/e^3*B*x^9*d+5/4*b^
9/e^2*A*x^8*a+1/10*b^10/e^2*B*x^10+1/9*b^10/e^2*A*x^9-1/e/(e*x+d)*a^10*A+1/e^2*ln(e*x+d)*B*a^10+9*b^10/e^10*A*
d^8*x-1/e^11/(e*x+d)*A*b^10*d^10+1/e^2/(e*x+d)*B*d*a^10+1/e^12/(e*x+d)*b^10*B*d^11+10/e^2*ln(e*x+d)*A*a^9*b-10
/e^11*ln(e*x+d)*A*b^10*d^9+11/e^12*ln(e*x+d)*b^10*B*d^10-1/4*b^10/e^3*A*x^8*d+45/8*b^8/e^2*B*x^8*a^2+3/8*b^10/
e^4*B*x^8*d^2+45/7*b^8/e^2*A*x^7*a^2+10/9*b^9/e^2*B*x^9*a+252/5*b^5/e^2*B*x^5*a^5-6/5*b^10/e^7*B*x^5*d^5+63*b^
5/e^2*A*x^4*a^5-3/2*b^10/e^7*A*x^4*d^5+105/2*b^4/e^2*B*x^4*a^6+b^10/e^6*A*x^5*d^4+3/7*b^10/e^4*A*x^7*d^2+120/7
*b^7/e^2*B*x^7*a^3-4/7*b^10/e^5*B*x^7*d^3+20*b^7/e^2*A*x^6*a^3-2/3*b^10/e^5*A*x^6*d^3+35*b^6/e^2*B*x^6*a^4+5/6
*b^10/e^6*B*x^6*d^4+42*b^6/e^2*A*x^5*a^4+7/4*b^10/e^8*B*x^4*d^6+70*b^4/e^2*A*x^3*a^6+7/3*b^10/e^8*A*x^3*d^6+10
*b/e^2*B*a^9*x-10*b^10/e^11*B*d^9*x+40*b^3/e^2*B*x^3*a^7-8/3*b^10/e^9*B*x^3*d^7+60*b^3/e^2*A*x^2*a^7-4*b^10/e^
9*A*x^2*d^7+45/2*b^2/e^2*B*x^2*a^8+45/e^4/(e*x+d)*B*a^8*b^2*d^3-120/e^5/(e*x+d)*B*a^7*b^3*d^4+210/e^6/(e*x+d)*
B*a^6*b^4*d^5-252/e^7/(e*x+d)*B*a^5*b^5*d^6+210/e^8/(e*x+d)*B*a^4*b^6*d^7-120/e^9/(e*x+d)*B*a^3*b^7*d^8+45/e^1
0/(e*x+d)*B*a^2*b^8*d^9-10/e^11/(e*x+d)*B*a*b^9*d^10-160*b^7/e^5*A*x^3*a^3*d^3+75*b^8/e^6*A*x^3*a^2*d^4-20*b^9
/e^7*A*x^3*a*d^5-140*b^4/e^3*B*x^3*a^6*d+252*b^5/e^4*B*x^3*a^5*d^2-280*b^6/e^5*B*x^3*a^4*d^3+200*b^7/e^6*B*x^3
*a^3*d^4-90*b^8/e^7*B*x^3*a^2*d^5+70/3*b^9/e^8*B*x^3*a*d^6-5/2*b^9/e^3*B*x^8*a*d-20/7*b^9/e^3*A*x^7*a*d-360*b^
8/e^9*B*a^2*d^7*x+90*b^9/e^10*B*a*d^8*x-126*b^5/e^3*B*x^4*a^5*d+315/2*b^6/e^4*B*x^4*a^4*d^2-120*b^7/e^5*B*x^4*
a^3*d^3-210*b^4/e^3*A*x^2*a^6*d+378*b^5/e^4*A*x^2*a^5*d^2-420*b^6/e^5*A*x^2*a^4*d^3+300*b^7/e^6*A*x^2*a^3*d^4-
135*b^8/e^7*A*x^2*a^2*d^5+35*b^9/e^8*A*x^2*a*d^6-120*b^3/e^3*B*x^2*a^7*d+315*b^4/e^4*B*x^2*a^6*d^2

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Maxima [B]  time = 1.26169, size = 2453, normalized size = 5.51 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^2,x, algorithm="maxima")

[Out]

(B*b^10*d^11 - A*a^10*e^11 - (10*B*a*b^9 + A*b^10)*d^10*e + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 - 15*(8*B*a^3*
b^7 + 3*A*a^2*b^8)*d^8*e^3 + 30*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4 - 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 +
 42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 - 30*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4*e^7 + 15*(3*B*a^8*b^2 + 8*A*a^7*b
^3)*d^3*e^8 - 5*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9 + (B*a^10 + 10*A*a^9*b)*d*e^10)/(e^13*x + d*e^12) + 1/2520*(
252*B*b^10*e^9*x^10 - 280*(2*B*b^10*d*e^8 - (10*B*a*b^9 + A*b^10)*e^9)*x^9 + 315*(3*B*b^10*d^2*e^7 - 2*(10*B*a
*b^9 + A*b^10)*d*e^8 + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*e^9)*x^8 - 360*(4*B*b^10*d^3*e^6 - 3*(10*B*a*b^9 + A*b^10)*
d^2*e^7 + 10*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^8 - 15*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^9)*x^7 + 420*(5*B*b^10*d^4*e^5
 - 4*(10*B*a*b^9 + A*b^10)*d^3*e^6 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^7 - 30*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e
^8 + 30*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^9)*x^6 - 504*(6*B*b^10*d^5*e^4 - 5*(10*B*a*b^9 + A*b^10)*d^4*e^5 + 20*(9
*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^6 - 45*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^7 + 60*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^
8 - 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^9)*x^5 + 630*(7*B*b^10*d^6*e^3 - 6*(10*B*a*b^9 + A*b^10)*d^5*e^4 + 25*(9*
B*a^2*b^8 + 2*A*a*b^9)*d^4*e^5 - 60*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^6 + 90*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e
^7 - 84*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^8 + 42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^9)*x^4 - 840*(8*B*b^10*d^7*e^2 -
7*(10*B*a*b^9 + A*b^10)*d^6*e^3 + 30*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^4 - 75*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^
5 + 120*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^6 - 126*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^7 + 84*(5*B*a^6*b^4 + 6*A*
a^5*b^5)*d*e^8 - 30*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^9)*x^3 + 1260*(9*B*b^10*d^8*e - 8*(10*B*a*b^9 + A*b^10)*d^7*
e^2 + 35*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^3 - 90*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^4 + 150*(7*B*a^4*b^6 + 4*A*a
^3*b^7)*d^4*e^5 - 168*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^6 + 126*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*e^7 - 60*(4*B*
a^7*b^3 + 7*A*a^6*b^4)*d*e^8 + 15*(3*B*a^8*b^2 + 8*A*a^7*b^3)*e^9)*x^2 - 2520*(10*B*b^10*d^9 - 9*(10*B*a*b^9 +
 A*b^10)*d^8*e + 40*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^2 - 105*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^3 + 180*(7*B*a^4
*b^6 + 4*A*a^3*b^7)*d^5*e^4 - 210*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*e^5 + 168*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*e^
6 - 90*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*e^7 + 30*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d*e^8 - 5*(2*B*a^9*b + 9*A*a^8*b^2
)*e^9)*x)/e^11 + (11*B*b^10*d^10 - 10*(10*B*a*b^9 + A*b^10)*d^9*e + 45*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^2 - 120
*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^3 + 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^4 - 252*(6*B*a^5*b^5 + 5*A*a^4*b^
6)*d^5*e^5 + 210*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*e^6 - 120*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*e^7 + 45*(3*B*a^8*b
^2 + 8*A*a^7*b^3)*d^2*e^8 - 10*(2*B*a^9*b + 9*A*a^8*b^2)*d*e^9 + (B*a^10 + 10*A*a^9*b)*e^10)*log(e*x + d)/e^12

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Fricas [B]  time = 2.19453, size = 5092, normalized size = 11.44 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^2,x, algorithm="fricas")

[Out]

1/2520*(252*B*b^10*e^11*x^11 + 2520*B*b^10*d^11 - 2520*A*a^10*e^11 - 2520*(10*B*a*b^9 + A*b^10)*d^10*e + 12600
*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 - 37800*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 75600*(7*B*a^4*b^6 + 4*A*a^3*
b^7)*d^7*e^4 - 105840*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 105840*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 - 75600
*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4*e^7 + 37800*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^8 - 12600*(2*B*a^9*b + 9*A*a^8*
b^2)*d^2*e^9 + 2520*(B*a^10 + 10*A*a^9*b)*d*e^10 - 28*(11*B*b^10*d*e^10 - 10*(10*B*a*b^9 + A*b^10)*e^11)*x^10
+ 35*(11*B*b^10*d^2*e^9 - 10*(10*B*a*b^9 + A*b^10)*d*e^10 + 45*(9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9 - 45*(11*B*
b^10*d^3*e^8 - 10*(10*B*a*b^9 + A*b^10)*d^2*e^9 + 45*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^10 - 120*(8*B*a^3*b^7 + 3*A
*a^2*b^8)*e^11)*x^8 + 60*(11*B*b^10*d^4*e^7 - 10*(10*B*a*b^9 + A*b^10)*d^3*e^8 + 45*(9*B*a^2*b^8 + 2*A*a*b^9)*
d^2*e^9 - 120*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^10 + 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 - 84*(11*B*b^10*d
^5*e^6 - 10*(10*B*a*b^9 + A*b^10)*d^4*e^7 + 45*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^8 - 120*(8*B*a^3*b^7 + 3*A*a^2*
b^8)*d^2*e^9 + 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^10 - 252*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 126*(11*B*
b^10*d^6*e^5 - 10*(10*B*a*b^9 + A*b^10)*d^5*e^6 + 45*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 - 120*(8*B*a^3*b^7 + 3*
A*a^2*b^8)*d^3*e^8 + 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^9 - 252*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 + 210*(5
*B*a^6*b^4 + 6*A*a^5*b^5)*e^11)*x^5 - 210*(11*B*b^10*d^7*e^4 - 10*(10*B*a*b^9 + A*b^10)*d^6*e^5 + 45*(9*B*a^2*
b^8 + 2*A*a*b^9)*d^5*e^6 - 120*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 + 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^8 -
 252*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^9 + 210*(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 + 420*(11*B*b^10*d^8*e^3 - 10*(10*B*a*b^9 + A*b^10)*d^7*e^4 + 45*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6
*e^5 - 120*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^6 + 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^7 - 252*(6*B*a^5*b^5 +
5*A*a^4*b^6)*d^3*e^8 + 210*(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 + 45*(
3*B*a^8*b^2 + 8*A*a^7*b^3)*e^11)*x^3 - 1260*(11*B*b^10*d^9*e^2 - 10*(10*B*a*b^9 + A*b^10)*d^8*e^3 + 45*(9*B*a^
2*b^8 + 2*A*a*b^9)*d^7*e^4 - 120*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^5 + 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*e^6
 - 252*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*e^7 + 210*(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 + 45*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d*e^10 - 10*(2*B*a^9*b + 9*A*a^8*b^2)*e^11)*x^2 - 2520*(10*B
*b^10*d^10*e - 9*(10*B*a*b^9 + A*b^10)*d^9*e^2 + 40*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^3 - 105*(8*B*a^3*b^7 + 3*A
*a^2*b^8)*d^7*e^4 + 180*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5 - 210*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 + 168*(5
*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*e^7 - 90*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*e^8 + 30*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^
2*e^9 - 5*(2*B*a^9*b + 9*A*a^8*b^2)*d*e^10)*x + 2520*(11*B*b^10*d^11 - 10*(10*B*a*b^9 + A*b^10)*d^10*e + 45*(9
*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 - 120*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^
7*e^4 - 252*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 210*(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 + 45*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^8 - 10*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9 + (B*a^1
0 + 10*A*a^9*b)*d*e^10 + (11*B*b^10*d^10*e - 10*(10*B*a*b^9 + A*b^10)*d^9*e^2 + 45*(9*B*a^2*b^8 + 2*A*a*b^9)*d
^8*e^3 - 120*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4 + 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5 - 252*(6*B*a^5*b^5
+ 5*A*a^4*b^6)*d^5*e^6 + 210*(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 + 4
5*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e^9 - 10*(2*B*a^9*b + 9*A*a^8*b^2)*d*e^10 + (B*a^10 + 10*A*a^9*b)*e^11)*x)*l
og(e*x + d))/(e^13*x + d*e^12)

________________________________________________________________________________________

Sympy [B]  time = 11.0026, size = 1904, normalized size = 4.28 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**10*(B*x+A)/(e*x+d)**2,x)

[Out]

B*b**10*x**10/(10*e**2) + (-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)/(d*e**12 + e**13*x) + x**9*(A*b**10*e + 10*B*a*b**9*e - 2*B*b**10*d)/(9*e**3) + x**8*
(10*A*a*b**9*e**2 - 2*A*b**10*d*e + 45*B*a**2*b**8*e**2 - 20*B*a*b**9*d*e + 3*B*b**10*d**2)/(8*e**4) + x**7*(4
5*A*a**2*b**8*e**3 - 20*A*a*b**9*d*e**2 + 3*A*b**10*d**2*e + 120*B*a**3*b**7*e**3 - 90*B*a**2*b**8*d*e**2 + 30
*B*a*b**9*d**2*e - 4*B*b**10*d**3)/(7*e**5) + x**6*(120*A*a**3*b**7*e**4 - 90*A*a**2*b**8*d*e**3 + 30*A*a*b**9
*d**2*e**2 - 4*A*b**10*d**3*e + 210*B*a**4*b**6*e**4 - 240*B*a**3*b**7*d*e**3 + 135*B*a**2*b**8*d**2*e**2 - 40
*B*a*b**9*d**3*e + 5*B*b**10*d**4)/(6*e**6) + x**5*(210*A*a**4*b**6*e**5 - 240*A*a**3*b**7*d*e**4 + 135*A*a**2
*b**8*d**2*e**3 - 40*A*a*b**9*d**3*e**2 + 5*A*b**10*d**4*e + 252*B*a**5*b**5*e**5 - 420*B*a**4*b**6*d*e**4 + 3
60*B*a**3*b**7*d**2*e**3 - 180*B*a**2*b**8*d**3*e**2 + 50*B*a*b**9*d**4*e - 6*B*b**10*d**5)/(5*e**7) + x**4*(2
52*A*a**5*b**5*e**6 - 420*A*a**4*b**6*d*e**5 + 360*A*a**3*b**7*d**2*e**4 - 180*A*a**2*b**8*d**3*e**3 + 50*A*a*
b**9*d**4*e**2 - 6*A*b**10*d**5*e + 210*B*a**6*b**4*e**6 - 504*B*a**5*b**5*d*e**5 + 630*B*a**4*b**6*d**2*e**4
- 480*B*a**3*b**7*d**3*e**3 + 225*B*a**2*b**8*d**4*e**2 - 60*B*a*b**9*d**5*e + 7*B*b**10*d**6)/(4*e**8) + x**3
*(210*A*a**6*b**4*e**7 - 504*A*a**5*b**5*d*e**6 + 630*A*a**4*b**6*d**2*e**5 - 480*A*a**3*b**7*d**3*e**4 + 225*
A*a**2*b**8*d**4*e**3 - 60*A*a*b**9*d**5*e**2 + 7*A*b**10*d**6*e + 120*B*a**7*b**3*e**7 - 420*B*a**6*b**4*d*e*
*6 + 756*B*a**5*b**5*d**2*e**5 - 840*B*a**4*b**6*d**3*e**4 + 600*B*a**3*b**7*d**4*e**3 - 270*B*a**2*b**8*d**5*
e**2 + 70*B*a*b**9*d**6*e - 8*B*b**10*d**7)/(3*e**9) + x**2*(120*A*a**7*b**3*e**8 - 420*A*a**6*b**4*d*e**7 + 7
56*A*a**5*b**5*d**2*e**6 - 840*A*a**4*b**6*d**3*e**5 + 600*A*a**3*b**7*d**4*e**4 - 270*A*a**2*b**8*d**5*e**3 +
 70*A*a*b**9*d**6*e**2 - 8*A*b**10*d**7*e + 45*B*a**8*b**2*e**8 - 240*B*a**7*b**3*d*e**7 + 630*B*a**6*b**4*d**
2*e**6 - 1008*B*a**5*b**5*d**3*e**5 + 1050*B*a**4*b**6*d**4*e**4 - 720*B*a**3*b**7*d**5*e**3 + 315*B*a**2*b**8
*d**6*e**2 - 80*B*a*b**9*d**7*e + 9*B*b**10*d**8)/(2*e**10) + x*(45*A*a**8*b**2*e**9 - 240*A*a**7*b**3*d*e**8
+ 630*A*a**6*b**4*d**2*e**7 - 1008*A*a**5*b**5*d**3*e**6 + 1050*A*a**4*b**6*d**4*e**5 - 720*A*a**3*b**7*d**5*e
**4 + 315*A*a**2*b**8*d**6*e**3 - 80*A*a*b**9*d**7*e**2 + 9*A*b**10*d**8*e + 10*B*a**9*b*e**9 - 90*B*a**8*b**2
*d*e**8 + 360*B*a**7*b**3*d**2*e**7 - 840*B*a**6*b**4*d**3*e**6 + 1260*B*a**5*b**5*d**4*e**5 - 1260*B*a**4*b**
6*d**5*e**4 + 840*B*a**3*b**7*d**6*e**3 - 360*B*a**2*b**8*d**7*e**2 + 90*B*a*b**9*d**8*e - 10*B*b**10*d**9)/e*
*11 + (a*e - b*d)**9*(10*A*b*e + B*a*e - 11*B*b*d)*log(d + e*x)/e**12

________________________________________________________________________________________

Giac [B]  time = 1.71976, size = 2824, normalized size = 6.35 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^2,x, algorithm="giac")

[Out]

1/2520*(252*B*b^10 - 280*(11*B*b^10*d*e - 10*B*a*b^9*e^2 - A*b^10*e^2)*e^(-1)/(x*e + d) + 1575*(11*B*b^10*d^2*
e^2 - 20*B*a*b^9*d*e^3 - 2*A*b^10*d*e^3 + 9*B*a^2*b^8*e^4 + 2*A*a*b^9*e^4)*e^(-2)/(x*e + d)^2 - 5400*(11*B*b^1
0*d^3*e^3 - 30*B*a*b^9*d^2*e^4 - 3*A*b^10*d^2*e^4 + 27*B*a^2*b^8*d*e^5 + 6*A*a*b^9*d*e^5 - 8*B*a^3*b^7*e^6 - 3
*A*a^2*b^8*e^6)*e^(-3)/(x*e + d)^3 + 12600*(11*B*b^10*d^4*e^4 - 40*B*a*b^9*d^3*e^5 - 4*A*b^10*d^3*e^5 + 54*B*a
^2*b^8*d^2*e^6 + 12*A*a*b^9*d^2*e^6 - 32*B*a^3*b^7*d*e^7 - 12*A*a^2*b^8*d*e^7 + 7*B*a^4*b^6*e^8 + 4*A*a^3*b^7*
e^8)*e^(-4)/(x*e + d)^4 - 21168*(11*B*b^10*d^5*e^5 - 50*B*a*b^9*d^4*e^6 - 5*A*b^10*d^4*e^6 + 90*B*a^2*b^8*d^3*
e^7 + 20*A*a*b^9*d^3*e^7 - 80*B*a^3*b^7*d^2*e^8 - 30*A*a^2*b^8*d^2*e^8 + 35*B*a^4*b^6*d*e^9 + 20*A*a^3*b^7*d*e
^9 - 6*B*a^5*b^5*e^10 - 5*A*a^4*b^6*e^10)*e^(-5)/(x*e + d)^5 + 26460*(11*B*b^10*d^6*e^6 - 60*B*a*b^9*d^5*e^7 -
 6*A*b^10*d^5*e^7 + 135*B*a^2*b^8*d^4*e^8 + 30*A*a*b^9*d^4*e^8 - 160*B*a^3*b^7*d^3*e^9 - 60*A*a^2*b^8*d^3*e^9
+ 105*B*a^4*b^6*d^2*e^10 + 60*A*a^3*b^7*d^2*e^10 - 36*B*a^5*b^5*d*e^11 - 30*A*a^4*b^6*d*e^11 + 5*B*a^6*b^4*e^1
2 + 6*A*a^5*b^5*e^12)*e^(-6)/(x*e + d)^6 - 25200*(11*B*b^10*d^7*e^7 - 70*B*a*b^9*d^6*e^8 - 7*A*b^10*d^6*e^8 +
189*B*a^2*b^8*d^5*e^9 + 42*A*a*b^9*d^5*e^9 - 280*B*a^3*b^7*d^4*e^10 - 105*A*a^2*b^8*d^4*e^10 + 245*B*a^4*b^6*d
^3*e^11 + 140*A*a^3*b^7*d^3*e^11 - 126*B*a^5*b^5*d^2*e^12 - 105*A*a^4*b^6*d^2*e^12 + 35*B*a^6*b^4*d*e^13 + 42*
A*a^5*b^5*d*e^13 - 4*B*a^7*b^3*e^14 - 7*A*a^6*b^4*e^14)*e^(-7)/(x*e + d)^7 + 18900*(11*B*b^10*d^8*e^8 - 80*B*a
*b^9*d^7*e^9 - 8*A*b^10*d^7*e^9 + 252*B*a^2*b^8*d^6*e^10 + 56*A*a*b^9*d^6*e^10 - 448*B*a^3*b^7*d^5*e^11 - 168*
A*a^2*b^8*d^5*e^11 + 490*B*a^4*b^6*d^4*e^12 + 280*A*a^3*b^7*d^4*e^12 - 336*B*a^5*b^5*d^3*e^13 - 280*A*a^4*b^6*
d^3*e^13 + 140*B*a^6*b^4*d^2*e^14 + 168*A*a^5*b^5*d^2*e^14 - 32*B*a^7*b^3*d*e^15 - 56*A*a^6*b^4*d*e^15 + 3*B*a
^8*b^2*e^16 + 8*A*a^7*b^3*e^16)*e^(-8)/(x*e + d)^8 - 12600*(11*B*b^10*d^9*e^9 - 90*B*a*b^9*d^8*e^10 - 9*A*b^10
*d^8*e^10 + 324*B*a^2*b^8*d^7*e^11 + 72*A*a*b^9*d^7*e^11 - 672*B*a^3*b^7*d^6*e^12 - 252*A*a^2*b^8*d^6*e^12 + 8
82*B*a^4*b^6*d^5*e^13 + 504*A*a^3*b^7*d^5*e^13 - 756*B*a^5*b^5*d^4*e^14 - 630*A*a^4*b^6*d^4*e^14 + 420*B*a^6*b
^4*d^3*e^15 + 504*A*a^5*b^5*d^3*e^15 - 144*B*a^7*b^3*d^2*e^16 - 252*A*a^6*b^4*d^2*e^16 + 27*B*a^8*b^2*d*e^17 +
 72*A*a^7*b^3*d*e^17 - 2*B*a^9*b*e^18 - 9*A*a^8*b^2*e^18)*e^(-9)/(x*e + d)^9)*(x*e + d)^10*e^(-12) - (11*B*b^1
0*d^10 - 100*B*a*b^9*d^9*e - 10*A*b^10*d^9*e + 405*B*a^2*b^8*d^8*e^2 + 90*A*a*b^9*d^8*e^2 - 960*B*a^3*b^7*d^7*
e^3 - 360*A*a^2*b^8*d^7*e^3 + 1470*B*a^4*b^6*d^6*e^4 + 840*A*a^3*b^7*d^6*e^4 - 1512*B*a^5*b^5*d^5*e^5 - 1260*A
*a^4*b^6*d^5*e^5 + 1050*B*a^6*b^4*d^4*e^6 + 1260*A*a^5*b^5*d^4*e^6 - 480*B*a^7*b^3*d^3*e^7 - 840*A*a^6*b^4*d^3
*e^7 + 135*B*a^8*b^2*d^2*e^8 + 360*A*a^7*b^3*d^2*e^8 - 20*B*a^9*b*d*e^9 - 90*A*a^8*b^2*d*e^9 + B*a^10*e^10 + 1
0*A*a^9*b*e^10)*e^(-12)*log(abs(x*e + d)*e^(-1)/(x*e + d)^2) + (B*b^10*d^11*e^10/(x*e + d) - 10*B*a*b^9*d^10*e
^11/(x*e + d) - A*b^10*d^10*e^11/(x*e + d) + 45*B*a^2*b^8*d^9*e^12/(x*e + d) + 10*A*a*b^9*d^9*e^12/(x*e + d) -
 120*B*a^3*b^7*d^8*e^13/(x*e + d) - 45*A*a^2*b^8*d^8*e^13/(x*e + d) + 210*B*a^4*b^6*d^7*e^14/(x*e + d) + 120*A
*a^3*b^7*d^7*e^14/(x*e + d) - 252*B*a^5*b^5*d^6*e^15/(x*e + d) - 210*A*a^4*b^6*d^6*e^15/(x*e + d) + 210*B*a^6*
b^4*d^5*e^16/(x*e + d) + 252*A*a^5*b^5*d^5*e^16/(x*e + d) - 120*B*a^7*b^3*d^4*e^17/(x*e + d) - 210*A*a^6*b^4*d
^4*e^17/(x*e + d) + 45*B*a^8*b^2*d^3*e^18/(x*e + d) + 120*A*a^7*b^3*d^3*e^18/(x*e + d) - 10*B*a^9*b*d^2*e^19/(
x*e + d) - 45*A*a^8*b^2*d^2*e^19/(x*e + d) + B*a^10*d*e^20/(x*e + d) + 10*A*a^9*b*d*e^20/(x*e + d) - A*a^10*e^
21/(x*e + d))*e^(-22)

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