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3.1091 \(\int \frac{(a+b x)^{10} (A+B x)}{(d+e x)^3} \, dx\)

Optimal. Leaf size=445 \[ -\frac{b^9 (d+e x)^8 (-10 a B e-A b e+11 b B d)}{8 e^{12}}+\frac{5 b^8 (d+e x)^7 (b d-a e) (-9 a B e-2 A b e+11 b B d)}{7 e^{12}}-\frac{5 b^7 (d+e x)^6 (b d-a e)^2 (-8 a B e-3 A b e+11 b B d)}{2 e^{12}}+\frac{6 b^6 (d+e x)^5 (b d-a e)^3 (-7 a B e-4 A b e+11 b B d)}{e^{12}}-\frac{21 b^5 (d+e x)^4 (b d-a e)^4 (-6 a B e-5 A b e+11 b B d)}{2 e^{12}}+\frac{14 b^4 (d+e x)^3 (b d-a e)^5 (-5 a B e-6 A b e+11 b B d)}{e^{12}}-\frac{15 b^3 (d+e x)^2 (b d-a e)^6 (-4 a B e-7 A b e+11 b B d)}{e^{12}}+\frac{15 b^2 x (b d-a e)^7 (-3 a B e-8 A b e+11 b B d)}{e^{11}}-\frac{(b d-a e)^9 (-a B e-10 A b e+11 b B d)}{e^{12} (d+e x)}+\frac{(b d-a e)^{10} (B d-A e)}{2 e^{12} (d+e x)^2}-\frac{5 b (b d-a e)^8 \log (d+e x) (-2 a B e-9 A b e+11 b B d)}{e^{12}}+\frac{b^{10} B (d+e x)^9}{9 e^{12}} \]

[Out]

(15*b^2*(b*d - a*e)^7*(11*b*B*d - 8*A*b*e - 3*a*B*e)*x)/e^11 + ((b*d - a*e)^10*(B*d - A*e))/(2*e^12*(d + e*x)^
2) - ((b*d - a*e)^9*(11*b*B*d - 10*A*b*e - a*B*e))/(e^12*(d + e*x)) - (15*b^3*(b*d - a*e)^6*(11*b*B*d - 7*A*b*
e - 4*a*B*e)*(d + e*x)^2)/e^12 + (14*b^4*(b*d - a*e)^5*(11*b*B*d - 6*A*b*e - 5*a*B*e)*(d + e*x)^3)/e^12 - (21*
b^5*(b*d - a*e)^4*(11*b*B*d - 5*A*b*e - 6*a*B*e)*(d + e*x)^4)/(2*e^12) + (6*b^6*(b*d - a*e)^3*(11*b*B*d - 4*A*
b*e - 7*a*B*e)*(d + e*x)^5)/e^12 - (5*b^7*(b*d - a*e)^2*(11*b*B*d - 3*A*b*e - 8*a*B*e)*(d + e*x)^6)/(2*e^12) +
 (5*b^8*(b*d - a*e)*(11*b*B*d - 2*A*b*e - 9*a*B*e)*(d + e*x)^7)/(7*e^12) - (b^9*(11*b*B*d - A*b*e - 10*a*B*e)*
(d + e*x)^8)/(8*e^12) + (b^10*B*(d + e*x)^9)/(9*e^12) - (5*b*(b*d - a*e)^8*(11*b*B*d - 9*A*b*e - 2*a*B*e)*Log[
d + e*x])/e^12

________________________________________________________________________________________

Rubi [A]  time = 1.48811, 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} \[ -\frac{b^9 (d+e x)^8 (-10 a B e-A b e+11 b B d)}{8 e^{12}}+\frac{5 b^8 (d+e x)^7 (b d-a e) (-9 a B e-2 A b e+11 b B d)}{7 e^{12}}-\frac{5 b^7 (d+e x)^6 (b d-a e)^2 (-8 a B e-3 A b e+11 b B d)}{2 e^{12}}+\frac{6 b^6 (d+e x)^5 (b d-a e)^3 (-7 a B e-4 A b e+11 b B d)}{e^{12}}-\frac{21 b^5 (d+e x)^4 (b d-a e)^4 (-6 a B e-5 A b e+11 b B d)}{2 e^{12}}+\frac{14 b^4 (d+e x)^3 (b d-a e)^5 (-5 a B e-6 A b e+11 b B d)}{e^{12}}-\frac{15 b^3 (d+e x)^2 (b d-a e)^6 (-4 a B e-7 A b e+11 b B d)}{e^{12}}+\frac{15 b^2 x (b d-a e)^7 (-3 a B e-8 A b e+11 b B d)}{e^{11}}-\frac{(b d-a e)^9 (-a B e-10 A b e+11 b B d)}{e^{12} (d+e x)}+\frac{(b d-a e)^{10} (B d-A e)}{2 e^{12} (d+e x)^2}-\frac{5 b (b d-a e)^8 \log (d+e x) (-2 a B e-9 A b e+11 b B d)}{e^{12}}+\frac{b^{10} B (d+e x)^9}{9 e^{12}} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^10*(A + B*x))/(d + e*x)^3,x]

[Out]

(15*b^2*(b*d - a*e)^7*(11*b*B*d - 8*A*b*e - 3*a*B*e)*x)/e^11 + ((b*d - a*e)^10*(B*d - A*e))/(2*e^12*(d + e*x)^
2) - ((b*d - a*e)^9*(11*b*B*d - 10*A*b*e - a*B*e))/(e^12*(d + e*x)) - (15*b^3*(b*d - a*e)^6*(11*b*B*d - 7*A*b*
e - 4*a*B*e)*(d + e*x)^2)/e^12 + (14*b^4*(b*d - a*e)^5*(11*b*B*d - 6*A*b*e - 5*a*B*e)*(d + e*x)^3)/e^12 - (21*
b^5*(b*d - a*e)^4*(11*b*B*d - 5*A*b*e - 6*a*B*e)*(d + e*x)^4)/(2*e^12) + (6*b^6*(b*d - a*e)^3*(11*b*B*d - 4*A*
b*e - 7*a*B*e)*(d + e*x)^5)/e^12 - (5*b^7*(b*d - a*e)^2*(11*b*B*d - 3*A*b*e - 8*a*B*e)*(d + e*x)^6)/(2*e^12) +
 (5*b^8*(b*d - a*e)*(11*b*B*d - 2*A*b*e - 9*a*B*e)*(d + e*x)^7)/(7*e^12) - (b^9*(11*b*B*d - A*b*e - 10*a*B*e)*
(d + e*x)^8)/(8*e^12) + (b^10*B*(d + e*x)^9)/(9*e^12) - (5*b*(b*d - a*e)^8*(11*b*B*d - 9*A*b*e - 2*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)^3} \, dx &=\int \left (-\frac{15 b^2 (b d-a e)^7 (-11 b B d+8 A b e+3 a B e)}{e^{11}}+\frac{(-b d+a e)^{10} (-B d+A e)}{e^{11} (d+e x)^3}+\frac{(-b d+a e)^9 (-11 b B d+10 A b e+a B e)}{e^{11} (d+e x)^2}+\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)}+\frac{30 b^3 (b d-a e)^6 (-11 b B d+7 A b e+4 a B e) (d+e x)}{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)^2}{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)^3}{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)^4}{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)^5}{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)^6}{e^{11}}+\frac{b^9 (-11 b B d+A b e+10 a B e) (d+e x)^7}{e^{11}}+\frac{b^{10} B (d+e x)^8}{e^{11}}\right ) \, dx\\ &=\frac{15 b^2 (b d-a e)^7 (11 b B d-8 A b e-3 a B e) x}{e^{11}}+\frac{(b d-a e)^{10} (B d-A e)}{2 e^{12} (d+e x)^2}-\frac{(b d-a e)^9 (11 b B d-10 A b e-a B e)}{e^{12} (d+e x)}-\frac{15 b^3 (b d-a e)^6 (11 b B d-7 A b e-4 a B e) (d+e x)^2}{e^{12}}+\frac{14 b^4 (b d-a e)^5 (11 b B d-6 A b e-5 a B e) (d+e x)^3}{e^{12}}-\frac{21 b^5 (b d-a e)^4 (11 b B d-5 A b e-6 a B e) (d+e x)^4}{2 e^{12}}+\frac{6 b^6 (b d-a e)^3 (11 b B d-4 A b e-7 a B e) (d+e x)^5}{e^{12}}-\frac{5 b^7 (b d-a e)^2 (11 b B d-3 A b e-8 a B e) (d+e x)^6}{2 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)^7}{7 e^{12}}-\frac{b^9 (11 b B d-A b e-10 a B e) (d+e x)^8}{8 e^{12}}+\frac{b^{10} B (d+e x)^9}{9 e^{12}}-\frac{5 b (b d-a e)^8 (11 b B d-9 A b e-2 a B e) \log (d+e x)}{e^{12}}\\ \end{align*}

Mathematica [B]  time = 0.751766, size = 1480, normalized size = 3.33 \[ \frac{\left (9 A e \left (532 d^{10}-1456 e x d^9-3248 e^2 x^2 d^8-840 e^3 x^3 d^7+210 e^4 x^4 d^6-84 e^5 x^5 d^5+42 e^6 x^6 d^4-24 e^7 x^7 d^3+15 e^8 x^8 d^2-10 e^9 x^9 d+7 e^{10} x^{10}\right )+B \left (-5292 d^{11}+17136 e x d^{10}+36288 e^2 x^2 d^9+9240 e^3 x^3 d^8-2310 e^4 x^4 d^7+924 e^5 x^5 d^6-462 e^6 x^6 d^5+264 e^7 x^7 d^4-165 e^8 x^8 d^3+110 e^9 x^9 d^2-77 e^{10} x^{10} d+56 e^{11} x^{11}\right )\right ) b^{10}+18 a e \left (4 A e \left (-595 d^9+1330 e x d^8+3185 e^2 x^2 d^7+840 e^3 x^3 d^6-210 e^4 x^4 d^5+84 e^5 x^5 d^4-42 e^6 x^6 d^3+24 e^7 x^7 d^2-15 e^8 x^8 d+10 e^9 x^9\right )+5 B \left (532 d^{10}-1456 e x d^9-3248 e^2 x^2 d^8-840 e^3 x^3 d^7+210 e^4 x^4 d^6-84 e^5 x^5 d^5+42 e^6 x^6 d^4-24 e^7 x^7 d^3+15 e^8 x^8 d^2-10 e^9 x^9 d+7 e^{10} x^{10}\right )\right ) b^9+108 a^2 e^2 \left (7 A e \left (225 d^8-390 e x d^7-1035 e^2 x^2 d^6-280 e^3 x^3 d^5+70 e^4 x^4 d^4-28 e^5 x^5 d^3+14 e^6 x^6 d^2-8 e^7 x^7 d+5 e^8 x^8\right )-3 B \left (595 d^9-1330 e x d^8-3185 e^2 x^2 d^7-840 e^3 x^3 d^6+210 e^4 x^4 d^5-84 e^5 x^5 d^4+42 e^6 x^6 d^3-24 e^7 x^7 d^2+15 e^8 x^8 d-10 e^9 x^9\right )\right ) b^8+1008 a^3 e^3 \left (3 A e \left (-130 d^7+160 e x d^6+500 e^2 x^2 d^5+140 e^3 x^3 d^4-35 e^4 x^4 d^3+14 e^5 x^5 d^2-7 e^6 x^6 d+4 e^7 x^7\right )+2 B \left (225 d^8-390 e x d^7-1035 e^2 x^2 d^6-280 e^3 x^3 d^5+70 e^4 x^4 d^4-28 e^5 x^5 d^3+14 e^6 x^6 d^2-8 e^7 x^7 d+5 e^8 x^8\right )\right ) b^7+5292 a^4 e^4 \left (5 A e \left (22 d^6-16 e x d^5-68 e^2 x^2 d^4-20 e^3 x^3 d^3+5 e^4 x^4 d^2-2 e^5 x^5 d+e^6 x^6\right )+B \left (-130 d^7+160 e x d^6+500 e^2 x^2 d^5+140 e^3 x^3 d^4-35 e^4 x^4 d^3+14 e^5 x^5 d^2-7 e^6 x^6 d+4 e^7 x^7\right )\right ) b^6+10584 a^5 e^5 \left (2 A e \left (-27 d^5+6 e x d^4+63 e^2 x^2 d^3+20 e^3 x^3 d^2-5 e^4 x^4 d+2 e^5 x^5\right )+3 B \left (22 d^6-16 e x d^5-68 e^2 x^2 d^4-20 e^3 x^3 d^3+5 e^4 x^4 d^2-2 e^5 x^5 d+e^6 x^6\right )\right ) b^5+17640 a^6 e^6 \left (3 A e \left (7 d^4+2 e x d^3-11 e^2 x^2 d^2-4 e^3 x^3 d+e^4 x^4\right )+B \left (-27 d^5+6 e x d^4+63 e^2 x^2 d^3+20 e^3 x^3 d^2-5 e^4 x^4 d+2 e^5 x^5\right )\right ) b^4+30240 a^7 e^7 \left (A e \left (-5 d^3-4 e x d^2+4 e^2 x^2 d+2 e^3 x^3\right )+B \left (7 d^4+2 e x d^3-11 e^2 x^2 d^2-4 e^3 x^3 d+e^4 x^4\right )\right ) b^3+11340 a^8 e^8 \left (A d e (3 d+4 e x)+B \left (-5 d^3-4 e x d^2+4 e^2 x^2 d+2 e^3 x^3\right )\right ) b^2-2520 a^9 e^9 (A e (d+2 e x)-B d (3 d+4 e x)) b-2520 (b d-a e)^8 (11 b B d-9 A b e-2 a B e) (d+e x)^2 \log (d+e x) b-252 a^{10} e^{10} (A e+B (d+2 e x))}{504 e^{12} (d+e x)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^10*(A + B*x))/(d + e*x)^3,x]

[Out]

(-252*a^10*e^10*(A*e + B*(d + 2*e*x)) - 2520*a^9*b*e^9*(A*e*(d + 2*e*x) - B*d*(3*d + 4*e*x)) + 11340*a^8*b^2*e
^8*(A*d*e*(3*d + 4*e*x) + B*(-5*d^3 - 4*d^2*e*x + 4*d*e^2*x^2 + 2*e^3*x^3)) + 30240*a^7*b^3*e^7*(A*e*(-5*d^3 -
 4*d^2*e*x + 4*d*e^2*x^2 + 2*e^3*x^3) + B*(7*d^4 + 2*d^3*e*x - 11*d^2*e^2*x^2 - 4*d*e^3*x^3 + e^4*x^4)) + 1764
0*a^6*b^4*e^6*(3*A*e*(7*d^4 + 2*d^3*e*x - 11*d^2*e^2*x^2 - 4*d*e^3*x^3 + e^4*x^4) + B*(-27*d^5 + 6*d^4*e*x + 6
3*d^3*e^2*x^2 + 20*d^2*e^3*x^3 - 5*d*e^4*x^4 + 2*e^5*x^5)) + 10584*a^5*b^5*e^5*(2*A*e*(-27*d^5 + 6*d^4*e*x + 6
3*d^3*e^2*x^2 + 20*d^2*e^3*x^3 - 5*d*e^4*x^4 + 2*e^5*x^5) + 3*B*(22*d^6 - 16*d^5*e*x - 68*d^4*e^2*x^2 - 20*d^3
*e^3*x^3 + 5*d^2*e^4*x^4 - 2*d*e^5*x^5 + e^6*x^6)) + 5292*a^4*b^6*e^4*(5*A*e*(22*d^6 - 16*d^5*e*x - 68*d^4*e^2
*x^2 - 20*d^3*e^3*x^3 + 5*d^2*e^4*x^4 - 2*d*e^5*x^5 + e^6*x^6) + B*(-130*d^7 + 160*d^6*e*x + 500*d^5*e^2*x^2 +
 140*d^4*e^3*x^3 - 35*d^3*e^4*x^4 + 14*d^2*e^5*x^5 - 7*d*e^6*x^6 + 4*e^7*x^7)) + 1008*a^3*b^7*e^3*(3*A*e*(-130
*d^7 + 160*d^6*e*x + 500*d^5*e^2*x^2 + 140*d^4*e^3*x^3 - 35*d^3*e^4*x^4 + 14*d^2*e^5*x^5 - 7*d*e^6*x^6 + 4*e^7
*x^7) + 2*B*(225*d^8 - 390*d^7*e*x - 1035*d^6*e^2*x^2 - 280*d^5*e^3*x^3 + 70*d^4*e^4*x^4 - 28*d^3*e^5*x^5 + 14
*d^2*e^6*x^6 - 8*d*e^7*x^7 + 5*e^8*x^8)) + 108*a^2*b^8*e^2*(7*A*e*(225*d^8 - 390*d^7*e*x - 1035*d^6*e^2*x^2 -
280*d^5*e^3*x^3 + 70*d^4*e^4*x^4 - 28*d^3*e^5*x^5 + 14*d^2*e^6*x^6 - 8*d*e^7*x^7 + 5*e^8*x^8) - 3*B*(595*d^9 -
 1330*d^8*e*x - 3185*d^7*e^2*x^2 - 840*d^6*e^3*x^3 + 210*d^5*e^4*x^4 - 84*d^4*e^5*x^5 + 42*d^3*e^6*x^6 - 24*d^
2*e^7*x^7 + 15*d*e^8*x^8 - 10*e^9*x^9)) + 18*a*b^9*e*(4*A*e*(-595*d^9 + 1330*d^8*e*x + 3185*d^7*e^2*x^2 + 840*
d^6*e^3*x^3 - 210*d^5*e^4*x^4 + 84*d^4*e^5*x^5 - 42*d^3*e^6*x^6 + 24*d^2*e^7*x^7 - 15*d*e^8*x^8 + 10*e^9*x^9)
+ 5*B*(532*d^10 - 1456*d^9*e*x - 3248*d^8*e^2*x^2 - 840*d^7*e^3*x^3 + 210*d^6*e^4*x^4 - 84*d^5*e^5*x^5 + 42*d^
4*e^6*x^6 - 24*d^3*e^7*x^7 + 15*d^2*e^8*x^8 - 10*d*e^9*x^9 + 7*e^10*x^10)) + b^10*(9*A*e*(532*d^10 - 1456*d^9*
e*x - 3248*d^8*e^2*x^2 - 840*d^7*e^3*x^3 + 210*d^6*e^4*x^4 - 84*d^5*e^5*x^5 + 42*d^4*e^6*x^6 - 24*d^3*e^7*x^7
+ 15*d^2*e^8*x^8 - 10*d*e^9*x^9 + 7*e^10*x^10) + B*(-5292*d^11 + 17136*d^10*e*x + 36288*d^9*e^2*x^2 + 9240*d^8
*e^3*x^3 - 2310*d^7*e^4*x^4 + 924*d^6*e^5*x^5 - 462*d^5*e^6*x^6 + 264*d^4*e^7*x^7 - 165*d^3*e^8*x^8 + 110*d^2*
e^9*x^9 - 77*d*e^10*x^10 + 56*e^11*x^11)) - 2520*b*(b*d - a*e)^8*(11*b*B*d - 9*A*b*e - 2*a*B*e)*(d + e*x)^2*Lo
g[d + e*x])/(504*e^12*(d + e*x)^2)

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Maple [B]  time = 0.028, size = 2532, normalized size = 5.7 \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)^3,x)

[Out]

63*b^5/e^3*B*x^4*a^5-21/4*b^10/e^8*B*x^4*d^5+84*b^5/e^3*A*x^3*a^5-7*b^10/e^8*A*x^3*d^5+70*b^4/e^3*B*x^3*a^6+28
/3*b^10/e^9*B*x^3*d^6+105*b^4/e^3*A*x^2*a^6+14*b^10/e^9*A*x^2*d^6+60*b^3/e^3*B*x^2*a^7-18*b^10/e^10*B*x^2*d^7+
120*b^3/e^3*A*a^7*x-36*b^10/e^10*A*d^7*x+10/7*b^9/e^3*A*x^7*a-3/7*b^10/e^4*A*x^7*d+45/7*b^8/e^3*B*x^7*a^2+6/7*
b^10/e^5*B*x^7*d^2+15/2*b^8/e^3*A*x^6*a^2+20*b^7/e^3*B*x^6*a^3+5/4*b^9/e^3*B*x^8*a-3/8*b^10/e^4*B*x^8*d+42*b^6
/e^3*B*x^5*a^4+1260*b^4/e^5*B*a^6*d^2*x-2520*b^5/e^6*B*a^5*d^3*x+3150*b^6/e^7*B*a^4*d^4*x-2520*b^7/e^8*B*a^3*d
^5*x+1260*b^8/e^9*B*a^2*d^6*x-360*b^9/e^10*B*a*d^7*x-25*b^9/e^6*A*x^4*a*d^3-315/2*b^6/e^4*B*x^4*a^4*d+180*b^7/
e^5*B*x^4*a^3*d^2-225/2*b^8/e^6*B*x^4*a^2*d^3+75/2*b^9/e^7*B*x^4*a*d^4-27*b^8/e^4*A*x^5*a^2*d-210*b^6/e^4*A*x^
3*a^4*d-90*b^7/e^4*A*x^4*a^3*d+135/2*b^8/e^5*A*x^4*a^2*d^2+240*b^7/e^5*A*x^3*a^3*d^2-150*b^8/e^6*A*x^3*a^2*d^3
+225*b^8/e^7*B*x^3*a^2*d^4-252*b^5/e^4*B*x^3*a^5*d+420*b^6/e^5*B*x^3*a^4*d^2-400*b^7/e^6*B*x^3*a^3*d^3+12*b^9/
e^5*A*x^5*a*d^2-72*b^7/e^4*B*x^5*a^3*d+54*b^8/e^5*B*x^5*a^2*d^2-20*b^9/e^6*B*x^5*a*d^3-30/7*b^9/e^4*B*x^7*a*d-
5*b^9/e^4*A*x^6*a*d-45/2*b^8/e^4*B*x^6*a^2*d+10*b^9/e^5*B*x^6*a*d^2-378*b^5/e^4*A*x^2*a^5*d+630*b^6/e^5*A*x^2*
a^4*d^2+100/e^11/(e*x+d)*B*a*b^9*d^9-360*b^3/e^4*ln(e*x+d)*A*a^7*d+1260*b^4/e^5*ln(e*x+d)*A*a^6*d^2-2520*b^5/e
^6*ln(e*x+d)*A*a^5*d^3+3150*b^6/e^7*ln(e*x+d)*A*a^4*d^4-2520*b^7/e^8*ln(e*x+d)*A*a^3*d^5+1260*b^8/e^9*ln(e*x+d
)*A*a^2*d^6-360*b^9/e^10*ln(e*x+d)*A*a*d^7-135*b^2/e^4*ln(e*x+d)*B*a^8*d+720*b^3/e^5*ln(e*x+d)*B*a^7*d^2-2100*
b^4/e^6*ln(e*x+d)*B*a^6*d^3+3780*b^5/e^7*ln(e*x+d)*B*a^5*d^4-4410*b^6/e^8*ln(e*x+d)*B*a^4*d^5+3360*b^7/e^9*ln(
e*x+d)*B*a^3*d^6-1620*b^8/e^10*ln(e*x+d)*B*a^2*d^7+450*b^9/e^11*ln(e*x+d)*B*a*d^8+50*b^9/e^7*A*x^3*a*d^4+756*b
^5/e^5*B*x^2*a^5*d^2-1050*b^6/e^6*B*x^2*a^4*d^3+900*b^7/e^7*B*x^2*a^3*d^4-945/2*b^8/e^8*B*x^2*a^2*d^5-600*b^7/
e^6*A*x^2*a^3*d^3+675/2*b^8/e^7*A*x^2*a^2*d^4-105*b^9/e^8*A*x^2*a*d^5-315*b^4/e^4*B*x^2*a^6*d-70*b^9/e^8*B*x^3
*a*d^5+5/e^2/(e*x+d)^2*A*d*a^9*b-45/2/e^3/(e*x+d)^2*A*d^2*a^8*b^2+60/e^4/(e*x+d)^2*A*a^7*b^3*d^3-105/e^5/(e*x+
d)^2*A*a^6*b^4*d^4+126/e^6/(e*x+d)^2*A*a^5*b^5*d^5-105/e^7/(e*x+d)^2*A*a^4*b^6*d^6+60/e^8/(e*x+d)^2*A*a^3*b^7*
d^7-45/2/e^9/(e*x+d)^2*A*a^2*b^8*d^8+5/e^10/(e*x+d)^2*A*a*b^9*d^9-5/e^3/(e*x+d)^2*B*d^2*a^9*b+45/2/e^4/(e*x+d)
^2*B*a^8*b^2*d^3-60/e^5/(e*x+d)^2*B*a^7*b^3*d^4+105/e^6/(e*x+d)^2*B*a^6*b^4*d^5-126/e^7/(e*x+d)^2*B*a^5*b^5*d^
6+105/e^8/(e*x+d)^2*B*a^4*b^6*d^7-60/e^9/(e*x+d)^2*B*a^3*b^7*d^8+45/2/e^10/(e*x+d)^2*B*a^2*b^8*d^9-5/e^11/(e*x
+d)^2*B*a*b^9*d^10+90/e^3/(e*x+d)*A*a^8*b^2*d-360/e^4/(e*x+d)*A*a^7*b^3*d^2+840/e^5/(e*x+d)*A*a^6*b^4*d^3-5/3*
b^10/e^6*B*x^6*d^3+24*b^7/e^3*A*x^5*a^3-2*b^10/e^6*A*x^5*d^3-1/2/e^11/(e*x+d)^2*A*b^10*d^10+1/2/e^2/(e*x+d)^2*
B*d*a^10+1/2/e^12/(e*x+d)^2*b^10*B*d^11-10/e^2/(e*x+d)*A*a^9*b+10/e^11/(e*x+d)*A*b^10*d^9-11/e^12/(e*x+d)*b^10
*B*d^10+45*b^2/e^3*ln(e*x+d)*A*a^8+45*b^10/e^11*ln(e*x+d)*A*d^8+10*b/e^3*ln(e*x+d)*B*a^9-55*b^10/e^12*ln(e*x+d
)*B*d^9+b^10/e^5*A*x^6*d^2+45*b^2/e^3*B*a^8*x+1/8*b^10/e^3*A*x^8+1/9*b^10/e^3*B*x^9-1/2/e/(e*x+d)^2*a^10*A-1/e
^2/(e*x+d)*B*a^10+140*b^9/e^9*B*x^2*a*d^6-630*b^4/e^4*A*a^6*d*x+1512*b^5/e^5*A*a^5*d^2*x-2100*b^6/e^6*A*a^4*d^
3*x+1800*b^7/e^7*A*a^3*d^4*x-945*b^8/e^8*A*a^2*d^5*x+280*b^9/e^9*A*a*d^6*x-360*b^3/e^4*B*a^7*d*x+45*b^10/e^11*
B*d^8*x+3*b^10/e^7*B*x^5*d^4+105/2*b^6/e^3*A*x^4*a^4+15/4*b^10/e^7*A*x^4*d^4-1260/e^6/(e*x+d)*A*a^5*b^5*d^4+12
60/e^7/(e*x+d)*A*a^4*b^6*d^5-840/e^8/(e*x+d)*A*a^3*b^7*d^6+360/e^9/(e*x+d)*A*a^2*b^8*d^7-90/e^10/(e*x+d)*A*a*b
^9*d^8+20/e^3/(e*x+d)*B*a^9*b*d-135/e^4/(e*x+d)*B*a^8*b^2*d^2+480/e^5/(e*x+d)*B*a^7*b^3*d^3-1050/e^6/(e*x+d)*B
*a^6*b^4*d^4+1512/e^7/(e*x+d)*B*a^5*b^5*d^5-1470/e^8/(e*x+d)*B*a^4*b^6*d^6+960/e^9/(e*x+d)*B*a^3*b^7*d^7-405/e
^10/(e*x+d)*B*a^2*b^8*d^8

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Maxima [B]  time = 1.35938, size = 2465, normalized size = 5.54 \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)^3,x, algorithm="maxima")

[Out]

-1/2*(21*B*b^10*d^11 + A*a^10*e^11 - 19*(10*B*a*b^9 + A*b^10)*d^10*e + 85*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 -
225*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 390*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4 - 462*(6*B*a^5*b^5 + 5*A*a^4
*b^6)*d^6*e^5 + 378*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 - 210*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4*e^7 + 75*(3*B*a^
8*b^2 + 8*A*a^7*b^3)*d^3*e^8 - 15*(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 + 2*(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 + 45*(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)/(e^14*x^2 + 2*d*e^13*x + d^2*e^12
) + 1/504*(56*B*b^10*e^8*x^9 - 63*(3*B*b^10*d*e^7 - (10*B*a*b^9 + A*b^10)*e^8)*x^8 + 72*(6*B*b^10*d^2*e^6 - 3*
(10*B*a*b^9 + A*b^10)*d*e^7 + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*e^8)*x^7 - 84*(10*B*b^10*d^3*e^5 - 6*(10*B*a*b^9 + A
*b^10)*d^2*e^6 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^7 - 15*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^8)*x^6 + 504*(3*B*b^10*
d^4*e^4 - 2*(10*B*a*b^9 + A*b^10)*d^3*e^5 + 6*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^6 - 9*(8*B*a^3*b^7 + 3*A*a^2*b^8
)*d*e^7 + 6*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^8)*x^5 - 126*(21*B*b^10*d^5*e^3 - 15*(10*B*a*b^9 + A*b^10)*d^4*e^4 +
 50*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^5 - 90*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^6 + 90*(7*B*a^4*b^6 + 4*A*a^3*b^7
)*d*e^7 - 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^8)*x^4 + 168*(28*B*b^10*d^6*e^2 - 21*(10*B*a*b^9 + A*b^10)*d^5*e^3
+ 75*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^4 - 150*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^5 + 180*(7*B*a^4*b^6 + 4*A*a^3*
b^7)*d^2*e^6 - 126*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^7 + 42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^8)*x^3 - 252*(36*B*b^1
0*d^7*e - 28*(10*B*a*b^9 + A*b^10)*d^6*e^2 + 105*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^3 - 225*(8*B*a^3*b^7 + 3*A*a^
2*b^8)*d^4*e^4 + 300*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^5 - 252*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^6 + 126*(5*B*
a^6*b^4 + 6*A*a^5*b^5)*d*e^7 - 30*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^8)*x^2 + 504*(45*B*b^10*d^8 - 36*(10*B*a*b^9 +
 A*b^10)*d^7*e + 140*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^2 - 315*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^3 + 450*(7*B*a^
4*b^6 + 4*A*a^3*b^7)*d^4*e^4 - 420*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^5 + 252*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*e
^6 - 90*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*e^7 + 15*(3*B*a^8*b^2 + 8*A*a^7*b^3)*e^8)*x)/e^11 - 5*(11*B*b^10*d^9 - 9
*(10*B*a*b^9 + A*b^10)*d^8*e + 36*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^2 - 84*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^3 +
 126*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*e^4 - 126*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*e^5 + 84*(5*B*a^6*b^4 + 6*A*a^5
*b^5)*d^3*e^6 - 36*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*e^7 + 9*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d*e^8 - (2*B*a^9*b + 9*
A*a^8*b^2)*e^9)*log(e*x + d)/e^12

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Fricas [B]  time = 2.46233, size = 5485, normalized size = 12.33 \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)^3,x, algorithm="fricas")

[Out]

1/504*(56*B*b^10*e^11*x^11 - 5292*B*b^10*d^11 - 252*A*a^10*e^11 + 4788*(10*B*a*b^9 + A*b^10)*d^10*e - 21420*(9
*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 + 56700*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 - 98280*(7*B*a^4*b^6 + 4*A*a^3*b^7
)*d^7*e^4 + 116424*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 - 95256*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 + 52920*(4*
B*a^7*b^3 + 7*A*a^6*b^4)*d^4*e^7 - 18900*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^8 + 3780*(2*B*a^9*b + 9*A*a^8*b^2)*
d^2*e^9 - 252*(B*a^10 + 10*A*a^9*b)*d*e^10 - 7*(11*B*b^10*d*e^10 - 9*(10*B*a*b^9 + A*b^10)*e^11)*x^10 + 10*(11
*B*b^10*d^2*e^9 - 9*(10*B*a*b^9 + A*b^10)*d*e^10 + 36*(9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9 - 15*(11*B*b^10*d^3*
e^8 - 9*(10*B*a*b^9 + A*b^10)*d^2*e^9 + 36*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^10 - 84*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e
^11)*x^8 + 24*(11*B*b^10*d^4*e^7 - 9*(10*B*a*b^9 + A*b^10)*d^3*e^8 + 36*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^9 - 84
*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^10 + 126*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 - 42*(11*B*b^10*d^5*e^6 - 9*(1
0*B*a*b^9 + A*b^10)*d^4*e^7 + 36*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^8 - 84*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^9 +
126*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^10 - 126*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 84*(11*B*b^10*d^6*e^5 - 9
*(10*B*a*b^9 + A*b^10)*d^5*e^6 + 36*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 - 84*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8
 + 126*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^9 - 126*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 + 84*(5*B*a^6*b^4 + 6*A*a^
5*b^5)*e^11)*x^5 - 210*(11*B*b^10*d^7*e^4 - 9*(10*B*a*b^9 + A*b^10)*d^6*e^5 + 36*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5
*e^6 - 84*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 + 126*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^8 - 126*(6*B*a^5*b^5 + 5
*A*a^4*b^6)*d^2*e^9 + 84*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^10 - 36*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^11)*x^4 + 840*(
11*B*b^10*d^8*e^3 - 9*(10*B*a*b^9 + A*b^10)*d^7*e^4 + 36*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5 - 84*(8*B*a^3*b^7 +
 3*A*a^2*b^8)*d^5*e^6 + 126*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^7 - 126*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^8 + 84
*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*e^9 - 36*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*e^10 + 9*(3*B*a^8*b^2 + 8*A*a^7*b^3)*e
^11)*x^3 + 252*(144*B*b^10*d^9*e^2 - 116*(10*B*a*b^9 + A*b^10)*d^8*e^3 + 455*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^4
 - 1035*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^5 + 1500*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*e^6 - 1428*(6*B*a^5*b^5 + 5
*A*a^4*b^6)*d^4*e^7 + 882*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*e^8 - 330*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*e^9 + 60*(
3*B*a^8*b^2 + 8*A*a^7*b^3)*d*e^10)*x^2 + 504*(34*B*b^10*d^10*e - 26*(10*B*a*b^9 + A*b^10)*d^9*e^2 + 95*(9*B*a^
2*b^8 + 2*A*a*b^9)*d^8*e^3 - 195*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4 + 240*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5
 - 168*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 + 42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*e^7 + 30*(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 + 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 - 2520*(11*B*b^10*d^11 - 9*(10*B*a*b^9 + A*b^10)*d^10*e + 36*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2
 - 84*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 126*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4 - 126*(6*B*a^5*b^5 + 5*A*a
^4*b^6)*d^6*e^5 + 84*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 - 36*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4*e^7 + 9*(3*B*a^8
*b^2 + 8*A*a^7*b^3)*d^3*e^8 - (2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9 + (11*B*b^10*d^9*e^2 - 9*(10*B*a*b^9 + A*b^10)
*d^8*e^3 + 36*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^4 - 84*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^5 + 126*(7*B*a^4*b^6 +
4*A*a^3*b^7)*d^5*e^6 - 126*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*e^7 + 84*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*e^8 - 36*(
4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*e^9 + 9*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d*e^10 - (2*B*a^9*b + 9*A*a^8*b^2)*e^11)*x^
2 + 2*(11*B*b^10*d^10*e - 9*(10*B*a*b^9 + A*b^10)*d^9*e^2 + 36*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^3 - 84*(8*B*a^3
*b^7 + 3*A*a^2*b^8)*d^7*e^4 + 126*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5 - 126*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^
6 + 84*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*e^7 - 36*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*e^8 + 9*(3*B*a^8*b^2 + 8*A*a^7
*b^3)*d^2*e^9 - (2*B*a^9*b + 9*A*a^8*b^2)*d*e^10)*x)*log(e*x + d))/(e^14*x^2 + 2*d*e^13*x + d^2*e^12)

________________________________________________________________________________________

Sympy [B]  time = 113.761, size = 1953, normalized size = 4.39 \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)**3,x)

[Out]

B*b**10*x**9/(9*e**3) + 5*b*(a*e - b*d)**8*(9*A*b*e + 2*B*a*e - 11*B*b*d)*log(d + e*x)/e**12 - (A*a**10*e**11
+ 10*A*a**9*b*d*e**10 - 135*A*a**8*b**2*d**2*e**9 + 600*A*a**7*b**3*d**3*e**8 - 1470*A*a**6*b**4*d**4*e**7 + 2
268*A*a**5*b**5*d**5*e**6 - 2310*A*a**4*b**6*d**6*e**5 + 1560*A*a**3*b**7*d**7*e**4 - 675*A*a**2*b**8*d**8*e**
3 + 170*A*a*b**9*d**9*e**2 - 19*A*b**10*d**10*e + B*a**10*d*e**10 - 30*B*a**9*b*d**2*e**9 + 225*B*a**8*b**2*d*
*3*e**8 - 840*B*a**7*b**3*d**4*e**7 + 1890*B*a**6*b**4*d**5*e**6 - 2772*B*a**5*b**5*d**6*e**5 + 2730*B*a**4*b*
*6*d**7*e**4 - 1800*B*a**3*b**7*d**8*e**3 + 765*B*a**2*b**8*d**9*e**2 - 190*B*a*b**9*d**10*e + 21*B*b**10*d**1
1 + x*(20*A*a**9*b*e**11 - 180*A*a**8*b**2*d*e**10 + 720*A*a**7*b**3*d**2*e**9 - 1680*A*a**6*b**4*d**3*e**8 +
2520*A*a**5*b**5*d**4*e**7 - 2520*A*a**4*b**6*d**5*e**6 + 1680*A*a**3*b**7*d**6*e**5 - 720*A*a**2*b**8*d**7*e*
*4 + 180*A*a*b**9*d**8*e**3 - 20*A*b**10*d**9*e**2 + 2*B*a**10*e**11 - 40*B*a**9*b*d*e**10 + 270*B*a**8*b**2*d
**2*e**9 - 960*B*a**7*b**3*d**3*e**8 + 2100*B*a**6*b**4*d**4*e**7 - 3024*B*a**5*b**5*d**5*e**6 + 2940*B*a**4*b
**6*d**6*e**5 - 1920*B*a**3*b**7*d**7*e**4 + 810*B*a**2*b**8*d**8*e**3 - 200*B*a*b**9*d**9*e**2 + 22*B*b**10*d
**10*e))/(2*d**2*e**12 + 4*d*e**13*x + 2*e**14*x**2) + x**8*(A*b**10*e + 10*B*a*b**9*e - 3*B*b**10*d)/(8*e**4)
 + x**7*(10*A*a*b**9*e**2 - 3*A*b**10*d*e + 45*B*a**2*b**8*e**2 - 30*B*a*b**9*d*e + 6*B*b**10*d**2)/(7*e**5) +
 x**6*(45*A*a**2*b**8*e**3 - 30*A*a*b**9*d*e**2 + 6*A*b**10*d**2*e + 120*B*a**3*b**7*e**3 - 135*B*a**2*b**8*d*
e**2 + 60*B*a*b**9*d**2*e - 10*B*b**10*d**3)/(6*e**6) + x**5*(24*A*a**3*b**7*e**4 - 27*A*a**2*b**8*d*e**3 + 12
*A*a*b**9*d**2*e**2 - 2*A*b**10*d**3*e + 42*B*a**4*b**6*e**4 - 72*B*a**3*b**7*d*e**3 + 54*B*a**2*b**8*d**2*e**
2 - 20*B*a*b**9*d**3*e + 3*B*b**10*d**4)/e**7 + x**4*(210*A*a**4*b**6*e**5 - 360*A*a**3*b**7*d*e**4 + 270*A*a*
*2*b**8*d**2*e**3 - 100*A*a*b**9*d**3*e**2 + 15*A*b**10*d**4*e + 252*B*a**5*b**5*e**5 - 630*B*a**4*b**6*d*e**4
 + 720*B*a**3*b**7*d**2*e**3 - 450*B*a**2*b**8*d**3*e**2 + 150*B*a*b**9*d**4*e - 21*B*b**10*d**5)/(4*e**8) + x
**3*(252*A*a**5*b**5*e**6 - 630*A*a**4*b**6*d*e**5 + 720*A*a**3*b**7*d**2*e**4 - 450*A*a**2*b**8*d**3*e**3 + 1
50*A*a*b**9*d**4*e**2 - 21*A*b**10*d**5*e + 210*B*a**6*b**4*e**6 - 756*B*a**5*b**5*d*e**5 + 1260*B*a**4*b**6*d
**2*e**4 - 1200*B*a**3*b**7*d**3*e**3 + 675*B*a**2*b**8*d**4*e**2 - 210*B*a*b**9*d**5*e + 28*B*b**10*d**6)/(3*
e**9) + x**2*(210*A*a**6*b**4*e**7 - 756*A*a**5*b**5*d*e**6 + 1260*A*a**4*b**6*d**2*e**5 - 1200*A*a**3*b**7*d*
*3*e**4 + 675*A*a**2*b**8*d**4*e**3 - 210*A*a*b**9*d**5*e**2 + 28*A*b**10*d**6*e + 120*B*a**7*b**3*e**7 - 630*
B*a**6*b**4*d*e**6 + 1512*B*a**5*b**5*d**2*e**5 - 2100*B*a**4*b**6*d**3*e**4 + 1800*B*a**3*b**7*d**4*e**3 - 94
5*B*a**2*b**8*d**5*e**2 + 280*B*a*b**9*d**6*e - 36*B*b**10*d**7)/(2*e**10) + x*(120*A*a**7*b**3*e**8 - 630*A*a
**6*b**4*d*e**7 + 1512*A*a**5*b**5*d**2*e**6 - 2100*A*a**4*b**6*d**3*e**5 + 1800*A*a**3*b**7*d**4*e**4 - 945*A
*a**2*b**8*d**5*e**3 + 280*A*a*b**9*d**6*e**2 - 36*A*b**10*d**7*e + 45*B*a**8*b**2*e**8 - 360*B*a**7*b**3*d*e*
*7 + 1260*B*a**6*b**4*d**2*e**6 - 2520*B*a**5*b**5*d**3*e**5 + 3150*B*a**4*b**6*d**4*e**4 - 2520*B*a**3*b**7*d
**5*e**3 + 1260*B*a**2*b**8*d**6*e**2 - 360*B*a*b**9*d**7*e + 45*B*b**10*d**8)/e**11

________________________________________________________________________________________

Giac [B]  time = 1.52853, size = 2703, normalized size = 6.07 \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)^3,x, algorithm="giac")

[Out]

-5*(11*B*b^10*d^9 - 90*B*a*b^9*d^8*e - 9*A*b^10*d^8*e + 324*B*a^2*b^8*d^7*e^2 + 72*A*a*b^9*d^7*e^2 - 672*B*a^3
*b^7*d^6*e^3 - 252*A*a^2*b^8*d^6*e^3 + 882*B*a^4*b^6*d^5*e^4 + 504*A*a^3*b^7*d^5*e^4 - 756*B*a^5*b^5*d^4*e^5 -
 630*A*a^4*b^6*d^4*e^5 + 420*B*a^6*b^4*d^3*e^6 + 504*A*a^5*b^5*d^3*e^6 - 144*B*a^7*b^3*d^2*e^7 - 252*A*a^6*b^4
*d^2*e^7 + 27*B*a^8*b^2*d*e^8 + 72*A*a^7*b^3*d*e^8 - 2*B*a^9*b*e^9 - 9*A*a^8*b^2*e^9)*e^(-12)*log(abs(x*e + d)
) + 1/504*(56*B*b^10*x^9*e^24 - 189*B*b^10*d*x^8*e^23 + 432*B*b^10*d^2*x^7*e^22 - 840*B*b^10*d^3*x^6*e^21 + 15
12*B*b^10*d^4*x^5*e^20 - 2646*B*b^10*d^5*x^4*e^19 + 4704*B*b^10*d^6*x^3*e^18 - 9072*B*b^10*d^7*x^2*e^17 + 2268
0*B*b^10*d^8*x*e^16 + 630*B*a*b^9*x^8*e^24 + 63*A*b^10*x^8*e^24 - 2160*B*a*b^9*d*x^7*e^23 - 216*A*b^10*d*x^7*e
^23 + 5040*B*a*b^9*d^2*x^6*e^22 + 504*A*b^10*d^2*x^6*e^22 - 10080*B*a*b^9*d^3*x^5*e^21 - 1008*A*b^10*d^3*x^5*e
^21 + 18900*B*a*b^9*d^4*x^4*e^20 + 1890*A*b^10*d^4*x^4*e^20 - 35280*B*a*b^9*d^5*x^3*e^19 - 3528*A*b^10*d^5*x^3
*e^19 + 70560*B*a*b^9*d^6*x^2*e^18 + 7056*A*b^10*d^6*x^2*e^18 - 181440*B*a*b^9*d^7*x*e^17 - 18144*A*b^10*d^7*x
*e^17 + 3240*B*a^2*b^8*x^7*e^24 + 720*A*a*b^9*x^7*e^24 - 11340*B*a^2*b^8*d*x^6*e^23 - 2520*A*a*b^9*d*x^6*e^23
+ 27216*B*a^2*b^8*d^2*x^5*e^22 + 6048*A*a*b^9*d^2*x^5*e^22 - 56700*B*a^2*b^8*d^3*x^4*e^21 - 12600*A*a*b^9*d^3*
x^4*e^21 + 113400*B*a^2*b^8*d^4*x^3*e^20 + 25200*A*a*b^9*d^4*x^3*e^20 - 238140*B*a^2*b^8*d^5*x^2*e^19 - 52920*
A*a*b^9*d^5*x^2*e^19 + 635040*B*a^2*b^8*d^6*x*e^18 + 141120*A*a*b^9*d^6*x*e^18 + 10080*B*a^3*b^7*x^6*e^24 + 37
80*A*a^2*b^8*x^6*e^24 - 36288*B*a^3*b^7*d*x^5*e^23 - 13608*A*a^2*b^8*d*x^5*e^23 + 90720*B*a^3*b^7*d^2*x^4*e^22
 + 34020*A*a^2*b^8*d^2*x^4*e^22 - 201600*B*a^3*b^7*d^3*x^3*e^21 - 75600*A*a^2*b^8*d^3*x^3*e^21 + 453600*B*a^3*
b^7*d^4*x^2*e^20 + 170100*A*a^2*b^8*d^4*x^2*e^20 - 1270080*B*a^3*b^7*d^5*x*e^19 - 476280*A*a^2*b^8*d^5*x*e^19
+ 21168*B*a^4*b^6*x^5*e^24 + 12096*A*a^3*b^7*x^5*e^24 - 79380*B*a^4*b^6*d*x^4*e^23 - 45360*A*a^3*b^7*d*x^4*e^2
3 + 211680*B*a^4*b^6*d^2*x^3*e^22 + 120960*A*a^3*b^7*d^2*x^3*e^22 - 529200*B*a^4*b^6*d^3*x^2*e^21 - 302400*A*a
^3*b^7*d^3*x^2*e^21 + 1587600*B*a^4*b^6*d^4*x*e^20 + 907200*A*a^3*b^7*d^4*x*e^20 + 31752*B*a^5*b^5*x^4*e^24 +
26460*A*a^4*b^6*x^4*e^24 - 127008*B*a^5*b^5*d*x^3*e^23 - 105840*A*a^4*b^6*d*x^3*e^23 + 381024*B*a^5*b^5*d^2*x^
2*e^22 + 317520*A*a^4*b^6*d^2*x^2*e^22 - 1270080*B*a^5*b^5*d^3*x*e^21 - 1058400*A*a^4*b^6*d^3*x*e^21 + 35280*B
*a^6*b^4*x^3*e^24 + 42336*A*a^5*b^5*x^3*e^24 - 158760*B*a^6*b^4*d*x^2*e^23 - 190512*A*a^5*b^5*d*x^2*e^23 + 635
040*B*a^6*b^4*d^2*x*e^22 + 762048*A*a^5*b^5*d^2*x*e^22 + 30240*B*a^7*b^3*x^2*e^24 + 52920*A*a^6*b^4*x^2*e^24 -
 181440*B*a^7*b^3*d*x*e^23 - 317520*A*a^6*b^4*d*x*e^23 + 22680*B*a^8*b^2*x*e^24 + 60480*A*a^7*b^3*x*e^24)*e^(-
27) - 1/2*(21*B*b^10*d^11 - 190*B*a*b^9*d^10*e - 19*A*b^10*d^10*e + 765*B*a^2*b^8*d^9*e^2 + 170*A*a*b^9*d^9*e^
2 - 1800*B*a^3*b^7*d^8*e^3 - 675*A*a^2*b^8*d^8*e^3 + 2730*B*a^4*b^6*d^7*e^4 + 1560*A*a^3*b^7*d^7*e^4 - 2772*B*
a^5*b^5*d^6*e^5 - 2310*A*a^4*b^6*d^6*e^5 + 1890*B*a^6*b^4*d^5*e^6 + 2268*A*a^5*b^5*d^5*e^6 - 840*B*a^7*b^3*d^4
*e^7 - 1470*A*a^6*b^4*d^4*e^7 + 225*B*a^8*b^2*d^3*e^8 + 600*A*a^7*b^3*d^3*e^8 - 30*B*a^9*b*d^2*e^9 - 135*A*a^8
*b^2*d^2*e^9 + B*a^10*d*e^10 + 10*A*a^9*b*d*e^10 + A*a^10*e^11 + 2*(11*B*b^10*d^10*e - 100*B*a*b^9*d^9*e^2 - 1
0*A*b^10*d^9*e^2 + 405*B*a^2*b^8*d^8*e^3 + 90*A*a*b^9*d^8*e^3 - 960*B*a^3*b^7*d^7*e^4 - 360*A*a^2*b^8*d^7*e^4
+ 1470*B*a^4*b^6*d^6*e^5 + 840*A*a^3*b^7*d^6*e^5 - 1512*B*a^5*b^5*d^5*e^6 - 1260*A*a^4*b^6*d^5*e^6 + 1050*B*a^
6*b^4*d^4*e^7 + 1260*A*a^5*b^5*d^4*e^7 - 480*B*a^7*b^3*d^3*e^8 - 840*A*a^6*b^4*d^3*e^8 + 135*B*a^8*b^2*d^2*e^9
 + 360*A*a^7*b^3*d^2*e^9 - 20*B*a^9*b*d*e^10 - 90*A*a^8*b^2*d*e^10 + B*a^10*e^11 + 10*A*a^9*b*e^11)*x)*e^(-12)
/(x*e + d)^2

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