\(\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^4} \, dx\) [1092]

Optimal result

Integrand size = 20, antiderivative size = 445 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^4} \, dx=-\frac {30 b^3 (b d-a e)^6 (11 b B d-7 A b e-4 a B e) x}{e^{11}}+\frac {(b d-a e)^{10} (B d-A e)}{3 e^{12} (d+e x)^3}-\frac {(b d-a e)^9 (11 b B d-10 A b e-a B e)}{2 e^{12} (d+e x)^2}+\frac {5 b (b d-a e)^8 (11 b B d-9 A b e-2 a B e)}{e^{12} (d+e x)}+\frac {21 b^4 (b d-a e)^5 (11 b B d-6 A b e-5 a B e) (d+e x)^2}{e^{12}}-\frac {14 b^5 (b d-a e)^4 (11 b B d-5 A b e-6 a B e) (d+e x)^3}{e^{12}}+\frac {15 b^6 (b d-a e)^3 (11 b B d-4 A b e-7 a B e) (d+e x)^4}{2 e^{12}}-\frac {3 b^7 (b d-a e)^2 (11 b B d-3 A b e-8 a B e) (d+e x)^5}{e^{12}}+\frac {5 b^8 (b d-a e) (11 b B d-2 A b e-9 a B e) (d+e x)^6}{6 e^{12}}-\frac {b^9 (11 b B d-A b e-10 a B e) (d+e x)^7}{7 e^{12}}+\frac {b^{10} B (d+e x)^8}{8 e^{12}}+\frac {15 b^2 (b d-a e)^7 (11 b B d-8 A b e-3 a B e) \log (d+e x)}{e^{12}} \]

[Out]

-30*b^3*(-a*e+b*d)^6*(-7*A*b*e-4*B*a*e+11*B*b*d)*x/e^11+1/3*(-a*e+b*d)^10*(-A*e+B*d)/e^12/(e*x+d)^3-1/2*(-a*e+
b*d)^9*(-10*A*b*e-B*a*e+11*B*b*d)/e^12/(e*x+d)^2+5*b*(-a*e+b*d)^8*(-9*A*b*e-2*B*a*e+11*B*b*d)/e^12/(e*x+d)+21*
b^4*(-a*e+b*d)^5*(-6*A*b*e-5*B*a*e+11*B*b*d)*(e*x+d)^2/e^12-14*b^5*(-a*e+b*d)^4*(-5*A*b*e-6*B*a*e+11*B*b*d)*(e
*x+d)^3/e^12+15/2*b^6*(-a*e+b*d)^3*(-4*A*b*e-7*B*a*e+11*B*b*d)*(e*x+d)^4/e^12-3*b^7*(-a*e+b*d)^2*(-3*A*b*e-8*B
*a*e+11*B*b*d)*(e*x+d)^5/e^12+5/6*b^8*(-a*e+b*d)*(-2*A*b*e-9*B*a*e+11*B*b*d)*(e*x+d)^6/e^12-1/7*b^9*(-A*b*e-10
*B*a*e+11*B*b*d)*(e*x+d)^7/e^12+1/8*b^10*B*(e*x+d)^8/e^12+15*b^2*(-a*e+b*d)^7*(-8*A*b*e-3*B*a*e+11*B*b*d)*ln(e
*x+d)/e^12

Rubi [A] (verified)

Time = 0.92 (sec) , antiderivative size = 445, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {78} \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^4} \, dx=-\frac {b^9 (d+e x)^7 (-10 a B e-A b e+11 b B d)}{7 e^{12}}+\frac {5 b^8 (d+e x)^6 (b d-a e) (-9 a B e-2 A b e+11 b B d)}{6 e^{12}}-\frac {3 b^7 (d+e x)^5 (b d-a e)^2 (-8 a B e-3 A b e+11 b B d)}{e^{12}}+\frac {15 b^6 (d+e x)^4 (b d-a e)^3 (-7 a B e-4 A b e+11 b B d)}{2 e^{12}}-\frac {14 b^5 (d+e x)^3 (b d-a e)^4 (-6 a B e-5 A b e+11 b B d)}{e^{12}}+\frac {21 b^4 (d+e x)^2 (b d-a e)^5 (-5 a B e-6 A b e+11 b B d)}{e^{12}}-\frac {30 b^3 x (b d-a e)^6 (-4 a B e-7 A b e+11 b B d)}{e^{11}}+\frac {15 b^2 (b d-a e)^7 \log (d+e x) (-3 a B e-8 A b e+11 b B d)}{e^{12}}+\frac {5 b (b d-a e)^8 (-2 a B e-9 A b e+11 b B d)}{e^{12} (d+e x)}-\frac {(b d-a e)^9 (-a B e-10 A b e+11 b B d)}{2 e^{12} (d+e x)^2}+\frac {(b d-a e)^{10} (B d-A e)}{3 e^{12} (d+e x)^3}+\frac {b^{10} B (d+e x)^8}{8 e^{12}} \]

[In]

Int[((a + b*x)^10*(A + B*x))/(d + e*x)^4,x]

[Out]

(-30*b^3*(b*d - a*e)^6*(11*b*B*d - 7*A*b*e - 4*a*B*e)*x)/e^11 + ((b*d - a*e)^10*(B*d - A*e))/(3*e^12*(d + e*x)
^3) - ((b*d - a*e)^9*(11*b*B*d - 10*A*b*e - a*B*e))/(2*e^12*(d + e*x)^2) + (5*b*(b*d - a*e)^8*(11*b*B*d - 9*A*
b*e - 2*a*B*e))/(e^12*(d + e*x)) + (21*b^4*(b*d - a*e)^5*(11*b*B*d - 6*A*b*e - 5*a*B*e)*(d + e*x)^2)/e^12 - (1
4*b^5*(b*d - a*e)^4*(11*b*B*d - 5*A*b*e - 6*a*B*e)*(d + e*x)^3)/e^12 + (15*b^6*(b*d - a*e)^3*(11*b*B*d - 4*A*b
*e - 7*a*B*e)*(d + e*x)^4)/(2*e^12) - (3*b^7*(b*d - a*e)^2*(11*b*B*d - 3*A*b*e - 8*a*B*e)*(d + e*x)^5)/e^12 +
(5*b^8*(b*d - a*e)*(11*b*B*d - 2*A*b*e - 9*a*B*e)*(d + e*x)^6)/(6*e^12) - (b^9*(11*b*B*d - A*b*e - 10*a*B*e)*(
d + e*x)^7)/(7*e^12) + (b^10*B*(d + e*x)^8)/(8*e^12) + (15*b^2*(b*d - a*e)^7*(11*b*B*d - 8*A*b*e - 3*a*B*e)*Lo
g[d + e*x])/e^12

Rule 78

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*} \text {integral}& = \int \left (\frac {30 b^3 (b d-a e)^6 (-11 b B d+7 A b e+4 a B e)}{e^{11}}+\frac {(-b d+a e)^{10} (-B d+A e)}{e^{11} (d+e x)^4}+\frac {(-b d+a e)^9 (-11 b B d+10 A b e+a B e)}{e^{11} (d+e x)^3}+\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)^2}-\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)}-\frac {42 b^4 (b d-a e)^5 (-11 b B d+6 A b e+5 a B e) (d+e x)}{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)^2}{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)^3}{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)^4}{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)^5}{e^{11}}+\frac {b^9 (-11 b B d+A b e+10 a B e) (d+e x)^6}{e^{11}}+\frac {b^{10} B (d+e x)^7}{e^{11}}\right ) \, dx \\ & = -\frac {30 b^3 (b d-a e)^6 (11 b B d-7 A b e-4 a B e) x}{e^{11}}+\frac {(b d-a e)^{10} (B d-A e)}{3 e^{12} (d+e x)^3}-\frac {(b d-a e)^9 (11 b B d-10 A b e-a B e)}{2 e^{12} (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^{12} (d+e x)}+\frac {21 b^4 (b d-a e)^5 (11 b B d-6 A b e-5 a B e) (d+e x)^2}{e^{12}}-\frac {14 b^5 (b d-a e)^4 (11 b B d-5 A b e-6 a B e) (d+e x)^3}{e^{12}}+\frac {15 b^6 (b d-a e)^3 (11 b B d-4 A b e-7 a B e) (d+e x)^4}{2 e^{12}}-\frac {3 b^7 (b d-a e)^2 (11 b B d-3 A b e-8 a B e) (d+e x)^5}{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)^6}{6 e^{12}}-\frac {b^9 (11 b B d-A b e-10 a B e) (d+e x)^7}{7 e^{12}}+\frac {b^{10} B (d+e x)^8}{8 e^{12}}+\frac {15 b^2 (b d-a e)^7 (11 b B d-8 A b e-3 a B e) \log (d+e x)}{e^{12}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 814, normalized size of antiderivative = 1.83 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^4} \, dx=\frac {168 b^3 e \left (120 a^7 B e^7-315 a^2 b^5 d^4 e^2 (8 B d-5 A e)+600 a^3 b^4 d^3 e^3 (7 B d-4 A e)+280 a b^6 d^5 e (3 B d-2 A e)+504 a^5 b^2 d e^5 (5 B d-2 A e)-2100 a^4 b^3 d^2 e^4 (2 B d-A e)+210 a^6 b e^6 (-4 B d+A e)+12 b^7 d^6 (-10 B d+7 A e)\right ) x-84 b^4 e^2 \left (-210 a^6 B e^6+70 a b^5 d^4 e (8 B d-5 A e)-225 a^2 b^4 d^3 e^2 (7 B d-4 A e)-420 a^4 b^2 d e^4 (5 B d-2 A e)+1200 a^3 b^3 d^2 e^3 (2 B d-A e)-252 a^5 b e^5 (-4 B d+A e)+28 b^6 d^5 (-3 B d+2 A e)\right ) x^2+56 b^5 e^3 \left (252 a^5 B e^5-7 b^5 d^4 (8 B d-5 A e)+50 a b^4 d^3 e (7 B d-4 A e)+240 a^3 b^2 d e^3 (5 B d-2 A e)-450 a^2 b^3 d^2 e^2 (2 B d-A e)+210 a^4 b e^4 (-4 B d+A e)\right ) x^3-210 b^6 e^4 \left (-42 a^4 B e^4+20 a b^3 d^2 e (2 B d-A e)-24 a^3 b e^3 (-4 B d+A e)+18 a^2 b^2 d e^2 (-5 B d+2 A e)+b^4 d^3 (-7 B d+4 A e)\right ) x^4+168 b^7 e^5 \left (24 a^3 B e^3+4 a b^2 d e (5 B d-2 A e)+9 a^2 b e^2 (-4 B d+A e)+2 b^3 d^2 (-2 B d+A e)\right ) x^5-28 b^8 e^6 \left (-45 a^2 B e^2-10 a b e (-4 B d+A e)+2 b^2 d (-5 B d+2 A e)\right ) x^6+24 b^9 e^7 (-4 b B d+A b e+10 a B e) x^7+21 b^{10} B e^8 x^8+\frac {56 (b d-a e)^{10} (B d-A e)}{(d+e x)^3}-\frac {84 (b d-a e)^9 (11 b B d-10 A b e-a B e)}{(d+e x)^2}+\frac {840 b (b d-a e)^8 (11 b B d-9 A b e-2 a B e)}{d+e x}+2520 b^2 (b d-a e)^7 (11 b B d-8 A b e-3 a B e) \log (d+e x)}{168 e^{12}} \]

[In]

Integrate[((a + b*x)^10*(A + B*x))/(d + e*x)^4,x]

[Out]

(168*b^3*e*(120*a^7*B*e^7 - 315*a^2*b^5*d^4*e^2*(8*B*d - 5*A*e) + 600*a^3*b^4*d^3*e^3*(7*B*d - 4*A*e) + 280*a*
b^6*d^5*e*(3*B*d - 2*A*e) + 504*a^5*b^2*d*e^5*(5*B*d - 2*A*e) - 2100*a^4*b^3*d^2*e^4*(2*B*d - A*e) + 210*a^6*b
*e^6*(-4*B*d + A*e) + 12*b^7*d^6*(-10*B*d + 7*A*e))*x - 84*b^4*e^2*(-210*a^6*B*e^6 + 70*a*b^5*d^4*e*(8*B*d - 5
*A*e) - 225*a^2*b^4*d^3*e^2*(7*B*d - 4*A*e) - 420*a^4*b^2*d*e^4*(5*B*d - 2*A*e) + 1200*a^3*b^3*d^2*e^3*(2*B*d
- A*e) - 252*a^5*b*e^5*(-4*B*d + A*e) + 28*b^6*d^5*(-3*B*d + 2*A*e))*x^2 + 56*b^5*e^3*(252*a^5*B*e^5 - 7*b^5*d
^4*(8*B*d - 5*A*e) + 50*a*b^4*d^3*e*(7*B*d - 4*A*e) + 240*a^3*b^2*d*e^3*(5*B*d - 2*A*e) - 450*a^2*b^3*d^2*e^2*
(2*B*d - A*e) + 210*a^4*b*e^4*(-4*B*d + A*e))*x^3 - 210*b^6*e^4*(-42*a^4*B*e^4 + 20*a*b^3*d^2*e*(2*B*d - A*e)
- 24*a^3*b*e^3*(-4*B*d + A*e) + 18*a^2*b^2*d*e^2*(-5*B*d + 2*A*e) + b^4*d^3*(-7*B*d + 4*A*e))*x^4 + 168*b^7*e^
5*(24*a^3*B*e^3 + 4*a*b^2*d*e*(5*B*d - 2*A*e) + 9*a^2*b*e^2*(-4*B*d + A*e) + 2*b^3*d^2*(-2*B*d + A*e))*x^5 - 2
8*b^8*e^6*(-45*a^2*B*e^2 - 10*a*b*e*(-4*B*d + A*e) + 2*b^2*d*(-5*B*d + 2*A*e))*x^6 + 24*b^9*e^7*(-4*b*B*d + A*
b*e + 10*a*B*e)*x^7 + 21*b^10*B*e^8*x^8 + (56*(b*d - a*e)^10*(B*d - A*e))/(d + e*x)^3 - (84*(b*d - a*e)^9*(11*
b*B*d - 10*A*b*e - a*B*e))/(d + e*x)^2 + (840*b*(b*d - a*e)^8*(11*b*B*d - 9*A*b*e - 2*a*B*e))/(d + e*x) + 2520
*b^2*(b*d - a*e)^7*(11*b*B*d - 8*A*b*e - 3*a*B*e)*Log[d + e*x])/(168*e^12)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1906\) vs. \(2(433)=866\).

Time = 2.79 (sec) , antiderivative size = 1907, normalized size of antiderivative = 4.29

method	result	size
norman	\(\text {Expression too large to display}\)	\(1907\)
default	\(\text {Expression too large to display}\)	\(2073\)
risch	\(\text {Expression too large to display}\)	\(2182\)
parallelrisch	\(\text {Expression too large to display}\)	\(3431\)

[In]

int((b*x+a)^10*(B*x+A)/(e*x+d)^4,x,method=_RETURNVERBOSE)

[Out]

(-1/6*(2*A*a^10*e^11+10*A*a^9*b*d*e^10+90*A*a^8*b^2*d^2*e^9-1320*A*a^7*b^3*d^3*e^8+9240*A*a^6*b^4*d^4*e^7-2772
0*A*a^5*b^5*d^5*e^6+46200*A*a^4*b^6*d^6*e^5-46200*A*a^3*b^7*d^7*e^4+27720*A*a^2*b^8*d^8*e^3-9240*A*a*b^9*d^9*e
^2+1320*A*b^10*d^10*e+B*a^10*d*e^10+20*B*a^9*b*d^2*e^9-495*B*a^8*b^2*d^3*e^8+5280*B*a^7*b^3*d^4*e^7-23100*B*a^
6*b^4*d^5*e^6+55440*B*a^5*b^5*d^6*e^5-80850*B*a^4*b^6*d^7*e^4+73920*B*a^3*b^7*d^8*e^3-41580*B*a^2*b^8*d^9*e^2+
13200*B*a*b^9*d^10*e-1815*B*b^10*d^11)/e^12-(45*A*a^8*b^2*e^9-360*A*a^7*b^3*d*e^8+2520*A*a^6*b^4*d^2*e^7-7560*
A*a^5*b^5*d^3*e^6+12600*A*a^4*b^6*d^4*e^5-12600*A*a^3*b^7*d^5*e^4+7560*A*a^2*b^8*d^6*e^3-2520*A*a*b^9*d^7*e^2+
360*A*b^10*d^8*e+10*B*a^9*b*e^9-135*B*a^8*b^2*d*e^8+1440*B*a^7*b^3*d^2*e^7-6300*B*a^6*b^4*d^3*e^6+15120*B*a^5*
b^5*d^4*e^5-22050*B*a^4*b^6*d^5*e^4+20160*B*a^3*b^7*d^6*e^3-11340*B*a^2*b^8*d^7*e^2+3600*B*a*b^9*d^8*e-495*B*b
^10*d^9)/e^10*x^2-1/2*(10*A*a^9*b*e^10+90*A*a^8*b^2*d*e^9-1080*A*a^7*b^3*d^2*e^8+7560*A*a^6*b^4*d^3*e^7-22680*
A*a^5*b^5*d^4*e^6+37800*A*a^4*b^6*d^5*e^5-37800*A*a^3*b^7*d^6*e^4+22680*A*a^2*b^8*d^7*e^3-7560*A*a*b^9*d^8*e^2
+1080*A*b^10*d^9*e+B*a^10*e^10+20*B*a^9*b*d*e^9-405*B*a^8*b^2*d^2*e^8+4320*B*a^7*b^3*d^3*e^7-18900*B*a^6*b^4*d
^4*e^6+45360*B*a^5*b^5*d^5*e^5-66150*B*a^4*b^6*d^6*e^4+60480*B*a^3*b^7*d^7*e^3-34020*B*a^2*b^8*d^8*e^2+10800*B
*a*b^9*d^9*e-1485*B*b^10*d^10)/e^11*x+15/4*b^3*(56*A*a^6*b*e^7-168*A*a^5*b^2*d*e^6+280*A*a^4*b^3*d^2*e^5-280*A
*a^3*b^4*d^3*e^4+168*A*a^2*b^5*d^4*e^3-56*A*a*b^6*d^5*e^2+8*A*b^7*d^6*e+32*B*a^7*e^7-140*B*a^6*b*d*e^6+336*B*a
^5*b^2*d^2*e^5-490*B*a^4*b^3*d^3*e^4+448*B*a^3*b^4*d^4*e^3-252*B*a^2*b^5*d^5*e^2+80*B*a*b^6*d^6*e-11*B*b^7*d^7
)/e^8*x^4+3/4*b^4*(168*A*a^5*b*e^6-280*A*a^4*b^2*d*e^5+280*A*a^3*b^3*d^2*e^4-168*A*a^2*b^4*d^3*e^3+56*A*a*b^5*
d^4*e^2-8*A*b^6*d^5*e+140*B*a^6*e^6-336*B*a^5*b*d*e^5+490*B*a^4*b^2*d^2*e^4-448*B*a^3*b^3*d^3*e^3+252*B*a^2*b^
4*d^4*e^2-80*B*a*b^5*d^5*e+11*B*b^6*d^6)/e^7*x^5+1/4*b^5*(280*A*a^4*b*e^5-280*A*a^3*b^2*d*e^4+168*A*a^2*b^3*d^
2*e^3-56*A*a*b^4*d^3*e^2+8*A*b^5*d^4*e+336*B*a^5*e^5-490*B*a^4*b*d*e^4+448*B*a^3*b^2*d^2*e^3-252*B*a^2*b^3*d^3
*e^2+80*B*a*b^4*d^4*e-11*B*b^5*d^5)/e^6*x^6+3/28*b^6*(280*A*a^3*b*e^4-168*A*a^2*b^2*d*e^3+56*A*a*b^3*d^2*e^2-8
*A*b^4*d^3*e+490*B*a^4*e^4-448*B*a^3*b*d*e^3+252*B*a^2*b^2*d^2*e^2-80*B*a*b^3*d^3*e+11*B*b^4*d^4)/e^5*x^7+3/56
*b^7*(168*A*a^2*b*e^3-56*A*a*b^2*d*e^2+8*A*b^3*d^2*e+448*B*a^3*e^3-252*B*a^2*b*d*e^2+80*B*a*b^2*d^2*e-11*B*b^3
*d^3)/e^4*x^8+5/168*b^8*(56*A*a*b*e^2-8*A*b^2*d*e+252*B*a^2*e^2-80*B*a*b*d*e+11*B*b^2*d^2)/e^3*x^9+1/56*b^9*(8
*A*b*e+80*B*a*e-11*B*b*d)/e^2*x^10+1/8*b^10*B/e*x^11)/(e*x+d)^3+15*b^2/e^12*(8*A*a^7*b*e^8-56*A*a^6*b^2*d*e^7+
168*A*a^5*b^3*d^2*e^6-280*A*a^4*b^4*d^3*e^5+280*A*a^3*b^5*d^4*e^4-168*A*a^2*b^6*d^5*e^3+56*A*a*b^7*d^6*e^2-8*A
*b^8*d^7*e+3*B*a^8*e^8-32*B*a^7*b*d*e^7+140*B*a^6*b^2*d^2*e^6-336*B*a^5*b^3*d^3*e^5+490*B*a^4*b^4*d^4*e^4-448*
B*a^3*b^5*d^5*e^3+252*B*a^2*b^6*d^6*e^2-80*B*a*b^7*d^7*e+11*B*b^8*d^8)*ln(e*x+d)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2702 vs. \(2 (433) = 866\).

Time = 0.28 (sec) , antiderivative size = 2702, normalized size of antiderivative = 6.07 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^4} \, dx=\text {Too large to display} \]

[In]

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^4,x, algorithm="fricas")

[Out]

1/168*(21*B*b^10*e^11*x^11 + 8372*B*b^10*d^11 - 56*A*a^10*e^11 - 6776*(10*B*a*b^9 + A*b^10)*d^10*e + 26740*(9*
B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 - 61320*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 89880*(7*B*a^4*b^6 + 4*A*a^3*b^7)
*d^7*e^4 - 87024*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 55272*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 - 21840*(4*B*
a^7*b^3 + 7*A*a^6*b^4)*d^4*e^7 + 4620*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^8 - 280*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*
e^9 - 28*(B*a^10 + 10*A*a^9*b)*d*e^10 - 3*(11*B*b^10*d*e^10 - 8*(10*B*a*b^9 + A*b^10)*e^11)*x^10 + 5*(11*B*b^1
0*d^2*e^9 - 8*(10*B*a*b^9 + A*b^10)*d*e^10 + 28*(9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9 - 9*(11*B*b^10*d^3*e^8 - 8
*(10*B*a*b^9 + A*b^10)*d^2*e^9 + 28*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^10 - 56*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^11)*x^
8 + 18*(11*B*b^10*d^4*e^7 - 8*(10*B*a*b^9 + A*b^10)*d^3*e^8 + 28*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^9 - 56*(8*B*a
^3*b^7 + 3*A*a^2*b^8)*d*e^10 + 70*(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 - 8*(10*B*a*b^
9 + A*b^10)*d^4*e^7 + 28*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^8 - 56*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^9 + 70*(7*B*
a^4*b^6 + 4*A*a^3*b^7)*d*e^10 - 56*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 126*(11*B*b^10*d^6*e^5 - 8*(10*B*a*
b^9 + A*b^10)*d^5*e^6 + 28*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 - 56*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8 + 70*(7*
B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^9 - 56*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 + 28*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^11
)*x^5 - 630*(11*B*b^10*d^7*e^4 - 8*(10*B*a*b^9 + A*b^10)*d^6*e^5 + 28*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^6 - 56*(
8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 + 70*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^8 - 56*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d
^2*e^9 + 28*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^10 - 8*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^11)*x^4 - 28*(1516*B*b^10*d^8
*e^3 - 1078*(10*B*a*b^9 + A*b^10)*d^7*e^4 + 3665*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5 - 7050*(8*B*a^3*b^7 + 3*A*a
^2*b^8)*d^5*e^6 + 8340*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^7 - 6132*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^8 + 2646*(
5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*e^9 - 540*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*e^10)*x^3 - 84*(526*B*b^10*d^9*e^2 - 35
8*(10*B*a*b^9 + A*b^10)*d^8*e^3 + 1145*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^4 - 2010*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^
6*e^5 + 2040*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*e^6 - 1092*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*e^7 + 126*(5*B*a^6*b^4
+ 6*A*a^5*b^5)*d^3*e^8 + 180*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*e^9 - 90*(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 - 84*(31*B*b^10*d^10*e + 2*(10*B*a*b^9 + A*b^10)*d^9*e^2 - 115*(9*B*a^2*b
^8 + 2*A*a*b^9)*d^8*e^3 + 510*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4 - 1110*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5 +
1428*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 - 1134*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*e^7 + 540*(4*B*a^7*b^3 + 7*A*
a^6*b^4)*d^3*e^8 - 135*(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 + 1
0*A*a^9*b)*e^11)*x + 2520*(11*B*b^10*d^11 - 8*(10*B*a*b^9 + A*b^10)*d^10*e + 28*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*
e^2 - 56*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 70*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4 - 56*(6*B*a^5*b^5 + 5*A*
a^4*b^6)*d^6*e^5 + 28*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 - 8*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4*e^7 + (3*B*a^8*b
^2 + 8*A*a^7*b^3)*d^3*e^8 + (11*B*b^10*d^8*e^3 - 8*(10*B*a*b^9 + A*b^10)*d^7*e^4 + 28*(9*B*a^2*b^8 + 2*A*a*b^9
)*d^6*e^5 - 56*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^6 + 70*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^7 - 56*(6*B*a^5*b^5
+ 5*A*a^4*b^6)*d^3*e^8 + 28*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*e^9 - 8*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*e^10 + (3*B*
a^8*b^2 + 8*A*a^7*b^3)*e^11)*x^3 + 3*(11*B*b^10*d^9*e^2 - 8*(10*B*a*b^9 + A*b^10)*d^8*e^3 + 28*(9*B*a^2*b^8 +
2*A*a*b^9)*d^7*e^4 - 56*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^5 + 70*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*e^6 - 56*(6*B
*a^5*b^5 + 5*A*a^4*b^6)*d^4*e^7 + 28*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*e^8 - 8*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*e
^9 + (3*B*a^8*b^2 + 8*A*a^7*b^3)*d*e^10)*x^2 + 3*(11*B*b^10*d^10*e - 8*(10*B*a*b^9 + A*b^10)*d^9*e^2 + 28*(9*B
*a^2*b^8 + 2*A*a*b^9)*d^8*e^3 - 56*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4 + 70*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^
5 - 56*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 + 28*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*e^7 - 8*(4*B*a^7*b^3 + 7*A*a^6
*b^4)*d^3*e^8 + (3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e^9)*x)*log(e*x + d))/(e^15*x^3 + 3*d*e^14*x^2 + 3*d^2*e^13*x
+ d^3*e^12)

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^4} \, dx=\text {Timed out} \]

[In]

integrate((b*x+a)**10*(B*x+A)/(e*x+d)**4,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1839 vs. \(2 (433) = 866\).

Time = 0.25 (sec) , antiderivative size = 1839, normalized size of antiderivative = 4.13 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^4} \, dx=\text {Too large to display} \]

[In]

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^4,x, algorithm="maxima")

[Out]

1/6*(299*B*b^10*d^11 - 2*A*a^10*e^11 - 242*(10*B*a*b^9 + A*b^10)*d^10*e + 955*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^
2 - 2190*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 3210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4 - 3108*(6*B*a^5*b^5 +
5*A*a^4*b^6)*d^6*e^5 + 1974*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 - 780*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4*e^7 + 16
5*(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^10 + 10*A*a^9*b)*d*e^10 +
30*(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 + 3*(209*B*b^10*d^10*e - 170*(10*B*a*b^9 + A*b^10)*d^9*e^2 +
675*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^3 - 1560*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4 + 2310*(7*B*a^4*b^6 + 4*A*a^3
*b^7)*d^6*e^5 - 2268*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 + 1470*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*e^7 - 600*(4*B
*a^7*b^3 + 7*A*a^6*b^4)*d^3*e^8 + 135*(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^1
0 - (B*a^10 + 10*A*a^9*b)*e^11)*x)/(e^15*x^3 + 3*d*e^14*x^2 + 3*d^2*e^13*x + d^3*e^12) + 1/168*(21*B*b^10*e^7*
x^8 - 24*(4*B*b^10*d*e^6 - (10*B*a*b^9 + A*b^10)*e^7)*x^7 + 28*(10*B*b^10*d^2*e^5 - 4*(10*B*a*b^9 + A*b^10)*d*
e^6 + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*e^7)*x^6 - 168*(4*B*b^10*d^3*e^4 - 2*(10*B*a*b^9 + A*b^10)*d^2*e^5 + 4*(9*B*
a^2*b^8 + 2*A*a*b^9)*d*e^6 - 3*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^7)*x^5 + 210*(7*B*b^10*d^4*e^3 - 4*(10*B*a*b^9 +
A*b^10)*d^3*e^4 + 10*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^5 - 12*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^6 + 6*(7*B*a^4*b^6
+ 4*A*a^3*b^7)*e^7)*x^4 - 56*(56*B*b^10*d^5*e^2 - 35*(10*B*a*b^9 + A*b^10)*d^4*e^3 + 100*(9*B*a^2*b^8 + 2*A*a
*b^9)*d^3*e^4 - 150*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^5 + 120*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^6 - 42*(6*B*a^5*
b^5 + 5*A*a^4*b^6)*e^7)*x^3 + 84*(84*B*b^10*d^6*e - 56*(10*B*a*b^9 + A*b^10)*d^5*e^2 + 175*(9*B*a^2*b^8 + 2*A*
a*b^9)*d^4*e^3 - 300*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^4 + 300*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^5 - 168*(6*B*
a^5*b^5 + 5*A*a^4*b^6)*d*e^6 + 42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^7)*x^2 - 168*(120*B*b^10*d^7 - 84*(10*B*a*b^9
+ A*b^10)*d^6*e + 280*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^2 - 525*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^3 + 600*(7*B*a
^4*b^6 + 4*A*a^3*b^7)*d^3*e^4 - 420*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^5 + 168*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^
6 - 30*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^7)*x)/e^11 + 15*(11*B*b^10*d^8 - 8*(10*B*a*b^9 + A*b^10)*d^7*e + 28*(9*B*
a^2*b^8 + 2*A*a*b^9)*d^6*e^2 - 56*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^3 + 70*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^4
- 56*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^5 + 28*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*e^6 - 8*(4*B*a^7*b^3 + 7*A*a^6*
b^4)*d*e^7 + (3*B*a^8*b^2 + 8*A*a^7*b^3)*e^8)*log(e*x + d)/e^12

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2096 vs. \(2 (433) = 866\).

Time = 0.29 (sec) , antiderivative size = 2096, normalized size of antiderivative = 4.71 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^4} \, dx=\text {Too large to display} \]

[In]

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^4,x, algorithm="giac")

[Out]

15*(11*B*b^10*d^8 - 80*B*a*b^9*d^7*e - 8*A*b^10*d^7*e + 252*B*a^2*b^8*d^6*e^2 + 56*A*a*b^9*d^6*e^2 - 448*B*a^3
*b^7*d^5*e^3 - 168*A*a^2*b^8*d^5*e^3 + 490*B*a^4*b^6*d^4*e^4 + 280*A*a^3*b^7*d^4*e^4 - 336*B*a^5*b^5*d^3*e^5 -
280*A*a^4*b^6*d^3*e^5 + 140*B*a^6*b^4*d^2*e^6 + 168*A*a^5*b^5*d^2*e^6 - 32*B*a^7*b^3*d*e^7 - 56*A*a^6*b^4*d*e
^7 + 3*B*a^8*b^2*e^8 + 8*A*a^7*b^3*e^8)*log(abs(e*x + d))/e^12 + 1/6*(299*B*b^10*d^11 - 2420*B*a*b^9*d^10*e -
242*A*b^10*d^10*e + 8595*B*a^2*b^8*d^9*e^2 + 1910*A*a*b^9*d^9*e^2 - 17520*B*a^3*b^7*d^8*e^3 - 6570*A*a^2*b^8*d
^8*e^3 + 22470*B*a^4*b^6*d^7*e^4 + 12840*A*a^3*b^7*d^7*e^4 - 18648*B*a^5*b^5*d^6*e^5 - 15540*A*a^4*b^6*d^6*e^5
+ 9870*B*a^6*b^4*d^5*e^6 + 11844*A*a^5*b^5*d^5*e^6 - 3120*B*a^7*b^3*d^4*e^7 - 5460*A*a^6*b^4*d^4*e^7 + 495*B*
a^8*b^2*d^3*e^8 + 1320*A*a^7*b^3*d^3*e^8 - 20*B*a^9*b*d^2*e^9 - 90*A*a^8*b^2*d^2*e^9 - B*a^10*d*e^10 - 10*A*a^
9*b*d*e^10 - 2*A*a^10*e^11 + 30*(11*B*b^10*d^9*e^2 - 90*B*a*b^9*d^8*e^3 - 9*A*b^10*d^8*e^3 + 324*B*a^2*b^8*d^7
*e^4 + 72*A*a*b^9*d^7*e^4 - 672*B*a^3*b^7*d^6*e^5 - 252*A*a^2*b^8*d^6*e^5 + 882*B*a^4*b^6*d^5*e^6 + 504*A*a^3*
b^7*d^5*e^6 - 756*B*a^5*b^5*d^4*e^7 - 630*A*a^4*b^6*d^4*e^7 + 420*B*a^6*b^4*d^3*e^8 + 504*A*a^5*b^5*d^3*e^8 -
144*B*a^7*b^3*d^2*e^9 - 252*A*a^6*b^4*d^2*e^9 + 27*B*a^8*b^2*d*e^10 + 72*A*a^7*b^3*d*e^10 - 2*B*a^9*b*e^11 - 9
*A*a^8*b^2*e^11)*x^2 + 3*(209*B*b^10*d^10*e - 1700*B*a*b^9*d^9*e^2 - 170*A*b^10*d^9*e^2 + 6075*B*a^2*b^8*d^8*e
^3 + 1350*A*a*b^9*d^8*e^3 - 12480*B*a^3*b^7*d^7*e^4 - 4680*A*a^2*b^8*d^7*e^4 + 16170*B*a^4*b^6*d^6*e^5 + 9240*
A*a^3*b^7*d^6*e^5 - 13608*B*a^5*b^5*d^5*e^6 - 11340*A*a^4*b^6*d^5*e^6 + 7350*B*a^6*b^4*d^4*e^7 + 8820*A*a^5*b^
5*d^4*e^7 - 2400*B*a^7*b^3*d^3*e^8 - 4200*A*a^6*b^4*d^3*e^8 + 405*B*a^8*b^2*d^2*e^9 + 1080*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*x + d)^3*e^12) + 1/168*(21*B*
b^10*e^28*x^8 - 96*B*b^10*d*e^27*x^7 + 240*B*a*b^9*e^28*x^7 + 24*A*b^10*e^28*x^7 + 280*B*b^10*d^2*e^26*x^6 - 1
120*B*a*b^9*d*e^27*x^6 - 112*A*b^10*d*e^27*x^6 + 1260*B*a^2*b^8*e^28*x^6 + 280*A*a*b^9*e^28*x^6 - 672*B*b^10*d
^3*e^25*x^5 + 3360*B*a*b^9*d^2*e^26*x^5 + 336*A*b^10*d^2*e^26*x^5 - 6048*B*a^2*b^8*d*e^27*x^5 - 1344*A*a*b^9*d
*e^27*x^5 + 4032*B*a^3*b^7*e^28*x^5 + 1512*A*a^2*b^8*e^28*x^5 + 1470*B*b^10*d^4*e^24*x^4 - 8400*B*a*b^9*d^3*e^
25*x^4 - 840*A*b^10*d^3*e^25*x^4 + 18900*B*a^2*b^8*d^2*e^26*x^4 + 4200*A*a*b^9*d^2*e^26*x^4 - 20160*B*a^3*b^7*
d*e^27*x^4 - 7560*A*a^2*b^8*d*e^27*x^4 + 8820*B*a^4*b^6*e^28*x^4 + 5040*A*a^3*b^7*e^28*x^4 - 3136*B*b^10*d^5*e
^23*x^3 + 19600*B*a*b^9*d^4*e^24*x^3 + 1960*A*b^10*d^4*e^24*x^3 - 50400*B*a^2*b^8*d^3*e^25*x^3 - 11200*A*a*b^9
*d^3*e^25*x^3 + 67200*B*a^3*b^7*d^2*e^26*x^3 + 25200*A*a^2*b^8*d^2*e^26*x^3 - 47040*B*a^4*b^6*d*e^27*x^3 - 268
80*A*a^3*b^7*d*e^27*x^3 + 14112*B*a^5*b^5*e^28*x^3 + 11760*A*a^4*b^6*e^28*x^3 + 7056*B*b^10*d^6*e^22*x^2 - 470
40*B*a*b^9*d^5*e^23*x^2 - 4704*A*b^10*d^5*e^23*x^2 + 132300*B*a^2*b^8*d^4*e^24*x^2 + 29400*A*a*b^9*d^4*e^24*x^
2 - 201600*B*a^3*b^7*d^3*e^25*x^2 - 75600*A*a^2*b^8*d^3*e^25*x^2 + 176400*B*a^4*b^6*d^2*e^26*x^2 + 100800*A*a^
3*b^7*d^2*e^26*x^2 - 84672*B*a^5*b^5*d*e^27*x^2 - 70560*A*a^4*b^6*d*e^27*x^2 + 17640*B*a^6*b^4*e^28*x^2 + 2116
8*A*a^5*b^5*e^28*x^2 - 20160*B*b^10*d^7*e^21*x + 141120*B*a*b^9*d^6*e^22*x + 14112*A*b^10*d^6*e^22*x - 423360*
B*a^2*b^8*d^5*e^23*x - 94080*A*a*b^9*d^5*e^23*x + 705600*B*a^3*b^7*d^4*e^24*x + 264600*A*a^2*b^8*d^4*e^24*x -
705600*B*a^4*b^6*d^3*e^25*x - 403200*A*a^3*b^7*d^3*e^25*x + 423360*B*a^5*b^5*d^2*e^26*x + 352800*A*a^4*b^6*d^2
*e^26*x - 141120*B*a^6*b^4*d*e^27*x - 169344*A*a^5*b^5*d*e^27*x + 20160*B*a^7*b^3*e^28*x + 35280*A*a^6*b^4*e^2
8*x)/e^32

Mupad [B] (verification not implemented)

Time = 1.67 (sec) , antiderivative size = 5544, normalized size of antiderivative = 12.46 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^4} \, dx=\text {Too large to display} \]

[In]

int(((A + B*x)*(a + b*x)^10)/(d + e*x)^4,x)

[Out]

x^2*((2*d^3*((6*d^2*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e^2 - (4*d*((4*d*((A*b^10 + 10*B*a*b^9)/e^
4 - (4*B*b^10*d)/e^5))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^4 + (6*B*b^10*d^2)/e^6))/e - (15*a^2*b^7*(3*A*b + 8*B*a
))/e^4 + (4*B*b^10*d^3)/e^7))/e^3 - (3*d^2*((4*d*((6*d^2*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e^2 -
(4*d*((4*d*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^4 + (6*B*b^10*d^2)
/e^6))/e - (15*a^2*b^7*(3*A*b + 8*B*a))/e^4 + (4*B*b^10*d^3)/e^7))/e - (4*d^3*((A*b^10 + 10*B*a*b^9)/e^4 - (4*
B*b^10*d)/e^5))/e^3 + (6*d^2*((4*d*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e - (5*a*b^8*(2*A*b + 9*B*a
))/e^4 + (6*B*b^10*d^2)/e^6))/e^2 + (30*a^3*b^6*(4*A*b + 7*B*a))/e^4 - (B*b^10*d^4)/e^8))/e^2 - (2*d*((6*d^2*(
(6*d^2*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e^2 - (4*d*((4*d*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10
*d)/e^5))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^4 + (6*B*b^10*d^2)/e^6))/e - (15*a^2*b^7*(3*A*b + 8*B*a))/e^4 + (4*B
*b^10*d^3)/e^7))/e^2 - (4*d*((4*d*((6*d^2*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e^2 - (4*d*((4*d*((A
*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^4 + (6*B*b^10*d^2)/e^6))/e - (15*
a^2*b^7*(3*A*b + 8*B*a))/e^4 + (4*B*b^10*d^3)/e^7))/e - (4*d^3*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))
/e^3 + (6*d^2*((4*d*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^4 + (6*B*b
^10*d^2)/e^6))/e^2 + (30*a^3*b^6*(4*A*b + 7*B*a))/e^4 - (B*b^10*d^4)/e^8))/e - (d^4*((A*b^10 + 10*B*a*b^9)/e^4
- (4*B*b^10*d)/e^5))/e^4 + (4*d^3*((4*d*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e - (5*a*b^8*(2*A*b +
9*B*a))/e^4 + (6*B*b^10*d^2)/e^6))/e^3 + (42*a^4*b^5*(5*A*b + 6*B*a))/e^4))/e + (d^4*((4*d*((A*b^10 + 10*B*a*
b^9)/e^4 - (4*B*b^10*d)/e^5))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^4 + (6*B*b^10*d^2)/e^6))/(2*e^4) + (21*a^5*b^4*(
6*A*b + 5*B*a))/e^4) - x^6*((2*d*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/(3*e) - (5*a*b^8*(2*A*b + 9*B
*a))/(6*e^4) + (B*b^10*d^2)/e^6) - ((2*A*a^10*e^11 - 299*B*b^10*d^11 + 242*A*b^10*d^10*e + B*a^10*d*e^10 - 191
0*A*a*b^9*d^9*e^2 + 20*B*a^9*b*d^2*e^9 + 6570*A*a^2*b^8*d^8*e^3 - 12840*A*a^3*b^7*d^7*e^4 + 15540*A*a^4*b^6*d^
6*e^5 - 11844*A*a^5*b^5*d^5*e^6 + 5460*A*a^6*b^4*d^4*e^7 - 1320*A*a^7*b^3*d^3*e^8 + 90*A*a^8*b^2*d^2*e^9 - 859
5*B*a^2*b^8*d^9*e^2 + 17520*B*a^3*b^7*d^8*e^3 - 22470*B*a^4*b^6*d^7*e^4 + 18648*B*a^5*b^5*d^6*e^5 - 9870*B*a^6
*b^4*d^5*e^6 + 3120*B*a^7*b^3*d^4*e^7 - 495*B*a^8*b^2*d^3*e^8 + 10*A*a^9*b*d*e^10 + 2420*B*a*b^9*d^10*e)/(6*e)
+ x*((B*a^10*e^10)/2 - (209*B*b^10*d^10)/2 + 5*A*a^9*b*e^10 + 85*A*b^10*d^9*e - 675*A*a*b^9*d^8*e^2 + 45*A*a^
8*b^2*d*e^9 + 2340*A*a^2*b^8*d^7*e^3 - 4620*A*a^3*b^7*d^6*e^4 + 5670*A*a^4*b^6*d^5*e^5 - 4410*A*a^5*b^5*d^4*e^
6 + 2100*A*a^6*b^4*d^3*e^7 - 540*A*a^7*b^3*d^2*e^8 - (6075*B*a^2*b^8*d^8*e^2)/2 + 6240*B*a^3*b^7*d^7*e^3 - 808
5*B*a^4*b^6*d^6*e^4 + 6804*B*a^5*b^5*d^5*e^5 - 3675*B*a^6*b^4*d^4*e^6 + 1200*B*a^7*b^3*d^3*e^7 - (405*B*a^8*b^
2*d^2*e^8)/2 + 850*B*a*b^9*d^9*e + 10*B*a^9*b*d*e^9) + x^2*(10*B*a^9*b*e^10 - 55*B*b^10*d^9*e + 45*A*a^8*b^2*e
^10 + 45*A*b^10*d^8*e^2 - 360*A*a*b^9*d^7*e^3 - 360*A*a^7*b^3*d*e^9 + 450*B*a*b^9*d^8*e^2 - 135*B*a^8*b^2*d*e^
9 + 1260*A*a^2*b^8*d^6*e^4 - 2520*A*a^3*b^7*d^5*e^5 + 3150*A*a^4*b^6*d^4*e^6 - 2520*A*a^5*b^5*d^3*e^7 + 1260*A
*a^6*b^4*d^2*e^8 - 1620*B*a^2*b^8*d^7*e^3 + 3360*B*a^3*b^7*d^6*e^4 - 4410*B*a^4*b^6*d^5*e^5 + 3780*B*a^5*b^5*d
^4*e^6 - 2100*B*a^6*b^4*d^3*e^7 + 720*B*a^7*b^3*d^2*e^8))/(d^3*e^11 + e^14*x^3 + 3*d^2*e^12*x + 3*d*e^13*x^2)
- x*((6*d^2*((6*d^2*((6*d^2*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e^2 - (4*d*((4*d*((A*b^10 + 10*B*a
*b^9)/e^4 - (4*B*b^10*d)/e^5))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^4 + (6*B*b^10*d^2)/e^6))/e - (15*a^2*b^7*(3*A*b
+ 8*B*a))/e^4 + (4*B*b^10*d^3)/e^7))/e^2 - (4*d*((4*d*((6*d^2*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))
/e^2 - (4*d*((4*d*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^4 + (6*B*b^1
0*d^2)/e^6))/e - (15*a^2*b^7*(3*A*b + 8*B*a))/e^4 + (4*B*b^10*d^3)/e^7))/e - (4*d^3*((A*b^10 + 10*B*a*b^9)/e^4
- (4*B*b^10*d)/e^5))/e^3 + (6*d^2*((4*d*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e - (5*a*b^8*(2*A*b +
9*B*a))/e^4 + (6*B*b^10*d^2)/e^6))/e^2 + (30*a^3*b^6*(4*A*b + 7*B*a))/e^4 - (B*b^10*d^4)/e^8))/e - (d^4*((A*b
^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e^4 + (4*d^3*((4*d*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/
e - (5*a*b^8*(2*A*b + 9*B*a))/e^4 + (6*B*b^10*d^2)/e^6))/e^3 + (42*a^4*b^5*(5*A*b + 6*B*a))/e^4))/e^2 - (d^4*(
(6*d^2*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e^2 - (4*d*((4*d*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10
*d)/e^5))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^4 + (6*B*b^10*d^2)/e^6))/e - (15*a^2*b^7*(3*A*b + 8*B*a))/e^4 + (4*B
*b^10*d^3)/e^7))/e^4 + (4*d^3*((4*d*((6*d^2*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e^2 - (4*d*((4*d*(
(A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^4 + (6*B*b^10*d^2)/e^6))/e - (1
5*a^2*b^7*(3*A*b + 8*B*a))/e^4 + (4*B*b^10*d^3)/e^7))/e - (4*d^3*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5
))/e^3 + (6*d^2*((4*d*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^4 + (6*B
*b^10*d^2)/e^6))/e^2 + (30*a^3*b^6*(4*A*b + 7*B*a))/e^4 - (B*b^10*d^4)/e^8))/e^3 + (4*d*((4*d^3*((6*d^2*((A*b^
10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e^2 - (4*d*((4*d*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e -
(5*a*b^8*(2*A*b + 9*B*a))/e^4 + (6*B*b^10*d^2)/e^6))/e - (15*a^2*b^7*(3*A*b + 8*B*a))/e^4 + (4*B*b^10*d^3)/e^
7))/e^3 - (6*d^2*((4*d*((6*d^2*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e^2 - (4*d*((4*d*((A*b^10 + 10*
B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^4 + (6*B*b^10*d^2)/e^6))/e - (15*a^2*b^7*(3*
A*b + 8*B*a))/e^4 + (4*B*b^10*d^3)/e^7))/e - (4*d^3*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e^3 + (6*d
^2*((4*d*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^4 + (6*B*b^10*d^2)/e^
6))/e^2 + (30*a^3*b^6*(4*A*b + 7*B*a))/e^4 - (B*b^10*d^4)/e^8))/e^2 - (4*d*((6*d^2*((6*d^2*((A*b^10 + 10*B*a*b
^9)/e^4 - (4*B*b^10*d)/e^5))/e^2 - (4*d*((4*d*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e - (5*a*b^8*(2*
A*b + 9*B*a))/e^4 + (6*B*b^10*d^2)/e^6))/e - (15*a^2*b^7*(3*A*b + 8*B*a))/e^4 + (4*B*b^10*d^3)/e^7))/e^2 - (4*
d*((4*d*((6*d^2*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e^2 - (4*d*((4*d*((A*b^10 + 10*B*a*b^9)/e^4 -
(4*B*b^10*d)/e^5))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^4 + (6*B*b^10*d^2)/e^6))/e - (15*a^2*b^7*(3*A*b + 8*B*a))/e
^4 + (4*B*b^10*d^3)/e^7))/e - (4*d^3*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e^3 + (6*d^2*((4*d*((A*b^
10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^4 + (6*B*b^10*d^2)/e^6))/e^2 + (30*a
^3*b^6*(4*A*b + 7*B*a))/e^4 - (B*b^10*d^4)/e^8))/e - (d^4*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e^4
+ (4*d^3*((4*d*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^4 + (6*B*b^10*d
^2)/e^6))/e^3 + (42*a^4*b^5*(5*A*b + 6*B*a))/e^4))/e + (d^4*((4*d*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^
5))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^4 + (6*B*b^10*d^2)/e^6))/e^4 + (42*a^5*b^4*(6*A*b + 5*B*a))/e^4))/e - (30*
a^6*b^3*(7*A*b + 4*B*a))/e^4) - x^5*((6*d^2*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/(5*e^2) - (4*d*((4
*d*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^4 + (6*B*b^10*d^2)/e^6))/(5
*e) - (3*a^2*b^7*(3*A*b + 8*B*a))/e^4 + (4*B*b^10*d^3)/(5*e^7)) + x^3*((2*d^2*((6*d^2*((A*b^10 + 10*B*a*b^9)/e
^4 - (4*B*b^10*d)/e^5))/e^2 - (4*d*((4*d*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e - (5*a*b^8*(2*A*b +
9*B*a))/e^4 + (6*B*b^10*d^2)/e^6))/e - (15*a^2*b^7*(3*A*b + 8*B*a))/e^4 + (4*B*b^10*d^3)/e^7))/e^2 - (4*d*((4
*d*((6*d^2*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e^2 - (4*d*((4*d*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*
b^10*d)/e^5))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^4 + (6*B*b^10*d^2)/e^6))/e - (15*a^2*b^7*(3*A*b + 8*B*a))/e^4 +
(4*B*b^10*d^3)/e^7))/e - (4*d^3*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e^3 + (6*d^2*((4*d*((A*b^10 +
10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^4 + (6*B*b^10*d^2)/e^6))/e^2 + (30*a^3*b^
6*(4*A*b + 7*B*a))/e^4 - (B*b^10*d^4)/e^8))/(3*e) - (d^4*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/(3*e^
4) + (4*d^3*((4*d*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^4 + (6*B*b^1
0*d^2)/e^6))/(3*e^3) + (14*a^4*b^5*(5*A*b + 6*B*a))/e^4) + x^7*((A*b^10 + 10*B*a*b^9)/(7*e^4) - (4*B*b^10*d)/(
7*e^5)) + x^4*((d*((6*d^2*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e^2 - (4*d*((4*d*((A*b^10 + 10*B*a*b
^9)/e^4 - (4*B*b^10*d)/e^5))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^4 + (6*B*b^10*d^2)/e^6))/e - (15*a^2*b^7*(3*A*b +
8*B*a))/e^4 + (4*B*b^10*d^3)/e^7))/e - (d^3*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e^3 + (3*d^2*((4*
d*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^4 + (6*B*b^10*d^2)/e^6))/(2*
e^2) + (15*a^3*b^6*(4*A*b + 7*B*a))/(2*e^4) - (B*b^10*d^4)/(4*e^8)) + (log(d + e*x)*(165*B*b^10*d^8 - 120*A*b^
10*d^7*e + 120*A*a^7*b^3*e^8 + 45*B*a^8*b^2*e^8 + 840*A*a*b^9*d^6*e^2 - 840*A*a^6*b^4*d*e^7 - 480*B*a^7*b^3*d*
e^7 - 2520*A*a^2*b^8*d^5*e^3 + 4200*A*a^3*b^7*d^4*e^4 - 4200*A*a^4*b^6*d^3*e^5 + 2520*A*a^5*b^5*d^2*e^6 + 3780
*B*a^2*b^8*d^6*e^2 - 6720*B*a^3*b^7*d^5*e^3 + 7350*B*a^4*b^6*d^4*e^4 - 5040*B*a^5*b^5*d^3*e^5 + 2100*B*a^6*b^4
*d^2*e^6 - 1200*B*a*b^9*d^7*e))/e^12 + (B*b^10*x^8)/(8*e^4)