3.11.0 ∫ (a+bx)10(A+Bx) (d+ex)6 dx [1094]

Integrand size = 20, antiderivative size = 447 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^6} \, dx=-\frac {42 b^5 (b d-a e)^4 (11 b B d-5 A b e-6 a B e) x}{e^{11}}+\frac {(b d-a e)^{10} (B d-A e)}{5 e^{12} (d+e x)^5}-\frac {(b d-a e)^9 (11 b B d-10 A b e-a B e)}{4 e^{12} (d+e x)^4}+\frac {5 b (b d-a e)^8 (11 b B d-9 A b e-2 a B e)}{3 e^{12} (d+e x)^3}-\frac {15 b^2 (b d-a e)^7 (11 b B d-8 A b e-3 a B e)}{2 e^{12} (d+e x)^2}+\frac {30 b^3 (b d-a e)^6 (11 b B d-7 A b e-4 a B e)}{e^{12} (d+e x)}+\frac {15 b^6 (b d-a e)^3 (11 b B d-4 A b e-7 a B e) (d+e x)^2}{e^{12}}-\frac {5 b^7 (b d-a e)^2 (11 b B d-3 A b e-8 a B e) (d+e x)^3}{e^{12}}+\frac {5 b^8 (b d-a e) (11 b B d-2 A b e-9 a B e) (d+e x)^4}{4 e^{12}}-\frac {b^9 (11 b B d-A b e-10 a B e) (d+e x)^5}{5 e^{12}}+\frac {b^{10} B (d+e x)^6}{6 e^{12}}+\frac {42 b^4 (b d-a e)^5 (11 b B d-6 A b e-5 a B e) \log (d+e x)}{e^{12}} \]

-42*b^5*(-a*e+b*d)^4*(-5*A*b*e-6*B*a*e+11*B*b*d)*x/e^11+1/5*(-a*e+b*d)^10*(-A*e+B*d)/e^12/(e*x+d)^5-1/4*(-a*e+

b*d)^9*(-10*A*b*e-B*a*e+11*B*b*d)/e^12/(e*x+d)^4+5/3*b*(-a*e+b*d)^8*(-9*A*b*e-2*B*a*e+11*B*b*d)/e^12/(e*x+d)^3

-15/2*b^2*(-a*e+b*d)^7*(-8*A*b*e-3*B*a*e+11*B*b*d)/e^12/(e*x+d)^2+30*b^3*(-a*e+b*d)^6*(-7*A*b*e-4*B*a*e+11*B*b

*d)/e^12/(e*x+d)+15*b^6*(-a*e+b*d)^3*(-4*A*b*e-7*B*a*e+11*B*b*d)*(e*x+d)^2/e^12-5*b^7*(-a*e+b*d)^2*(-3*A*b*e-8

*B*a*e+11*B*b*d)*(e*x+d)^3/e^12+5/4*b^8*(-a*e+b*d)*(-2*A*b*e-9*B*a*e+11*B*b*d)*(e*x+d)^4/e^12-1/5*b^9*(-A*b*e-

10*B*a*e+11*B*b*d)*(e*x+d)^5/e^12+1/6*b^10*B*(e*x+d)^6/e^12+42*b^4*(-a*e+b*d)^5*(-6*A*b*e-5*B*a*e+11*B*b*d)*ln

Rubi [A] (veriﬁed)

Time = 0.83 (sec) , antiderivative size = 447, 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} \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^6} \, dx=-\frac {b^9 (d+e x)^5 (-10 a B e-A b e+11 b B d)}{5 e^{12}}+\frac {5 b^8 (d+e x)^4 (b d-a e) (-9 a B e-2 A b e+11 b B d)}{4 e^{12}}-\frac {5 b^7 (d+e x)^3 (b d-a e)^2 (-8 a B e-3 A b e+11 b B d)}{e^{12}}+\frac {15 b^6 (d+e x)^2 (b d-a e)^3 (-7 a B e-4 A b e+11 b B d)}{e^{12}}-\frac {42 b^5 x (b d-a e)^4 (-6 a B e-5 A b e+11 b B d)}{e^{11}}+\frac {42 b^4 (b d-a e)^5 \log (d+e x) (-5 a B e-6 A b e+11 b B d)}{e^{12}}+\frac {30 b^3 (b d-a e)^6 (-4 a B e-7 A b e+11 b B d)}{e^{12} (d+e x)}-\frac {15 b^2 (b d-a e)^7 (-3 a B e-8 A b e+11 b B d)}{2 e^{12} (d+e x)^2}+\frac {5 b (b d-a e)^8 (-2 a B e-9 A b e+11 b B d)}{3 e^{12} (d+e x)^3}-\frac {(b d-a e)^9 (-a B e-10 A b e+11 b B d)}{4 e^{12} (d+e x)^4}+\frac {(b d-a e)^{10} (B d-A e)}{5 e^{12} (d+e x)^5}+\frac {b^{10} B (d+e x)^6}{6 e^{12}} \]

(-42*b^5*(b*d - a*e)^4*(11*b*B*d - 5*A*b*e - 6*a*B*e)*x)/e^11 + ((b*d - a*e)^10*(B*d - A*e))/(5*e^12*(d + e*x)

^5) - ((b*d - a*e)^9*(11*b*B*d - 10*A*b*e - a*B*e))/(4*e^12*(d + e*x)^4) + (5*b*(b*d - a*e)^8*(11*b*B*d - 9*A*

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

^2) + (30*b^3*(b*d - a*e)^6*(11*b*B*d - 7*A*b*e - 4*a*B*e))/(e^12*(d + e*x)) + (15*b^6*(b*d - a*e)^3*(11*b*B*d

- 4*A*b*e - 7*a*B*e)*(d + e*x)^2)/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)^3)/e^1

2 + (5*b^8*(b*d - a*e)*(11*b*B*d - 2*A*b*e - 9*a*B*e)*(d + e*x)^4)/(4*e^12) - (b^9*(11*b*B*d - A*b*e - 10*a*B*

e)*(d + e*x)^5)/(5*e^12) + (b^10*B*(d + e*x)^6)/(6*e^12) + (42*b^4*(b*d - a*e)^5*(11*b*B*d - 6*A*b*e - 5*a*B*e

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

Mathematica [A] (veriﬁed)

Time = 0.29 (sec) , antiderivative size = 587, normalized size of antiderivative = 1.31 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^6} \, dx=\frac {60 b^5 e \left (252 a^5 B e^5+140 a b^4 d^3 e (9 B d-4 A e)-315 a^2 b^3 d^2 e^2 (8 B d-3 A e)+360 a^3 b^2 d e^3 (7 B d-2 A e)-126 b^5 d^4 (2 B d-A e)+210 a^4 b e^4 (-6 B d+A e)\right ) x-30 b^6 e^2 \left (-210 a^4 B e^4-14 b^4 d^3 (9 B d-4 A e)+70 a b^3 d^2 e (8 B d-3 A e)-135 a^2 b^2 d e^2 (7 B d-2 A e)-120 a^3 b e^3 (-6 B d+A e)\right ) x^2+20 b^7 e^3 \left (120 a^3 B e^3-7 b^3 d^2 (8 B d-3 A e)+30 a b^2 d e (7 B d-2 A e)+45 a^2 b e^2 (-6 B d+A e)\right ) x^3-15 b^8 e^4 \left (-45 a^2 B e^2-10 a b e (-6 B d+A e)+3 b^2 d (-7 B d+2 A e)\right ) x^4+12 b^9 e^5 (-6 b B d+A b e+10 a B e) x^5+10 b^{10} B e^6 x^6+\frac {12 (b d-a e)^{10} (B d-A e)}{(d+e x)^5}-\frac {15 (b d-a e)^9 (11 b B d-10 A b e-a B e)}{(d+e x)^4}+\frac {100 b (b d-a e)^8 (11 b B d-9 A b e-2 a B e)}{(d+e x)^3}-\frac {450 b^2 (b d-a e)^7 (11 b B d-8 A b e-3 a B e)}{(d+e x)^2}+\frac {1800 b^3 (b d-a e)^6 (11 b B d-7 A b e-4 a B e)}{d+e x}+2520 b^4 (b d-a e)^5 (11 b B d-6 A b e-5 a B e) \log (d+e x)}{60 e^{12}} \]

(60*b^5*e*(252*a^5*B*e^5 + 140*a*b^4*d^3*e*(9*B*d - 4*A*e) - 315*a^2*b^3*d^2*e^2*(8*B*d - 3*A*e) + 360*a^3*b^2

*d*e^3*(7*B*d - 2*A*e) - 126*b^5*d^4*(2*B*d - A*e) + 210*a^4*b*e^4*(-6*B*d + A*e))*x - 30*b^6*e^2*(-210*a^4*B*

e^4 - 14*b^4*d^3*(9*B*d - 4*A*e) + 70*a*b^3*d^2*e*(8*B*d - 3*A*e) - 135*a^2*b^2*d*e^2*(7*B*d - 2*A*e) - 120*a^

3*b*e^3*(-6*B*d + A*e))*x^2 + 20*b^7*e^3*(120*a^3*B*e^3 - 7*b^3*d^2*(8*B*d - 3*A*e) + 30*a*b^2*d*e*(7*B*d - 2*

A*e) + 45*a^2*b*e^2*(-6*B*d + A*e))*x^3 - 15*b^8*e^4*(-45*a^2*B*e^2 - 10*a*b*e*(-6*B*d + A*e) + 3*b^2*d*(-7*B*

d + 2*A*e))*x^4 + 12*b^9*e^5*(-6*b*B*d + A*b*e + 10*a*B*e)*x^5 + 10*b^10*B*e^6*x^6 + (12*(b*d - a*e)^10*(B*d -

A*e))/(d + e*x)^5 - (15*(b*d - a*e)^9*(11*b*B*d - 10*A*b*e - a*B*e))/(d + e*x)^4 + (100*b*(b*d - a*e)^8*(11*b

*B*d - 9*A*b*e - 2*a*B*e))/(d + e*x)^3 - (450*b^2*(b*d - a*e)^7*(11*b*B*d - 8*A*b*e - 3*a*B*e))/(d + e*x)^2 +

(1800*b^3*(b*d - a*e)^6*(11*b*B*d - 7*A*b*e - 4*a*B*e))/(d + e*x) + 2520*b^4*(b*d - a*e)^5*(11*b*B*d - 6*A*b*e

Maple [B] (veriﬁed)

Leaf count of result is larger than twice the leaf count of optimal. \(1915\) vs. \(2(433)=866\).

Time = 2.10 (sec) , antiderivative size = 1916, normalized size of antiderivative = 4.29


method	result	size

norman	\(\text {Expression too large to display}\)	\(1916\)
default	\(\text {Expression too large to display}\)	\(2001\)
risch	\(\text {Expression too large to display}\)	\(2079\)
parallelrisch	\(\text {Expression too large to display}\)	\(3743\)

int((b*x+a)^10*(B*x+A)/(e*x+d)^6,x,method=_RETURNVERBOSE)

(b^6*(60*A*a^3*b*e^4-60*A*a^2*b^2*d*e^3+30*A*a*b^3*d^2*e^2-6*A*b^4*d^3*e+105*B*a^4*e^4-160*B*a^3*b*d*e^3+135*B

*a^2*b^2*d^2*e^2-60*B*a*b^3*d^3*e+11*B*b^4*d^4)/e^5*x^7-1/60*(12*A*a^10*e^11+30*A*a^9*b*d*e^10+90*A*a^8*b^2*d^

2*e^9+360*A*a^7*b^3*d^3*e^8+2520*A*a^6*b^4*d^4*e^7-34524*A*a^5*b^5*d^5*e^6+172620*A*a^4*b^6*d^6*e^5-345240*A*a

^3*b^7*d^7*e^4+345240*A*a^2*b^8*d^8*e^3-172620*A*a*b^9*d^9*e^2+34524*A*b^10*d^10*e+3*B*a^10*d*e^10+20*B*a^9*b*

d^2*e^9+135*B*a^8*b^2*d^3*e^8+1440*B*a^7*b^3*d^4*e^7-28770*B*a^6*b^4*d^5*e^6+207144*B*a^5*b^5*d^6*e^5-604170*B

*a^4*b^6*d^7*e^4+920640*B*a^3*b^7*d^8*e^3-776790*B*a^2*b^8*d^9*e^2+345240*B*a*b^9*d^10*e-63294*B*b^10*d^11)/e^

12-5*(42*A*a^6*b^4*e^7-252*A*a^5*b^5*d*e^6+1260*A*a^4*b^6*d^2*e^5-2520*A*a^3*b^7*d^3*e^4+2520*A*a^2*b^8*d^4*e^

3-1260*A*a*b^9*d^5*e^2+252*A*b^10*d^6*e+24*B*a^7*b^3*e^7-210*B*a^6*b^4*d*e^6+1512*B*a^5*b^5*d^2*e^5-4410*B*a^4

*b^6*d^3*e^4+6720*B*a^3*b^7*d^4*e^3-5670*B*a^2*b^8*d^5*e^2+2520*B*a*b^9*d^6*e-462*B*b^10*d^7)/e^8*x^4-5/2*(24*

A*a^7*b^3*e^8+168*A*a^6*b^4*d*e^7-1512*A*a^5*b^5*d^2*e^6+7560*A*a^4*b^6*d^3*e^5-15120*A*a^3*b^7*d^4*e^4+15120*

A*a^2*b^8*d^5*e^3-7560*A*a*b^9*d^6*e^2+1512*A*b^10*d^7*e+9*B*a^8*b^2*e^8+96*B*a^7*b^3*d*e^7-1260*B*a^6*b^4*d^2

*e^6+9072*B*a^5*b^5*d^3*e^5-26460*B*a^4*b^6*d^4*e^4+40320*B*a^3*b^7*d^5*e^3-34020*B*a^2*b^8*d^6*e^2+15120*B*a*

b^9*d^7*e-2772*B*b^10*d^8)/e^9*x^3-5/6*(18*A*a^8*b^2*e^9+72*A*a^7*b^3*d*e^8+504*A*a^6*b^4*d^2*e^7-5544*A*a^5*b

^5*d^3*e^6+27720*A*a^4*b^6*d^4*e^5-55440*A*a^3*b^7*d^5*e^4+55440*A*a^2*b^8*d^6*e^3-27720*A*a*b^9*d^7*e^2+5544*

A*b^10*d^8*e+4*B*a^9*b*e^9+27*B*a^8*b^2*d*e^8+288*B*a^7*b^3*d^2*e^7-4620*B*a^6*b^4*d^3*e^6+33264*B*a^5*b^5*d^4

*e^5-97020*B*a^4*b^6*d^5*e^4+147840*B*a^3*b^7*d^6*e^3-124740*B*a^2*b^8*d^7*e^2+55440*B*a*b^9*d^8*e-10164*B*b^1

0*d^9)/e^10*x^2-1/12*(30*A*a^9*b*e^10+90*A*a^8*b^2*d*e^9+360*A*a^7*b^3*d^2*e^8+2520*A*a^6*b^4*d^3*e^7-31500*A*

a^5*b^5*d^4*e^6+157500*A*a^4*b^6*d^5*e^5-315000*A*a^3*b^7*d^6*e^4+315000*A*a^2*b^8*d^7*e^3-157500*A*a*b^9*d^8*

e^2+31500*A*b^10*d^9*e+3*B*a^10*e^10+20*B*a^9*b*d*e^9+135*B*a^8*b^2*d^2*e^8+1440*B*a^7*b^3*d^3*e^7-26250*B*a^6

*b^4*d^4*e^6+189000*B*a^5*b^5*d^5*e^5-551250*B*a^4*b^6*d^6*e^4+840000*B*a^3*b^7*d^7*e^3-708750*B*a^2*b^8*d^8*e

^2+315000*B*a*b^9*d^9*e-57750*B*b^10*d^10)/e^11*x+7*b^5*(30*A*a^4*b*e^5-60*A*a^3*b^2*d*e^4+60*A*a^2*b^3*d^2*e^

3-30*A*a*b^4*d^3*e^2+6*A*b^5*d^4*e+36*B*a^5*e^5-105*B*a^4*b*d*e^4+160*B*a^3*b^2*d^2*e^3-135*B*a^2*b^3*d^3*e^2+

60*B*a*b^4*d^4*e-11*B*b^5*d^5)/e^6*x^6+1/4*b^7*(60*A*a^2*b*e^3-30*A*a*b^2*d*e^2+6*A*b^3*d^2*e+160*B*a^3*e^3-13

5*B*a^2*b*d*e^2+60*B*a*b^2*d^2*e-11*B*b^3*d^3)/e^4*x^8+1/12*b^8*(30*A*a*b*e^2-6*A*b^2*d*e+135*B*a^2*e^2-60*B*a

*b*d*e+11*B*b^2*d^2)/e^3*x^9+1/30*b^9*(6*A*b*e+60*B*a*e-11*B*b*d)/e^2*x^10+1/6*b^10*B/e*x^11)/(e*x+d)^5+42*b^4

/e^12*(6*A*a^5*b*e^6-30*A*a^4*b^2*d*e^5+60*A*a^3*b^3*d^2*e^4-60*A*a^2*b^4*d^3*e^3+30*A*a*b^5*d^4*e^2-6*A*b^6*d

^5*e+5*B*a^6*e^6-36*B*a^5*b*d*e^5+105*B*a^4*b^2*d^2*e^4-160*B*a^3*b^3*d^3*e^3+135*B*a^2*b^4*d^4*e^2-60*B*a*b^5

Fricas [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2851 vs. \(2 (433) = 866\).

Time = 0.30 (sec) , antiderivative size = 2851, normalized size of antiderivative = 6.38 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^6} \, dx=\text {Too large to display} \]

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^6,x, algorithm="fricas")

1/60*(10*B*b^10*e^11*x^11 + 15797*B*b^10*d^11 - 12*A*a^10*e^11 - 9762*(10*B*a*b^9 + A*b^10)*d^10*e + 28185*(9*

B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 - 44580*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 41310*(7*B*a^4*b^6 + 4*A*a^3*b^7)

*d^7*e^4 - 21924*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 5754*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 - 360*(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 -

3*(B*a^10 + 10*A*a^9*b)*d*e^10 - 2*(11*B*b^10*d*e^10 - 6*(10*B*a*b^9 + A*b^10)*e^11)*x^10 + 5*(11*B*b^10*d^2*e

^9 - 6*(10*B*a*b^9 + A*b^10)*d*e^10 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9 - 15*(11*B*b^10*d^3*e^8 - 6*(10*B

*a*b^9 + A*b^10)*d^2*e^9 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^10 - 20*(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 - 6*(10*B*a*b^9 + A*b^10)*d^3*e^8 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^9 - 20*(8*B*a^3*b^7

+ 3*A*a^2*b^8)*d*e^10 + 15*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 - 420*(11*B*b^10*d^5*e^6 - 6*(10*B*a*b^9 + A

*b^10)*d^4*e^7 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^8 - 20*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^9 + 15*(7*B*a^4*b

^6 + 4*A*a^3*b^7)*d*e^10 - 6*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 - (47497*B*b^10*d^6*e^5 - 24762*(10*B*a*b^9

+ A*b^10)*d^5*e^6 + 58125*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 - 70500*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8 + 450

00*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^9 - 12600*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10)*x^5 - 5*(19777*B*b^10*d^7*e

^4 - 9642*(10*B*a*b^9 + A*b^10)*d^6*e^5 + 20325*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^6 - 20100*(8*B*a^3*b^7 + 3*A*a

^2*b^8)*d^4*e^7 + 7200*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^8 + 2520*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^9 - 2520*(

5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^10 + 360*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^11)*x^4 - 10*(5917*B*b^10*d^8*e^3 - 2082

*(10*B*a*b^9 + A*b^10)*d^7*e^4 + 1425*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5 + 5100*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5

*e^6 - 11700*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^7 + 10080*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^8 - 3780*(5*B*a^6*b

^4 + 6*A*a^5*b^5)*d^2*e^9 + 360*(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

+ 10*(3323*B*b^10*d^9*e^2 - 2958*(10*B*a*b^9 + A*b^10)*d^8*e^3 + 11175*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^4 - 219

00*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^5 + 24300*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*e^6 - 15120*(6*B*a^5*b^5 + 5*A*

a^4*b^6)*d^4*e^7 + 4620*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*e^8 - 360*(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 + 5*(10253*B*b^10*d^10*e - 6738*(10*B

*a*b^9 + A*b^10)*d^9*e^2 + 20625*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^3 - 34500*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4

+ 33750*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5 - 18900*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 + 5250*(5*B*a^6*b^4 +

6*A*a^5*b^5)*d^4*e^7 - 360*(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 - 3*(B*a^10 + 10*A*a^9*b)*e^11)*x + 2520*(11*B*b^10*d^11 - 6*(10*B*a*b^9 + A*

b^10)*d^10*e + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 - 20*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 15*(7*B*a^4*b^6

+ 4*A*a^3*b^7)*d^7*e^4 - 6*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + (5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 + (11*B*

b^10*d^6*e^5 - 6*(10*B*a*b^9 + A*b^10)*d^5*e^6 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 - 20*(8*B*a^3*b^7 + 3*A*

a^2*b^8)*d^3*e^8 + 15*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^9 - 6*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 + (5*B*a^6*b^

4 + 6*A*a^5*b^5)*e^11)*x^5 + 5*(11*B*b^10*d^7*e^4 - 6*(10*B*a*b^9 + A*b^10)*d^6*e^5 + 15*(9*B*a^2*b^8 + 2*A*a*

b^9)*d^5*e^6 - 20*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 + 15*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^8 - 6*(6*B*a^5*b^

5 + 5*A*a^4*b^6)*d^2*e^9 + (5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^10)*x^4 + 10*(11*B*b^10*d^8*e^3 - 6*(10*B*a*b^9 + A

*b^10)*d^7*e^4 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5 - 20*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^6 + 15*(7*B*a^4*b

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

10*(11*B*b^10*d^9*e^2 - 6*(10*B*a*b^9 + A*b^10)*d^8*e^3 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^4 - 20*(8*B*a^3*

b^7 + 3*A*a^2*b^8)*d^6*e^5 + 15*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*e^6 - 6*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*e^7 +

(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*e^8)*x^2 + 5*(11*B*b^10*d^10*e - 6*(10*B*a*b^9 + A*b^10)*d^9*e^2 + 15*(9*B*a^2

*b^8 + 2*A*a*b^9)*d^8*e^3 - 20*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4 + 15*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5 -

6*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 + (5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*e^7)*x)*log(e*x + d))/(e^17*x^5 + 5*d*

e^16*x^4 + 10*d^2*e^15*x^3 + 10*d^3*e^14*x^2 + 5*d^4*e^13*x + d^5*e^12)

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^6} \, dx=\text {Timed out} \]

Maxima [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1861 vs. \(2 (433) = 866\).

Time = 0.31 (sec) , antiderivative size = 1861, normalized size of antiderivative = 4.16 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^6} \, dx=\text {Too large to display} \]

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^6,x, algorithm="maxima")

1/60*(15797*B*b^10*d^11 - 12*A*a^10*e^11 - 9762*(10*B*a*b^9 + A*b^10)*d^10*e + 28185*(9*B*a^2*b^8 + 2*A*a*b^9)

*d^9*e^2 - 44580*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 41310*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4 - 21924*(6*B*

a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 5754*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 - 360*(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 - 3*(B*a^10 + 10*A*a^9*b

)*d*e^10 + 1800*(11*B*b^10*d^7*e^4 - 7*(10*B*a*b^9 + A*b^10)*d^6*e^5 + 21*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^6 -

35*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 + 35*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^8 - 21*(6*B*a^5*b^5 + 5*A*a^4*b^

6)*d^2*e^9 + 7*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^10 - (4*B*a^7*b^3 + 7*A*a^6*b^4)*e^11)*x^4 + 450*(165*B*b^10*d^

8*e^3 - 104*(10*B*a*b^9 + A*b^10)*d^7*e^4 + 308*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5 - 504*(8*B*a^3*b^7 + 3*A*a^2

*b^8)*d^5*e^6 + 490*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^7 - 280*(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 - 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 +

50*(2101*B*b^10*d^9*e^2 - 1314*(10*B*a*b^9 + A*b^10)*d^8*e^3 + 3852*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^4 - 6216*(

8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^5 + 5922*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*e^6 - 3276*(6*B*a^5*b^5 + 5*A*a^4*b^

6)*d^4*e^7 + 924*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*e^8 - 72*(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*(2*B*a^9*b + 9*A*a^8*b^2)*e^11)*x^2 + 5*(13277*B*b^10*d^10*e - 8250*(10*B*a*b^9 + A

*b^10)*d^9*e^2 + 23985*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^3 - 38280*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4 + 35910*(

7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5 - 19404*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 + 5250*(5*B*a^6*b^4 + 6*A*a^5*b

^5)*d^4*e^7 - 360*(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 - 3*(B*a^10 + 10*A*a^9*b)*e^11)*x)/(e^17*x^5 + 5*d*e^16*x^4 + 10*d^2*e^15*x^3 + 10*d^3*

e^14*x^2 + 5*d^4*e^13*x + d^5*e^12) + 1/60*(10*B*b^10*e^5*x^6 - 12*(6*B*b^10*d*e^4 - (10*B*a*b^9 + A*b^10)*e^5

)*x^5 + 15*(21*B*b^10*d^2*e^3 - 6*(10*B*a*b^9 + A*b^10)*d*e^4 + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*e^5)*x^4 - 20*(56*

B*b^10*d^3*e^2 - 21*(10*B*a*b^9 + A*b^10)*d^2*e^3 + 30*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^4 - 15*(8*B*a^3*b^7 + 3*A

*a^2*b^8)*e^5)*x^3 + 30*(126*B*b^10*d^4*e - 56*(10*B*a*b^9 + A*b^10)*d^3*e^2 + 105*(9*B*a^2*b^8 + 2*A*a*b^9)*d

^2*e^3 - 90*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^4 + 30*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^5)*x^2 - 60*(252*B*b^10*d^5 -

126*(10*B*a*b^9 + A*b^10)*d^4*e + 280*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^2 - 315*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2

*e^3 + 180*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^4 - 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^5)*x)/e^11 + 42*(11*B*b^10*d^6

- 6*(10*B*a*b^9 + A*b^10)*d^5*e + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^2 - 20*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e

^3 + 15*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^4 - 6*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^5 + (5*B*a^6*b^4 + 6*A*a^5*b^5

Giac [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2032 vs. \(2 (433) = 866\).

Time = 0.30 (sec) , antiderivative size = 2032, normalized size of antiderivative = 4.55 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^6} \, dx=\text {Too large to display} \]

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^6,x, algorithm="giac")

42*(11*B*b^10*d^6 - 60*B*a*b^9*d^5*e - 6*A*b^10*d^5*e + 135*B*a^2*b^8*d^4*e^2 + 30*A*a*b^9*d^4*e^2 - 160*B*a^3

*b^7*d^3*e^3 - 60*A*a^2*b^8*d^3*e^3 + 105*B*a^4*b^6*d^2*e^4 + 60*A*a^3*b^7*d^2*e^4 - 36*B*a^5*b^5*d*e^5 - 30*A

*a^4*b^6*d*e^5 + 5*B*a^6*b^4*e^6 + 6*A*a^5*b^5*e^6)*log(abs(e*x + d))/e^12 + 1/60*(15797*B*b^10*d^11 - 97620*B

*a*b^9*d^10*e - 9762*A*b^10*d^10*e + 253665*B*a^2*b^8*d^9*e^2 + 56370*A*a*b^9*d^9*e^2 - 356640*B*a^3*b^7*d^8*e

^3 - 133740*A*a^2*b^8*d^8*e^3 + 289170*B*a^4*b^6*d^7*e^4 + 165240*A*a^3*b^7*d^7*e^4 - 131544*B*a^5*b^5*d^6*e^5

- 109620*A*a^4*b^6*d^6*e^5 + 28770*B*a^6*b^4*d^5*e^6 + 34524*A*a^5*b^5*d^5*e^6 - 1440*B*a^7*b^3*d^4*e^7 - 252

0*A*a^6*b^4*d^4*e^7 - 135*B*a^8*b^2*d^3*e^8 - 360*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 - 3*B*a^10*d*e^10 - 30*A*a^9*b*d*e^10 - 12*A*a^10*e^11 + 1800*(11*B*b^10*d^7*e^4 - 70*B*a*b^9*d^6*e^5 - 7*A*

b^10*d^6*e^5 + 189*B*a^2*b^8*d^5*e^6 + 42*A*a*b^9*d^5*e^6 - 280*B*a^3*b^7*d^4*e^7 - 105*A*a^2*b^8*d^4*e^7 + 24

5*B*a^4*b^6*d^3*e^8 + 140*A*a^3*b^7*d^3*e^8 - 126*B*a^5*b^5*d^2*e^9 - 105*A*a^4*b^6*d^2*e^9 + 35*B*a^6*b^4*d*e

^10 + 42*A*a^5*b^5*d*e^10 - 4*B*a^7*b^3*e^11 - 7*A*a^6*b^4*e^11)*x^4 + 450*(165*B*b^10*d^8*e^3 - 1040*B*a*b^9*

d^7*e^4 - 104*A*b^10*d^7*e^4 + 2772*B*a^2*b^8*d^6*e^5 + 616*A*a*b^9*d^6*e^5 - 4032*B*a^3*b^7*d^5*e^6 - 1512*A*

a^2*b^8*d^5*e^6 + 3430*B*a^4*b^6*d^4*e^7 + 1960*A*a^3*b^7*d^4*e^7 - 1680*B*a^5*b^5*d^3*e^8 - 1400*A*a^4*b^6*d^

3*e^8 + 420*B*a^6*b^4*d^2*e^9 + 504*A*a^5*b^5*d^2*e^9 - 32*B*a^7*b^3*d*e^10 - 56*A*a^6*b^4*d*e^10 - 3*B*a^8*b^

2*e^11 - 8*A*a^7*b^3*e^11)*x^3 + 50*(2101*B*b^10*d^9*e^2 - 13140*B*a*b^9*d^8*e^3 - 1314*A*b^10*d^8*e^3 + 34668

*B*a^2*b^8*d^7*e^4 + 7704*A*a*b^9*d^7*e^4 - 49728*B*a^3*b^7*d^6*e^5 - 18648*A*a^2*b^8*d^6*e^5 + 41454*B*a^4*b^

6*d^5*e^6 + 23688*A*a^3*b^7*d^5*e^6 - 19656*B*a^5*b^5*d^4*e^7 - 16380*A*a^4*b^6*d^4*e^7 + 4620*B*a^6*b^4*d^3*e

^8 + 5544*A*a^5*b^5*d^3*e^8 - 288*B*a^7*b^3*d^2*e^9 - 504*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 - 4*B*a^9*b*e^11 - 18*A*a^8*b^2*e^11)*x^2 + 5*(13277*B*b^10*d^10*e - 82500*B*a*b^9*d^9*e^2 - 8250*A*

b^10*d^9*e^2 + 215865*B*a^2*b^8*d^8*e^3 + 47970*A*a*b^9*d^8*e^3 - 306240*B*a^3*b^7*d^7*e^4 - 114840*A*a^2*b^8*

d^7*e^4 + 251370*B*a^4*b^6*d^6*e^5 + 143640*A*a^3*b^7*d^6*e^5 - 116424*B*a^5*b^5*d^5*e^6 - 97020*A*a^4*b^6*d^5

*e^6 + 26250*B*a^6*b^4*d^4*e^7 + 31500*A*a^5*b^5*d^4*e^7 - 1440*B*a^7*b^3*d^3*e^8 - 2520*A*a^6*b^4*d^3*e^8 - 1

35*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 - 3*B*a^10*e^11 - 30*A*

a^9*b*e^11)*x)/((e*x + d)^5*e^12) + 1/60*(10*B*b^10*e^30*x^6 - 72*B*b^10*d*e^29*x^5 + 120*B*a*b^9*e^30*x^5 + 1

2*A*b^10*e^30*x^5 + 315*B*b^10*d^2*e^28*x^4 - 900*B*a*b^9*d*e^29*x^4 - 90*A*b^10*d*e^29*x^4 + 675*B*a^2*b^8*e^

30*x^4 + 150*A*a*b^9*e^30*x^4 - 1120*B*b^10*d^3*e^27*x^3 + 4200*B*a*b^9*d^2*e^28*x^3 + 420*A*b^10*d^2*e^28*x^3

- 5400*B*a^2*b^8*d*e^29*x^3 - 1200*A*a*b^9*d*e^29*x^3 + 2400*B*a^3*b^7*e^30*x^3 + 900*A*a^2*b^8*e^30*x^3 + 37

80*B*b^10*d^4*e^26*x^2 - 16800*B*a*b^9*d^3*e^27*x^2 - 1680*A*b^10*d^3*e^27*x^2 + 28350*B*a^2*b^8*d^2*e^28*x^2

+ 6300*A*a*b^9*d^2*e^28*x^2 - 21600*B*a^3*b^7*d*e^29*x^2 - 8100*A*a^2*b^8*d*e^29*x^2 + 6300*B*a^4*b^6*e^30*x^2

+ 3600*A*a^3*b^7*e^30*x^2 - 15120*B*b^10*d^5*e^25*x + 75600*B*a*b^9*d^4*e^26*x + 7560*A*b^10*d^4*e^26*x - 151

200*B*a^2*b^8*d^3*e^27*x - 33600*A*a*b^9*d^3*e^27*x + 151200*B*a^3*b^7*d^2*e^28*x + 56700*A*a^2*b^8*d^2*e^28*x

- 75600*B*a^4*b^6*d*e^29*x - 43200*A*a^3*b^7*d*e^29*x + 15120*B*a^5*b^5*e^30*x + 12600*A*a^4*b^6*e^30*x)/e^36

Mupad [B] (veriﬁcation not implemented)

Time = 0.45 (sec) , antiderivative size = 2681, normalized size of antiderivative = 6.00 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^6} \, dx=\text {Too large to display} \]

x^5*((A*b^10 + 10*B*a*b^9)/(5*e^6) - (6*B*b^10*d)/(5*e^7)) - x^4*((3*d*((A*b^10 + 10*B*a*b^9)/e^6 - (6*B*b^10*

d)/e^7))/(2*e) - (5*a*b^8*(2*A*b + 9*B*a))/(4*e^6) + (15*B*b^10*d^2)/(4*e^8)) - (x^4*(210*A*a^6*b^4*e^10 + 120

*B*a^7*b^3*e^10 + 210*A*b^10*d^6*e^4 - 330*B*b^10*d^7*e^3 - 1260*A*a*b^9*d^5*e^5 - 1260*A*a^5*b^5*d*e^9 + 2100

*B*a*b^9*d^6*e^4 - 1050*B*a^6*b^4*d*e^9 + 3150*A*a^2*b^8*d^4*e^6 - 4200*A*a^3*b^7*d^3*e^7 + 3150*A*a^4*b^6*d^2

*e^8 - 5670*B*a^2*b^8*d^5*e^5 + 8400*B*a^3*b^7*d^4*e^6 - 7350*B*a^4*b^6*d^3*e^7 + 3780*B*a^5*b^5*d^2*e^8) + x^

3*(60*A*a^7*b^3*e^10 + (45*B*a^8*b^2*e^10)/2 + 780*A*b^10*d^7*e^3 - (2475*B*b^10*d^8*e^2)/2 - 4620*A*a*b^9*d^6

*e^4 + 420*A*a^6*b^4*d*e^9 + 7800*B*a*b^9*d^7*e^3 + 240*B*a^7*b^3*d*e^9 + 11340*A*a^2*b^8*d^5*e^5 - 14700*A*a^

3*b^7*d^4*e^6 + 10500*A*a^4*b^6*d^3*e^7 - 3780*A*a^5*b^5*d^2*e^8 - 20790*B*a^2*b^8*d^6*e^4 + 30240*B*a^3*b^7*d

^5*e^5 - 25725*B*a^4*b^6*d^4*e^6 + 12600*B*a^5*b^5*d^3*e^7 - 3150*B*a^6*b^4*d^2*e^8) + (12*A*a^10*e^11 - 15797

*B*b^10*d^11 + 9762*A*b^10*d^10*e + 3*B*a^10*d*e^10 - 56370*A*a*b^9*d^9*e^2 + 20*B*a^9*b*d^2*e^9 + 133740*A*a^

2*b^8*d^8*e^3 - 165240*A*a^3*b^7*d^7*e^4 + 109620*A*a^4*b^6*d^6*e^5 - 34524*A*a^5*b^5*d^5*e^6 + 2520*A*a^6*b^4

*d^4*e^7 + 360*A*a^7*b^3*d^3*e^8 + 90*A*a^8*b^2*d^2*e^9 - 253665*B*a^2*b^8*d^9*e^2 + 356640*B*a^3*b^7*d^8*e^3

- 289170*B*a^4*b^6*d^7*e^4 + 131544*B*a^5*b^5*d^6*e^5 - 28770*B*a^6*b^4*d^5*e^6 + 1440*B*a^7*b^3*d^4*e^7 + 135

*B*a^8*b^2*d^3*e^8 + 30*A*a^9*b*d*e^10 + 97620*B*a*b^9*d^10*e)/(60*e) + x*((B*a^10*e^10)/4 - (13277*B*b^10*d^1

0)/12 + (5*A*a^9*b*e^10)/2 + (1375*A*b^10*d^9*e)/2 - (7995*A*a*b^9*d^8*e^2)/2 + (15*A*a^8*b^2*d*e^9)/2 + 9570*

A*a^2*b^8*d^7*e^3 - 11970*A*a^3*b^7*d^6*e^4 + 8085*A*a^4*b^6*d^5*e^5 - 2625*A*a^5*b^5*d^4*e^6 + 210*A*a^6*b^4*

d^3*e^7 + 30*A*a^7*b^3*d^2*e^8 - (71955*B*a^2*b^8*d^8*e^2)/4 + 25520*B*a^3*b^7*d^7*e^3 - (41895*B*a^4*b^6*d^6*

e^4)/2 + 9702*B*a^5*b^5*d^5*e^5 - (4375*B*a^6*b^4*d^4*e^6)/2 + 120*B*a^7*b^3*d^3*e^7 + (45*B*a^8*b^2*d^2*e^8)/

4 + 6875*B*a*b^9*d^9*e + (5*B*a^9*b*d*e^9)/3) + x^2*((10*B*a^9*b*e^10)/3 - (10505*B*b^10*d^9*e)/6 + 15*A*a^8*b

^2*e^10 + 1095*A*b^10*d^8*e^2 - 6420*A*a*b^9*d^7*e^3 + 60*A*a^7*b^3*d*e^9 + 10950*B*a*b^9*d^8*e^2 + (45*B*a^8*

b^2*d*e^9)/2 + 15540*A*a^2*b^8*d^6*e^4 - 19740*A*a^3*b^7*d^5*e^5 + 13650*A*a^4*b^6*d^4*e^6 - 4620*A*a^5*b^5*d^

3*e^7 + 420*A*a^6*b^4*d^2*e^8 - 28890*B*a^2*b^8*d^7*e^3 + 41440*B*a^3*b^7*d^6*e^4 - 34545*B*a^4*b^6*d^5*e^5 +

16380*B*a^5*b^5*d^4*e^6 - 3850*B*a^6*b^4*d^3*e^7 + 240*B*a^7*b^3*d^2*e^8))/(d^5*e^11 + e^16*x^5 + 5*d^4*e^12*x

+ 5*d*e^15*x^4 + 10*d^3*e^13*x^2 + 10*d^2*e^14*x^3) - x^3*((5*d^2*((A*b^10 + 10*B*a*b^9)/e^6 - (6*B*b^10*d)/e

^7))/e^2 - (2*d*((6*d*((A*b^10 + 10*B*a*b^9)/e^6 - (6*B*b^10*d)/e^7))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^6 + (15*

B*b^10*d^2)/e^8))/e - (5*a^2*b^7*(3*A*b + 8*B*a))/e^6 + (20*B*b^10*d^3)/(3*e^9)) - x*((6*d*((6*d*((15*d^2*((A*

b^10 + 10*B*a*b^9)/e^6 - (6*B*b^10*d)/e^7))/e^2 - (6*d*((6*d*((A*b^10 + 10*B*a*b^9)/e^6 - (6*B*b^10*d)/e^7))/e

- (5*a*b^8*(2*A*b + 9*B*a))/e^6 + (15*B*b^10*d^2)/e^8))/e - (15*a^2*b^7*(3*A*b + 8*B*a))/e^6 + (20*B*b^10*d^3

)/e^9))/e - (20*d^3*((A*b^10 + 10*B*a*b^9)/e^6 - (6*B*b^10*d)/e^7))/e^3 + (15*d^2*((6*d*((A*b^10 + 10*B*a*b^9)

/e^6 - (6*B*b^10*d)/e^7))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^6 + (15*B*b^10*d^2)/e^8))/e^2 + (30*a^3*b^6*(4*A*b +

7*B*a))/e^6 - (15*B*b^10*d^4)/e^10))/e - (15*d^2*((15*d^2*((A*b^10 + 10*B*a*b^9)/e^6 - (6*B*b^10*d)/e^7))/e^2

- (6*d*((6*d*((A*b^10 + 10*B*a*b^9)/e^6 - (6*B*b^10*d)/e^7))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^6 + (15*B*b^10*d

^2)/e^8))/e - (15*a^2*b^7*(3*A*b + 8*B*a))/e^6 + (20*B*b^10*d^3)/e^9))/e^2 + (15*d^4*((A*b^10 + 10*B*a*b^9)/e^

6 - (6*B*b^10*d)/e^7))/e^4 - (20*d^3*((6*d*((A*b^10 + 10*B*a*b^9)/e^6 - (6*B*b^10*d)/e^7))/e - (5*a*b^8*(2*A*b

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

*d*((15*d^2*((A*b^10 + 10*B*a*b^9)/e^6 - (6*B*b^10*d)/e^7))/e^2 - (6*d*((6*d*((A*b^10 + 10*B*a*b^9)/e^6 - (6*B

*b^10*d)/e^7))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^6 + (15*B*b^10*d^2)/e^8))/e - (15*a^2*b^7*(3*A*b + 8*B*a))/e^6

+ (20*B*b^10*d^3)/e^9))/e - (10*d^3*((A*b^10 + 10*B*a*b^9)/e^6 - (6*B*b^10*d)/e^7))/e^3 + (15*d^2*((6*d*((A*b^

10 + 10*B*a*b^9)/e^6 - (6*B*b^10*d)/e^7))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^6 + (15*B*b^10*d^2)/e^8))/(2*e^2) +

(15*a^3*b^6*(4*A*b + 7*B*a))/e^6 - (15*B*b^10*d^4)/(2*e^10)) + (log(d + e*x)*(462*B*b^10*d^6 - 252*A*b^10*d^5*

e + 252*A*a^5*b^5*e^6 + 210*B*a^6*b^4*e^6 + 1260*A*a*b^9*d^4*e^2 - 1260*A*a^4*b^6*d*e^5 - 1512*B*a^5*b^5*d*e^5

- 2520*A*a^2*b^8*d^3*e^3 + 2520*A*a^3*b^7*d^2*e^4 + 5670*B*a^2*b^8*d^4*e^2 - 6720*B*a^3*b^7*d^3*e^3 + 4410*B*

a^4*b^6*d^2*e^4 - 2520*B*a*b^9*d^5*e))/e^12 + (B*b^10*x^6)/(6*e^6)

\(\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^6} \, dx\) [1094]

Optimal result