3.11.0 ∫ (a+bx)10(A+Bx) (d+ex)8 dx [1096]

Integrand size = 20, antiderivative size = 444 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^8} \, dx=-\frac {15 b^7 (b d-a e)^2 (11 b B d-3 A b e-8 a B e) x}{e^{11}}+\frac {(b d-a e)^{10} (B d-A e)}{7 e^{12} (d+e x)^7}-\frac {(b d-a e)^9 (11 b B d-10 A b e-a B e)}{6 e^{12} (d+e x)^6}+\frac {b (b d-a e)^8 (11 b B d-9 A b e-2 a B e)}{e^{12} (d+e x)^5}-\frac {15 b^2 (b d-a e)^7 (11 b B d-8 A b e-3 a B e)}{4 e^{12} (d+e x)^4}+\frac {10 b^3 (b d-a e)^6 (11 b B d-7 A b e-4 a B e)}{e^{12} (d+e x)^3}-\frac {21 b^4 (b d-a e)^5 (11 b B d-6 A b e-5 a B e)}{e^{12} (d+e x)^2}+\frac {42 b^5 (b d-a e)^4 (11 b B d-5 A b e-6 a B e)}{e^{12} (d+e x)}+\frac {5 b^8 (b d-a e) (11 b B d-2 A b e-9 a B e) (d+e x)^2}{2 e^{12}}-\frac {b^9 (11 b B d-A b e-10 a B e) (d+e x)^3}{3 e^{12}}+\frac {b^{10} B (d+e x)^4}{4 e^{12}}+\frac {30 b^6 (b d-a e)^3 (11 b B d-4 A b e-7 a B e) \log (d+e x)}{e^{12}} \]

-15*b^7*(-a*e+b*d)^2*(-3*A*b*e-8*B*a*e+11*B*b*d)*x/e^11+1/7*(-a*e+b*d)^10*(-A*e+B*d)/e^12/(e*x+d)^7-1/6*(-a*e+

b*d)^9*(-10*A*b*e-B*a*e+11*B*b*d)/e^12/(e*x+d)^6+b*(-a*e+b*d)^8*(-9*A*b*e-2*B*a*e+11*B*b*d)/e^12/(e*x+d)^5-15/

4*b^2*(-a*e+b*d)^7*(-8*A*b*e-3*B*a*e+11*B*b*d)/e^12/(e*x+d)^4+10*b^3*(-a*e+b*d)^6*(-7*A*b*e-4*B*a*e+11*B*b*d)/

e^12/(e*x+d)^3-21*b^4*(-a*e+b*d)^5*(-6*A*b*e-5*B*a*e+11*B*b*d)/e^12/(e*x+d)^2+42*b^5*(-a*e+b*d)^4*(-5*A*b*e-6*

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

B*a*e+11*B*b*d)*(e*x+d)^3/e^12+1/4*b^10*B*(e*x+d)^4/e^12+30*b^6*(-a*e+b*d)^3*(-4*A*b*e-7*B*a*e+11*B*b*d)*ln(e*

Rubi [A] (veriﬁed)

Time = 0.67 (sec) , antiderivative size = 444, 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} \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^8} \, dx=-\frac {b^9 (d+e x)^3 (-10 a B e-A b e+11 b B d)}{3 e^{12}}+\frac {5 b^8 (d+e x)^2 (b d-a e) (-9 a B e-2 A b e+11 b B d)}{2 e^{12}}-\frac {15 b^7 x (b d-a e)^2 (-8 a B e-3 A b e+11 b B d)}{e^{11}}+\frac {30 b^6 (b d-a e)^3 \log (d+e x) (-7 a B e-4 A b e+11 b B d)}{e^{12}}+\frac {42 b^5 (b d-a e)^4 (-6 a B e-5 A b e+11 b B d)}{e^{12} (d+e x)}-\frac {21 b^4 (b d-a e)^5 (-5 a B e-6 A b e+11 b B d)}{e^{12} (d+e x)^2}+\frac {10 b^3 (b d-a e)^6 (-4 a B e-7 A b e+11 b B d)}{e^{12} (d+e x)^3}-\frac {15 b^2 (b d-a e)^7 (-3 a B e-8 A b e+11 b B d)}{4 e^{12} (d+e x)^4}+\frac {b (b d-a e)^8 (-2 a B e-9 A b e+11 b B d)}{e^{12} (d+e x)^5}-\frac {(b d-a e)^9 (-a B e-10 A b e+11 b B d)}{6 e^{12} (d+e x)^6}+\frac {(b d-a e)^{10} (B d-A e)}{7 e^{12} (d+e x)^7}+\frac {b^{10} B (d+e x)^4}{4 e^{12}} \]

(-15*b^7*(b*d - a*e)^2*(11*b*B*d - 3*A*b*e - 8*a*B*e)*x)/e^11 + ((b*d - a*e)^10*(B*d - A*e))/(7*e^12*(d + e*x)

^7) - ((b*d - a*e)^9*(11*b*B*d - 10*A*b*e - a*B*e))/(6*e^12*(d + e*x)^6) + (b*(b*d - a*e)^8*(11*b*B*d - 9*A*b*

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

+ (10*b^3*(b*d - a*e)^6*(11*b*B*d - 7*A*b*e - 4*a*B*e))/(e^12*(d + e*x)^3) - (21*b^4*(b*d - a*e)^5*(11*b*B*d -

6*A*b*e - 5*a*B*e))/(e^12*(d + e*x)^2) + (42*b^5*(b*d - a*e)^4*(11*b*B*d - 5*A*b*e - 6*a*B*e))/(e^12*(d + e*x

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

*e)*(d + e*x)^3)/(3*e^12) + (b^10*B*(d + e*x)^4)/(4*e^12) + (30*b^6*(b*d - a*e)^3*(11*b*B*d - 4*A*b*e - 7*a*B*

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

Mathematica [A] (veriﬁed)

Time = 0.33 (sec) , antiderivative size = 450, normalized size of antiderivative = 1.01 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^8} \, dx=\frac {84 b^7 e \left (120 a^3 B e^3+40 a b^2 d e (9 B d-2 A e)+45 a^2 b e^2 (-8 B d+A e)+12 b^3 d^2 (-10 B d+3 A e)\right ) x-42 b^8 e^2 \left (-45 a^2 B e^2-10 a b e (-8 B d+A e)+4 b^2 d (-9 B d+2 A e)\right ) x^2+28 b^9 e^3 (-8 b B d+A b e+10 a B e) x^3+21 b^{10} B e^4 x^4+\frac {12 (b d-a e)^{10} (B d-A e)}{(d+e x)^7}-\frac {14 (b d-a e)^9 (11 b B d-10 A b e-a B e)}{(d+e x)^6}+\frac {84 b (b d-a e)^8 (11 b B d-9 A b e-2 a B e)}{(d+e x)^5}-\frac {315 b^2 (b d-a e)^7 (11 b B d-8 A b e-3 a B e)}{(d+e x)^4}+\frac {840 b^3 (b d-a e)^6 (11 b B d-7 A b e-4 a B e)}{(d+e x)^3}-\frac {1764 b^4 (b d-a e)^5 (11 b B d-6 A b e-5 a B e)}{(d+e x)^2}+\frac {3528 b^5 (b d-a e)^4 (11 b B d-5 A b e-6 a B e)}{d+e x}+2520 b^6 (b d-a e)^3 (11 b B d-4 A b e-7 a B e) \log (d+e x)}{84 e^{12}} \]

(84*b^7*e*(120*a^3*B*e^3 + 40*a*b^2*d*e*(9*B*d - 2*A*e) + 45*a^2*b*e^2*(-8*B*d + A*e) + 12*b^3*d^2*(-10*B*d +

3*A*e))*x - 42*b^8*e^2*(-45*a^2*B*e^2 - 10*a*b*e*(-8*B*d + A*e) + 4*b^2*d*(-9*B*d + 2*A*e))*x^2 + 28*b^9*e^3*(

-8*b*B*d + A*b*e + 10*a*B*e)*x^3 + 21*b^10*B*e^4*x^4 + (12*(b*d - a*e)^10*(B*d - A*e))/(d + e*x)^7 - (14*(b*d

- a*e)^9*(11*b*B*d - 10*A*b*e - a*B*e))/(d + e*x)^6 + (84*b*(b*d - a*e)^8*(11*b*B*d - 9*A*b*e - 2*a*B*e))/(d +

e*x)^5 - (315*b^2*(b*d - a*e)^7*(11*b*B*d - 8*A*b*e - 3*a*B*e))/(d + e*x)^4 + (840*b^3*(b*d - a*e)^6*(11*b*B*

d - 7*A*b*e - 4*a*B*e))/(d + e*x)^3 - (1764*b^4*(b*d - a*e)^5*(11*b*B*d - 6*A*b*e - 5*a*B*e))/(d + e*x)^2 + (3

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

Maple [B] (veriﬁed)

Leaf count of result is larger than twice the leaf count of optimal. \(1924\) vs. \(2(432)=864\).

Time = 2.12 (sec) , antiderivative size = 1925, normalized size of antiderivative = 4.34


method	result	size

norman	\(\text {Expression too large to display}\)	\(1925\)
default	\(\text {Expression too large to display}\)	\(1953\)
risch	\(\text {Expression too large to display}\)	\(1999\)
parallelrisch	\(\text {Expression too large to display}\)	\(3671\)

int((b*x+a)^10*(B*x+A)/(e*x+d)^8,x,method=_RETURNVERBOSE)

(-1/84*(12*A*a^10*e^11+20*A*a^9*b*d*e^10+36*A*a^8*b^2*d^2*e^9+72*A*a^7*b^3*d^3*e^8+168*A*a^6*b^4*d^4*e^7+504*A

*a^5*b^5*d^5*e^6+2520*A*a^4*b^6*d^6*e^5-26136*A*a^3*b^7*d^7*e^4+78408*A*a^2*b^8*d^8*e^3-78408*A*a*b^9*d^9*e^2+

26136*A*b^10*d^10*e+2*B*a^10*d*e^10+8*B*a^9*b*d^2*e^9+27*B*a^8*b^2*d^3*e^8+96*B*a^7*b^3*d^4*e^7+420*B*a^6*b^4*

d^5*e^6+3024*B*a^5*b^5*d^6*e^5-45738*B*a^4*b^6*d^7*e^4+209088*B*a^3*b^7*d^8*e^3-352836*B*a^2*b^8*d^9*e^2+26136

0*B*a*b^9*d^10*e-71874*B*b^10*d^11)/e^12-7*(30*A*a^4*b^6*e^5-120*A*a^3*b^7*d*e^4+360*A*a^2*b^8*d^2*e^3-360*A*a

*b^9*d^3*e^2+120*A*b^10*d^4*e+36*B*a^5*b^5*e^5-210*B*a^4*b^6*d*e^4+960*B*a^3*b^7*d^2*e^3-1620*B*a^2*b^8*d^3*e^

2+1200*B*a*b^9*d^4*e-330*B*b^10*d^5)/e^6*x^6-21*(6*A*a^5*b^5*e^6+30*A*a^4*b^6*d*e^5-180*A*a^3*b^7*d^2*e^4+540*

A*a^2*b^8*d^3*e^3-540*A*a*b^9*d^4*e^2+180*A*b^10*d^5*e+5*B*a^6*b^4*e^6+36*B*a^5*b^5*d*e^5-315*B*a^4*b^6*d^2*e^

4+1440*B*a^3*b^7*d^3*e^3-2430*B*a^2*b^8*d^4*e^2+1800*B*a*b^9*d^5*e-495*B*b^10*d^6)/e^7*x^5-5*(14*A*a^6*b^4*e^7

+42*A*a^5*b^5*d*e^6+210*A*a^4*b^6*d^2*e^5-1540*A*a^3*b^7*d^3*e^4+4620*A*a^2*b^8*d^4*e^3-4620*A*a*b^9*d^5*e^2+1

540*A*b^10*d^6*e+8*B*a^7*b^3*e^7+35*B*a^6*b^4*d*e^6+252*B*a^5*b^5*d^2*e^5-2695*B*a^4*b^6*d^3*e^4+12320*B*a^3*b

^7*d^4*e^3-20790*B*a^2*b^8*d^5*e^2+15400*B*a*b^9*d^6*e-4235*B*b^10*d^7)/e^8*x^4-5/4*(24*A*a^7*b^3*e^8+56*A*a^6

*b^4*d*e^7+168*A*a^5*b^5*d^2*e^6+840*A*a^4*b^6*d^3*e^5-7000*A*a^3*b^7*d^4*e^4+21000*A*a^2*b^8*d^5*e^3-21000*A*

a*b^9*d^6*e^2+7000*A*b^10*d^7*e+9*B*a^8*b^2*e^8+32*B*a^7*b^3*d*e^7+140*B*a^6*b^4*d^2*e^6+1008*B*a^5*b^5*d^3*e^

5-12250*B*a^4*b^6*d^4*e^4+56000*B*a^3*b^7*d^5*e^3-94500*B*a^2*b^8*d^6*e^2+70000*B*a*b^9*d^7*e-19250*B*b^10*d^8

)/e^9*x^3-1/4*(36*A*a^8*b^2*e^9+72*A*a^7*b^3*d*e^8+168*A*a^6*b^4*d^2*e^7+504*A*a^5*b^5*d^3*e^6+2520*A*a^4*b^6*

d^4*e^5-23016*A*a^3*b^7*d^5*e^4+69048*A*a^2*b^8*d^6*e^3-69048*A*a*b^9*d^7*e^2+23016*A*b^10*d^8*e+8*B*a^9*b*e^9

+27*B*a^8*b^2*d*e^8+96*B*a^7*b^3*d^2*e^7+420*B*a^6*b^4*d^3*e^6+3024*B*a^5*b^5*d^4*e^5-40278*B*a^4*b^6*d^5*e^4+

184128*B*a^3*b^7*d^6*e^3-310716*B*a^2*b^8*d^7*e^2+230160*B*a*b^9*d^8*e-63294*B*b^10*d^9)/e^10*x^2-1/12*(20*A*a

^9*b*e^10+36*A*a^8*b^2*d*e^9+72*A*a^7*b^3*d^2*e^8+168*A*a^6*b^4*d^3*e^7+504*A*a^5*b^5*d^4*e^6+2520*A*a^4*b^6*d

^5*e^5-24696*A*a^3*b^7*d^6*e^4+74088*A*a^2*b^8*d^7*e^3-74088*A*a*b^9*d^8*e^2+24696*A*b^10*d^9*e+2*B*a^10*e^10+

8*B*a^9*b*d*e^9+27*B*a^8*b^2*d^2*e^8+96*B*a^7*b^3*d^3*e^7+420*B*a^6*b^4*d^4*e^6+3024*B*a^5*b^5*d^5*e^5-43218*B

*a^4*b^6*d^6*e^4+197568*B*a^3*b^7*d^7*e^3-333396*B*a^2*b^8*d^8*e^2+246960*B*a*b^9*d^9*e-67914*B*b^10*d^10)/e^1

1*x+15/4*b^7*(12*A*a^2*b*e^3-12*A*a*b^2*d*e^2+4*A*b^3*d^2*e+32*B*a^3*e^3-54*B*a^2*b*d*e^2+40*B*a*b^2*d^2*e-11*

B*b^3*d^3)/e^4*x^8+5/12*b^8*(12*A*a*b*e^2-4*A*b^2*d*e+54*B*a^2*e^2-40*B*a*b*d*e+11*B*b^2*d^2)/e^3*x^9+1/12*b^9

*(4*A*b*e+40*B*a*e-11*B*b*d)/e^2*x^10+1/4*b^10*B/e*x^11)/(e*x+d)^7+30*b^6/e^12*(4*A*a^3*b*e^4-12*A*a^2*b^2*d*e

^3+12*A*a*b^3*d^2*e^2-4*A*b^4*d^3*e+7*B*a^4*e^4-32*B*a^3*b*d*e^3+54*B*a^2*b^2*d^2*e^2-40*B*a*b^3*d^3*e+11*B*b^

Fricas [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2783 vs. \(2 (432) = 864\).

Time = 0.31 (sec) , antiderivative size = 2783, normalized size of antiderivative = 6.27 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^8} \, dx=\text {Too large to display} \]

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^8,x, algorithm="fricas")

1/84*(21*B*b^10*e^11*x^11 + 25961*B*b^10*d^11 - 12*A*a^10*e^11 - 11044*(10*B*a*b^9 + A*b^10)*d^10*e + 20094*(9

*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 - 17316*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 6534*(7*B*a^4*b^6 + 4*A*a^3*b^7)

*d^7*e^4 - 504*(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 - 24*(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 - 4*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9 - 2*(B*a^

10 + 10*A*a^9*b)*d*e^10 - 7*(11*B*b^10*d*e^10 - 4*(10*B*a*b^9 + A*b^10)*e^11)*x^10 + 35*(11*B*b^10*d^2*e^9 - 4

*(10*B*a*b^9 + A*b^10)*d*e^10 + 6*(9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9 - 315*(11*B*b^10*d^3*e^8 - 4*(10*B*a*b^9

+ A*b^10)*d^2*e^9 + 6*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^10 - 4*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^11)*x^8 - 49*(937*B*

b^10*d^4*e^7 - 308*(10*B*a*b^9 + A*b^10)*d^3*e^8 + 390*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^9 - 180*(8*B*a^3*b^7 +

3*A*a^2*b^8)*d*e^10)*x^7 - 49*(2599*B*b^10*d^5*e^6 - 716*(10*B*a*b^9 + A*b^10)*d^4*e^7 + 570*(9*B*a^2*b^8 + 2*

A*a*b^9)*d^3*e^8 + 180*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^9 - 360*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^10 + 72*(6*B*

a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 - 147*(619*B*b^10*d^6*e^5 + 4*(10*B*a*b^9 + A*b^10)*d^5*e^6 - 510*(9*B*a^2*b^

8 + 2*A*a*b^9)*d^4*e^7 + 900*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8 - 540*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^9 + 7

2*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 + 12*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^11)*x^5 + 35*(4907*B*b^10*d^7*e^4 - 33

88*(10*B*a*b^9 + A*b^10)*d^6*e^5 + 8610*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^6 - 9660*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d

^4*e^7 + 4620*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^8 - 504*(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 - 24*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^11)*x^4 + 35*(11837*B*b^10*d^8*e^3 - 5908*(10*B*a*b^9

+ A*b^10)*d^7*e^4 + 12390*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5 - 12180*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^6 + 525

0*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^7 - 504*(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 - 24*(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 + 21*(17381*B*b^1

0*d^9*e^2 - 7924*(10*B*a*b^9 + A*b^10)*d^8*e^3 + 15414*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^4 - 14196*(8*B*a^3*b^7

+ 3*A*a^2*b^8)*d^6*e^5 + 5754*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*e^6 - 504*(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 - 24*(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 - 4*(2*B*a^9*b + 9*A*a^8*b^2)*e^11)*x^2 + 7*(22001*B*b^10*d^10*e - 9604*(10*B*a*b^9 + A*b^10)*d^9*e^2

+ 17934*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^3 - 15876*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4 + 6174*(7*B*a^4*b^6 + 4

*A*a^3*b^7)*d^6*e^5 - 504*(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 - 24*(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 - 4*(2*B*a^9*b + 9*A*a^8*b^2)*d*e^10

- 2*(B*a^10 + 10*A*a^9*b)*e^11)*x + 2520*(11*B*b^10*d^11 - 4*(10*B*a*b^9 + A*b^10)*d^10*e + 6*(9*B*a^2*b^8 +

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

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

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

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

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

*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8 + (7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^9)*x^5 + 35*(11*B*b^10*d^7*e^4 - 4*(10*B*a

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

4*b^6 + 4*A*a^3*b^7)*d^3*e^8)*x^4 + 35*(11*B*b^10*d^8*e^3 - 4*(10*B*a*b^9 + A*b^10)*d^7*e^4 + 6*(9*B*a^2*b^8 +

2*A*a*b^9)*d^6*e^5 - 4*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^6 + (7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^7)*x^3 + 21*(1

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

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

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

*b^7)*d^6*e^5)*x)*log(e*x + d))/(e^19*x^7 + 7*d*e^18*x^6 + 21*d^2*e^17*x^5 + 35*d^3*e^16*x^4 + 35*d^4*e^15*x^3

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^8} \, dx=\text {Timed out} \]

Maxima [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1883 vs. \(2 (432) = 864\).

Time = 0.35 (sec) , antiderivative size = 1883, normalized size of antiderivative = 4.24 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^8} \, dx=\text {Too large to display} \]

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^8,x, algorithm="maxima")

1/84*(25961*B*b^10*d^11 - 12*A*a^10*e^11 - 11044*(10*B*a*b^9 + A*b^10)*d^10*e + 20094*(9*B*a^2*b^8 + 2*A*a*b^9

)*d^9*e^2 - 17316*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 6534*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4 - 504*(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 - 24*(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 - 4*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9 - 2*(B*a^10 + 10*A*a^9*b)*d*e^1

0 + 3528*(11*B*b^10*d^5*e^6 - 5*(10*B*a*b^9 + A*b^10)*d^4*e^7 + 10*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^8 - 10*(8*B

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

+ 1764*(121*B*b^10*d^6*e^5 - 54*(10*B*a*b^9 + A*b^10)*d^5*e^6 + 105*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 - 100*(

8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8 + 45*(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 + 420*(1177*B*b^10*d^7*e^4 - 518*(10*B*a*b^9 + A*b^10)*d^6*e^5 +

987*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^6 - 910*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 + 385*(7*B*a^4*b^6 + 4*A*a^3*b

^7)*d^3*e^8 - 42*(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 - 2*(4*B*a^7*b^3 +

7*A*a^6*b^4)*e^11)*x^4 + 105*(5863*B*b^10*d^8*e^3 - 2552*(10*B*a*b^9 + A*b^10)*d^7*e^4 + 4788*(9*B*a^2*b^8 +

2*A*a*b^9)*d^6*e^5 - 4312*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^6 + 1750*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^7 - 168

*(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*(3*B*a^8*b^2 + 8*A*a^7*b^3)*e^11)*x^3 + 21*(20669*B*b^10*d^9*e^2 - 8916*(10*B*a*b^9 + A*b^10)*d^8*e

^3 + 16524*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^4 - 14616*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^5 + 5754*(7*B*a^4*b^6 +

4*A*a^3*b^7)*d^5*e^6 - 504*(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 - 24*

(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 - 4*(2*B*a^9*b + 9*A*a^8*b^2)*e^11)

*x^2 + 7*(23441*B*b^10*d^10*e - 10036*(10*B*a*b^9 + A*b^10)*d^9*e^2 + 18414*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^3

- 16056*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4 + 6174*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5 - 504*(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 - 24*(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 - 4*(2*B*a^9*b + 9*A*a^8*b^2)*d*e^10 - 2*(B*a^10 + 10*A*a^9*b)*e^11)*x)/(e^19*x

^7 + 7*d*e^18*x^6 + 21*d^2*e^17*x^5 + 35*d^3*e^16*x^4 + 35*d^4*e^15*x^3 + 21*d^5*e^14*x^2 + 7*d^6*e^13*x + d^7

*e^12) + 1/12*(3*B*b^10*e^3*x^4 - 4*(8*B*b^10*d*e^2 - (10*B*a*b^9 + A*b^10)*e^3)*x^3 + 6*(36*B*b^10*d^2*e - 8*

(10*B*a*b^9 + A*b^10)*d*e^2 + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*e^3)*x^2 - 12*(120*B*b^10*d^3 - 36*(10*B*a*b^9 + A*b

^10)*d^2*e + 40*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^2 - 15*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^3)*x)/e^11 + 30*(11*B*b^10*

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

3 + (7*B*a^4*b^6 + 4*A*a^3*b^7)*e^4)*log(e*x + d)/e^12

Giac [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1992 vs. \(2 (432) = 864\).

Time = 0.30 (sec) , antiderivative size = 1992, normalized size of antiderivative = 4.49 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^8} \, dx=\text {Too large to display} \]

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^8,x, algorithm="giac")

30*(11*B*b^10*d^4 - 40*B*a*b^9*d^3*e - 4*A*b^10*d^3*e + 54*B*a^2*b^8*d^2*e^2 + 12*A*a*b^9*d^2*e^2 - 32*B*a^3*b

^7*d*e^3 - 12*A*a^2*b^8*d*e^3 + 7*B*a^4*b^6*e^4 + 4*A*a^3*b^7*e^4)*log(abs(e*x + d))/e^12 + 1/84*(25961*B*b^10

*d^11 - 110440*B*a*b^9*d^10*e - 11044*A*b^10*d^10*e + 180846*B*a^2*b^8*d^9*e^2 + 40188*A*a*b^9*d^9*e^2 - 13852

8*B*a^3*b^7*d^8*e^3 - 51948*A*a^2*b^8*d^8*e^3 + 45738*B*a^4*b^6*d^7*e^4 + 26136*A*a^3*b^7*d^7*e^4 - 3024*B*a^5

*b^5*d^6*e^5 - 2520*A*a^4*b^6*d^6*e^5 - 420*B*a^6*b^4*d^5*e^6 - 504*A*a^5*b^5*d^5*e^6 - 96*B*a^7*b^3*d^4*e^7 -

168*A*a^6*b^4*d^4*e^7 - 27*B*a^8*b^2*d^3*e^8 - 72*A*a^7*b^3*d^3*e^8 - 8*B*a^9*b*d^2*e^9 - 36*A*a^8*b^2*d^2*e^

9 - 2*B*a^10*d*e^10 - 20*A*a^9*b*d*e^10 - 12*A*a^10*e^11 + 3528*(11*B*b^10*d^5*e^6 - 50*B*a*b^9*d^4*e^7 - 5*A*

b^10*d^4*e^7 + 90*B*a^2*b^8*d^3*e^8 + 20*A*a*b^9*d^3*e^8 - 80*B*a^3*b^7*d^2*e^9 - 30*A*a^2*b^8*d^2*e^9 + 35*B*

a^4*b^6*d*e^10 + 20*A*a^3*b^7*d*e^10 - 6*B*a^5*b^5*e^11 - 5*A*a^4*b^6*e^11)*x^6 + 1764*(121*B*b^10*d^6*e^5 - 5

40*B*a*b^9*d^5*e^6 - 54*A*b^10*d^5*e^6 + 945*B*a^2*b^8*d^4*e^7 + 210*A*a*b^9*d^4*e^7 - 800*B*a^3*b^7*d^3*e^8 -

300*A*a^2*b^8*d^3*e^8 + 315*B*a^4*b^6*d^2*e^9 + 180*A*a^3*b^7*d^2*e^9 - 36*B*a^5*b^5*d*e^10 - 30*A*a^4*b^6*d*

e^10 - 5*B*a^6*b^4*e^11 - 6*A*a^5*b^5*e^11)*x^5 + 420*(1177*B*b^10*d^7*e^4 - 5180*B*a*b^9*d^6*e^5 - 518*A*b^10

*d^6*e^5 + 8883*B*a^2*b^8*d^5*e^6 + 1974*A*a*b^9*d^5*e^6 - 7280*B*a^3*b^7*d^4*e^7 - 2730*A*a^2*b^8*d^4*e^7 + 2

695*B*a^4*b^6*d^3*e^8 + 1540*A*a^3*b^7*d^3*e^8 - 252*B*a^5*b^5*d^2*e^9 - 210*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 - 8*B*a^7*b^3*e^11 - 14*A*a^6*b^4*e^11)*x^4 + 105*(5863*B*b^10*d^8*e^3 - 25520*B*

a*b^9*d^7*e^4 - 2552*A*b^10*d^7*e^4 + 43092*B*a^2*b^8*d^6*e^5 + 9576*A*a*b^9*d^6*e^5 - 34496*B*a^3*b^7*d^5*e^6

- 12936*A*a^2*b^8*d^5*e^6 + 12250*B*a^4*b^6*d^4*e^7 + 7000*A*a^3*b^7*d^4*e^7 - 1008*B*a^5*b^5*d^3*e^8 - 840*A

*a^4*b^6*d^3*e^8 - 140*B*a^6*b^4*d^2*e^9 - 168*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 -

9*B*a^8*b^2*e^11 - 24*A*a^7*b^3*e^11)*x^3 + 21*(20669*B*b^10*d^9*e^2 - 89160*B*a*b^9*d^8*e^3 - 8916*A*b^10*d^

8*e^3 + 148716*B*a^2*b^8*d^7*e^4 + 33048*A*a*b^9*d^7*e^4 - 116928*B*a^3*b^7*d^6*e^5 - 43848*A*a^2*b^8*d^6*e^5

+ 40278*B*a^4*b^6*d^5*e^6 + 23016*A*a^3*b^7*d^5*e^6 - 3024*B*a^5*b^5*d^4*e^7 - 2520*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 - 96*B*a^7*b^3*d^2*e^9 - 168*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 - 8*B*a^9*b*e^11 - 36*A*a^8*b^2*e^11)*x^2 + 7*(23441*B*b^10*d^10*e - 100360*B*a*b^9*d^9*e

^2 - 10036*A*b^10*d^9*e^2 + 165726*B*a^2*b^8*d^8*e^3 + 36828*A*a*b^9*d^8*e^3 - 128448*B*a^3*b^7*d^7*e^4 - 4816

8*A*a^2*b^8*d^7*e^4 + 43218*B*a^4*b^6*d^6*e^5 + 24696*A*a^3*b^7*d^6*e^5 - 3024*B*a^5*b^5*d^5*e^6 - 2520*A*a^4*

b^6*d^5*e^6 - 420*B*a^6*b^4*d^4*e^7 - 504*A*a^5*b^5*d^4*e^7 - 96*B*a^7*b^3*d^3*e^8 - 168*A*a^6*b^4*d^3*e^8 - 2

7*B*a^8*b^2*d^2*e^9 - 72*A*a^7*b^3*d^2*e^9 - 8*B*a^9*b*d*e^10 - 36*A*a^8*b^2*d*e^10 - 2*B*a^10*e^11 - 20*A*a^9

*b*e^11)*x)/((e*x + d)^7*e^12) + 1/12*(3*B*b^10*e^24*x^4 - 32*B*b^10*d*e^23*x^3 + 40*B*a*b^9*e^24*x^3 + 4*A*b^

10*e^24*x^3 + 216*B*b^10*d^2*e^22*x^2 - 480*B*a*b^9*d*e^23*x^2 - 48*A*b^10*d*e^23*x^2 + 270*B*a^2*b^8*e^24*x^2

+ 60*A*a*b^9*e^24*x^2 - 1440*B*b^10*d^3*e^21*x + 4320*B*a*b^9*d^2*e^22*x + 432*A*b^10*d^2*e^22*x - 4320*B*a^2

*b^8*d*e^23*x - 960*A*a*b^9*d*e^23*x + 1440*B*a^3*b^7*e^24*x + 540*A*a^2*b^8*e^24*x)/e^32

Mupad [B] (veriﬁcation not implemented)

Time = 1.85 (sec) , antiderivative size = 2092, normalized size of antiderivative = 4.71 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^8} \, dx=\text {Too large to display} \]

x^3*((A*b^10 + 10*B*a*b^9)/(3*e^8) - (8*B*b^10*d)/(3*e^9)) - x*((28*d^2*((A*b^10 + 10*B*a*b^9)/e^8 - (8*B*b^10

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

(28*B*b^10*d^2)/e^10))/e - (15*a^2*b^7*(3*A*b + 8*B*a))/e^8 + (56*B*b^10*d^3)/e^11) - x^2*((4*d*((A*b^10 + 10

*B*a*b^9)/e^8 - (8*B*b^10*d)/e^9))/e - (5*a*b^8*(2*A*b + 9*B*a))/(2*e^8) + (14*B*b^10*d^2)/e^10) - (x^4*(70*A*

a^6*b^4*e^10 + 40*B*a^7*b^3*e^10 + 2590*A*b^10*d^6*e^4 - 5885*B*b^10*d^7*e^3 - 9870*A*a*b^9*d^5*e^5 + 210*A*a^

5*b^5*d*e^9 + 25900*B*a*b^9*d^6*e^4 + 175*B*a^6*b^4*d*e^9 + 13650*A*a^2*b^8*d^4*e^6 - 7700*A*a^3*b^7*d^3*e^7 +

1050*A*a^4*b^6*d^2*e^8 - 44415*B*a^2*b^8*d^5*e^5 + 36400*B*a^3*b^7*d^4*e^6 - 13475*B*a^4*b^6*d^3*e^7 + 1260*B

*a^5*b^5*d^2*e^8) + x^6*(210*A*a^4*b^6*e^10 + 252*B*a^5*b^5*e^10 + 210*A*b^10*d^4*e^6 - 462*B*b^10*d^5*e^5 - 8

40*A*a*b^9*d^3*e^7 - 840*A*a^3*b^7*d*e^9 + 2100*B*a*b^9*d^4*e^6 - 1470*B*a^4*b^6*d*e^9 + 1260*A*a^2*b^8*d^2*e^

8 - 3780*B*a^2*b^8*d^3*e^7 + 3360*B*a^3*b^7*d^2*e^8) + x^3*(30*A*a^7*b^3*e^10 + (45*B*a^8*b^2*e^10)/4 + 3190*A

*b^10*d^7*e^3 - (29315*B*b^10*d^8*e^2)/4 - 11970*A*a*b^9*d^6*e^4 + 70*A*a^6*b^4*d*e^9 + 31900*B*a*b^9*d^7*e^3

+ 40*B*a^7*b^3*d*e^9 + 16170*A*a^2*b^8*d^5*e^5 - 8750*A*a^3*b^7*d^4*e^6 + 1050*A*a^4*b^6*d^3*e^7 + 210*A*a^5*b

^5*d^2*e^8 - 53865*B*a^2*b^8*d^6*e^4 + 43120*B*a^3*b^7*d^5*e^5 - (30625*B*a^4*b^6*d^4*e^6)/2 + 1260*B*a^5*b^5*

d^3*e^7 + 175*B*a^6*b^4*d^2*e^8) + (12*A*a^10*e^11 - 25961*B*b^10*d^11 + 11044*A*b^10*d^10*e + 2*B*a^10*d*e^10

- 40188*A*a*b^9*d^9*e^2 + 8*B*a^9*b*d^2*e^9 + 51948*A*a^2*b^8*d^8*e^3 - 26136*A*a^3*b^7*d^7*e^4 + 2520*A*a^4*

b^6*d^6*e^5 + 504*A*a^5*b^5*d^5*e^6 + 168*A*a^6*b^4*d^4*e^7 + 72*A*a^7*b^3*d^3*e^8 + 36*A*a^8*b^2*d^2*e^9 - 18

0846*B*a^2*b^8*d^9*e^2 + 138528*B*a^3*b^7*d^8*e^3 - 45738*B*a^4*b^6*d^7*e^4 + 3024*B*a^5*b^5*d^6*e^5 + 420*B*a

^6*b^4*d^5*e^6 + 96*B*a^7*b^3*d^4*e^7 + 27*B*a^8*b^2*d^3*e^8 + 20*A*a^9*b*d*e^10 + 110440*B*a*b^9*d^10*e)/(84*

e) + x*((B*a^10*e^10)/6 - (23441*B*b^10*d^10)/12 + (5*A*a^9*b*e^10)/3 + (2509*A*b^10*d^9*e)/3 - 3069*A*a*b^9*d

^8*e^2 + 3*A*a^8*b^2*d*e^9 + 4014*A*a^2*b^8*d^7*e^3 - 2058*A*a^3*b^7*d^6*e^4 + 210*A*a^4*b^6*d^5*e^5 + 42*A*a^

5*b^5*d^4*e^6 + 14*A*a^6*b^4*d^3*e^7 + 6*A*a^7*b^3*d^2*e^8 - (27621*B*a^2*b^8*d^8*e^2)/2 + 10704*B*a^3*b^7*d^7

*e^3 - (7203*B*a^4*b^6*d^6*e^4)/2 + 252*B*a^5*b^5*d^5*e^5 + 35*B*a^6*b^4*d^4*e^6 + 8*B*a^7*b^3*d^3*e^7 + (9*B*

a^8*b^2*d^2*e^8)/4 + (25090*B*a*b^9*d^9*e)/3 + (2*B*a^9*b*d*e^9)/3) + x^5*(126*A*a^5*b^5*e^10 + 105*B*a^6*b^4*

e^10 + 1134*A*b^10*d^5*e^5 - 2541*B*b^10*d^6*e^4 - 4410*A*a*b^9*d^4*e^6 + 630*A*a^4*b^6*d*e^9 + 11340*B*a*b^9*

d^5*e^5 + 756*B*a^5*b^5*d*e^9 + 6300*A*a^2*b^8*d^3*e^7 - 3780*A*a^3*b^7*d^2*e^8 - 19845*B*a^2*b^8*d^4*e^6 + 16

800*B*a^3*b^7*d^3*e^7 - 6615*B*a^4*b^6*d^2*e^8) + x^2*(2*B*a^9*b*e^10 - (20669*B*b^10*d^9*e)/4 + 9*A*a^8*b^2*e

^10 + 2229*A*b^10*d^8*e^2 - 8262*A*a*b^9*d^7*e^3 + 18*A*a^7*b^3*d*e^9 + 22290*B*a*b^9*d^8*e^2 + (27*B*a^8*b^2*

d*e^9)/4 + 10962*A*a^2*b^8*d^6*e^4 - 5754*A*a^3*b^7*d^5*e^5 + 630*A*a^4*b^6*d^4*e^6 + 126*A*a^5*b^5*d^3*e^7 +

42*A*a^6*b^4*d^2*e^8 - 37179*B*a^2*b^8*d^7*e^3 + 29232*B*a^3*b^7*d^6*e^4 - (20139*B*a^4*b^6*d^5*e^5)/2 + 756*B

*a^5*b^5*d^4*e^6 + 105*B*a^6*b^4*d^3*e^7 + 24*B*a^7*b^3*d^2*e^8))/(d^7*e^11 + e^18*x^7 + 7*d^6*e^12*x + 7*d*e^

17*x^6 + 21*d^5*e^13*x^2 + 35*d^4*e^14*x^3 + 35*d^3*e^15*x^4 + 21*d^2*e^16*x^5) + (log(d + e*x)*(330*B*b^10*d^

4 - 120*A*b^10*d^3*e + 120*A*a^3*b^7*e^4 + 210*B*a^4*b^6*e^4 + 360*A*a*b^9*d^2*e^2 - 360*A*a^2*b^8*d*e^3 - 960

*B*a^3*b^7*d*e^3 + 1620*B*a^2*b^8*d^2*e^2 - 1200*B*a*b^9*d^3*e))/e^12 + (B*b^10*x^4)/(4*e^8)

\(\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^8} \, dx\) [1096]

Optimal result