3.11.0 ∫ (a+bx)10(A+Bx) (d+ex)9 dx [1097]

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

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

^9*(-10*A*b*e-B*a*e+11*B*b*d)/e^12/(e*x+d)^7+5/6*b*(-a*e+b*d)^8*(-9*A*b*e-2*B*a*e+11*B*b*d)/e^12/(e*x+d)^6-3*b

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

^12/(e*x+d)^4-14*b^4*(-a*e+b*d)^5*(-6*A*b*e-5*B*a*e+11*B*b*d)/e^12/(e*x+d)^3+21*b^5*(-a*e+b*d)^4*(-5*A*b*e-6*B

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

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

Rubi [A] (veriﬁed)

Time = 0.62 (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} \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^9} \, dx=-\frac {b^9 (d+e x)^2 (-10 a B e-A b e+11 b B d)}{2 e^{12}}+\frac {5 b^8 x (b d-a e) (-9 a B e-2 A b e+11 b B d)}{e^{11}}-\frac {15 b^7 (b d-a e)^2 \log (d+e x) (-8 a B e-3 A b e+11 b B d)}{e^{12}}-\frac {30 b^6 (b d-a e)^3 (-7 a B e-4 A b e+11 b B d)}{e^{12} (d+e x)}+\frac {21 b^5 (b d-a e)^4 (-6 a B e-5 A b e+11 b B d)}{e^{12} (d+e x)^2}-\frac {14 b^4 (b d-a e)^5 (-5 a B e-6 A b e+11 b B d)}{e^{12} (d+e x)^3}+\frac {15 b^3 (b d-a e)^6 (-4 a B e-7 A b e+11 b B d)}{2 e^{12} (d+e x)^4}-\frac {3 b^2 (b d-a e)^7 (-3 a B e-8 A b e+11 b B d)}{e^{12} (d+e x)^5}+\frac {5 b (b d-a e)^8 (-2 a B e-9 A b e+11 b B d)}{6 e^{12} (d+e x)^6}-\frac {(b d-a e)^9 (-a B e-10 A b e+11 b B d)}{7 e^{12} (d+e x)^7}+\frac {(b d-a e)^{10} (B d-A e)}{8 e^{12} (d+e x)^8}+\frac {b^{10} B (d+e x)^3}{3 e^{12}} \]

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

- ((b*d - a*e)^9*(11*b*B*d - 10*A*b*e - a*B*e))/(7*e^12*(d + e*x)^7) + (5*b*(b*d - a*e)^8*(11*b*B*d - 9*A*b*e

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

15*b^3*(b*d - a*e)^6*(11*b*B*d - 7*A*b*e - 4*a*B*e))/(2*e^12*(d + e*x)^4) - (14*b^4*(b*d - a*e)^5*(11*b*B*d -

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

^2) - (30*b^6*(b*d - a*e)^3*(11*b*B*d - 4*A*b*e - 7*a*B*e))/(e^12*(d + e*x)) - (b^9*(11*b*B*d - A*b*e - 10*a*B

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

Mathematica [A] (veriﬁed)

Time = 0.34 (sec) , antiderivative size = 415, normalized size of antiderivative = 0.93 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^9} \, dx=\frac {-168 b^8 e \left (-45 a^2 B e^2-10 a b e (-9 B d+A e)+9 b^2 d (-5 B d+A e)\right ) x+84 b^9 e^2 (-9 b B d+A b e+10 a B e) x^2+56 b^{10} B e^3 x^3+\frac {21 (b d-a e)^{10} (B d-A e)}{(d+e x)^8}-\frac {24 (b d-a e)^9 (11 b B d-10 A b e-a B e)}{(d+e x)^7}+\frac {140 b (b d-a e)^8 (11 b B d-9 A b e-2 a B e)}{(d+e x)^6}-\frac {504 b^2 (b d-a e)^7 (11 b B d-8 A b e-3 a B e)}{(d+e x)^5}+\frac {1260 b^3 (b d-a e)^6 (11 b B d-7 A b e-4 a B e)}{(d+e x)^4}-\frac {2352 b^4 (b d-a e)^5 (11 b B d-6 A b e-5 a B e)}{(d+e x)^3}+\frac {3528 b^5 (b d-a e)^4 (11 b B d-5 A b e-6 a B e)}{(d+e x)^2}-\frac {5040 b^6 (b d-a e)^3 (11 b B d-4 A b e-7 a B e)}{d+e x}-2520 b^7 (b d-a e)^2 (11 b B d-3 A b e-8 a B e) \log (d+e x)}{168 e^{12}} \]

(-168*b^8*e*(-45*a^2*B*e^2 - 10*a*b*e*(-9*B*d + A*e) + 9*b^2*d*(-5*B*d + A*e))*x + 84*b^9*e^2*(-9*b*B*d + A*b*

e + 10*a*B*e)*x^2 + 56*b^10*B*e^3*x^3 + (21*(b*d - a*e)^10*(B*d - A*e))/(d + e*x)^8 - (24*(b*d - a*e)^9*(11*b*

B*d - 10*A*b*e - a*B*e))/(d + e*x)^7 + (140*b*(b*d - a*e)^8*(11*b*B*d - 9*A*b*e - 2*a*B*e))/(d + e*x)^6 - (504

*b^2*(b*d - a*e)^7*(11*b*B*d - 8*A*b*e - 3*a*B*e))/(d + e*x)^5 + (1260*b^3*(b*d - a*e)^6*(11*b*B*d - 7*A*b*e -

4*a*B*e))/(d + e*x)^4 - (2352*b^4*(b*d - a*e)^5*(11*b*B*d - 6*A*b*e - 5*a*B*e))/(d + e*x)^3 + (3528*b^5*(b*d

- a*e)^4*(11*b*B*d - 5*A*b*e - 6*a*B*e))/(d + e*x)^2 - (5040*b^6*(b*d - a*e)^3*(11*b*B*d - 4*A*b*e - 7*a*B*e))

/(d + e*x) - 2520*b^7*(b*d - a*e)^2*(11*b*B*d - 3*A*b*e - 8*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. \(1928\) vs. \(2(433)=866\).

Time = 2.48 (sec) , antiderivative size = 1929, normalized size of antiderivative = 4.33


method	result	size

norman	\(\text {Expression too large to display}\)	\(1929\)
default	\(\text {Expression too large to display}\)	\(1938\)
risch	\(\text {Expression too large to display}\)	\(1968\)
parallelrisch	\(\text {Expression too large to display}\)	\(3491\)

int((b*x+a)^10*(B*x+A)/(e*x+d)^9,x,method=_RETURNVERBOSE)

(-1/168*(21*A*a^10*e^11+30*A*a^9*b*d*e^10+45*A*a^8*b^2*d^2*e^9+72*A*a^7*b^3*d^3*e^8+126*A*a^6*b^4*d^4*e^7+252*

A*a^5*b^5*d^5*e^6+630*A*a^4*b^6*d^6*e^5+2520*A*a^3*b^7*d^7*e^4-20547*A*a^2*b^8*d^8*e^3+41094*A*a*b^9*d^9*e^2-2

0547*A*b^10*d^10*e+3*B*a^10*d*e^10+10*B*a^9*b*d^2*e^9+27*B*a^8*b^2*d^3*e^8+72*B*a^7*b^3*d^4*e^7+210*B*a^6*b^4*

d^5*e^6+756*B*a^5*b^5*d^6*e^5+4410*B*a^4*b^6*d^7*e^4-54792*B*a^3*b^7*d^8*e^3+184923*B*a^2*b^8*d^9*e^2-205470*B

*a*b^9*d^10*e+75339*B*b^10*d^11)/e^12-2*(60*A*a^3*b^7*e^4-180*A*a^2*b^8*d*e^3+360*A*a*b^9*d^2*e^2-180*A*b^10*d

^3*e+105*B*a^4*b^6*e^4-480*B*a^3*b^7*d*e^3+1620*B*a^2*b^8*d^2*e^2-1800*B*a*b^9*d^3*e+660*B*b^10*d^4)/e^5*x^7-7

*(15*A*a^4*b^6*e^5+60*A*a^3*b^7*d*e^4-270*A*a^2*b^8*d^2*e^3+540*A*a*b^9*d^3*e^2-270*A*b^10*d^4*e+18*B*a^5*b^5*

e^5+105*B*a^4*b^6*d*e^4-720*B*a^3*b^7*d^2*e^3+2430*B*a^2*b^8*d^3*e^2-2700*B*a*b^9*d^4*e+990*B*b^10*d^5)/e^6*x^

6-14*(6*A*a^5*b^5*e^6+15*A*a^4*b^6*d*e^5+60*A*a^3*b^7*d^2*e^4-330*A*a^2*b^8*d^3*e^3+660*A*a*b^9*d^4*e^2-330*A*

b^10*d^5*e+5*B*a^6*b^4*e^6+18*B*a^5*b^5*d*e^5+105*B*a^4*b^6*d^2*e^4-880*B*a^3*b^7*d^3*e^3+2970*B*a^2*b^8*d^4*e

^2-3300*B*a*b^9*d^5*e+1210*B*b^10*d^6)/e^7*x^5-5/2*(21*A*a^6*b^4*e^7+42*A*a^5*b^5*d*e^6+105*A*a^4*b^6*d^2*e^5+

420*A*a^3*b^7*d^3*e^4-2625*A*a^2*b^8*d^4*e^3+5250*A*a*b^9*d^5*e^2-2625*A*b^10*d^6*e+12*B*a^7*b^3*e^7+35*B*a^6*

b^4*d*e^6+126*B*a^5*b^5*d^2*e^5+735*B*a^4*b^6*d^3*e^4-7000*B*a^3*b^7*d^4*e^3+23625*B*a^2*b^8*d^5*e^2-26250*B*a

*b^9*d^6*e+9625*B*b^10*d^7)/e^8*x^4-(24*A*a^7*b^3*e^8+42*A*a^6*b^4*d*e^7+84*A*a^5*b^5*d^2*e^6+210*A*a^4*b^6*d^

3*e^5+840*A*a^3*b^7*d^4*e^4-5754*A*a^2*b^8*d^5*e^3+11508*A*a*b^9*d^6*e^2-5754*A*b^10*d^7*e+9*B*a^8*b^2*e^8+24*

B*a^7*b^3*d*e^7+70*B*a^6*b^4*d^2*e^6+252*B*a^5*b^5*d^3*e^5+1470*B*a^4*b^6*d^4*e^4-15344*B*a^3*b^7*d^5*e^3+5178

6*B*a^2*b^8*d^6*e^2-57540*B*a*b^9*d^7*e+21098*B*b^10*d^8)/e^9*x^3-1/6*(45*A*a^8*b^2*e^9+72*A*a^7*b^3*d*e^8+126

*A*a^6*b^4*d^2*e^7+252*A*a^5*b^5*d^3*e^6+630*A*a^4*b^6*d^4*e^5+2520*A*a^3*b^7*d^5*e^4-18522*A*a^2*b^8*d^6*e^3+

37044*A*a*b^9*d^7*e^2-18522*A*b^10*d^8*e+10*B*a^9*b*e^9+27*B*a^8*b^2*d*e^8+72*B*a^7*b^3*d^2*e^7+210*B*a^6*b^4*

d^3*e^6+756*B*a^5*b^5*d^4*e^5+4410*B*a^4*b^6*d^5*e^4-49392*B*a^3*b^7*d^6*e^3+166698*B*a^2*b^8*d^7*e^2-185220*B

*a*b^9*d^8*e+67914*B*b^10*d^9)/e^10*x^2-1/21*(30*A*a^9*b*e^10+45*A*a^8*b^2*d*e^9+72*A*a^7*b^3*d^2*e^8+126*A*a^

6*b^4*d^3*e^7+252*A*a^5*b^5*d^4*e^6+630*A*a^4*b^6*d^5*e^5+2520*A*a^3*b^7*d^6*e^4-19602*A*a^2*b^8*d^7*e^3+39204

*A*a*b^9*d^8*e^2-19602*A*b^10*d^9*e+3*B*a^10*e^10+10*B*a^9*b*d*e^9+27*B*a^8*b^2*d^2*e^8+72*B*a^7*b^3*d^3*e^7+2

10*B*a^6*b^4*d^4*e^6+756*B*a^5*b^5*d^5*e^5+4410*B*a^4*b^6*d^6*e^4-52272*B*a^3*b^7*d^7*e^3+176418*B*a^2*b^8*d^8

*e^2-196020*B*a*b^9*d^9*e+71874*B*b^10*d^10)/e^11*x+5/3*b^8*(6*A*a*b*e^2-3*A*b^2*d*e+27*B*a^2*e^2-30*B*a*b*d*e

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

3*A*a^2*b*e^3-6*A*a*b^2*d*e^2+3*A*b^3*d^2*e+8*B*a^3*e^3-27*B*a^2*b*d*e^2+30*B*a*b^2*d^2*e-11*B*b^3*d^3)*ln(e*x

Fricas [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2677 vs. \(2 (433) = 866\).

Time = 0.33 (sec) , antiderivative size = 2677, normalized size of antiderivative = 6.02 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^9} \, dx=\text {Too large to display} \]

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^9,x, algorithm="fricas")

1/168*(56*B*b^10*e^11*x^11 - 32891*B*b^10*d^11 - 21*A*a^10*e^11 + 10803*(10*B*a*b^9 + A*b^10)*d^10*e - 13827*(

9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 + 6849*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 - 630*(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 - 42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 - 18*(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 - 5*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9 - 3*(B*a^1

0 + 10*A*a^9*b)*d*e^10 - 28*(11*B*b^10*d*e^10 - 3*(10*B*a*b^9 + A*b^10)*e^11)*x^10 + 280*(11*B*b^10*d^2*e^9 -

3*(10*B*a*b^9 + A*b^10)*d*e^10 + 3*(9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9 + 112*(379*B*b^10*d^3*e^8 - 87*(10*B*a*

b^9 + A*b^10)*d^2*e^9 + 60*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^10)*x^8 + 112*(1052*B*b^10*d^4*e^7 - 156*(10*B*a*b^9

+ A*b^10)*d^3*e^8 - 60*(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 - 45*(7*B*a^

4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 + 392*(62*B*b^10*d^5*e^6 + 114*(10*B*a*b^9 + A*b^10)*d^4*e^7 - 330*(9*B*a^2*b^8

+ 2*A*a*b^9)*d^3*e^8 + 270*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^9 - 45*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^10 - 9*(6

*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 - 784*(598*B*b^10*d^6*e^5 - 294*(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 - 330*(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

+ 9*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 + 3*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^11)*x^5 - 140*(7651*B*b^10*d^7*e^4 -

3003*(10*B*a*b^9 + A*b^10)*d^6*e^5 + 4515*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^6 - 2625*(8*B*a^3*b^7 + 3*A*a^2*b^8

)*d^4*e^7 + 315*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^8 + 63*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^9 + 21*(5*B*a^6*b^4

+ 6*A*a^5*b^5)*d*e^10 + 9*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^11)*x^4 - 56*(20846*B*b^10*d^8*e^3 - 7518*(10*B*a*b^9

+ A*b^10)*d^7*e^4 + 10542*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5 - 5754*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^6 + 630*

(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 + 42*(5*B*a^6*b^4 + 6*A*a^5*b^5)

*d^2*e^9 + 18*(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 - 28*(25466*B*b^10*

d^9*e^2 - 8778*(10*B*a*b^9 + A*b^10)*d^8*e^3 + 11802*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^4 - 6174*(8*B*a^3*b^7 + 3

*A*a^2*b^8)*d^6*e^5 + 630*(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 + 42*(

5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*e^8 + 18*(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 + 5*(2*B*a^9*b + 9*A*a^8*b^2)*e^11)*x^2 - 8*(29426*B*b^10*d^10*e - 9858*(10*B*a*b^9 + A*b^10)*d^9*e^2 + 1

2882*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^3 - 6534*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4 + 630*(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 + 42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*e^7 + 18*(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 + 5*(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 - 3*(10*B*a*b^9 + A*b^10)*d^10*e + 3*(9*B*a^2*b^8 + 2*A*a*

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

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

9 + A*b^10)*d^3*e^8 + 3*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^9 - (8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^10)*x^7 + 28*(11*B

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

*b^8)*d^2*e^9)*x^6 + 56*(11*B*b^10*d^6*e^5 - 3*(10*B*a*b^9 + A*b^10)*d^5*e^6 + 3*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4

*e^7 - (8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8)*x^5 + 70*(11*B*b^10*d^7*e^4 - 3*(10*B*a*b^9 + A*b^10)*d^6*e^5 + 3*

(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^6 - (8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7)*x^4 + 56*(11*B*b^10*d^8*e^3 - 3*(10*B

*a*b^9 + A*b^10)*d^7*e^4 + 3*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5 - (8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^6)*x^3 + 28

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

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

*d^8*e^3 - (8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4)*x)*log(e*x + d))/(e^20*x^8 + 8*d*e^19*x^7 + 28*d^2*e^18*x^6 +

56*d^3*e^17*x^5 + 70*d^4*e^16*x^4 + 56*d^5*e^15*x^3 + 28*d^6*e^14*x^2 + 8*d^7*e^13*x + d^8*e^12)

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^9} \, dx=\text {Timed out} \]

Maxima [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1892 vs. \(2 (433) = 866\).

Time = 0.36 (sec) , antiderivative size = 1892, normalized size of antiderivative = 4.25 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^9} \, dx=\text {Too large to display} \]

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^9,x, algorithm="maxima")

-1/168*(32891*B*b^10*d^11 + 21*A*a^10*e^11 - 10803*(10*B*a*b^9 + A*b^10)*d^10*e + 13827*(9*B*a^2*b^8 + 2*A*a*b

^9)*d^9*e^2 - 6849*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 630*(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 + 42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 + 18*(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 + 5*(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^1

0 + 5040*(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 + 3528*(99*B*b^10*d^5*e^6 - 35*(10*B*a*b^

9 + A*b^10)*d^4*e^7 + 50*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^8 - 30*(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 + 2352*(407*B*b^10*d^6*e^5 - 141*(10*B*a*

b^9 + A*b^10)*d^5*e^6 + 195*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 - 110*(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 + 3*(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*(3509*B*b^10*d^7*e^4 - 1197*(10*B*a*b^9 + A*b^10)*d^6*e^5 + 1617*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^6 -

875*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 + 105*(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 + 3*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^11)*x^4 + 168*(8173*B*b

^10*d^8*e^3 - 2754*(10*B*a*b^9 + A*b^10)*d^7*e^4 + 3654*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5 - 1918*(8*B*a^3*b^7

+ 3*A*a^2*b^8)*d^5*e^6 + 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^7 + 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^8 + 14

*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*e^9 + 6*(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 + 28*(27599*B*b^10*d^9*e^2 - 9207*(10*B*a*b^9 + A*b^10)*d^8*e^3 + 12042*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*

e^4 - 6174*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^5 + 630*(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 + 42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*e^8 + 18*(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 + 5*(2*B*a^9*b + 9*A*a^8*b^2)*e^11)*x^2 + 8*(30371*B*b^10*d^10*e - 10047*(10*B

*a*b^9 + A*b^10)*d^9*e^2 + 12987*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^3 - 6534*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4

+ 630*(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 + 42*(5*B*a^6*b^4 + 6*A*a^

5*b^5)*d^4*e^7 + 18*(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 + 5*(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^20*x^8 + 8*d*e^19*x^7 + 28*d^2*e^18*x^6 + 56*d^3*

e^17*x^5 + 70*d^4*e^16*x^4 + 56*d^5*e^15*x^3 + 28*d^6*e^14*x^2 + 8*d^7*e^13*x + d^8*e^12) + 1/6*(2*B*b^10*e^2*

x^3 - 3*(9*B*b^10*d*e - (10*B*a*b^9 + A*b^10)*e^2)*x^2 + 6*(45*B*b^10*d^2 - 9*(10*B*a*b^9 + A*b^10)*d*e + 5*(9

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

A*a*b^9)*d*e^2 - (8*B*a^3*b^7 + 3*A*a^2*b^8)*e^3)*log(e*x + d)/e^12

Giac [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1981 vs. \(2 (433) = 866\).

Time = 0.30 (sec) , antiderivative size = 1981, normalized size of antiderivative = 4.45 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^9} \, dx=\text {Too large to display} \]

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^9,x, algorithm="giac")

-15*(11*B*b^10*d^3 - 30*B*a*b^9*d^2*e - 3*A*b^10*d^2*e + 27*B*a^2*b^8*d*e^2 + 6*A*a*b^9*d*e^2 - 8*B*a^3*b^7*e^

3 - 3*A*a^2*b^8*e^3)*log(abs(e*x + d))/e^12 - 1/168*(32891*B*b^10*d^11 - 108030*B*a*b^9*d^10*e - 10803*A*b^10*

d^10*e + 124443*B*a^2*b^8*d^9*e^2 + 27654*A*a*b^9*d^9*e^2 - 54792*B*a^3*b^7*d^8*e^3 - 20547*A*a^2*b^8*d^8*e^3

+ 4410*B*a^4*b^6*d^7*e^4 + 2520*A*a^3*b^7*d^7*e^4 + 756*B*a^5*b^5*d^6*e^5 + 630*A*a^4*b^6*d^6*e^5 + 210*B*a^6*

b^4*d^5*e^6 + 252*A*a^5*b^5*d^5*e^6 + 72*B*a^7*b^3*d^4*e^7 + 126*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 + 10*B*a^9*b*d^2*e^9 + 45*A*a^8*b^2*d^2*e^9 + 3*B*a^10*d*e^10 + 30*A*a^9*b*d*e^10 + 21*A*a^

10*e^11 + 5040*(11*B*b^10*d^4*e^7 - 40*B*a*b^9*d^3*e^8 - 4*A*b^10*d^3*e^8 + 54*B*a^2*b^8*d^2*e^9 + 12*A*a*b^9*

d^2*e^9 - 32*B*a^3*b^7*d*e^10 - 12*A*a^2*b^8*d*e^10 + 7*B*a^4*b^6*e^11 + 4*A*a^3*b^7*e^11)*x^7 + 3528*(99*B*b^

10*d^5*e^6 - 350*B*a*b^9*d^4*e^7 - 35*A*b^10*d^4*e^7 + 450*B*a^2*b^8*d^3*e^8 + 100*A*a*b^9*d^3*e^8 - 240*B*a^3

*b^7*d^2*e^9 - 90*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 + 2352*(407*B*b^10*d^6*e^5 - 1410*B*a*b^9*d^5*e^6 - 141*A*b^10*d^5*e^6 + 1755*B*a^2*b^8*d^4*e^7 +

390*A*a*b^9*d^4*e^7 - 880*B*a^3*b^7*d^3*e^8 - 330*A*a^2*b^8*d^3*e^8 + 105*B*a^4*b^6*d^2*e^9 + 60*A*a^3*b^7*d^

2*e^9 + 18*B*a^5*b^5*d*e^10 + 15*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*(3509*B*b^1

0*d^7*e^4 - 11970*B*a*b^9*d^6*e^5 - 1197*A*b^10*d^6*e^5 + 14553*B*a^2*b^8*d^5*e^6 + 3234*A*a*b^9*d^5*e^6 - 700

0*B*a^3*b^7*d^4*e^7 - 2625*A*a^2*b^8*d^4*e^7 + 735*B*a^4*b^6*d^3*e^8 + 420*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 + 12*B*a^7*b^3*e^11 + 21*A*a^6*b^4*

e^11)*x^4 + 168*(8173*B*b^10*d^8*e^3 - 27540*B*a*b^9*d^7*e^4 - 2754*A*b^10*d^7*e^4 + 32886*B*a^2*b^8*d^6*e^5 +

7308*A*a*b^9*d^6*e^5 - 15344*B*a^3*b^7*d^5*e^6 - 5754*A*a^2*b^8*d^5*e^6 + 1470*B*a^4*b^6*d^4*e^7 + 840*A*a^3*

b^7*d^4*e^7 + 252*B*a^5*b^5*d^3*e^8 + 210*A*a^4*b^6*d^3*e^8 + 70*B*a^6*b^4*d^2*e^9 + 84*A*a^5*b^5*d^2*e^9 + 24

*B*a^7*b^3*d*e^10 + 42*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 + 28*(27599*B*b^10*d^9*e^2

- 92070*B*a*b^9*d^8*e^3 - 9207*A*b^10*d^8*e^3 + 108378*B*a^2*b^8*d^7*e^4 + 24084*A*a*b^9*d^7*e^4 - 49392*B*a^

3*b^7*d^6*e^5 - 18522*A*a^2*b^8*d^6*e^5 + 4410*B*a^4*b^6*d^5*e^6 + 2520*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 + 210*B*a^6*b^4*d^3*e^8 + 252*A*a^5*b^5*d^3*e^8 + 72*B*a^7*b^3*d^2*e^9 + 126*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 + 10*B*a^9*b*e^11 + 45*A*a^8*b^2*e^11)*x^2 + 8*(30371

*B*b^10*d^10*e - 100470*B*a*b^9*d^9*e^2 - 10047*A*b^10*d^9*e^2 + 116883*B*a^2*b^8*d^8*e^3 + 25974*A*a*b^9*d^8*

e^3 - 52272*B*a^3*b^7*d^7*e^4 - 19602*A*a^2*b^8*d^7*e^4 + 4410*B*a^4*b^6*d^6*e^5 + 2520*A*a^3*b^7*d^6*e^5 + 75

6*B*a^5*b^5*d^5*e^6 + 630*A*a^4*b^6*d^5*e^6 + 210*B*a^6*b^4*d^4*e^7 + 252*A*a^5*b^5*d^4*e^7 + 72*B*a^7*b^3*d^3

*e^8 + 126*A*a^6*b^4*d^3*e^8 + 27*B*a^8*b^2*d^2*e^9 + 72*A*a^7*b^3*d^2*e^9 + 10*B*a^9*b*d*e^10 + 45*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)^8*e^12) + 1/6*(2*B*b^10*e^18*x^3 - 27*B*b^10*d*e^17*x^

2 + 30*B*a*b^9*e^18*x^2 + 3*A*b^10*e^18*x^2 + 270*B*b^10*d^2*e^16*x - 540*B*a*b^9*d*e^17*x - 54*A*b^10*d*e^17*

Mupad [B] (veriﬁcation not implemented)

Time = 0.57 (sec) , antiderivative size = 2048, normalized size of antiderivative = 4.60 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^9} \, dx=\text {Too large to display} \]

x^2*((A*b^10 + 10*B*a*b^9)/(2*e^9) - (9*B*b^10*d)/(2*e^10)) - (x^7*(120*A*a^3*b^7*e^10 + 210*B*a^4*b^6*e^10 -

120*A*b^10*d^3*e^7 + 330*B*b^10*d^4*e^6 + 360*A*a*b^9*d^2*e^8 - 360*A*a^2*b^8*d*e^9 - 1200*B*a*b^9*d^3*e^7 - 9

60*B*a^3*b^7*d*e^9 + 1620*B*a^2*b^8*d^2*e^8) + x^4*((105*A*a^6*b^4*e^10)/2 + 30*B*a^7*b^3*e^10 - (5985*A*b^10*

d^6*e^4)/2 + (17545*B*b^10*d^7*e^3)/2 + 8085*A*a*b^9*d^5*e^5 + 105*A*a^5*b^5*d*e^9 - 29925*B*a*b^9*d^6*e^4 + (

175*B*a^6*b^4*d*e^9)/2 - (13125*A*a^2*b^8*d^4*e^6)/2 + 1050*A*a^3*b^7*d^3*e^7 + (525*A*a^4*b^6*d^2*e^8)/2 + (7

2765*B*a^2*b^8*d^5*e^5)/2 - 17500*B*a^3*b^7*d^4*e^6 + (3675*B*a^4*b^6*d^3*e^7)/2 + 315*B*a^5*b^5*d^2*e^8) + x^

6*(105*A*a^4*b^6*e^10 + 126*B*a^5*b^5*e^10 - 735*A*b^10*d^4*e^6 + 2079*B*b^10*d^5*e^5 + 2100*A*a*b^9*d^3*e^7 +

420*A*a^3*b^7*d*e^9 - 7350*B*a*b^9*d^4*e^6 + 735*B*a^4*b^6*d*e^9 - 1890*A*a^2*b^8*d^2*e^8 + 9450*B*a^2*b^8*d^

3*e^7 - 5040*B*a^3*b^7*d^2*e^8) + x^3*(24*A*a^7*b^3*e^10 + 9*B*a^8*b^2*e^10 - 2754*A*b^10*d^7*e^3 + 8173*B*b^1

0*d^8*e^2 + 7308*A*a*b^9*d^6*e^4 + 42*A*a^6*b^4*d*e^9 - 27540*B*a*b^9*d^7*e^3 + 24*B*a^7*b^3*d*e^9 - 5754*A*a^

2*b^8*d^5*e^5 + 840*A*a^3*b^7*d^4*e^6 + 210*A*a^4*b^6*d^3*e^7 + 84*A*a^5*b^5*d^2*e^8 + 32886*B*a^2*b^8*d^6*e^4

- 15344*B*a^3*b^7*d^5*e^5 + 1470*B*a^4*b^6*d^4*e^6 + 252*B*a^5*b^5*d^3*e^7 + 70*B*a^6*b^4*d^2*e^8) + (21*A*a^

10*e^11 + 32891*B*b^10*d^11 - 10803*A*b^10*d^10*e + 3*B*a^10*d*e^10 + 27654*A*a*b^9*d^9*e^2 + 10*B*a^9*b*d^2*e

^9 - 20547*A*a^2*b^8*d^8*e^3 + 2520*A*a^3*b^7*d^7*e^4 + 630*A*a^4*b^6*d^6*e^5 + 252*A*a^5*b^5*d^5*e^6 + 126*A*

a^6*b^4*d^4*e^7 + 72*A*a^7*b^3*d^3*e^8 + 45*A*a^8*b^2*d^2*e^9 + 124443*B*a^2*b^8*d^9*e^2 - 54792*B*a^3*b^7*d^8

*e^3 + 4410*B*a^4*b^6*d^7*e^4 + 756*B*a^5*b^5*d^6*e^5 + 210*B*a^6*b^4*d^5*e^6 + 72*B*a^7*b^3*d^4*e^7 + 27*B*a^

8*b^2*d^3*e^8 + 30*A*a^9*b*d*e^10 - 108030*B*a*b^9*d^10*e)/(168*e) + x*((B*a^10*e^10)/7 + (30371*B*b^10*d^10)/

21 + (10*A*a^9*b*e^10)/7 - (3349*A*b^10*d^9*e)/7 + (8658*A*a*b^9*d^8*e^2)/7 + (15*A*a^8*b^2*d*e^9)/7 - (6534*A

*a^2*b^8*d^7*e^3)/7 + 120*A*a^3*b^7*d^6*e^4 + 30*A*a^4*b^6*d^5*e^5 + 12*A*a^5*b^5*d^4*e^6 + 6*A*a^6*b^4*d^3*e^

7 + (24*A*a^7*b^3*d^2*e^8)/7 + (38961*B*a^2*b^8*d^8*e^2)/7 - (17424*B*a^3*b^7*d^7*e^3)/7 + 210*B*a^4*b^6*d^6*e

^4 + 36*B*a^5*b^5*d^5*e^5 + 10*B*a^6*b^4*d^4*e^6 + (24*B*a^7*b^3*d^3*e^7)/7 + (9*B*a^8*b^2*d^2*e^8)/7 - (33490

*B*a*b^9*d^9*e)/7 + (10*B*a^9*b*d*e^9)/21) + x^5*(84*A*a^5*b^5*e^10 + 70*B*a^6*b^4*e^10 - 1974*A*b^10*d^5*e^5

+ 5698*B*b^10*d^6*e^4 + 5460*A*a*b^9*d^4*e^6 + 210*A*a^4*b^6*d*e^9 - 19740*B*a*b^9*d^5*e^5 + 252*B*a^5*b^5*d*e

^9 - 4620*A*a^2*b^8*d^3*e^7 + 840*A*a^3*b^7*d^2*e^8 + 24570*B*a^2*b^8*d^4*e^6 - 12320*B*a^3*b^7*d^3*e^7 + 1470

*B*a^4*b^6*d^2*e^8) + x^2*((5*B*a^9*b*e^10)/3 + (27599*B*b^10*d^9*e)/6 + (15*A*a^8*b^2*e^10)/2 - (3069*A*b^10*

d^8*e^2)/2 + 4014*A*a*b^9*d^7*e^3 + 12*A*a^7*b^3*d*e^9 - 15345*B*a*b^9*d^8*e^2 + (9*B*a^8*b^2*d*e^9)/2 - 3087*

A*a^2*b^8*d^6*e^4 + 420*A*a^3*b^7*d^5*e^5 + 105*A*a^4*b^6*d^4*e^6 + 42*A*a^5*b^5*d^3*e^7 + 21*A*a^6*b^4*d^2*e^

8 + 18063*B*a^2*b^8*d^7*e^3 - 8232*B*a^3*b^7*d^6*e^4 + 735*B*a^4*b^6*d^5*e^5 + 126*B*a^5*b^5*d^4*e^6 + 35*B*a^

6*b^4*d^3*e^7 + 12*B*a^7*b^3*d^2*e^8))/(d^8*e^11 + e^19*x^8 + 8*d^7*e^12*x + 8*d*e^18*x^7 + 28*d^6*e^13*x^2 +

56*d^5*e^14*x^3 + 70*d^4*e^15*x^4 + 56*d^3*e^16*x^5 + 28*d^2*e^17*x^6) - x*((9*d*((A*b^10 + 10*B*a*b^9)/e^9 -

(9*B*b^10*d)/e^10))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^9 + (36*B*b^10*d^2)/e^11) + (log(d + e*x)*(45*A*b^10*d^2*e

- 165*B*b^10*d^3 + 45*A*a^2*b^8*e^3 + 120*B*a^3*b^7*e^3 - 405*B*a^2*b^8*d*e^2 - 90*A*a*b^9*d*e^2 + 450*B*a*b^

\(\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^9} \, dx\) [1097]

Optimal result