3.11.0 ∫ (a+bx)10(A+Bx) d+ex dx [1089]

Integrand size = 20, antiderivative size = 348 \[ \int \frac {(a+b x)^{10} (A+B x)}{d+e x} \, dx=\frac {b (b d-a e)^9 (B d-A e) x}{e^{11}}-\frac {(b d-a e)^8 (B d-A e) (a+b x)^2}{2 e^{10}}+\frac {(b d-a e)^7 (B d-A e) (a+b x)^3}{3 e^9}-\frac {(b d-a e)^6 (B d-A e) (a+b x)^4}{4 e^8}+\frac {(b d-a e)^5 (B d-A e) (a+b x)^5}{5 e^7}-\frac {(b d-a e)^4 (B d-A e) (a+b x)^6}{6 e^6}+\frac {(b d-a e)^3 (B d-A e) (a+b x)^7}{7 e^5}-\frac {(b d-a e)^2 (B d-A e) (a+b x)^8}{8 e^4}+\frac {(b d-a e) (B d-A e) (a+b x)^9}{9 e^3}-\frac {(B d-A e) (a+b x)^{10}}{10 e^2}+\frac {B (a+b x)^{11}}{11 b e}-\frac {(b d-a e)^{10} (B d-A e) \log (d+e x)}{e^{12}} \]

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

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

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

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

Rubi [A] (veriﬁed)

Time = 0.24 (sec) , antiderivative size = 348, 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} \[ \int \frac {(a+b x)^{10} (A+B x)}{d+e x} \, dx=-\frac {(b d-a e)^{10} (B d-A e) \log (d+e x)}{e^{12}}+\frac {b x (b d-a e)^9 (B d-A e)}{e^{11}}-\frac {(a+b x)^2 (b d-a e)^8 (B d-A e)}{2 e^{10}}+\frac {(a+b x)^3 (b d-a e)^7 (B d-A e)}{3 e^9}-\frac {(a+b x)^4 (b d-a e)^6 (B d-A e)}{4 e^8}+\frac {(a+b x)^5 (b d-a e)^5 (B d-A e)}{5 e^7}-\frac {(a+b x)^6 (b d-a e)^4 (B d-A e)}{6 e^6}+\frac {(a+b x)^7 (b d-a e)^3 (B d-A e)}{7 e^5}-\frac {(a+b x)^8 (b d-a e)^2 (B d-A e)}{8 e^4}+\frac {(a+b x)^9 (b d-a e) (B d-A e)}{9 e^3}-\frac {(a+b x)^{10} (B d-A e)}{10 e^2}+\frac {B (a+b x)^{11}}{11 b e} \]

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

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

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

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

B*d - A*e)*(a + b*x)^10)/(10*e^2) + (B*(a + b*x)^11)/(11*b*e) - ((b*d - a*e)^10*(B*d - A*e)*Log[d + e*x])/e^12

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

Mathematica [B] (veriﬁed)

Leaf count is larger than twice the leaf count of optimal. \(1252\) vs. \(2(348)=696\).

Time = 0.80 (sec) , antiderivative size = 1252, normalized size of antiderivative = 3.60 \[ \int \frac {(a+b x)^{10} (A+B x)}{d+e x} \, dx=\frac {x \left (27720 a^{10} B e^{10}+138600 a^9 b e^9 (-2 B d+2 A e+B e x)+207900 a^8 b^2 e^8 \left (3 A e (-2 d+e x)+B \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )+277200 a^7 b^3 e^7 \left (2 A e \left (6 d^2-3 d e x+2 e^2 x^2\right )+B \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )\right )+97020 a^6 b^4 e^6 \left (5 A e \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )+B \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )\right )+116424 a^5 b^5 e^5 \left (A e \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )+B \left (-60 d^5+30 d^4 e x-20 d^3 e^2 x^2+15 d^2 e^3 x^3-12 d e^4 x^4+10 e^5 x^5\right )\right )+13860 a^4 b^6 e^4 \left (7 A e \left (-60 d^5+30 d^4 e x-20 d^3 e^2 x^2+15 d^2 e^3 x^3-12 d e^4 x^4+10 e^5 x^5\right )+B \left (420 d^6-210 d^5 e x+140 d^4 e^2 x^2-105 d^3 e^3 x^3+84 d^2 e^4 x^4-70 d e^5 x^5+60 e^6 x^6\right )\right )+3960 a^3 b^7 e^3 \left (2 A e \left (420 d^6-210 d^5 e x+140 d^4 e^2 x^2-105 d^3 e^3 x^3+84 d^2 e^4 x^4-70 d e^5 x^5+60 e^6 x^6\right )+B \left (-840 d^7+420 d^6 e x-280 d^5 e^2 x^2+210 d^4 e^3 x^3-168 d^3 e^4 x^4+140 d^2 e^5 x^5-120 d e^6 x^6+105 e^7 x^7\right )\right )+495 a^2 b^8 e^2 \left (3 A e \left (-840 d^7+420 d^6 e x-280 d^5 e^2 x^2+210 d^4 e^3 x^3-168 d^3 e^4 x^4+140 d^2 e^5 x^5-120 d e^6 x^6+105 e^7 x^7\right )+B \left (2520 d^8-1260 d^7 e x+840 d^6 e^2 x^2-630 d^5 e^3 x^3+504 d^4 e^4 x^4-420 d^3 e^5 x^5+360 d^2 e^6 x^6-315 d e^7 x^7+280 e^8 x^8\right )\right )+110 a b^9 e \left (A e \left (2520 d^8-1260 d^7 e x+840 d^6 e^2 x^2-630 d^5 e^3 x^3+504 d^4 e^4 x^4-420 d^3 e^5 x^5+360 d^2 e^6 x^6-315 d e^7 x^7+280 e^8 x^8\right )+B \left (-2520 d^9+1260 d^8 e x-840 d^7 e^2 x^2+630 d^6 e^3 x^3-504 d^5 e^4 x^4+420 d^4 e^5 x^5-360 d^3 e^6 x^6+315 d^2 e^7 x^7-280 d e^8 x^8+252 e^9 x^9\right )\right )+b^{10} \left (11 A e \left (-2520 d^9+1260 d^8 e x-840 d^7 e^2 x^2+630 d^6 e^3 x^3-504 d^5 e^4 x^4+420 d^4 e^5 x^5-360 d^3 e^6 x^6+315 d^2 e^7 x^7-280 d e^8 x^8+252 e^9 x^9\right )+B \left (27720 d^{10}-13860 d^9 e x+9240 d^8 e^2 x^2-6930 d^7 e^3 x^3+5544 d^6 e^4 x^4-4620 d^5 e^5 x^5+3960 d^4 e^6 x^6-3465 d^3 e^7 x^7+3080 d^2 e^8 x^8-2772 d e^9 x^9+2520 e^{10} x^{10}\right )\right )\right )}{27720 e^{11}}+\frac {(b d-a e)^{10} (-B d+A e) \log (d+e x)}{e^{12}} \]

(x*(27720*a^10*B*e^10 + 138600*a^9*b*e^9*(-2*B*d + 2*A*e + B*e*x) + 207900*a^8*b^2*e^8*(3*A*e*(-2*d + e*x) + B

*(6*d^2 - 3*d*e*x + 2*e^2*x^2)) + 277200*a^7*b^3*e^7*(2*A*e*(6*d^2 - 3*d*e*x + 2*e^2*x^2) + B*(-12*d^3 + 6*d^2

*e*x - 4*d*e^2*x^2 + 3*e^3*x^3)) + 97020*a^6*b^4*e^6*(5*A*e*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3) +

B*(60*d^4 - 30*d^3*e*x + 20*d^2*e^2*x^2 - 15*d*e^3*x^3 + 12*e^4*x^4)) + 116424*a^5*b^5*e^5*(A*e*(60*d^4 - 30*d

^3*e*x + 20*d^2*e^2*x^2 - 15*d*e^3*x^3 + 12*e^4*x^4) + B*(-60*d^5 + 30*d^4*e*x - 20*d^3*e^2*x^2 + 15*d^2*e^3*x

^3 - 12*d*e^4*x^4 + 10*e^5*x^5)) + 13860*a^4*b^6*e^4*(7*A*e*(-60*d^5 + 30*d^4*e*x - 20*d^3*e^2*x^2 + 15*d^2*e^

3*x^3 - 12*d*e^4*x^4 + 10*e^5*x^5) + B*(420*d^6 - 210*d^5*e*x + 140*d^4*e^2*x^2 - 105*d^3*e^3*x^3 + 84*d^2*e^4

*x^4 - 70*d*e^5*x^5 + 60*e^6*x^6)) + 3960*a^3*b^7*e^3*(2*A*e*(420*d^6 - 210*d^5*e*x + 140*d^4*e^2*x^2 - 105*d^

3*e^3*x^3 + 84*d^2*e^4*x^4 - 70*d*e^5*x^5 + 60*e^6*x^6) + B*(-840*d^7 + 420*d^6*e*x - 280*d^5*e^2*x^2 + 210*d^

4*e^3*x^3 - 168*d^3*e^4*x^4 + 140*d^2*e^5*x^5 - 120*d*e^6*x^6 + 105*e^7*x^7)) + 495*a^2*b^8*e^2*(3*A*e*(-840*d

^7 + 420*d^6*e*x - 280*d^5*e^2*x^2 + 210*d^4*e^3*x^3 - 168*d^3*e^4*x^4 + 140*d^2*e^5*x^5 - 120*d*e^6*x^6 + 105

*e^7*x^7) + B*(2520*d^8 - 1260*d^7*e*x + 840*d^6*e^2*x^2 - 630*d^5*e^3*x^3 + 504*d^4*e^4*x^4 - 420*d^3*e^5*x^5

+ 360*d^2*e^6*x^6 - 315*d*e^7*x^7 + 280*e^8*x^8)) + 110*a*b^9*e*(A*e*(2520*d^8 - 1260*d^7*e*x + 840*d^6*e^2*x

^2 - 630*d^5*e^3*x^3 + 504*d^4*e^4*x^4 - 420*d^3*e^5*x^5 + 360*d^2*e^6*x^6 - 315*d*e^7*x^7 + 280*e^8*x^8) + B*

(-2520*d^9 + 1260*d^8*e*x - 840*d^7*e^2*x^2 + 630*d^6*e^3*x^3 - 504*d^5*e^4*x^4 + 420*d^4*e^5*x^5 - 360*d^3*e^

6*x^6 + 315*d^2*e^7*x^7 - 280*d*e^8*x^8 + 252*e^9*x^9)) + b^10*(11*A*e*(-2520*d^9 + 1260*d^8*e*x - 840*d^7*e^2

*x^2 + 630*d^6*e^3*x^3 - 504*d^5*e^4*x^4 + 420*d^4*e^5*x^5 - 360*d^3*e^6*x^6 + 315*d^2*e^7*x^7 - 280*d*e^8*x^8

+ 252*e^9*x^9) + B*(27720*d^10 - 13860*d^9*e*x + 9240*d^8*e^2*x^2 - 6930*d^7*e^3*x^3 + 5544*d^6*e^4*x^4 - 462

0*d^5*e^5*x^5 + 3960*d^4*e^6*x^6 - 3465*d^3*e^7*x^7 + 3080*d^2*e^8*x^8 - 2772*d*e^9*x^9 + 2520*e^10*x^10))))/(

27720*e^11) + ((b*d - a*e)^10*(-(B*d) + A*e)*Log[d + e*x])/e^12

Maple [B] (veriﬁed)

Leaf count of result is larger than twice the leaf count of optimal. \(1883\) vs. \(2(328)=656\).

Time = 0.72 (sec) , antiderivative size = 1884, normalized size of antiderivative = 5.41


method	result	size

norman	\(\text {Expression too large to display}\)	\(1884\)
default	\(\text {Expression too large to display}\)	\(2225\)
risch	\(\text {Expression too large to display}\)	\(2357\)
parallelrisch	\(\text {Expression too large to display}\)	\(2358\)

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

(10*A*a^9*b*e^10-45*A*a^8*b^2*d*e^9+120*A*a^7*b^3*d^2*e^8-210*A*a^6*b^4*d^3*e^7+252*A*a^5*b^5*d^4*e^6-210*A*a^

4*b^6*d^5*e^5+120*A*a^3*b^7*d^6*e^4-45*A*a^2*b^8*d^7*e^3+10*A*a*b^9*d^8*e^2-A*b^10*d^9*e+B*a^10*e^10-10*B*a^9*

b*d*e^9+45*B*a^8*b^2*d^2*e^8-120*B*a^7*b^3*d^3*e^7+210*B*a^6*b^4*d^4*e^6-252*B*a^5*b^5*d^5*e^5+210*B*a^4*b^6*d

^6*e^4-120*B*a^3*b^7*d^7*e^3+45*B*a^2*b^8*d^8*e^2-10*B*a*b^9*d^9*e+B*b^10*d^10)/e^11*x+1/2*b/e^10*(45*A*a^8*b*

e^9-120*A*a^7*b^2*d*e^8+210*A*a^6*b^3*d^2*e^7-252*A*a^5*b^4*d^3*e^6+210*A*a^4*b^5*d^4*e^5-120*A*a^3*b^6*d^5*e^

4+45*A*a^2*b^7*d^6*e^3-10*A*a*b^8*d^7*e^2+A*b^9*d^8*e+10*B*a^9*e^9-45*B*a^8*b*d*e^8+120*B*a^7*b^2*d^2*e^7-210*

B*a^6*b^3*d^3*e^6+252*B*a^5*b^4*d^4*e^5-210*B*a^4*b^5*d^5*e^4+120*B*a^3*b^6*d^6*e^3-45*B*a^2*b^7*d^7*e^2+10*B*

a*b^8*d^8*e-B*b^9*d^9)*x^2+1/3*b^2/e^9*(120*A*a^7*b*e^8-210*A*a^6*b^2*d*e^7+252*A*a^5*b^3*d^2*e^6-210*A*a^4*b^

4*d^3*e^5+120*A*a^3*b^5*d^4*e^4-45*A*a^2*b^6*d^5*e^3+10*A*a*b^7*d^6*e^2-A*b^8*d^7*e+45*B*a^8*e^8-120*B*a^7*b*d

*e^7+210*B*a^6*b^2*d^2*e^6-252*B*a^5*b^3*d^3*e^5+210*B*a^4*b^4*d^4*e^4-120*B*a^3*b^5*d^5*e^3+45*B*a^2*b^6*d^6*

e^2-10*B*a*b^7*d^7*e+B*b^8*d^8)*x^3+1/4*b^3/e^8*(210*A*a^6*b*e^7-252*A*a^5*b^2*d*e^6+210*A*a^4*b^3*d^2*e^5-120

*A*a^3*b^4*d^3*e^4+45*A*a^2*b^5*d^4*e^3-10*A*a*b^6*d^5*e^2+A*b^7*d^6*e+120*B*a^7*e^7-210*B*a^6*b*d*e^6+252*B*a

^5*b^2*d^2*e^5-210*B*a^4*b^3*d^3*e^4+120*B*a^3*b^4*d^4*e^3-45*B*a^2*b^5*d^5*e^2+10*B*a*b^6*d^6*e-B*b^7*d^7)*x^

4+1/5*b^4/e^7*(252*A*a^5*b*e^6-210*A*a^4*b^2*d*e^5+120*A*a^3*b^3*d^2*e^4-45*A*a^2*b^4*d^3*e^3+10*A*a*b^5*d^4*e

^2-A*b^6*d^5*e+210*B*a^6*e^6-252*B*a^5*b*d*e^5+210*B*a^4*b^2*d^2*e^4-120*B*a^3*b^3*d^3*e^3+45*B*a^2*b^4*d^4*e^

2-10*B*a*b^5*d^5*e+B*b^6*d^6)*x^5+1/6*b^5/e^6*(210*A*a^4*b*e^5-120*A*a^3*b^2*d*e^4+45*A*a^2*b^3*d^2*e^3-10*A*a

*b^4*d^3*e^2+A*b^5*d^4*e+252*B*a^5*e^5-210*B*a^4*b*d*e^4+120*B*a^3*b^2*d^2*e^3-45*B*a^2*b^3*d^3*e^2+10*B*a*b^4

*d^4*e-B*b^5*d^5)*x^6+1/7*b^6/e^5*(120*A*a^3*b*e^4-45*A*a^2*b^2*d*e^3+10*A*a*b^3*d^2*e^2-A*b^4*d^3*e+210*B*a^4

*e^4-120*B*a^3*b*d*e^3+45*B*a^2*b^2*d^2*e^2-10*B*a*b^3*d^3*e+B*b^4*d^4)*x^7+1/8*b^7/e^4*(45*A*a^2*b*e^3-10*A*a

*b^2*d*e^2+A*b^3*d^2*e+120*B*a^3*e^3-45*B*a^2*b*d*e^2+10*B*a*b^2*d^2*e-B*b^3*d^3)*x^8+1/9*b^8/e^3*(10*A*a*b*e^

2-A*b^2*d*e+45*B*a^2*e^2-10*B*a*b*d*e+B*b^2*d^2)*x^9+1/10*b^9/e^2*(A*b*e+10*B*a*e-B*b*d)*x^10+1/11*b^10*B/e*x^

11+(A*a^10*e^11-10*A*a^9*b*d*e^10+45*A*a^8*b^2*d^2*e^9-120*A*a^7*b^3*d^3*e^8+210*A*a^6*b^4*d^4*e^7-252*A*a^5*b

^5*d^5*e^6+210*A*a^4*b^6*d^6*e^5-120*A*a^3*b^7*d^7*e^4+45*A*a^2*b^8*d^8*e^3-10*A*a*b^9*d^9*e^2+A*b^10*d^10*e-B

*a^10*d*e^10+10*B*a^9*b*d^2*e^9-45*B*a^8*b^2*d^3*e^8+120*B*a^7*b^3*d^4*e^7-210*B*a^6*b^4*d^5*e^6+252*B*a^5*b^5

*d^6*e^5-210*B*a^4*b^6*d^7*e^4+120*B*a^3*b^7*d^8*e^3-45*B*a^2*b^8*d^9*e^2+10*B*a*b^9*d^10*e-B*b^10*d^11)/e^12*

Fricas [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1805 vs. \(2 (328) = 656\).

Time = 0.23 (sec) , antiderivative size = 1805, normalized size of antiderivative = 5.19 \[ \int \frac {(a+b x)^{10} (A+B x)}{d+e x} \, dx=\text {Too large to display} \]

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

1/27720*(2520*B*b^10*e^11*x^11 - 2772*(B*b^10*d*e^10 - (10*B*a*b^9 + A*b^10)*e^11)*x^10 + 3080*(B*b^10*d^2*e^9

- (10*B*a*b^9 + A*b^10)*d*e^10 + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9 - 3465*(B*b^10*d^3*e^8 - (10*B*a*b^9 +

A*b^10)*d^2*e^9 + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^10 - 15*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^11)*x^8 + 3960*(B*b^1

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

8)*d*e^10 + 30*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 - 4620*(B*b^10*d^5*e^6 - (10*B*a*b^9 + A*b^10)*d^4*e^7 +

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

d*e^10 - 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 5544*(B*b^10*d^6*e^5 - (10*B*a*b^9 + A*b^10)*d^5*e^6 + 5*(

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

*e^9 - 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 + 42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^11)*x^5 - 6930*(B*b^10*d^7*e^4

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

7 + 30*(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 + 42*(5*B*a^6*b^4 + 6*A*a^

5*b^5)*d*e^10 - 30*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^11)*x^4 + 9240*(B*b^10*d^8*e^3 - (10*B*a*b^9 + A*b^10)*d^7*e^

4 + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5 - 15*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^6 + 30*(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 + 42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*e^9 - 30*(4*B*a^7*b^

3 + 7*A*a^6*b^4)*d*e^10 + 15*(3*B*a^8*b^2 + 8*A*a^7*b^3)*e^11)*x^3 - 13860*(B*b^10*d^9*e^2 - (10*B*a*b^9 + A*b

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

+ 4*A*a^3*b^7)*d^5*e^6 - 42*(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 - 30*

(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*e^9 + 15*(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 + 27720*(B*b^10*d^10*e - (10*B*a*b^9 + A*b^10)*d^9*e^2 + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^3 - 15*(8*B*a

^3*b^7 + 3*A*a^2*b^8)*d^7*e^4 + 30*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5 - 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^

6 + 42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*e^7 - 30*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*e^8 + 15*(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 + (B*a^10 + 10*A*a^9*b)*e^11)*x - 27720*(B*b^10*d^11 - A*a

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

*d^8*e^3 + 30*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4 - 42*(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 - 30*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4*e^7 + 15*(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 + (B*a^10 + 10*A*a^9*b)*d*e^10)*log(e*x + d))/e^12

Sympy [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1912 vs. \(2 (298) = 596\).

Time = 1.96 (sec) , antiderivative size = 1912, normalized size of antiderivative = 5.49 \[ \int \frac {(a+b x)^{10} (A+B x)}{d+e x} \, dx=\text {Too large to display} \]

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

*b**10*d/(9*e**2) + 5*B*a**2*b**8/e - 10*B*a*b**9*d/(9*e**2) + B*b**10*d**2/(9*e**3)) + x**8*(45*A*a**2*b**8/(

8*e) - 5*A*a*b**9*d/(4*e**2) + A*b**10*d**2/(8*e**3) + 15*B*a**3*b**7/e - 45*B*a**2*b**8*d/(8*e**2) + 5*B*a*b*

*9*d**2/(4*e**3) - B*b**10*d**3/(8*e**4)) + x**7*(120*A*a**3*b**7/(7*e) - 45*A*a**2*b**8*d/(7*e**2) + 10*A*a*b

**9*d**2/(7*e**3) - A*b**10*d**3/(7*e**4) + 30*B*a**4*b**6/e - 120*B*a**3*b**7*d/(7*e**2) + 45*B*a**2*b**8*d**

2/(7*e**3) - 10*B*a*b**9*d**3/(7*e**4) + B*b**10*d**4/(7*e**5)) + x**6*(35*A*a**4*b**6/e - 20*A*a**3*b**7*d/e*

*2 + 15*A*a**2*b**8*d**2/(2*e**3) - 5*A*a*b**9*d**3/(3*e**4) + A*b**10*d**4/(6*e**5) + 42*B*a**5*b**5/e - 35*B

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

10*d**5/(6*e**6)) + x**5*(252*A*a**5*b**5/(5*e) - 42*A*a**4*b**6*d/e**2 + 24*A*a**3*b**7*d**2/e**3 - 9*A*a**2*

b**8*d**3/e**4 + 2*A*a*b**9*d**4/e**5 - A*b**10*d**5/(5*e**6) + 42*B*a**6*b**4/e - 252*B*a**5*b**5*d/(5*e**2)

+ 42*B*a**4*b**6*d**2/e**3 - 24*B*a**3*b**7*d**3/e**4 + 9*B*a**2*b**8*d**4/e**5 - 2*B*a*b**9*d**5/e**6 + B*b**

10*d**6/(5*e**7)) + x**4*(105*A*a**6*b**4/(2*e) - 63*A*a**5*b**5*d/e**2 + 105*A*a**4*b**6*d**2/(2*e**3) - 30*A

*a**3*b**7*d**3/e**4 + 45*A*a**2*b**8*d**4/(4*e**5) - 5*A*a*b**9*d**5/(2*e**6) + A*b**10*d**6/(4*e**7) + 30*B*

a**7*b**3/e - 105*B*a**6*b**4*d/(2*e**2) + 63*B*a**5*b**5*d**2/e**3 - 105*B*a**4*b**6*d**3/(2*e**4) + 30*B*a**

3*b**7*d**4/e**5 - 45*B*a**2*b**8*d**5/(4*e**6) + 5*B*a*b**9*d**6/(2*e**7) - B*b**10*d**7/(4*e**8)) + x**3*(40

*A*a**7*b**3/e - 70*A*a**6*b**4*d/e**2 + 84*A*a**5*b**5*d**2/e**3 - 70*A*a**4*b**6*d**3/e**4 + 40*A*a**3*b**7*

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

40*B*a**7*b**3*d/e**2 + 70*B*a**6*b**4*d**2/e**3 - 84*B*a**5*b**5*d**3/e**4 + 70*B*a**4*b**6*d**4/e**5 - 40*B*

a**3*b**7*d**5/e**6 + 15*B*a**2*b**8*d**6/e**7 - 10*B*a*b**9*d**7/(3*e**8) + B*b**10*d**8/(3*e**9)) + x**2*(45

*A*a**8*b**2/(2*e) - 60*A*a**7*b**3*d/e**2 + 105*A*a**6*b**4*d**2/e**3 - 126*A*a**5*b**5*d**3/e**4 + 105*A*a**

4*b**6*d**4/e**5 - 60*A*a**3*b**7*d**5/e**6 + 45*A*a**2*b**8*d**6/(2*e**7) - 5*A*a*b**9*d**7/e**8 + A*b**10*d*

*8/(2*e**9) + 5*B*a**9*b/e - 45*B*a**8*b**2*d/(2*e**2) + 60*B*a**7*b**3*d**2/e**3 - 105*B*a**6*b**4*d**3/e**4

+ 126*B*a**5*b**5*d**4/e**5 - 105*B*a**4*b**6*d**5/e**6 + 60*B*a**3*b**7*d**6/e**7 - 45*B*a**2*b**8*d**7/(2*e*

*8) + 5*B*a*b**9*d**8/e**9 - B*b**10*d**9/(2*e**10)) + x*(10*A*a**9*b/e - 45*A*a**8*b**2*d/e**2 + 120*A*a**7*b

**3*d**2/e**3 - 210*A*a**6*b**4*d**3/e**4 + 252*A*a**5*b**5*d**4/e**5 - 210*A*a**4*b**6*d**5/e**6 + 120*A*a**3

*b**7*d**6/e**7 - 45*A*a**2*b**8*d**7/e**8 + 10*A*a*b**9*d**8/e**9 - A*b**10*d**9/e**10 + B*a**10/e - 10*B*a**

9*b*d/e**2 + 45*B*a**8*b**2*d**2/e**3 - 120*B*a**7*b**3*d**3/e**4 + 210*B*a**6*b**4*d**4/e**5 - 252*B*a**5*b**

5*d**5/e**6 + 210*B*a**4*b**6*d**6/e**7 - 120*B*a**3*b**7*d**7/e**8 + 45*B*a**2*b**8*d**8/e**9 - 10*B*a*b**9*d

**9/e**10 + B*b**10*d**10/e**11) - (-A*e + B*d)*(a*e - b*d)**10*log(d + e*x)/e**12

Maxima [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1804 vs. \(2 (328) = 656\).

Time = 0.22 (sec) , antiderivative size = 1804, normalized size of antiderivative = 5.18 \[ \int \frac {(a+b x)^{10} (A+B x)}{d+e x} \, dx=\text {Too large to display} \]

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

1/27720*(2520*B*b^10*e^10*x^11 - 2772*(B*b^10*d*e^9 - (10*B*a*b^9 + A*b^10)*e^10)*x^10 + 3080*(B*b^10*d^2*e^8

- (10*B*a*b^9 + A*b^10)*d*e^9 + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*e^10)*x^9 - 3465*(B*b^10*d^3*e^7 - (10*B*a*b^9 + A

*b^10)*d^2*e^8 + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^9 - 15*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^10)*x^8 + 3960*(B*b^10*d

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

d*e^9 + 30*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^10)*x^7 - 4620*(B*b^10*d^5*e^5 - (10*B*a*b^9 + A*b^10)*d^4*e^6 + 5*(9

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

9 - 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^10)*x^6 + 5544*(B*b^10*d^6*e^4 - (10*B*a*b^9 + A*b^10)*d^5*e^5 + 5*(9*B*a

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

- 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^9 + 42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^10)*x^5 - 6930*(B*b^10*d^7*e^3 - (10

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

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

*d*e^9 - 30*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^10)*x^4 + 9240*(B*b^10*d^8*e^2 - (10*B*a*b^9 + A*b^10)*d^7*e^3 + 5*(

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

*e^6 - 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^7 + 42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*e^8 - 30*(4*B*a^7*b^3 + 7*A

*a^6*b^4)*d*e^9 + 15*(3*B*a^8*b^2 + 8*A*a^7*b^3)*e^10)*x^3 - 13860*(B*b^10*d^9*e - (10*B*a*b^9 + A*b^10)*d^8*e

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

b^7)*d^5*e^5 - 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*e^6 + 42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*e^7 - 30*(4*B*a^7*b

^3 + 7*A*a^6*b^4)*d^2*e^8 + 15*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d*e^9 - 5*(2*B*a^9*b + 9*A*a^8*b^2)*e^10)*x^2 + 277

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

2*b^8)*d^7*e^3 + 30*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^4 - 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^5 + 42*(5*B*a^6

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

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

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

*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4 - 42*(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 - 30*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4*e^7 + 15*(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 + (B*a^10 + 10*A*a^9*b)*d*e^10)*log(e*x + d)/e^12

Giac [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2230 vs. \(2 (328) = 656\).

Time = 0.30 (sec) , antiderivative size = 2230, normalized size of antiderivative = 6.41 \[ \int \frac {(a+b x)^{10} (A+B x)}{d+e x} \, dx=\text {Too large to display} \]

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

1/27720*(2520*B*b^10*e^10*x^11 - 2772*B*b^10*d*e^9*x^10 + 27720*B*a*b^9*e^10*x^10 + 2772*A*b^10*e^10*x^10 + 30

80*B*b^10*d^2*e^8*x^9 - 30800*B*a*b^9*d*e^9*x^9 - 3080*A*b^10*d*e^9*x^9 + 138600*B*a^2*b^8*e^10*x^9 + 30800*A*

a*b^9*e^10*x^9 - 3465*B*b^10*d^3*e^7*x^8 + 34650*B*a*b^9*d^2*e^8*x^8 + 3465*A*b^10*d^2*e^8*x^8 - 155925*B*a^2*

b^8*d*e^9*x^8 - 34650*A*a*b^9*d*e^9*x^8 + 415800*B*a^3*b^7*e^10*x^8 + 155925*A*a^2*b^8*e^10*x^8 + 3960*B*b^10*

d^4*e^6*x^7 - 39600*B*a*b^9*d^3*e^7*x^7 - 3960*A*b^10*d^3*e^7*x^7 + 178200*B*a^2*b^8*d^2*e^8*x^7 + 39600*A*a*b

^9*d^2*e^8*x^7 - 475200*B*a^3*b^7*d*e^9*x^7 - 178200*A*a^2*b^8*d*e^9*x^7 + 831600*B*a^4*b^6*e^10*x^7 + 475200*

A*a^3*b^7*e^10*x^7 - 4620*B*b^10*d^5*e^5*x^6 + 46200*B*a*b^9*d^4*e^6*x^6 + 4620*A*b^10*d^4*e^6*x^6 - 207900*B*

a^2*b^8*d^3*e^7*x^6 - 46200*A*a*b^9*d^3*e^7*x^6 + 554400*B*a^3*b^7*d^2*e^8*x^6 + 207900*A*a^2*b^8*d^2*e^8*x^6

- 970200*B*a^4*b^6*d*e^9*x^6 - 554400*A*a^3*b^7*d*e^9*x^6 + 1164240*B*a^5*b^5*e^10*x^6 + 970200*A*a^4*b^6*e^10

*x^6 + 5544*B*b^10*d^6*e^4*x^5 - 55440*B*a*b^9*d^5*e^5*x^5 - 5544*A*b^10*d^5*e^5*x^5 + 249480*B*a^2*b^8*d^4*e^

6*x^5 + 55440*A*a*b^9*d^4*e^6*x^5 - 665280*B*a^3*b^7*d^3*e^7*x^5 - 249480*A*a^2*b^8*d^3*e^7*x^5 + 1164240*B*a^

4*b^6*d^2*e^8*x^5 + 665280*A*a^3*b^7*d^2*e^8*x^5 - 1397088*B*a^5*b^5*d*e^9*x^5 - 1164240*A*a^4*b^6*d*e^9*x^5 +

1164240*B*a^6*b^4*e^10*x^5 + 1397088*A*a^5*b^5*e^10*x^5 - 6930*B*b^10*d^7*e^3*x^4 + 69300*B*a*b^9*d^6*e^4*x^4

+ 6930*A*b^10*d^6*e^4*x^4 - 311850*B*a^2*b^8*d^5*e^5*x^4 - 69300*A*a*b^9*d^5*e^5*x^4 + 831600*B*a^3*b^7*d^4*e

^6*x^4 + 311850*A*a^2*b^8*d^4*e^6*x^4 - 1455300*B*a^4*b^6*d^3*e^7*x^4 - 831600*A*a^3*b^7*d^3*e^7*x^4 + 1746360

*B*a^5*b^5*d^2*e^8*x^4 + 1455300*A*a^4*b^6*d^2*e^8*x^4 - 1455300*B*a^6*b^4*d*e^9*x^4 - 1746360*A*a^5*b^5*d*e^9

*x^4 + 831600*B*a^7*b^3*e^10*x^4 + 1455300*A*a^6*b^4*e^10*x^4 + 9240*B*b^10*d^8*e^2*x^3 - 92400*B*a*b^9*d^7*e^

3*x^3 - 9240*A*b^10*d^7*e^3*x^3 + 415800*B*a^2*b^8*d^6*e^4*x^3 + 92400*A*a*b^9*d^6*e^4*x^3 - 1108800*B*a^3*b^7

*d^5*e^5*x^3 - 415800*A*a^2*b^8*d^5*e^5*x^3 + 1940400*B*a^4*b^6*d^4*e^6*x^3 + 1108800*A*a^3*b^7*d^4*e^6*x^3 -

2328480*B*a^5*b^5*d^3*e^7*x^3 - 1940400*A*a^4*b^6*d^3*e^7*x^3 + 1940400*B*a^6*b^4*d^2*e^8*x^3 + 2328480*A*a^5*

b^5*d^2*e^8*x^3 - 1108800*B*a^7*b^3*d*e^9*x^3 - 1940400*A*a^6*b^4*d*e^9*x^3 + 415800*B*a^8*b^2*e^10*x^3 + 1108

800*A*a^7*b^3*e^10*x^3 - 13860*B*b^10*d^9*e*x^2 + 138600*B*a*b^9*d^8*e^2*x^2 + 13860*A*b^10*d^8*e^2*x^2 - 6237

00*B*a^2*b^8*d^7*e^3*x^2 - 138600*A*a*b^9*d^7*e^3*x^2 + 1663200*B*a^3*b^7*d^6*e^4*x^2 + 623700*A*a^2*b^8*d^6*e

^4*x^2 - 2910600*B*a^4*b^6*d^5*e^5*x^2 - 1663200*A*a^3*b^7*d^5*e^5*x^2 + 3492720*B*a^5*b^5*d^4*e^6*x^2 + 29106

00*A*a^4*b^6*d^4*e^6*x^2 - 2910600*B*a^6*b^4*d^3*e^7*x^2 - 3492720*A*a^5*b^5*d^3*e^7*x^2 + 1663200*B*a^7*b^3*d

^2*e^8*x^2 + 2910600*A*a^6*b^4*d^2*e^8*x^2 - 623700*B*a^8*b^2*d*e^9*x^2 - 1663200*A*a^7*b^3*d*e^9*x^2 + 138600

*B*a^9*b*e^10*x^2 + 623700*A*a^8*b^2*e^10*x^2 + 27720*B*b^10*d^10*x - 277200*B*a*b^9*d^9*e*x - 27720*A*b^10*d^

9*e*x + 1247400*B*a^2*b^8*d^8*e^2*x + 277200*A*a*b^9*d^8*e^2*x - 3326400*B*a^3*b^7*d^7*e^3*x - 1247400*A*a^2*b

^8*d^7*e^3*x + 5821200*B*a^4*b^6*d^6*e^4*x + 3326400*A*a^3*b^7*d^6*e^4*x - 6985440*B*a^5*b^5*d^5*e^5*x - 58212

00*A*a^4*b^6*d^5*e^5*x + 5821200*B*a^6*b^4*d^4*e^6*x + 6985440*A*a^5*b^5*d^4*e^6*x - 3326400*B*a^7*b^3*d^3*e^7

*x - 5821200*A*a^6*b^4*d^3*e^7*x + 1247400*B*a^8*b^2*d^2*e^8*x + 3326400*A*a^7*b^3*d^2*e^8*x - 277200*B*a^9*b*

d*e^9*x - 1247400*A*a^8*b^2*d*e^9*x + 27720*B*a^10*e^10*x + 277200*A*a^9*b*e^10*x)/e^11 - (B*b^10*d^11 - 10*B*

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

8*d^8*e^3 + 210*B*a^4*b^6*d^7*e^4 + 120*A*a^3*b^7*d^7*e^4 - 252*B*a^5*b^5*d^6*e^5 - 210*A*a^4*b^6*d^6*e^5 + 21

0*B*a^6*b^4*d^5*e^6 + 252*A*a^5*b^5*d^5*e^6 - 120*B*a^7*b^3*d^4*e^7 - 210*A*a^6*b^4*d^4*e^7 + 45*B*a^8*b^2*d^3

*e^8 + 120*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 + B*a^10*d*e^10 + 10*A*a^9*b*d*e^10 -

Mupad [B] (veriﬁcation not implemented)

Time = 1.54 (sec) , antiderivative size = 1795, normalized size of antiderivative = 5.16 \[ \int \frac {(a+b x)^{10} (A+B x)}{d+e x} \, dx=\text {Too large to display} \]

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

(15*a^2*b^7*(3*A*b + 8*B*a))/e))/e - (30*a^3*b^6*(4*A*b + 7*B*a))/e))/e + (42*a^4*b^5*(5*A*b + 6*B*a))/e))/e -

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

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

))/e))/e - (30*a^3*b^6*(4*A*b + 7*B*a))/e))/e + (42*a^4*b^5*(5*A*b + 6*B*a))/e))/e - (42*a^5*b^4*(6*A*b + 5*B*

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

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

/e))/e - (30*a^3*b^6*(4*A*b + 7*B*a))/e))/e + (42*a^4*b^5*(5*A*b + 6*B*a))/e))/(5*e) - (42*a^5*b^4*(6*A*b + 5*

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

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

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

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

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

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

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

*b + 8*B*a))/e))/e - (30*a^3*b^6*(4*A*b + 7*B*a))/e))/e + (42*a^4*b^5*(5*A*b + 6*B*a))/e))/e - (42*a^5*b^4*(6*

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

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

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

(4*A*b + 7*B*a))/e))/e + (42*a^4*b^5*(5*A*b + 6*B*a))/e))/e - (42*a^5*b^4*(6*A*b + 5*B*a))/e))/e + (30*a^6*b^3

*(7*A*b + 4*B*a))/e))/e - (15*a^7*b^2*(8*A*b + 3*B*a))/e))/e + (5*a^8*b*(9*A*b + 2*B*a))/e))/e) + x^10*((A*b^1

0 + 10*B*a*b^9)/(10*e) - (B*b^10*d)/(10*e^2)) + (log(d + e*x)*(A*a^10*e^11 - B*b^10*d^11 + A*b^10*d^10*e - B*a

^10*d*e^10 - 10*A*a*b^9*d^9*e^2 + 10*B*a^9*b*d^2*e^9 + 45*A*a^2*b^8*d^8*e^3 - 120*A*a^3*b^7*d^7*e^4 + 210*A*a^

4*b^6*d^6*e^5 - 252*A*a^5*b^5*d^5*e^6 + 210*A*a^6*b^4*d^4*e^7 - 120*A*a^7*b^3*d^3*e^8 + 45*A*a^8*b^2*d^2*e^9 -

45*B*a^2*b^8*d^9*e^2 + 120*B*a^3*b^7*d^8*e^3 - 210*B*a^4*b^6*d^7*e^4 + 252*B*a^5*b^5*d^6*e^5 - 210*B*a^6*b^4*

d^5*e^6 + 120*B*a^7*b^3*d^4*e^7 - 45*B*a^8*b^2*d^3*e^8 - 10*A*a^9*b*d*e^10 + 10*B*a*b^9*d^10*e))/e^12 + (B*b^1

\(\int \frac {(a+b x)^{10} (A+B x)}{d+e x} \, dx\) [1089]

Optimal result