[next] [prev] [prev-tail] [tail] [up]

3.1097 \(\int \frac{(a+b x)^{10} (A+B x)}{(d+e x)^9} \, dx\)

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

[Out]

(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*
e)*Log[d + e*x])/e^12

________________________________________________________________________________________

Rubi [A]  time = 0.967882, antiderivative size = 445, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {77} \[ -\frac{b^9 (d+e x)^2 (-10 a B e-A b e+11 b B d)}{2 e^{12}}+\frac{5 b^8 x (b d-a e) (-9 a B e-2 A b e+11 b B d)}{e^{11}}-\frac{30 b^6 (b d-a e)^3 (-7 a B e-4 A b e+11 b B d)}{e^{12} (d+e x)}+\frac{21 b^5 (b d-a e)^4 (-6 a B e-5 A b e+11 b B d)}{e^{12} (d+e x)^2}-\frac{14 b^4 (b d-a e)^5 (-5 a B e-6 A b e+11 b B d)}{e^{12} (d+e x)^3}+\frac{15 b^3 (b d-a e)^6 (-4 a B e-7 A b e+11 b B d)}{2 e^{12} (d+e x)^4}-\frac{3 b^2 (b d-a e)^7 (-3 a B e-8 A b e+11 b B d)}{e^{12} (d+e x)^5}-\frac{15 b^7 (b d-a e)^2 \log (d+e x) (-8 a B e-3 A b e+11 b B d)}{e^{12}}+\frac{5 b (b d-a e)^8 (-2 a B e-9 A b e+11 b B d)}{6 e^{12} (d+e x)^6}-\frac{(b d-a e)^9 (-a B e-10 A b e+11 b B d)}{7 e^{12} (d+e x)^7}+\frac{(b d-a e)^{10} (B d-A e)}{8 e^{12} (d+e x)^8}+\frac{b^{10} B (d+e x)^3}{3 e^{12}} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^10*(A + B*x))/(d + e*x)^9,x]

[Out]

(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*
e)*Log[d + e*x])/e^12

Rule 77

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*} \int \frac{(a+b x)^{10} (A+B x)}{(d+e x)^9} \, dx &=\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]  time = 0.630268, size = 415, normalized size = 0.93 \[ \frac{-168 b^8 e x \left (-45 a^2 B e^2-10 a b e (A e-9 B d)+9 b^2 d (A e-5 B d)\right )+84 b^9 e^2 x^2 (10 a B e+A b e-9 b B d)-\frac{5040 b^6 (b d-a e)^3 (-7 a B e-4 A b e+11 b B d)}{d+e x}+\frac{3528 b^5 (b d-a e)^4 (-6 a B e-5 A b e+11 b B d)}{(d+e x)^2}-\frac{2352 b^4 (b d-a e)^5 (-5 a B e-6 A b e+11 b B d)}{(d+e x)^3}+\frac{1260 b^3 (b d-a e)^6 (-4 a B e-7 A b e+11 b B d)}{(d+e x)^4}-\frac{504 b^2 (b d-a e)^7 (-3 a B e-8 A b e+11 b B d)}{(d+e x)^5}-2520 b^7 (b d-a e)^2 \log (d+e x) (-8 a B e-3 A b e+11 b B d)+\frac{140 b (b d-a e)^8 (-2 a B e-9 A b e+11 b B d)}{(d+e x)^6}-\frac{24 (b d-a e)^9 (-a B e-10 A b e+11 b B d)}{(d+e x)^7}+\frac{21 (b d-a e)^{10} (B d-A e)}{(d+e x)^8}+56 b^{10} B e^3 x^3}{168 e^{12}} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^10*(A + B*x))/(d + e*x)^9,x]

[Out]

(-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]  time = 0.031, size = 2857, normalized size = 6.4 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9*b^2/e^4/(e*x+d)^5*B*a^8-33*b^10/e^12/(e*x+d)^5*B*d^8-1/7/e^2/(e*x+d)^7*B*a^10-1/8/e/(e*x+d)^8*a^10*A+1/3*b^
10/e^9*B*x^3+1/2*b^10/e^9*A*x^2-1470*b^6/e^8/(e*x+d)^5*B*a^4*d^4+1344*b^7/e^9/(e*x+d)^5*B*a^3*d^5-756*b^8/e^10
/(e*x+d)^5*B*a^2*d^6+240*b^9/e^11/(e*x+d)^5*B*a*d^7+5/4/e^2/(e*x+d)^8*A*d*a^9*b-45/8/e^3/(e*x+d)^8*A*d^2*a^8*b
^2+480/7/e^5/(e*x+d)^7*B*a^7*b^3*d^3-150/e^6/(e*x+d)^7*B*a^6*b^4*d^4+216/e^7/(e*x+d)^7*B*a^5*b^5*d^5-210/e^8/(
e*x+d)^7*B*a^4*b^6*d^6+960/7/e^9/(e*x+d)^7*B*a^3*b^7*d^7-405/7/e^10/(e*x+d)^7*B*a^2*b^8*d^8+100/7/e^11/(e*x+d)
^7*B*a*b^9*d^9+420*b^7/e^8/(e*x+d)^2*A*a^3*d-630*b^8/e^9/(e*x+d)^2*A*a^2*d^2+420*b^9/e^10/(e*x+d)^2*A*a*d^3+73
5*b^6/e^8/(e*x+d)^2*B*a^4*d-1680*b^7/e^9/(e*x+d)^2*B*a^3*d^2+1890*b^8/e^10/(e*x+d)^2*B*a^2*d^3-1050*b^9/e^11/(
e*x+d)^2*B*a*d^4+360*b^8/e^9/(e*x+d)*A*a^2*d+10*b^9/e^9*A*a*x-9*b^10/e^10*A*d*x+45*b^8/e^9*a^2*B*x+45*b^10/e^1
1*B*d^2*x+1/8/e^12/(e*x+d)^8*b^10*B*d^11-10/7/e^2/(e*x+d)^7*A*a^9*b+10/7/e^11/(e*x+d)^7*A*b^10*d^9-11/7/e^12/(
e*x+d)^7*b^10*B*d^10-105*b^6/e^7/(e*x+d)^2*A*a^4-105*b^10/e^11/(e*x+d)^2*A*d^4-126*b^5/e^7/(e*x+d)^2*B*a^5+231
*b^10/e^12/(e*x+d)^2*B*d^5-120*b^7/e^8/(e*x+d)*A*a^3+120*b^10/e^11/(e*x+d)*A*d^3-210*b^6/e^8/(e*x+d)*B*a^4-330
*b^10/e^12/(e*x+d)*B*d^4-24*b^3/e^4/(e*x+d)^5*A*a^7+24*b^10/e^11/(e*x+d)^5*A*d^7-1/8/e^11/(e*x+d)^8*A*b^10*d^1
0+1/8/e^2/(e*x+d)^8*B*d*a^10-105/2*b^4/e^5/(e*x+d)^4*A*a^6-105/2*b^10/e^11/(e*x+d)^4*A*d^6-30*b^3/e^5/(e*x+d)^
4*B*a^7+165/2*b^10/e^12/(e*x+d)^4*B*d^7-84*b^5/e^6/(e*x+d)^3*A*a^5+84*b^10/e^11/(e*x+d)^3*A*d^5-70*b^4/e^6/(e*
x+d)^3*B*a^6-154*b^10/e^12/(e*x+d)^3*B*d^6+5*b^9/e^9*B*x^2*a-9/2*b^10/e^10*B*x^2*d-15/2*b^2/e^3/(e*x+d)^6*A*a^
8-15/2*b^10/e^11/(e*x+d)^6*A*d^8-5/3*b/e^3/(e*x+d)^6*B*a^9+55/6*b^10/e^12/(e*x+d)^6*B*d^9+45*b^8/e^9*ln(e*x+d)
*A*a^2+45*b^10/e^11*ln(e*x+d)*A*d^2+120*b^7/e^9*ln(e*x+d)*B*a^3-165*b^10/e^12*ln(e*x+d)*B*d^3-168*b^9/e^10/(e*
x+d)^5*A*a*d^6+96*b^3/e^5/(e*x+d)^5*B*a^7*d-420*b^4/e^6/(e*x+d)^5*B*a^6*d^2+1008*b^5/e^7/(e*x+d)^5*B*a^5*d^3-9
0*b^9/e^10*B*a*d*x+60*b^3/e^4/(e*x+d)^6*A*a^7*d-210*b^4/e^5/(e*x+d)^6*A*a^6*d^2+420*b^5/e^6/(e*x+d)^6*A*a^5*d^
3-525*b^6/e^7/(e*x+d)^6*A*a^4*d^4+420*b^7/e^8/(e*x+d)^6*A*a^3*d^5-210*b^8/e^9/(e*x+d)^6*A*a^2*d^6+60*b^9/e^10/
(e*x+d)^6*A*a*d^7+45/2*b^2/e^4/(e*x+d)^6*B*a^8*d-120*b^3/e^5/(e*x+d)^6*B*a^7*d^2+350*b^4/e^6/(e*x+d)^6*B*a^6*d
^3-630*b^5/e^7/(e*x+d)^6*B*a^5*d^4+735*b^6/e^8/(e*x+d)^6*B*a^4*d^5-560*b^7/e^9/(e*x+d)^6*B*a^3*d^6+270*b^8/e^1
0/(e*x+d)^6*B*a^2*d^7-75*b^9/e^11/(e*x+d)^6*B*a*d^8+420*b^6/e^7/(e*x+d)^3*A*a^4*d-840*b^7/e^8/(e*x+d)^3*A*a^3*
d^2+840*b^8/e^9/(e*x+d)^3*A*a^2*d^3-420*b^9/e^10/(e*x+d)^3*A*a*d^4+504*b^5/e^7/(e*x+d)^3*B*a^5*d-1470*b^6/e^8/
(e*x+d)^3*B*a^4*d^2+2240*b^7/e^9/(e*x+d)^3*B*a^3*d^3-1890*b^8/e^10/(e*x+d)^3*B*a^2*d^4+840*b^9/e^11/(e*x+d)^3*
B*a*d^5+90/7/e^3/(e*x+d)^7*A*a^8*b^2*d-360/7/e^4/(e*x+d)^7*A*a^7*b^3*d^2+120/e^5/(e*x+d)^7*A*a^6*b^4*d^3-180/e
^6/(e*x+d)^7*A*a^5*b^5*d^4+180/e^7/(e*x+d)^7*A*a^4*b^6*d^5-120/e^8/(e*x+d)^7*A*a^3*b^7*d^6+360/7/e^9/(e*x+d)^7
*A*a^2*b^8*d^7-90/7/e^10/(e*x+d)^7*A*a*b^9*d^8+20/7/e^3/(e*x+d)^7*B*a^9*b*d-135/7/e^4/(e*x+d)^7*B*a^8*b^2*d^2+
15/e^4/(e*x+d)^8*A*d^3*a^7*b^3-105/4/e^5/(e*x+d)^8*A*d^4*a^6*b^4+63/2/e^6/(e*x+d)^8*A*d^5*a^5*b^5-105/4/e^7/(e
*x+d)^8*A*d^6*a^4*b^6+15/e^8/(e*x+d)^8*A*d^7*a^3*b^7-45/8/e^9/(e*x+d)^8*A*d^8*a^2*b^8+5/4/e^10/(e*x+d)^8*A*a*b
^9*d^9-5/4/e^3/(e*x+d)^8*B*d^2*a^9*b+45/8/e^4/(e*x+d)^8*B*d^3*a^8*b^2-15/e^5/(e*x+d)^8*B*d^4*a^7*b^3+105/4/e^6
/(e*x+d)^8*B*d^5*a^6*b^4-63/2/e^7/(e*x+d)^8*B*d^6*a^5*b^5+105/4/e^8/(e*x+d)^8*B*d^7*a^4*b^6-15/e^9/(e*x+d)^8*B
*d^8*a^3*b^7+45/8/e^10/(e*x+d)^8*B*a^2*b^8*d^9-5/4/e^11/(e*x+d)^8*B*a*b^9*d^10+315*b^5/e^6/(e*x+d)^4*A*a^5*d-1
575/2*b^6/e^7/(e*x+d)^4*A*a^4*d^2+1050*b^7/e^8/(e*x+d)^4*A*a^3*d^3-1575/2*b^8/e^9/(e*x+d)^4*A*a^2*d^4+315*b^9/
e^10/(e*x+d)^4*A*a*d^5+525/2*b^4/e^6/(e*x+d)^4*B*a^6*d-945*b^5/e^7/(e*x+d)^4*B*a^5*d^2+3675/2*b^6/e^8/(e*x+d)^
4*B*a^4*d^3-2100*b^7/e^9/(e*x+d)^4*B*a^3*d^4+2835/2*b^8/e^10/(e*x+d)^4*B*a^2*d^5-525*b^9/e^11/(e*x+d)^4*B*a*d^
6-360*b^9/e^10/(e*x+d)*A*a*d^2+960*b^7/e^9/(e*x+d)*B*a^3*d-1620*b^8/e^10/(e*x+d)*B*a^2*d^2+1200*b^9/e^11/(e*x+
d)*B*a*d^3-90*b^9/e^10*ln(e*x+d)*A*a*d-405*b^8/e^10*ln(e*x+d)*B*a^2*d+450*b^9/e^11*ln(e*x+d)*B*a*d^2+168*b^4/e
^5/(e*x+d)^5*A*a^6*d-504*b^5/e^6/(e*x+d)^5*A*a^5*d^2+840*b^6/e^7/(e*x+d)^5*A*a^4*d^3-840*b^7/e^8/(e*x+d)^5*A*a
^3*d^4+504*b^8/e^9/(e*x+d)^5*A*a^2*d^5

________________________________________________________________________________________

Maxima [B]  time = 1.94541, size = 2554, normalized size = 5.74 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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

________________________________________________________________________________________

Fricas [B]  time = 2.16176, size = 5775, normalized size = 12.98 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**10*(B*x+A)/(e*x+d)**9,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 2.34641, size = 2496, normalized size = 5.61 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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)*e^(-12)*log(abs(x*e + d)) + 1/6*(2*B*b^10*x^3*e^18 - 27*B*b^10*d*x^2*e^17 + 270*B*b^10*d^
2*x*e^16 + 30*B*a*b^9*x^2*e^18 + 3*A*b^10*x^2*e^18 - 540*B*a*b^9*d*x*e^17 - 54*A*b^10*d*x*e^17 + 270*B*a^2*b^8
*x*e^18 + 60*A*a*b^9*x*e^18)*e^(-27) - 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^10*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 - 7000*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 - 9207
0*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 + 6
30*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 - 5
2272*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 + 756*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^(-12)/(x*e + d)^8

[next] [prev] [prev-tail] [front] [up]