3.1066 \(\int (a+b x)^{10} (A+B x) (d+e x)^5 \, dx\)

Optimal. Leaf size=243 \[ \frac{e^4 (a+b x)^{16} (-6 a B e+A b e+5 b B d)}{16 b^7}+\frac{e^3 (a+b x)^{15} (b d-a e) (-3 a B e+A b e+2 b B d)}{3 b^7}+\frac{5 e^2 (a+b x)^{14} (b d-a e)^2 (-2 a B e+A b e+b B d)}{7 b^7}+\frac{5 e (a+b x)^{13} (b d-a e)^3 (-3 a B e+2 A b e+b B d)}{13 b^7}+\frac{(a+b x)^{12} (b d-a e)^4 (-6 a B e+5 A b e+b B d)}{12 b^7}+\frac{(a+b x)^{11} (A b-a B) (b d-a e)^5}{11 b^7}+\frac{B e^5 (a+b x)^{17}}{17 b^7} \]

[Out]

((A*b - a*B)*(b*d - a*e)^5*(a + b*x)^11)/(11*b^7) + ((b*d - a*e)^4*(b*B*d + 5*A*
b*e - 6*a*B*e)*(a + b*x)^12)/(12*b^7) + (5*e*(b*d - a*e)^3*(b*B*d + 2*A*b*e - 3*
a*B*e)*(a + b*x)^13)/(13*b^7) + (5*e^2*(b*d - a*e)^2*(b*B*d + A*b*e - 2*a*B*e)*(
a + b*x)^14)/(7*b^7) + (e^3*(b*d - a*e)*(2*b*B*d + A*b*e - 3*a*B*e)*(a + b*x)^15
)/(3*b^7) + (e^4*(5*b*B*d + A*b*e - 6*a*B*e)*(a + b*x)^16)/(16*b^7) + (B*e^5*(a
+ b*x)^17)/(17*b^7)

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Rubi [A]  time = 5.17353, antiderivative size = 243, 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 \[ \frac{e^4 (a+b x)^{16} (-6 a B e+A b e+5 b B d)}{16 b^7}+\frac{e^3 (a+b x)^{15} (b d-a e) (-3 a B e+A b e+2 b B d)}{3 b^7}+\frac{5 e^2 (a+b x)^{14} (b d-a e)^2 (-2 a B e+A b e+b B d)}{7 b^7}+\frac{5 e (a+b x)^{13} (b d-a e)^3 (-3 a B e+2 A b e+b B d)}{13 b^7}+\frac{(a+b x)^{12} (b d-a e)^4 (-6 a B e+5 A b e+b B d)}{12 b^7}+\frac{(a+b x)^{11} (A b-a B) (b d-a e)^5}{11 b^7}+\frac{B e^5 (a+b x)^{17}}{17 b^7} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^10*(A + B*x)*(d + e*x)^5,x]

[Out]

((A*b - a*B)*(b*d - a*e)^5*(a + b*x)^11)/(11*b^7) + ((b*d - a*e)^4*(b*B*d + 5*A*
b*e - 6*a*B*e)*(a + b*x)^12)/(12*b^7) + (5*e*(b*d - a*e)^3*(b*B*d + 2*A*b*e - 3*
a*B*e)*(a + b*x)^13)/(13*b^7) + (5*e^2*(b*d - a*e)^2*(b*B*d + A*b*e - 2*a*B*e)*(
a + b*x)^14)/(7*b^7) + (e^3*(b*d - a*e)*(2*b*B*d + A*b*e - 3*a*B*e)*(a + b*x)^15
)/(3*b^7) + (e^4*(5*b*B*d + A*b*e - 6*a*B*e)*(a + b*x)^16)/(16*b^7) + (B*e^5*(a
+ b*x)^17)/(17*b^7)

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**10*(B*x+A)*(e*x+d)**5,x)

[Out]

Timed out

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Mathematica [B]  time = 1.07009, size = 1509, normalized size = 6.21 \[ \frac{1}{17} b^{10} B e^5 x^{17}+\frac{1}{16} b^9 e^4 (5 b B d+A b e+10 a B e) x^{16}+\frac{1}{3} b^8 e^3 \left (d (2 B d+A e) b^2+2 a e (5 B d+A e) b+9 a^2 B e^2\right ) x^{15}+\frac{5}{14} b^7 e^2 \left (2 d^2 (B d+A e) b^3+10 a d e (2 B d+A e) b^2+9 a^2 e^2 (5 B d+A e) b+24 a^3 B e^3\right ) x^{14}+\frac{5}{13} b^6 e \left (d^3 (B d+2 A e) b^4+20 a d^2 e (B d+A e) b^3+45 a^2 d e^2 (2 B d+A e) b^2+24 a^3 e^3 (5 B d+A e) b+42 a^4 B e^4\right ) x^{13}+\frac{1}{12} b^5 \left (d^4 (B d+5 A e) b^5+50 a d^3 e (B d+2 A e) b^4+450 a^2 d^2 e^2 (B d+A e) b^3+600 a^3 d e^3 (2 B d+A e) b^2+210 a^4 e^4 (5 B d+A e) b+252 a^5 B e^5\right ) x^{12}+\frac{1}{11} b^4 \left (5 a B \left (2 b^5 d^5+45 a b^4 e d^4+240 a^2 b^3 e^2 d^3+420 a^3 b^2 e^3 d^2+252 a^4 b e^4 d+42 a^5 e^5\right )+A b \left (b^5 d^5+50 a b^4 e d^4+450 a^2 b^3 e^2 d^3+1200 a^3 b^2 e^3 d^2+1050 a^4 b e^4 d+252 a^5 e^5\right )\right ) x^{11}+\frac{1}{2} a b^3 \left (3 a B \left (3 b^5 d^5+40 a b^4 e d^4+140 a^2 b^3 e^2 d^3+168 a^3 b^2 e^3 d^2+70 a^4 b e^4 d+8 a^5 e^5\right )+A b \left (2 b^5 d^5+45 a b^4 e d^4+240 a^2 b^3 e^2 d^3+420 a^3 b^2 e^3 d^2+252 a^4 b e^4 d+42 a^5 e^5\right )\right ) x^{10}+\frac{5}{3} a^2 b^2 \left (a B \left (8 b^5 d^5+70 a b^4 e d^4+168 a^2 b^3 e^2 d^3+140 a^3 b^2 e^3 d^2+40 a^4 b e^4 d+3 a^5 e^5\right )+A b \left (3 b^5 d^5+40 a b^4 e d^4+140 a^2 b^3 e^2 d^3+168 a^3 b^2 e^3 d^2+70 a^4 b e^4 d+8 a^5 e^5\right )\right ) x^9+\frac{5}{8} a^3 b \left (a B \left (42 b^5 d^5+252 a b^4 e d^4+420 a^2 b^3 e^2 d^3+240 a^3 b^2 e^3 d^2+45 a^4 b e^4 d+2 a^5 e^5\right )+3 A b \left (8 b^5 d^5+70 a b^4 e d^4+168 a^2 b^3 e^2 d^3+140 a^3 b^2 e^3 d^2+40 a^4 b e^4 d+3 a^5 e^5\right )\right ) x^8+\frac{1}{7} a^4 \left (a B \left (252 b^5 d^5+1050 a b^4 e d^4+1200 a^2 b^3 e^2 d^3+450 a^3 b^2 e^3 d^2+50 a^4 b e^4 d+a^5 e^5\right )+5 A b \left (42 b^5 d^5+252 a b^4 e d^4+420 a^2 b^3 e^2 d^3+240 a^3 b^2 e^3 d^2+45 a^4 b e^4 d+2 a^5 e^5\right )\right ) x^7+\frac{1}{6} a^5 \left (5 a B d \left (42 b^4 d^4+120 a b^3 e d^3+90 a^2 b^2 e^2 d^2+20 a^3 b e^3 d+a^4 e^4\right )+A \left (252 b^5 d^5+1050 a b^4 e d^4+1200 a^2 b^3 e^2 d^3+450 a^3 b^2 e^3 d^2+50 a^4 b e^4 d+a^5 e^5\right )\right ) x^6+a^6 d \left (a B d \left (24 b^3 d^3+45 a b^2 e d^2+20 a^2 b e^2 d+2 a^3 e^3\right )+A \left (42 b^4 d^4+120 a b^3 e d^3+90 a^2 b^2 e^2 d^2+20 a^3 b e^3 d+a^4 e^4\right )\right ) x^5+\frac{5}{4} a^7 d^2 \left (a B d \left (9 b^2 d^2+10 a b e d+2 a^2 e^2\right )+A \left (24 b^3 d^3+45 a b^2 e d^2+20 a^2 b e^2 d+2 a^3 e^3\right )\right ) x^4+\frac{5}{3} a^8 d^3 \left (a B d (2 b d+a e)+A \left (9 b^2 d^2+10 a b e d+2 a^2 e^2\right )\right ) x^3+\frac{1}{2} a^9 d^4 (a B d+5 A (2 b d+a e)) x^2+a^{10} A d^5 x \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^10*(A + B*x)*(d + e*x)^5,x]

[Out]

a^10*A*d^5*x + (a^9*d^4*(a*B*d + 5*A*(2*b*d + a*e))*x^2)/2 + (5*a^8*d^3*(a*B*d*(
2*b*d + a*e) + A*(9*b^2*d^2 + 10*a*b*d*e + 2*a^2*e^2))*x^3)/3 + (5*a^7*d^2*(a*B*
d*(9*b^2*d^2 + 10*a*b*d*e + 2*a^2*e^2) + A*(24*b^3*d^3 + 45*a*b^2*d^2*e + 20*a^2
*b*d*e^2 + 2*a^3*e^3))*x^4)/4 + a^6*d*(a*B*d*(24*b^3*d^3 + 45*a*b^2*d^2*e + 20*a
^2*b*d*e^2 + 2*a^3*e^3) + A*(42*b^4*d^4 + 120*a*b^3*d^3*e + 90*a^2*b^2*d^2*e^2 +
20*a^3*b*d*e^3 + a^4*e^4))*x^5 + (a^5*(5*a*B*d*(42*b^4*d^4 + 120*a*b^3*d^3*e +
90*a^2*b^2*d^2*e^2 + 20*a^3*b*d*e^3 + a^4*e^4) + A*(252*b^5*d^5 + 1050*a*b^4*d^4
*e + 1200*a^2*b^3*d^3*e^2 + 450*a^3*b^2*d^2*e^3 + 50*a^4*b*d*e^4 + a^5*e^5))*x^6
)/6 + (a^4*(a*B*(252*b^5*d^5 + 1050*a*b^4*d^4*e + 1200*a^2*b^3*d^3*e^2 + 450*a^3
*b^2*d^2*e^3 + 50*a^4*b*d*e^4 + a^5*e^5) + 5*A*b*(42*b^5*d^5 + 252*a*b^4*d^4*e +
420*a^2*b^3*d^3*e^2 + 240*a^3*b^2*d^2*e^3 + 45*a^4*b*d*e^4 + 2*a^5*e^5))*x^7)/7
+ (5*a^3*b*(a*B*(42*b^5*d^5 + 252*a*b^4*d^4*e + 420*a^2*b^3*d^3*e^2 + 240*a^3*b
^2*d^2*e^3 + 45*a^4*b*d*e^4 + 2*a^5*e^5) + 3*A*b*(8*b^5*d^5 + 70*a*b^4*d^4*e + 1
68*a^2*b^3*d^3*e^2 + 140*a^3*b^2*d^2*e^3 + 40*a^4*b*d*e^4 + 3*a^5*e^5))*x^8)/8 +
(5*a^2*b^2*(a*B*(8*b^5*d^5 + 70*a*b^4*d^4*e + 168*a^2*b^3*d^3*e^2 + 140*a^3*b^2
*d^2*e^3 + 40*a^4*b*d*e^4 + 3*a^5*e^5) + A*b*(3*b^5*d^5 + 40*a*b^4*d^4*e + 140*a
^2*b^3*d^3*e^2 + 168*a^3*b^2*d^2*e^3 + 70*a^4*b*d*e^4 + 8*a^5*e^5))*x^9)/3 + (a*
b^3*(3*a*B*(3*b^5*d^5 + 40*a*b^4*d^4*e + 140*a^2*b^3*d^3*e^2 + 168*a^3*b^2*d^2*e
^3 + 70*a^4*b*d*e^4 + 8*a^5*e^5) + A*b*(2*b^5*d^5 + 45*a*b^4*d^4*e + 240*a^2*b^3
*d^3*e^2 + 420*a^3*b^2*d^2*e^3 + 252*a^4*b*d*e^4 + 42*a^5*e^5))*x^10)/2 + (b^4*(
5*a*B*(2*b^5*d^5 + 45*a*b^4*d^4*e + 240*a^2*b^3*d^3*e^2 + 420*a^3*b^2*d^2*e^3 +
252*a^4*b*d*e^4 + 42*a^5*e^5) + A*b*(b^5*d^5 + 50*a*b^4*d^4*e + 450*a^2*b^3*d^3*
e^2 + 1200*a^3*b^2*d^2*e^3 + 1050*a^4*b*d*e^4 + 252*a^5*e^5))*x^11)/11 + (b^5*(2
52*a^5*B*e^5 + 450*a^2*b^3*d^2*e^2*(B*d + A*e) + 600*a^3*b^2*d*e^3*(2*B*d + A*e)
+ 210*a^4*b*e^4*(5*B*d + A*e) + 50*a*b^4*d^3*e*(B*d + 2*A*e) + b^5*d^4*(B*d + 5
*A*e))*x^12)/12 + (5*b^6*e*(42*a^4*B*e^4 + 20*a*b^3*d^2*e*(B*d + A*e) + 45*a^2*b
^2*d*e^2*(2*B*d + A*e) + 24*a^3*b*e^3*(5*B*d + A*e) + b^4*d^3*(B*d + 2*A*e))*x^1
3)/13 + (5*b^7*e^2*(24*a^3*B*e^3 + 2*b^3*d^2*(B*d + A*e) + 10*a*b^2*d*e*(2*B*d +
A*e) + 9*a^2*b*e^2*(5*B*d + A*e))*x^14)/14 + (b^8*e^3*(9*a^2*B*e^2 + b^2*d*(2*B
*d + A*e) + 2*a*b*e*(5*B*d + A*e))*x^15)/3 + (b^9*e^4*(5*b*B*d + A*b*e + 10*a*B*
e)*x^16)/16 + (b^10*B*e^5*x^17)/17

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Maple [B]  time = 0.004, size = 1621, normalized size = 6.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^10*(B*x+A)*(e*x+d)^5,x)

[Out]

1/17*b^10*B*e^5*x^17+1/16*((A*b^10+10*B*a*b^9)*e^5+5*b^10*B*d*e^4)*x^16+1/15*((1
0*A*a*b^9+45*B*a^2*b^8)*e^5+5*(A*b^10+10*B*a*b^9)*d*e^4+10*b^10*B*d^2*e^3)*x^15+
1/14*((45*A*a^2*b^8+120*B*a^3*b^7)*e^5+5*(10*A*a*b^9+45*B*a^2*b^8)*d*e^4+10*(A*b
^10+10*B*a*b^9)*d^2*e^3+10*b^10*B*d^3*e^2)*x^14+1/13*((120*A*a^3*b^7+210*B*a^4*b
^6)*e^5+5*(45*A*a^2*b^8+120*B*a^3*b^7)*d*e^4+10*(10*A*a*b^9+45*B*a^2*b^8)*d^2*e^
3+10*(A*b^10+10*B*a*b^9)*d^3*e^2+5*b^10*B*d^4*e)*x^13+1/12*((210*A*a^4*b^6+252*B
*a^5*b^5)*e^5+5*(120*A*a^3*b^7+210*B*a^4*b^6)*d*e^4+10*(45*A*a^2*b^8+120*B*a^3*b
^7)*d^2*e^3+10*(10*A*a*b^9+45*B*a^2*b^8)*d^3*e^2+5*(A*b^10+10*B*a*b^9)*d^4*e+b^1
0*B*d^5)*x^12+1/11*((252*A*a^5*b^5+210*B*a^6*b^4)*e^5+5*(210*A*a^4*b^6+252*B*a^5
*b^5)*d*e^4+10*(120*A*a^3*b^7+210*B*a^4*b^6)*d^2*e^3+10*(45*A*a^2*b^8+120*B*a^3*
b^7)*d^3*e^2+5*(10*A*a*b^9+45*B*a^2*b^8)*d^4*e+(A*b^10+10*B*a*b^9)*d^5)*x^11+1/1
0*((210*A*a^6*b^4+120*B*a^7*b^3)*e^5+5*(252*A*a^5*b^5+210*B*a^6*b^4)*d*e^4+10*(2
10*A*a^4*b^6+252*B*a^5*b^5)*d^2*e^3+10*(120*A*a^3*b^7+210*B*a^4*b^6)*d^3*e^2+5*(
45*A*a^2*b^8+120*B*a^3*b^7)*d^4*e+(10*A*a*b^9+45*B*a^2*b^8)*d^5)*x^10+1/9*((120*
A*a^7*b^3+45*B*a^8*b^2)*e^5+5*(210*A*a^6*b^4+120*B*a^7*b^3)*d*e^4+10*(252*A*a^5*
b^5+210*B*a^6*b^4)*d^2*e^3+10*(210*A*a^4*b^6+252*B*a^5*b^5)*d^3*e^2+5*(120*A*a^3
*b^7+210*B*a^4*b^6)*d^4*e+(45*A*a^2*b^8+120*B*a^3*b^7)*d^5)*x^9+1/8*((45*A*a^8*b
^2+10*B*a^9*b)*e^5+5*(120*A*a^7*b^3+45*B*a^8*b^2)*d*e^4+10*(210*A*a^6*b^4+120*B*
a^7*b^3)*d^2*e^3+10*(252*A*a^5*b^5+210*B*a^6*b^4)*d^3*e^2+5*(210*A*a^4*b^6+252*B
*a^5*b^5)*d^4*e+(120*A*a^3*b^7+210*B*a^4*b^6)*d^5)*x^8+1/7*((10*A*a^9*b+B*a^10)*
e^5+5*(45*A*a^8*b^2+10*B*a^9*b)*d*e^4+10*(120*A*a^7*b^3+45*B*a^8*b^2)*d^2*e^3+10
*(210*A*a^6*b^4+120*B*a^7*b^3)*d^3*e^2+5*(252*A*a^5*b^5+210*B*a^6*b^4)*d^4*e+(21
0*A*a^4*b^6+252*B*a^5*b^5)*d^5)*x^7+1/6*(a^10*A*e^5+5*(10*A*a^9*b+B*a^10)*d*e^4+
10*(45*A*a^8*b^2+10*B*a^9*b)*d^2*e^3+10*(120*A*a^7*b^3+45*B*a^8*b^2)*d^3*e^2+5*(
210*A*a^6*b^4+120*B*a^7*b^3)*d^4*e+(252*A*a^5*b^5+210*B*a^6*b^4)*d^5)*x^6+1/5*(5
*a^10*A*d*e^4+10*(10*A*a^9*b+B*a^10)*d^2*e^3+10*(45*A*a^8*b^2+10*B*a^9*b)*d^3*e^
2+5*(120*A*a^7*b^3+45*B*a^8*b^2)*d^4*e+(210*A*a^6*b^4+120*B*a^7*b^3)*d^5)*x^5+1/
4*(10*a^10*A*d^2*e^3+10*(10*A*a^9*b+B*a^10)*d^3*e^2+5*(45*A*a^8*b^2+10*B*a^9*b)*
d^4*e+(120*A*a^7*b^3+45*B*a^8*b^2)*d^5)*x^4+1/3*(10*a^10*A*d^3*e^2+5*(10*A*a^9*b
+B*a^10)*d^4*e+(45*A*a^8*b^2+10*B*a^9*b)*d^5)*x^3+1/2*(5*a^10*A*d^4*e+(10*A*a^9*
b+B*a^10)*d^5)*x^2+a^10*A*d^5*x

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Maxima [A]  time = 1.39617, size = 2194, normalized size = 9.03 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10*(e*x + d)^5,x, algorithm="maxima")

[Out]

1/17*B*b^10*e^5*x^17 + A*a^10*d^5*x + 1/16*(5*B*b^10*d*e^4 + (10*B*a*b^9 + A*b^1
0)*e^5)*x^16 + 1/3*(2*B*b^10*d^2*e^3 + (10*B*a*b^9 + A*b^10)*d*e^4 + (9*B*a^2*b^
8 + 2*A*a*b^9)*e^5)*x^15 + 5/14*(2*B*b^10*d^3*e^2 + 2*(10*B*a*b^9 + A*b^10)*d^2*
e^3 + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^4 + 3*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^5)*x^1
4 + 5/13*(B*b^10*d^4*e + 2*(10*B*a*b^9 + A*b^10)*d^3*e^2 + 10*(9*B*a^2*b^8 + 2*A
*a*b^9)*d^2*e^3 + 15*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^4 + 6*(7*B*a^4*b^6 + 4*A*a^
3*b^7)*e^5)*x^13 + 1/12*(B*b^10*d^5 + 5*(10*B*a*b^9 + A*b^10)*d^4*e + 50*(9*B*a^
2*b^8 + 2*A*a*b^9)*d^3*e^2 + 150*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^3 + 150*(7*B*
a^4*b^6 + 4*A*a^3*b^7)*d*e^4 + 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^5)*x^12 + 1/11*(
(10*B*a*b^9 + A*b^10)*d^5 + 25*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e + 150*(8*B*a^3*b^
7 + 3*A*a^2*b^8)*d^3*e^2 + 300*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^3 + 210*(6*B*a^
5*b^5 + 5*A*a^4*b^6)*d*e^4 + 42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^5)*x^11 + 1/2*((9*
B*a^2*b^8 + 2*A*a*b^9)*d^5 + 15*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e + 60*(7*B*a^4*
b^6 + 4*A*a^3*b^7)*d^3*e^2 + 84*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^3 + 42*(5*B*a^
6*b^4 + 6*A*a^5*b^5)*d*e^4 + 6*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^5)*x^10 + 5/3*((8*B
*a^3*b^7 + 3*A*a^2*b^8)*d^5 + 10*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e + 28*(6*B*a^5
*b^5 + 5*A*a^4*b^6)*d^3*e^2 + 28*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*e^3 + 10*(4*B*a
^7*b^3 + 7*A*a^6*b^4)*d*e^4 + (3*B*a^8*b^2 + 8*A*a^7*b^3)*e^5)*x^9 + 5/8*(6*(7*B
*a^4*b^6 + 4*A*a^3*b^7)*d^5 + 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*e + 84*(5*B*a^6
*b^4 + 6*A*a^5*b^5)*d^3*e^2 + 60*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*e^3 + 15*(3*B*a
^8*b^2 + 8*A*a^7*b^3)*d*e^4 + (2*B*a^9*b + 9*A*a^8*b^2)*e^5)*x^8 + 1/7*(42*(6*B*
a^5*b^5 + 5*A*a^4*b^6)*d^5 + 210*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*e + 300*(4*B*a^
7*b^3 + 7*A*a^6*b^4)*d^3*e^2 + 150*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e^3 + 25*(2*B
*a^9*b + 9*A*a^8*b^2)*d*e^4 + (B*a^10 + 10*A*a^9*b)*e^5)*x^7 + 1/6*(A*a^10*e^5 +
42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5 + 150*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4*e + 15
0*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^2 + 50*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^3 + 5
*(B*a^10 + 10*A*a^9*b)*d*e^4)*x^6 + (A*a^10*d*e^4 + 6*(4*B*a^7*b^3 + 7*A*a^6*b^4
)*d^5 + 15*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^4*e + 10*(2*B*a^9*b + 9*A*a^8*b^2)*d^3*
e^2 + 2*(B*a^10 + 10*A*a^9*b)*d^2*e^3)*x^5 + 5/4*(2*A*a^10*d^2*e^3 + 3*(3*B*a^8*
b^2 + 8*A*a^7*b^3)*d^5 + 5*(2*B*a^9*b + 9*A*a^8*b^2)*d^4*e + 2*(B*a^10 + 10*A*a^
9*b)*d^3*e^2)*x^4 + 5/3*(2*A*a^10*d^3*e^2 + (2*B*a^9*b + 9*A*a^8*b^2)*d^5 + (B*a
^10 + 10*A*a^9*b)*d^4*e)*x^3 + 1/2*(5*A*a^10*d^4*e + (B*a^10 + 10*A*a^9*b)*d^5)*
x^2

_______________________________________________________________________________________

Fricas [A]  time = 0.207581, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10*(e*x + d)^5,x, algorithm="fricas")

[Out]

1/17*x^17*e^5*b^10*B + 5/16*x^16*e^4*d*b^10*B + 5/8*x^16*e^5*b^9*a*B + 1/16*x^16
*e^5*b^10*A + 2/3*x^15*e^3*d^2*b^10*B + 10/3*x^15*e^4*d*b^9*a*B + 3*x^15*e^5*b^8
*a^2*B + 1/3*x^15*e^4*d*b^10*A + 2/3*x^15*e^5*b^9*a*A + 5/7*x^14*e^2*d^3*b^10*B
+ 50/7*x^14*e^3*d^2*b^9*a*B + 225/14*x^14*e^4*d*b^8*a^2*B + 60/7*x^14*e^5*b^7*a^
3*B + 5/7*x^14*e^3*d^2*b^10*A + 25/7*x^14*e^4*d*b^9*a*A + 45/14*x^14*e^5*b^8*a^2
*A + 5/13*x^13*e*d^4*b^10*B + 100/13*x^13*e^2*d^3*b^9*a*B + 450/13*x^13*e^3*d^2*
b^8*a^2*B + 600/13*x^13*e^4*d*b^7*a^3*B + 210/13*x^13*e^5*b^6*a^4*B + 10/13*x^13
*e^2*d^3*b^10*A + 100/13*x^13*e^3*d^2*b^9*a*A + 225/13*x^13*e^4*d*b^8*a^2*A + 12
0/13*x^13*e^5*b^7*a^3*A + 1/12*x^12*d^5*b^10*B + 25/6*x^12*e*d^4*b^9*a*B + 75/2*
x^12*e^2*d^3*b^8*a^2*B + 100*x^12*e^3*d^2*b^7*a^3*B + 175/2*x^12*e^4*d*b^6*a^4*B
+ 21*x^12*e^5*b^5*a^5*B + 5/12*x^12*e*d^4*b^10*A + 25/3*x^12*e^2*d^3*b^9*a*A +
75/2*x^12*e^3*d^2*b^8*a^2*A + 50*x^12*e^4*d*b^7*a^3*A + 35/2*x^12*e^5*b^6*a^4*A
+ 10/11*x^11*d^5*b^9*a*B + 225/11*x^11*e*d^4*b^8*a^2*B + 1200/11*x^11*e^2*d^3*b^
7*a^3*B + 2100/11*x^11*e^3*d^2*b^6*a^4*B + 1260/11*x^11*e^4*d*b^5*a^5*B + 210/11
*x^11*e^5*b^4*a^6*B + 1/11*x^11*d^5*b^10*A + 50/11*x^11*e*d^4*b^9*a*A + 450/11*x
^11*e^2*d^3*b^8*a^2*A + 1200/11*x^11*e^3*d^2*b^7*a^3*A + 1050/11*x^11*e^4*d*b^6*
a^4*A + 252/11*x^11*e^5*b^5*a^5*A + 9/2*x^10*d^5*b^8*a^2*B + 60*x^10*e*d^4*b^7*a
^3*B + 210*x^10*e^2*d^3*b^6*a^4*B + 252*x^10*e^3*d^2*b^5*a^5*B + 105*x^10*e^4*d*
b^4*a^6*B + 12*x^10*e^5*b^3*a^7*B + x^10*d^5*b^9*a*A + 45/2*x^10*e*d^4*b^8*a^2*A
+ 120*x^10*e^2*d^3*b^7*a^3*A + 210*x^10*e^3*d^2*b^6*a^4*A + 126*x^10*e^4*d*b^5*
a^5*A + 21*x^10*e^5*b^4*a^6*A + 40/3*x^9*d^5*b^7*a^3*B + 350/3*x^9*e*d^4*b^6*a^4
*B + 280*x^9*e^2*d^3*b^5*a^5*B + 700/3*x^9*e^3*d^2*b^4*a^6*B + 200/3*x^9*e^4*d*b
^3*a^7*B + 5*x^9*e^5*b^2*a^8*B + 5*x^9*d^5*b^8*a^2*A + 200/3*x^9*e*d^4*b^7*a^3*A
+ 700/3*x^9*e^2*d^3*b^6*a^4*A + 280*x^9*e^3*d^2*b^5*a^5*A + 350/3*x^9*e^4*d*b^4
*a^6*A + 40/3*x^9*e^5*b^3*a^7*A + 105/4*x^8*d^5*b^6*a^4*B + 315/2*x^8*e*d^4*b^5*
a^5*B + 525/2*x^8*e^2*d^3*b^4*a^6*B + 150*x^8*e^3*d^2*b^3*a^7*B + 225/8*x^8*e^4*
d*b^2*a^8*B + 5/4*x^8*e^5*b*a^9*B + 15*x^8*d^5*b^7*a^3*A + 525/4*x^8*e*d^4*b^6*a
^4*A + 315*x^8*e^2*d^3*b^5*a^5*A + 525/2*x^8*e^3*d^2*b^4*a^6*A + 75*x^8*e^4*d*b^
3*a^7*A + 45/8*x^8*e^5*b^2*a^8*A + 36*x^7*d^5*b^5*a^5*B + 150*x^7*e*d^4*b^4*a^6*
B + 1200/7*x^7*e^2*d^3*b^3*a^7*B + 450/7*x^7*e^3*d^2*b^2*a^8*B + 50/7*x^7*e^4*d*
b*a^9*B + 1/7*x^7*e^5*a^10*B + 30*x^7*d^5*b^6*a^4*A + 180*x^7*e*d^4*b^5*a^5*A +
300*x^7*e^2*d^3*b^4*a^6*A + 1200/7*x^7*e^3*d^2*b^3*a^7*A + 225/7*x^7*e^4*d*b^2*a
^8*A + 10/7*x^7*e^5*b*a^9*A + 35*x^6*d^5*b^4*a^6*B + 100*x^6*e*d^4*b^3*a^7*B + 7
5*x^6*e^2*d^3*b^2*a^8*B + 50/3*x^6*e^3*d^2*b*a^9*B + 5/6*x^6*e^4*d*a^10*B + 42*x
^6*d^5*b^5*a^5*A + 175*x^6*e*d^4*b^4*a^6*A + 200*x^6*e^2*d^3*b^3*a^7*A + 75*x^6*
e^3*d^2*b^2*a^8*A + 25/3*x^6*e^4*d*b*a^9*A + 1/6*x^6*e^5*a^10*A + 24*x^5*d^5*b^3
*a^7*B + 45*x^5*e*d^4*b^2*a^8*B + 20*x^5*e^2*d^3*b*a^9*B + 2*x^5*e^3*d^2*a^10*B
+ 42*x^5*d^5*b^4*a^6*A + 120*x^5*e*d^4*b^3*a^7*A + 90*x^5*e^2*d^3*b^2*a^8*A + 20
*x^5*e^3*d^2*b*a^9*A + x^5*e^4*d*a^10*A + 45/4*x^4*d^5*b^2*a^8*B + 25/2*x^4*e*d^
4*b*a^9*B + 5/2*x^4*e^2*d^3*a^10*B + 30*x^4*d^5*b^3*a^7*A + 225/4*x^4*e*d^4*b^2*
a^8*A + 25*x^4*e^2*d^3*b*a^9*A + 5/2*x^4*e^3*d^2*a^10*A + 10/3*x^3*d^5*b*a^9*B +
5/3*x^3*e*d^4*a^10*B + 15*x^3*d^5*b^2*a^8*A + 50/3*x^3*e*d^4*b*a^9*A + 10/3*x^3
*e^2*d^3*a^10*A + 1/2*x^2*d^5*a^10*B + 5*x^2*d^5*b*a^9*A + 5/2*x^2*e*d^4*a^10*A
+ x*d^5*a^10*A

_______________________________________________________________________________________

Sympy [A]  time = 0.930112, size = 2076, normalized size = 8.54 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**10*(B*x+A)*(e*x+d)**5,x)

[Out]

A*a**10*d**5*x + B*b**10*e**5*x**17/17 + x**16*(A*b**10*e**5/16 + 5*B*a*b**9*e**
5/8 + 5*B*b**10*d*e**4/16) + x**15*(2*A*a*b**9*e**5/3 + A*b**10*d*e**4/3 + 3*B*a
**2*b**8*e**5 + 10*B*a*b**9*d*e**4/3 + 2*B*b**10*d**2*e**3/3) + x**14*(45*A*a**2
*b**8*e**5/14 + 25*A*a*b**9*d*e**4/7 + 5*A*b**10*d**2*e**3/7 + 60*B*a**3*b**7*e*
*5/7 + 225*B*a**2*b**8*d*e**4/14 + 50*B*a*b**9*d**2*e**3/7 + 5*B*b**10*d**3*e**2
/7) + x**13*(120*A*a**3*b**7*e**5/13 + 225*A*a**2*b**8*d*e**4/13 + 100*A*a*b**9*
d**2*e**3/13 + 10*A*b**10*d**3*e**2/13 + 210*B*a**4*b**6*e**5/13 + 600*B*a**3*b*
*7*d*e**4/13 + 450*B*a**2*b**8*d**2*e**3/13 + 100*B*a*b**9*d**3*e**2/13 + 5*B*b*
*10*d**4*e/13) + x**12*(35*A*a**4*b**6*e**5/2 + 50*A*a**3*b**7*d*e**4 + 75*A*a**
2*b**8*d**2*e**3/2 + 25*A*a*b**9*d**3*e**2/3 + 5*A*b**10*d**4*e/12 + 21*B*a**5*b
**5*e**5 + 175*B*a**4*b**6*d*e**4/2 + 100*B*a**3*b**7*d**2*e**3 + 75*B*a**2*b**8
*d**3*e**2/2 + 25*B*a*b**9*d**4*e/6 + B*b**10*d**5/12) + x**11*(252*A*a**5*b**5*
e**5/11 + 1050*A*a**4*b**6*d*e**4/11 + 1200*A*a**3*b**7*d**2*e**3/11 + 450*A*a**
2*b**8*d**3*e**2/11 + 50*A*a*b**9*d**4*e/11 + A*b**10*d**5/11 + 210*B*a**6*b**4*
e**5/11 + 1260*B*a**5*b**5*d*e**4/11 + 2100*B*a**4*b**6*d**2*e**3/11 + 1200*B*a*
*3*b**7*d**3*e**2/11 + 225*B*a**2*b**8*d**4*e/11 + 10*B*a*b**9*d**5/11) + x**10*
(21*A*a**6*b**4*e**5 + 126*A*a**5*b**5*d*e**4 + 210*A*a**4*b**6*d**2*e**3 + 120*
A*a**3*b**7*d**3*e**2 + 45*A*a**2*b**8*d**4*e/2 + A*a*b**9*d**5 + 12*B*a**7*b**3
*e**5 + 105*B*a**6*b**4*d*e**4 + 252*B*a**5*b**5*d**2*e**3 + 210*B*a**4*b**6*d**
3*e**2 + 60*B*a**3*b**7*d**4*e + 9*B*a**2*b**8*d**5/2) + x**9*(40*A*a**7*b**3*e*
*5/3 + 350*A*a**6*b**4*d*e**4/3 + 280*A*a**5*b**5*d**2*e**3 + 700*A*a**4*b**6*d*
*3*e**2/3 + 200*A*a**3*b**7*d**4*e/3 + 5*A*a**2*b**8*d**5 + 5*B*a**8*b**2*e**5 +
200*B*a**7*b**3*d*e**4/3 + 700*B*a**6*b**4*d**2*e**3/3 + 280*B*a**5*b**5*d**3*e
**2 + 350*B*a**4*b**6*d**4*e/3 + 40*B*a**3*b**7*d**5/3) + x**8*(45*A*a**8*b**2*e
**5/8 + 75*A*a**7*b**3*d*e**4 + 525*A*a**6*b**4*d**2*e**3/2 + 315*A*a**5*b**5*d*
*3*e**2 + 525*A*a**4*b**6*d**4*e/4 + 15*A*a**3*b**7*d**5 + 5*B*a**9*b*e**5/4 + 2
25*B*a**8*b**2*d*e**4/8 + 150*B*a**7*b**3*d**2*e**3 + 525*B*a**6*b**4*d**3*e**2/
2 + 315*B*a**5*b**5*d**4*e/2 + 105*B*a**4*b**6*d**5/4) + x**7*(10*A*a**9*b*e**5/
7 + 225*A*a**8*b**2*d*e**4/7 + 1200*A*a**7*b**3*d**2*e**3/7 + 300*A*a**6*b**4*d*
*3*e**2 + 180*A*a**5*b**5*d**4*e + 30*A*a**4*b**6*d**5 + B*a**10*e**5/7 + 50*B*a
**9*b*d*e**4/7 + 450*B*a**8*b**2*d**2*e**3/7 + 1200*B*a**7*b**3*d**3*e**2/7 + 15
0*B*a**6*b**4*d**4*e + 36*B*a**5*b**5*d**5) + x**6*(A*a**10*e**5/6 + 25*A*a**9*b
*d*e**4/3 + 75*A*a**8*b**2*d**2*e**3 + 200*A*a**7*b**3*d**3*e**2 + 175*A*a**6*b*
*4*d**4*e + 42*A*a**5*b**5*d**5 + 5*B*a**10*d*e**4/6 + 50*B*a**9*b*d**2*e**3/3 +
75*B*a**8*b**2*d**3*e**2 + 100*B*a**7*b**3*d**4*e + 35*B*a**6*b**4*d**5) + x**5
*(A*a**10*d*e**4 + 20*A*a**9*b*d**2*e**3 + 90*A*a**8*b**2*d**3*e**2 + 120*A*a**7
*b**3*d**4*e + 42*A*a**6*b**4*d**5 + 2*B*a**10*d**2*e**3 + 20*B*a**9*b*d**3*e**2
+ 45*B*a**8*b**2*d**4*e + 24*B*a**7*b**3*d**5) + x**4*(5*A*a**10*d**2*e**3/2 +
25*A*a**9*b*d**3*e**2 + 225*A*a**8*b**2*d**4*e/4 + 30*A*a**7*b**3*d**5 + 5*B*a**
10*d**3*e**2/2 + 25*B*a**9*b*d**4*e/2 + 45*B*a**8*b**2*d**5/4) + x**3*(10*A*a**1
0*d**3*e**2/3 + 50*A*a**9*b*d**4*e/3 + 15*A*a**8*b**2*d**5 + 5*B*a**10*d**4*e/3
+ 10*B*a**9*b*d**5/3) + x**2*(5*A*a**10*d**4*e/2 + 5*A*a**9*b*d**5 + B*a**10*d**
5/2)

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GIAC/XCAS [A]  time = 0.210135, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10*(e*x + d)^5,x, algorithm="giac")

[Out]

Done