3.1064 \(\int (a+b x)^{10} (A+B x) (d+e x)^7 \, dx\)

Optimal. Leaf size=329 \[ \frac{e^6 (a+b x)^{18} (-8 a B e+A b e+7 b B d)}{18 b^9}+\frac{7 e^5 (a+b x)^{17} (b d-a e) (-4 a B e+A b e+3 b B d)}{17 b^9}+\frac{7 e^4 (a+b x)^{16} (b d-a e)^2 (-8 a B e+3 A b e+5 b B d)}{16 b^9}+\frac{7 e^3 (a+b x)^{15} (b d-a e)^3 (-2 a B e+A b e+b B d)}{3 b^9}+\frac{e^2 (a+b x)^{14} (b d-a e)^4 (-8 a B e+5 A b e+3 b B d)}{2 b^9}+\frac{7 e (a+b x)^{13} (b d-a e)^5 (-4 a B e+3 A b e+b B d)}{13 b^9}+\frac{(a+b x)^{12} (b d-a e)^6 (-8 a B e+7 A b e+b B d)}{12 b^9}+\frac{(a+b x)^{11} (A b-a B) (b d-a e)^7}{11 b^9}+\frac{B e^7 (a+b x)^{19}}{19 b^9} \]

[Out]

((A*b - a*B)*(b*d - a*e)^7*(a + b*x)^11)/(11*b^9) + ((b*d - a*e)^6*(b*B*d + 7*A*
b*e - 8*a*B*e)*(a + b*x)^12)/(12*b^9) + (7*e*(b*d - a*e)^5*(b*B*d + 3*A*b*e - 4*
a*B*e)*(a + b*x)^13)/(13*b^9) + (e^2*(b*d - a*e)^4*(3*b*B*d + 5*A*b*e - 8*a*B*e)
*(a + b*x)^14)/(2*b^9) + (7*e^3*(b*d - a*e)^3*(b*B*d + A*b*e - 2*a*B*e)*(a + b*x
)^15)/(3*b^9) + (7*e^4*(b*d - a*e)^2*(5*b*B*d + 3*A*b*e - 8*a*B*e)*(a + b*x)^16)
/(16*b^9) + (7*e^5*(b*d - a*e)*(3*b*B*d + A*b*e - 4*a*B*e)*(a + b*x)^17)/(17*b^9
) + (e^6*(7*b*B*d + A*b*e - 8*a*B*e)*(a + b*x)^18)/(18*b^9) + (B*e^7*(a + b*x)^1
9)/(19*b^9)

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Rubi [A]  time = 8.31475, antiderivative size = 329, 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 \[ \frac{e^6 (a+b x)^{18} (-8 a B e+A b e+7 b B d)}{18 b^9}+\frac{7 e^5 (a+b x)^{17} (b d-a e) (-4 a B e+A b e+3 b B d)}{17 b^9}+\frac{7 e^4 (a+b x)^{16} (b d-a e)^2 (-8 a B e+3 A b e+5 b B d)}{16 b^9}+\frac{7 e^3 (a+b x)^{15} (b d-a e)^3 (-2 a B e+A b e+b B d)}{3 b^9}+\frac{e^2 (a+b x)^{14} (b d-a e)^4 (-8 a B e+5 A b e+3 b B d)}{2 b^9}+\frac{7 e (a+b x)^{13} (b d-a e)^5 (-4 a B e+3 A b e+b B d)}{13 b^9}+\frac{(a+b x)^{12} (b d-a e)^6 (-8 a B e+7 A b e+b B d)}{12 b^9}+\frac{(a+b x)^{11} (A b-a B) (b d-a e)^7}{11 b^9}+\frac{B e^7 (a+b x)^{19}}{19 b^9} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^10*(A + B*x)*(d + e*x)^7,x]

[Out]

((A*b - a*B)*(b*d - a*e)^7*(a + b*x)^11)/(11*b^9) + ((b*d - a*e)^6*(b*B*d + 7*A*
b*e - 8*a*B*e)*(a + b*x)^12)/(12*b^9) + (7*e*(b*d - a*e)^5*(b*B*d + 3*A*b*e - 4*
a*B*e)*(a + b*x)^13)/(13*b^9) + (e^2*(b*d - a*e)^4*(3*b*B*d + 5*A*b*e - 8*a*B*e)
*(a + b*x)^14)/(2*b^9) + (7*e^3*(b*d - a*e)^3*(b*B*d + A*b*e - 2*a*B*e)*(a + b*x
)^15)/(3*b^9) + (7*e^4*(b*d - a*e)^2*(5*b*B*d + 3*A*b*e - 8*a*B*e)*(a + b*x)^16)
/(16*b^9) + (7*e^5*(b*d - a*e)*(3*b*B*d + A*b*e - 4*a*B*e)*(a + b*x)^17)/(17*b^9
) + (e^6*(7*b*B*d + A*b*e - 8*a*B*e)*(a + b*x)^18)/(18*b^9) + (B*e^7*(a + b*x)^1
9)/(19*b^9)

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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)**7,x)

[Out]

Timed out

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Mathematica [B]  time = 1.51416, size = 2034, normalized size = 6.18 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^10*(A + B*x)*(d + e*x)^7,x]

[Out]

a^10*A*d^7*x + (a^9*d^6*(10*A*b*d + a*B*d + 7*a*A*e)*x^2)/2 + (a^8*d^5*(a*B*d*(1
0*b*d + 7*a*e) + A*(45*b^2*d^2 + 70*a*b*d*e + 21*a^2*e^2))*x^3)/3 + (a^7*d^4*(a*
B*d*(45*b^2*d^2 + 70*a*b*d*e + 21*a^2*e^2) + 5*A*(24*b^3*d^3 + 63*a*b^2*d^2*e +
42*a^2*b*d*e^2 + 7*a^3*e^3))*x^4)/4 + a^6*d^3*(a*B*d*(24*b^3*d^3 + 63*a*b^2*d^2*
e + 42*a^2*b*d*e^2 + 7*a^3*e^3) + 7*A*(6*b^4*d^4 + 24*a*b^3*d^3*e + 27*a^2*b^2*d
^2*e^2 + 10*a^3*b*d*e^3 + a^4*e^4))*x^5 + (7*a^5*d^2*(5*a*B*d*(6*b^4*d^4 + 24*a*
b^3*d^3*e + 27*a^2*b^2*d^2*e^2 + 10*a^3*b*d*e^3 + a^4*e^4) + A*(36*b^5*d^5 + 210
*a*b^4*d^4*e + 360*a^2*b^3*d^3*e^2 + 225*a^3*b^2*d^2*e^3 + 50*a^4*b*d*e^4 + 3*a^
5*e^5))*x^6)/6 + a^4*d*(a*B*d*(36*b^5*d^5 + 210*a*b^4*d^4*e + 360*a^2*b^3*d^3*e^
2 + 225*a^3*b^2*d^2*e^3 + 50*a^4*b*d*e^4 + 3*a^5*e^5) + A*(30*b^6*d^6 + 252*a*b^
5*d^5*e + 630*a^2*b^4*d^4*e^2 + 600*a^3*b^3*d^3*e^3 + 225*a^4*b^2*d^2*e^4 + 30*a
^5*b*d*e^5 + a^6*e^6))*x^7 + (a^3*(7*a*B*d*(30*b^6*d^6 + 252*a*b^5*d^5*e + 630*a
^2*b^4*d^4*e^2 + 600*a^3*b^3*d^3*e^3 + 225*a^4*b^2*d^2*e^4 + 30*a^5*b*d*e^5 + a^
6*e^6) + A*(120*b^7*d^7 + 1470*a*b^6*d^6*e + 5292*a^2*b^5*d^5*e^2 + 7350*a^3*b^4
*d^4*e^3 + 4200*a^4*b^3*d^3*e^4 + 945*a^5*b^2*d^2*e^5 + 70*a^6*b*d*e^6 + a^7*e^7
))*x^8)/8 + (a^2*(a*B*(120*b^7*d^7 + 1470*a*b^6*d^6*e + 5292*a^2*b^5*d^5*e^2 + 7
350*a^3*b^4*d^4*e^3 + 4200*a^4*b^3*d^3*e^4 + 945*a^5*b^2*d^2*e^5 + 70*a^6*b*d*e^
6 + a^7*e^7) + 5*A*b*(9*b^7*d^7 + 168*a*b^6*d^6*e + 882*a^2*b^5*d^5*e^2 + 1764*a
^3*b^4*d^4*e^3 + 1470*a^4*b^3*d^3*e^4 + 504*a^5*b^2*d^2*e^5 + 63*a^6*b*d*e^6 + 2
*a^7*e^7))*x^9)/9 + (a*b*(a*B*(9*b^7*d^7 + 168*a*b^6*d^6*e + 882*a^2*b^5*d^5*e^2
+ 1764*a^3*b^4*d^4*e^3 + 1470*a^4*b^3*d^3*e^4 + 504*a^5*b^2*d^2*e^5 + 63*a^6*b*
d*e^6 + 2*a^7*e^7) + A*b*(2*b^7*d^7 + 63*a*b^6*d^6*e + 504*a^2*b^5*d^5*e^2 + 147
0*a^3*b^4*d^4*e^3 + 1764*a^4*b^3*d^3*e^4 + 882*a^5*b^2*d^2*e^5 + 168*a^6*b*d*e^6
+ 9*a^7*e^7))*x^10)/2 + (b^2*(5*a*B*(2*b^7*d^7 + 63*a*b^6*d^6*e + 504*a^2*b^5*d
^5*e^2 + 1470*a^3*b^4*d^4*e^3 + 1764*a^4*b^3*d^3*e^4 + 882*a^5*b^2*d^2*e^5 + 168
*a^6*b*d*e^6 + 9*a^7*e^7) + A*b*(b^7*d^7 + 70*a*b^6*d^6*e + 945*a^2*b^5*d^5*e^2
+ 4200*a^3*b^4*d^4*e^3 + 7350*a^4*b^3*d^3*e^4 + 5292*a^5*b^2*d^2*e^5 + 1470*a^6*
b*d*e^6 + 120*a^7*e^7))*x^11)/11 + (b^3*(120*a^7*B*e^7 + 4200*a^3*b^4*d^3*e^3*(B
*d + A*e) + 1764*a^5*b^2*d*e^5*(3*B*d + A*e) + 210*a^6*b*e^6*(7*B*d + A*e) + 70*
a*b^6*d^5*e*(B*d + 3*A*e) + 1470*a^4*b^3*d^2*e^4*(5*B*d + 3*A*e) + 315*a^2*b^5*d
^4*e^2*(3*B*d + 5*A*e) + b^7*d^6*(B*d + 7*A*e))*x^12)/12 + (7*b^4*e*(30*a^6*B*e^
6 + 225*a^2*b^4*d^3*e^2*(B*d + A*e) + 210*a^4*b^2*d*e^4*(3*B*d + A*e) + 36*a^5*b
*e^5*(7*B*d + A*e) + b^6*d^5*(B*d + 3*A*e) + 120*a^3*b^3*d^2*e^3*(5*B*d + 3*A*e)
+ 10*a*b^5*d^4*e*(3*B*d + 5*A*e))*x^13)/13 + (b^5*e^2*(36*a^5*B*e^5 + 50*a*b^4*
d^3*e*(B*d + A*e) + 120*a^3*b^2*d*e^3*(3*B*d + A*e) + 30*a^4*b*e^4*(7*B*d + A*e)
+ 45*a^2*b^3*d^2*e^2*(5*B*d + 3*A*e) + b^5*d^4*(3*B*d + 5*A*e))*x^14)/2 + (b^6*
e^3*(42*a^4*B*e^4 + 7*b^4*d^3*(B*d + A*e) + 63*a^2*b^2*d*e^2*(3*B*d + A*e) + 24*
a^3*b*e^3*(7*B*d + A*e) + 14*a*b^3*d^2*e*(5*B*d + 3*A*e))*x^15)/3 + (b^7*e^4*(12
0*a^3*B*e^3 + 70*a*b^2*d*e*(3*B*d + A*e) + 45*a^2*b*e^2*(7*B*d + A*e) + 7*b^3*d^
2*(5*B*d + 3*A*e))*x^16)/16 + (b^8*e^5*(45*a^2*B*e^2 + 7*b^2*d*(3*B*d + A*e) + 1
0*a*b*e*(7*B*d + A*e))*x^17)/17 + (b^9*e^6*(7*b*B*d + A*b*e + 10*a*B*e)*x^18)/18
+ (b^10*B*e^7*x^19)/19

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Maple [B]  time = 0.004, size = 2189, 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)^7,x)

[Out]

1/19*b^10*B*e^7*x^19+1/18*((A*b^10+10*B*a*b^9)*e^7+7*b^10*B*d*e^6)*x^18+1/17*((1
0*A*a*b^9+45*B*a^2*b^8)*e^7+7*(A*b^10+10*B*a*b^9)*d*e^6+21*b^10*B*d^2*e^5)*x^17+
1/16*((45*A*a^2*b^8+120*B*a^3*b^7)*e^7+7*(10*A*a*b^9+45*B*a^2*b^8)*d*e^6+21*(A*b
^10+10*B*a*b^9)*d^2*e^5+35*b^10*B*d^3*e^4)*x^16+1/15*((120*A*a^3*b^7+210*B*a^4*b
^6)*e^7+7*(45*A*a^2*b^8+120*B*a^3*b^7)*d*e^6+21*(10*A*a*b^9+45*B*a^2*b^8)*d^2*e^
5+35*(A*b^10+10*B*a*b^9)*d^3*e^4+35*b^10*B*d^4*e^3)*x^15+1/14*((210*A*a^4*b^6+25
2*B*a^5*b^5)*e^7+7*(120*A*a^3*b^7+210*B*a^4*b^6)*d*e^6+21*(45*A*a^2*b^8+120*B*a^
3*b^7)*d^2*e^5+35*(10*A*a*b^9+45*B*a^2*b^8)*d^3*e^4+35*(A*b^10+10*B*a*b^9)*d^4*e
^3+21*b^10*B*d^5*e^2)*x^14+1/13*((252*A*a^5*b^5+210*B*a^6*b^4)*e^7+7*(210*A*a^4*
b^6+252*B*a^5*b^5)*d*e^6+21*(120*A*a^3*b^7+210*B*a^4*b^6)*d^2*e^5+35*(45*A*a^2*b
^8+120*B*a^3*b^7)*d^3*e^4+35*(10*A*a*b^9+45*B*a^2*b^8)*d^4*e^3+21*(A*b^10+10*B*a
*b^9)*d^5*e^2+7*b^10*B*d^6*e)*x^13+1/12*((210*A*a^6*b^4+120*B*a^7*b^3)*e^7+7*(25
2*A*a^5*b^5+210*B*a^6*b^4)*d*e^6+21*(210*A*a^4*b^6+252*B*a^5*b^5)*d^2*e^5+35*(12
0*A*a^3*b^7+210*B*a^4*b^6)*d^3*e^4+35*(45*A*a^2*b^8+120*B*a^3*b^7)*d^4*e^3+21*(1
0*A*a*b^9+45*B*a^2*b^8)*d^5*e^2+7*(A*b^10+10*B*a*b^9)*d^6*e+b^10*B*d^7)*x^12+1/1
1*((120*A*a^7*b^3+45*B*a^8*b^2)*e^7+7*(210*A*a^6*b^4+120*B*a^7*b^3)*d*e^6+21*(25
2*A*a^5*b^5+210*B*a^6*b^4)*d^2*e^5+35*(210*A*a^4*b^6+252*B*a^5*b^5)*d^3*e^4+35*(
120*A*a^3*b^7+210*B*a^4*b^6)*d^4*e^3+21*(45*A*a^2*b^8+120*B*a^3*b^7)*d^5*e^2+7*(
10*A*a*b^9+45*B*a^2*b^8)*d^6*e+(A*b^10+10*B*a*b^9)*d^7)*x^11+1/10*((45*A*a^8*b^2
+10*B*a^9*b)*e^7+7*(120*A*a^7*b^3+45*B*a^8*b^2)*d*e^6+21*(210*A*a^6*b^4+120*B*a^
7*b^3)*d^2*e^5+35*(252*A*a^5*b^5+210*B*a^6*b^4)*d^3*e^4+35*(210*A*a^4*b^6+252*B*
a^5*b^5)*d^4*e^3+21*(120*A*a^3*b^7+210*B*a^4*b^6)*d^5*e^2+7*(45*A*a^2*b^8+120*B*
a^3*b^7)*d^6*e+(10*A*a*b^9+45*B*a^2*b^8)*d^7)*x^10+1/9*((10*A*a^9*b+B*a^10)*e^7+
7*(45*A*a^8*b^2+10*B*a^9*b)*d*e^6+21*(120*A*a^7*b^3+45*B*a^8*b^2)*d^2*e^5+35*(21
0*A*a^6*b^4+120*B*a^7*b^3)*d^3*e^4+35*(252*A*a^5*b^5+210*B*a^6*b^4)*d^4*e^3+21*(
210*A*a^4*b^6+252*B*a^5*b^5)*d^5*e^2+7*(120*A*a^3*b^7+210*B*a^4*b^6)*d^6*e+(45*A
*a^2*b^8+120*B*a^3*b^7)*d^7)*x^9+1/8*(a^10*A*e^7+7*(10*A*a^9*b+B*a^10)*d*e^6+21*
(45*A*a^8*b^2+10*B*a^9*b)*d^2*e^5+35*(120*A*a^7*b^3+45*B*a^8*b^2)*d^3*e^4+35*(21
0*A*a^6*b^4+120*B*a^7*b^3)*d^4*e^3+21*(252*A*a^5*b^5+210*B*a^6*b^4)*d^5*e^2+7*(2
10*A*a^4*b^6+252*B*a^5*b^5)*d^6*e+(120*A*a^3*b^7+210*B*a^4*b^6)*d^7)*x^8+1/7*(7*
a^10*A*d*e^6+21*(10*A*a^9*b+B*a^10)*d^2*e^5+35*(45*A*a^8*b^2+10*B*a^9*b)*d^3*e^4
+35*(120*A*a^7*b^3+45*B*a^8*b^2)*d^4*e^3+21*(210*A*a^6*b^4+120*B*a^7*b^3)*d^5*e^
2+7*(252*A*a^5*b^5+210*B*a^6*b^4)*d^6*e+(210*A*a^4*b^6+252*B*a^5*b^5)*d^7)*x^7+1
/6*(21*a^10*A*d^2*e^5+35*(10*A*a^9*b+B*a^10)*d^3*e^4+35*(45*A*a^8*b^2+10*B*a^9*b
)*d^4*e^3+21*(120*A*a^7*b^3+45*B*a^8*b^2)*d^5*e^2+7*(210*A*a^6*b^4+120*B*a^7*b^3
)*d^6*e+(252*A*a^5*b^5+210*B*a^6*b^4)*d^7)*x^6+1/5*(35*a^10*A*d^3*e^4+35*(10*A*a
^9*b+B*a^10)*d^4*e^3+21*(45*A*a^8*b^2+10*B*a^9*b)*d^5*e^2+7*(120*A*a^7*b^3+45*B*
a^8*b^2)*d^6*e+(210*A*a^6*b^4+120*B*a^7*b^3)*d^7)*x^5+1/4*(35*a^10*A*d^4*e^3+21*
(10*A*a^9*b+B*a^10)*d^5*e^2+7*(45*A*a^8*b^2+10*B*a^9*b)*d^6*e+(120*A*a^7*b^3+45*
B*a^8*b^2)*d^7)*x^4+1/3*(21*a^10*A*d^5*e^2+7*(10*A*a^9*b+B*a^10)*d^6*e+(45*A*a^8
*b^2+10*B*a^9*b)*d^7)*x^3+1/2*(7*a^10*A*d^6*e+(10*A*a^9*b+B*a^10)*d^7)*x^2+a^10*
A*d^7*x

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Maxima [A]  time = 1.39962, size = 2967, normalized size = 9.02 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10*(e*x + d)^7,x, algorithm="maxima")

[Out]

1/19*B*b^10*e^7*x^19 + A*a^10*d^7*x + 1/18*(7*B*b^10*d*e^6 + (10*B*a*b^9 + A*b^1
0)*e^7)*x^18 + 1/17*(21*B*b^10*d^2*e^5 + 7*(10*B*a*b^9 + A*b^10)*d*e^6 + 5*(9*B*
a^2*b^8 + 2*A*a*b^9)*e^7)*x^17 + 1/16*(35*B*b^10*d^3*e^4 + 21*(10*B*a*b^9 + A*b^
10)*d^2*e^5 + 35*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^6 + 15*(8*B*a^3*b^7 + 3*A*a^2*b^8
)*e^7)*x^16 + 1/3*(7*B*b^10*d^4*e^3 + 7*(10*B*a*b^9 + A*b^10)*d^3*e^4 + 21*(9*B*
a^2*b^8 + 2*A*a*b^9)*d^2*e^5 + 21*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^6 + 6*(7*B*a^4
*b^6 + 4*A*a^3*b^7)*e^7)*x^15 + 1/2*(3*B*b^10*d^5*e^2 + 5*(10*B*a*b^9 + A*b^10)*
d^4*e^3 + 25*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^4 + 45*(8*B*a^3*b^7 + 3*A*a^2*b^8)*
d^2*e^5 + 30*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^6 + 6*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e
^7)*x^14 + 7/13*(B*b^10*d^6*e + 3*(10*B*a*b^9 + A*b^10)*d^5*e^2 + 25*(9*B*a^2*b^
8 + 2*A*a*b^9)*d^4*e^3 + 75*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^4 + 90*(7*B*a^4*b^
6 + 4*A*a^3*b^7)*d^2*e^5 + 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^6 + 6*(5*B*a^6*b^4
+ 6*A*a^5*b^5)*e^7)*x^13 + 1/12*(B*b^10*d^7 + 7*(10*B*a*b^9 + A*b^10)*d^6*e + 1
05*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^2 + 525*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^3 +
1050*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^4 + 882*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*
e^5 + 294*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^6 + 30*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^7
)*x^12 + 1/11*((10*B*a*b^9 + A*b^10)*d^7 + 35*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e +
315*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^2 + 1050*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e
^3 + 1470*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^4 + 882*(5*B*a^6*b^4 + 6*A*a^5*b^5)*
d^2*e^5 + 210*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*e^6 + 15*(3*B*a^8*b^2 + 8*A*a^7*b^3)
*e^7)*x^11 + 1/2*((9*B*a^2*b^8 + 2*A*a*b^9)*d^7 + 21*(8*B*a^3*b^7 + 3*A*a^2*b^8)
*d^6*e + 126*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*e^2 + 294*(6*B*a^5*b^5 + 5*A*a^4*b^
6)*d^4*e^3 + 294*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*e^4 + 126*(4*B*a^7*b^3 + 7*A*a^
6*b^4)*d^2*e^5 + 21*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d*e^6 + (2*B*a^9*b + 9*A*a^8*b^2
)*e^7)*x^10 + 1/9*(15*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7 + 210*(7*B*a^4*b^6 + 4*A*a
^3*b^7)*d^6*e + 882*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^2 + 1470*(5*B*a^6*b^4 + 6*
A*a^5*b^5)*d^4*e^3 + 1050*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*e^4 + 315*(3*B*a^8*b^2
+ 8*A*a^7*b^3)*d^2*e^5 + 35*(2*B*a^9*b + 9*A*a^8*b^2)*d*e^6 + (B*a^10 + 10*A*a^
9*b)*e^7)*x^9 + 1/8*(A*a^10*e^7 + 30*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7 + 294*(6*B*
a^5*b^5 + 5*A*a^4*b^6)*d^6*e + 882*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^2 + 1050*(4
*B*a^7*b^3 + 7*A*a^6*b^4)*d^4*e^3 + 525*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^4 + 10
5*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^5 + 7*(B*a^10 + 10*A*a^9*b)*d*e^6)*x^8 + (A*a^
10*d*e^6 + 6*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^7 + 42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^
6*e + 90*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^5*e^2 + 75*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^
4*e^3 + 25*(2*B*a^9*b + 9*A*a^8*b^2)*d^3*e^4 + 3*(B*a^10 + 10*A*a^9*b)*d^2*e^5)*
x^7 + 7/6*(3*A*a^10*d^2*e^5 + 6*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^7 + 30*(4*B*a^7*b^
3 + 7*A*a^6*b^4)*d^6*e + 45*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^5*e^2 + 25*(2*B*a^9*b
+ 9*A*a^8*b^2)*d^4*e^3 + 5*(B*a^10 + 10*A*a^9*b)*d^3*e^4)*x^6 + (7*A*a^10*d^3*e^
4 + 6*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^7 + 21*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^6*e + 2
1*(2*B*a^9*b + 9*A*a^8*b^2)*d^5*e^2 + 7*(B*a^10 + 10*A*a^9*b)*d^4*e^3)*x^5 + 1/4
*(35*A*a^10*d^4*e^3 + 15*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^7 + 35*(2*B*a^9*b + 9*A*a
^8*b^2)*d^6*e + 21*(B*a^10 + 10*A*a^9*b)*d^5*e^2)*x^4 + 1/3*(21*A*a^10*d^5*e^2 +
5*(2*B*a^9*b + 9*A*a^8*b^2)*d^7 + 7*(B*a^10 + 10*A*a^9*b)*d^6*e)*x^3 + 1/2*(7*A
*a^10*d^6*e + (B*a^10 + 10*A*a^9*b)*d^7)*x^2

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Fricas [A]  time = 0.19273, 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)^7,x, algorithm="fricas")

[Out]

1/19*x^19*e^7*b^10*B + 7/18*x^18*e^6*d*b^10*B + 5/9*x^18*e^7*b^9*a*B + 1/18*x^18
*e^7*b^10*A + 21/17*x^17*e^5*d^2*b^10*B + 70/17*x^17*e^6*d*b^9*a*B + 45/17*x^17*
e^7*b^8*a^2*B + 7/17*x^17*e^6*d*b^10*A + 10/17*x^17*e^7*b^9*a*A + 35/16*x^16*e^4
*d^3*b^10*B + 105/8*x^16*e^5*d^2*b^9*a*B + 315/16*x^16*e^6*d*b^8*a^2*B + 15/2*x^
16*e^7*b^7*a^3*B + 21/16*x^16*e^5*d^2*b^10*A + 35/8*x^16*e^6*d*b^9*a*A + 45/16*x
^16*e^7*b^8*a^2*A + 7/3*x^15*e^3*d^4*b^10*B + 70/3*x^15*e^4*d^3*b^9*a*B + 63*x^1
5*e^5*d^2*b^8*a^2*B + 56*x^15*e^6*d*b^7*a^3*B + 14*x^15*e^7*b^6*a^4*B + 7/3*x^15
*e^4*d^3*b^10*A + 14*x^15*e^5*d^2*b^9*a*A + 21*x^15*e^6*d*b^8*a^2*A + 8*x^15*e^7
*b^7*a^3*A + 3/2*x^14*e^2*d^5*b^10*B + 25*x^14*e^3*d^4*b^9*a*B + 225/2*x^14*e^4*
d^3*b^8*a^2*B + 180*x^14*e^5*d^2*b^7*a^3*B + 105*x^14*e^6*d*b^6*a^4*B + 18*x^14*
e^7*b^5*a^5*B + 5/2*x^14*e^3*d^4*b^10*A + 25*x^14*e^4*d^3*b^9*a*A + 135/2*x^14*e
^5*d^2*b^8*a^2*A + 60*x^14*e^6*d*b^7*a^3*A + 15*x^14*e^7*b^6*a^4*A + 7/13*x^13*e
*d^6*b^10*B + 210/13*x^13*e^2*d^5*b^9*a*B + 1575/13*x^13*e^3*d^4*b^8*a^2*B + 420
0/13*x^13*e^4*d^3*b^7*a^3*B + 4410/13*x^13*e^5*d^2*b^6*a^4*B + 1764/13*x^13*e^6*
d*b^5*a^5*B + 210/13*x^13*e^7*b^4*a^6*B + 21/13*x^13*e^2*d^5*b^10*A + 350/13*x^1
3*e^3*d^4*b^9*a*A + 1575/13*x^13*e^4*d^3*b^8*a^2*A + 2520/13*x^13*e^5*d^2*b^7*a^
3*A + 1470/13*x^13*e^6*d*b^6*a^4*A + 252/13*x^13*e^7*b^5*a^5*A + 1/12*x^12*d^7*b
^10*B + 35/6*x^12*e*d^6*b^9*a*B + 315/4*x^12*e^2*d^5*b^8*a^2*B + 350*x^12*e^3*d^
4*b^7*a^3*B + 1225/2*x^12*e^4*d^3*b^6*a^4*B + 441*x^12*e^5*d^2*b^5*a^5*B + 245/2
*x^12*e^6*d*b^4*a^6*B + 10*x^12*e^7*b^3*a^7*B + 7/12*x^12*e*d^6*b^10*A + 35/2*x^
12*e^2*d^5*b^9*a*A + 525/4*x^12*e^3*d^4*b^8*a^2*A + 350*x^12*e^4*d^3*b^7*a^3*A +
735/2*x^12*e^5*d^2*b^6*a^4*A + 147*x^12*e^6*d*b^5*a^5*A + 35/2*x^12*e^7*b^4*a^6
*A + 10/11*x^11*d^7*b^9*a*B + 315/11*x^11*e*d^6*b^8*a^2*B + 2520/11*x^11*e^2*d^5
*b^7*a^3*B + 7350/11*x^11*e^3*d^4*b^6*a^4*B + 8820/11*x^11*e^4*d^3*b^5*a^5*B + 4
410/11*x^11*e^5*d^2*b^4*a^6*B + 840/11*x^11*e^6*d*b^3*a^7*B + 45/11*x^11*e^7*b^2
*a^8*B + 1/11*x^11*d^7*b^10*A + 70/11*x^11*e*d^6*b^9*a*A + 945/11*x^11*e^2*d^5*b
^8*a^2*A + 4200/11*x^11*e^3*d^4*b^7*a^3*A + 7350/11*x^11*e^4*d^3*b^6*a^4*A + 529
2/11*x^11*e^5*d^2*b^5*a^5*A + 1470/11*x^11*e^6*d*b^4*a^6*A + 120/11*x^11*e^7*b^3
*a^7*A + 9/2*x^10*d^7*b^8*a^2*B + 84*x^10*e*d^6*b^7*a^3*B + 441*x^10*e^2*d^5*b^6
*a^4*B + 882*x^10*e^3*d^4*b^5*a^5*B + 735*x^10*e^4*d^3*b^4*a^6*B + 252*x^10*e^5*
d^2*b^3*a^7*B + 63/2*x^10*e^6*d*b^2*a^8*B + x^10*e^7*b*a^9*B + x^10*d^7*b^9*a*A
+ 63/2*x^10*e*d^6*b^8*a^2*A + 252*x^10*e^2*d^5*b^7*a^3*A + 735*x^10*e^3*d^4*b^6*
a^4*A + 882*x^10*e^4*d^3*b^5*a^5*A + 441*x^10*e^5*d^2*b^4*a^6*A + 84*x^10*e^6*d*
b^3*a^7*A + 9/2*x^10*e^7*b^2*a^8*A + 40/3*x^9*d^7*b^7*a^3*B + 490/3*x^9*e*d^6*b^
6*a^4*B + 588*x^9*e^2*d^5*b^5*a^5*B + 2450/3*x^9*e^3*d^4*b^4*a^6*B + 1400/3*x^9*
e^4*d^3*b^3*a^7*B + 105*x^9*e^5*d^2*b^2*a^8*B + 70/9*x^9*e^6*d*b*a^9*B + 1/9*x^9
*e^7*a^10*B + 5*x^9*d^7*b^8*a^2*A + 280/3*x^9*e*d^6*b^7*a^3*A + 490*x^9*e^2*d^5*
b^6*a^4*A + 980*x^9*e^3*d^4*b^5*a^5*A + 2450/3*x^9*e^4*d^3*b^4*a^6*A + 280*x^9*e
^5*d^2*b^3*a^7*A + 35*x^9*e^6*d*b^2*a^8*A + 10/9*x^9*e^7*b*a^9*A + 105/4*x^8*d^7
*b^6*a^4*B + 441/2*x^8*e*d^6*b^5*a^5*B + 2205/4*x^8*e^2*d^5*b^4*a^6*B + 525*x^8*
e^3*d^4*b^3*a^7*B + 1575/8*x^8*e^4*d^3*b^2*a^8*B + 105/4*x^8*e^5*d^2*b*a^9*B + 7
/8*x^8*e^6*d*a^10*B + 15*x^8*d^7*b^7*a^3*A + 735/4*x^8*e*d^6*b^6*a^4*A + 1323/2*
x^8*e^2*d^5*b^5*a^5*A + 3675/4*x^8*e^3*d^4*b^4*a^6*A + 525*x^8*e^4*d^3*b^3*a^7*A
+ 945/8*x^8*e^5*d^2*b^2*a^8*A + 35/4*x^8*e^6*d*b*a^9*A + 1/8*x^8*e^7*a^10*A + 3
6*x^7*d^7*b^5*a^5*B + 210*x^7*e*d^6*b^4*a^6*B + 360*x^7*e^2*d^5*b^3*a^7*B + 225*
x^7*e^3*d^4*b^2*a^8*B + 50*x^7*e^4*d^3*b*a^9*B + 3*x^7*e^5*d^2*a^10*B + 30*x^7*d
^7*b^6*a^4*A + 252*x^7*e*d^6*b^5*a^5*A + 630*x^7*e^2*d^5*b^4*a^6*A + 600*x^7*e^3
*d^4*b^3*a^7*A + 225*x^7*e^4*d^3*b^2*a^8*A + 30*x^7*e^5*d^2*b*a^9*A + x^7*e^6*d*
a^10*A + 35*x^6*d^7*b^4*a^6*B + 140*x^6*e*d^6*b^3*a^7*B + 315/2*x^6*e^2*d^5*b^2*
a^8*B + 175/3*x^6*e^3*d^4*b*a^9*B + 35/6*x^6*e^4*d^3*a^10*B + 42*x^6*d^7*b^5*a^5
*A + 245*x^6*e*d^6*b^4*a^6*A + 420*x^6*e^2*d^5*b^3*a^7*A + 525/2*x^6*e^3*d^4*b^2
*a^8*A + 175/3*x^6*e^4*d^3*b*a^9*A + 7/2*x^6*e^5*d^2*a^10*A + 24*x^5*d^7*b^3*a^7
*B + 63*x^5*e*d^6*b^2*a^8*B + 42*x^5*e^2*d^5*b*a^9*B + 7*x^5*e^3*d^4*a^10*B + 42
*x^5*d^7*b^4*a^6*A + 168*x^5*e*d^6*b^3*a^7*A + 189*x^5*e^2*d^5*b^2*a^8*A + 70*x^
5*e^3*d^4*b*a^9*A + 7*x^5*e^4*d^3*a^10*A + 45/4*x^4*d^7*b^2*a^8*B + 35/2*x^4*e*d
^6*b*a^9*B + 21/4*x^4*e^2*d^5*a^10*B + 30*x^4*d^7*b^3*a^7*A + 315/4*x^4*e*d^6*b^
2*a^8*A + 105/2*x^4*e^2*d^5*b*a^9*A + 35/4*x^4*e^3*d^4*a^10*A + 10/3*x^3*d^7*b*a
^9*B + 7/3*x^3*e*d^6*a^10*B + 15*x^3*d^7*b^2*a^8*A + 70/3*x^3*e*d^6*b*a^9*A + 7*
x^3*e^2*d^5*a^10*A + 1/2*x^2*d^7*a^10*B + 5*x^2*d^7*b*a^9*A + 7/2*x^2*e*d^6*a^10
*A + x*d^7*a^10*A

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Sympy [A]  time = 1.19447, size = 2824, normalized size = 8.58 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**10*(B*x+A)*(e*x+d)**7,x)

[Out]

A*a**10*d**7*x + B*b**10*e**7*x**19/19 + x**18*(A*b**10*e**7/18 + 5*B*a*b**9*e**
7/9 + 7*B*b**10*d*e**6/18) + x**17*(10*A*a*b**9*e**7/17 + 7*A*b**10*d*e**6/17 +
45*B*a**2*b**8*e**7/17 + 70*B*a*b**9*d*e**6/17 + 21*B*b**10*d**2*e**5/17) + x**1
6*(45*A*a**2*b**8*e**7/16 + 35*A*a*b**9*d*e**6/8 + 21*A*b**10*d**2*e**5/16 + 15*
B*a**3*b**7*e**7/2 + 315*B*a**2*b**8*d*e**6/16 + 105*B*a*b**9*d**2*e**5/8 + 35*B
*b**10*d**3*e**4/16) + x**15*(8*A*a**3*b**7*e**7 + 21*A*a**2*b**8*d*e**6 + 14*A*
a*b**9*d**2*e**5 + 7*A*b**10*d**3*e**4/3 + 14*B*a**4*b**6*e**7 + 56*B*a**3*b**7*
d*e**6 + 63*B*a**2*b**8*d**2*e**5 + 70*B*a*b**9*d**3*e**4/3 + 7*B*b**10*d**4*e**
3/3) + x**14*(15*A*a**4*b**6*e**7 + 60*A*a**3*b**7*d*e**6 + 135*A*a**2*b**8*d**2
*e**5/2 + 25*A*a*b**9*d**3*e**4 + 5*A*b**10*d**4*e**3/2 + 18*B*a**5*b**5*e**7 +
105*B*a**4*b**6*d*e**6 + 180*B*a**3*b**7*d**2*e**5 + 225*B*a**2*b**8*d**3*e**4/2
+ 25*B*a*b**9*d**4*e**3 + 3*B*b**10*d**5*e**2/2) + x**13*(252*A*a**5*b**5*e**7/
13 + 1470*A*a**4*b**6*d*e**6/13 + 2520*A*a**3*b**7*d**2*e**5/13 + 1575*A*a**2*b*
*8*d**3*e**4/13 + 350*A*a*b**9*d**4*e**3/13 + 21*A*b**10*d**5*e**2/13 + 210*B*a*
*6*b**4*e**7/13 + 1764*B*a**5*b**5*d*e**6/13 + 4410*B*a**4*b**6*d**2*e**5/13 + 4
200*B*a**3*b**7*d**3*e**4/13 + 1575*B*a**2*b**8*d**4*e**3/13 + 210*B*a*b**9*d**5
*e**2/13 + 7*B*b**10*d**6*e/13) + x**12*(35*A*a**6*b**4*e**7/2 + 147*A*a**5*b**5
*d*e**6 + 735*A*a**4*b**6*d**2*e**5/2 + 350*A*a**3*b**7*d**3*e**4 + 525*A*a**2*b
**8*d**4*e**3/4 + 35*A*a*b**9*d**5*e**2/2 + 7*A*b**10*d**6*e/12 + 10*B*a**7*b**3
*e**7 + 245*B*a**6*b**4*d*e**6/2 + 441*B*a**5*b**5*d**2*e**5 + 1225*B*a**4*b**6*
d**3*e**4/2 + 350*B*a**3*b**7*d**4*e**3 + 315*B*a**2*b**8*d**5*e**2/4 + 35*B*a*b
**9*d**6*e/6 + B*b**10*d**7/12) + x**11*(120*A*a**7*b**3*e**7/11 + 1470*A*a**6*b
**4*d*e**6/11 + 5292*A*a**5*b**5*d**2*e**5/11 + 7350*A*a**4*b**6*d**3*e**4/11 +
4200*A*a**3*b**7*d**4*e**3/11 + 945*A*a**2*b**8*d**5*e**2/11 + 70*A*a*b**9*d**6*
e/11 + A*b**10*d**7/11 + 45*B*a**8*b**2*e**7/11 + 840*B*a**7*b**3*d*e**6/11 + 44
10*B*a**6*b**4*d**2*e**5/11 + 8820*B*a**5*b**5*d**3*e**4/11 + 7350*B*a**4*b**6*d
**4*e**3/11 + 2520*B*a**3*b**7*d**5*e**2/11 + 315*B*a**2*b**8*d**6*e/11 + 10*B*a
*b**9*d**7/11) + x**10*(9*A*a**8*b**2*e**7/2 + 84*A*a**7*b**3*d*e**6 + 441*A*a**
6*b**4*d**2*e**5 + 882*A*a**5*b**5*d**3*e**4 + 735*A*a**4*b**6*d**4*e**3 + 252*A
*a**3*b**7*d**5*e**2 + 63*A*a**2*b**8*d**6*e/2 + A*a*b**9*d**7 + B*a**9*b*e**7 +
63*B*a**8*b**2*d*e**6/2 + 252*B*a**7*b**3*d**2*e**5 + 735*B*a**6*b**4*d**3*e**4
+ 882*B*a**5*b**5*d**4*e**3 + 441*B*a**4*b**6*d**5*e**2 + 84*B*a**3*b**7*d**6*e
+ 9*B*a**2*b**8*d**7/2) + x**9*(10*A*a**9*b*e**7/9 + 35*A*a**8*b**2*d*e**6 + 28
0*A*a**7*b**3*d**2*e**5 + 2450*A*a**6*b**4*d**3*e**4/3 + 980*A*a**5*b**5*d**4*e*
*3 + 490*A*a**4*b**6*d**5*e**2 + 280*A*a**3*b**7*d**6*e/3 + 5*A*a**2*b**8*d**7 +
B*a**10*e**7/9 + 70*B*a**9*b*d*e**6/9 + 105*B*a**8*b**2*d**2*e**5 + 1400*B*a**7
*b**3*d**3*e**4/3 + 2450*B*a**6*b**4*d**4*e**3/3 + 588*B*a**5*b**5*d**5*e**2 + 4
90*B*a**4*b**6*d**6*e/3 + 40*B*a**3*b**7*d**7/3) + x**8*(A*a**10*e**7/8 + 35*A*a
**9*b*d*e**6/4 + 945*A*a**8*b**2*d**2*e**5/8 + 525*A*a**7*b**3*d**3*e**4 + 3675*
A*a**6*b**4*d**4*e**3/4 + 1323*A*a**5*b**5*d**5*e**2/2 + 735*A*a**4*b**6*d**6*e/
4 + 15*A*a**3*b**7*d**7 + 7*B*a**10*d*e**6/8 + 105*B*a**9*b*d**2*e**5/4 + 1575*B
*a**8*b**2*d**3*e**4/8 + 525*B*a**7*b**3*d**4*e**3 + 2205*B*a**6*b**4*d**5*e**2/
4 + 441*B*a**5*b**5*d**6*e/2 + 105*B*a**4*b**6*d**7/4) + x**7*(A*a**10*d*e**6 +
30*A*a**9*b*d**2*e**5 + 225*A*a**8*b**2*d**3*e**4 + 600*A*a**7*b**3*d**4*e**3 +
630*A*a**6*b**4*d**5*e**2 + 252*A*a**5*b**5*d**6*e + 30*A*a**4*b**6*d**7 + 3*B*a
**10*d**2*e**5 + 50*B*a**9*b*d**3*e**4 + 225*B*a**8*b**2*d**4*e**3 + 360*B*a**7*
b**3*d**5*e**2 + 210*B*a**6*b**4*d**6*e + 36*B*a**5*b**5*d**7) + x**6*(7*A*a**10
*d**2*e**5/2 + 175*A*a**9*b*d**3*e**4/3 + 525*A*a**8*b**2*d**4*e**3/2 + 420*A*a*
*7*b**3*d**5*e**2 + 245*A*a**6*b**4*d**6*e + 42*A*a**5*b**5*d**7 + 35*B*a**10*d*
*3*e**4/6 + 175*B*a**9*b*d**4*e**3/3 + 315*B*a**8*b**2*d**5*e**2/2 + 140*B*a**7*
b**3*d**6*e + 35*B*a**6*b**4*d**7) + x**5*(7*A*a**10*d**3*e**4 + 70*A*a**9*b*d**
4*e**3 + 189*A*a**8*b**2*d**5*e**2 + 168*A*a**7*b**3*d**6*e + 42*A*a**6*b**4*d**
7 + 7*B*a**10*d**4*e**3 + 42*B*a**9*b*d**5*e**2 + 63*B*a**8*b**2*d**6*e + 24*B*a
**7*b**3*d**7) + x**4*(35*A*a**10*d**4*e**3/4 + 105*A*a**9*b*d**5*e**2/2 + 315*A
*a**8*b**2*d**6*e/4 + 30*A*a**7*b**3*d**7 + 21*B*a**10*d**5*e**2/4 + 35*B*a**9*b
*d**6*e/2 + 45*B*a**8*b**2*d**7/4) + x**3*(7*A*a**10*d**5*e**2 + 70*A*a**9*b*d**
6*e/3 + 15*A*a**8*b**2*d**7 + 7*B*a**10*d**6*e/3 + 10*B*a**9*b*d**7/3) + x**2*(7
*A*a**10*d**6*e/2 + 5*A*a**9*b*d**7 + B*a**10*d**7/2)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.212046, 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)^7,x, algorithm="giac")

[Out]

Done