3.1084 \(\int (a+b x)^{10} (A+B x) (d+e x)^4 \, dx\)

Optimal. Leaf size=204 \[ \frac {e^3 (a+b x)^{15} (-5 a B e+A b e+4 b B d)}{15 b^6}+\frac {e^2 (a+b x)^{14} (b d-a e) (-5 a B e+2 A b e+3 b B d)}{7 b^6}+\frac {2 e (a+b x)^{13} (b d-a e)^2 (-5 a B e+3 A b e+2 b B d)}{13 b^6}+\frac {(a+b x)^{12} (b d-a e)^3 (-5 a B e+4 A b e+b B d)}{12 b^6}+\frac {(a+b x)^{11} (A b-a B) (b d-a e)^4}{11 b^6}+\frac {B e^4 (a+b x)^{16}}{16 b^6} \]

[Out]

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

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Rubi [A]  time = 1.17, antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {77} \[ \frac {e^3 (a+b x)^{15} (-5 a B e+A b e+4 b B d)}{15 b^6}+\frac {e^2 (a+b x)^{14} (b d-a e) (-5 a B e+2 A b e+3 b B d)}{7 b^6}+\frac {2 e (a+b x)^{13} (b d-a e)^2 (-5 a B e+3 A b e+2 b B d)}{13 b^6}+\frac {(a+b x)^{12} (b d-a e)^3 (-5 a B e+4 A b e+b B d)}{12 b^6}+\frac {(a+b x)^{11} (A b-a B) (b d-a e)^4}{11 b^6}+\frac {B e^4 (a+b x)^{16}}{16 b^6} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^10*(A + B*x)*(d + e*x)^4,x]

[Out]

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

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 (a+b x)^{10} (A+B x) (d+e x)^4 \, dx &=\int \left (\frac {(A b-a B) (b d-a e)^4 (a+b x)^{10}}{b^5}+\frac {(b d-a e)^3 (b B d+4 A b e-5 a B e) (a+b x)^{11}}{b^5}+\frac {2 e (b d-a e)^2 (2 b B d+3 A b e-5 a B e) (a+b x)^{12}}{b^5}+\frac {2 e^2 (b d-a e) (3 b B d+2 A b e-5 a B e) (a+b x)^{13}}{b^5}+\frac {e^3 (4 b B d+A b e-5 a B e) (a+b x)^{14}}{b^5}+\frac {B e^4 (a+b x)^{15}}{b^5}\right ) \, dx\\ &=\frac {(A b-a B) (b d-a e)^4 (a+b x)^{11}}{11 b^6}+\frac {(b d-a e)^3 (b B d+4 A b e-5 a B e) (a+b x)^{12}}{12 b^6}+\frac {2 e (b d-a e)^2 (2 b B d+3 A b e-5 a B e) (a+b x)^{13}}{13 b^6}+\frac {e^2 (b d-a e) (3 b B d+2 A b e-5 a B e) (a+b x)^{14}}{7 b^6}+\frac {e^3 (4 b B d+A b e-5 a B e) (a+b x)^{15}}{15 b^6}+\frac {B e^4 (a+b x)^{16}}{16 b^6}\\ \end {align*}

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Mathematica [B]  time = 0.70, size = 1098, normalized size = 5.38 \[ \frac {x \left (8008 \left (6 A \left (5 d^4+10 e x d^3+10 e^2 x^2 d^2+5 e^3 x^3 d+e^4 x^4\right )+B x \left (15 d^4+40 e x d^3+45 e^2 x^2 d^2+24 e^3 x^3 d+5 e^4 x^4\right )\right ) a^{10}+11440 b x \left (7 A \left (15 d^4+40 e x d^3+45 e^2 x^2 d^2+24 e^3 x^3 d+5 e^4 x^4\right )+2 B x \left (35 d^4+105 e x d^3+126 e^2 x^2 d^2+70 e^3 x^3 d+15 e^4 x^4\right )\right ) a^9+12870 b^2 x^2 \left (8 A \left (35 d^4+105 e x d^3+126 e^2 x^2 d^2+70 e^3 x^3 d+15 e^4 x^4\right )+3 B x \left (70 d^4+224 e x d^3+280 e^2 x^2 d^2+160 e^3 x^3 d+35 e^4 x^4\right )\right ) a^8+11440 b^3 x^3 \left (9 A \left (70 d^4+224 e x d^3+280 e^2 x^2 d^2+160 e^3 x^3 d+35 e^4 x^4\right )+4 B x \left (126 d^4+420 e x d^3+540 e^2 x^2 d^2+315 e^3 x^3 d+70 e^4 x^4\right )\right ) a^7+40040 b^4 x^4 \left (2 A \left (126 d^4+420 e x d^3+540 e^2 x^2 d^2+315 e^3 x^3 d+70 e^4 x^4\right )+B x \left (210 d^4+720 e x d^3+945 e^2 x^2 d^2+560 e^3 x^3 d+126 e^4 x^4\right )\right ) a^6+4368 b^5 x^5 \left (11 A \left (210 d^4+720 e x d^3+945 e^2 x^2 d^2+560 e^3 x^3 d+126 e^4 x^4\right )+6 B x \left (330 d^4+1155 e x d^3+1540 e^2 x^2 d^2+924 e^3 x^3 d+210 e^4 x^4\right )\right ) a^5+1820 b^6 x^6 \left (12 A \left (330 d^4+1155 e x d^3+1540 e^2 x^2 d^2+924 e^3 x^3 d+210 e^4 x^4\right )+7 B x \left (495 d^4+1760 e x d^3+2376 e^2 x^2 d^2+1440 e^3 x^3 d+330 e^4 x^4\right )\right ) a^4+560 b^7 x^7 \left (13 A \left (495 d^4+1760 e x d^3+2376 e^2 x^2 d^2+1440 e^3 x^3 d+330 e^4 x^4\right )+8 B x \left (715 d^4+2574 e x d^3+3510 e^2 x^2 d^2+2145 e^3 x^3 d+495 e^4 x^4\right )\right ) a^3+120 b^8 x^8 \left (14 A \left (715 d^4+2574 e x d^3+3510 e^2 x^2 d^2+2145 e^3 x^3 d+495 e^4 x^4\right )+9 B x \left (1001 d^4+3640 e x d^3+5005 e^2 x^2 d^2+3080 e^3 x^3 d+715 e^4 x^4\right )\right ) a^2+80 b^9 x^9 \left (3 A \left (1001 d^4+3640 e x d^3+5005 e^2 x^2 d^2+3080 e^3 x^3 d+715 e^4 x^4\right )+2 B x \left (1365 d^4+5005 e x d^3+6930 e^2 x^2 d^2+4290 e^3 x^3 d+1001 e^4 x^4\right )\right ) a+b^{10} x^{10} \left (16 A \left (1365 d^4+5005 e x d^3+6930 e^2 x^2 d^2+4290 e^3 x^3 d+1001 e^4 x^4\right )+11 B x \left (1820 d^4+6720 e x d^3+9360 e^2 x^2 d^2+5824 e^3 x^3 d+1365 e^4 x^4\right )\right )\right )}{240240} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^10*(A + B*x)*(d + e*x)^4,x]

[Out]

(x*(8008*a^10*(6*A*(5*d^4 + 10*d^3*e*x + 10*d^2*e^2*x^2 + 5*d*e^3*x^3 + e^4*x^4) + B*x*(15*d^4 + 40*d^3*e*x +
45*d^2*e^2*x^2 + 24*d*e^3*x^3 + 5*e^4*x^4)) + 11440*a^9*b*x*(7*A*(15*d^4 + 40*d^3*e*x + 45*d^2*e^2*x^2 + 24*d*
e^3*x^3 + 5*e^4*x^4) + 2*B*x*(35*d^4 + 105*d^3*e*x + 126*d^2*e^2*x^2 + 70*d*e^3*x^3 + 15*e^4*x^4)) + 12870*a^8
*b^2*x^2*(8*A*(35*d^4 + 105*d^3*e*x + 126*d^2*e^2*x^2 + 70*d*e^3*x^3 + 15*e^4*x^4) + 3*B*x*(70*d^4 + 224*d^3*e
*x + 280*d^2*e^2*x^2 + 160*d*e^3*x^3 + 35*e^4*x^4)) + 11440*a^7*b^3*x^3*(9*A*(70*d^4 + 224*d^3*e*x + 280*d^2*e
^2*x^2 + 160*d*e^3*x^3 + 35*e^4*x^4) + 4*B*x*(126*d^4 + 420*d^3*e*x + 540*d^2*e^2*x^2 + 315*d*e^3*x^3 + 70*e^4
*x^4)) + 40040*a^6*b^4*x^4*(2*A*(126*d^4 + 420*d^3*e*x + 540*d^2*e^2*x^2 + 315*d*e^3*x^3 + 70*e^4*x^4) + B*x*(
210*d^4 + 720*d^3*e*x + 945*d^2*e^2*x^2 + 560*d*e^3*x^3 + 126*e^4*x^4)) + 4368*a^5*b^5*x^5*(11*A*(210*d^4 + 72
0*d^3*e*x + 945*d^2*e^2*x^2 + 560*d*e^3*x^3 + 126*e^4*x^4) + 6*B*x*(330*d^4 + 1155*d^3*e*x + 1540*d^2*e^2*x^2
+ 924*d*e^3*x^3 + 210*e^4*x^4)) + 1820*a^4*b^6*x^6*(12*A*(330*d^4 + 1155*d^3*e*x + 1540*d^2*e^2*x^2 + 924*d*e^
3*x^3 + 210*e^4*x^4) + 7*B*x*(495*d^4 + 1760*d^3*e*x + 2376*d^2*e^2*x^2 + 1440*d*e^3*x^3 + 330*e^4*x^4)) + 560
*a^3*b^7*x^7*(13*A*(495*d^4 + 1760*d^3*e*x + 2376*d^2*e^2*x^2 + 1440*d*e^3*x^3 + 330*e^4*x^4) + 8*B*x*(715*d^4
+ 2574*d^3*e*x + 3510*d^2*e^2*x^2 + 2145*d*e^3*x^3 + 495*e^4*x^4)) + 120*a^2*b^8*x^8*(14*A*(715*d^4 + 2574*d^
3*e*x + 3510*d^2*e^2*x^2 + 2145*d*e^3*x^3 + 495*e^4*x^4) + 9*B*x*(1001*d^4 + 3640*d^3*e*x + 5005*d^2*e^2*x^2 +
3080*d*e^3*x^3 + 715*e^4*x^4)) + 80*a*b^9*x^9*(3*A*(1001*d^4 + 3640*d^3*e*x + 5005*d^2*e^2*x^2 + 3080*d*e^3*x
^3 + 715*e^4*x^4) + 2*B*x*(1365*d^4 + 5005*d^3*e*x + 6930*d^2*e^2*x^2 + 4290*d*e^3*x^3 + 1001*e^4*x^4)) + b^10
*x^10*(16*A*(1365*d^4 + 5005*d^3*e*x + 6930*d^2*e^2*x^2 + 4290*d*e^3*x^3 + 1001*e^4*x^4) + 11*B*x*(1820*d^4 +
6720*d^3*e*x + 9360*d^2*e^2*x^2 + 5824*d*e^3*x^3 + 1365*e^4*x^4))))/240240

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fricas [B]  time = 0.75, size = 1656, normalized size = 8.12 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)*(e*x+d)^4,x, algorithm="fricas")

[Out]

1/16*x^16*e^4*b^10*B + 4/15*x^15*e^3*d*b^10*B + 2/3*x^15*e^4*b^9*a*B + 1/15*x^15*e^4*b^10*A + 3/7*x^14*e^2*d^2
*b^10*B + 20/7*x^14*e^3*d*b^9*a*B + 45/14*x^14*e^4*b^8*a^2*B + 2/7*x^14*e^3*d*b^10*A + 5/7*x^14*e^4*b^9*a*A +
4/13*x^13*e*d^3*b^10*B + 60/13*x^13*e^2*d^2*b^9*a*B + 180/13*x^13*e^3*d*b^8*a^2*B + 120/13*x^13*e^4*b^7*a^3*B
+ 6/13*x^13*e^2*d^2*b^10*A + 40/13*x^13*e^3*d*b^9*a*A + 45/13*x^13*e^4*b^8*a^2*A + 1/12*x^12*d^4*b^10*B + 10/3
*x^12*e*d^3*b^9*a*B + 45/2*x^12*e^2*d^2*b^8*a^2*B + 40*x^12*e^3*d*b^7*a^3*B + 35/2*x^12*e^4*b^6*a^4*B + 1/3*x^
12*e*d^3*b^10*A + 5*x^12*e^2*d^2*b^9*a*A + 15*x^12*e^3*d*b^8*a^2*A + 10*x^12*e^4*b^7*a^3*A + 10/11*x^11*d^4*b^
9*a*B + 180/11*x^11*e*d^3*b^8*a^2*B + 720/11*x^11*e^2*d^2*b^7*a^3*B + 840/11*x^11*e^3*d*b^6*a^4*B + 252/11*x^1
1*e^4*b^5*a^5*B + 1/11*x^11*d^4*b^10*A + 40/11*x^11*e*d^3*b^9*a*A + 270/11*x^11*e^2*d^2*b^8*a^2*A + 480/11*x^1
1*e^3*d*b^7*a^3*A + 210/11*x^11*e^4*b^6*a^4*A + 9/2*x^10*d^4*b^8*a^2*B + 48*x^10*e*d^3*b^7*a^3*B + 126*x^10*e^
2*d^2*b^6*a^4*B + 504/5*x^10*e^3*d*b^5*a^5*B + 21*x^10*e^4*b^4*a^6*B + x^10*d^4*b^9*a*A + 18*x^10*e*d^3*b^8*a^
2*A + 72*x^10*e^2*d^2*b^7*a^3*A + 84*x^10*e^3*d*b^6*a^4*A + 126/5*x^10*e^4*b^5*a^5*A + 40/3*x^9*d^4*b^7*a^3*B
+ 280/3*x^9*e*d^3*b^6*a^4*B + 168*x^9*e^2*d^2*b^5*a^5*B + 280/3*x^9*e^3*d*b^4*a^6*B + 40/3*x^9*e^4*b^3*a^7*B +
5*x^9*d^4*b^8*a^2*A + 160/3*x^9*e*d^3*b^7*a^3*A + 140*x^9*e^2*d^2*b^6*a^4*A + 112*x^9*e^3*d*b^5*a^5*A + 70/3*
x^9*e^4*b^4*a^6*A + 105/4*x^8*d^4*b^6*a^4*B + 126*x^8*e*d^3*b^5*a^5*B + 315/2*x^8*e^2*d^2*b^4*a^6*B + 60*x^8*e
^3*d*b^3*a^7*B + 45/8*x^8*e^4*b^2*a^8*B + 15*x^8*d^4*b^7*a^3*A + 105*x^8*e*d^3*b^6*a^4*A + 189*x^8*e^2*d^2*b^5
*a^5*A + 105*x^8*e^3*d*b^4*a^6*A + 15*x^8*e^4*b^3*a^7*A + 36*x^7*d^4*b^5*a^5*B + 120*x^7*e*d^3*b^4*a^6*B + 720
/7*x^7*e^2*d^2*b^3*a^7*B + 180/7*x^7*e^3*d*b^2*a^8*B + 10/7*x^7*e^4*b*a^9*B + 30*x^7*d^4*b^6*a^4*A + 144*x^7*e
*d^3*b^5*a^5*A + 180*x^7*e^2*d^2*b^4*a^6*A + 480/7*x^7*e^3*d*b^3*a^7*A + 45/7*x^7*e^4*b^2*a^8*A + 35*x^6*d^4*b
^4*a^6*B + 80*x^6*e*d^3*b^3*a^7*B + 45*x^6*e^2*d^2*b^2*a^8*B + 20/3*x^6*e^3*d*b*a^9*B + 1/6*x^6*e^4*a^10*B + 4
2*x^6*d^4*b^5*a^5*A + 140*x^6*e*d^3*b^4*a^6*A + 120*x^6*e^2*d^2*b^3*a^7*A + 30*x^6*e^3*d*b^2*a^8*A + 5/3*x^6*e
^4*b*a^9*A + 24*x^5*d^4*b^3*a^7*B + 36*x^5*e*d^3*b^2*a^8*B + 12*x^5*e^2*d^2*b*a^9*B + 4/5*x^5*e^3*d*a^10*B + 4
2*x^5*d^4*b^4*a^6*A + 96*x^5*e*d^3*b^3*a^7*A + 54*x^5*e^2*d^2*b^2*a^8*A + 8*x^5*e^3*d*b*a^9*A + 1/5*x^5*e^4*a^
10*A + 45/4*x^4*d^4*b^2*a^8*B + 10*x^4*e*d^3*b*a^9*B + 3/2*x^4*e^2*d^2*a^10*B + 30*x^4*d^4*b^3*a^7*A + 45*x^4*
e*d^3*b^2*a^8*A + 15*x^4*e^2*d^2*b*a^9*A + x^4*e^3*d*a^10*A + 10/3*x^3*d^4*b*a^9*B + 4/3*x^3*e*d^3*a^10*B + 15
*x^3*d^4*b^2*a^8*A + 40/3*x^3*e*d^3*b*a^9*A + 2*x^3*e^2*d^2*a^10*A + 1/2*x^2*d^4*a^10*B + 5*x^2*d^4*b*a^9*A +
2*x^2*e*d^3*a^10*A + x*d^4*a^10*A

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giac [B]  time = 1.23, size = 1612, normalized size = 7.90 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)*(e*x+d)^4,x, algorithm="giac")

[Out]

1/16*B*b^10*x^16*e^4 + 4/15*B*b^10*d*x^15*e^3 + 3/7*B*b^10*d^2*x^14*e^2 + 4/13*B*b^10*d^3*x^13*e + 1/12*B*b^10
*d^4*x^12 + 2/3*B*a*b^9*x^15*e^4 + 1/15*A*b^10*x^15*e^4 + 20/7*B*a*b^9*d*x^14*e^3 + 2/7*A*b^10*d*x^14*e^3 + 60
/13*B*a*b^9*d^2*x^13*e^2 + 6/13*A*b^10*d^2*x^13*e^2 + 10/3*B*a*b^9*d^3*x^12*e + 1/3*A*b^10*d^3*x^12*e + 10/11*
B*a*b^9*d^4*x^11 + 1/11*A*b^10*d^4*x^11 + 45/14*B*a^2*b^8*x^14*e^4 + 5/7*A*a*b^9*x^14*e^4 + 180/13*B*a^2*b^8*d
*x^13*e^3 + 40/13*A*a*b^9*d*x^13*e^3 + 45/2*B*a^2*b^8*d^2*x^12*e^2 + 5*A*a*b^9*d^2*x^12*e^2 + 180/11*B*a^2*b^8
*d^3*x^11*e + 40/11*A*a*b^9*d^3*x^11*e + 9/2*B*a^2*b^8*d^4*x^10 + A*a*b^9*d^4*x^10 + 120/13*B*a^3*b^7*x^13*e^4
+ 45/13*A*a^2*b^8*x^13*e^4 + 40*B*a^3*b^7*d*x^12*e^3 + 15*A*a^2*b^8*d*x^12*e^3 + 720/11*B*a^3*b^7*d^2*x^11*e^
2 + 270/11*A*a^2*b^8*d^2*x^11*e^2 + 48*B*a^3*b^7*d^3*x^10*e + 18*A*a^2*b^8*d^3*x^10*e + 40/3*B*a^3*b^7*d^4*x^9
+ 5*A*a^2*b^8*d^4*x^9 + 35/2*B*a^4*b^6*x^12*e^4 + 10*A*a^3*b^7*x^12*e^4 + 840/11*B*a^4*b^6*d*x^11*e^3 + 480/1
1*A*a^3*b^7*d*x^11*e^3 + 126*B*a^4*b^6*d^2*x^10*e^2 + 72*A*a^3*b^7*d^2*x^10*e^2 + 280/3*B*a^4*b^6*d^3*x^9*e +
160/3*A*a^3*b^7*d^3*x^9*e + 105/4*B*a^4*b^6*d^4*x^8 + 15*A*a^3*b^7*d^4*x^8 + 252/11*B*a^5*b^5*x^11*e^4 + 210/1
1*A*a^4*b^6*x^11*e^4 + 504/5*B*a^5*b^5*d*x^10*e^3 + 84*A*a^4*b^6*d*x^10*e^3 + 168*B*a^5*b^5*d^2*x^9*e^2 + 140*
A*a^4*b^6*d^2*x^9*e^2 + 126*B*a^5*b^5*d^3*x^8*e + 105*A*a^4*b^6*d^3*x^8*e + 36*B*a^5*b^5*d^4*x^7 + 30*A*a^4*b^
6*d^4*x^7 + 21*B*a^6*b^4*x^10*e^4 + 126/5*A*a^5*b^5*x^10*e^4 + 280/3*B*a^6*b^4*d*x^9*e^3 + 112*A*a^5*b^5*d*x^9
*e^3 + 315/2*B*a^6*b^4*d^2*x^8*e^2 + 189*A*a^5*b^5*d^2*x^8*e^2 + 120*B*a^6*b^4*d^3*x^7*e + 144*A*a^5*b^5*d^3*x
^7*e + 35*B*a^6*b^4*d^4*x^6 + 42*A*a^5*b^5*d^4*x^6 + 40/3*B*a^7*b^3*x^9*e^4 + 70/3*A*a^6*b^4*x^9*e^4 + 60*B*a^
7*b^3*d*x^8*e^3 + 105*A*a^6*b^4*d*x^8*e^3 + 720/7*B*a^7*b^3*d^2*x^7*e^2 + 180*A*a^6*b^4*d^2*x^7*e^2 + 80*B*a^7
*b^3*d^3*x^6*e + 140*A*a^6*b^4*d^3*x^6*e + 24*B*a^7*b^3*d^4*x^5 + 42*A*a^6*b^4*d^4*x^5 + 45/8*B*a^8*b^2*x^8*e^
4 + 15*A*a^7*b^3*x^8*e^4 + 180/7*B*a^8*b^2*d*x^7*e^3 + 480/7*A*a^7*b^3*d*x^7*e^3 + 45*B*a^8*b^2*d^2*x^6*e^2 +
120*A*a^7*b^3*d^2*x^6*e^2 + 36*B*a^8*b^2*d^3*x^5*e + 96*A*a^7*b^3*d^3*x^5*e + 45/4*B*a^8*b^2*d^4*x^4 + 30*A*a^
7*b^3*d^4*x^4 + 10/7*B*a^9*b*x^7*e^4 + 45/7*A*a^8*b^2*x^7*e^4 + 20/3*B*a^9*b*d*x^6*e^3 + 30*A*a^8*b^2*d*x^6*e^
3 + 12*B*a^9*b*d^2*x^5*e^2 + 54*A*a^8*b^2*d^2*x^5*e^2 + 10*B*a^9*b*d^3*x^4*e + 45*A*a^8*b^2*d^3*x^4*e + 10/3*B
*a^9*b*d^4*x^3 + 15*A*a^8*b^2*d^4*x^3 + 1/6*B*a^10*x^6*e^4 + 5/3*A*a^9*b*x^6*e^4 + 4/5*B*a^10*d*x^5*e^3 + 8*A*
a^9*b*d*x^5*e^3 + 3/2*B*a^10*d^2*x^4*e^2 + 15*A*a^9*b*d^2*x^4*e^2 + 4/3*B*a^10*d^3*x^3*e + 40/3*A*a^9*b*d^3*x^
3*e + 1/2*B*a^10*d^4*x^2 + 5*A*a^9*b*d^4*x^2 + 1/5*A*a^10*x^5*e^4 + A*a^10*d*x^4*e^3 + 2*A*a^10*d^2*x^3*e^2 +
2*A*a^10*d^3*x^2*e + A*a^10*d^4*x

________________________________________________________________________________________

maple [B]  time = 0.00, size = 1337, normalized size = 6.55 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^10*(B*x+A)*(e*x+d)^4,x)

[Out]

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

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maxima [B]  time = 0.63, size = 1352, normalized size = 6.63 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)*(e*x+d)^4,x, algorithm="maxima")

[Out]

1/16*B*b^10*e^4*x^16 + A*a^10*d^4*x + 1/15*(4*B*b^10*d*e^3 + (10*B*a*b^9 + A*b^10)*e^4)*x^15 + 1/14*(6*B*b^10*
d^2*e^2 + 4*(10*B*a*b^9 + A*b^10)*d*e^3 + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*e^4)*x^14 + 1/13*(4*B*b^10*d^3*e + 6*(10
*B*a*b^9 + A*b^10)*d^2*e^2 + 20*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^3 + 15*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^4)*x^13 + 1
/12*(B*b^10*d^4 + 4*(10*B*a*b^9 + A*b^10)*d^3*e + 30*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^2 + 60*(8*B*a^3*b^7 + 3*A
*a^2*b^8)*d*e^3 + 30*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^4)*x^12 + 1/11*((10*B*a*b^9 + A*b^10)*d^4 + 20*(9*B*a^2*b^8
+ 2*A*a*b^9)*d^3*e + 90*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^2 + 120*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^3 + 42*(6*B
*a^5*b^5 + 5*A*a^4*b^6)*e^4)*x^11 + 1/10*(5*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4 + 60*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3
*e + 180*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^2 + 168*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^3 + 42*(5*B*a^6*b^4 + 6*A*a
^5*b^5)*e^4)*x^10 + 1/3*(5*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4 + 40*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e + 84*(6*B*a^
5*b^5 + 5*A*a^4*b^6)*d^2*e^2 + 56*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^3 + 10*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^4)*x^9
+ 3/8*(10*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4 + 56*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e + 84*(5*B*a^6*b^4 + 6*A*a^5*b
^5)*d^2*e^2 + 40*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*e^3 + 5*(3*B*a^8*b^2 + 8*A*a^7*b^3)*e^4)*x^8 + 1/7*(42*(6*B*a^5
*b^5 + 5*A*a^4*b^6)*d^4 + 168*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*e + 180*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*e^2 + 60
*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d*e^3 + 5*(2*B*a^9*b + 9*A*a^8*b^2)*e^4)*x^7 + 1/6*(42*(5*B*a^6*b^4 + 6*A*a^5*b^5
)*d^4 + 120*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*e + 90*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e^2 + 20*(2*B*a^9*b + 9*A*a
^8*b^2)*d*e^3 + (B*a^10 + 10*A*a^9*b)*e^4)*x^6 + 1/5*(A*a^10*e^4 + 30*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4 + 60*(3*
B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e + 30*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^2 + 4*(B*a^10 + 10*A*a^9*b)*d*e^3)*x^5 + 1
/4*(4*A*a^10*d*e^3 + 15*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^4 + 20*(2*B*a^9*b + 9*A*a^8*b^2)*d^3*e + 6*(B*a^10 + 10*
A*a^9*b)*d^2*e^2)*x^4 + 1/3*(6*A*a^10*d^2*e^2 + 5*(2*B*a^9*b + 9*A*a^8*b^2)*d^4 + 4*(B*a^10 + 10*A*a^9*b)*d^3*
e)*x^3 + 1/2*(4*A*a^10*d^3*e + (B*a^10 + 10*A*a^9*b)*d^4)*x^2

________________________________________________________________________________________

mupad [B]  time = 0.53, size = 1386, normalized size = 6.79 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)*(a + b*x)^10*(d + e*x)^4,x)

[Out]

x^6*((B*a^10*e^4)/6 + (5*A*a^9*b*e^4)/3 + 42*A*a^5*b^5*d^4 + 35*B*a^6*b^4*d^4 + 140*A*a^6*b^4*d^3*e + 30*A*a^8
*b^2*d*e^3 + 80*B*a^7*b^3*d^3*e + 120*A*a^7*b^3*d^2*e^2 + 45*B*a^8*b^2*d^2*e^2 + (20*B*a^9*b*d*e^3)/3) + x^11*
((A*b^10*d^4)/11 + (10*B*a*b^9*d^4)/11 + (210*A*a^4*b^6*e^4)/11 + (252*B*a^5*b^5*e^4)/11 + (480*A*a^3*b^7*d*e^
3)/11 + (180*B*a^2*b^8*d^3*e)/11 + (840*B*a^4*b^6*d*e^3)/11 + (270*A*a^2*b^8*d^2*e^2)/11 + (720*B*a^3*b^7*d^2*
e^2)/11 + (40*A*a*b^9*d^3*e)/11) + x^10*(A*a*b^9*d^4 + (126*A*a^5*b^5*e^4)/5 + (9*B*a^2*b^8*d^4)/2 + 21*B*a^6*
b^4*e^4 + 18*A*a^2*b^8*d^3*e + 84*A*a^4*b^6*d*e^3 + 48*B*a^3*b^7*d^3*e + (504*B*a^5*b^5*d*e^3)/5 + 72*A*a^3*b^
7*d^2*e^2 + 126*B*a^4*b^6*d^2*e^2) + x^7*((10*B*a^9*b*e^4)/7 + 30*A*a^4*b^6*d^4 + (45*A*a^8*b^2*e^4)/7 + 36*B*
a^5*b^5*d^4 + 144*A*a^5*b^5*d^3*e + (480*A*a^7*b^3*d*e^3)/7 + 120*B*a^6*b^4*d^3*e + (180*B*a^8*b^2*d*e^3)/7 +
180*A*a^6*b^4*d^2*e^2 + (720*B*a^7*b^3*d^2*e^2)/7) + x^4*(A*a^10*d*e^3 + 30*A*a^7*b^3*d^4 + (45*B*a^8*b^2*d^4)
/4 + (3*B*a^10*d^2*e^2)/2 + 45*A*a^8*b^2*d^3*e + 15*A*a^9*b*d^2*e^2 + 10*B*a^9*b*d^3*e) + x^13*((4*B*b^10*d^3*
e)/13 + (45*A*a^2*b^8*e^4)/13 + (120*B*a^3*b^7*e^4)/13 + (6*A*b^10*d^2*e^2)/13 + (60*B*a*b^9*d^2*e^2)/13 + (18
0*B*a^2*b^8*d*e^3)/13 + (40*A*a*b^9*d*e^3)/13) + x^3*((10*B*a^9*b*d^4)/3 + (4*B*a^10*d^3*e)/3 + 15*A*a^8*b^2*d
^4 + 2*A*a^10*d^2*e^2 + (40*A*a^9*b*d^3*e)/3) + x^14*((5*A*a*b^9*e^4)/7 + (2*A*b^10*d*e^3)/7 + (45*B*a^2*b^8*e
^4)/14 + (3*B*b^10*d^2*e^2)/7 + (20*B*a*b^9*d*e^3)/7) + x^5*((A*a^10*e^4)/5 + (4*B*a^10*d*e^3)/5 + 42*A*a^6*b^
4*d^4 + 24*B*a^7*b^3*d^4 + 96*A*a^7*b^3*d^3*e + 36*B*a^8*b^2*d^3*e + 12*B*a^9*b*d^2*e^2 + 54*A*a^8*b^2*d^2*e^2
+ 8*A*a^9*b*d*e^3) + x^12*((B*b^10*d^4)/12 + (A*b^10*d^3*e)/3 + 10*A*a^3*b^7*e^4 + (35*B*a^4*b^6*e^4)/2 + 5*A
*a*b^9*d^2*e^2 + 15*A*a^2*b^8*d*e^3 + 40*B*a^3*b^7*d*e^3 + (45*B*a^2*b^8*d^2*e^2)/2 + (10*B*a*b^9*d^3*e)/3) +
x^8*(15*A*a^3*b^7*d^4 + 15*A*a^7*b^3*e^4 + (105*B*a^4*b^6*d^4)/4 + (45*B*a^8*b^2*e^4)/8 + 105*A*a^4*b^6*d^3*e
+ 105*A*a^6*b^4*d*e^3 + 126*B*a^5*b^5*d^3*e + 60*B*a^7*b^3*d*e^3 + 189*A*a^5*b^5*d^2*e^2 + (315*B*a^6*b^4*d^2*
e^2)/2) + x^9*(5*A*a^2*b^8*d^4 + (70*A*a^6*b^4*e^4)/3 + (40*B*a^3*b^7*d^4)/3 + (40*B*a^7*b^3*e^4)/3 + (160*A*a
^3*b^7*d^3*e)/3 + 112*A*a^5*b^5*d*e^3 + (280*B*a^4*b^6*d^3*e)/3 + (280*B*a^6*b^4*d*e^3)/3 + 140*A*a^4*b^6*d^2*
e^2 + 168*B*a^5*b^5*d^2*e^2) + (a^9*d^3*x^2*(4*A*a*e + 10*A*b*d + B*a*d))/2 + (b^9*e^3*x^15*(A*b*e + 10*B*a*e
+ 4*B*b*d))/15 + A*a^10*d^4*x + (B*b^10*e^4*x^16)/16

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sympy [B]  time = 0.29, size = 1676, normalized size = 8.22 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**10*(B*x+A)*(e*x+d)**4,x)

[Out]

A*a**10*d**4*x + B*b**10*e**4*x**16/16 + x**15*(A*b**10*e**4/15 + 2*B*a*b**9*e**4/3 + 4*B*b**10*d*e**3/15) + x
**14*(5*A*a*b**9*e**4/7 + 2*A*b**10*d*e**3/7 + 45*B*a**2*b**8*e**4/14 + 20*B*a*b**9*d*e**3/7 + 3*B*b**10*d**2*
e**2/7) + x**13*(45*A*a**2*b**8*e**4/13 + 40*A*a*b**9*d*e**3/13 + 6*A*b**10*d**2*e**2/13 + 120*B*a**3*b**7*e**
4/13 + 180*B*a**2*b**8*d*e**3/13 + 60*B*a*b**9*d**2*e**2/13 + 4*B*b**10*d**3*e/13) + x**12*(10*A*a**3*b**7*e**
4 + 15*A*a**2*b**8*d*e**3 + 5*A*a*b**9*d**2*e**2 + A*b**10*d**3*e/3 + 35*B*a**4*b**6*e**4/2 + 40*B*a**3*b**7*d
*e**3 + 45*B*a**2*b**8*d**2*e**2/2 + 10*B*a*b**9*d**3*e/3 + B*b**10*d**4/12) + x**11*(210*A*a**4*b**6*e**4/11
+ 480*A*a**3*b**7*d*e**3/11 + 270*A*a**2*b**8*d**2*e**2/11 + 40*A*a*b**9*d**3*e/11 + A*b**10*d**4/11 + 252*B*a
**5*b**5*e**4/11 + 840*B*a**4*b**6*d*e**3/11 + 720*B*a**3*b**7*d**2*e**2/11 + 180*B*a**2*b**8*d**3*e/11 + 10*B
*a*b**9*d**4/11) + x**10*(126*A*a**5*b**5*e**4/5 + 84*A*a**4*b**6*d*e**3 + 72*A*a**3*b**7*d**2*e**2 + 18*A*a**
2*b**8*d**3*e + A*a*b**9*d**4 + 21*B*a**6*b**4*e**4 + 504*B*a**5*b**5*d*e**3/5 + 126*B*a**4*b**6*d**2*e**2 + 4
8*B*a**3*b**7*d**3*e + 9*B*a**2*b**8*d**4/2) + x**9*(70*A*a**6*b**4*e**4/3 + 112*A*a**5*b**5*d*e**3 + 140*A*a*
*4*b**6*d**2*e**2 + 160*A*a**3*b**7*d**3*e/3 + 5*A*a**2*b**8*d**4 + 40*B*a**7*b**3*e**4/3 + 280*B*a**6*b**4*d*
e**3/3 + 168*B*a**5*b**5*d**2*e**2 + 280*B*a**4*b**6*d**3*e/3 + 40*B*a**3*b**7*d**4/3) + x**8*(15*A*a**7*b**3*
e**4 + 105*A*a**6*b**4*d*e**3 + 189*A*a**5*b**5*d**2*e**2 + 105*A*a**4*b**6*d**3*e + 15*A*a**3*b**7*d**4 + 45*
B*a**8*b**2*e**4/8 + 60*B*a**7*b**3*d*e**3 + 315*B*a**6*b**4*d**2*e**2/2 + 126*B*a**5*b**5*d**3*e + 105*B*a**4
*b**6*d**4/4) + x**7*(45*A*a**8*b**2*e**4/7 + 480*A*a**7*b**3*d*e**3/7 + 180*A*a**6*b**4*d**2*e**2 + 144*A*a**
5*b**5*d**3*e + 30*A*a**4*b**6*d**4 + 10*B*a**9*b*e**4/7 + 180*B*a**8*b**2*d*e**3/7 + 720*B*a**7*b**3*d**2*e**
2/7 + 120*B*a**6*b**4*d**3*e + 36*B*a**5*b**5*d**4) + x**6*(5*A*a**9*b*e**4/3 + 30*A*a**8*b**2*d*e**3 + 120*A*
a**7*b**3*d**2*e**2 + 140*A*a**6*b**4*d**3*e + 42*A*a**5*b**5*d**4 + B*a**10*e**4/6 + 20*B*a**9*b*d*e**3/3 + 4
5*B*a**8*b**2*d**2*e**2 + 80*B*a**7*b**3*d**3*e + 35*B*a**6*b**4*d**4) + x**5*(A*a**10*e**4/5 + 8*A*a**9*b*d*e
**3 + 54*A*a**8*b**2*d**2*e**2 + 96*A*a**7*b**3*d**3*e + 42*A*a**6*b**4*d**4 + 4*B*a**10*d*e**3/5 + 12*B*a**9*
b*d**2*e**2 + 36*B*a**8*b**2*d**3*e + 24*B*a**7*b**3*d**4) + x**4*(A*a**10*d*e**3 + 15*A*a**9*b*d**2*e**2 + 45
*A*a**8*b**2*d**3*e + 30*A*a**7*b**3*d**4 + 3*B*a**10*d**2*e**2/2 + 10*B*a**9*b*d**3*e + 45*B*a**8*b**2*d**4/4
) + x**3*(2*A*a**10*d**2*e**2 + 40*A*a**9*b*d**3*e/3 + 15*A*a**8*b**2*d**4 + 4*B*a**10*d**3*e/3 + 10*B*a**9*b*
d**4/3) + x**2*(2*A*a**10*d**3*e + 5*A*a**9*b*d**4 + B*a**10*d**4/2)

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