\(\int (a+b x)^{10} (A+B x) (d+e x)^4 \, dx\) [1084]

Optimal result

Integrand size = 20, antiderivative size = 204 \[ \int (a+b x)^{10} (A+B x) (d+e x)^4 \, dx=\frac {(A b-a B) (b d-a e)^4 (a+b x)^{11}}{11 b^6}+\frac {(b d-a e)^3 (b B d+4 A b e-5 a B e) (a+b x)^{12}}{12 b^6}+\frac {2 e (b d-a e)^2 (2 b B d+3 A b e-5 a B e) (a+b x)^{13}}{13 b^6}+\frac {e^2 (b d-a e) (3 b B d+2 A b e-5 a B e) (a+b x)^{14}}{7 b^6}+\frac {e^3 (4 b B d+A b e-5 a B e) (a+b x)^{15}}{15 b^6}+\frac {B e^4 (a+b x)^{16}}{16 b^6} \]

[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

Rubi [A] (verified)

Time = 0.73 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {78} \[ \int (a+b x)^{10} (A+B x) (d+e x)^4 \, dx=\frac {e^3 (a+b x)^{15} (-5 a B e+A b e+4 b B d)}{15 b^6}+\frac {e^2 (a+b x)^{14} (b d-a e) (-5 a B e+2 A b e+3 b B d)}{7 b^6}+\frac {2 e (a+b x)^{13} (b d-a e)^2 (-5 a B e+3 A b e+2 b B d)}{13 b^6}+\frac {(a+b x)^{12} (b d-a e)^3 (-5 a B e+4 A b e+b B d)}{12 b^6}+\frac {(a+b x)^{11} (A b-a B) (b d-a e)^4}{11 b^6}+\frac {B e^4 (a+b x)^{16}}{16 b^6} \]

[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 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(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*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1098\) vs. \(2(204)=408\).

Time = 0.47 (sec) , antiderivative size = 1098, normalized size of antiderivative = 5.38 \[ \int (a+b x)^{10} (A+B x) (d+e x)^4 \, dx=\frac {x \left (8008 a^{10} \left (6 A \left (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\right )+B x \left (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\right )\right )+11440 a^9 b x \left (7 A \left (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\right )+2 B x \left (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\right )\right )+12870 a^8 b^2 x^2 \left (8 A \left (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\right )+3 B x \left (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\right )\right )+11440 a^7 b^3 x^3 \left (9 A \left (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\right )+4 B x \left (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\right )\right )+40040 a^6 b^4 x^4 \left (2 A \left (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\right )+B x \left (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\right )\right )+4368 a^5 b^5 x^5 \left (11 A \left (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\right )+6 B x \left (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\right )\right )+1820 a^4 b^6 x^6 \left (12 A \left (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\right )+7 B x \left (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\right )\right )+560 a^3 b^7 x^7 \left (13 A \left (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\right )+8 B x \left (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\right )\right )+120 a^2 b^8 x^8 \left (14 A \left (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\right )+9 B x \left (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\right )\right )+80 a b^9 x^9 \left (3 A \left (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\right )+2 B x \left (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\right )\right )+b^{10} x^{10} \left (16 A \left (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\right )+11 B x \left (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\right )\right )\right )}{240240} \]

[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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1336\) vs. \(2(192)=384\).

Time = 2.08 (sec) , antiderivative size = 1337, normalized size of antiderivative = 6.55

method	result	size
default	\(\text {Expression too large to display}\)	\(1337\)
norman	\(\text {Expression too large to display}\)	\(1403\)
gosper	\(\text {Expression too large to display}\)	\(1657\)
risch	\(\text {Expression too large to display}\)	\(1657\)
parallelrisch	\(\text {Expression too large to display}\)	\(1657\)

[In]

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

[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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1352 vs. \(2 (192) = 384\).

Time = 0.23 (sec) , antiderivative size = 1352, normalized size of antiderivative = 6.63 \[ \int (a+b x)^{10} (A+B x) (d+e x)^4 \, dx=\text {Too large to display} \]

[In]

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

[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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1676 vs. \(2 (204) = 408\).

Time = 0.11 (sec) , antiderivative size = 1676, normalized size of antiderivative = 8.22 \[ \int (a+b x)^{10} (A+B x) (d+e x)^4 \, dx=\text {Too large to display} \]

[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)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1352 vs. \(2 (192) = 384\).

Time = 0.20 (sec) , antiderivative size = 1352, normalized size of antiderivative = 6.63 \[ \int (a+b x)^{10} (A+B x) (d+e x)^4 \, dx=\text {Too large to display} \]

[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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1656 vs. \(2 (192) = 384\).

Time = 0.31 (sec) , antiderivative size = 1656, normalized size of antiderivative = 8.12 \[ \int (a+b x)^{10} (A+B x) (d+e x)^4 \, dx=\text {Too large to display} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.77 (sec) , antiderivative size = 1386, normalized size of antiderivative = 6.79 \[ \int (a+b x)^{10} (A+B x) (d+e x)^4 \, dx=\text {Too large to display} \]

[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