3.1275 \(\int (a+b x)^7 (c+d x)^7 \, dx\)

Optimal. Leaf size=200 \[ \frac{d^6 (a+b x)^{14} (b c-a d)}{2 b^8}+\frac{21 d^5 (a+b x)^{13} (b c-a d)^2}{13 b^8}+\frac{35 d^4 (a+b x)^{12} (b c-a d)^3}{12 b^8}+\frac{35 d^3 (a+b x)^{11} (b c-a d)^4}{11 b^8}+\frac{21 d^2 (a+b x)^{10} (b c-a d)^5}{10 b^8}+\frac{7 d (a+b x)^9 (b c-a d)^6}{9 b^8}+\frac{(a+b x)^8 (b c-a d)^7}{8 b^8}+\frac{d^7 (a+b x)^{15}}{15 b^8} \]

[Out]

((b*c - a*d)^7*(a + b*x)^8)/(8*b^8) + (7*d*(b*c - a*d)^6*(a + b*x)^9)/(9*b^8) + (21*d^2*(b*c - a*d)^5*(a + b*x
)^10)/(10*b^8) + (35*d^3*(b*c - a*d)^4*(a + b*x)^11)/(11*b^8) + (35*d^4*(b*c - a*d)^3*(a + b*x)^12)/(12*b^8) +
(21*d^5*(b*c - a*d)^2*(a + b*x)^13)/(13*b^8) + (d^6*(b*c - a*d)*(a + b*x)^14)/(2*b^8) + (d^7*(a + b*x)^15)/(1
5*b^8)

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Rubi [A]  time = 0.453697, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {43} \[ \frac{d^6 (a+b x)^{14} (b c-a d)}{2 b^8}+\frac{21 d^5 (a+b x)^{13} (b c-a d)^2}{13 b^8}+\frac{35 d^4 (a+b x)^{12} (b c-a d)^3}{12 b^8}+\frac{35 d^3 (a+b x)^{11} (b c-a d)^4}{11 b^8}+\frac{21 d^2 (a+b x)^{10} (b c-a d)^5}{10 b^8}+\frac{7 d (a+b x)^9 (b c-a d)^6}{9 b^8}+\frac{(a+b x)^8 (b c-a d)^7}{8 b^8}+\frac{d^7 (a+b x)^{15}}{15 b^8} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^7*(c + d*x)^7,x]

[Out]

((b*c - a*d)^7*(a + b*x)^8)/(8*b^8) + (7*d*(b*c - a*d)^6*(a + b*x)^9)/(9*b^8) + (21*d^2*(b*c - a*d)^5*(a + b*x
)^10)/(10*b^8) + (35*d^3*(b*c - a*d)^4*(a + b*x)^11)/(11*b^8) + (35*d^4*(b*c - a*d)^3*(a + b*x)^12)/(12*b^8) +
(21*d^5*(b*c - a*d)^2*(a + b*x)^13)/(13*b^8) + (d^6*(b*c - a*d)*(a + b*x)^14)/(2*b^8) + (d^7*(a + b*x)^15)/(1
5*b^8)

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int (a+b x)^7 (c+d x)^7 \, dx &=\int \left (\frac{(b c-a d)^7 (a+b x)^7}{b^7}+\frac{7 d (b c-a d)^6 (a+b x)^8}{b^7}+\frac{21 d^2 (b c-a d)^5 (a+b x)^9}{b^7}+\frac{35 d^3 (b c-a d)^4 (a+b x)^{10}}{b^7}+\frac{35 d^4 (b c-a d)^3 (a+b x)^{11}}{b^7}+\frac{21 d^5 (b c-a d)^2 (a+b x)^{12}}{b^7}+\frac{7 d^6 (b c-a d) (a+b x)^{13}}{b^7}+\frac{d^7 (a+b x)^{14}}{b^7}\right ) \, dx\\ &=\frac{(b c-a d)^7 (a+b x)^8}{8 b^8}+\frac{7 d (b c-a d)^6 (a+b x)^9}{9 b^8}+\frac{21 d^2 (b c-a d)^5 (a+b x)^{10}}{10 b^8}+\frac{35 d^3 (b c-a d)^4 (a+b x)^{11}}{11 b^8}+\frac{35 d^4 (b c-a d)^3 (a+b x)^{12}}{12 b^8}+\frac{21 d^5 (b c-a d)^2 (a+b x)^{13}}{13 b^8}+\frac{d^6 (b c-a d) (a+b x)^{14}}{2 b^8}+\frac{d^7 (a+b x)^{15}}{15 b^8}\\ \end{align*}

Mathematica [B]  time = 0.0799013, size = 785, normalized size = 3.92 \[ \frac{7}{13} b^5 d^5 x^{13} \left (3 a^2 d^2+7 a b c d+3 b^2 c^2\right )+\frac{7}{12} b^4 d^4 x^{12} \left (21 a^2 b c d^2+5 a^3 d^3+21 a b^2 c^2 d+5 b^3 c^3\right )+\frac{7}{11} b^3 d^3 x^{11} \left (63 a^2 b^2 c^2 d^2+35 a^3 b c d^3+5 a^4 d^4+35 a b^3 c^3 d+5 b^4 c^4\right )+\frac{7}{10} b^2 d^2 x^{10} \left (105 a^2 b^3 c^3 d^2+105 a^3 b^2 c^2 d^3+35 a^4 b c d^4+3 a^5 d^5+35 a b^4 c^4 d+3 b^5 c^5\right )+\frac{7}{9} b d x^9 \left (105 a^2 b^4 c^4 d^2+175 a^3 b^3 c^3 d^3+105 a^4 b^2 c^2 d^4+21 a^5 b c d^5+a^6 d^6+21 a b^5 c^5 d+b^6 c^6\right )+\frac{1}{8} x^8 \left (441 a^2 b^5 c^5 d^2+1225 a^3 b^4 c^4 d^3+1225 a^4 b^3 c^3 d^4+441 a^5 b^2 c^2 d^5+49 a^6 b c d^6+a^7 d^7+49 a b^6 c^6 d+b^7 c^7\right )+a c x^7 \left (105 a^2 b^4 c^4 d^2+175 a^3 b^3 c^3 d^3+105 a^4 b^2 c^2 d^4+21 a^5 b c d^5+a^6 d^6+21 a b^5 c^5 d+b^6 c^6\right )+\frac{7}{6} a^2 c^2 x^6 \left (105 a^2 b^3 c^3 d^2+105 a^3 b^2 c^2 d^3+35 a^4 b c d^4+3 a^5 d^5+35 a b^4 c^4 d+3 b^5 c^5\right )+\frac{7}{5} a^3 c^3 x^5 \left (63 a^2 b^2 c^2 d^2+35 a^3 b c d^3+5 a^4 d^4+35 a b^3 c^3 d+5 b^4 c^4\right )+\frac{7}{4} a^4 c^4 x^4 \left (21 a^2 b c d^2+5 a^3 d^3+21 a b^2 c^2 d+5 b^3 c^3\right )+\frac{7}{3} a^5 c^5 x^3 \left (3 a^2 d^2+7 a b c d+3 b^2 c^2\right )+\frac{7}{2} a^6 c^6 x^2 (a d+b c)+a^7 c^7 x+\frac{1}{2} b^6 d^6 x^{14} (a d+b c)+\frac{1}{15} b^7 d^7 x^{15} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^7*(c + d*x)^7,x]

[Out]

a^7*c^7*x + (7*a^6*c^6*(b*c + a*d)*x^2)/2 + (7*a^5*c^5*(3*b^2*c^2 + 7*a*b*c*d + 3*a^2*d^2)*x^3)/3 + (7*a^4*c^4
*(5*b^3*c^3 + 21*a*b^2*c^2*d + 21*a^2*b*c*d^2 + 5*a^3*d^3)*x^4)/4 + (7*a^3*c^3*(5*b^4*c^4 + 35*a*b^3*c^3*d + 6
3*a^2*b^2*c^2*d^2 + 35*a^3*b*c*d^3 + 5*a^4*d^4)*x^5)/5 + (7*a^2*c^2*(3*b^5*c^5 + 35*a*b^4*c^4*d + 105*a^2*b^3*
c^3*d^2 + 105*a^3*b^2*c^2*d^3 + 35*a^4*b*c*d^4 + 3*a^5*d^5)*x^6)/6 + a*c*(b^6*c^6 + 21*a*b^5*c^5*d + 105*a^2*b
^4*c^4*d^2 + 175*a^3*b^3*c^3*d^3 + 105*a^4*b^2*c^2*d^4 + 21*a^5*b*c*d^5 + a^6*d^6)*x^7 + ((b^7*c^7 + 49*a*b^6*
c^6*d + 441*a^2*b^5*c^5*d^2 + 1225*a^3*b^4*c^4*d^3 + 1225*a^4*b^3*c^3*d^4 + 441*a^5*b^2*c^2*d^5 + 49*a^6*b*c*d
^6 + a^7*d^7)*x^8)/8 + (7*b*d*(b^6*c^6 + 21*a*b^5*c^5*d + 105*a^2*b^4*c^4*d^2 + 175*a^3*b^3*c^3*d^3 + 105*a^4*
b^2*c^2*d^4 + 21*a^5*b*c*d^5 + a^6*d^6)*x^9)/9 + (7*b^2*d^2*(3*b^5*c^5 + 35*a*b^4*c^4*d + 105*a^2*b^3*c^3*d^2
+ 105*a^3*b^2*c^2*d^3 + 35*a^4*b*c*d^4 + 3*a^5*d^5)*x^10)/10 + (7*b^3*d^3*(5*b^4*c^4 + 35*a*b^3*c^3*d + 63*a^2
*b^2*c^2*d^2 + 35*a^3*b*c*d^3 + 5*a^4*d^4)*x^11)/11 + (7*b^4*d^4*(5*b^3*c^3 + 21*a*b^2*c^2*d + 21*a^2*b*c*d^2
+ 5*a^3*d^3)*x^12)/12 + (7*b^5*d^5*(3*b^2*c^2 + 7*a*b*c*d + 3*a^2*d^2)*x^13)/13 + (b^6*d^6*(b*c + a*d)*x^14)/2
+ (b^7*d^7*x^15)/15

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Maple [B]  time = 0.002, size = 817, normalized size = 4.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^7*(d*x+c)^7,x)

[Out]

1/15*b^7*d^7*x^15+1/14*(7*a*b^6*d^7+7*b^7*c*d^6)*x^14+1/13*(21*a^2*b^5*d^7+49*a*b^6*c*d^6+21*b^7*c^2*d^5)*x^13
+1/12*(35*a^3*b^4*d^7+147*a^2*b^5*c*d^6+147*a*b^6*c^2*d^5+35*b^7*c^3*d^4)*x^12+1/11*(35*a^4*b^3*d^7+245*a^3*b^
4*c*d^6+441*a^2*b^5*c^2*d^5+245*a*b^6*c^3*d^4+35*b^7*c^4*d^3)*x^11+1/10*(21*a^5*b^2*d^7+245*a^4*b^3*c*d^6+735*
a^3*b^4*c^2*d^5+735*a^2*b^5*c^3*d^4+245*a*b^6*c^4*d^3+21*b^7*c^5*d^2)*x^10+1/9*(7*a^6*b*d^7+147*a^5*b^2*c*d^6+
735*a^4*b^3*c^2*d^5+1225*a^3*b^4*c^3*d^4+735*a^2*b^5*c^4*d^3+147*a*b^6*c^5*d^2+7*b^7*c^6*d)*x^9+1/8*(a^7*d^7+4
9*a^6*b*c*d^6+441*a^5*b^2*c^2*d^5+1225*a^4*b^3*c^3*d^4+1225*a^3*b^4*c^4*d^3+441*a^2*b^5*c^5*d^2+49*a*b^6*c^6*d
+b^7*c^7)*x^8+1/7*(7*a^7*c*d^6+147*a^6*b*c^2*d^5+735*a^5*b^2*c^3*d^4+1225*a^4*b^3*c^4*d^3+735*a^3*b^4*c^5*d^2+
147*a^2*b^5*c^6*d+7*a*b^6*c^7)*x^7+1/6*(21*a^7*c^2*d^5+245*a^6*b*c^3*d^4+735*a^5*b^2*c^4*d^3+735*a^4*b^3*c^5*d
^2+245*a^3*b^4*c^6*d+21*a^2*b^5*c^7)*x^6+1/5*(35*a^7*c^3*d^4+245*a^6*b*c^4*d^3+441*a^5*b^2*c^5*d^2+245*a^4*b^3
*c^6*d+35*a^3*b^4*c^7)*x^5+1/4*(35*a^7*c^4*d^3+147*a^6*b*c^5*d^2+147*a^5*b^2*c^6*d+35*a^4*b^3*c^7)*x^4+1/3*(21
*a^7*c^5*d^2+49*a^6*b*c^6*d+21*a^5*b^2*c^7)*x^3+1/2*(7*a^7*c^6*d+7*a^6*b*c^7)*x^2+a^7*c^7*x

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Maxima [B]  time = 0.994162, size = 1089, normalized size = 5.44 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^7*(d*x+c)^7,x, algorithm="maxima")

[Out]

1/15*b^7*d^7*x^15 + a^7*c^7*x + 1/2*(b^7*c*d^6 + a*b^6*d^7)*x^14 + 7/13*(3*b^7*c^2*d^5 + 7*a*b^6*c*d^6 + 3*a^2
*b^5*d^7)*x^13 + 7/12*(5*b^7*c^3*d^4 + 21*a*b^6*c^2*d^5 + 21*a^2*b^5*c*d^6 + 5*a^3*b^4*d^7)*x^12 + 7/11*(5*b^7
*c^4*d^3 + 35*a*b^6*c^3*d^4 + 63*a^2*b^5*c^2*d^5 + 35*a^3*b^4*c*d^6 + 5*a^4*b^3*d^7)*x^11 + 7/10*(3*b^7*c^5*d^
2 + 35*a*b^6*c^4*d^3 + 105*a^2*b^5*c^3*d^4 + 105*a^3*b^4*c^2*d^5 + 35*a^4*b^3*c*d^6 + 3*a^5*b^2*d^7)*x^10 + 7/
9*(b^7*c^6*d + 21*a*b^6*c^5*d^2 + 105*a^2*b^5*c^4*d^3 + 175*a^3*b^4*c^3*d^4 + 105*a^4*b^3*c^2*d^5 + 21*a^5*b^2
*c*d^6 + a^6*b*d^7)*x^9 + 1/8*(b^7*c^7 + 49*a*b^6*c^6*d + 441*a^2*b^5*c^5*d^2 + 1225*a^3*b^4*c^4*d^3 + 1225*a^
4*b^3*c^3*d^4 + 441*a^5*b^2*c^2*d^5 + 49*a^6*b*c*d^6 + a^7*d^7)*x^8 + (a*b^6*c^7 + 21*a^2*b^5*c^6*d + 105*a^3*
b^4*c^5*d^2 + 175*a^4*b^3*c^4*d^3 + 105*a^5*b^2*c^3*d^4 + 21*a^6*b*c^2*d^5 + a^7*c*d^6)*x^7 + 7/6*(3*a^2*b^5*c
^7 + 35*a^3*b^4*c^6*d + 105*a^4*b^3*c^5*d^2 + 105*a^5*b^2*c^4*d^3 + 35*a^6*b*c^3*d^4 + 3*a^7*c^2*d^5)*x^6 + 7/
5*(5*a^3*b^4*c^7 + 35*a^4*b^3*c^6*d + 63*a^5*b^2*c^5*d^2 + 35*a^6*b*c^4*d^3 + 5*a^7*c^3*d^4)*x^5 + 7/4*(5*a^4*
b^3*c^7 + 21*a^5*b^2*c^6*d + 21*a^6*b*c^5*d^2 + 5*a^7*c^4*d^3)*x^4 + 7/3*(3*a^5*b^2*c^7 + 7*a^6*b*c^6*d + 3*a^
7*c^5*d^2)*x^3 + 7/2*(a^6*b*c^7 + a^7*c^6*d)*x^2

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Fricas [B]  time = 1.95135, size = 2079, normalized size = 10.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^7*(d*x+c)^7,x, algorithm="fricas")

[Out]

1/15*x^15*d^7*b^7 + 1/2*x^14*d^6*c*b^7 + 1/2*x^14*d^7*b^6*a + 21/13*x^13*d^5*c^2*b^7 + 49/13*x^13*d^6*c*b^6*a
+ 21/13*x^13*d^7*b^5*a^2 + 35/12*x^12*d^4*c^3*b^7 + 49/4*x^12*d^5*c^2*b^6*a + 49/4*x^12*d^6*c*b^5*a^2 + 35/12*
x^12*d^7*b^4*a^3 + 35/11*x^11*d^3*c^4*b^7 + 245/11*x^11*d^4*c^3*b^6*a + 441/11*x^11*d^5*c^2*b^5*a^2 + 245/11*x
^11*d^6*c*b^4*a^3 + 35/11*x^11*d^7*b^3*a^4 + 21/10*x^10*d^2*c^5*b^7 + 49/2*x^10*d^3*c^4*b^6*a + 147/2*x^10*d^4
*c^3*b^5*a^2 + 147/2*x^10*d^5*c^2*b^4*a^3 + 49/2*x^10*d^6*c*b^3*a^4 + 21/10*x^10*d^7*b^2*a^5 + 7/9*x^9*d*c^6*b
^7 + 49/3*x^9*d^2*c^5*b^6*a + 245/3*x^9*d^3*c^4*b^5*a^2 + 1225/9*x^9*d^4*c^3*b^4*a^3 + 245/3*x^9*d^5*c^2*b^3*a
^4 + 49/3*x^9*d^6*c*b^2*a^5 + 7/9*x^9*d^7*b*a^6 + 1/8*x^8*c^7*b^7 + 49/8*x^8*d*c^6*b^6*a + 441/8*x^8*d^2*c^5*b
^5*a^2 + 1225/8*x^8*d^3*c^4*b^4*a^3 + 1225/8*x^8*d^4*c^3*b^3*a^4 + 441/8*x^8*d^5*c^2*b^2*a^5 + 49/8*x^8*d^6*c*
b*a^6 + 1/8*x^8*d^7*a^7 + x^7*c^7*b^6*a + 21*x^7*d*c^6*b^5*a^2 + 105*x^7*d^2*c^5*b^4*a^3 + 175*x^7*d^3*c^4*b^3
*a^4 + 105*x^7*d^4*c^3*b^2*a^5 + 21*x^7*d^5*c^2*b*a^6 + x^7*d^6*c*a^7 + 7/2*x^6*c^7*b^5*a^2 + 245/6*x^6*d*c^6*
b^4*a^3 + 245/2*x^6*d^2*c^5*b^3*a^4 + 245/2*x^6*d^3*c^4*b^2*a^5 + 245/6*x^6*d^4*c^3*b*a^6 + 7/2*x^6*d^5*c^2*a^
7 + 7*x^5*c^7*b^4*a^3 + 49*x^5*d*c^6*b^3*a^4 + 441/5*x^5*d^2*c^5*b^2*a^5 + 49*x^5*d^3*c^4*b*a^6 + 7*x^5*d^4*c^
3*a^7 + 35/4*x^4*c^7*b^3*a^4 + 147/4*x^4*d*c^6*b^2*a^5 + 147/4*x^4*d^2*c^5*b*a^6 + 35/4*x^4*d^3*c^4*a^7 + 7*x^
3*c^7*b^2*a^5 + 49/3*x^3*d*c^6*b*a^6 + 7*x^3*d^2*c^5*a^7 + 7/2*x^2*c^7*b*a^6 + 7/2*x^2*d*c^6*a^7 + x*c^7*a^7

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Sympy [B]  time = 0.195297, size = 935, normalized size = 4.68 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**7*(d*x+c)**7,x)

[Out]

a**7*c**7*x + b**7*d**7*x**15/15 + x**14*(a*b**6*d**7/2 + b**7*c*d**6/2) + x**13*(21*a**2*b**5*d**7/13 + 49*a*
b**6*c*d**6/13 + 21*b**7*c**2*d**5/13) + x**12*(35*a**3*b**4*d**7/12 + 49*a**2*b**5*c*d**6/4 + 49*a*b**6*c**2*
d**5/4 + 35*b**7*c**3*d**4/12) + x**11*(35*a**4*b**3*d**7/11 + 245*a**3*b**4*c*d**6/11 + 441*a**2*b**5*c**2*d*
*5/11 + 245*a*b**6*c**3*d**4/11 + 35*b**7*c**4*d**3/11) + x**10*(21*a**5*b**2*d**7/10 + 49*a**4*b**3*c*d**6/2
+ 147*a**3*b**4*c**2*d**5/2 + 147*a**2*b**5*c**3*d**4/2 + 49*a*b**6*c**4*d**3/2 + 21*b**7*c**5*d**2/10) + x**9
*(7*a**6*b*d**7/9 + 49*a**5*b**2*c*d**6/3 + 245*a**4*b**3*c**2*d**5/3 + 1225*a**3*b**4*c**3*d**4/9 + 245*a**2*
b**5*c**4*d**3/3 + 49*a*b**6*c**5*d**2/3 + 7*b**7*c**6*d/9) + x**8*(a**7*d**7/8 + 49*a**6*b*c*d**6/8 + 441*a**
5*b**2*c**2*d**5/8 + 1225*a**4*b**3*c**3*d**4/8 + 1225*a**3*b**4*c**4*d**3/8 + 441*a**2*b**5*c**5*d**2/8 + 49*
a*b**6*c**6*d/8 + b**7*c**7/8) + x**7*(a**7*c*d**6 + 21*a**6*b*c**2*d**5 + 105*a**5*b**2*c**3*d**4 + 175*a**4*
b**3*c**4*d**3 + 105*a**3*b**4*c**5*d**2 + 21*a**2*b**5*c**6*d + a*b**6*c**7) + x**6*(7*a**7*c**2*d**5/2 + 245
*a**6*b*c**3*d**4/6 + 245*a**5*b**2*c**4*d**3/2 + 245*a**4*b**3*c**5*d**2/2 + 245*a**3*b**4*c**6*d/6 + 7*a**2*
b**5*c**7/2) + x**5*(7*a**7*c**3*d**4 + 49*a**6*b*c**4*d**3 + 441*a**5*b**2*c**5*d**2/5 + 49*a**4*b**3*c**6*d
+ 7*a**3*b**4*c**7) + x**4*(35*a**7*c**4*d**3/4 + 147*a**6*b*c**5*d**2/4 + 147*a**5*b**2*c**6*d/4 + 35*a**4*b*
*3*c**7/4) + x**3*(7*a**7*c**5*d**2 + 49*a**6*b*c**6*d/3 + 7*a**5*b**2*c**7) + x**2*(7*a**7*c**6*d/2 + 7*a**6*
b*c**7/2)

________________________________________________________________________________________

Giac [B]  time = 1.05949, size = 1247, normalized size = 6.24 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^7*(d*x+c)^7,x, algorithm="giac")

[Out]

1/15*b^7*d^7*x^15 + 1/2*b^7*c*d^6*x^14 + 1/2*a*b^6*d^7*x^14 + 21/13*b^7*c^2*d^5*x^13 + 49/13*a*b^6*c*d^6*x^13
+ 21/13*a^2*b^5*d^7*x^13 + 35/12*b^7*c^3*d^4*x^12 + 49/4*a*b^6*c^2*d^5*x^12 + 49/4*a^2*b^5*c*d^6*x^12 + 35/12*
a^3*b^4*d^7*x^12 + 35/11*b^7*c^4*d^3*x^11 + 245/11*a*b^6*c^3*d^4*x^11 + 441/11*a^2*b^5*c^2*d^5*x^11 + 245/11*a
^3*b^4*c*d^6*x^11 + 35/11*a^4*b^3*d^7*x^11 + 21/10*b^7*c^5*d^2*x^10 + 49/2*a*b^6*c^4*d^3*x^10 + 147/2*a^2*b^5*
c^3*d^4*x^10 + 147/2*a^3*b^4*c^2*d^5*x^10 + 49/2*a^4*b^3*c*d^6*x^10 + 21/10*a^5*b^2*d^7*x^10 + 7/9*b^7*c^6*d*x
^9 + 49/3*a*b^6*c^5*d^2*x^9 + 245/3*a^2*b^5*c^4*d^3*x^9 + 1225/9*a^3*b^4*c^3*d^4*x^9 + 245/3*a^4*b^3*c^2*d^5*x
^9 + 49/3*a^5*b^2*c*d^6*x^9 + 7/9*a^6*b*d^7*x^9 + 1/8*b^7*c^7*x^8 + 49/8*a*b^6*c^6*d*x^8 + 441/8*a^2*b^5*c^5*d
^2*x^8 + 1225/8*a^3*b^4*c^4*d^3*x^8 + 1225/8*a^4*b^3*c^3*d^4*x^8 + 441/8*a^5*b^2*c^2*d^5*x^8 + 49/8*a^6*b*c*d^
6*x^8 + 1/8*a^7*d^7*x^8 + a*b^6*c^7*x^7 + 21*a^2*b^5*c^6*d*x^7 + 105*a^3*b^4*c^5*d^2*x^7 + 175*a^4*b^3*c^4*d^3
*x^7 + 105*a^5*b^2*c^3*d^4*x^7 + 21*a^6*b*c^2*d^5*x^7 + a^7*c*d^6*x^7 + 7/2*a^2*b^5*c^7*x^6 + 245/6*a^3*b^4*c^
6*d*x^6 + 245/2*a^4*b^3*c^5*d^2*x^6 + 245/2*a^5*b^2*c^4*d^3*x^6 + 245/6*a^6*b*c^3*d^4*x^6 + 7/2*a^7*c^2*d^5*x^
6 + 7*a^3*b^4*c^7*x^5 + 49*a^4*b^3*c^6*d*x^5 + 441/5*a^5*b^2*c^5*d^2*x^5 + 49*a^6*b*c^4*d^3*x^5 + 7*a^7*c^3*d^
4*x^5 + 35/4*a^4*b^3*c^7*x^4 + 147/4*a^5*b^2*c^6*d*x^4 + 147/4*a^6*b*c^5*d^2*x^4 + 35/4*a^7*c^4*d^3*x^4 + 7*a^
5*b^2*c^7*x^3 + 49/3*a^6*b*c^6*d*x^3 + 7*a^7*c^5*d^2*x^3 + 7/2*a^6*b*c^7*x^2 + 7/2*a^7*c^6*d*x^2 + a^7*c^7*x