3.192 \(\int (b+2 c x+3 d x^2) (a+b x+c x^2+d x^3)^7 \, dx\)
Optimal. Leaf size=21 \[ \frac{1}{8} \left (a+b x+c x^2+d x^3\right )^8 \]
[Out]
(a + b*x + c*x^2 + d*x^3)^8/8
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Rubi [A] time = 0.125167, antiderivative size = 21, normalized size of antiderivative = 1.,
number of steps used = 1, number of rules used = 1, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.033, Rules used =
{1588} \[ \frac{1}{8} \left (a+b x+c x^2+d x^3\right )^8 \]
Antiderivative was successfully verified.
[In]
Int[(b + 2*c*x + 3*d*x^2)*(a + b*x + c*x^2 + d*x^3)^7,x]
[Out]
(a + b*x + c*x^2 + d*x^3)^8/8
Rule 1588
Int[(Pp_)*(Qq_)^(m_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*x^(p - q
+ 1)*Qq^(m + 1))/((p + m*q + 1)*Coeff[Qq, x, q]), x] /; NeQ[p + m*q + 1, 0] && EqQ[(p + m*q + 1)*Coeff[Qq, x,
q]*Pp, Coeff[Pp, x, p]*x^(p - q)*((p - q + 1)*Qq + (m + 1)*x*D[Qq, x])]] /; FreeQ[m, x] && PolyQ[Pp, x] && Pol
yQ[Qq, x] && NeQ[m, -1]
Rubi steps
\begin{align*} \int \left (b+2 c x+3 d x^2\right ) \left (a+b x+c x^2+d x^3\right )^7 \, dx &=\frac{1}{8} \left (a+b x+c x^2+d x^3\right )^8\\ \end{align*}
Mathematica [B] time = 0.146059, size = 143, normalized size = 6.81 \[ \frac{1}{8} x (b+x (c+d x)) \left (56 a^5 x^2 (b+x (c+d x))^2+70 a^4 x^3 (b+x (c+d x))^3+56 a^3 x^4 (b+x (c+d x))^4+28 a^2 x^5 (b+x (c+d x))^5+28 a^6 x (b+x (c+d x))+8 a^7+8 a x^6 (b+x (c+d x))^6+x^7 (b+x (c+d x))^7\right ) \]
Antiderivative was successfully verified.
[In]
Integrate[(b + 2*c*x + 3*d*x^2)*(a + b*x + c*x^2 + d*x^3)^7,x]
[Out]
(x*(b + x*(c + d*x))*(8*a^7 + 28*a^6*x*(b + x*(c + d*x)) + 56*a^5*x^2*(b + x*(c + d*x))^2 + 70*a^4*x^3*(b + x*
(c + d*x))^3 + 56*a^3*x^4*(b + x*(c + d*x))^4 + 28*a^2*x^5*(b + x*(c + d*x))^5 + 8*a*x^6*(b + x*(c + d*x))^6 +
x^7*(b + x*(c + d*x))^7))/8
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Maple [B] time = 0.005, size = 25686, normalized size = 1223.1 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((3*d*x^2+2*c*x+b)*(d*x^3+c*x^2+b*x+a)^7,x)
[Out]
result too large to display
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Maxima [A] time = 1.00193, size = 26, normalized size = 1.24 \begin{align*} \frac{1}{8} \,{\left (d x^{3} + c x^{2} + b x + a\right )}^{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((3*d*x^2+2*c*x+b)*(d*x^3+c*x^2+b*x+a)^7,x, algorithm="maxima")
[Out]
1/8*(d*x^3 + c*x^2 + b*x + a)^8
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Fricas [B] time = 1.15867, size = 4385, normalized size = 208.81 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((3*d*x^2+2*c*x+b)*(d*x^3+c*x^2+b*x+a)^7,x, algorithm="fricas")
[Out]
1/8*x^24*d^8 + x^23*d^7*c + 7/2*x^22*d^6*c^2 + x^22*d^7*b + 7*x^21*d^5*c^3 + 7*x^21*d^6*c*b + x^21*d^7*a + 35/
4*x^20*d^4*c^4 + 21*x^20*d^5*c^2*b + 7/2*x^20*d^6*b^2 + 7*x^20*d^6*c*a + 7*x^19*d^3*c^5 + 35*x^19*d^4*c^3*b +
21*x^19*d^5*c*b^2 + 21*x^19*d^5*c^2*a + 7*x^19*d^6*b*a + 7/2*x^18*d^2*c^6 + 35*x^18*d^3*c^4*b + 105/2*x^18*d^4
*c^2*b^2 + 7*x^18*d^5*b^3 + 35*x^18*d^4*c^3*a + 42*x^18*d^5*c*b*a + 7/2*x^18*d^6*a^2 + x^17*d*c^7 + 21*x^17*d^
2*c^5*b + 70*x^17*d^3*c^3*b^2 + 35*x^17*d^4*c*b^3 + 35*x^17*d^3*c^4*a + 105*x^17*d^4*c^2*b*a + 21*x^17*d^5*b^2
*a + 21*x^17*d^5*c*a^2 + 1/8*x^16*c^8 + 7*x^16*d*c^6*b + 105/2*x^16*d^2*c^4*b^2 + 70*x^16*d^3*c^2*b^3 + 35/4*x
^16*d^4*b^4 + 21*x^16*d^2*c^5*a + 140*x^16*d^3*c^3*b*a + 105*x^16*d^4*c*b^2*a + 105/2*x^16*d^4*c^2*a^2 + 21*x^
16*d^5*b*a^2 + x^15*c^7*b + 21*x^15*d*c^5*b^2 + 70*x^15*d^2*c^3*b^3 + 35*x^15*d^3*c*b^4 + 7*x^15*d*c^6*a + 105
*x^15*d^2*c^4*b*a + 210*x^15*d^3*c^2*b^2*a + 35*x^15*d^4*b^3*a + 70*x^15*d^3*c^3*a^2 + 105*x^15*d^4*c*b*a^2 +
7*x^15*d^5*a^3 + 7/2*x^14*c^6*b^2 + 35*x^14*d*c^4*b^3 + 105/2*x^14*d^2*c^2*b^4 + 7*x^14*d^3*b^5 + x^14*c^7*a +
42*x^14*d*c^5*b*a + 210*x^14*d^2*c^3*b^2*a + 140*x^14*d^3*c*b^3*a + 105/2*x^14*d^2*c^4*a^2 + 210*x^14*d^3*c^2
*b*a^2 + 105/2*x^14*d^4*b^2*a^2 + 35*x^14*d^4*c*a^3 + 7*x^13*c^5*b^3 + 35*x^13*d*c^3*b^4 + 21*x^13*d^2*c*b^5 +
7*x^13*c^6*b*a + 105*x^13*d*c^4*b^2*a + 210*x^13*d^2*c^2*b^3*a + 35*x^13*d^3*b^4*a + 21*x^13*d*c^5*a^2 + 210*
x^13*d^2*c^3*b*a^2 + 210*x^13*d^3*c*b^2*a^2 + 70*x^13*d^3*c^2*a^3 + 35*x^13*d^4*b*a^3 + 35/4*x^12*c^4*b^4 + 21
*x^12*d*c^2*b^5 + 7/2*x^12*d^2*b^6 + 21*x^12*c^5*b^2*a + 140*x^12*d*c^3*b^3*a + 105*x^12*d^2*c*b^4*a + 7/2*x^1
2*c^6*a^2 + 105*x^12*d*c^4*b*a^2 + 315*x^12*d^2*c^2*b^2*a^2 + 70*x^12*d^3*b^3*a^2 + 70*x^12*d^2*c^3*a^3 + 140*
x^12*d^3*c*b*a^3 + 35/4*x^12*d^4*a^4 + 7*x^11*c^3*b^5 + 7*x^11*d*c*b^6 + 35*x^11*c^4*b^3*a + 105*x^11*d*c^2*b^
4*a + 21*x^11*d^2*b^5*a + 21*x^11*c^5*b*a^2 + 210*x^11*d*c^3*b^2*a^2 + 210*x^11*d^2*c*b^3*a^2 + 35*x^11*d*c^4*
a^3 + 210*x^11*d^2*c^2*b*a^3 + 70*x^11*d^3*b^2*a^3 + 35*x^11*d^3*c*a^4 + 7/2*x^10*c^2*b^6 + x^10*d*b^7 + 35*x^
10*c^3*b^4*a + 42*x^10*d*c*b^5*a + 105/2*x^10*c^4*b^2*a^2 + 210*x^10*d*c^2*b^3*a^2 + 105/2*x^10*d^2*b^4*a^2 +
7*x^10*c^5*a^3 + 140*x^10*d*c^3*b*a^3 + 210*x^10*d^2*c*b^2*a^3 + 105/2*x^10*d^2*c^2*a^4 + 35*x^10*d^3*b*a^4 +
x^9*c*b^7 + 21*x^9*c^2*b^5*a + 7*x^9*d*b^6*a + 70*x^9*c^3*b^3*a^2 + 105*x^9*d*c*b^4*a^2 + 35*x^9*c^4*b*a^3 + 2
10*x^9*d*c^2*b^2*a^3 + 70*x^9*d^2*b^3*a^3 + 35*x^9*d*c^3*a^4 + 105*x^9*d^2*c*b*a^4 + 7*x^9*d^3*a^5 + 1/8*x^8*b
^8 + 7*x^8*c*b^6*a + 105/2*x^8*c^2*b^4*a^2 + 21*x^8*d*b^5*a^2 + 70*x^8*c^3*b^2*a^3 + 140*x^8*d*c*b^3*a^3 + 35/
4*x^8*c^4*a^4 + 105*x^8*d*c^2*b*a^4 + 105/2*x^8*d^2*b^2*a^4 + 21*x^8*d^2*c*a^5 + x^7*b^7*a + 21*x^7*c*b^5*a^2
+ 70*x^7*c^2*b^3*a^3 + 35*x^7*d*b^4*a^3 + 35*x^7*c^3*b*a^4 + 105*x^7*d*c*b^2*a^4 + 21*x^7*d*c^2*a^5 + 21*x^7*d
^2*b*a^5 + 7/2*x^6*b^6*a^2 + 35*x^6*c*b^4*a^3 + 105/2*x^6*c^2*b^2*a^4 + 35*x^6*d*b^3*a^4 + 7*x^6*c^3*a^5 + 42*
x^6*d*c*b*a^5 + 7/2*x^6*d^2*a^6 + 7*x^5*b^5*a^3 + 35*x^5*c*b^3*a^4 + 21*x^5*c^2*b*a^5 + 21*x^5*d*b^2*a^5 + 7*x
^5*d*c*a^6 + 35/4*x^4*b^4*a^4 + 21*x^4*c*b^2*a^5 + 7/2*x^4*c^2*a^6 + 7*x^4*d*b*a^6 + 7*x^3*b^3*a^5 + 7*x^3*c*b
*a^6 + x^3*d*a^7 + 7/2*x^2*b^2*a^6 + x^2*c*a^7 + x*b*a^7
________________________________________________________________________________________
Sympy [B] time = 0.344554, size = 1771, normalized size = 84.33 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((3*d*x**2+2*c*x+b)*(d*x**3+c*x**2+b*x+a)**7,x)
[Out]
a**7*b*x + c*d**7*x**23 + d**8*x**24/8 + x**22*(b*d**7 + 7*c**2*d**6/2) + x**21*(a*d**7 + 7*b*c*d**6 + 7*c**3*
d**5) + x**20*(7*a*c*d**6 + 7*b**2*d**6/2 + 21*b*c**2*d**5 + 35*c**4*d**4/4) + x**19*(7*a*b*d**6 + 21*a*c**2*d
**5 + 21*b**2*c*d**5 + 35*b*c**3*d**4 + 7*c**5*d**3) + x**18*(7*a**2*d**6/2 + 42*a*b*c*d**5 + 35*a*c**3*d**4 +
7*b**3*d**5 + 105*b**2*c**2*d**4/2 + 35*b*c**4*d**3 + 7*c**6*d**2/2) + x**17*(21*a**2*c*d**5 + 21*a*b**2*d**5
+ 105*a*b*c**2*d**4 + 35*a*c**4*d**3 + 35*b**3*c*d**4 + 70*b**2*c**3*d**3 + 21*b*c**5*d**2 + c**7*d) + x**16*
(21*a**2*b*d**5 + 105*a**2*c**2*d**4/2 + 105*a*b**2*c*d**4 + 140*a*b*c**3*d**3 + 21*a*c**5*d**2 + 35*b**4*d**4
/4 + 70*b**3*c**2*d**3 + 105*b**2*c**4*d**2/2 + 7*b*c**6*d + c**8/8) + x**15*(7*a**3*d**5 + 105*a**2*b*c*d**4
+ 70*a**2*c**3*d**3 + 35*a*b**3*d**4 + 210*a*b**2*c**2*d**3 + 105*a*b*c**4*d**2 + 7*a*c**6*d + 35*b**4*c*d**3
+ 70*b**3*c**3*d**2 + 21*b**2*c**5*d + b*c**7) + x**14*(35*a**3*c*d**4 + 105*a**2*b**2*d**4/2 + 210*a**2*b*c**
2*d**3 + 105*a**2*c**4*d**2/2 + 140*a*b**3*c*d**3 + 210*a*b**2*c**3*d**2 + 42*a*b*c**5*d + a*c**7 + 7*b**5*d**
3 + 105*b**4*c**2*d**2/2 + 35*b**3*c**4*d + 7*b**2*c**6/2) + x**13*(35*a**3*b*d**4 + 70*a**3*c**2*d**3 + 210*a
**2*b**2*c*d**3 + 210*a**2*b*c**3*d**2 + 21*a**2*c**5*d + 35*a*b**4*d**3 + 210*a*b**3*c**2*d**2 + 105*a*b**2*c
**4*d + 7*a*b*c**6 + 21*b**5*c*d**2 + 35*b**4*c**3*d + 7*b**3*c**5) + x**12*(35*a**4*d**4/4 + 140*a**3*b*c*d**
3 + 70*a**3*c**3*d**2 + 70*a**2*b**3*d**3 + 315*a**2*b**2*c**2*d**2 + 105*a**2*b*c**4*d + 7*a**2*c**6/2 + 105*
a*b**4*c*d**2 + 140*a*b**3*c**3*d + 21*a*b**2*c**5 + 7*b**6*d**2/2 + 21*b**5*c**2*d + 35*b**4*c**4/4) + x**11*
(35*a**4*c*d**3 + 70*a**3*b**2*d**3 + 210*a**3*b*c**2*d**2 + 35*a**3*c**4*d + 210*a**2*b**3*c*d**2 + 210*a**2*
b**2*c**3*d + 21*a**2*b*c**5 + 21*a*b**5*d**2 + 105*a*b**4*c**2*d + 35*a*b**3*c**4 + 7*b**6*c*d + 7*b**5*c**3)
+ x**10*(35*a**4*b*d**3 + 105*a**4*c**2*d**2/2 + 210*a**3*b**2*c*d**2 + 140*a**3*b*c**3*d + 7*a**3*c**5 + 105
*a**2*b**4*d**2/2 + 210*a**2*b**3*c**2*d + 105*a**2*b**2*c**4/2 + 42*a*b**5*c*d + 35*a*b**4*c**3 + b**7*d + 7*
b**6*c**2/2) + x**9*(7*a**5*d**3 + 105*a**4*b*c*d**2 + 35*a**4*c**3*d + 70*a**3*b**3*d**2 + 210*a**3*b**2*c**2
*d + 35*a**3*b*c**4 + 105*a**2*b**4*c*d + 70*a**2*b**3*c**3 + 7*a*b**6*d + 21*a*b**5*c**2 + b**7*c) + x**8*(21
*a**5*c*d**2 + 105*a**4*b**2*d**2/2 + 105*a**4*b*c**2*d + 35*a**4*c**4/4 + 140*a**3*b**3*c*d + 70*a**3*b**2*c*
*3 + 21*a**2*b**5*d + 105*a**2*b**4*c**2/2 + 7*a*b**6*c + b**8/8) + x**7*(21*a**5*b*d**2 + 21*a**5*c**2*d + 10
5*a**4*b**2*c*d + 35*a**4*b*c**3 + 35*a**3*b**4*d + 70*a**3*b**3*c**2 + 21*a**2*b**5*c + a*b**7) + x**6*(7*a**
6*d**2/2 + 42*a**5*b*c*d + 7*a**5*c**3 + 35*a**4*b**3*d + 105*a**4*b**2*c**2/2 + 35*a**3*b**4*c + 7*a**2*b**6/
2) + x**5*(7*a**6*c*d + 21*a**5*b**2*d + 21*a**5*b*c**2 + 35*a**4*b**3*c + 7*a**3*b**5) + x**4*(7*a**6*b*d + 7
*a**6*c**2/2 + 21*a**5*b**2*c + 35*a**4*b**4/4) + x**3*(a**7*d + 7*a**6*b*c + 7*a**5*b**3) + x**2*(a**7*c + 7*
a**6*b**2/2)
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Giac [B] time = 1.22099, size = 2641, normalized size = 125.76 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((3*d*x^2+2*c*x+b)*(d*x^3+c*x^2+b*x+a)^7,x, algorithm="giac")
[Out]
1/8*d^8*x^24 + c*d^7*x^23 + 7/2*c^2*d^6*x^22 + b*d^7*x^22 + 7*c^3*d^5*x^21 + 7*b*c*d^6*x^21 + a*d^7*x^21 + 35/
4*c^4*d^4*x^20 + 21*b*c^2*d^5*x^20 + 7/2*b^2*d^6*x^20 + 7*a*c*d^6*x^20 + 7*c^5*d^3*x^19 + 35*b*c^3*d^4*x^19 +
21*b^2*c*d^5*x^19 + 21*a*c^2*d^5*x^19 + 7*a*b*d^6*x^19 + 7/2*c^6*d^2*x^18 + 35*b*c^4*d^3*x^18 + 105/2*b^2*c^2*
d^4*x^18 + 35*a*c^3*d^4*x^18 + 7*b^3*d^5*x^18 + 42*a*b*c*d^5*x^18 + 7/2*a^2*d^6*x^18 + c^7*d*x^17 + 21*b*c^5*d
^2*x^17 + 70*b^2*c^3*d^3*x^17 + 35*a*c^4*d^3*x^17 + 35*b^3*c*d^4*x^17 + 105*a*b*c^2*d^4*x^17 + 21*a*b^2*d^5*x^
17 + 21*a^2*c*d^5*x^17 + 1/8*c^8*x^16 + 7*b*c^6*d*x^16 + 105/2*b^2*c^4*d^2*x^16 + 21*a*c^5*d^2*x^16 + 70*b^3*c
^2*d^3*x^16 + 140*a*b*c^3*d^3*x^16 + 35/4*b^4*d^4*x^16 + 105*a*b^2*c*d^4*x^16 + 105/2*a^2*c^2*d^4*x^16 + 21*a^
2*b*d^5*x^16 + b*c^7*x^15 + 21*b^2*c^5*d*x^15 + 7*a*c^6*d*x^15 + 70*b^3*c^3*d^2*x^15 + 105*a*b*c^4*d^2*x^15 +
35*b^4*c*d^3*x^15 + 210*a*b^2*c^2*d^3*x^15 + 70*a^2*c^3*d^3*x^15 + 35*a*b^3*d^4*x^15 + 105*a^2*b*c*d^4*x^15 +
7*a^3*d^5*x^15 + 7/2*b^2*c^6*x^14 + a*c^7*x^14 + 35*b^3*c^4*d*x^14 + 42*a*b*c^5*d*x^14 + 105/2*b^4*c^2*d^2*x^1
4 + 210*a*b^2*c^3*d^2*x^14 + 105/2*a^2*c^4*d^2*x^14 + 7*b^5*d^3*x^14 + 140*a*b^3*c*d^3*x^14 + 210*a^2*b*c^2*d^
3*x^14 + 105/2*a^2*b^2*d^4*x^14 + 35*a^3*c*d^4*x^14 + 7*b^3*c^5*x^13 + 7*a*b*c^6*x^13 + 35*b^4*c^3*d*x^13 + 10
5*a*b^2*c^4*d*x^13 + 21*a^2*c^5*d*x^13 + 21*b^5*c*d^2*x^13 + 210*a*b^3*c^2*d^2*x^13 + 210*a^2*b*c^3*d^2*x^13 +
35*a*b^4*d^3*x^13 + 210*a^2*b^2*c*d^3*x^13 + 70*a^3*c^2*d^3*x^13 + 35*a^3*b*d^4*x^13 + 35/4*b^4*c^4*x^12 + 21
*a*b^2*c^5*x^12 + 7/2*a^2*c^6*x^12 + 21*b^5*c^2*d*x^12 + 140*a*b^3*c^3*d*x^12 + 105*a^2*b*c^4*d*x^12 + 7/2*b^6
*d^2*x^12 + 105*a*b^4*c*d^2*x^12 + 315*a^2*b^2*c^2*d^2*x^12 + 70*a^3*c^3*d^2*x^12 + 70*a^2*b^3*d^3*x^12 + 140*
a^3*b*c*d^3*x^12 + 35/4*a^4*d^4*x^12 + 7*b^5*c^3*x^11 + 35*a*b^3*c^4*x^11 + 21*a^2*b*c^5*x^11 + 7*b^6*c*d*x^11
+ 105*a*b^4*c^2*d*x^11 + 210*a^2*b^2*c^3*d*x^11 + 35*a^3*c^4*d*x^11 + 21*a*b^5*d^2*x^11 + 210*a^2*b^3*c*d^2*x
^11 + 210*a^3*b*c^2*d^2*x^11 + 70*a^3*b^2*d^3*x^11 + 35*a^4*c*d^3*x^11 + 7/2*b^6*c^2*x^10 + 35*a*b^4*c^3*x^10
+ 105/2*a^2*b^2*c^4*x^10 + 7*a^3*c^5*x^10 + b^7*d*x^10 + 42*a*b^5*c*d*x^10 + 210*a^2*b^3*c^2*d*x^10 + 140*a^3*
b*c^3*d*x^10 + 105/2*a^2*b^4*d^2*x^10 + 210*a^3*b^2*c*d^2*x^10 + 105/2*a^4*c^2*d^2*x^10 + 35*a^4*b*d^3*x^10 +
b^7*c*x^9 + 21*a*b^5*c^2*x^9 + 70*a^2*b^3*c^3*x^9 + 35*a^3*b*c^4*x^9 + 7*a*b^6*d*x^9 + 105*a^2*b^4*c*d*x^9 + 2
10*a^3*b^2*c^2*d*x^9 + 35*a^4*c^3*d*x^9 + 70*a^3*b^3*d^2*x^9 + 105*a^4*b*c*d^2*x^9 + 7*a^5*d^3*x^9 + 1/8*b^8*x
^8 + 7*a*b^6*c*x^8 + 105/2*a^2*b^4*c^2*x^8 + 70*a^3*b^2*c^3*x^8 + 35/4*a^4*c^4*x^8 + 21*a^2*b^5*d*x^8 + 140*a^
3*b^3*c*d*x^8 + 105*a^4*b*c^2*d*x^8 + 105/2*a^4*b^2*d^2*x^8 + 21*a^5*c*d^2*x^8 + a*b^7*x^7 + 21*a^2*b^5*c*x^7
+ 70*a^3*b^3*c^2*x^7 + 35*a^4*b*c^3*x^7 + 35*a^3*b^4*d*x^7 + 105*a^4*b^2*c*d*x^7 + 21*a^5*c^2*d*x^7 + 21*a^5*b
*d^2*x^7 + 7/2*a^2*b^6*x^6 + 35*a^3*b^4*c*x^6 + 105/2*a^4*b^2*c^2*x^6 + 7*a^5*c^3*x^6 + 35*a^4*b^3*d*x^6 + 42*
a^5*b*c*d*x^6 + 7/2*a^6*d^2*x^6 + 7*a^3*b^5*x^5 + 35*a^4*b^3*c*x^5 + 21*a^5*b*c^2*x^5 + 21*a^5*b^2*d*x^5 + 7*a
^6*c*d*x^5 + 35/4*a^4*b^4*x^4 + 21*a^5*b^2*c*x^4 + 7/2*a^6*c^2*x^4 + 7*a^6*b*d*x^4 + 7*a^5*b^3*x^3 + 7*a^6*b*c
*x^3 + a^7*d*x^3 + 7/2*a^6*b^2*x^2 + a^7*c*x^2 + a^7*b*x