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]
1/8*(d*x^3+c*x^2+b*x+a)^8
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Rubi [A] time = 0.13, antiderivative size = 21, normalized size of antiderivative = 1.00,
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*}
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Mathematica [B] time = 0.15, size = 143, normalized size = 6.81 \[ \frac {1}{8} x (b+x (c+d x)) \left (8 a^7+28 a^6 x (b+x (c+d x))+56 a^5 x^2 (b+x (c+d x))^2+70 a^4 x^3 (b+x (c+d x))^3+56 a^3 x^4 (b+x (c+d x))^4+28 a^2 x^5 (b+x (c+d x))^5+8 a x^6 (b+x (c+d x))^6+x^7 (b+x (c+d x))^7\right ) \]
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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fricas [B] time = 0.56, size = 1956, normalized size = 93.14 \[ \text {result too large to display} \]
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
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giac [B] time = 0.30, size = 160, normalized size = 7.62 \[ \frac {1}{8} \, {\left (d x^{3} + c x^{2} + b x\right )}^{8} + {\left (d x^{3} + c x^{2} + b x\right )}^{7} a + \frac {7}{2} \, {\left (d x^{3} + c x^{2} + b x\right )}^{6} a^{2} + 7 \, {\left (d x^{3} + c x^{2} + b x\right )}^{5} a^{3} + \frac {35}{4} \, {\left (d x^{3} + c x^{2} + b x\right )}^{4} a^{4} + 7 \, {\left (d x^{3} + c x^{2} + b x\right )}^{3} a^{5} + \frac {7}{2} \, {\left (d x^{3} + c x^{2} + b x\right )}^{2} a^{6} + {\left (d x^{3} + c x^{2} + b x\right )} a^{7} \]
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*x^3 + c*x^2 + b*x)^8 + (d*x^3 + c*x^2 + b*x)^7*a + 7/2*(d*x^3 + c*x^2 + b*x)^6*a^2 + 7*(d*x^3 + c*x^2 +
b*x)^5*a^3 + 35/4*(d*x^3 + c*x^2 + b*x)^4*a^4 + 7*(d*x^3 + c*x^2 + b*x)^3*a^5 + 7/2*(d*x^3 + c*x^2 + b*x)^2*a
^6 + (d*x^3 + c*x^2 + b*x)*a^7
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maple [B] time = 0.00, size = 25686, normalized size = 1223.14 \[ \text {output too large to display} \]
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 = 0.74, size = 19, normalized size = 0.90 \[ \frac {1}{8} \, {\left (d x^{3} + c x^{2} + b x + a\right )}^{8} \]
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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mupad [B] time = 2.98, size = 1576, normalized size = 75.05 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b + 2*c*x + 3*d*x^2)*(a + b*x + c*x^2 + d*x^3)^7,x)
[Out]
x^12*((7*a^2*c^6)/2 + (35*a^4*d^4)/4 + (35*b^4*c^4)/4 + (7*b^6*d^2)/2 + 21*a*b^2*c^5 + 21*b^5*c^2*d + 70*a^2*b
^3*d^3 + 70*a^3*c^3*d^2 + 315*a^2*b^2*c^2*d^2 + 140*a*b^3*c^3*d + 105*a*b^4*c*d^2 + 105*a^2*b*c^4*d + 140*a^3*
b*c*d^3) + x^11*(7*b^5*c^3 + 35*a*b^3*c^4 + 21*a^2*b*c^5 + 21*a*b^5*d^2 + 35*a^3*c^4*d + 35*a^4*c*d^3 + 70*a^3
*b^2*d^3 + 7*b^6*c*d + 210*a^2*b^2*c^3*d + 210*a^2*b^3*c*d^2 + 210*a^3*b*c^2*d^2 + 105*a*b^4*c^2*d) + x^13*(7*
b^3*c^5 + 35*a*b^4*d^3 + 35*a^3*b*d^4 + 21*a^2*c^5*d + 35*b^4*c^3*d + 21*b^5*c*d^2 + 70*a^3*c^2*d^3 + 7*a*b*c^
6 + 210*a*b^3*c^2*d^2 + 210*a^2*b*c^3*d^2 + 210*a^2*b^2*c*d^3 + 105*a*b^2*c^4*d) + x^5*(7*a^3*b^5 + 35*a^4*b^3
*c + 21*a^5*b*c^2 + 21*a^5*b^2*d + 7*a^6*c*d) + x^19*(7*c^5*d^3 + 21*a*c^2*d^5 + 35*b*c^3*d^4 + 21*b^2*c*d^5 +
7*a*b*d^6) + x^8*(b^8/8 + (35*a^4*c^4)/4 + 21*a^2*b^5*d + 21*a^5*c*d^2 + (105*a^2*b^4*c^2)/2 + 70*a^3*b^2*c^3
+ (105*a^4*b^2*d^2)/2 + 7*a*b^6*c + 140*a^3*b^3*c*d + 105*a^4*b*c^2*d) + x^9*(b^7*c + 7*a^5*d^3 + 21*a*b^5*c^
2 + 35*a^3*b*c^4 + 35*a^4*c^3*d + 70*a^2*b^3*c^3 + 70*a^3*b^3*d^2 + 7*a*b^6*d + 210*a^3*b^2*c^2*d + 105*a^2*b^
4*c*d + 105*a^4*b*c*d^2) + x^16*(c^8/8 + (35*b^4*d^4)/4 + 21*a^2*b*d^5 + 21*a*c^5*d^2 + (105*a^2*c^2*d^4)/2 +
(105*b^2*c^4*d^2)/2 + 70*b^3*c^2*d^3 + 7*b*c^6*d + 140*a*b*c^3*d^3 + 105*a*b^2*c*d^4) + x^10*(b^7*d + 7*a^3*c^
5 + (7*b^6*c^2)/2 + 35*a*b^4*c^3 + 35*a^4*b*d^3 + (105*a^2*b^2*c^4)/2 + (105*a^2*b^4*d^2)/2 + (105*a^4*c^2*d^2
)/2 + 210*a^2*b^3*c^2*d + 210*a^3*b^2*c*d^2 + 42*a*b^5*c*d + 140*a^3*b*c^3*d) + x^15*(b*c^7 + 7*a^3*d^5 + 35*a
*b^3*d^4 + 21*b^2*c^5*d + 35*b^4*c*d^3 + 70*a^2*c^3*d^3 + 70*b^3*c^3*d^2 + 7*a*c^6*d + 210*a*b^2*c^2*d^3 + 105
*a*b*c^4*d^2 + 105*a^2*b*c*d^4) + x^14*(a*c^7 + (7*b^2*c^6)/2 + 7*b^5*d^3 + 35*a^3*c*d^4 + 35*b^3*c^4*d + (105
*a^2*b^2*d^4)/2 + (105*a^2*c^4*d^2)/2 + (105*b^4*c^2*d^2)/2 + 210*a*b^2*c^3*d^2 + 210*a^2*b*c^2*d^3 + 42*a*b*c
^5*d + 140*a*b^3*c*d^3) + x^4*((35*a^4*b^4)/4 + (7*a^6*c^2)/2 + 21*a^5*b^2*c + 7*a^6*b*d) + x^20*((7*b^2*d^6)/
2 + (35*c^4*d^4)/4 + 21*b*c^2*d^5 + 7*a*c*d^6) + x^6*((7*a^2*b^6)/2 + 7*a^5*c^3 + (7*a^6*d^2)/2 + 35*a^3*b^4*c
+ 35*a^4*b^3*d + (105*a^4*b^2*c^2)/2 + 42*a^5*b*c*d) + x^7*(a*b^7 + 21*a^2*b^5*c + 35*a^4*b*c^3 + 35*a^3*b^4*
d + 21*a^5*b*d^2 + 21*a^5*c^2*d + 70*a^3*b^3*c^2 + 105*a^4*b^2*c*d) + x^18*((7*a^2*d^6)/2 + 7*b^3*d^5 + (7*c^6
*d^2)/2 + 35*a*c^3*d^4 + 35*b*c^4*d^3 + (105*b^2*c^2*d^4)/2 + 42*a*b*c*d^5) + x^17*(c^7*d + 21*a*b^2*d^5 + 35*
a*c^4*d^3 + 21*a^2*c*d^5 + 21*b*c^5*d^2 + 35*b^3*c*d^4 + 70*b^2*c^3*d^3 + 105*a*b*c^2*d^4) + x^3*(a^7*d + 7*a^
5*b^3 + 7*a^6*b*c) + (d^8*x^24)/8 + x^2*(a^7*c + (7*a^6*b^2)/2) + c*d^7*x^23 + d^5*x^21*(a*d^2 + 7*c^3 + 7*b*c
*d) + (d^6*x^22*(2*b*d + 7*c^2))/2 + a^7*b*x
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sympy [B] time = 0.42, size = 1771, normalized size = 84.33 \[ \text {result too large to display} \]
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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