\(\int (b+2 c x+3 d x^2) (a+b x+c x^2+d x^3)^7 \, dx\) [192]
Optimal result
Integrand size = 30, antiderivative size = 21 \[
\int \left (b+2 c x+3 d x^2\right ) \left (a+b x+c x^2+d x^3\right )^7 \, dx=\frac {1}{8} \left (a+b x+c x^2+d x^3\right )^8
\]
[Out]
1/8*(d*x^3+c*x^2+b*x+a)^8
Rubi [A] (verified)
Time = 0.08 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00,
number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {1602}
\[
\int \left (b+2 c x+3 d x^2\right ) \left (a+b x+c x^2+d x^3\right )^7 \, dx=\frac {1}{8} \left (a+b x+c x^2+d x^3\right )^8
\]
[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 1602
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*}
\text {integral}& = \frac {1}{8} \left (a+b x+c x^2+d x^3\right )^8 \\
\end{align*}
Mathematica [B] (verified)
Leaf count is larger than twice the leaf count of optimal. \(143\) vs. \(2(21)=42\).
Time = 0.09 (sec) , antiderivative size = 143, normalized size of antiderivative = 6.81
\[
\int \left (b+2 c x+3 d x^2\right ) \left (a+b x+c x^2+d x^3\right )^7 \, dx=\frac {1}{8} x (b+x (c+d x)) \left (8 a^7+28 a^6 x (b+x (c+d x))+56 a^5 x^2 (b+x (c+d x))^2+70 a^4 x^3 (b+x (c+d x))^3+56 a^3 x^4 (b+x (c+d x))^4+28 a^2 x^5 (b+x (c+d x))^5+8 a x^6 (b+x (c+d x))^6+x^7 (b+x (c+d x))^7\right )
\]
[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
Maple [A] (verified)
Time = 0.16 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95
| | |
| method | result | size |
| | |
| default |
\(\frac {\left (x^{3} d +c \,x^{2}+b x +a \right )^{8}}{8}\) |
\(20\) |
| norman |
\(\text {Expression too large to display}\) |
\(1579\) |
| gosper |
\(\text {Expression too large to display}\) |
\(1957\) |
| parallelrisch |
\(\text {Expression too large to display}\) |
\(1957\) |
| risch |
\(\text {Expression too large to display}\) |
\(1962\) |
| | |
|
|
|
|
[In]
int((3*d*x^2+2*c*x+b)*(d*x^3+c*x^2+b*x+a)^7,x,method=_RETURNVERBOSE)
[Out]
1/8*(d*x^3+c*x^2+b*x+a)^8
Fricas [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 1528 vs. \(2 (19) = 38\).
Time = 0.26 (sec) , antiderivative size = 1528, normalized size of antiderivative = 72.76
\[
\int \left (b+2 c x+3 d x^2\right ) \left (a+b x+c x^2+d x^3\right )^7 \, dx=\text {Too large to display}
\]
[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*d^8*x^24 + c*d^7*x^23 + 1/2*(7*c^2*d^6 + 2*b*d^7)*x^22 + (7*c^3*d^5 + 7*b*c*d^6 + a*d^7)*x^21 + 7/4*(5*c^4
*d^4 + 12*b*c^2*d^5 + 2*(b^2 + 2*a*c)*d^6)*x^20 + 7*(c^5*d^3 + 5*b*c^3*d^4 + a*b*d^6 + 3*(b^2*c + a*c^2)*d^5)*
x^19 + 7/2*(c^6*d^2 + 10*b*c^4*d^3 + a^2*d^6 + 2*(b^3 + 6*a*b*c)*d^5 + 5*(3*b^2*c^2 + 2*a*c^3)*d^4)*x^18 + (c^
7*d + 21*b*c^5*d^2 + 21*(a*b^2 + a^2*c)*d^5 + 35*(b^3*c + 3*a*b*c^2)*d^4 + 35*(2*b^2*c^3 + a*c^4)*d^3)*x^17 +
1/8*(c^8 + 56*b*c^6*d + 168*a^2*b*d^5 + 70*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4 + 560*(b^3*c^2 + 2*a*b*c^3)*d^3
+ 84*(5*b^2*c^4 + 2*a*c^5)*d^2)*x^16 + (b*c^7 + 7*a^3*d^5 + 35*(a*b^3 + 3*a^2*b*c)*d^4 + 35*(b^4*c + 6*a*b^2*c
^2 + 2*a^2*c^3)*d^3 + 35*(2*b^3*c^3 + 3*a*b*c^4)*d^2 + 7*(3*b^2*c^5 + a*c^6)*d)*x^15 + 1/2*(7*b^2*c^6 + 2*a*c^
7 + 35*(3*a^2*b^2 + 2*a^3*c)*d^4 + 14*(b^5 + 20*a*b^3*c + 30*a^2*b*c^2)*d^3 + 105*(b^4*c^2 + 4*a*b^2*c^3 + a^2
*c^4)*d^2 + 14*(5*b^3*c^4 + 6*a*b*c^5)*d)*x^14 + 7*(b^3*c^5 + a*b*c^6 + 5*a^3*b*d^4 + 5*(a*b^4 + 6*a^2*b^2*c +
2*a^3*c^2)*d^3 + 3*(b^5*c + 10*a*b^3*c^2 + 10*a^2*b*c^3)*d^2 + (5*b^4*c^3 + 15*a*b^2*c^4 + 3*a^2*c^5)*d)*x^13
+ 7/4*(5*b^4*c^4 + 12*a*b^2*c^5 + 2*a^2*c^6 + 5*a^4*d^4 + 40*(a^2*b^3 + 2*a^3*b*c)*d^3 + 2*(b^6 + 30*a*b^4*c
+ 90*a^2*b^2*c^2 + 20*a^3*c^3)*d^2 + 4*(3*b^5*c^2 + 20*a*b^3*c^3 + 15*a^2*b*c^4)*d)*x^12 + 7*(b^5*c^3 + 5*a*b^
3*c^4 + 3*a^2*b*c^5 + 5*(2*a^3*b^2 + a^4*c)*d^3 + 3*(a*b^5 + 10*a^2*b^3*c + 10*a^3*b*c^2)*d^2 + (b^6*c + 15*a*
b^4*c^2 + 30*a^2*b^2*c^3 + 5*a^3*c^4)*d)*x^11 + 1/2*(7*b^6*c^2 + 70*a*b^4*c^3 + 105*a^2*b^2*c^4 + 14*a^3*c^5 +
70*a^4*b*d^3 + 105*(a^2*b^4 + 4*a^3*b^2*c + a^4*c^2)*d^2 + 2*(b^7 + 42*a*b^5*c + 210*a^2*b^3*c^2 + 140*a^3*b*
c^3)*d)*x^10 + (b^7*c + 21*a*b^5*c^2 + 70*a^2*b^3*c^3 + 35*a^3*b*c^4 + 7*a^5*d^3 + 35*(2*a^3*b^3 + 3*a^4*b*c)*
d^2 + 7*(a*b^6 + 15*a^2*b^4*c + 30*a^3*b^2*c^2 + 5*a^4*c^3)*d)*x^9 + a^7*b*x + 1/8*(b^8 + 56*a*b^6*c + 420*a^2
*b^4*c^2 + 560*a^3*b^2*c^3 + 70*a^4*c^4 + 84*(5*a^4*b^2 + 2*a^5*c)*d^2 + 56*(3*a^2*b^5 + 20*a^3*b^3*c + 15*a^4
*b*c^2)*d)*x^8 + (a*b^7 + 21*a^2*b^5*c + 70*a^3*b^3*c^2 + 35*a^4*b*c^3 + 21*a^5*b*d^2 + 7*(5*a^3*b^4 + 15*a^4*
b^2*c + 3*a^5*c^2)*d)*x^7 + 7/2*(a^2*b^6 + 10*a^3*b^4*c + 15*a^4*b^2*c^2 + 2*a^5*c^3 + a^6*d^2 + 2*(5*a^4*b^3
+ 6*a^5*b*c)*d)*x^6 + 7*(a^3*b^5 + 5*a^4*b^3*c + 3*a^5*b*c^2 + (3*a^5*b^2 + a^6*c)*d)*x^5 + 7/4*(5*a^4*b^4 + 1
2*a^5*b^2*c + 2*a^6*c^2 + 4*a^6*b*d)*x^4 + (7*a^5*b^3 + 7*a^6*b*c + a^7*d)*x^3 + 1/2*(7*a^6*b^2 + 2*a^7*c)*x^2
Sympy [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 1771 vs. \(2 (17) = 34\).
Time = 0.18 (sec) , antiderivative size = 1771, normalized size of antiderivative = 84.33
\[
\int \left (b+2 c x+3 d x^2\right ) \left (a+b x+c x^2+d x^3\right )^7 \, dx=\text {Too large to display}
\]
[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)
Maxima [A] (verification not implemented)
none
Time = 0.19 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90
\[
\int \left (b+2 c x+3 d x^2\right ) \left (a+b x+c x^2+d x^3\right )^7 \, dx=\frac {1}{8} \, {\left (d x^{3} + c x^{2} + b x + a\right )}^{8}
\]
[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
Giac [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 160 vs. \(2 (19) = 38\).
Time = 0.30 (sec) , antiderivative size = 160, normalized size of antiderivative = 7.62
\[
\int \left (b+2 c x+3 d x^2\right ) \left (a+b x+c x^2+d x^3\right )^7 \, dx=\frac {1}{8} \, {\left (d x^{3} + c x^{2} + b x\right )}^{8} + {\left (d x^{3} + c x^{2} + b x\right )}^{7} a + \frac {7}{2} \, {\left (d x^{3} + c x^{2} + b x\right )}^{6} a^{2} + 7 \, {\left (d x^{3} + c x^{2} + b x\right )}^{5} a^{3} + \frac {35}{4} \, {\left (d x^{3} + c x^{2} + b x\right )}^{4} a^{4} + 7 \, {\left (d x^{3} + c x^{2} + b x\right )}^{3} a^{5} + \frac {7}{2} \, {\left (d x^{3} + c x^{2} + b x\right )}^{2} a^{6} + {\left (d x^{3} + c x^{2} + b x\right )} a^{7}
\]
[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
Mupad [B] (verification not implemented)
Time = 9.90 (sec) , antiderivative size = 1576, normalized size of antiderivative = 75.05
\[
\int \left (b+2 c x+3 d x^2\right ) \left (a+b x+c x^2+d x^3\right )^7 \, dx=\text {Too large to display}
\]
[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