3.28 \(\int (c+d x)^n (A+B x+C x^2+D x^3) \, dx\)
Optimal. Leaf size=126 \[ \frac {(c+d x)^{n+1} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^4 (n+1)}-\frac {(c+d x)^{n+2} \left (-B d^2-3 c^2 D+2 c C d\right )}{d^4 (n+2)}+\frac {(C d-3 c D) (c+d x)^{n+3}}{d^4 (n+3)}+\frac {D (c+d x)^{n+4}}{d^4 (n+4)} \]
[Out]
(A*d^3-B*c*d^2+C*c^2*d-D*c^3)*(d*x+c)^(1+n)/d^4/(1+n)-(-B*d^2+2*C*c*d-3*D*c^2)*(d*x+c)^(2+n)/d^4/(2+n)+(C*d-3*
D*c)*(d*x+c)^(3+n)/d^4/(3+n)+D*(d*x+c)^(4+n)/d^4/(4+n)
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Rubi [A] time = 0.07, antiderivative size = 126, normalized size of antiderivative = 1.00,
number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used =
{1850} \[ \frac {(c+d x)^{n+1} \left (A d^3-B c d^2+c^2 C d+c^3 (-D)\right )}{d^4 (n+1)}-\frac {(c+d x)^{n+2} \left (-B d^2-3 c^2 D+2 c C d\right )}{d^4 (n+2)}+\frac {(C d-3 c D) (c+d x)^{n+3}}{d^4 (n+3)}+\frac {D (c+d x)^{n+4}}{d^4 (n+4)} \]
Antiderivative was successfully verified.
[In]
Int[(c + d*x)^n*(A + B*x + C*x^2 + D*x^3),x]
[Out]
((c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)*(c + d*x)^(1 + n))/(d^4*(1 + n)) - ((2*c*C*d - B*d^2 - 3*c^2*D)*(c + d*x)
^(2 + n))/(d^4*(2 + n)) + ((C*d - 3*c*D)*(c + d*x)^(3 + n))/(d^4*(3 + n)) + (D*(c + d*x)^(4 + n))/(d^4*(4 + n)
)
Rule 1850
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])
Rubi steps
\begin {align*} \int (c+d x)^n \left (A+B x+C x^2+D x^3\right ) \, dx &=\int \left (\frac {\left (c^2 C d-B c d^2+A d^3-c^3 D\right ) (c+d x)^n}{d^3}+\frac {\left (-2 c C d+B d^2+3 c^2 D\right ) (c+d x)^{1+n}}{d^3}+\frac {(C d-3 c D) (c+d x)^{2+n}}{d^3}+\frac {D (c+d x)^{3+n}}{d^3}\right ) \, dx\\ &=\frac {\left (c^2 C d-B c d^2+A d^3-c^3 D\right ) (c+d x)^{1+n}}{d^4 (1+n)}-\frac {\left (2 c C d-B d^2-3 c^2 D\right ) (c+d x)^{2+n}}{d^4 (2+n)}+\frac {(C d-3 c D) (c+d x)^{3+n}}{d^4 (3+n)}+\frac {D (c+d x)^{4+n}}{d^4 (4+n)}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 108, normalized size = 0.86 \[ \frac {(c+d x)^{n+1} \left (\frac {A d^3-B c d^2+c^3 (-D)+c^2 C d}{n+1}+\frac {(c+d x) \left (B d^2+3 c^2 D-2 c C d\right )}{n+2}+\frac {(c+d x)^2 (C d-3 c D)}{n+3}+\frac {D (c+d x)^3}{n+4}\right )}{d^4} \]
Antiderivative was successfully verified.
[In]
Integrate[(c + d*x)^n*(A + B*x + C*x^2 + D*x^3),x]
[Out]
((c + d*x)^(1 + n)*((c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)/(1 + n) + ((-2*c*C*d + B*d^2 + 3*c^2*D)*(c + d*x))/(2
+ n) + ((C*d - 3*c*D)*(c + d*x)^2)/(3 + n) + (D*(c + d*x)^3)/(4 + n)))/d^4
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fricas [B] time = 0.94, size = 394, normalized size = 3.13 \[ \frac {{\left (A c d^{3} n^{3} - 6 \, D c^{4} + 8 \, C c^{3} d - 12 \, B c^{2} d^{2} + 24 \, A c d^{3} + {\left (D d^{4} n^{3} + 6 \, D d^{4} n^{2} + 11 \, D d^{4} n + 6 \, D d^{4}\right )} x^{4} + {\left (8 \, C d^{4} + {\left (D c d^{3} + C d^{4}\right )} n^{3} + {\left (3 \, D c d^{3} + 7 \, C d^{4}\right )} n^{2} + 2 \, {\left (D c d^{3} + 7 \, C d^{4}\right )} n\right )} x^{3} - {\left (B c^{2} d^{2} - 9 \, A c d^{3}\right )} n^{2} + {\left (12 \, B d^{4} + {\left (C c d^{3} + B d^{4}\right )} n^{3} - {\left (3 \, D c^{2} d^{2} - 5 \, C c d^{3} - 8 \, B d^{4}\right )} n^{2} - {\left (3 \, D c^{2} d^{2} - 4 \, C c d^{3} - 19 \, B d^{4}\right )} n\right )} x^{2} + {\left (2 \, C c^{3} d - 7 \, B c^{2} d^{2} + 26 \, A c d^{3}\right )} n + {\left (24 \, A d^{4} + {\left (B c d^{3} + A d^{4}\right )} n^{3} - {\left (2 \, C c^{2} d^{2} - 7 \, B c d^{3} - 9 \, A d^{4}\right )} n^{2} + 2 \, {\left (3 \, D c^{3} d - 4 \, C c^{2} d^{2} + 6 \, B c d^{3} + 13 \, A d^{4}\right )} n\right )} x\right )} {\left (d x + c\right )}^{n}}{d^{4} n^{4} + 10 \, d^{4} n^{3} + 35 \, d^{4} n^{2} + 50 \, d^{4} n + 24 \, d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)^n*(D*x^3+C*x^2+B*x+A),x, algorithm="fricas")
[Out]
(A*c*d^3*n^3 - 6*D*c^4 + 8*C*c^3*d - 12*B*c^2*d^2 + 24*A*c*d^3 + (D*d^4*n^3 + 6*D*d^4*n^2 + 11*D*d^4*n + 6*D*d
^4)*x^4 + (8*C*d^4 + (D*c*d^3 + C*d^4)*n^3 + (3*D*c*d^3 + 7*C*d^4)*n^2 + 2*(D*c*d^3 + 7*C*d^4)*n)*x^3 - (B*c^2
*d^2 - 9*A*c*d^3)*n^2 + (12*B*d^4 + (C*c*d^3 + B*d^4)*n^3 - (3*D*c^2*d^2 - 5*C*c*d^3 - 8*B*d^4)*n^2 - (3*D*c^2
*d^2 - 4*C*c*d^3 - 19*B*d^4)*n)*x^2 + (2*C*c^3*d - 7*B*c^2*d^2 + 26*A*c*d^3)*n + (24*A*d^4 + (B*c*d^3 + A*d^4)
*n^3 - (2*C*c^2*d^2 - 7*B*c*d^3 - 9*A*d^4)*n^2 + 2*(3*D*c^3*d - 4*C*c^2*d^2 + 6*B*c*d^3 + 13*A*d^4)*n)*x)*(d*x
+ c)^n/(d^4*n^4 + 10*d^4*n^3 + 35*d^4*n^2 + 50*d^4*n + 24*d^4)
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giac [B] time = 1.32, size = 728, normalized size = 5.78 \[ \frac {{\left (d x + c\right )}^{n} D d^{4} n^{3} x^{4} + {\left (d x + c\right )}^{n} D c d^{3} n^{3} x^{3} + {\left (d x + c\right )}^{n} C d^{4} n^{3} x^{3} + 6 \, {\left (d x + c\right )}^{n} D d^{4} n^{2} x^{4} + {\left (d x + c\right )}^{n} C c d^{3} n^{3} x^{2} + {\left (d x + c\right )}^{n} B d^{4} n^{3} x^{2} + 3 \, {\left (d x + c\right )}^{n} D c d^{3} n^{2} x^{3} + 7 \, {\left (d x + c\right )}^{n} C d^{4} n^{2} x^{3} + 11 \, {\left (d x + c\right )}^{n} D d^{4} n x^{4} + {\left (d x + c\right )}^{n} B c d^{3} n^{3} x + {\left (d x + c\right )}^{n} A d^{4} n^{3} x - 3 \, {\left (d x + c\right )}^{n} D c^{2} d^{2} n^{2} x^{2} + 5 \, {\left (d x + c\right )}^{n} C c d^{3} n^{2} x^{2} + 8 \, {\left (d x + c\right )}^{n} B d^{4} n^{2} x^{2} + 2 \, {\left (d x + c\right )}^{n} D c d^{3} n x^{3} + 14 \, {\left (d x + c\right )}^{n} C d^{4} n x^{3} + 6 \, {\left (d x + c\right )}^{n} D d^{4} x^{4} + {\left (d x + c\right )}^{n} A c d^{3} n^{3} - 2 \, {\left (d x + c\right )}^{n} C c^{2} d^{2} n^{2} x + 7 \, {\left (d x + c\right )}^{n} B c d^{3} n^{2} x + 9 \, {\left (d x + c\right )}^{n} A d^{4} n^{2} x - 3 \, {\left (d x + c\right )}^{n} D c^{2} d^{2} n x^{2} + 4 \, {\left (d x + c\right )}^{n} C c d^{3} n x^{2} + 19 \, {\left (d x + c\right )}^{n} B d^{4} n x^{2} + 8 \, {\left (d x + c\right )}^{n} C d^{4} x^{3} - {\left (d x + c\right )}^{n} B c^{2} d^{2} n^{2} + 9 \, {\left (d x + c\right )}^{n} A c d^{3} n^{2} + 6 \, {\left (d x + c\right )}^{n} D c^{3} d n x - 8 \, {\left (d x + c\right )}^{n} C c^{2} d^{2} n x + 12 \, {\left (d x + c\right )}^{n} B c d^{3} n x + 26 \, {\left (d x + c\right )}^{n} A d^{4} n x + 12 \, {\left (d x + c\right )}^{n} B d^{4} x^{2} + 2 \, {\left (d x + c\right )}^{n} C c^{3} d n - 7 \, {\left (d x + c\right )}^{n} B c^{2} d^{2} n + 26 \, {\left (d x + c\right )}^{n} A c d^{3} n + 24 \, {\left (d x + c\right )}^{n} A d^{4} x - 6 \, {\left (d x + c\right )}^{n} D c^{4} + 8 \, {\left (d x + c\right )}^{n} C c^{3} d - 12 \, {\left (d x + c\right )}^{n} B c^{2} d^{2} + 24 \, {\left (d x + c\right )}^{n} A c d^{3}}{d^{4} n^{4} + 10 \, d^{4} n^{3} + 35 \, d^{4} n^{2} + 50 \, d^{4} n + 24 \, d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)^n*(D*x^3+C*x^2+B*x+A),x, algorithm="giac")
[Out]
((d*x + c)^n*D*d^4*n^3*x^4 + (d*x + c)^n*D*c*d^3*n^3*x^3 + (d*x + c)^n*C*d^4*n^3*x^3 + 6*(d*x + c)^n*D*d^4*n^2
*x^4 + (d*x + c)^n*C*c*d^3*n^3*x^2 + (d*x + c)^n*B*d^4*n^3*x^2 + 3*(d*x + c)^n*D*c*d^3*n^2*x^3 + 7*(d*x + c)^n
*C*d^4*n^2*x^3 + 11*(d*x + c)^n*D*d^4*n*x^4 + (d*x + c)^n*B*c*d^3*n^3*x + (d*x + c)^n*A*d^4*n^3*x - 3*(d*x + c
)^n*D*c^2*d^2*n^2*x^2 + 5*(d*x + c)^n*C*c*d^3*n^2*x^2 + 8*(d*x + c)^n*B*d^4*n^2*x^2 + 2*(d*x + c)^n*D*c*d^3*n*
x^3 + 14*(d*x + c)^n*C*d^4*n*x^3 + 6*(d*x + c)^n*D*d^4*x^4 + (d*x + c)^n*A*c*d^3*n^3 - 2*(d*x + c)^n*C*c^2*d^2
*n^2*x + 7*(d*x + c)^n*B*c*d^3*n^2*x + 9*(d*x + c)^n*A*d^4*n^2*x - 3*(d*x + c)^n*D*c^2*d^2*n*x^2 + 4*(d*x + c)
^n*C*c*d^3*n*x^2 + 19*(d*x + c)^n*B*d^4*n*x^2 + 8*(d*x + c)^n*C*d^4*x^3 - (d*x + c)^n*B*c^2*d^2*n^2 + 9*(d*x +
c)^n*A*c*d^3*n^2 + 6*(d*x + c)^n*D*c^3*d*n*x - 8*(d*x + c)^n*C*c^2*d^2*n*x + 12*(d*x + c)^n*B*c*d^3*n*x + 26*
(d*x + c)^n*A*d^4*n*x + 12*(d*x + c)^n*B*d^4*x^2 + 2*(d*x + c)^n*C*c^3*d*n - 7*(d*x + c)^n*B*c^2*d^2*n + 26*(d
*x + c)^n*A*c*d^3*n + 24*(d*x + c)^n*A*d^4*x - 6*(d*x + c)^n*D*c^4 + 8*(d*x + c)^n*C*c^3*d - 12*(d*x + c)^n*B*
c^2*d^2 + 24*(d*x + c)^n*A*c*d^3)/(d^4*n^4 + 10*d^4*n^3 + 35*d^4*n^2 + 50*d^4*n + 24*d^4)
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maple [B] time = 0.01, size = 308, normalized size = 2.44 \[ \frac {\left (D d^{3} n^{3} x^{3}+C \,d^{3} n^{3} x^{2}+6 D d^{3} n^{2} x^{3}+B \,d^{3} n^{3} x +7 C \,d^{3} n^{2} x^{2}-3 D c \,d^{2} n^{2} x^{2}+11 D d^{3} n \,x^{3}+A \,d^{3} n^{3}+8 B \,d^{3} n^{2} x -2 C c \,d^{2} n^{2} x +14 C \,d^{3} n \,x^{2}-9 D c \,d^{2} n \,x^{2}+6 D x^{3} d^{3}+9 A \,d^{3} n^{2}-B c \,d^{2} n^{2}+19 B \,d^{3} n x -10 C c \,d^{2} n x +8 C \,d^{3} x^{2}+6 D c^{2} d n x -6 D c \,d^{2} x^{2}+26 A \,d^{3} n -7 B c \,d^{2} n +12 B \,d^{3} x +2 C \,c^{2} d n -8 C c \,d^{2} x +6 D c^{2} d x +24 A \,d^{3}-12 B c \,d^{2}+8 C \,c^{2} d -6 D c^{3}\right ) \left (d x +c \right )^{n +1}}{\left (n^{4}+10 n^{3}+35 n^{2}+50 n +24\right ) d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d*x+c)^n*(D*x^3+C*x^2+B*x+A),x)
[Out]
(d*x+c)^(n+1)*(D*d^3*n^3*x^3+C*d^3*n^3*x^2+6*D*d^3*n^2*x^3+B*d^3*n^3*x+7*C*d^3*n^2*x^2-3*D*c*d^2*n^2*x^2+11*D*
d^3*n*x^3+A*d^3*n^3+8*B*d^3*n^2*x-2*C*c*d^2*n^2*x+14*C*d^3*n*x^2-9*D*c*d^2*n*x^2+6*D*d^3*x^3+9*A*d^3*n^2-B*c*d
^2*n^2+19*B*d^3*n*x-10*C*c*d^2*n*x+8*C*d^3*x^2+6*D*c^2*d*n*x-6*D*c*d^2*x^2+26*A*d^3*n-7*B*c*d^2*n+12*B*d^3*x+2
*C*c^2*d*n-8*C*c*d^2*x+6*D*c^2*d*x+24*A*d^3-12*B*c*d^2+8*C*c^2*d-6*D*c^3)/d^4/(n^4+10*n^3+35*n^2+50*n+24)
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maxima [A] time = 0.49, size = 234, normalized size = 1.86 \[ \frac {{\left (d^{2} {\left (n + 1\right )} x^{2} + c d n x - c^{2}\right )} {\left (d x + c\right )}^{n} B}{{\left (n^{2} + 3 \, n + 2\right )} d^{2}} + \frac {{\left (d x + c\right )}^{n + 1} A}{d {\left (n + 1\right )}} + \frac {{\left ({\left (n^{2} + 3 \, n + 2\right )} d^{3} x^{3} + {\left (n^{2} + n\right )} c d^{2} x^{2} - 2 \, c^{2} d n x + 2 \, c^{3}\right )} {\left (d x + c\right )}^{n} C}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} d^{3}} + \frac {{\left ({\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} d^{4} x^{4} + {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} c d^{3} x^{3} - 3 \, {\left (n^{2} + n\right )} c^{2} d^{2} x^{2} + 6 \, c^{3} d n x - 6 \, c^{4}\right )} {\left (d x + c\right )}^{n} D}{{\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)^n*(D*x^3+C*x^2+B*x+A),x, algorithm="maxima")
[Out]
(d^2*(n + 1)*x^2 + c*d*n*x - c^2)*(d*x + c)^n*B/((n^2 + 3*n + 2)*d^2) + (d*x + c)^(n + 1)*A/(d*(n + 1)) + ((n^
2 + 3*n + 2)*d^3*x^3 + (n^2 + n)*c*d^2*x^2 - 2*c^2*d*n*x + 2*c^3)*(d*x + c)^n*C/((n^3 + 6*n^2 + 11*n + 6)*d^3)
+ ((n^3 + 6*n^2 + 11*n + 6)*d^4*x^4 + (n^3 + 3*n^2 + 2*n)*c*d^3*x^3 - 3*(n^2 + n)*c^2*d^2*x^2 + 6*c^3*d*n*x -
6*c^4)*(d*x + c)^n*D/((n^4 + 10*n^3 + 35*n^2 + 50*n + 24)*d^4)
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (c+d\,x\right )}^n\,\left (A+B\,x+C\,x^2+x^3\,D\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((c + d*x)^n*(A + B*x + C*x^2 + x^3*D),x)
[Out]
int((c + d*x)^n*(A + B*x + C*x^2 + x^3*D), x)
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sympy [A] time = 4.32, size = 3798, normalized size = 30.14 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)**n*(D*x**3+C*x**2+B*x+A),x)
[Out]
Piecewise((c**n*(A*x + B*x**2/2 + C*x**3/3 + D*x**4/4), Eq(d, 0)), (-2*A*d**3/(6*c**3*d**4 + 18*c**2*d**5*x +
18*c*d**6*x**2 + 6*d**7*x**3) - B*c*d**2/(6*c**3*d**4 + 18*c**2*d**5*x + 18*c*d**6*x**2 + 6*d**7*x**3) - 3*B*d
**3*x/(6*c**3*d**4 + 18*c**2*d**5*x + 18*c*d**6*x**2 + 6*d**7*x**3) - 2*C*c**2*d/(6*c**3*d**4 + 18*c**2*d**5*x
+ 18*c*d**6*x**2 + 6*d**7*x**3) - 6*C*c*d**2*x/(6*c**3*d**4 + 18*c**2*d**5*x + 18*c*d**6*x**2 + 6*d**7*x**3)
- 6*C*d**3*x**2/(6*c**3*d**4 + 18*c**2*d**5*x + 18*c*d**6*x**2 + 6*d**7*x**3) + 6*D*c**3*log(c/d + x)/(6*c**3*
d**4 + 18*c**2*d**5*x + 18*c*d**6*x**2 + 6*d**7*x**3) + 11*D*c**3/(6*c**3*d**4 + 18*c**2*d**5*x + 18*c*d**6*x*
*2 + 6*d**7*x**3) + 18*D*c**2*d*x*log(c/d + x)/(6*c**3*d**4 + 18*c**2*d**5*x + 18*c*d**6*x**2 + 6*d**7*x**3) +
27*D*c**2*d*x/(6*c**3*d**4 + 18*c**2*d**5*x + 18*c*d**6*x**2 + 6*d**7*x**3) + 18*D*c*d**2*x**2*log(c/d + x)/(
6*c**3*d**4 + 18*c**2*d**5*x + 18*c*d**6*x**2 + 6*d**7*x**3) + 18*D*c*d**2*x**2/(6*c**3*d**4 + 18*c**2*d**5*x
+ 18*c*d**6*x**2 + 6*d**7*x**3) + 6*D*d**3*x**3*log(c/d + x)/(6*c**3*d**4 + 18*c**2*d**5*x + 18*c*d**6*x**2 +
6*d**7*x**3), Eq(n, -4)), (-A*d**3/(2*c**2*d**4 + 4*c*d**5*x + 2*d**6*x**2) - B*c*d**2/(2*c**2*d**4 + 4*c*d**5
*x + 2*d**6*x**2) - 2*B*d**3*x/(2*c**2*d**4 + 4*c*d**5*x + 2*d**6*x**2) + 2*C*c**2*d*log(c/d + x)/(2*c**2*d**4
+ 4*c*d**5*x + 2*d**6*x**2) + 3*C*c**2*d/(2*c**2*d**4 + 4*c*d**5*x + 2*d**6*x**2) + 4*C*c*d**2*x*log(c/d + x)
/(2*c**2*d**4 + 4*c*d**5*x + 2*d**6*x**2) + 4*C*c*d**2*x/(2*c**2*d**4 + 4*c*d**5*x + 2*d**6*x**2) + 2*C*d**3*x
**2*log(c/d + x)/(2*c**2*d**4 + 4*c*d**5*x + 2*d**6*x**2) - 6*D*c**3*log(c/d + x)/(2*c**2*d**4 + 4*c*d**5*x +
2*d**6*x**2) - 9*D*c**3/(2*c**2*d**4 + 4*c*d**5*x + 2*d**6*x**2) - 12*D*c**2*d*x*log(c/d + x)/(2*c**2*d**4 + 4
*c*d**5*x + 2*d**6*x**2) - 12*D*c**2*d*x/(2*c**2*d**4 + 4*c*d**5*x + 2*d**6*x**2) - 6*D*c*d**2*x**2*log(c/d +
x)/(2*c**2*d**4 + 4*c*d**5*x + 2*d**6*x**2) + 2*D*d**3*x**3/(2*c**2*d**4 + 4*c*d**5*x + 2*d**6*x**2), Eq(n, -3
)), (-2*A*d**3/(2*c*d**4 + 2*d**5*x) + 2*B*c*d**2*log(c/d + x)/(2*c*d**4 + 2*d**5*x) + 2*B*c*d**2/(2*c*d**4 +
2*d**5*x) + 2*B*d**3*x*log(c/d + x)/(2*c*d**4 + 2*d**5*x) - 4*C*c**2*d*log(c/d + x)/(2*c*d**4 + 2*d**5*x) - 4*
C*c**2*d/(2*c*d**4 + 2*d**5*x) - 4*C*c*d**2*x*log(c/d + x)/(2*c*d**4 + 2*d**5*x) + 2*C*d**3*x**2/(2*c*d**4 + 2
*d**5*x) + 6*D*c**3*log(c/d + x)/(2*c*d**4 + 2*d**5*x) + 6*D*c**3/(2*c*d**4 + 2*d**5*x) + 6*D*c**2*d*x*log(c/d
+ x)/(2*c*d**4 + 2*d**5*x) - 3*D*c*d**2*x**2/(2*c*d**4 + 2*d**5*x) + D*d**3*x**3/(2*c*d**4 + 2*d**5*x), Eq(n,
-2)), (A*log(c/d + x)/d - B*c*log(c/d + x)/d**2 + B*x/d + C*c**2*log(c/d + x)/d**3 - C*c*x/d**2 + C*x**2/(2*d
) - D*c**3*log(c/d + x)/d**4 + D*c**2*x/d**3 - D*c*x**2/(2*d**2) + D*x**3/(3*d), Eq(n, -1)), (A*c*d**3*n**3*(c
+ d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) + 9*A*c*d**3*n**2*(c + d*x)**n/(d**
4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) + 26*A*c*d**3*n*(c + d*x)**n/(d**4*n**4 + 10*d**4*
n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) + 24*A*c*d**3*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2
+ 50*d**4*n + 24*d**4) + A*d**4*n**3*x*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24
*d**4) + 9*A*d**4*n**2*x*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) + 26*A*d
**4*n*x*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) + 24*A*d**4*x*(c + d*x)**
n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) - B*c**2*d**2*n**2*(c + d*x)**n/(d**4*n**4 +
10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) - 7*B*c**2*d**2*n*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 +
35*d**4*n**2 + 50*d**4*n + 24*d**4) - 12*B*c**2*d**2*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 +
50*d**4*n + 24*d**4) + B*c*d**3*n**3*x*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*
d**4) + 7*B*c*d**3*n**2*x*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) + 12*B*
c*d**3*n*x*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) + B*d**4*n**3*x**2*(c
+ d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) + 8*B*d**4*n**2*x**2*(c + d*x)**n/(d
**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) + 19*B*d**4*n*x**2*(c + d*x)**n/(d**4*n**4 + 10*
d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) + 12*B*d**4*x**2*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d
**4*n**2 + 50*d**4*n + 24*d**4) + 2*C*c**3*d*n*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4
*n + 24*d**4) + 8*C*c**3*d*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) - 2*C*
c**2*d**2*n**2*x*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) - 8*C*c**2*d**2*
n*x*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) + C*c*d**3*n**3*x**2*(c + d*x
)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) + 5*C*c*d**3*n**2*x**2*(c + d*x)**n/(d**4
*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) + 4*C*c*d**3*n*x**2*(c + d*x)**n/(d**4*n**4 + 10*d*
*4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) + C*d**4*n**3*x**3*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d
**4*n**2 + 50*d**4*n + 24*d**4) + 7*C*d**4*n**2*x**3*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 5
0*d**4*n + 24*d**4) + 14*C*d**4*n*x**3*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*
d**4) + 8*C*d**4*x**3*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) - 6*D*c**4*
(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) + 6*D*c**3*d*n*x*(c + d*x)**n/(d*
*4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) - 3*D*c**2*d**2*n**2*x**2*(c + d*x)**n/(d**4*n**4
+ 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) - 3*D*c**2*d**2*n*x**2*(c + d*x)**n/(d**4*n**4 + 10*d**4
*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) + D*c*d**3*n**3*x**3*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d
**4*n**2 + 50*d**4*n + 24*d**4) + 3*D*c*d**3*n**2*x**3*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 +
50*d**4*n + 24*d**4) + 2*D*c*d**3*n*x**3*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n +
24*d**4) + D*d**4*n**3*x**4*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) + 6*D
*d**4*n**2*x**4*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) + 11*D*d**4*n*x**
4*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) + 6*D*d**4*x**4*(c + d*x)**n/(d
**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4), True))
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