3.924 \(\int x (a+b x)^n (c+d x)^2 \, dx\)
Optimal. Leaf size=114 \[ -\frac {a (b c-a d)^2 (a+b x)^{n+1}}{b^4 (n+1)}+\frac {(b c-3 a d) (b c-a d) (a+b x)^{n+2}}{b^4 (n+2)}+\frac {d (2 b c-3 a d) (a+b x)^{n+3}}{b^4 (n+3)}+\frac {d^2 (a+b x)^{n+4}}{b^4 (n+4)} \]
[Out]
-a*(-a*d+b*c)^2*(b*x+a)^(1+n)/b^4/(1+n)+(-3*a*d+b*c)*(-a*d+b*c)*(b*x+a)^(2+n)/b^4/(2+n)+d*(-3*a*d+2*b*c)*(b*x+
a)^(3+n)/b^4/(3+n)+d^2*(b*x+a)^(4+n)/b^4/(4+n)
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Rubi [A] time = 0.06, antiderivative size = 114, normalized size of antiderivative = 1.00,
number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used =
{77} \[ -\frac {a (b c-a d)^2 (a+b x)^{n+1}}{b^4 (n+1)}+\frac {(b c-3 a d) (b c-a d) (a+b x)^{n+2}}{b^4 (n+2)}+\frac {d (2 b c-3 a d) (a+b x)^{n+3}}{b^4 (n+3)}+\frac {d^2 (a+b x)^{n+4}}{b^4 (n+4)} \]
Antiderivative was successfully verified.
[In]
Int[x*(a + b*x)^n*(c + d*x)^2,x]
[Out]
-((a*(b*c - a*d)^2*(a + b*x)^(1 + n))/(b^4*(1 + n))) + ((b*c - 3*a*d)*(b*c - a*d)*(a + b*x)^(2 + n))/(b^4*(2 +
n)) + (d*(2*b*c - 3*a*d)*(a + b*x)^(3 + n))/(b^4*(3 + n)) + (d^2*(a + b*x)^(4 + n))/(b^4*(4 + n))
Rule 77
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Rubi steps
\begin {align*} \int x (a+b x)^n (c+d x)^2 \, dx &=\int \left (-\frac {a (-b c+a d)^2 (a+b x)^n}{b^3}+\frac {(b c-3 a d) (b c-a d) (a+b x)^{1+n}}{b^3}+\frac {d (2 b c-3 a d) (a+b x)^{2+n}}{b^3}+\frac {d^2 (a+b x)^{3+n}}{b^3}\right ) \, dx\\ &=-\frac {a (b c-a d)^2 (a+b x)^{1+n}}{b^4 (1+n)}+\frac {(b c-3 a d) (b c-a d) (a+b x)^{2+n}}{b^4 (2+n)}+\frac {d (2 b c-3 a d) (a+b x)^{3+n}}{b^4 (3+n)}+\frac {d^2 (a+b x)^{4+n}}{b^4 (4+n)}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 98, normalized size = 0.86 \[ \frac {(a+b x)^{n+1} \left (\frac {d (a+b x)^2 (2 b c-3 a d)}{n+3}+\frac {(a+b x) (b c-3 a d) (b c-a d)}{n+2}-\frac {a (b c-a d)^2}{n+1}+\frac {d^2 (a+b x)^3}{n+4}\right )}{b^4} \]
Antiderivative was successfully verified.
[In]
Integrate[x*(a + b*x)^n*(c + d*x)^2,x]
[Out]
((a + b*x)^(1 + n)*(-((a*(b*c - a*d)^2)/(1 + n)) + ((b*c - 3*a*d)*(b*c - a*d)*(a + b*x))/(2 + n) + (d*(2*b*c -
3*a*d)*(a + b*x)^2)/(3 + n) + (d^2*(a + b*x)^3)/(4 + n)))/b^4
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fricas [B] time = 0.77, size = 393, normalized size = 3.45 \[ -\frac {{\left (a^{2} b^{2} c^{2} n^{2} + 12 \, a^{2} b^{2} c^{2} - 16 \, a^{3} b c d + 6 \, a^{4} d^{2} - {\left (b^{4} d^{2} n^{3} + 6 \, b^{4} d^{2} n^{2} + 11 \, b^{4} d^{2} n + 6 \, b^{4} d^{2}\right )} x^{4} - {\left (16 \, b^{4} c d + {\left (2 \, b^{4} c d + a b^{3} d^{2}\right )} n^{3} + {\left (14 \, b^{4} c d + 3 \, a b^{3} d^{2}\right )} n^{2} + 2 \, {\left (14 \, b^{4} c d + a b^{3} d^{2}\right )} n\right )} x^{3} - {\left (12 \, b^{4} c^{2} + {\left (b^{4} c^{2} + 2 \, a b^{3} c d\right )} n^{3} + {\left (8 \, b^{4} c^{2} + 10 \, a b^{3} c d - 3 \, a^{2} b^{2} d^{2}\right )} n^{2} + {\left (19 \, b^{4} c^{2} + 8 \, a b^{3} c d - 3 \, a^{2} b^{2} d^{2}\right )} n\right )} x^{2} + {\left (7 \, a^{2} b^{2} c^{2} - 4 \, a^{3} b c d\right )} n - {\left (a b^{3} c^{2} n^{3} + {\left (7 \, a b^{3} c^{2} - 4 \, a^{2} b^{2} c d\right )} n^{2} + 2 \, {\left (6 \, a b^{3} c^{2} - 8 \, a^{2} b^{2} c d + 3 \, a^{3} b d^{2}\right )} n\right )} x\right )} {\left (b x + a\right )}^{n}}{b^{4} n^{4} + 10 \, b^{4} n^{3} + 35 \, b^{4} n^{2} + 50 \, b^{4} n + 24 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*(b*x+a)^n*(d*x+c)^2,x, algorithm="fricas")
[Out]
-(a^2*b^2*c^2*n^2 + 12*a^2*b^2*c^2 - 16*a^3*b*c*d + 6*a^4*d^2 - (b^4*d^2*n^3 + 6*b^4*d^2*n^2 + 11*b^4*d^2*n +
6*b^4*d^2)*x^4 - (16*b^4*c*d + (2*b^4*c*d + a*b^3*d^2)*n^3 + (14*b^4*c*d + 3*a*b^3*d^2)*n^2 + 2*(14*b^4*c*d +
a*b^3*d^2)*n)*x^3 - (12*b^4*c^2 + (b^4*c^2 + 2*a*b^3*c*d)*n^3 + (8*b^4*c^2 + 10*a*b^3*c*d - 3*a^2*b^2*d^2)*n^2
+ (19*b^4*c^2 + 8*a*b^3*c*d - 3*a^2*b^2*d^2)*n)*x^2 + (7*a^2*b^2*c^2 - 4*a^3*b*c*d)*n - (a*b^3*c^2*n^3 + (7*a
*b^3*c^2 - 4*a^2*b^2*c*d)*n^2 + 2*(6*a*b^3*c^2 - 8*a^2*b^2*c*d + 3*a^3*b*d^2)*n)*x)*(b*x + a)^n/(b^4*n^4 + 10*
b^4*n^3 + 35*b^4*n^2 + 50*b^4*n + 24*b^4)
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giac [B] time = 1.23, size = 659, normalized size = 5.78 \[ \frac {{\left (b x + a\right )}^{n} b^{4} d^{2} n^{3} x^{4} + 2 \, {\left (b x + a\right )}^{n} b^{4} c d n^{3} x^{3} + {\left (b x + a\right )}^{n} a b^{3} d^{2} n^{3} x^{3} + 6 \, {\left (b x + a\right )}^{n} b^{4} d^{2} n^{2} x^{4} + {\left (b x + a\right )}^{n} b^{4} c^{2} n^{3} x^{2} + 2 \, {\left (b x + a\right )}^{n} a b^{3} c d n^{3} x^{2} + 14 \, {\left (b x + a\right )}^{n} b^{4} c d n^{2} x^{3} + 3 \, {\left (b x + a\right )}^{n} a b^{3} d^{2} n^{2} x^{3} + 11 \, {\left (b x + a\right )}^{n} b^{4} d^{2} n x^{4} + {\left (b x + a\right )}^{n} a b^{3} c^{2} n^{3} x + 8 \, {\left (b x + a\right )}^{n} b^{4} c^{2} n^{2} x^{2} + 10 \, {\left (b x + a\right )}^{n} a b^{3} c d n^{2} x^{2} - 3 \, {\left (b x + a\right )}^{n} a^{2} b^{2} d^{2} n^{2} x^{2} + 28 \, {\left (b x + a\right )}^{n} b^{4} c d n x^{3} + 2 \, {\left (b x + a\right )}^{n} a b^{3} d^{2} n x^{3} + 6 \, {\left (b x + a\right )}^{n} b^{4} d^{2} x^{4} + 7 \, {\left (b x + a\right )}^{n} a b^{3} c^{2} n^{2} x - 4 \, {\left (b x + a\right )}^{n} a^{2} b^{2} c d n^{2} x + 19 \, {\left (b x + a\right )}^{n} b^{4} c^{2} n x^{2} + 8 \, {\left (b x + a\right )}^{n} a b^{3} c d n x^{2} - 3 \, {\left (b x + a\right )}^{n} a^{2} b^{2} d^{2} n x^{2} + 16 \, {\left (b x + a\right )}^{n} b^{4} c d x^{3} - {\left (b x + a\right )}^{n} a^{2} b^{2} c^{2} n^{2} + 12 \, {\left (b x + a\right )}^{n} a b^{3} c^{2} n x - 16 \, {\left (b x + a\right )}^{n} a^{2} b^{2} c d n x + 6 \, {\left (b x + a\right )}^{n} a^{3} b d^{2} n x + 12 \, {\left (b x + a\right )}^{n} b^{4} c^{2} x^{2} - 7 \, {\left (b x + a\right )}^{n} a^{2} b^{2} c^{2} n + 4 \, {\left (b x + a\right )}^{n} a^{3} b c d n - 12 \, {\left (b x + a\right )}^{n} a^{2} b^{2} c^{2} + 16 \, {\left (b x + a\right )}^{n} a^{3} b c d - 6 \, {\left (b x + a\right )}^{n} a^{4} d^{2}}{b^{4} n^{4} + 10 \, b^{4} n^{3} + 35 \, b^{4} n^{2} + 50 \, b^{4} n + 24 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*(b*x+a)^n*(d*x+c)^2,x, algorithm="giac")
[Out]
((b*x + a)^n*b^4*d^2*n^3*x^4 + 2*(b*x + a)^n*b^4*c*d*n^3*x^3 + (b*x + a)^n*a*b^3*d^2*n^3*x^3 + 6*(b*x + a)^n*b
^4*d^2*n^2*x^4 + (b*x + a)^n*b^4*c^2*n^3*x^2 + 2*(b*x + a)^n*a*b^3*c*d*n^3*x^2 + 14*(b*x + a)^n*b^4*c*d*n^2*x^
3 + 3*(b*x + a)^n*a*b^3*d^2*n^2*x^3 + 11*(b*x + a)^n*b^4*d^2*n*x^4 + (b*x + a)^n*a*b^3*c^2*n^3*x + 8*(b*x + a)
^n*b^4*c^2*n^2*x^2 + 10*(b*x + a)^n*a*b^3*c*d*n^2*x^2 - 3*(b*x + a)^n*a^2*b^2*d^2*n^2*x^2 + 28*(b*x + a)^n*b^4
*c*d*n*x^3 + 2*(b*x + a)^n*a*b^3*d^2*n*x^3 + 6*(b*x + a)^n*b^4*d^2*x^4 + 7*(b*x + a)^n*a*b^3*c^2*n^2*x - 4*(b*
x + a)^n*a^2*b^2*c*d*n^2*x + 19*(b*x + a)^n*b^4*c^2*n*x^2 + 8*(b*x + a)^n*a*b^3*c*d*n*x^2 - 3*(b*x + a)^n*a^2*
b^2*d^2*n*x^2 + 16*(b*x + a)^n*b^4*c*d*x^3 - (b*x + a)^n*a^2*b^2*c^2*n^2 + 12*(b*x + a)^n*a*b^3*c^2*n*x - 16*(
b*x + a)^n*a^2*b^2*c*d*n*x + 6*(b*x + a)^n*a^3*b*d^2*n*x + 12*(b*x + a)^n*b^4*c^2*x^2 - 7*(b*x + a)^n*a^2*b^2*
c^2*n + 4*(b*x + a)^n*a^3*b*c*d*n - 12*(b*x + a)^n*a^2*b^2*c^2 + 16*(b*x + a)^n*a^3*b*c*d - 6*(b*x + a)^n*a^4*
d^2)/(b^4*n^4 + 10*b^4*n^3 + 35*b^4*n^2 + 50*b^4*n + 24*b^4)
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maple [B] time = 0.01, size = 324, normalized size = 2.84 \[ -\frac {\left (-b^{3} d^{2} n^{3} x^{3}-2 b^{3} c d \,n^{3} x^{2}-6 b^{3} d^{2} n^{2} x^{3}+3 a \,b^{2} d^{2} n^{2} x^{2}-b^{3} c^{2} n^{3} x -14 b^{3} c d \,n^{2} x^{2}-11 b^{3} d^{2} n \,x^{3}+4 a \,b^{2} c d \,n^{2} x +9 a \,b^{2} d^{2} n \,x^{2}-8 b^{3} c^{2} n^{2} x -28 b^{3} c d n \,x^{2}-6 d^{2} x^{3} b^{3}-6 a^{2} b \,d^{2} n x +a \,b^{2} c^{2} n^{2}+20 a \,b^{2} c d n x +6 a \,b^{2} d^{2} x^{2}-19 b^{3} c^{2} n x -16 b^{3} c d \,x^{2}-4 a^{2} b c d n -6 a^{2} b \,d^{2} x +7 a \,b^{2} c^{2} n +16 a \,b^{2} c d x -12 b^{3} c^{2} x +6 a^{3} d^{2}-16 a^{2} b c d +12 a \,b^{2} c^{2}\right ) \left (b x +a \right )^{n +1}}{\left (n^{4}+10 n^{3}+35 n^{2}+50 n +24\right ) b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x*(b*x+a)^n*(d*x+c)^2,x)
[Out]
-(b*x+a)^(n+1)*(-b^3*d^2*n^3*x^3-2*b^3*c*d*n^3*x^2-6*b^3*d^2*n^2*x^3+3*a*b^2*d^2*n^2*x^2-b^3*c^2*n^3*x-14*b^3*
c*d*n^2*x^2-11*b^3*d^2*n*x^3+4*a*b^2*c*d*n^2*x+9*a*b^2*d^2*n*x^2-8*b^3*c^2*n^2*x-28*b^3*c*d*n*x^2-6*b^3*d^2*x^
3-6*a^2*b*d^2*n*x+a*b^2*c^2*n^2+20*a*b^2*c*d*n*x+6*a*b^2*d^2*x^2-19*b^3*c^2*n*x-16*b^3*c*d*x^2-4*a^2*b*c*d*n-6
*a^2*b*d^2*x+7*a*b^2*c^2*n+16*a*b^2*c*d*x-12*b^3*c^2*x+6*a^3*d^2-16*a^2*b*c*d+12*a*b^2*c^2)/b^4/(n^4+10*n^3+35
*n^2+50*n+24)
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maxima [A] time = 0.65, size = 221, normalized size = 1.94 \[ \frac {{\left (b^{2} {\left (n + 1\right )} x^{2} + a b n x - a^{2}\right )} {\left (b x + a\right )}^{n} c^{2}}{{\left (n^{2} + 3 \, n + 2\right )} b^{2}} + \frac {2 \, {\left ({\left (n^{2} + 3 \, n + 2\right )} b^{3} x^{3} + {\left (n^{2} + n\right )} a b^{2} x^{2} - 2 \, a^{2} b n x + 2 \, a^{3}\right )} {\left (b x + a\right )}^{n} c d}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{3}} + \frac {{\left ({\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{4} x^{4} + {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a b^{3} x^{3} - 3 \, {\left (n^{2} + n\right )} a^{2} b^{2} x^{2} + 6 \, a^{3} b n x - 6 \, a^{4}\right )} {\left (b x + a\right )}^{n} d^{2}}{{\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*(b*x+a)^n*(d*x+c)^2,x, algorithm="maxima")
[Out]
(b^2*(n + 1)*x^2 + a*b*n*x - a^2)*(b*x + a)^n*c^2/((n^2 + 3*n + 2)*b^2) + 2*((n^2 + 3*n + 2)*b^3*x^3 + (n^2 +
n)*a*b^2*x^2 - 2*a^2*b*n*x + 2*a^3)*(b*x + a)^n*c*d/((n^3 + 6*n^2 + 11*n + 6)*b^3) + ((n^3 + 6*n^2 + 11*n + 6)
*b^4*x^4 + (n^3 + 3*n^2 + 2*n)*a*b^3*x^3 - 3*(n^2 + n)*a^2*b^2*x^2 + 6*a^3*b*n*x - 6*a^4)*(b*x + a)^n*d^2/((n^
4 + 10*n^3 + 35*n^2 + 50*n + 24)*b^4)
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mupad [B] time = 1.22, size = 335, normalized size = 2.94 \[ {\left (a+b\,x\right )}^n\,\left (\frac {d^2\,x^4\,\left (n^3+6\,n^2+11\,n+6\right )}{n^4+10\,n^3+35\,n^2+50\,n+24}-\frac {a^2\,\left (6\,a^2\,d^2-4\,a\,b\,c\,d\,n-16\,a\,b\,c\,d+b^2\,c^2\,n^2+7\,b^2\,c^2\,n+12\,b^2\,c^2\right )}{b^4\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {x^2\,\left (n+1\right )\,\left (-3\,a^2\,d^2\,n+2\,a\,b\,c\,d\,n^2+8\,a\,b\,c\,d\,n+b^2\,c^2\,n^2+7\,b^2\,c^2\,n+12\,b^2\,c^2\right )}{b^2\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {a\,n\,x\,\left (6\,a^2\,d^2-4\,a\,b\,c\,d\,n-16\,a\,b\,c\,d+b^2\,c^2\,n^2+7\,b^2\,c^2\,n+12\,b^2\,c^2\right )}{b^3\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {d\,x^3\,\left (8\,b\,c+a\,d\,n+2\,b\,c\,n\right )\,\left (n^2+3\,n+2\right )}{b\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x*(a + b*x)^n*(c + d*x)^2,x)
[Out]
(a + b*x)^n*((d^2*x^4*(11*n + 6*n^2 + n^3 + 6))/(50*n + 35*n^2 + 10*n^3 + n^4 + 24) - (a^2*(6*a^2*d^2 + 12*b^2
*c^2 + 7*b^2*c^2*n + b^2*c^2*n^2 - 16*a*b*c*d - 4*a*b*c*d*n))/(b^4*(50*n + 35*n^2 + 10*n^3 + n^4 + 24)) + (x^2
*(n + 1)*(12*b^2*c^2 - 3*a^2*d^2*n + 7*b^2*c^2*n + b^2*c^2*n^2 + 8*a*b*c*d*n + 2*a*b*c*d*n^2))/(b^2*(50*n + 35
*n^2 + 10*n^3 + n^4 + 24)) + (a*n*x*(6*a^2*d^2 + 12*b^2*c^2 + 7*b^2*c^2*n + b^2*c^2*n^2 - 16*a*b*c*d - 4*a*b*c
*d*n))/(b^3*(50*n + 35*n^2 + 10*n^3 + n^4 + 24)) + (d*x^3*(8*b*c + a*d*n + 2*b*c*n)*(3*n + n^2 + 2))/(b*(50*n
+ 35*n^2 + 10*n^3 + n^4 + 24)))
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sympy [A] time = 4.16, size = 3412, normalized size = 29.93 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*(b*x+a)**n*(d*x+c)**2,x)
[Out]
Piecewise((a**n*(c**2*x**2/2 + 2*c*d*x**3/3 + d**2*x**4/4), Eq(b, 0)), (6*a**3*d**2*log(a/b + x)/(6*a**3*b**4
+ 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) + 11*a**3*d**2/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2
+ 6*b**7*x**3) - 4*a**2*b*c*d/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) + 18*a**2*b*d**2*
x*log(a/b + x)/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) + 27*a**2*b*d**2*x/(6*a**3*b**4 +
18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) - a*b**2*c**2/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 +
6*b**7*x**3) - 12*a*b**2*c*d*x/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) + 18*a*b**2*d**2
*x**2*log(a/b + x)/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) + 18*a*b**2*d**2*x**2/(6*a**3
*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) - 3*b**3*c**2*x/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b*
*6*x**2 + 6*b**7*x**3) - 12*b**3*c*d*x**2/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) + 6*b*
*3*d**2*x**3*log(a/b + x)/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3), Eq(n, -4)), (-6*a**3*
d**2*log(a/b + x)/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) - 9*a**3*d**2/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x*
*2) + 4*a**2*b*c*d*log(a/b + x)/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) + 6*a**2*b*c*d/(2*a**2*b**4 + 4*a*b**
5*x + 2*b**6*x**2) - 12*a**2*b*d**2*x*log(a/b + x)/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) - 12*a**2*b*d**2*x
/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) - a*b**2*c**2/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) + 8*a*b**2*c*
d*x*log(a/b + x)/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) + 8*a*b**2*c*d*x/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*
x**2) - 6*a*b**2*d**2*x**2*log(a/b + x)/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) - 2*b**3*c**2*x/(2*a**2*b**4
+ 4*a*b**5*x + 2*b**6*x**2) + 4*b**3*c*d*x**2*log(a/b + x)/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) + 2*b**3*d
**2*x**3/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2), Eq(n, -3)), (6*a**3*d**2*log(a/b + x)/(2*a*b**4 + 2*b**5*x)
+ 6*a**3*d**2/(2*a*b**4 + 2*b**5*x) - 8*a**2*b*c*d*log(a/b + x)/(2*a*b**4 + 2*b**5*x) - 8*a**2*b*c*d/(2*a*b**
4 + 2*b**5*x) + 6*a**2*b*d**2*x*log(a/b + x)/(2*a*b**4 + 2*b**5*x) + 2*a*b**2*c**2*log(a/b + x)/(2*a*b**4 + 2*
b**5*x) + 2*a*b**2*c**2/(2*a*b**4 + 2*b**5*x) - 8*a*b**2*c*d*x*log(a/b + x)/(2*a*b**4 + 2*b**5*x) - 3*a*b**2*d
**2*x**2/(2*a*b**4 + 2*b**5*x) + 2*b**3*c**2*x*log(a/b + x)/(2*a*b**4 + 2*b**5*x) + 4*b**3*c*d*x**2/(2*a*b**4
+ 2*b**5*x) + b**3*d**2*x**3/(2*a*b**4 + 2*b**5*x), Eq(n, -2)), (-a**3*d**2*log(a/b + x)/b**4 + 2*a**2*c*d*log
(a/b + x)/b**3 + a**2*d**2*x/b**3 - a*c**2*log(a/b + x)/b**2 - 2*a*c*d*x/b**2 - a*d**2*x**2/(2*b**2) + c**2*x/
b + c*d*x**2/b + d**2*x**3/(3*b), Eq(n, -1)), (-6*a**4*d**2*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n
**2 + 50*b**4*n + 24*b**4) + 4*a**3*b*c*d*n*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n
+ 24*b**4) + 16*a**3*b*c*d*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 6*a*
*3*b*d**2*n*x*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) - a**2*b**2*c**2*n*
*2*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) - 7*a**2*b**2*c**2*n*(a + b*x)
**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) - 12*a**2*b**2*c**2*(a + b*x)**n/(b**4*n**
4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) - 4*a**2*b**2*c*d*n**2*x*(a + b*x)**n/(b**4*n**4 + 10*b
**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) - 16*a**2*b**2*c*d*n*x*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 +
35*b**4*n**2 + 50*b**4*n + 24*b**4) - 3*a**2*b**2*d**2*n**2*x**2*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*
b**4*n**2 + 50*b**4*n + 24*b**4) - 3*a**2*b**2*d**2*n*x**2*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n*
*2 + 50*b**4*n + 24*b**4) + a*b**3*c**2*n**3*x*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4
*n + 24*b**4) + 7*a*b**3*c**2*n**2*x*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b*
*4) + 12*a*b**3*c**2*n*x*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 2*a*b*
*3*c*d*n**3*x**2*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 10*a*b**3*c*d*
n**2*x**2*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 8*a*b**3*c*d*n*x**2*(
a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + a*b**3*d**2*n**3*x**3*(a + b*x)*
*n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 3*a*b**3*d**2*n**2*x**3*(a + b*x)**n/(b**
4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 2*a*b**3*d**2*n*x**3*(a + b*x)**n/(b**4*n**4 + 1
0*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + b**4*c**2*n**3*x**2*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3
+ 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 8*b**4*c**2*n**2*x**2*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**
4*n**2 + 50*b**4*n + 24*b**4) + 19*b**4*c**2*n*x**2*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50
*b**4*n + 24*b**4) + 12*b**4*c**2*x**2*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*
b**4) + 2*b**4*c*d*n**3*x**3*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 14
*b**4*c*d*n**2*x**3*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 28*b**4*c*d
*n*x**3*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 16*b**4*c*d*x**3*(a + b
*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + b**4*d**2*n**3*x**4*(a + b*x)**n/(b**
4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 6*b**4*d**2*n**2*x**4*(a + b*x)**n/(b**4*n**4 +
10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 11*b**4*d**2*n*x**4*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**
3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 6*b**4*d**2*x**4*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n*
*2 + 50*b**4*n + 24*b**4), True))
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