3.3.62 \(\int x^2 (a+b x)^n (c+d x^2) \, dx\)
Optimal. Leaf size=135 \[ \frac {a^2 \left (a^2 d+b^2 c\right ) (a+b x)^{n+1}}{b^5 (n+1)}-\frac {2 a \left (2 a^2 d+b^2 c\right ) (a+b x)^{n+2}}{b^5 (n+2)}+\frac {\left (6 a^2 d+b^2 c\right ) (a+b x)^{n+3}}{b^5 (n+3)}-\frac {4 a d (a+b x)^{n+4}}{b^5 (n+4)}+\frac {d (a+b x)^{n+5}}{b^5 (n+5)} \]
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Rubi [A] time = 0.08, antiderivative size = 135, normalized size of antiderivative = 1.00,
number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used =
{948} \begin {gather*} \frac {a^2 \left (a^2 d+b^2 c\right ) (a+b x)^{n+1}}{b^5 (n+1)}-\frac {2 a \left (2 a^2 d+b^2 c\right ) (a+b x)^{n+2}}{b^5 (n+2)}+\frac {\left (6 a^2 d+b^2 c\right ) (a+b x)^{n+3}}{b^5 (n+3)}-\frac {4 a d (a+b x)^{n+4}}{b^5 (n+4)}+\frac {d (a+b x)^{n+5}}{b^5 (n+5)} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[x^2*(a + b*x)^n*(c + d*x^2),x]
[Out]
(a^2*(b^2*c + a^2*d)*(a + b*x)^(1 + n))/(b^5*(1 + n)) - (2*a*(b^2*c + 2*a^2*d)*(a + b*x)^(2 + n))/(b^5*(2 + n)
) + ((b^2*c + 6*a^2*d)*(a + b*x)^(3 + n))/(b^5*(3 + n)) - (4*a*d*(a + b*x)^(4 + n))/(b^5*(4 + n)) + (d*(a + b*
x)^(5 + n))/(b^5*(5 + n))
Rule 948
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIn
tegrand[(d + e*x)^m*(f + g*x)^n*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] &&
NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0] && (IGtQ[m, 0] || (EqQ[m, -2] && EqQ[p, 1] && EqQ[d, 0]))
Rubi steps
\begin {align*} \int x^2 (a+b x)^n \left (c+d x^2\right ) \, dx &=\int \left (\frac {\left (a^2 b^2 c+a^4 d\right ) (a+b x)^n}{b^4}-\frac {2 \left (a b^2 c+2 a^3 d\right ) (a+b x)^{1+n}}{b^4}+\frac {\left (b^2 c+6 a^2 d\right ) (a+b x)^{2+n}}{b^4}-\frac {4 a d (a+b x)^{3+n}}{b^4}+\frac {d (a+b x)^{4+n}}{b^4}\right ) \, dx\\ &=\frac {a^2 \left (b^2 c+a^2 d\right ) (a+b x)^{1+n}}{b^5 (1+n)}-\frac {2 a \left (b^2 c+2 a^2 d\right ) (a+b x)^{2+n}}{b^5 (2+n)}+\frac {\left (b^2 c+6 a^2 d\right ) (a+b x)^{3+n}}{b^5 (3+n)}-\frac {4 a d (a+b x)^{4+n}}{b^5 (4+n)}+\frac {d (a+b x)^{5+n}}{b^5 (5+n)}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 114, normalized size = 0.84 \begin {gather*} \frac {(a+b x)^{n+1} \left (\frac {(a+b x)^2 \left (6 a^2 d+b^2 c\right )}{n+3}-\frac {2 a (a+b x) \left (2 a^2 d+b^2 c\right )}{n+2}+\frac {a^4 d+a^2 b^2 c}{n+1}+\frac {d (a+b x)^4}{n+5}-\frac {4 a d (a+b x)^3}{n+4}\right )}{b^5} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[x^2*(a + b*x)^n*(c + d*x^2),x]
[Out]
((a + b*x)^(1 + n)*((a^2*b^2*c + a^4*d)/(1 + n) - (2*a*(b^2*c + 2*a^2*d)*(a + b*x))/(2 + n) + ((b^2*c + 6*a^2*
d)*(a + b*x)^2)/(3 + n) - (4*a*d*(a + b*x)^3)/(4 + n) + (d*(a + b*x)^4)/(5 + n)))/b^5
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IntegrateAlgebraic [F] time = 0.06, size = 0, normalized size = 0.00 \begin {gather*} \int x^2 (a+b x)^n \left (c+d x^2\right ) \, dx \end {gather*}
Verification is not applicable to the result.
[In]
IntegrateAlgebraic[x^2*(a + b*x)^n*(c + d*x^2),x]
[Out]
Defer[IntegrateAlgebraic][x^2*(a + b*x)^n*(c + d*x^2), x]
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fricas [B] time = 0.42, size = 368, normalized size = 2.73 \begin {gather*} \frac {{\left (2 \, a^{3} b^{2} c n^{2} + 18 \, a^{3} b^{2} c n + 40 \, a^{3} b^{2} c + 24 \, a^{5} d + {\left (b^{5} d n^{4} + 10 \, b^{5} d n^{3} + 35 \, b^{5} d n^{2} + 50 \, b^{5} d n + 24 \, b^{5} d\right )} x^{5} + {\left (a b^{4} d n^{4} + 6 \, a b^{4} d n^{3} + 11 \, a b^{4} d n^{2} + 6 \, a b^{4} d n\right )} x^{4} + {\left (b^{5} c n^{4} + 40 \, b^{5} c + 4 \, {\left (3 \, b^{5} c - a^{2} b^{3} d\right )} n^{3} + {\left (49 \, b^{5} c - 12 \, a^{2} b^{3} d\right )} n^{2} + 2 \, {\left (39 \, b^{5} c - 4 \, a^{2} b^{3} d\right )} n\right )} x^{3} + {\left (a b^{4} c n^{4} + 10 \, a b^{4} c n^{3} + {\left (29 \, a b^{4} c + 12 \, a^{3} b^{2} d\right )} n^{2} + 4 \, {\left (5 \, a b^{4} c + 3 \, a^{3} b^{2} d\right )} n\right )} x^{2} - 2 \, {\left (a^{2} b^{3} c n^{3} + 9 \, a^{2} b^{3} c n^{2} + 4 \, {\left (5 \, a^{2} b^{3} c + 3 \, a^{4} b d\right )} n\right )} x\right )} {\left (b x + a\right )}^{n}}{b^{5} n^{5} + 15 \, b^{5} n^{4} + 85 \, b^{5} n^{3} + 225 \, b^{5} n^{2} + 274 \, b^{5} n + 120 \, b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2*(b*x+a)^n*(d*x^2+c),x, algorithm="fricas")
[Out]
(2*a^3*b^2*c*n^2 + 18*a^3*b^2*c*n + 40*a^3*b^2*c + 24*a^5*d + (b^5*d*n^4 + 10*b^5*d*n^3 + 35*b^5*d*n^2 + 50*b^
5*d*n + 24*b^5*d)*x^5 + (a*b^4*d*n^4 + 6*a*b^4*d*n^3 + 11*a*b^4*d*n^2 + 6*a*b^4*d*n)*x^4 + (b^5*c*n^4 + 40*b^5
*c + 4*(3*b^5*c - a^2*b^3*d)*n^3 + (49*b^5*c - 12*a^2*b^3*d)*n^2 + 2*(39*b^5*c - 4*a^2*b^3*d)*n)*x^3 + (a*b^4*
c*n^4 + 10*a*b^4*c*n^3 + (29*a*b^4*c + 12*a^3*b^2*d)*n^2 + 4*(5*a*b^4*c + 3*a^3*b^2*d)*n)*x^2 - 2*(a^2*b^3*c*n
^3 + 9*a^2*b^3*c*n^2 + 4*(5*a^2*b^3*c + 3*a^4*b*d)*n)*x)*(b*x + a)^n/(b^5*n^5 + 15*b^5*n^4 + 85*b^5*n^3 + 225*
b^5*n^2 + 274*b^5*n + 120*b^5)
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giac [B] time = 0.19, size = 624, normalized size = 4.62 \begin {gather*} \frac {{\left (b x + a\right )}^{n} b^{5} d n^{4} x^{5} + {\left (b x + a\right )}^{n} a b^{4} d n^{4} x^{4} + 10 \, {\left (b x + a\right )}^{n} b^{5} d n^{3} x^{5} + {\left (b x + a\right )}^{n} b^{5} c n^{4} x^{3} + 6 \, {\left (b x + a\right )}^{n} a b^{4} d n^{3} x^{4} + 35 \, {\left (b x + a\right )}^{n} b^{5} d n^{2} x^{5} + {\left (b x + a\right )}^{n} a b^{4} c n^{4} x^{2} + 12 \, {\left (b x + a\right )}^{n} b^{5} c n^{3} x^{3} - 4 \, {\left (b x + a\right )}^{n} a^{2} b^{3} d n^{3} x^{3} + 11 \, {\left (b x + a\right )}^{n} a b^{4} d n^{2} x^{4} + 50 \, {\left (b x + a\right )}^{n} b^{5} d n x^{5} + 10 \, {\left (b x + a\right )}^{n} a b^{4} c n^{3} x^{2} + 49 \, {\left (b x + a\right )}^{n} b^{5} c n^{2} x^{3} - 12 \, {\left (b x + a\right )}^{n} a^{2} b^{3} d n^{2} x^{3} + 6 \, {\left (b x + a\right )}^{n} a b^{4} d n x^{4} + 24 \, {\left (b x + a\right )}^{n} b^{5} d x^{5} - 2 \, {\left (b x + a\right )}^{n} a^{2} b^{3} c n^{3} x + 29 \, {\left (b x + a\right )}^{n} a b^{4} c n^{2} x^{2} + 12 \, {\left (b x + a\right )}^{n} a^{3} b^{2} d n^{2} x^{2} + 78 \, {\left (b x + a\right )}^{n} b^{5} c n x^{3} - 8 \, {\left (b x + a\right )}^{n} a^{2} b^{3} d n x^{3} - 18 \, {\left (b x + a\right )}^{n} a^{2} b^{3} c n^{2} x + 20 \, {\left (b x + a\right )}^{n} a b^{4} c n x^{2} + 12 \, {\left (b x + a\right )}^{n} a^{3} b^{2} d n x^{2} + 40 \, {\left (b x + a\right )}^{n} b^{5} c x^{3} + 2 \, {\left (b x + a\right )}^{n} a^{3} b^{2} c n^{2} - 40 \, {\left (b x + a\right )}^{n} a^{2} b^{3} c n x - 24 \, {\left (b x + a\right )}^{n} a^{4} b d n x + 18 \, {\left (b x + a\right )}^{n} a^{3} b^{2} c n + 40 \, {\left (b x + a\right )}^{n} a^{3} b^{2} c + 24 \, {\left (b x + a\right )}^{n} a^{5} d}{b^{5} n^{5} + 15 \, b^{5} n^{4} + 85 \, b^{5} n^{3} + 225 \, b^{5} n^{2} + 274 \, b^{5} n + 120 \, b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2*(b*x+a)^n*(d*x^2+c),x, algorithm="giac")
[Out]
((b*x + a)^n*b^5*d*n^4*x^5 + (b*x + a)^n*a*b^4*d*n^4*x^4 + 10*(b*x + a)^n*b^5*d*n^3*x^5 + (b*x + a)^n*b^5*c*n^
4*x^3 + 6*(b*x + a)^n*a*b^4*d*n^3*x^4 + 35*(b*x + a)^n*b^5*d*n^2*x^5 + (b*x + a)^n*a*b^4*c*n^4*x^2 + 12*(b*x +
a)^n*b^5*c*n^3*x^3 - 4*(b*x + a)^n*a^2*b^3*d*n^3*x^3 + 11*(b*x + a)^n*a*b^4*d*n^2*x^4 + 50*(b*x + a)^n*b^5*d*
n*x^5 + 10*(b*x + a)^n*a*b^4*c*n^3*x^2 + 49*(b*x + a)^n*b^5*c*n^2*x^3 - 12*(b*x + a)^n*a^2*b^3*d*n^2*x^3 + 6*(
b*x + a)^n*a*b^4*d*n*x^4 + 24*(b*x + a)^n*b^5*d*x^5 - 2*(b*x + a)^n*a^2*b^3*c*n^3*x + 29*(b*x + a)^n*a*b^4*c*n
^2*x^2 + 12*(b*x + a)^n*a^3*b^2*d*n^2*x^2 + 78*(b*x + a)^n*b^5*c*n*x^3 - 8*(b*x + a)^n*a^2*b^3*d*n*x^3 - 18*(b
*x + a)^n*a^2*b^3*c*n^2*x + 20*(b*x + a)^n*a*b^4*c*n*x^2 + 12*(b*x + a)^n*a^3*b^2*d*n*x^2 + 40*(b*x + a)^n*b^5
*c*x^3 + 2*(b*x + a)^n*a^3*b^2*c*n^2 - 40*(b*x + a)^n*a^2*b^3*c*n*x - 24*(b*x + a)^n*a^4*b*d*n*x + 18*(b*x + a
)^n*a^3*b^2*c*n + 40*(b*x + a)^n*a^3*b^2*c + 24*(b*x + a)^n*a^5*d)/(b^5*n^5 + 15*b^5*n^4 + 85*b^5*n^3 + 225*b^
5*n^2 + 274*b^5*n + 120*b^5)
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maple [B] time = 0.01, size = 328, normalized size = 2.43 \begin {gather*} \frac {\left (b^{4} d \,n^{4} x^{4}+10 b^{4} d \,n^{3} x^{4}-4 a \,b^{3} d \,n^{3} x^{3}+b^{4} c \,n^{4} x^{2}+35 b^{4} d \,n^{2} x^{4}-24 a \,b^{3} d \,n^{2} x^{3}+12 b^{4} c \,n^{3} x^{2}+50 b^{4} d n \,x^{4}+12 a^{2} b^{2} d \,n^{2} x^{2}-2 a \,b^{3} c \,n^{3} x -44 a \,b^{3} d n \,x^{3}+49 b^{4} c \,n^{2} x^{2}+24 d \,x^{4} b^{4}+36 a^{2} b^{2} d n \,x^{2}-20 a \,b^{3} c \,n^{2} x -24 d a \,x^{3} b^{3}+78 b^{4} c n \,x^{2}-24 a^{3} b d n x +2 a^{2} b^{2} c \,n^{2}+24 a^{2} b^{2} d \,x^{2}-58 a \,b^{3} c n x +40 b^{4} c \,x^{2}-24 a^{3} b d x +18 a^{2} b^{2} c n -40 a \,b^{3} c x +24 a^{4} d +40 a^{2} b^{2} c \right ) \left (b x +a \right )^{n +1}}{\left (n^{5}+15 n^{4}+85 n^{3}+225 n^{2}+274 n +120\right ) b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^2*(b*x+a)^n*(d*x^2+c),x)
[Out]
(b*x+a)^(1+n)*(b^4*d*n^4*x^4+10*b^4*d*n^3*x^4-4*a*b^3*d*n^3*x^3+b^4*c*n^4*x^2+35*b^4*d*n^2*x^4-24*a*b^3*d*n^2*
x^3+12*b^4*c*n^3*x^2+50*b^4*d*n*x^4+12*a^2*b^2*d*n^2*x^2-2*a*b^3*c*n^3*x-44*a*b^3*d*n*x^3+49*b^4*c*n^2*x^2+24*
b^4*d*x^4+36*a^2*b^2*d*n*x^2-20*a*b^3*c*n^2*x-24*a*b^3*d*x^3+78*b^4*c*n*x^2-24*a^3*b*d*n*x+2*a^2*b^2*c*n^2+24*
a^2*b^2*d*x^2-58*a*b^3*c*n*x+40*b^4*c*x^2-24*a^3*b*d*x+18*a^2*b^2*c*n-40*a*b^3*c*x+24*a^4*d+40*a^2*b^2*c)/b^5/
(n^5+15*n^4+85*n^3+225*n^2+274*n+120)
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maxima [A] time = 0.47, size = 210, normalized size = 1.56 \begin {gather*} \frac {{\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}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{3}} + \frac {{\left ({\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{5} x^{5} + {\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a b^{4} x^{4} - 4 \, {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{2} b^{3} x^{3} + 12 \, {\left (n^{2} + n\right )} a^{3} b^{2} x^{2} - 24 \, a^{4} b n x + 24 \, a^{5}\right )} {\left (b x + a\right )}^{n} d}{{\left (n^{5} + 15 \, n^{4} + 85 \, n^{3} + 225 \, n^{2} + 274 \, n + 120\right )} b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2*(b*x+a)^n*(d*x^2+c),x, algorithm="maxima")
[Out]
((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/((n^3 + 6*n^2 + 11*n + 6)*
b^3) + ((n^4 + 10*n^3 + 35*n^2 + 50*n + 24)*b^5*x^5 + (n^4 + 6*n^3 + 11*n^2 + 6*n)*a*b^4*x^4 - 4*(n^3 + 3*n^2
+ 2*n)*a^2*b^3*x^3 + 12*(n^2 + n)*a^3*b^2*x^2 - 24*a^4*b*n*x + 24*a^5)*(b*x + a)^n*d/((n^5 + 15*n^4 + 85*n^3 +
225*n^2 + 274*n + 120)*b^5)
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mupad [B] time = 2.82, size = 363, normalized size = 2.69 \begin {gather*} {\left (a+b\,x\right )}^n\,\left (\frac {2\,a^3\,\left (12\,d\,a^2+c\,b^2\,n^2+9\,c\,b^2\,n+20\,c\,b^2\right )}{b^5\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}+\frac {d\,x^5\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}{n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120}+\frac {x^3\,\left (n^2+3\,n+2\right )\,\left (-4\,d\,a^2\,n+c\,b^2\,n^2+9\,c\,b^2\,n+20\,c\,b^2\right )}{b^2\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}-\frac {2\,a^2\,n\,x\,\left (12\,d\,a^2+c\,b^2\,n^2+9\,c\,b^2\,n+20\,c\,b^2\right )}{b^4\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}+\frac {a\,n\,x^2\,\left (n+1\right )\,\left (12\,d\,a^2+c\,b^2\,n^2+9\,c\,b^2\,n+20\,c\,b^2\right )}{b^3\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}+\frac {a\,d\,n\,x^4\,\left (n^3+6\,n^2+11\,n+6\right )}{b\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^2*(c + d*x^2)*(a + b*x)^n,x)
[Out]
(a + b*x)^n*((2*a^3*(12*a^2*d + 20*b^2*c + b^2*c*n^2 + 9*b^2*c*n))/(b^5*(274*n + 225*n^2 + 85*n^3 + 15*n^4 + n
^5 + 120)) + (d*x^5*(50*n + 35*n^2 + 10*n^3 + n^4 + 24))/(274*n + 225*n^2 + 85*n^3 + 15*n^4 + n^5 + 120) + (x^
3*(3*n + n^2 + 2)*(20*b^2*c + b^2*c*n^2 - 4*a^2*d*n + 9*b^2*c*n))/(b^2*(274*n + 225*n^2 + 85*n^3 + 15*n^4 + n^
5 + 120)) - (2*a^2*n*x*(12*a^2*d + 20*b^2*c + b^2*c*n^2 + 9*b^2*c*n))/(b^4*(274*n + 225*n^2 + 85*n^3 + 15*n^4
+ n^5 + 120)) + (a*n*x^2*(n + 1)*(12*a^2*d + 20*b^2*c + b^2*c*n^2 + 9*b^2*c*n))/(b^3*(274*n + 225*n^2 + 85*n^3
+ 15*n^4 + n^5 + 120)) + (a*d*n*x^4*(11*n + 6*n^2 + n^3 + 6))/(b*(274*n + 225*n^2 + 85*n^3 + 15*n^4 + n^5 + 1
20)))
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sympy [A] time = 6.70, size = 4134, normalized size = 30.62
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**2*(b*x+a)**n*(d*x**2+c),x)
[Out]
Piecewise((a**n*(c*x**3/3 + d*x**5/5), Eq(b, 0)), (12*a**4*d*log(a/b + x)/(12*a**4*b**5 + 48*a**3*b**6*x + 72*
a**2*b**7*x**2 + 48*a*b**8*x**3 + 12*b**9*x**4) + 25*a**4*d/(12*a**4*b**5 + 48*a**3*b**6*x + 72*a**2*b**7*x**2
+ 48*a*b**8*x**3 + 12*b**9*x**4) + 48*a**3*b*d*x*log(a/b + x)/(12*a**4*b**5 + 48*a**3*b**6*x + 72*a**2*b**7*x
**2 + 48*a*b**8*x**3 + 12*b**9*x**4) + 88*a**3*b*d*x/(12*a**4*b**5 + 48*a**3*b**6*x + 72*a**2*b**7*x**2 + 48*a
*b**8*x**3 + 12*b**9*x**4) - a**2*b**2*c/(12*a**4*b**5 + 48*a**3*b**6*x + 72*a**2*b**7*x**2 + 48*a*b**8*x**3 +
12*b**9*x**4) + 72*a**2*b**2*d*x**2*log(a/b + x)/(12*a**4*b**5 + 48*a**3*b**6*x + 72*a**2*b**7*x**2 + 48*a*b*
*8*x**3 + 12*b**9*x**4) + 108*a**2*b**2*d*x**2/(12*a**4*b**5 + 48*a**3*b**6*x + 72*a**2*b**7*x**2 + 48*a*b**8*
x**3 + 12*b**9*x**4) - 4*a*b**3*c*x/(12*a**4*b**5 + 48*a**3*b**6*x + 72*a**2*b**7*x**2 + 48*a*b**8*x**3 + 12*b
**9*x**4) + 48*a*b**3*d*x**3*log(a/b + x)/(12*a**4*b**5 + 48*a**3*b**6*x + 72*a**2*b**7*x**2 + 48*a*b**8*x**3
+ 12*b**9*x**4) + 48*a*b**3*d*x**3/(12*a**4*b**5 + 48*a**3*b**6*x + 72*a**2*b**7*x**2 + 48*a*b**8*x**3 + 12*b*
*9*x**4) - 6*b**4*c*x**2/(12*a**4*b**5 + 48*a**3*b**6*x + 72*a**2*b**7*x**2 + 48*a*b**8*x**3 + 12*b**9*x**4) +
12*b**4*d*x**4*log(a/b + x)/(12*a**4*b**5 + 48*a**3*b**6*x + 72*a**2*b**7*x**2 + 48*a*b**8*x**3 + 12*b**9*x**
4), Eq(n, -5)), (-12*a**4*d*log(a/b + x)/(3*a**3*b**5 + 9*a**2*b**6*x + 9*a*b**7*x**2 + 3*b**8*x**3) - 22*a**4
*d/(3*a**3*b**5 + 9*a**2*b**6*x + 9*a*b**7*x**2 + 3*b**8*x**3) - 36*a**3*b*d*x*log(a/b + x)/(3*a**3*b**5 + 9*a
**2*b**6*x + 9*a*b**7*x**2 + 3*b**8*x**3) - 54*a**3*b*d*x/(3*a**3*b**5 + 9*a**2*b**6*x + 9*a*b**7*x**2 + 3*b**
8*x**3) - a**2*b**2*c/(3*a**3*b**5 + 9*a**2*b**6*x + 9*a*b**7*x**2 + 3*b**8*x**3) - 36*a**2*b**2*d*x**2*log(a/
b + x)/(3*a**3*b**5 + 9*a**2*b**6*x + 9*a*b**7*x**2 + 3*b**8*x**3) - 36*a**2*b**2*d*x**2/(3*a**3*b**5 + 9*a**2
*b**6*x + 9*a*b**7*x**2 + 3*b**8*x**3) - 3*a*b**3*c*x/(3*a**3*b**5 + 9*a**2*b**6*x + 9*a*b**7*x**2 + 3*b**8*x*
*3) - 12*a*b**3*d*x**3*log(a/b + x)/(3*a**3*b**5 + 9*a**2*b**6*x + 9*a*b**7*x**2 + 3*b**8*x**3) - 3*b**4*c*x**
2/(3*a**3*b**5 + 9*a**2*b**6*x + 9*a*b**7*x**2 + 3*b**8*x**3) + 3*b**4*d*x**4/(3*a**3*b**5 + 9*a**2*b**6*x + 9
*a*b**7*x**2 + 3*b**8*x**3), Eq(n, -4)), (12*a**4*d*log(a/b + x)/(2*a**2*b**5 + 4*a*b**6*x + 2*b**7*x**2) + 18
*a**4*d/(2*a**2*b**5 + 4*a*b**6*x + 2*b**7*x**2) + 24*a**3*b*d*x*log(a/b + x)/(2*a**2*b**5 + 4*a*b**6*x + 2*b*
*7*x**2) + 24*a**3*b*d*x/(2*a**2*b**5 + 4*a*b**6*x + 2*b**7*x**2) + 2*a**2*b**2*c*log(a/b + x)/(2*a**2*b**5 +
4*a*b**6*x + 2*b**7*x**2) + 3*a**2*b**2*c/(2*a**2*b**5 + 4*a*b**6*x + 2*b**7*x**2) + 12*a**2*b**2*d*x**2*log(a
/b + x)/(2*a**2*b**5 + 4*a*b**6*x + 2*b**7*x**2) + 4*a*b**3*c*x*log(a/b + x)/(2*a**2*b**5 + 4*a*b**6*x + 2*b**
7*x**2) + 4*a*b**3*c*x/(2*a**2*b**5 + 4*a*b**6*x + 2*b**7*x**2) - 4*a*b**3*d*x**3/(2*a**2*b**5 + 4*a*b**6*x +
2*b**7*x**2) + 2*b**4*c*x**2*log(a/b + x)/(2*a**2*b**5 + 4*a*b**6*x + 2*b**7*x**2) + b**4*d*x**4/(2*a**2*b**5
+ 4*a*b**6*x + 2*b**7*x**2), Eq(n, -3)), (-12*a**4*d*log(a/b + x)/(3*a*b**5 + 3*b**6*x) - 12*a**4*d/(3*a*b**5
+ 3*b**6*x) - 12*a**3*b*d*x*log(a/b + x)/(3*a*b**5 + 3*b**6*x) - 6*a**2*b**2*c*log(a/b + x)/(3*a*b**5 + 3*b**6
*x) - 6*a**2*b**2*c/(3*a*b**5 + 3*b**6*x) + 6*a**2*b**2*d*x**2/(3*a*b**5 + 3*b**6*x) - 6*a*b**3*c*x*log(a/b +
x)/(3*a*b**5 + 3*b**6*x) - 2*a*b**3*d*x**3/(3*a*b**5 + 3*b**6*x) + 3*b**4*c*x**2/(3*a*b**5 + 3*b**6*x) + b**4*
d*x**4/(3*a*b**5 + 3*b**6*x), Eq(n, -2)), (a**4*d*log(a/b + x)/b**5 - a**3*d*x/b**4 + a**2*c*log(a/b + x)/b**3
+ a**2*d*x**2/(2*b**3) - a*c*x/b**2 - a*d*x**3/(3*b**2) + c*x**2/(2*b) + d*x**4/(4*b), Eq(n, -1)), (24*a**5*d
*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) - 24*a**4*b*d*
n*x*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) + 2*a**3*b*
*2*c*n**2*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) + 18*
a**3*b**2*c*n*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) +
40*a**3*b**2*c*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5)
+ 12*a**3*b**2*d*n**2*x**2*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n
+ 120*b**5) + 12*a**3*b**2*d*n*x**2*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 2
74*b**5*n + 120*b**5) - 2*a**2*b**3*c*n**3*x*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*
n**2 + 274*b**5*n + 120*b**5) - 18*a**2*b**3*c*n**2*x*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 +
225*b**5*n**2 + 274*b**5*n + 120*b**5) - 40*a**2*b**3*c*n*x*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n
**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) - 4*a**2*b**3*d*n**3*x**3*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4
+ 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) - 12*a**2*b**3*d*n**2*x**3*(a + b*x)**n/(b**5*n**5 + 1
5*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) - 8*a**2*b**3*d*n*x**3*(a + b*x)**n/(b**5*
n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) + a*b**4*c*n**4*x**2*(a + b*x)**n/
(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) + 10*a*b**4*c*n**3*x**2*(a +
b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) + 29*a*b**4*c*n**2*
x**2*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) + 20*a*b**
4*c*n*x**2*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) + a*
b**4*d*n**4*x**4*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5
) + 6*a*b**4*d*n**3*x**4*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n +
120*b**5) + 11*a*b**4*d*n**2*x**4*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*
b**5*n + 120*b**5) + 6*a*b**4*d*n*x**4*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 +
274*b**5*n + 120*b**5) + b**5*c*n**4*x**3*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n*
*2 + 274*b**5*n + 120*b**5) + 12*b**5*c*n**3*x**3*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*
b**5*n**2 + 274*b**5*n + 120*b**5) + 49*b**5*c*n**2*x**3*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3
+ 225*b**5*n**2 + 274*b**5*n + 120*b**5) + 78*b**5*c*n*x**3*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*
n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) + 40*b**5*c*x**3*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**
5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) + b**5*d*n**4*x**5*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85
*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) + 10*b**5*d*n**3*x**5*(a + b*x)**n/(b**5*n**5 + 15*b**5*n*
*4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) + 35*b**5*d*n**2*x**5*(a + b*x)**n/(b**5*n**5 + 15*
b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) + 50*b**5*d*n*x**5*(a + b*x)**n/(b**5*n**5 +
15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) + 24*b**5*d*x**5*(a + b*x)**n/(b**5*n**5
+ 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5), True))
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