3.175 \(\int x (a+b x)^n (c+d x^3) \, dx\)
Optimal. Leaf size=126 \[ -\frac {a \left (b^3 c-a^3 d\right ) (a+b x)^{n+1}}{b^5 (n+1)}+\frac {\left (b^3 c-4 a^3 d\right ) (a+b x)^{n+2}}{b^5 (n+2)}+\frac {6 a^2 d (a+b x)^{n+3}}{b^5 (n+3)}-\frac {4 a d (a+b x)^{n+4}}{b^5 (n+4)}+\frac {d (a+b x)^{n+5}}{b^5 (n+5)} \]
[Out]
-a*(-a^3*d+b^3*c)*(b*x+a)^(1+n)/b^5/(1+n)+(-4*a^3*d+b^3*c)*(b*x+a)^(2+n)/b^5/(2+n)+6*a^2*d*(b*x+a)^(3+n)/b^5/(
3+n)-4*a*d*(b*x+a)^(4+n)/b^5/(4+n)+d*(b*x+a)^(5+n)/b^5/(5+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 = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used =
{1620} \[ -\frac {a \left (b^3 c-a^3 d\right ) (a+b x)^{n+1}}{b^5 (n+1)}+\frac {\left (b^3 c-4 a^3 d\right ) (a+b x)^{n+2}}{b^5 (n+2)}+\frac {6 a^2 d (a+b x)^{n+3}}{b^5 (n+3)}-\frac {4 a d (a+b x)^{n+4}}{b^5 (n+4)}+\frac {d (a+b x)^{n+5}}{b^5 (n+5)} \]
Antiderivative was successfully verified.
[In]
Int[x*(a + b*x)^n*(c + d*x^3),x]
[Out]
-((a*(b^3*c - a^3*d)*(a + b*x)^(1 + n))/(b^5*(1 + n))) + ((b^3*c - 4*a^3*d)*(a + b*x)^(2 + n))/(b^5*(2 + n)) +
(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 1620
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]
Rubi steps
\begin {align*} \int x (a+b x)^n \left (c+d x^3\right ) \, dx &=\int \left (\frac {a \left (-b^3 c+a^3 d\right ) (a+b x)^n}{b^4}+\frac {\left (b^3 c-4 a^3 d\right ) (a+b x)^{1+n}}{b^4}+\frac {6 a^2 d (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 \left (b^3 c-a^3 d\right ) (a+b x)^{1+n}}{b^5 (1+n)}+\frac {\left (b^3 c-4 a^3 d\right ) (a+b x)^{2+n}}{b^5 (2+n)}+\frac {6 a^2 d (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 = 0.08, size = 104, normalized size = 0.83 \[ \frac {(a+b x)^{n+1} \left (\frac {(a+b x) \left (b^3 c-4 a^3 d\right )}{n+2}+\frac {a \left (a^3 d-b^3 c\right )}{n+1}+\frac {6 a^2 d (a+b x)^2}{n+3}+\frac {d (a+b x)^4}{n+5}-\frac {4 a d (a+b x)^3}{n+4}\right )}{b^5} \]
Antiderivative was successfully verified.
[In]
Integrate[x*(a + b*x)^n*(c + d*x^3),x]
[Out]
((a + b*x)^(1 + n)*((a*(-(b^3*c) + a^3*d))/(1 + n) + ((b^3*c - 4*a^3*d)*(a + b*x))/(2 + n) + (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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fricas [B] time = 0.72, size = 348, normalized size = 2.76 \[ -\frac {{\left (a^{2} b^{3} c n^{3} + 12 \, a^{2} b^{3} c n^{2} + 47 \, a^{2} b^{3} c n + 60 \, a^{2} b^{3} c - 24 \, a^{5} d - {\left (b^{5} d n^{4} + 10 \, b^{5} d n^{3} + 35 \, b^{5} d n^{2} + 50 \, b^{5} d n + 24 \, b^{5} d\right )} x^{5} - {\left (a b^{4} d n^{4} + 6 \, a b^{4} d n^{3} + 11 \, a b^{4} d n^{2} + 6 \, a b^{4} d n\right )} x^{4} + 4 \, {\left (a^{2} b^{3} d n^{3} + 3 \, a^{2} b^{3} d n^{2} + 2 \, a^{2} b^{3} d n\right )} x^{3} - {\left (b^{5} c n^{4} + 13 \, b^{5} c n^{3} + 60 \, b^{5} c + {\left (59 \, b^{5} c + 12 \, a^{3} b^{2} d\right )} n^{2} + {\left (107 \, b^{5} c + 12 \, a^{3} b^{2} d\right )} n\right )} x^{2} - {\left (a b^{4} c n^{4} + 12 \, a b^{4} c n^{3} + 47 \, a b^{4} c n^{2} + 12 \, {\left (5 \, a b^{4} c - 2 \, a^{4} b d\right )} n\right )} x\right )} {\left (b x + a\right )}^{n}}{b^{5} n^{5} + 15 \, b^{5} n^{4} + 85 \, b^{5} n^{3} + 225 \, b^{5} n^{2} + 274 \, b^{5} n + 120 \, b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*(b*x+a)^n*(d*x^3+c),x, algorithm="fricas")
[Out]
-(a^2*b^3*c*n^3 + 12*a^2*b^3*c*n^2 + 47*a^2*b^3*c*n + 60*a^2*b^3*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 + 4
*(a^2*b^3*d*n^3 + 3*a^2*b^3*d*n^2 + 2*a^2*b^3*d*n)*x^3 - (b^5*c*n^4 + 13*b^5*c*n^3 + 60*b^5*c + (59*b^5*c + 12
*a^3*b^2*d)*n^2 + (107*b^5*c + 12*a^3*b^2*d)*n)*x^2 - (a*b^4*c*n^4 + 12*a*b^4*c*n^3 + 47*a*b^4*c*n^2 + 12*(5*a
*b^4*c - 2*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 = 0.38, size = 577, normalized size = 4.58 \[ \frac {{\left (b x + a\right )}^{n} b^{5} d n^{4} x^{5} + {\left (b x + a\right )}^{n} a b^{4} d n^{4} x^{4} + 10 \, {\left (b x + a\right )}^{n} b^{5} d n^{3} x^{5} + 6 \, {\left (b x + a\right )}^{n} a b^{4} d n^{3} x^{4} + 35 \, {\left (b x + a\right )}^{n} b^{5} d n^{2} x^{5} + {\left (b x + a\right )}^{n} b^{5} c n^{4} x^{2} - 4 \, {\left (b x + a\right )}^{n} a^{2} b^{3} d n^{3} x^{3} + 11 \, {\left (b x + a\right )}^{n} a b^{4} d n^{2} x^{4} + 50 \, {\left (b x + a\right )}^{n} b^{5} d n x^{5} + {\left (b x + a\right )}^{n} a b^{4} c n^{4} x + 13 \, {\left (b x + a\right )}^{n} b^{5} c n^{3} x^{2} - 12 \, {\left (b x + a\right )}^{n} a^{2} b^{3} d n^{2} x^{3} + 6 \, {\left (b x + a\right )}^{n} a b^{4} d n x^{4} + 24 \, {\left (b x + a\right )}^{n} b^{5} d x^{5} + 12 \, {\left (b x + a\right )}^{n} a b^{4} c n^{3} x + 59 \, {\left (b x + a\right )}^{n} b^{5} c n^{2} x^{2} + 12 \, {\left (b x + a\right )}^{n} a^{3} b^{2} d n^{2} x^{2} - 8 \, {\left (b x + a\right )}^{n} a^{2} b^{3} d n x^{3} - {\left (b x + a\right )}^{n} a^{2} b^{3} c n^{3} + 47 \, {\left (b x + a\right )}^{n} a b^{4} c n^{2} x + 107 \, {\left (b x + a\right )}^{n} b^{5} c n x^{2} + 12 \, {\left (b x + a\right )}^{n} a^{3} b^{2} d n x^{2} - 12 \, {\left (b x + a\right )}^{n} a^{2} b^{3} c n^{2} + 60 \, {\left (b x + a\right )}^{n} a b^{4} c n x - 24 \, {\left (b x + a\right )}^{n} a^{4} b d n x + 60 \, {\left (b x + a\right )}^{n} b^{5} c x^{2} - 47 \, {\left (b x + a\right )}^{n} a^{2} b^{3} c n - 60 \, {\left (b x + a\right )}^{n} a^{2} b^{3} c + 24 \, {\left (b x + a\right )}^{n} a^{5} d}{b^{5} n^{5} + 15 \, b^{5} n^{4} + 85 \, b^{5} n^{3} + 225 \, b^{5} n^{2} + 274 \, b^{5} n + 120 \, b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*(b*x+a)^n*(d*x^3+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 + 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*b^5*c*n^4*x^2 - 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 + (b*x + a)^n*a*b^4*c*n^4*x + 13*(b*x + a)^n*b^5*c*n^3*
x^2 - 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 + 12*(b*x + a)
^n*a*b^4*c*n^3*x + 59*(b*x + a)^n*b^5*c*n^2*x^2 + 12*(b*x + a)^n*a^3*b^2*d*n^2*x^2 - 8*(b*x + a)^n*a^2*b^3*d*n
*x^3 - (b*x + a)^n*a^2*b^3*c*n^3 + 47*(b*x + a)^n*a*b^4*c*n^2*x + 107*(b*x + a)^n*b^5*c*n*x^2 + 12*(b*x + a)^n
*a^3*b^2*d*n*x^2 - 12*(b*x + a)^n*a^2*b^3*c*n^2 + 60*(b*x + a)^n*a*b^4*c*n*x - 24*(b*x + a)^n*a^4*b*d*n*x + 60
*(b*x + a)^n*b^5*c*x^2 - 47*(b*x + a)^n*a^2*b^3*c*n - 60*(b*x + a)^n*a^2*b^3*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 = 283, normalized size = 2.25 \[ \frac {\left (b^{4} d \,n^{4} x^{4}+10 b^{4} d \,n^{3} x^{4}-4 a \,b^{3} d \,n^{3} x^{3}+35 b^{4} d \,n^{2} x^{4}-24 a \,b^{3} d \,n^{2} x^{3}+b^{4} c \,n^{4} x +50 b^{4} d n \,x^{4}+12 a^{2} b^{2} d \,n^{2} x^{2}-44 a \,b^{3} d n \,x^{3}+13 b^{4} c \,n^{3} x +24 d \,x^{4} b^{4}+36 a^{2} b^{2} d n \,x^{2}-a \,b^{3} c \,n^{3}-24 a d \,x^{3} b^{3}+59 b^{4} c \,n^{2} x -24 a^{3} b d n x +24 d \,a^{2} x^{2} b^{2}-12 a \,b^{3} c \,n^{2}+107 b^{4} c n x -24 a^{3} b d x -47 a \,b^{3} c n +60 b^{4} c x +24 a^{4} d -60 a \,b^{3} c \right ) \left (b x +a \right )^{n +1}}{\left (n^{5}+15 n^{4}+85 n^{3}+225 n^{2}+274 n +120\right ) b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x*(b*x+a)^n*(d*x^3+c),x)
[Out]
(b*x+a)^(n+1)*(b^4*d*n^4*x^4+10*b^4*d*n^3*x^4-4*a*b^3*d*n^3*x^3+35*b^4*d*n^2*x^4-24*a*b^3*d*n^2*x^3+b^4*c*n^4*
x+50*b^4*d*n*x^4+12*a^2*b^2*d*n^2*x^2-44*a*b^3*d*n*x^3+13*b^4*c*n^3*x+24*b^4*d*x^4+36*a^2*b^2*d*n*x^2-a*b^3*c*
n^3-24*a*b^3*d*x^3+59*b^4*c*n^2*x-24*a^3*b*d*n*x+24*a^2*b^2*d*x^2-12*a*b^3*c*n^2+107*b^4*c*n*x-24*a^3*b*d*x-47
*a*b^3*c*n+60*b^4*c*x+24*a^4*d-60*a*b^3*c)/b^5/(n^5+15*n^4+85*n^3+225*n^2+274*n+120)
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maxima [A] time = 1.02, size = 184, normalized size = 1.46 \[ \frac {{\left (b^{2} {\left (n + 1\right )} x^{2} + a b n x - a^{2}\right )} {\left (b x + a\right )}^{n} c}{{\left (n^{2} + 3 \, n + 2\right )} b^{2}} + \frac {{\left ({\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{5} x^{5} + {\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a b^{4} x^{4} - 4 \, {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{2} b^{3} x^{3} + 12 \, {\left (n^{2} + n\right )} a^{3} b^{2} x^{2} - 24 \, a^{4} b n x + 24 \, a^{5}\right )} {\left (b x + a\right )}^{n} d}{{\left (n^{5} + 15 \, n^{4} + 85 \, n^{3} + 225 \, n^{2} + 274 \, n + 120\right )} b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*(b*x+a)^n*(d*x^3+c),x, algorithm="maxima")
[Out]
(b^2*(n + 1)*x^2 + a*b*n*x - a^2)*(b*x + a)^n*c/((n^2 + 3*n + 2)*b^2) + ((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 = 2.95, size = 363, normalized size = 2.88 \[ {\left (a+b\,x\right )}^n\,\left (\frac {d\,x^5\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}{n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120}-\frac {a^2\,\left (-24\,d\,a^3+c\,b^3\,n^3+12\,c\,b^3\,n^2+47\,c\,b^3\,n+60\,c\,b^3\right )}{b^5\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}+\frac {x^2\,\left (n+1\right )\,\left (12\,d\,a^3\,n+c\,b^3\,n^3+12\,c\,b^3\,n^2+47\,c\,b^3\,n+60\,c\,b^3\right )}{b^3\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}+\frac {a\,n\,x\,\left (-24\,d\,a^3+c\,b^3\,n^3+12\,c\,b^3\,n^2+47\,c\,b^3\,n+60\,c\,b^3\right )}{b^4\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}+\frac {a\,d\,n\,x^4\,\left (n^3+6\,n^2+11\,n+6\right )}{b\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}-\frac {4\,a^2\,d\,n\,x^3\,\left (n^2+3\,n+2\right )}{b^2\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x*(c + d*x^3)*(a + b*x)^n,x)
[Out]
(a + b*x)^n*((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) - (a^
2*(60*b^3*c - 24*a^3*d + 12*b^3*c*n^2 + b^3*c*n^3 + 47*b^3*c*n))/(b^5*(274*n + 225*n^2 + 85*n^3 + 15*n^4 + n^5
+ 120)) + (x^2*(n + 1)*(60*b^3*c + 12*b^3*c*n^2 + b^3*c*n^3 + 12*a^3*d*n + 47*b^3*c*n))/(b^3*(274*n + 225*n^2
+ 85*n^3 + 15*n^4 + n^5 + 120)) + (a*n*x*(60*b^3*c - 24*a^3*d + 12*b^3*c*n^2 + b^3*c*n^3 + 47*b^3*c*n))/(b^4*
(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 + 120)) - (4*a^2*d*n*x^3*(3*n + n^2 + 2))/(b^2*(274*n + 225*n^2 + 85*n^3 + 15*n^4 + n^5
+ 120)))
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sympy [A] time = 4.86, size = 3704, normalized size = 29.40 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*(b*x+a)**n*(d*x**3+c),x)
[Out]
Piecewise((a**n*(c*x**2/2 + 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) + 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) - a*b**3*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) + 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) - 4*b**4*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) + 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)), (-24*a**4*d*log(a/b + x)/(6*a**3*b**5 + 18*a**2*b**6*x + 18*a*b**7*x**2 + 6*b**8*x**3) -
44*a**4*d/(6*a**3*b**5 + 18*a**2*b**6*x + 18*a*b**7*x**2 + 6*b**8*x**3) - 72*a**3*b*d*x*log(a/b + x)/(6*a**3*b
**5 + 18*a**2*b**6*x + 18*a*b**7*x**2 + 6*b**8*x**3) - 108*a**3*b*d*x/(6*a**3*b**5 + 18*a**2*b**6*x + 18*a*b**
7*x**2 + 6*b**8*x**3) - 72*a**2*b**2*d*x**2*log(a/b + x)/(6*a**3*b**5 + 18*a**2*b**6*x + 18*a*b**7*x**2 + 6*b*
*8*x**3) - 72*a**2*b**2*d*x**2/(6*a**3*b**5 + 18*a**2*b**6*x + 18*a*b**7*x**2 + 6*b**8*x**3) - a*b**3*c/(6*a**
3*b**5 + 18*a**2*b**6*x + 18*a*b**7*x**2 + 6*b**8*x**3) - 24*a*b**3*d*x**3*log(a/b + x)/(6*a**3*b**5 + 18*a**2
*b**6*x + 18*a*b**7*x**2 + 6*b**8*x**3) - 3*b**4*c*x/(6*a**3*b**5 + 18*a**2*b**6*x + 18*a*b**7*x**2 + 6*b**8*x
**3) + 6*b**4*d*x**4/(6*a**3*b**5 + 18*a**2*b**6*x + 18*a*b**7*x**2 + 6*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) + 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) - a*b**3*c/(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*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*d*x**2/(3*a*b**5 + 3*b**6*x) + 3*a*b**3*c*log(a/b + x)/(3*a*b**5 + 3*b**6*x) + 3*a*b
**3*c/(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*log(a/b + x)/(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*d*x**2
/(2*b**3) - a*c*log(a/b + x)/b**2 - a*d*x**3/(3*b**2) + c*x/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) + 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 + 274*b**5*n + 120*b**5)
- a**2*b**3*c*n**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*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) - 47*a**2*b**3*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) - 60*a**2*b**3*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) - 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 + 15*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*(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*b**4*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) + 47*a*b**4*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) + 60*a*b**4*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) + 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*
*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) + 13*b**5*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) + 59*
b**5*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
) + 107*b**5*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) + 60*b**5*c*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 + 1
20*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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