3.2672 \(\int \frac {x^m}{(a+b x^n)^3} \, dx\)
Optimal. Leaf size=40 \[ \frac {x^{m+1} \, _2F_1\left (3,\frac {m+1}{n};\frac {m+n+1}{n};-\frac {b x^n}{a}\right )}{a^3 (m+1)} \]
[Out]
x^(1+m)*hypergeom([3, (1+m)/n],[(1+m+n)/n],-b*x^n/a)/a^3/(1+m)
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Rubi [A] time = 0.01, antiderivative size = 40, normalized size of antiderivative = 1.00,
number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used =
{364} \[ \frac {x^{m+1} \, _2F_1\left (3,\frac {m+1}{n};\frac {m+n+1}{n};-\frac {b x^n}{a}\right )}{a^3 (m+1)} \]
Antiderivative was successfully verified.
[In]
Int[x^m/(a + b*x^n)^3,x]
[Out]
(x^(1 + m)*Hypergeometric2F1[3, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)])/(a^3*(1 + m))
Rule 364
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])
Rubi steps
\begin {align*} \int \frac {x^m}{\left (a+b x^n\right )^3} \, dx &=\frac {x^{1+m} \, _2F_1\left (3,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b x^n}{a}\right )}{a^3 (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 41, normalized size = 1.02 \[ \frac {x^{m+1} \, _2F_1\left (3,\frac {m+1}{n};\frac {m+1}{n}+1;-\frac {b x^n}{a}\right )}{a^3 (m+1)} \]
Antiderivative was successfully verified.
[In]
Integrate[x^m/(a + b*x^n)^3,x]
[Out]
(x^(1 + m)*Hypergeometric2F1[3, (1 + m)/n, 1 + (1 + m)/n, -((b*x^n)/a)])/(a^3*(1 + m))
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fricas [F] time = 0.78, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{m}}{b^{3} x^{3 \, n} + 3 \, a b^{2} x^{2 \, n} + 3 \, a^{2} b x^{n} + a^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^m/(a+b*x^n)^3,x, algorithm="fricas")
[Out]
integral(x^m/(b^3*x^(3*n) + 3*a*b^2*x^(2*n) + 3*a^2*b*x^n + a^3), x)
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{{\left (b x^{n} + a\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^m/(a+b*x^n)^3,x, algorithm="giac")
[Out]
integrate(x^m/(b*x^n + a)^3, x)
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maple [F] time = 0.22, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{\left (b \,x^{n}+a \right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^m/(b*x^n+a)^3,x)
[Out]
int(x^m/(b*x^n+a)^3,x)
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ {\left (m^{2} - m {\left (3 \, n - 2\right )} + 2 \, n^{2} - 3 \, n + 1\right )} \int \frac {x^{m}}{2 \, {\left (a^{2} b n^{2} x^{n} + a^{3} n^{2}\right )}}\,{d x} - \frac {a {\left (m - 3 \, n + 1\right )} x x^{m} + b {\left (m - 2 \, n + 1\right )} x e^{\left (m \log \relax (x) + n \log \relax (x)\right )}}{2 \, {\left (a^{2} b^{2} n^{2} x^{2 \, n} + 2 \, a^{3} b n^{2} x^{n} + a^{4} n^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^m/(a+b*x^n)^3,x, algorithm="maxima")
[Out]
(m^2 - m*(3*n - 2) + 2*n^2 - 3*n + 1)*integrate(1/2*x^m/(a^2*b*n^2*x^n + a^3*n^2), x) - 1/2*(a*(m - 3*n + 1)*x
*x^m + b*(m - 2*n + 1)*x*e^(m*log(x) + n*log(x)))/(a^2*b^2*n^2*x^(2*n) + 2*a^3*b*n^2*x^n + a^4*n^2)
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x^m}{{\left (a+b\,x^n\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^m/(a + b*x^n)^3,x)
[Out]
int(x^m/(a + b*x^n)^3, x)
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sympy [C] time = 4.75, size = 6803, normalized size = 170.08 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**m/(a+b*x**n)**3,x)
[Out]
a*m**3*x*x**m*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1/n)*gamma(m/n + 1/n)/(2*a**4*n**4*gamma(m/n + 1 + 1
/n) + 6*a**3*b*n**4*x**n*gamma(m/n + 1 + 1/n) + 6*a**2*b**2*n**4*x**(2*n)*gamma(m/n + 1 + 1/n) + 2*a*b**3*n**4
*x**(3*n)*gamma(m/n + 1 + 1/n)) - 3*a*m**2*n*x*x**m*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1/n)*gamma(m/n
+ 1/n)/(2*a**4*n**4*gamma(m/n + 1 + 1/n) + 6*a**3*b*n**4*x**n*gamma(m/n + 1 + 1/n) + 6*a**2*b**2*n**4*x**(2*n
)*gamma(m/n + 1 + 1/n) + 2*a*b**3*n**4*x**(3*n)*gamma(m/n + 1 + 1/n)) - a*m**2*n*x*x**m*gamma(m/n + 1/n)/(2*a*
*4*n**4*gamma(m/n + 1 + 1/n) + 6*a**3*b*n**4*x**n*gamma(m/n + 1 + 1/n) + 6*a**2*b**2*n**4*x**(2*n)*gamma(m/n +
1 + 1/n) + 2*a*b**3*n**4*x**(3*n)*gamma(m/n + 1 + 1/n)) + 3*a*m**2*x*x**m*lerchphi(b*x**n*exp_polar(I*pi)/a,
1, m/n + 1/n)*gamma(m/n + 1/n)/(2*a**4*n**4*gamma(m/n + 1 + 1/n) + 6*a**3*b*n**4*x**n*gamma(m/n + 1 + 1/n) + 6
*a**2*b**2*n**4*x**(2*n)*gamma(m/n + 1 + 1/n) + 2*a*b**3*n**4*x**(3*n)*gamma(m/n + 1 + 1/n)) + 2*a*m*n**2*x*x*
*m*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1/n)*gamma(m/n + 1/n)/(2*a**4*n**4*gamma(m/n + 1 + 1/n) + 6*a**
3*b*n**4*x**n*gamma(m/n + 1 + 1/n) + 6*a**2*b**2*n**4*x**(2*n)*gamma(m/n + 1 + 1/n) + 2*a*b**3*n**4*x**(3*n)*g
amma(m/n + 1 + 1/n)) + 3*a*m*n**2*x*x**m*gamma(m/n + 1/n)/(2*a**4*n**4*gamma(m/n + 1 + 1/n) + 6*a**3*b*n**4*x*
*n*gamma(m/n + 1 + 1/n) + 6*a**2*b**2*n**4*x**(2*n)*gamma(m/n + 1 + 1/n) + 2*a*b**3*n**4*x**(3*n)*gamma(m/n +
1 + 1/n)) - 6*a*m*n*x*x**m*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1/n)*gamma(m/n + 1/n)/(2*a**4*n**4*gamm
a(m/n + 1 + 1/n) + 6*a**3*b*n**4*x**n*gamma(m/n + 1 + 1/n) + 6*a**2*b**2*n**4*x**(2*n)*gamma(m/n + 1 + 1/n) +
2*a*b**3*n**4*x**(3*n)*gamma(m/n + 1 + 1/n)) - 2*a*m*n*x*x**m*gamma(m/n + 1/n)/(2*a**4*n**4*gamma(m/n + 1 + 1/
n) + 6*a**3*b*n**4*x**n*gamma(m/n + 1 + 1/n) + 6*a**2*b**2*n**4*x**(2*n)*gamma(m/n + 1 + 1/n) + 2*a*b**3*n**4*
x**(3*n)*gamma(m/n + 1 + 1/n)) + 3*a*m*x*x**m*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1/n)*gamma(m/n + 1/n
)/(2*a**4*n**4*gamma(m/n + 1 + 1/n) + 6*a**3*b*n**4*x**n*gamma(m/n + 1 + 1/n) + 6*a**2*b**2*n**4*x**(2*n)*gamm
a(m/n + 1 + 1/n) + 2*a*b**3*n**4*x**(3*n)*gamma(m/n + 1 + 1/n)) + 2*a*n**2*x*x**m*lerchphi(b*x**n*exp_polar(I*
pi)/a, 1, m/n + 1/n)*gamma(m/n + 1/n)/(2*a**4*n**4*gamma(m/n + 1 + 1/n) + 6*a**3*b*n**4*x**n*gamma(m/n + 1 + 1
/n) + 6*a**2*b**2*n**4*x**(2*n)*gamma(m/n + 1 + 1/n) + 2*a*b**3*n**4*x**(3*n)*gamma(m/n + 1 + 1/n)) + 3*a*n**2
*x*x**m*gamma(m/n + 1/n)/(2*a**4*n**4*gamma(m/n + 1 + 1/n) + 6*a**3*b*n**4*x**n*gamma(m/n + 1 + 1/n) + 6*a**2*
b**2*n**4*x**(2*n)*gamma(m/n + 1 + 1/n) + 2*a*b**3*n**4*x**(3*n)*gamma(m/n + 1 + 1/n)) - 3*a*n*x*x**m*lerchphi
(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1/n)*gamma(m/n + 1/n)/(2*a**4*n**4*gamma(m/n + 1 + 1/n) + 6*a**3*b*n**4*x*
*n*gamma(m/n + 1 + 1/n) + 6*a**2*b**2*n**4*x**(2*n)*gamma(m/n + 1 + 1/n) + 2*a*b**3*n**4*x**(3*n)*gamma(m/n +
1 + 1/n)) - a*n*x*x**m*gamma(m/n + 1/n)/(2*a**4*n**4*gamma(m/n + 1 + 1/n) + 6*a**3*b*n**4*x**n*gamma(m/n + 1 +
1/n) + 6*a**2*b**2*n**4*x**(2*n)*gamma(m/n + 1 + 1/n) + 2*a*b**3*n**4*x**(3*n)*gamma(m/n + 1 + 1/n)) + a*x*x*
*m*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1/n)*gamma(m/n + 1/n)/(2*a**4*n**4*gamma(m/n + 1 + 1/n) + 6*a**
3*b*n**4*x**n*gamma(m/n + 1 + 1/n) + 6*a**2*b**2*n**4*x**(2*n)*gamma(m/n + 1 + 1/n) + 2*a*b**3*n**4*x**(3*n)*g
amma(m/n + 1 + 1/n)) + 3*b*m**3*x*x**m*x**n*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1/n)*gamma(m/n + 1/n)/
(2*a**4*n**4*gamma(m/n + 1 + 1/n) + 6*a**3*b*n**4*x**n*gamma(m/n + 1 + 1/n) + 6*a**2*b**2*n**4*x**(2*n)*gamma(
m/n + 1 + 1/n) + 2*a*b**3*n**4*x**(3*n)*gamma(m/n + 1 + 1/n)) - 9*b*m**2*n*x*x**m*x**n*lerchphi(b*x**n*exp_pol
ar(I*pi)/a, 1, m/n + 1/n)*gamma(m/n + 1/n)/(2*a**4*n**4*gamma(m/n + 1 + 1/n) + 6*a**3*b*n**4*x**n*gamma(m/n +
1 + 1/n) + 6*a**2*b**2*n**4*x**(2*n)*gamma(m/n + 1 + 1/n) + 2*a*b**3*n**4*x**(3*n)*gamma(m/n + 1 + 1/n)) - 2*b
*m**2*n*x*x**m*x**n*gamma(m/n + 1/n)/(2*a**4*n**4*gamma(m/n + 1 + 1/n) + 6*a**3*b*n**4*x**n*gamma(m/n + 1 + 1/
n) + 6*a**2*b**2*n**4*x**(2*n)*gamma(m/n + 1 + 1/n) + 2*a*b**3*n**4*x**(3*n)*gamma(m/n + 1 + 1/n)) + 9*b*m**2*
x*x**m*x**n*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1/n)*gamma(m/n + 1/n)/(2*a**4*n**4*gamma(m/n + 1 + 1/n
) + 6*a**3*b*n**4*x**n*gamma(m/n + 1 + 1/n) + 6*a**2*b**2*n**4*x**(2*n)*gamma(m/n + 1 + 1/n) + 2*a*b**3*n**4*x
**(3*n)*gamma(m/n + 1 + 1/n)) + 6*b*m*n**2*x*x**m*x**n*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1/n)*gamma(
m/n + 1/n)/(2*a**4*n**4*gamma(m/n + 1 + 1/n) + 6*a**3*b*n**4*x**n*gamma(m/n + 1 + 1/n) + 6*a**2*b**2*n**4*x**(
2*n)*gamma(m/n + 1 + 1/n) + 2*a*b**3*n**4*x**(3*n)*gamma(m/n + 1 + 1/n)) + 5*b*m*n**2*x*x**m*x**n*gamma(m/n +
1/n)/(2*a**4*n**4*gamma(m/n + 1 + 1/n) + 6*a**3*b*n**4*x**n*gamma(m/n + 1 + 1/n) + 6*a**2*b**2*n**4*x**(2*n)*g
amma(m/n + 1 + 1/n) + 2*a*b**3*n**4*x**(3*n)*gamma(m/n + 1 + 1/n)) - 18*b*m*n*x*x**m*x**n*lerchphi(b*x**n*exp_
polar(I*pi)/a, 1, m/n + 1/n)*gamma(m/n + 1/n)/(2*a**4*n**4*gamma(m/n + 1 + 1/n) + 6*a**3*b*n**4*x**n*gamma(m/n
+ 1 + 1/n) + 6*a**2*b**2*n**4*x**(2*n)*gamma(m/n + 1 + 1/n) + 2*a*b**3*n**4*x**(3*n)*gamma(m/n + 1 + 1/n)) -
4*b*m*n*x*x**m*x**n*gamma(m/n + 1/n)/(2*a**4*n**4*gamma(m/n + 1 + 1/n) + 6*a**3*b*n**4*x**n*gamma(m/n + 1 + 1/
n) + 6*a**2*b**2*n**4*x**(2*n)*gamma(m/n + 1 + 1/n) + 2*a*b**3*n**4*x**(3*n)*gamma(m/n + 1 + 1/n)) + 9*b*m*x*x
**m*x**n*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1/n)*gamma(m/n + 1/n)/(2*a**4*n**4*gamma(m/n + 1 + 1/n) +
6*a**3*b*n**4*x**n*gamma(m/n + 1 + 1/n) + 6*a**2*b**2*n**4*x**(2*n)*gamma(m/n + 1 + 1/n) + 2*a*b**3*n**4*x**(
3*n)*gamma(m/n + 1 + 1/n)) + 6*b*n**2*x*x**m*x**n*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1/n)*gamma(m/n +
1/n)/(2*a**4*n**4*gamma(m/n + 1 + 1/n) + 6*a**3*b*n**4*x**n*gamma(m/n + 1 + 1/n) + 6*a**2*b**2*n**4*x**(2*n)*
gamma(m/n + 1 + 1/n) + 2*a*b**3*n**4*x**(3*n)*gamma(m/n + 1 + 1/n)) + 5*b*n**2*x*x**m*x**n*gamma(m/n + 1/n)/(2
*a**4*n**4*gamma(m/n + 1 + 1/n) + 6*a**3*b*n**4*x**n*gamma(m/n + 1 + 1/n) + 6*a**2*b**2*n**4*x**(2*n)*gamma(m/
n + 1 + 1/n) + 2*a*b**3*n**4*x**(3*n)*gamma(m/n + 1 + 1/n)) - 9*b*n*x*x**m*x**n*lerchphi(b*x**n*exp_polar(I*pi
)/a, 1, m/n + 1/n)*gamma(m/n + 1/n)/(2*a**4*n**4*gamma(m/n + 1 + 1/n) + 6*a**3*b*n**4*x**n*gamma(m/n + 1 + 1/n
) + 6*a**2*b**2*n**4*x**(2*n)*gamma(m/n + 1 + 1/n) + 2*a*b**3*n**4*x**(3*n)*gamma(m/n + 1 + 1/n)) - 2*b*n*x*x*
*m*x**n*gamma(m/n + 1/n)/(2*a**4*n**4*gamma(m/n + 1 + 1/n) + 6*a**3*b*n**4*x**n*gamma(m/n + 1 + 1/n) + 6*a**2*
b**2*n**4*x**(2*n)*gamma(m/n + 1 + 1/n) + 2*a*b**3*n**4*x**(3*n)*gamma(m/n + 1 + 1/n)) + 3*b*x*x**m*x**n*lerch
phi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1/n)*gamma(m/n + 1/n)/(2*a**4*n**4*gamma(m/n + 1 + 1/n) + 6*a**3*b*n**4
*x**n*gamma(m/n + 1 + 1/n) + 6*a**2*b**2*n**4*x**(2*n)*gamma(m/n + 1 + 1/n) + 2*a*b**3*n**4*x**(3*n)*gamma(m/n
+ 1 + 1/n)) + 3*b**2*m**3*x*x**m*x**(2*n)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1/n)*gamma(m/n + 1/n)/(
a*(2*a**4*n**4*gamma(m/n + 1 + 1/n) + 6*a**3*b*n**4*x**n*gamma(m/n + 1 + 1/n) + 6*a**2*b**2*n**4*x**(2*n)*gamm
a(m/n + 1 + 1/n) + 2*a*b**3*n**4*x**(3*n)*gamma(m/n + 1 + 1/n))) - 9*b**2*m**2*n*x*x**m*x**(2*n)*lerchphi(b*x*
*n*exp_polar(I*pi)/a, 1, m/n + 1/n)*gamma(m/n + 1/n)/(a*(2*a**4*n**4*gamma(m/n + 1 + 1/n) + 6*a**3*b*n**4*x**n
*gamma(m/n + 1 + 1/n) + 6*a**2*b**2*n**4*x**(2*n)*gamma(m/n + 1 + 1/n) + 2*a*b**3*n**4*x**(3*n)*gamma(m/n + 1
+ 1/n))) - b**2*m**2*n*x*x**m*x**(2*n)*gamma(m/n + 1/n)/(a*(2*a**4*n**4*gamma(m/n + 1 + 1/n) + 6*a**3*b*n**4*x
**n*gamma(m/n + 1 + 1/n) + 6*a**2*b**2*n**4*x**(2*n)*gamma(m/n + 1 + 1/n) + 2*a*b**3*n**4*x**(3*n)*gamma(m/n +
1 + 1/n))) + 9*b**2*m**2*x*x**m*x**(2*n)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1/n)*gamma(m/n + 1/n)/(a
*(2*a**4*n**4*gamma(m/n + 1 + 1/n) + 6*a**3*b*n**4*x**n*gamma(m/n + 1 + 1/n) + 6*a**2*b**2*n**4*x**(2*n)*gamma
(m/n + 1 + 1/n) + 2*a*b**3*n**4*x**(3*n)*gamma(m/n + 1 + 1/n))) + 6*b**2*m*n**2*x*x**m*x**(2*n)*lerchphi(b*x**
n*exp_polar(I*pi)/a, 1, m/n + 1/n)*gamma(m/n + 1/n)/(a*(2*a**4*n**4*gamma(m/n + 1 + 1/n) + 6*a**3*b*n**4*x**n*
gamma(m/n + 1 + 1/n) + 6*a**2*b**2*n**4*x**(2*n)*gamma(m/n + 1 + 1/n) + 2*a*b**3*n**4*x**(3*n)*gamma(m/n + 1 +
1/n))) + 2*b**2*m*n**2*x*x**m*x**(2*n)*gamma(m/n + 1/n)/(a*(2*a**4*n**4*gamma(m/n + 1 + 1/n) + 6*a**3*b*n**4*
x**n*gamma(m/n + 1 + 1/n) + 6*a**2*b**2*n**4*x**(2*n)*gamma(m/n + 1 + 1/n) + 2*a*b**3*n**4*x**(3*n)*gamma(m/n
+ 1 + 1/n))) - 18*b**2*m*n*x*x**m*x**(2*n)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1/n)*gamma(m/n + 1/n)/(
a*(2*a**4*n**4*gamma(m/n + 1 + 1/n) + 6*a**3*b*n**4*x**n*gamma(m/n + 1 + 1/n) + 6*a**2*b**2*n**4*x**(2*n)*gamm
a(m/n + 1 + 1/n) + 2*a*b**3*n**4*x**(3*n)*gamma(m/n + 1 + 1/n))) - 2*b**2*m*n*x*x**m*x**(2*n)*gamma(m/n + 1/n)
/(a*(2*a**4*n**4*gamma(m/n + 1 + 1/n) + 6*a**3*b*n**4*x**n*gamma(m/n + 1 + 1/n) + 6*a**2*b**2*n**4*x**(2*n)*ga
mma(m/n + 1 + 1/n) + 2*a*b**3*n**4*x**(3*n)*gamma(m/n + 1 + 1/n))) + 9*b**2*m*x*x**m*x**(2*n)*lerchphi(b*x**n*
exp_polar(I*pi)/a, 1, m/n + 1/n)*gamma(m/n + 1/n)/(a*(2*a**4*n**4*gamma(m/n + 1 + 1/n) + 6*a**3*b*n**4*x**n*ga
mma(m/n + 1 + 1/n) + 6*a**2*b**2*n**4*x**(2*n)*gamma(m/n + 1 + 1/n) + 2*a*b**3*n**4*x**(3*n)*gamma(m/n + 1 + 1
/n))) + 6*b**2*n**2*x*x**m*x**(2*n)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1/n)*gamma(m/n + 1/n)/(a*(2*a*
*4*n**4*gamma(m/n + 1 + 1/n) + 6*a**3*b*n**4*x**n*gamma(m/n + 1 + 1/n) + 6*a**2*b**2*n**4*x**(2*n)*gamma(m/n +
1 + 1/n) + 2*a*b**3*n**4*x**(3*n)*gamma(m/n + 1 + 1/n))) + 2*b**2*n**2*x*x**m*x**(2*n)*gamma(m/n + 1/n)/(a*(2
*a**4*n**4*gamma(m/n + 1 + 1/n) + 6*a**3*b*n**4*x**n*gamma(m/n + 1 + 1/n) + 6*a**2*b**2*n**4*x**(2*n)*gamma(m/
n + 1 + 1/n) + 2*a*b**3*n**4*x**(3*n)*gamma(m/n + 1 + 1/n))) - 9*b**2*n*x*x**m*x**(2*n)*lerchphi(b*x**n*exp_po
lar(I*pi)/a, 1, m/n + 1/n)*gamma(m/n + 1/n)/(a*(2*a**4*n**4*gamma(m/n + 1 + 1/n) + 6*a**3*b*n**4*x**n*gamma(m/
n + 1 + 1/n) + 6*a**2*b**2*n**4*x**(2*n)*gamma(m/n + 1 + 1/n) + 2*a*b**3*n**4*x**(3*n)*gamma(m/n + 1 + 1/n)))
- b**2*n*x*x**m*x**(2*n)*gamma(m/n + 1/n)/(a*(2*a**4*n**4*gamma(m/n + 1 + 1/n) + 6*a**3*b*n**4*x**n*gamma(m/n
+ 1 + 1/n) + 6*a**2*b**2*n**4*x**(2*n)*gamma(m/n + 1 + 1/n) + 2*a*b**3*n**4*x**(3*n)*gamma(m/n + 1 + 1/n))) +
3*b**2*x*x**m*x**(2*n)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1/n)*gamma(m/n + 1/n)/(a*(2*a**4*n**4*gamma
(m/n + 1 + 1/n) + 6*a**3*b*n**4*x**n*gamma(m/n + 1 + 1/n) + 6*a**2*b**2*n**4*x**(2*n)*gamma(m/n + 1 + 1/n) + 2
*a*b**3*n**4*x**(3*n)*gamma(m/n + 1 + 1/n))) + b**3*m**3*x*x**m*x**(3*n)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1,
m/n + 1/n)*gamma(m/n + 1/n)/(a**2*(2*a**4*n**4*gamma(m/n + 1 + 1/n) + 6*a**3*b*n**4*x**n*gamma(m/n + 1 + 1/n)
+ 6*a**2*b**2*n**4*x**(2*n)*gamma(m/n + 1 + 1/n) + 2*a*b**3*n**4*x**(3*n)*gamma(m/n + 1 + 1/n))) - 3*b**3*m**
2*n*x*x**m*x**(3*n)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1/n)*gamma(m/n + 1/n)/(a**2*(2*a**4*n**4*gamma
(m/n + 1 + 1/n) + 6*a**3*b*n**4*x**n*gamma(m/n + 1 + 1/n) + 6*a**2*b**2*n**4*x**(2*n)*gamma(m/n + 1 + 1/n) + 2
*a*b**3*n**4*x**(3*n)*gamma(m/n + 1 + 1/n))) + 3*b**3*m**2*x*x**m*x**(3*n)*lerchphi(b*x**n*exp_polar(I*pi)/a,
1, m/n + 1/n)*gamma(m/n + 1/n)/(a**2*(2*a**4*n**4*gamma(m/n + 1 + 1/n) + 6*a**3*b*n**4*x**n*gamma(m/n + 1 + 1/
n) + 6*a**2*b**2*n**4*x**(2*n)*gamma(m/n + 1 + 1/n) + 2*a*b**3*n**4*x**(3*n)*gamma(m/n + 1 + 1/n))) + 2*b**3*m
*n**2*x*x**m*x**(3*n)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1/n)*gamma(m/n + 1/n)/(a**2*(2*a**4*n**4*gam
ma(m/n + 1 + 1/n) + 6*a**3*b*n**4*x**n*gamma(m/n + 1 + 1/n) + 6*a**2*b**2*n**4*x**(2*n)*gamma(m/n + 1 + 1/n) +
2*a*b**3*n**4*x**(3*n)*gamma(m/n + 1 + 1/n))) - 6*b**3*m*n*x*x**m*x**(3*n)*lerchphi(b*x**n*exp_polar(I*pi)/a,
1, m/n + 1/n)*gamma(m/n + 1/n)/(a**2*(2*a**4*n**4*gamma(m/n + 1 + 1/n) + 6*a**3*b*n**4*x**n*gamma(m/n + 1 + 1
/n) + 6*a**2*b**2*n**4*x**(2*n)*gamma(m/n + 1 + 1/n) + 2*a*b**3*n**4*x**(3*n)*gamma(m/n + 1 + 1/n))) + 3*b**3*
m*x*x**m*x**(3*n)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1/n)*gamma(m/n + 1/n)/(a**2*(2*a**4*n**4*gamma(m
/n + 1 + 1/n) + 6*a**3*b*n**4*x**n*gamma(m/n + 1 + 1/n) + 6*a**2*b**2*n**4*x**(2*n)*gamma(m/n + 1 + 1/n) + 2*a
*b**3*n**4*x**(3*n)*gamma(m/n + 1 + 1/n))) + 2*b**3*n**2*x*x**m*x**(3*n)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1,
m/n + 1/n)*gamma(m/n + 1/n)/(a**2*(2*a**4*n**4*gamma(m/n + 1 + 1/n) + 6*a**3*b*n**4*x**n*gamma(m/n + 1 + 1/n)
+ 6*a**2*b**2*n**4*x**(2*n)*gamma(m/n + 1 + 1/n) + 2*a*b**3*n**4*x**(3*n)*gamma(m/n + 1 + 1/n))) - 3*b**3*n*x
*x**m*x**(3*n)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1/n)*gamma(m/n + 1/n)/(a**2*(2*a**4*n**4*gamma(m/n
+ 1 + 1/n) + 6*a**3*b*n**4*x**n*gamma(m/n + 1 + 1/n) + 6*a**2*b**2*n**4*x**(2*n)*gamma(m/n + 1 + 1/n) + 2*a*b*
*3*n**4*x**(3*n)*gamma(m/n + 1 + 1/n))) + b**3*x*x**m*x**(3*n)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1/n
)*gamma(m/n + 1/n)/(a**2*(2*a**4*n**4*gamma(m/n + 1 + 1/n) + 6*a**3*b*n**4*x**n*gamma(m/n + 1 + 1/n) + 6*a**2*
b**2*n**4*x**(2*n)*gamma(m/n + 1 + 1/n) + 2*a*b**3*n**4*x**(3*n)*gamma(m/n + 1 + 1/n)))
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