3.6 \(\int \frac {(e x)^m (A+B x^n) (c+d x^n)}{(a+b x^n)^2} \, dx\)
Optimal. Leaf size=177 \[ \frac {(e x)^{m+1} \, _2F_1\left (1,\frac {m+1}{n};\frac {m+n+1}{n};-\frac {b x^n}{a}\right ) (b c (a B (m+1)-A b (m-n+1))+a d (A b (m+1)-a B (m+n+1)))}{a^2 b^2 e (m+1) n}-\frac {d (e x)^{m+1} (A b (m+1)-a B (m+n+1))}{a b^2 e (m+1) n}+\frac {(e x)^{m+1} (A b-a B) \left (c+d x^n\right )}{a b e n \left (a+b x^n\right )} \]
[Out]
-d*(A*b*(1+m)-a*B*(1+m+n))*(e*x)^(1+m)/a/b^2/e/(1+m)/n+(A*b-B*a)*(e*x)^(1+m)*(c+d*x^n)/a/b/e/n/(a+b*x^n)+(b*c*
(a*B*(1+m)-A*b*(1+m-n))+a*d*(A*b*(1+m)-a*B*(1+m+n)))*(e*x)^(1+m)*hypergeom([1, (1+m)/n],[(1+m+n)/n],-b*x^n/a)/
a^2/b^2/e/(1+m)/n
________________________________________________________________________________________
Rubi [A] time = 0.26, antiderivative size = 177, normalized size of antiderivative = 1.00,
number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used =
{594, 459, 364} \[ \frac {(e x)^{m+1} \, _2F_1\left (1,\frac {m+1}{n};\frac {m+n+1}{n};-\frac {b x^n}{a}\right ) (b c (a B (m+1)-A b (m-n+1))+a d (A b (m+1)-a B (m+n+1)))}{a^2 b^2 e (m+1) n}-\frac {d (e x)^{m+1} (A b (m+1)-a B (m+n+1))}{a b^2 e (m+1) n}+\frac {(e x)^{m+1} (A b-a B) \left (c+d x^n\right )}{a b e n \left (a+b x^n\right )} \]
Antiderivative was successfully verified.
[In]
Int[((e*x)^m*(A + B*x^n)*(c + d*x^n))/(a + b*x^n)^2,x]
[Out]
-((d*(A*b*(1 + m) - a*B*(1 + m + n))*(e*x)^(1 + m))/(a*b^2*e*(1 + m)*n)) + ((A*b - a*B)*(e*x)^(1 + m)*(c + d*x
^n))/(a*b*e*n*(a + b*x^n)) + ((b*c*(a*B*(1 + m) - A*b*(1 + m - n)) + a*d*(A*b*(1 + m) - a*B*(1 + m + n)))*(e*x
)^(1 + m)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)])/(a^2*b^2*e*(1 + m)*n)
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])
Rule 459
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(d*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(b*e*(m + n*(p + 1) + 1)), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m +
n*(p + 1) + 1)), Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0]
&& NeQ[m + n*(p + 1) + 1, 0]
Rule 594
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> -Simp[((b*e - a*f)*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q)/(a*b*g*n*(p + 1)), x] + Dist[
1/(a*b*n*(p + 1)), Int[(g*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1)*Simp[c*(b*e*n*(p + 1) + (b*e - a*f)*(m
+ 1)) + d*(b*e*n*(p + 1) + (b*e - a*f)*(m + n*q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x]
&& LtQ[p, -1] && GtQ[q, 0] && !(EqQ[q, 1] && SimplerQ[b*c - a*d, b*e - a*f])
Rubi steps
\begin {align*} \int \frac {(e x)^m \left (A+B x^n\right ) \left (c+d x^n\right )}{\left (a+b x^n\right )^2} \, dx &=\frac {(A b-a B) (e x)^{1+m} \left (c+d x^n\right )}{a b e n \left (a+b x^n\right )}-\frac {\int \frac {(e x)^m \left (-c (a B (1+m)-A b (1+m-n))+d (A b (1+m)-a B (1+m+n)) x^n\right )}{a+b x^n} \, dx}{a b n}\\ &=-\frac {d (A b (1+m)-a B (1+m+n)) (e x)^{1+m}}{a b^2 e (1+m) n}+\frac {(A b-a B) (e x)^{1+m} \left (c+d x^n\right )}{a b e n \left (a+b x^n\right )}+\frac {(b c (a B (1+m)-A b (1+m-n))+a d (A b (1+m)-a B (1+m+n))) \int \frac {(e x)^m}{a+b x^n} \, dx}{a b^2 n}\\ &=-\frac {d (A b (1+m)-a B (1+m+n)) (e x)^{1+m}}{a b^2 e (1+m) n}+\frac {(A b-a B) (e x)^{1+m} \left (c+d x^n\right )}{a b e n \left (a+b x^n\right )}+\frac {(b c (a B (1+m)-A b (1+m-n))+a d (A b (1+m)-a B (1+m+n))) (e x)^{1+m} \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b x^n}{a}\right )}{a^2 b^2 e (1+m) n}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.24, size = 110, normalized size = 0.62 \[ \frac {x (e x)^m \left (a^2 B d+a (-2 a B d+A b d+b B c) \, _2F_1\left (1,\frac {m+1}{n};\frac {m+n+1}{n};-\frac {b x^n}{a}\right )+(A b-a B) (b c-a d) \, _2F_1\left (2,\frac {m+1}{n};\frac {m+n+1}{n};-\frac {b x^n}{a}\right )\right )}{a^2 b^2 (m+1)} \]
Antiderivative was successfully verified.
[In]
Integrate[((e*x)^m*(A + B*x^n)*(c + d*x^n))/(a + b*x^n)^2,x]
[Out]
(x*(e*x)^m*(a^2*B*d + a*(b*B*c + A*b*d - 2*a*B*d)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)]
+ (A*b - a*B)*(b*c - a*d)*Hypergeometric2F1[2, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)]))/(a^2*b^2*(1 + m))
________________________________________________________________________________________
fricas [F] time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B d x^{2 \, n} + A c + {\left (B c + A d\right )} x^{n}\right )} \left (e x\right )^{m}}{b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x)^m*(A+B*x^n)*(c+d*x^n)/(a+b*x^n)^2,x, algorithm="fricas")
[Out]
integral((B*d*x^(2*n) + A*c + (B*c + A*d)*x^n)*(e*x)^m/(b^2*x^(2*n) + 2*a*b*x^n + a^2), x)
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B x^{n} + A\right )} {\left (d x^{n} + c\right )} \left (e x\right )^{m}}{{\left (b x^{n} + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x)^m*(A+B*x^n)*(c+d*x^n)/(a+b*x^n)^2,x, algorithm="giac")
[Out]
integrate((B*x^n + A)*(d*x^n + c)*(e*x)^m/(b*x^n + a)^2, x)
________________________________________________________________________________________
maple [F] time = 0.67, size = 0, normalized size = 0.00 \[ \int \frac {\left (B \,x^{n}+A \right ) \left (d \,x^{n}+c \right ) \left (e x \right )^{m}}{\left (b \,x^{n}+a \right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((e*x)^m*(B*x^n+A)*(d*x^n+c)/(b*x^n+a)^2,x)
[Out]
int((e*x)^m*(B*x^n+A)*(d*x^n+c)/(b*x^n+a)^2,x)
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -{\left ({\left (b^{2} c e^{m} {\left (m - n + 1\right )} - a b d e^{m} {\left (m + 1\right )}\right )} A + {\left (a^{2} d e^{m} {\left (m + n + 1\right )} - a b c e^{m} {\left (m + 1\right )}\right )} B\right )} \int \frac {x^{m}}{a b^{3} n x^{n} + a^{2} b^{2} n}\,{d x} + \frac {B a b d e^{m} n x e^{\left (m \log \relax (x) + n \log \relax (x)\right )} + {\left ({\left (b^{2} c e^{m} {\left (m + 1\right )} - a b d e^{m} {\left (m + 1\right )}\right )} A + {\left (a^{2} d e^{m} {\left (m + n + 1\right )} - a b c e^{m} {\left (m + 1\right )}\right )} B\right )} x x^{m}}{{\left (m n + n\right )} a b^{3} x^{n} + {\left (m n + n\right )} a^{2} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x)^m*(A+B*x^n)*(c+d*x^n)/(a+b*x^n)^2,x, algorithm="maxima")
[Out]
-((b^2*c*e^m*(m - n + 1) - a*b*d*e^m*(m + 1))*A + (a^2*d*e^m*(m + n + 1) - a*b*c*e^m*(m + 1))*B)*integrate(x^m
/(a*b^3*n*x^n + a^2*b^2*n), x) + (B*a*b*d*e^m*n*x*e^(m*log(x) + n*log(x)) + ((b^2*c*e^m*(m + 1) - a*b*d*e^m*(m
+ 1))*A + (a^2*d*e^m*(m + n + 1) - a*b*c*e^m*(m + 1))*B)*x*x^m)/((m*n + n)*a*b^3*x^n + (m*n + n)*a^2*b^2)
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (e\,x\right )}^m\,\left (A+B\,x^n\right )\,\left (c+d\,x^n\right )}{{\left (a+b\,x^n\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((e*x)^m*(A + B*x^n)*(c + d*x^n))/(a + b*x^n)^2,x)
[Out]
int(((e*x)^m*(A + B*x^n)*(c + d*x^n))/(a + b*x^n)^2, x)
________________________________________________________________________________________
sympy [C] time = 48.65, size = 4129, normalized size = 23.33 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x)**m*(A+B*x**n)*(c+d*x**n)/(a+b*x**n)**2,x)
[Out]
A*c*(-e**m*m**2*x*x**m*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1/n)*gamma(m/n + 1/n)/(a*(a*n**3*gamma(m/n
+ 1 + 1/n) + b*n**3*x**n*gamma(m/n + 1 + 1/n))) + e**m*m*n*x*x**m*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n +
1/n)*gamma(m/n + 1/n)/(a*(a*n**3*gamma(m/n + 1 + 1/n) + b*n**3*x**n*gamma(m/n + 1 + 1/n))) + e**m*m*n*x*x**m*g
amma(m/n + 1/n)/(a*(a*n**3*gamma(m/n + 1 + 1/n) + b*n**3*x**n*gamma(m/n + 1 + 1/n))) - 2*e**m*m*x*x**m*lerchph
i(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1/n)*gamma(m/n + 1/n)/(a*(a*n**3*gamma(m/n + 1 + 1/n) + b*n**3*x**n*gamma
(m/n + 1 + 1/n))) + e**m*n*x*x**m*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1/n)*gamma(m/n + 1/n)/(a*(a*n**3
*gamma(m/n + 1 + 1/n) + b*n**3*x**n*gamma(m/n + 1 + 1/n))) + e**m*n*x*x**m*gamma(m/n + 1/n)/(a*(a*n**3*gamma(m
/n + 1 + 1/n) + b*n**3*x**n*gamma(m/n + 1 + 1/n))) - e**m*x*x**m*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1
/n)*gamma(m/n + 1/n)/(a*(a*n**3*gamma(m/n + 1 + 1/n) + b*n**3*x**n*gamma(m/n + 1 + 1/n))) - b*e**m*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)/(a**2*(a*n**3*gamma(m/n + 1 + 1/n) + b
*n**3*x**n*gamma(m/n + 1 + 1/n))) + b*e**m*m*n*x*x**m*x**n*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1/n)*ga
mma(m/n + 1/n)/(a**2*(a*n**3*gamma(m/n + 1 + 1/n) + b*n**3*x**n*gamma(m/n + 1 + 1/n))) - 2*b*e**m*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)/(a**2*(a*n**3*gamma(m/n + 1 + 1/n) + b*n**
3*x**n*gamma(m/n + 1 + 1/n))) + b*e**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)/(a**2*(a*n**3*gamma(m/n + 1 + 1/n) + b*n**3*x**n*gamma(m/n + 1 + 1/n))) - b*e**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)/(a**2*(a*n**3*gamma(m/n + 1 + 1/n) + b*n**3*x**n*gam
ma(m/n + 1 + 1/n)))) + A*d*(-e**m*m**2*x*x**m*x**n*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1 + 1/n)*gamma(
m/n + 1 + 1/n)/(a*(a*n**3*gamma(m/n + 2 + 1/n) + b*n**3*x**n*gamma(m/n + 2 + 1/n))) - e**m*m*n*x*x**m*x**n*ler
chphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1 + 1/n)*gamma(m/n + 1 + 1/n)/(a*(a*n**3*gamma(m/n + 2 + 1/n) + b*n**
3*x**n*gamma(m/n + 2 + 1/n))) + e**m*m*n*x*x**m*x**n*gamma(m/n + 1 + 1/n)/(a*(a*n**3*gamma(m/n + 2 + 1/n) + b*
n**3*x**n*gamma(m/n + 2 + 1/n))) - 2*e**m*m*x*x**m*x**n*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1 + 1/n)*g
amma(m/n + 1 + 1/n)/(a*(a*n**3*gamma(m/n + 2 + 1/n) + b*n**3*x**n*gamma(m/n + 2 + 1/n))) + e**m*n**2*x*x**m*x*
*n*gamma(m/n + 1 + 1/n)/(a*(a*n**3*gamma(m/n + 2 + 1/n) + b*n**3*x**n*gamma(m/n + 2 + 1/n))) - e**m*n*x*x**m*x
**n*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1 + 1/n)*gamma(m/n + 1 + 1/n)/(a*(a*n**3*gamma(m/n + 2 + 1/n)
+ b*n**3*x**n*gamma(m/n + 2 + 1/n))) + e**m*n*x*x**m*x**n*gamma(m/n + 1 + 1/n)/(a*(a*n**3*gamma(m/n + 2 + 1/n)
+ b*n**3*x**n*gamma(m/n + 2 + 1/n))) - e**m*x*x**m*x**n*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1 + 1/n)*
gamma(m/n + 1 + 1/n)/(a*(a*n**3*gamma(m/n + 2 + 1/n) + b*n**3*x**n*gamma(m/n + 2 + 1/n))) - b*e**m*m**2*x*x**m
*x**(2*n)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1 + 1/n)*gamma(m/n + 1 + 1/n)/(a**2*(a*n**3*gamma(m/n +
2 + 1/n) + b*n**3*x**n*gamma(m/n + 2 + 1/n))) - b*e**m*m*n*x*x**m*x**(2*n)*lerchphi(b*x**n*exp_polar(I*pi)/a,
1, m/n + 1 + 1/n)*gamma(m/n + 1 + 1/n)/(a**2*(a*n**3*gamma(m/n + 2 + 1/n) + b*n**3*x**n*gamma(m/n + 2 + 1/n)))
- 2*b*e**m*m*x*x**m*x**(2*n)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1 + 1/n)*gamma(m/n + 1 + 1/n)/(a**2*
(a*n**3*gamma(m/n + 2 + 1/n) + b*n**3*x**n*gamma(m/n + 2 + 1/n))) - b*e**m*n*x*x**m*x**(2*n)*lerchphi(b*x**n*e
xp_polar(I*pi)/a, 1, m/n + 1 + 1/n)*gamma(m/n + 1 + 1/n)/(a**2*(a*n**3*gamma(m/n + 2 + 1/n) + b*n**3*x**n*gamm
a(m/n + 2 + 1/n))) - b*e**m*x*x**m*x**(2*n)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1 + 1/n)*gamma(m/n + 1
+ 1/n)/(a**2*(a*n**3*gamma(m/n + 2 + 1/n) + b*n**3*x**n*gamma(m/n + 2 + 1/n)))) + B*c*(-e**m*m**2*x*x**m*x**n
*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1 + 1/n)*gamma(m/n + 1 + 1/n)/(a*(a*n**3*gamma(m/n + 2 + 1/n) + b
*n**3*x**n*gamma(m/n + 2 + 1/n))) - e**m*m*n*x*x**m*x**n*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1 + 1/n)*
gamma(m/n + 1 + 1/n)/(a*(a*n**3*gamma(m/n + 2 + 1/n) + b*n**3*x**n*gamma(m/n + 2 + 1/n))) + e**m*m*n*x*x**m*x*
*n*gamma(m/n + 1 + 1/n)/(a*(a*n**3*gamma(m/n + 2 + 1/n) + b*n**3*x**n*gamma(m/n + 2 + 1/n))) - 2*e**m*m*x*x**m
*x**n*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1 + 1/n)*gamma(m/n + 1 + 1/n)/(a*(a*n**3*gamma(m/n + 2 + 1/n
) + b*n**3*x**n*gamma(m/n + 2 + 1/n))) + e**m*n**2*x*x**m*x**n*gamma(m/n + 1 + 1/n)/(a*(a*n**3*gamma(m/n + 2 +
1/n) + b*n**3*x**n*gamma(m/n + 2 + 1/n))) - e**m*n*x*x**m*x**n*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1
+ 1/n)*gamma(m/n + 1 + 1/n)/(a*(a*n**3*gamma(m/n + 2 + 1/n) + b*n**3*x**n*gamma(m/n + 2 + 1/n))) + e**m*n*x*x*
*m*x**n*gamma(m/n + 1 + 1/n)/(a*(a*n**3*gamma(m/n + 2 + 1/n) + b*n**3*x**n*gamma(m/n + 2 + 1/n))) - e**m*x*x**
m*x**n*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1 + 1/n)*gamma(m/n + 1 + 1/n)/(a*(a*n**3*gamma(m/n + 2 + 1/
n) + b*n**3*x**n*gamma(m/n + 2 + 1/n))) - b*e**m*m**2*x*x**m*x**(2*n)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/
n + 1 + 1/n)*gamma(m/n + 1 + 1/n)/(a**2*(a*n**3*gamma(m/n + 2 + 1/n) + b*n**3*x**n*gamma(m/n + 2 + 1/n))) - b*
e**m*m*n*x*x**m*x**(2*n)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1 + 1/n)*gamma(m/n + 1 + 1/n)/(a**2*(a*n*
*3*gamma(m/n + 2 + 1/n) + b*n**3*x**n*gamma(m/n + 2 + 1/n))) - 2*b*e**m*m*x*x**m*x**(2*n)*lerchphi(b*x**n*exp_
polar(I*pi)/a, 1, m/n + 1 + 1/n)*gamma(m/n + 1 + 1/n)/(a**2*(a*n**3*gamma(m/n + 2 + 1/n) + b*n**3*x**n*gamma(m
/n + 2 + 1/n))) - b*e**m*n*x*x**m*x**(2*n)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1 + 1/n)*gamma(m/n + 1
+ 1/n)/(a**2*(a*n**3*gamma(m/n + 2 + 1/n) + b*n**3*x**n*gamma(m/n + 2 + 1/n))) - b*e**m*x*x**m*x**(2*n)*lerchp
hi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1 + 1/n)*gamma(m/n + 1 + 1/n)/(a**2*(a*n**3*gamma(m/n + 2 + 1/n) + b*n**
3*x**n*gamma(m/n + 2 + 1/n)))) + B*d*(-e**m*m**2*x*x**m*x**(2*n)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 2
+ 1/n)*gamma(m/n + 2 + 1/n)/(a*(a*n**3*gamma(m/n + 3 + 1/n) + b*n**3*x**n*gamma(m/n + 3 + 1/n))) - 3*e**m*m*n
*x*x**m*x**(2*n)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 2 + 1/n)*gamma(m/n + 2 + 1/n)/(a*(a*n**3*gamma(m/
n + 3 + 1/n) + b*n**3*x**n*gamma(m/n + 3 + 1/n))) + e**m*m*n*x*x**m*x**(2*n)*gamma(m/n + 2 + 1/n)/(a*(a*n**3*g
amma(m/n + 3 + 1/n) + b*n**3*x**n*gamma(m/n + 3 + 1/n))) - 2*e**m*m*x*x**m*x**(2*n)*lerchphi(b*x**n*exp_polar(
I*pi)/a, 1, m/n + 2 + 1/n)*gamma(m/n + 2 + 1/n)/(a*(a*n**3*gamma(m/n + 3 + 1/n) + b*n**3*x**n*gamma(m/n + 3 +
1/n))) - 2*e**m*n**2*x*x**m*x**(2*n)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 2 + 1/n)*gamma(m/n + 2 + 1/n)
/(a*(a*n**3*gamma(m/n + 3 + 1/n) + b*n**3*x**n*gamma(m/n + 3 + 1/n))) + 2*e**m*n**2*x*x**m*x**(2*n)*gamma(m/n
+ 2 + 1/n)/(a*(a*n**3*gamma(m/n + 3 + 1/n) + b*n**3*x**n*gamma(m/n + 3 + 1/n))) - 3*e**m*n*x*x**m*x**(2*n)*ler
chphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 2 + 1/n)*gamma(m/n + 2 + 1/n)/(a*(a*n**3*gamma(m/n + 3 + 1/n) + b*n**
3*x**n*gamma(m/n + 3 + 1/n))) + e**m*n*x*x**m*x**(2*n)*gamma(m/n + 2 + 1/n)/(a*(a*n**3*gamma(m/n + 3 + 1/n) +
b*n**3*x**n*gamma(m/n + 3 + 1/n))) - e**m*x*x**m*x**(2*n)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 2 + 1/n)
*gamma(m/n + 2 + 1/n)/(a*(a*n**3*gamma(m/n + 3 + 1/n) + b*n**3*x**n*gamma(m/n + 3 + 1/n))) - b*e**m*m**2*x*x**
m*x**(3*n)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 2 + 1/n)*gamma(m/n + 2 + 1/n)/(a**2*(a*n**3*gamma(m/n +
3 + 1/n) + b*n**3*x**n*gamma(m/n + 3 + 1/n))) - 3*b*e**m*m*n*x*x**m*x**(3*n)*lerchphi(b*x**n*exp_polar(I*pi)/
a, 1, m/n + 2 + 1/n)*gamma(m/n + 2 + 1/n)/(a**2*(a*n**3*gamma(m/n + 3 + 1/n) + b*n**3*x**n*gamma(m/n + 3 + 1/n
))) - 2*b*e**m*m*x*x**m*x**(3*n)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 2 + 1/n)*gamma(m/n + 2 + 1/n)/(a*
*2*(a*n**3*gamma(m/n + 3 + 1/n) + b*n**3*x**n*gamma(m/n + 3 + 1/n))) - 2*b*e**m*n**2*x*x**m*x**(3*n)*lerchphi(
b*x**n*exp_polar(I*pi)/a, 1, m/n + 2 + 1/n)*gamma(m/n + 2 + 1/n)/(a**2*(a*n**3*gamma(m/n + 3 + 1/n) + b*n**3*x
**n*gamma(m/n + 3 + 1/n))) - 3*b*e**m*n*x*x**m*x**(3*n)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 2 + 1/n)*g
amma(m/n + 2 + 1/n)/(a**2*(a*n**3*gamma(m/n + 3 + 1/n) + b*n**3*x**n*gamma(m/n + 3 + 1/n))) - b*e**m*x*x**m*x*
*(3*n)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 2 + 1/n)*gamma(m/n + 2 + 1/n)/(a**2*(a*n**3*gamma(m/n + 3 +
1/n) + b*n**3*x**n*gamma(m/n + 3 + 1/n))))
________________________________________________________________________________________