3.61 \(\int \frac{(e x)^m}{(2-2 a x)^4 (1+a x)^3} \, dx\)
Optimal. Leaf size=86 \[ \frac{a (e x)^{m+2} \, _2F_1\left (4,\frac{m+2}{2};\frac{m+4}{2};a^2 x^2\right )}{16 e^2 (m+2)}+\frac{(e x)^{m+1} \, _2F_1\left (4,\frac{m+1}{2};\frac{m+3}{2};a^2 x^2\right )}{16 e (m+1)} \]
[Out]
((e*x)^(1 + m)*Hypergeometric2F1[4, (1 + m)/2, (3 + m)/2, a^2*x^2])/(16*e*(1 + m)) + (a*(e*x)^(2 + m)*Hypergeo
metric2F1[4, (2 + m)/2, (4 + m)/2, a^2*x^2])/(16*e^2*(2 + m))
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Rubi [A] time = 0.0236572, antiderivative size = 86, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used =
{82, 73, 364} \[ \frac{a (e x)^{m+2} \, _2F_1\left (4,\frac{m+2}{2};\frac{m+4}{2};a^2 x^2\right )}{16 e^2 (m+2)}+\frac{(e x)^{m+1} \, _2F_1\left (4,\frac{m+1}{2};\frac{m+3}{2};a^2 x^2\right )}{16 e (m+1)} \]
Antiderivative was successfully verified.
[In]
Int[(e*x)^m/((2 - 2*a*x)^4*(1 + a*x)^3),x]
[Out]
((e*x)^(1 + m)*Hypergeometric2F1[4, (1 + m)/2, (3 + m)/2, a^2*x^2])/(16*e*(1 + m)) + (a*(e*x)^(2 + m)*Hypergeo
metric2F1[4, (2 + m)/2, (4 + m)/2, a^2*x^2])/(16*e^2*(2 + m))
Rule 82
Int[((f_.)*(x_))^(p_.)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Dist[a, Int[(a + b*
x)^n*(c + d*x)^n*(f*x)^p, x], x] + Dist[b/f, Int[(a + b*x)^n*(c + d*x)^n*(f*x)^(p + 1), x], x] /; FreeQ[{a, b,
c, d, f, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[m - n - 1, 0] && !RationalQ[p] && !IGtQ[m, 0] && NeQ[m +
n + p + 2, 0]
Rule 73
Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[(a*c + b*
d*x^2)^m*(e + f*x)^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[n, m] && Integer
Q[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{(e x)^m}{(2-2 a x)^4 (1+a x)^3} \, dx &=\frac{a \int \frac{(e x)^{1+m}}{(2-2 a x)^4 (1+a x)^4} \, dx}{e}+\int \frac{(e x)^m}{(2-2 a x)^4 (1+a x)^4} \, dx\\ &=\frac{a \int \frac{(e x)^{1+m}}{\left (2-2 a^2 x^2\right )^4} \, dx}{e}+\int \frac{(e x)^m}{\left (2-2 a^2 x^2\right )^4} \, dx\\ &=\frac{(e x)^{1+m} \, _2F_1\left (4,\frac{1+m}{2};\frac{3+m}{2};a^2 x^2\right )}{16 e (1+m)}+\frac{a (e x)^{2+m} \, _2F_1\left (4,\frac{2+m}{2};\frac{4+m}{2};a^2 x^2\right )}{16 e^2 (2+m)}\\ \end{align*}
Mathematica [A] time = 0.0268563, size = 77, normalized size = 0.9 \[ \frac{x (e x)^m \left (a (m+1) x \, _2F_1\left (4,\frac{m}{2}+1;\frac{m}{2}+2;a^2 x^2\right )+(m+2) \, _2F_1\left (4,\frac{m+1}{2};\frac{m+3}{2};a^2 x^2\right )\right )}{16 (m+1) (m+2)} \]
Antiderivative was successfully verified.
[In]
Integrate[(e*x)^m/((2 - 2*a*x)^4*(1 + a*x)^3),x]
[Out]
(x*(e*x)^m*(a*(1 + m)*x*Hypergeometric2F1[4, 1 + m/2, 2 + m/2, a^2*x^2] + (2 + m)*Hypergeometric2F1[4, (1 + m)
/2, (3 + m)/2, a^2*x^2]))/(16*(1 + m)*(2 + m))
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Maple [F] time = 0.089, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex \right ) ^{m}}{ \left ( -2\,ax+2 \right ) ^{4} \left ( ax+1 \right ) ^{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((e*x)^m/(-2*a*x+2)^4/(a*x+1)^3,x)
[Out]
int((e*x)^m/(-2*a*x+2)^4/(a*x+1)^3,x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{16} \, \int \frac{\left (e x\right )^{m}}{{\left (a x + 1\right )}^{3}{\left (a x - 1\right )}^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x)^m/(-2*a*x+2)^4/(a*x+1)^3,x, algorithm="maxima")
[Out]
1/16*integrate((e*x)^m/((a*x + 1)^3*(a*x - 1)^4), x)
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\left (e x\right )^{m}}{16 \,{\left (a^{7} x^{7} - a^{6} x^{6} - 3 \, a^{5} x^{5} + 3 \, a^{4} x^{4} + 3 \, a^{3} x^{3} - 3 \, a^{2} x^{2} - a x + 1\right )}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x)^m/(-2*a*x+2)^4/(a*x+1)^3,x, algorithm="fricas")
[Out]
integral(1/16*(e*x)^m/(a^7*x^7 - a^6*x^6 - 3*a^5*x^5 + 3*a^4*x^4 + 3*a^3*x^3 - 3*a^2*x^2 - a*x + 1), x)
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Sympy [C] time = 10.0606, size = 5872, normalized size = 68.28 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x)**m/(-2*a*x+2)**4/(a*x+1)**3,x)
[Out]
2*a**5*e**m*m**4*x**5*x**m*lerchphi(1/(a*x), 1, m*exp_polar(I*pi))*gamma(-m)/(1536*a**6*x**5*gamma(1 - m) - 15
36*a**5*x**4*gamma(1 - m) - 3072*a**4*x**3*gamma(1 - m) + 3072*a**3*x**2*gamma(1 - m) + 1536*a**2*x*gamma(1 -
m) - 1536*a*gamma(1 - m)) - 15*a**5*e**m*m**3*x**5*x**m*lerchphi(1/(a*x), 1, m*exp_polar(I*pi))*gamma(-m)/(153
6*a**6*x**5*gamma(1 - m) - 1536*a**5*x**4*gamma(1 - m) - 3072*a**4*x**3*gamma(1 - m) + 3072*a**3*x**2*gamma(1
- m) + 1536*a**2*x*gamma(1 - m) - 1536*a*gamma(1 - m)) + 3*a**5*e**m*m**3*x**5*x**m*lerchphi(exp_polar(I*pi)/(
a*x), 1, m*exp_polar(I*pi))*gamma(-m)/(1536*a**6*x**5*gamma(1 - m) - 1536*a**5*x**4*gamma(1 - m) - 3072*a**4*x
**3*gamma(1 - m) + 3072*a**3*x**2*gamma(1 - m) + 1536*a**2*x*gamma(1 - m) - 1536*a*gamma(1 - m)) + 2*a**5*e**m
*m**3*x**5*x**m*gamma(-m)/(1536*a**6*x**5*gamma(1 - m) - 1536*a**5*x**4*gamma(1 - m) - 3072*a**4*x**3*gamma(1
- m) + 3072*a**3*x**2*gamma(1 - m) + 1536*a**2*x*gamma(1 - m) - 1536*a*gamma(1 - m)) + 31*a**5*e**m*m**2*x**5*
x**m*lerchphi(1/(a*x), 1, m*exp_polar(I*pi))*gamma(-m)/(1536*a**6*x**5*gamma(1 - m) - 1536*a**5*x**4*gamma(1 -
m) - 3072*a**4*x**3*gamma(1 - m) + 3072*a**3*x**2*gamma(1 - m) + 1536*a**2*x*gamma(1 - m) - 1536*a*gamma(1 -
m)) - 15*a**5*e**m*m**2*x**5*x**m*lerchphi(exp_polar(I*pi)/(a*x), 1, m*exp_polar(I*pi))*gamma(-m)/(1536*a**6*x
**5*gamma(1 - m) - 1536*a**5*x**4*gamma(1 - m) - 3072*a**4*x**3*gamma(1 - m) + 3072*a**3*x**2*gamma(1 - m) + 1
536*a**2*x*gamma(1 - m) - 1536*a*gamma(1 - m)) - 12*a**5*e**m*m**2*x**5*x**m*gamma(-m)/(1536*a**6*x**5*gamma(1
- m) - 1536*a**5*x**4*gamma(1 - m) - 3072*a**4*x**3*gamma(1 - m) + 3072*a**3*x**2*gamma(1 - m) + 1536*a**2*x*
gamma(1 - m) - 1536*a*gamma(1 - m)) - 15*a**5*e**m*m*x**5*x**m*lerchphi(1/(a*x), 1, m*exp_polar(I*pi))*gamma(-
m)/(1536*a**6*x**5*gamma(1 - m) - 1536*a**5*x**4*gamma(1 - m) - 3072*a**4*x**3*gamma(1 - m) + 3072*a**3*x**2*g
amma(1 - m) + 1536*a**2*x*gamma(1 - m) - 1536*a*gamma(1 - m)) + 15*a**5*e**m*m*x**5*x**m*lerchphi(exp_polar(I*
pi)/(a*x), 1, m*exp_polar(I*pi))*gamma(-m)/(1536*a**6*x**5*gamma(1 - m) - 1536*a**5*x**4*gamma(1 - m) - 3072*a
**4*x**3*gamma(1 - m) + 3072*a**3*x**2*gamma(1 - m) + 1536*a**2*x*gamma(1 - m) - 1536*a*gamma(1 - m)) + 16*a**
5*e**m*m*x**5*x**m*gamma(-m)/(1536*a**6*x**5*gamma(1 - m) - 1536*a**5*x**4*gamma(1 - m) - 3072*a**4*x**3*gamma
(1 - m) + 3072*a**3*x**2*gamma(1 - m) + 1536*a**2*x*gamma(1 - m) - 1536*a*gamma(1 - m)) - 2*a**4*e**m*m**4*x**
4*x**m*lerchphi(1/(a*x), 1, m*exp_polar(I*pi))*gamma(-m)/(1536*a**6*x**5*gamma(1 - m) - 1536*a**5*x**4*gamma(1
- m) - 3072*a**4*x**3*gamma(1 - m) + 3072*a**3*x**2*gamma(1 - m) + 1536*a**2*x*gamma(1 - m) - 1536*a*gamma(1
- m)) + 15*a**4*e**m*m**3*x**4*x**m*lerchphi(1/(a*x), 1, m*exp_polar(I*pi))*gamma(-m)/(1536*a**6*x**5*gamma(1
- m) - 1536*a**5*x**4*gamma(1 - m) - 3072*a**4*x**3*gamma(1 - m) + 3072*a**3*x**2*gamma(1 - m) + 1536*a**2*x*g
amma(1 - m) - 1536*a*gamma(1 - m)) - 3*a**4*e**m*m**3*x**4*x**m*lerchphi(exp_polar(I*pi)/(a*x), 1, m*exp_polar
(I*pi))*gamma(-m)/(1536*a**6*x**5*gamma(1 - m) - 1536*a**5*x**4*gamma(1 - m) - 3072*a**4*x**3*gamma(1 - m) + 3
072*a**3*x**2*gamma(1 - m) + 1536*a**2*x*gamma(1 - m) - 1536*a*gamma(1 - m)) - 31*a**4*e**m*m**2*x**4*x**m*ler
chphi(1/(a*x), 1, m*exp_polar(I*pi))*gamma(-m)/(1536*a**6*x**5*gamma(1 - m) - 1536*a**5*x**4*gamma(1 - m) - 30
72*a**4*x**3*gamma(1 - m) + 3072*a**3*x**2*gamma(1 - m) + 1536*a**2*x*gamma(1 - m) - 1536*a*gamma(1 - m)) + 15
*a**4*e**m*m**2*x**4*x**m*lerchphi(exp_polar(I*pi)/(a*x), 1, m*exp_polar(I*pi))*gamma(-m)/(1536*a**6*x**5*gamm
a(1 - m) - 1536*a**5*x**4*gamma(1 - m) - 3072*a**4*x**3*gamma(1 - m) + 3072*a**3*x**2*gamma(1 - m) + 1536*a**2
*x*gamma(1 - m) - 1536*a*gamma(1 - m)) - 4*a**4*e**m*m**2*x**4*x**m*gamma(-m)/(1536*a**6*x**5*gamma(1 - m) - 1
536*a**5*x**4*gamma(1 - m) - 3072*a**4*x**3*gamma(1 - m) + 3072*a**3*x**2*gamma(1 - m) + 1536*a**2*x*gamma(1 -
m) - 1536*a*gamma(1 - m)) + 15*a**4*e**m*m*x**4*x**m*lerchphi(1/(a*x), 1, m*exp_polar(I*pi))*gamma(-m)/(1536*
a**6*x**5*gamma(1 - m) - 1536*a**5*x**4*gamma(1 - m) - 3072*a**4*x**3*gamma(1 - m) + 3072*a**3*x**2*gamma(1 -
m) + 1536*a**2*x*gamma(1 - m) - 1536*a*gamma(1 - m)) - 15*a**4*e**m*m*x**4*x**m*lerchphi(exp_polar(I*pi)/(a*x)
, 1, m*exp_polar(I*pi))*gamma(-m)/(1536*a**6*x**5*gamma(1 - m) - 1536*a**5*x**4*gamma(1 - m) - 3072*a**4*x**3*
gamma(1 - m) + 3072*a**3*x**2*gamma(1 - m) + 1536*a**2*x*gamma(1 - m) - 1536*a*gamma(1 - m)) + 14*a**4*e**m*m*
x**4*x**m*gamma(-m)/(1536*a**6*x**5*gamma(1 - m) - 1536*a**5*x**4*gamma(1 - m) - 3072*a**4*x**3*gamma(1 - m) +
3072*a**3*x**2*gamma(1 - m) + 1536*a**2*x*gamma(1 - m) - 1536*a*gamma(1 - m)) - 4*a**3*e**m*m**4*x**3*x**m*le
rchphi(1/(a*x), 1, m*exp_polar(I*pi))*gamma(-m)/(1536*a**6*x**5*gamma(1 - m) - 1536*a**5*x**4*gamma(1 - m) - 3
072*a**4*x**3*gamma(1 - m) + 3072*a**3*x**2*gamma(1 - m) + 1536*a**2*x*gamma(1 - m) - 1536*a*gamma(1 - m)) + 3
0*a**3*e**m*m**3*x**3*x**m*lerchphi(1/(a*x), 1, m*exp_polar(I*pi))*gamma(-m)/(1536*a**6*x**5*gamma(1 - m) - 15
36*a**5*x**4*gamma(1 - m) - 3072*a**4*x**3*gamma(1 - m) + 3072*a**3*x**2*gamma(1 - m) + 1536*a**2*x*gamma(1 -
m) - 1536*a*gamma(1 - m)) - 6*a**3*e**m*m**3*x**3*x**m*lerchphi(exp_polar(I*pi)/(a*x), 1, m*exp_polar(I*pi))*g
amma(-m)/(1536*a**6*x**5*gamma(1 - m) - 1536*a**5*x**4*gamma(1 - m) - 3072*a**4*x**3*gamma(1 - m) + 3072*a**3*
x**2*gamma(1 - m) + 1536*a**2*x*gamma(1 - m) - 1536*a*gamma(1 - m)) - 4*a**3*e**m*m**3*x**3*x**m*gamma(-m)/(15
36*a**6*x**5*gamma(1 - m) - 1536*a**5*x**4*gamma(1 - m) - 3072*a**4*x**3*gamma(1 - m) + 3072*a**3*x**2*gamma(1
- m) + 1536*a**2*x*gamma(1 - m) - 1536*a*gamma(1 - m)) - 62*a**3*e**m*m**2*x**3*x**m*lerchphi(1/(a*x), 1, m*e
xp_polar(I*pi))*gamma(-m)/(1536*a**6*x**5*gamma(1 - m) - 1536*a**5*x**4*gamma(1 - m) - 3072*a**4*x**3*gamma(1
- m) + 3072*a**3*x**2*gamma(1 - m) + 1536*a**2*x*gamma(1 - m) - 1536*a*gamma(1 - m)) + 30*a**3*e**m*m**2*x**3*
x**m*lerchphi(exp_polar(I*pi)/(a*x), 1, m*exp_polar(I*pi))*gamma(-m)/(1536*a**6*x**5*gamma(1 - m) - 1536*a**5*
x**4*gamma(1 - m) - 3072*a**4*x**3*gamma(1 - m) + 3072*a**3*x**2*gamma(1 - m) + 1536*a**2*x*gamma(1 - m) - 153
6*a*gamma(1 - m)) + 32*a**3*e**m*m**2*x**3*x**m*gamma(-m)/(1536*a**6*x**5*gamma(1 - m) - 1536*a**5*x**4*gamma(
1 - m) - 3072*a**4*x**3*gamma(1 - m) + 3072*a**3*x**2*gamma(1 - m) + 1536*a**2*x*gamma(1 - m) - 1536*a*gamma(1
- m)) + 30*a**3*e**m*m*x**3*x**m*lerchphi(1/(a*x), 1, m*exp_polar(I*pi))*gamma(-m)/(1536*a**6*x**5*gamma(1 -
m) - 1536*a**5*x**4*gamma(1 - m) - 3072*a**4*x**3*gamma(1 - m) + 3072*a**3*x**2*gamma(1 - m) + 1536*a**2*x*gam
ma(1 - m) - 1536*a*gamma(1 - m)) - 30*a**3*e**m*m*x**3*x**m*lerchphi(exp_polar(I*pi)/(a*x), 1, m*exp_polar(I*p
i))*gamma(-m)/(1536*a**6*x**5*gamma(1 - m) - 1536*a**5*x**4*gamma(1 - m) - 3072*a**4*x**3*gamma(1 - m) + 3072*
a**3*x**2*gamma(1 - m) + 1536*a**2*x*gamma(1 - m) - 1536*a*gamma(1 - m)) - 62*a**3*e**m*m*x**3*x**m*gamma(-m)/
(1536*a**6*x**5*gamma(1 - m) - 1536*a**5*x**4*gamma(1 - m) - 3072*a**4*x**3*gamma(1 - m) + 3072*a**3*x**2*gamm
a(1 - m) + 1536*a**2*x*gamma(1 - m) - 1536*a*gamma(1 - m)) + 4*a**2*e**m*m**4*x**2*x**m*lerchphi(1/(a*x), 1, m
*exp_polar(I*pi))*gamma(-m)/(1536*a**6*x**5*gamma(1 - m) - 1536*a**5*x**4*gamma(1 - m) - 3072*a**4*x**3*gamma(
1 - m) + 3072*a**3*x**2*gamma(1 - m) + 1536*a**2*x*gamma(1 - m) - 1536*a*gamma(1 - m)) - 30*a**2*e**m*m**3*x**
2*x**m*lerchphi(1/(a*x), 1, m*exp_polar(I*pi))*gamma(-m)/(1536*a**6*x**5*gamma(1 - m) - 1536*a**5*x**4*gamma(1
- m) - 3072*a**4*x**3*gamma(1 - m) + 3072*a**3*x**2*gamma(1 - m) + 1536*a**2*x*gamma(1 - m) - 1536*a*gamma(1
- m)) + 6*a**2*e**m*m**3*x**2*x**m*lerchphi(exp_polar(I*pi)/(a*x), 1, m*exp_polar(I*pi))*gamma(-m)/(1536*a**6*
x**5*gamma(1 - m) - 1536*a**5*x**4*gamma(1 - m) - 3072*a**4*x**3*gamma(1 - m) + 3072*a**3*x**2*gamma(1 - m) +
1536*a**2*x*gamma(1 - m) - 1536*a*gamma(1 - m)) + 62*a**2*e**m*m**2*x**2*x**m*lerchphi(1/(a*x), 1, m*exp_polar
(I*pi))*gamma(-m)/(1536*a**6*x**5*gamma(1 - m) - 1536*a**5*x**4*gamma(1 - m) - 3072*a**4*x**3*gamma(1 - m) + 3
072*a**3*x**2*gamma(1 - m) + 1536*a**2*x*gamma(1 - m) - 1536*a*gamma(1 - m)) - 30*a**2*e**m*m**2*x**2*x**m*ler
chphi(exp_polar(I*pi)/(a*x), 1, m*exp_polar(I*pi))*gamma(-m)/(1536*a**6*x**5*gamma(1 - m) - 1536*a**5*x**4*gam
ma(1 - m) - 3072*a**4*x**3*gamma(1 - m) + 3072*a**3*x**2*gamma(1 - m) + 1536*a**2*x*gamma(1 - m) - 1536*a*gamm
a(1 - m)) + 4*a**2*e**m*m**2*x**2*x**m*gamma(-m)/(1536*a**6*x**5*gamma(1 - m) - 1536*a**5*x**4*gamma(1 - m) -
3072*a**4*x**3*gamma(1 - m) + 3072*a**3*x**2*gamma(1 - m) + 1536*a**2*x*gamma(1 - m) - 1536*a*gamma(1 - m)) -
30*a**2*e**m*m*x**2*x**m*lerchphi(1/(a*x), 1, m*exp_polar(I*pi))*gamma(-m)/(1536*a**6*x**5*gamma(1 - m) - 1536
*a**5*x**4*gamma(1 - m) - 3072*a**4*x**3*gamma(1 - m) + 3072*a**3*x**2*gamma(1 - m) + 1536*a**2*x*gamma(1 - m)
- 1536*a*gamma(1 - m)) + 30*a**2*e**m*m*x**2*x**m*lerchphi(exp_polar(I*pi)/(a*x), 1, m*exp_polar(I*pi))*gamma
(-m)/(1536*a**6*x**5*gamma(1 - m) - 1536*a**5*x**4*gamma(1 - m) - 3072*a**4*x**3*gamma(1 - m) + 3072*a**3*x**2
*gamma(1 - m) + 1536*a**2*x*gamma(1 - m) - 1536*a*gamma(1 - m)) - 18*a**2*e**m*m*x**2*x**m*gamma(-m)/(1536*a**
6*x**5*gamma(1 - m) - 1536*a**5*x**4*gamma(1 - m) - 3072*a**4*x**3*gamma(1 - m) + 3072*a**3*x**2*gamma(1 - m)
+ 1536*a**2*x*gamma(1 - m) - 1536*a*gamma(1 - m)) + 2*a*e**m*m**4*x*x**m*lerchphi(1/(a*x), 1, m*exp_polar(I*pi
))*gamma(-m)/(1536*a**6*x**5*gamma(1 - m) - 1536*a**5*x**4*gamma(1 - m) - 3072*a**4*x**3*gamma(1 - m) + 3072*a
**3*x**2*gamma(1 - m) + 1536*a**2*x*gamma(1 - m) - 1536*a*gamma(1 - m)) - 15*a*e**m*m**3*x*x**m*lerchphi(1/(a*
x), 1, m*exp_polar(I*pi))*gamma(-m)/(1536*a**6*x**5*gamma(1 - m) - 1536*a**5*x**4*gamma(1 - m) - 3072*a**4*x**
3*gamma(1 - m) + 3072*a**3*x**2*gamma(1 - m) + 1536*a**2*x*gamma(1 - m) - 1536*a*gamma(1 - m)) + 3*a*e**m*m**3
*x*x**m*lerchphi(exp_polar(I*pi)/(a*x), 1, m*exp_polar(I*pi))*gamma(-m)/(1536*a**6*x**5*gamma(1 - m) - 1536*a*
*5*x**4*gamma(1 - m) - 3072*a**4*x**3*gamma(1 - m) + 3072*a**3*x**2*gamma(1 - m) + 1536*a**2*x*gamma(1 - m) -
1536*a*gamma(1 - m)) + 2*a*e**m*m**3*x*x**m*gamma(-m)/(1536*a**6*x**5*gamma(1 - m) - 1536*a**5*x**4*gamma(1 -
m) - 3072*a**4*x**3*gamma(1 - m) + 3072*a**3*x**2*gamma(1 - m) + 1536*a**2*x*gamma(1 - m) - 1536*a*gamma(1 - m
)) + 31*a*e**m*m**2*x*x**m*lerchphi(1/(a*x), 1, m*exp_polar(I*pi))*gamma(-m)/(1536*a**6*x**5*gamma(1 - m) - 15
36*a**5*x**4*gamma(1 - m) - 3072*a**4*x**3*gamma(1 - m) + 3072*a**3*x**2*gamma(1 - m) + 1536*a**2*x*gamma(1 -
m) - 1536*a*gamma(1 - m)) - 15*a*e**m*m**2*x*x**m*lerchphi(exp_polar(I*pi)/(a*x), 1, m*exp_polar(I*pi))*gamma(
-m)/(1536*a**6*x**5*gamma(1 - m) - 1536*a**5*x**4*gamma(1 - m) - 3072*a**4*x**3*gamma(1 - m) + 3072*a**3*x**2*
gamma(1 - m) + 1536*a**2*x*gamma(1 - m) - 1536*a*gamma(1 - m)) - 20*a*e**m*m**2*x*x**m*gamma(-m)/(1536*a**6*x*
*5*gamma(1 - m) - 1536*a**5*x**4*gamma(1 - m) - 3072*a**4*x**3*gamma(1 - m) + 3072*a**3*x**2*gamma(1 - m) + 15
36*a**2*x*gamma(1 - m) - 1536*a*gamma(1 - m)) - 15*a*e**m*m*x*x**m*lerchphi(1/(a*x), 1, m*exp_polar(I*pi))*gam
ma(-m)/(1536*a**6*x**5*gamma(1 - m) - 1536*a**5*x**4*gamma(1 - m) - 3072*a**4*x**3*gamma(1 - m) + 3072*a**3*x*
*2*gamma(1 - m) + 1536*a**2*x*gamma(1 - m) - 1536*a*gamma(1 - m)) + 15*a*e**m*m*x*x**m*lerchphi(exp_polar(I*pi
)/(a*x), 1, m*exp_polar(I*pi))*gamma(-m)/(1536*a**6*x**5*gamma(1 - m) - 1536*a**5*x**4*gamma(1 - m) - 3072*a**
4*x**3*gamma(1 - m) + 3072*a**3*x**2*gamma(1 - m) + 1536*a**2*x*gamma(1 - m) - 1536*a*gamma(1 - m)) + 66*a*e**
m*m*x*x**m*gamma(-m)/(1536*a**6*x**5*gamma(1 - m) - 1536*a**5*x**4*gamma(1 - m) - 3072*a**4*x**3*gamma(1 - m)
+ 3072*a**3*x**2*gamma(1 - m) + 1536*a**2*x*gamma(1 - m) - 1536*a*gamma(1 - m)) - 2*e**m*m**4*x**m*lerchphi(1/
(a*x), 1, m*exp_polar(I*pi))*gamma(-m)/(1536*a**6*x**5*gamma(1 - m) - 1536*a**5*x**4*gamma(1 - m) - 3072*a**4*
x**3*gamma(1 - m) + 3072*a**3*x**2*gamma(1 - m) + 1536*a**2*x*gamma(1 - m) - 1536*a*gamma(1 - m)) + 15*e**m*m*
*3*x**m*lerchphi(1/(a*x), 1, m*exp_polar(I*pi))*gamma(-m)/(1536*a**6*x**5*gamma(1 - m) - 1536*a**5*x**4*gamma(
1 - m) - 3072*a**4*x**3*gamma(1 - m) + 3072*a**3*x**2*gamma(1 - m) + 1536*a**2*x*gamma(1 - m) - 1536*a*gamma(1
- m)) - 3*e**m*m**3*x**m*lerchphi(exp_polar(I*pi)/(a*x), 1, m*exp_polar(I*pi))*gamma(-m)/(1536*a**6*x**5*gamm
a(1 - m) - 1536*a**5*x**4*gamma(1 - m) - 3072*a**4*x**3*gamma(1 - m) + 3072*a**3*x**2*gamma(1 - m) + 1536*a**2
*x*gamma(1 - m) - 1536*a*gamma(1 - m)) - 31*e**m*m**2*x**m*lerchphi(1/(a*x), 1, m*exp_polar(I*pi))*gamma(-m)/(
1536*a**6*x**5*gamma(1 - m) - 1536*a**5*x**4*gamma(1 - m) - 3072*a**4*x**3*gamma(1 - m) + 3072*a**3*x**2*gamma
(1 - m) + 1536*a**2*x*gamma(1 - m) - 1536*a*gamma(1 - m)) + 15*e**m*m**2*x**m*lerchphi(exp_polar(I*pi)/(a*x),
1, m*exp_polar(I*pi))*gamma(-m)/(1536*a**6*x**5*gamma(1 - m) - 1536*a**5*x**4*gamma(1 - m) - 3072*a**4*x**3*ga
mma(1 - m) + 3072*a**3*x**2*gamma(1 - m) + 1536*a**2*x*gamma(1 - m) - 1536*a*gamma(1 - m)) + 15*e**m*m*x**m*le
rchphi(1/(a*x), 1, m*exp_polar(I*pi))*gamma(-m)/(1536*a**6*x**5*gamma(1 - m) - 1536*a**5*x**4*gamma(1 - m) - 3
072*a**4*x**3*gamma(1 - m) + 3072*a**3*x**2*gamma(1 - m) + 1536*a**2*x*gamma(1 - m) - 1536*a*gamma(1 - m)) - 1
5*e**m*m*x**m*lerchphi(exp_polar(I*pi)/(a*x), 1, m*exp_polar(I*pi))*gamma(-m)/(1536*a**6*x**5*gamma(1 - m) - 1
536*a**5*x**4*gamma(1 - m) - 3072*a**4*x**3*gamma(1 - m) + 3072*a**3*x**2*gamma(1 - m) + 1536*a**2*x*gamma(1 -
m) - 1536*a*gamma(1 - m))
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (e x\right )^{m}}{16 \,{\left (a x + 1\right )}^{3}{\left (a x - 1\right )}^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x)^m/(-2*a*x+2)^4/(a*x+1)^3,x, algorithm="giac")
[Out]
integrate(1/16*(e*x)^m/((a*x + 1)^3*(a*x - 1)^4), x)