\(\int \frac {(e x)^m}{(2-2 a x)^3 (1+a x)^2} \, dx\) [60]
Optimal result
Integrand size = 21, antiderivative size = 86 \[
\int \frac {(e x)^m}{(2-2 a x)^3 (1+a x)^2} \, dx=\frac {(e x)^{1+m} \operatorname {Hypergeometric2F1}\left (3,\frac {1+m}{2},\frac {3+m}{2},a^2 x^2\right )}{8 e (1+m)}+\frac {a (e x)^{2+m} \operatorname {Hypergeometric2F1}\left (3,\frac {2+m}{2},\frac {4+m}{2},a^2 x^2\right )}{8 e^2 (2+m)}
\]
[Out]
1/8*(e*x)^(1+m)*hypergeom([3, 1/2+1/2*m],[3/2+1/2*m],a^2*x^2)/e/(1+m)+1/8*a*(e*x)^(2+m)*hypergeom([3, 1+1/2*m]
,[2+1/2*m],a^2*x^2)/e^2/(2+m)
Rubi [A] (verified)
Time = 0.02 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number
of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {83, 74, 371}
\[
\int \frac {(e x)^m}{(2-2 a x)^3 (1+a x)^2} \, dx=\frac {a (e x)^{m+2} \operatorname {Hypergeometric2F1}\left (3,\frac {m+2}{2},\frac {m+4}{2},a^2 x^2\right )}{8 e^2 (m+2)}+\frac {(e x)^{m+1} \operatorname {Hypergeometric2F1}\left (3,\frac {m+1}{2},\frac {m+3}{2},a^2 x^2\right )}{8 e (m+1)}
\]
[In]
Int[(e*x)^m/((2 - 2*a*x)^3*(1 + a*x)^2),x]
[Out]
((e*x)^(1 + m)*Hypergeometric2F1[3, (1 + m)/2, (3 + m)/2, a^2*x^2])/(8*e*(1 + m)) + (a*(e*x)^(2 + m)*Hypergeom
etric2F1[3, (2 + m)/2, (4 + m)/2, a^2*x^2])/(8*e^2*(2 + m))
Rule 74
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] && (NeQ[m, -1] || (EqQ[e, 0] && (EqQ[p, 1] || !IntegerQ[p])))
Rule 83
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 371
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] && !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])
Rubi steps \begin{align*}
\text {integral}& = \frac {a \int \frac {(e x)^{1+m}}{(2-2 a x)^3 (1+a x)^3} \, dx}{e}+\int \frac {(e x)^m}{(2-2 a x)^3 (1+a x)^3} \, dx \\ & = \frac {a \int \frac {(e x)^{1+m}}{\left (2-2 a^2 x^2\right )^3} \, dx}{e}+\int \frac {(e x)^m}{\left (2-2 a^2 x^2\right )^3} \, dx \\ & = \frac {(e x)^{1+m} \, _2F_1\left (3,\frac {1+m}{2};\frac {3+m}{2};a^2 x^2\right )}{8 e (1+m)}+\frac {a (e x)^{2+m} \, _2F_1\left (3,\frac {2+m}{2};\frac {4+m}{2};a^2 x^2\right )}{8 e^2 (2+m)} \\
\end{align*}
Mathematica [A] (verified)
Time = 0.07 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.90
\[
\int \frac {(e x)^m}{(2-2 a x)^3 (1+a x)^2} \, dx=\frac {x (e x)^m \left (a (1+m) x \operatorname {Hypergeometric2F1}\left (3,1+\frac {m}{2},2+\frac {m}{2},a^2 x^2\right )+(2+m) \operatorname {Hypergeometric2F1}\left (3,\frac {1+m}{2},\frac {3+m}{2},a^2 x^2\right )\right )}{8 (1+m) (2+m)}
\]
[In]
Integrate[(e*x)^m/((2 - 2*a*x)^3*(1 + a*x)^2),x]
[Out]
(x*(e*x)^m*(a*(1 + m)*x*Hypergeometric2F1[3, 1 + m/2, 2 + m/2, a^2*x^2] + (2 + m)*Hypergeometric2F1[3, (1 + m)
/2, (3 + m)/2, a^2*x^2]))/(8*(1 + m)*(2 + m))
Maple [F]
\[\int \frac {\left (e x \right )^{m}}{\left (-2 a x +2\right )^{3} \left (a x +1\right )^{2}}d x\]
[In]
int((e*x)^m/(-2*a*x+2)^3/(a*x+1)^2,x)
[Out]
int((e*x)^m/(-2*a*x+2)^3/(a*x+1)^2,x)
Fricas [F]
\[
\int \frac {(e x)^m}{(2-2 a x)^3 (1+a x)^2} \, dx=\int { -\frac {\left (e x\right )^{m}}{8 \, {\left (a x + 1\right )}^{2} {\left (a x - 1\right )}^{3}} \,d x }
\]
[In]
integrate((e*x)^m/(-2*a*x+2)^3/(a*x+1)^2,x, algorithm="fricas")
[Out]
integral(-1/8*(e*x)^m/(a^5*x^5 - a^4*x^4 - 2*a^3*x^3 + 2*a^2*x^2 + a*x - 1), x)
Sympy [C] (verification not implemented)
Result contains complex when optimal does not.
Time = 2.36 (sec) , antiderivative size = 1972, normalized size of antiderivative = 22.93
\[
\int \frac {(e x)^m}{(2-2 a x)^3 (1+a x)^2} \, dx=\text {Too large to display}
\]
[In]
integrate((e*x)**m/(-2*a*x+2)**3/(a*x+1)**2,x)
[Out]
-2*a**3*e**m*m**3*x**3*x**m*lerchphi(1/(a*x), 1, m*exp_polar(I*pi))*gamma(-m)/(128*a**4*x**3*gamma(1 - m) - 12
8*a**3*x**2*gamma(1 - m) - 128*a**2*x*gamma(1 - m) + 128*a*gamma(1 - m)) + 6*a**3*e**m*m**2*x**3*x**m*lerchphi
(1/(a*x), 1, m*exp_polar(I*pi))*gamma(-m)/(128*a**4*x**3*gamma(1 - m) - 128*a**3*x**2*gamma(1 - m) - 128*a**2*
x*gamma(1 - m) + 128*a*gamma(1 - m)) - 2*a**3*e**m*m**2*x**3*x**m*lerchphi(exp_polar(I*pi)/(a*x), 1, m*exp_pol
ar(I*pi))*gamma(-m)/(128*a**4*x**3*gamma(1 - m) - 128*a**3*x**2*gamma(1 - m) - 128*a**2*x*gamma(1 - m) + 128*a
*gamma(1 - m)) - 2*a**3*e**m*m**2*x**3*x**m*gamma(-m)/(128*a**4*x**3*gamma(1 - m) - 128*a**3*x**2*gamma(1 - m)
- 128*a**2*x*gamma(1 - m) + 128*a*gamma(1 - m)) - 3*a**3*e**m*m*x**3*x**m*lerchphi(1/(a*x), 1, m*exp_polar(I*
pi))*gamma(-m)/(128*a**4*x**3*gamma(1 - m) - 128*a**3*x**2*gamma(1 - m) - 128*a**2*x*gamma(1 - m) + 128*a*gamm
a(1 - m)) + 3*a**3*e**m*m*x**3*x**m*lerchphi(exp_polar(I*pi)/(a*x), 1, m*exp_polar(I*pi))*gamma(-m)/(128*a**4*
x**3*gamma(1 - m) - 128*a**3*x**2*gamma(1 - m) - 128*a**2*x*gamma(1 - m) + 128*a*gamma(1 - m)) + 4*a**3*e**m*m
*x**3*x**m*gamma(-m)/(128*a**4*x**3*gamma(1 - m) - 128*a**3*x**2*gamma(1 - m) - 128*a**2*x*gamma(1 - m) + 128*
a*gamma(1 - m)) + 2*a**2*e**m*m**3*x**2*x**m*lerchphi(1/(a*x), 1, m*exp_polar(I*pi))*gamma(-m)/(128*a**4*x**3*
gamma(1 - m) - 128*a**3*x**2*gamma(1 - m) - 128*a**2*x*gamma(1 - m) + 128*a*gamma(1 - m)) - 6*a**2*e**m*m**2*x
**2*x**m*lerchphi(1/(a*x), 1, m*exp_polar(I*pi))*gamma(-m)/(128*a**4*x**3*gamma(1 - m) - 128*a**3*x**2*gamma(1
- m) - 128*a**2*x*gamma(1 - m) + 128*a*gamma(1 - m)) + 2*a**2*e**m*m**2*x**2*x**m*lerchphi(exp_polar(I*pi)/(a
*x), 1, m*exp_polar(I*pi))*gamma(-m)/(128*a**4*x**3*gamma(1 - m) - 128*a**3*x**2*gamma(1 - m) - 128*a**2*x*gam
ma(1 - m) + 128*a*gamma(1 - m)) + 3*a**2*e**m*m*x**2*x**m*lerchphi(1/(a*x), 1, m*exp_polar(I*pi))*gamma(-m)/(1
28*a**4*x**3*gamma(1 - m) - 128*a**3*x**2*gamma(1 - m) - 128*a**2*x*gamma(1 - m) + 128*a*gamma(1 - m)) - 3*a**
2*e**m*m*x**2*x**m*lerchphi(exp_polar(I*pi)/(a*x), 1, m*exp_polar(I*pi))*gamma(-m)/(128*a**4*x**3*gamma(1 - m)
- 128*a**3*x**2*gamma(1 - m) - 128*a**2*x*gamma(1 - m) + 128*a*gamma(1 - m)) + 2*a**2*e**m*m*x**2*x**m*gamma(
-m)/(128*a**4*x**3*gamma(1 - m) - 128*a**3*x**2*gamma(1 - m) - 128*a**2*x*gamma(1 - m) + 128*a*gamma(1 - m)) +
2*a*e**m*m**3*x*x**m*lerchphi(1/(a*x), 1, m*exp_polar(I*pi))*gamma(-m)/(128*a**4*x**3*gamma(1 - m) - 128*a**3
*x**2*gamma(1 - m) - 128*a**2*x*gamma(1 - m) + 128*a*gamma(1 - m)) - 6*a*e**m*m**2*x*x**m*lerchphi(1/(a*x), 1,
m*exp_polar(I*pi))*gamma(-m)/(128*a**4*x**3*gamma(1 - m) - 128*a**3*x**2*gamma(1 - m) - 128*a**2*x*gamma(1 -
m) + 128*a*gamma(1 - m)) + 2*a*e**m*m**2*x*x**m*lerchphi(exp_polar(I*pi)/(a*x), 1, m*exp_polar(I*pi))*gamma(-m
)/(128*a**4*x**3*gamma(1 - m) - 128*a**3*x**2*gamma(1 - m) - 128*a**2*x*gamma(1 - m) + 128*a*gamma(1 - m)) + 2
*a*e**m*m**2*x*x**m*gamma(-m)/(128*a**4*x**3*gamma(1 - m) - 128*a**3*x**2*gamma(1 - m) - 128*a**2*x*gamma(1 -
m) + 128*a*gamma(1 - m)) + 3*a*e**m*m*x*x**m*lerchphi(1/(a*x), 1, m*exp_polar(I*pi))*gamma(-m)/(128*a**4*x**3*
gamma(1 - m) - 128*a**3*x**2*gamma(1 - m) - 128*a**2*x*gamma(1 - m) + 128*a*gamma(1 - m)) - 3*a*e**m*m*x*x**m*
lerchphi(exp_polar(I*pi)/(a*x), 1, m*exp_polar(I*pi))*gamma(-m)/(128*a**4*x**3*gamma(1 - m) - 128*a**3*x**2*ga
mma(1 - m) - 128*a**2*x*gamma(1 - m) + 128*a*gamma(1 - m)) - 10*a*e**m*m*x*x**m*gamma(-m)/(128*a**4*x**3*gamma
(1 - m) - 128*a**3*x**2*gamma(1 - m) - 128*a**2*x*gamma(1 - m) + 128*a*gamma(1 - m)) - 2*e**m*m**3*x**m*lerchp
hi(1/(a*x), 1, m*exp_polar(I*pi))*gamma(-m)/(128*a**4*x**3*gamma(1 - m) - 128*a**3*x**2*gamma(1 - m) - 128*a**
2*x*gamma(1 - m) + 128*a*gamma(1 - m)) + 6*e**m*m**2*x**m*lerchphi(1/(a*x), 1, m*exp_polar(I*pi))*gamma(-m)/(1
28*a**4*x**3*gamma(1 - m) - 128*a**3*x**2*gamma(1 - m) - 128*a**2*x*gamma(1 - m) + 128*a*gamma(1 - m)) - 2*e**
m*m**2*x**m*lerchphi(exp_polar(I*pi)/(a*x), 1, m*exp_polar(I*pi))*gamma(-m)/(128*a**4*x**3*gamma(1 - m) - 128*
a**3*x**2*gamma(1 - m) - 128*a**2*x*gamma(1 - m) + 128*a*gamma(1 - m)) - 3*e**m*m*x**m*lerchphi(1/(a*x), 1, m*
exp_polar(I*pi))*gamma(-m)/(128*a**4*x**3*gamma(1 - m) - 128*a**3*x**2*gamma(1 - m) - 128*a**2*x*gamma(1 - m)
+ 128*a*gamma(1 - m)) + 3*e**m*m*x**m*lerchphi(exp_polar(I*pi)/(a*x), 1, m*exp_polar(I*pi))*gamma(-m)/(128*a**
4*x**3*gamma(1 - m) - 128*a**3*x**2*gamma(1 - m) - 128*a**2*x*gamma(1 - m) + 128*a*gamma(1 - m))
Maxima [F]
\[
\int \frac {(e x)^m}{(2-2 a x)^3 (1+a x)^2} \, dx=\int { -\frac {\left (e x\right )^{m}}{8 \, {\left (a x + 1\right )}^{2} {\left (a x - 1\right )}^{3}} \,d x }
\]
[In]
integrate((e*x)^m/(-2*a*x+2)^3/(a*x+1)^2,x, algorithm="maxima")
[Out]
-1/8*integrate((e*x)^m/((a*x + 1)^2*(a*x - 1)^3), x)
Giac [F]
\[
\int \frac {(e x)^m}{(2-2 a x)^3 (1+a x)^2} \, dx=\int { -\frac {\left (e x\right )^{m}}{8 \, {\left (a x + 1\right )}^{2} {\left (a x - 1\right )}^{3}} \,d x }
\]
[In]
integrate((e*x)^m/(-2*a*x+2)^3/(a*x+1)^2,x, algorithm="giac")
[Out]
integrate(-1/8*(e*x)^m/((a*x + 1)^2*(a*x - 1)^3), x)
Mupad [F(-1)]
Timed out. \[
\int \frac {(e x)^m}{(2-2 a x)^3 (1+a x)^2} \, dx=\int -\frac {{\left (e\,x\right )}^m}{{\left (a\,x+1\right )}^2\,{\left (2\,a\,x-2\right )}^3} \,d x
\]
[In]
int(-(e*x)^m/((a*x + 1)^2*(2*a*x - 2)^3),x)
[Out]
int(-(e*x)^m/((a*x + 1)^2*(2*a*x - 2)^3), x)