3.60 \(\int \frac{(e x)^m}{(2-2 a x)^3 (1+a x)^2} \, dx\)
Optimal. Leaf size=86 \[ \frac{a (e x)^{m+2} \, _2F_1\left (3,\frac{m+2}{2};\frac{m+4}{2};a^2 x^2\right )}{8 e^2 (m+2)}+\frac{(e x)^{m+1} \, _2F_1\left (3,\frac{m+1}{2};\frac{m+3}{2};a^2 x^2\right )}{8 e (m+1)} \]
[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))
________________________________________________________________________________________
Rubi [A] time = 0.0234094, 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} \[ \frac{a (e x)^{m+2} \, _2F_1\left (3,\frac{m+2}{2};\frac{m+4}{2};a^2 x^2\right )}{8 e^2 (m+2)}+\frac{(e x)^{m+1} \, _2F_1\left (3,\frac{m+1}{2};\frac{m+3}{2};a^2 x^2\right )}{8 e (m+1)} \]
Antiderivative was successfully verified.
[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 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)^3 (1+a x)^2} \, dx &=\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] time = 0.0267156, size = 77, normalized size = 0.9 \[ \frac{x (e x)^m \left (a (m+1) x \, _2F_1\left (3,\frac{m}{2}+1;\frac{m}{2}+2;a^2 x^2\right )+(m+2) \, _2F_1\left (3,\frac{m+1}{2};\frac{m+3}{2};a^2 x^2\right )\right )}{8 (m+1) (m+2)} \]
Antiderivative was successfully verified.
[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] time = 0.062, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex \right ) ^{m}}{ \left ( -2\,ax+2 \right ) ^{3} \left ( ax+1 \right ) ^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{8} \, \int \frac{\left (e x\right )^{m}}{{\left (a x + 1\right )}^{2}{\left (a x - 1\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\left (e x\right )^{m}}{8 \,{\left (a^{5} x^{5} - a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, 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)^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] time = 4.80476, size = 1972, normalized size = 22.93 \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)**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))
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{\left (e x\right )^{m}}{8 \,{\left (a x + 1\right )}^{2}{\left (a x - 1\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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)