3.359 \(\int \frac{(e x)^m}{(2-2 a x)^3 (1+a x)^2} \, dx\)
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)*Hypergeometric2F1[3, (2 + m)/2, (4 + m)/2, a^2*x^2])/(8*e^2
*(2 + m))
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Rubi [A] time = 0.130722, 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
\[ \frac{a (e x)^{m+2} \, _2F_1\left (3,\frac{m+2}{2};\frac{m+4}{2};a^2 x^2\right )}{8 e^2 (m+2)}+\frac{(e x)^{m+1} \, _2F_1\left (3,\frac{m+1}{2};\frac{m+3}{2};a^2 x^2\right )}{8 e (m+1)} \]
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)*Hypergeometric2F1[3, (2 + m)/2, (4 + m)/2, a^2*x^2])/(8*e^2
*(2 + m))
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Rubi in Sympy [A] time = 14.6483, size = 63, normalized size = 0.73 \[ \frac{a \left (e x\right )^{m + 2}{{}_{2}F_{1}\left (\begin{matrix} 3, \frac{m}{2} + 1 \\ \frac{m}{2} + 2 \end{matrix}\middle |{a^{2} x^{2}} \right )}}{8 e^{2} \left (m + 2\right )} + \frac{\left (e x\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 3, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{a^{2} x^{2}} \right )}}{8 e \left (m + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x)**m/(-2*a*x+2)**3/(a*x+1)**2,x)
[Out]
a*(e*x)**(m + 2)*hyper((3, m/2 + 1), (m/2 + 2,), a**2*x**2)/(8*e**2*(m + 2)) + (
e*x)**(m + 1)*hyper((3, m/2 + 1/2), (m/2 + 3/2,), a**2*x**2)/(8*e*(m + 1))
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Mathematica [A] time = 0.0776352, size = 83, normalized size = 0.97 \[ \frac{x (e x)^m (3 \, _2F_1(1,m+1;m+2;-a x)+3 \, _2F_1(1,m+1;m+2;a x)+2 \, _2F_1(2,m+1;m+2;-a x)+4 \, _2F_1(2,m+1;m+2;a x)+4 \, _2F_1(3,m+1;m+2;a x))}{128 (m+1)} \]
Antiderivative was successfully verified.
[In] Integrate[(e*x)^m/((2 - 2*a*x)^3*(1 + a*x)^2),x]
[Out]
(x*(e*x)^m*(3*Hypergeometric2F1[1, 1 + m, 2 + m, -(a*x)] + 3*Hypergeometric2F1[1
, 1 + m, 2 + m, a*x] + 2*Hypergeometric2F1[2, 1 + m, 2 + m, -(a*x)] + 4*Hypergeo
metric2F1[2, 1 + m, 2 + m, a*x] + 4*Hypergeometric2F1[3, 1 + m, 2 + m, a*x]))/(1
28*(1 + m))
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Maple [F] time = 0.092, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex \right ) ^{m}}{ \left ( -2\,ax+2 \right ) ^{3} \left ( ax+1 \right ) ^{2}}}\, dx \]
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)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\frac{1}{8} \, \int \frac{\left (e x\right )^{m}}{{\left (a x + 1\right )}^{2}{\left (a x - 1\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/8*(e*x)^m/((a*x + 1)^2*(a*x - 1)^3),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. \[{\rm integral}\left (-\frac{\left (e x\right )^{m}}{8 \,{\left (a^{5} x^{5} - a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a^{2} x^{2} + a x - 1\right )}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/8*(e*x)^m/((a*x + 1)^2*(a*x - 1)^3),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 [A] time = 14.518, size = 1972, normalized size = 22.93 \[ \text{result too large to display} \]
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)/(1
28*a**4*x**3*gamma(-m + 1) - 128*a**3*x**2*gamma(-m + 1) - 128*a**2*x*gamma(-m +
1) + 128*a*gamma(-m + 1)) + 6*a**3*e**m*m**2*x**3*x**m*lerchphi(1/(a*x), 1, m*e
xp_polar(I*pi))*gamma(-m)/(128*a**4*x**3*gamma(-m + 1) - 128*a**3*x**2*gamma(-m
+ 1) - 128*a**2*x*gamma(-m + 1) + 128*a*gamma(-m + 1)) - 2*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)/(128*a**4*x*
*3*gamma(-m + 1) - 128*a**3*x**2*gamma(-m + 1) - 128*a**2*x*gamma(-m + 1) + 128*
a*gamma(-m + 1)) - 2*a**3*e**m*m**2*x**3*x**m*gamma(-m)/(128*a**4*x**3*gamma(-m
+ 1) - 128*a**3*x**2*gamma(-m + 1) - 128*a**2*x*gamma(-m + 1) + 128*a*gamma(-m +
1)) - 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(-m + 1) - 128*a**3*x**2*gamma(-m + 1) - 128*a**2*x*gamma(-
m + 1) + 128*a*gamma(-m + 1)) + 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(-m + 1) - 128*a**3*
x**2*gamma(-m + 1) - 128*a**2*x*gamma(-m + 1) + 128*a*gamma(-m + 1)) + 4*a**3*e*
*m*m*x**3*x**m*gamma(-m)/(128*a**4*x**3*gamma(-m + 1) - 128*a**3*x**2*gamma(-m +
1) - 128*a**2*x*gamma(-m + 1) + 128*a*gamma(-m + 1)) + 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(-m + 1
) - 128*a**3*x**2*gamma(-m + 1) - 128*a**2*x*gamma(-m + 1) + 128*a*gamma(-m + 1)
) - 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(-m + 1) - 128*a**3*x**2*gamma(-m + 1) - 128*a**2*x*gamma(-
m + 1) + 128*a*gamma(-m + 1)) + 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(-m + 1) - 128*a*
*3*x**2*gamma(-m + 1) - 128*a**2*x*gamma(-m + 1) + 128*a*gamma(-m + 1)) + 3*a**2
*e**m*m*x**2*x**m*lerchphi(1/(a*x), 1, m*exp_polar(I*pi))*gamma(-m)/(128*a**4*x*
*3*gamma(-m + 1) - 128*a**3*x**2*gamma(-m + 1) - 128*a**2*x*gamma(-m + 1) + 128*
a*gamma(-m + 1)) - 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(-m + 1) - 128*a**3*x**2*gamma(-m
+ 1) - 128*a**2*x*gamma(-m + 1) + 128*a*gamma(-m + 1)) + 2*a**2*e**m*m*x**2*x**
m*gamma(-m)/(128*a**4*x**3*gamma(-m + 1) - 128*a**3*x**2*gamma(-m + 1) - 128*a**
2*x*gamma(-m + 1) + 128*a*gamma(-m + 1)) + 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(-m + 1) - 128*a**3*x**2*g
amma(-m + 1) - 128*a**2*x*gamma(-m + 1) + 128*a*gamma(-m + 1)) - 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(-m
+ 1) - 128*a**3*x**2*gamma(-m + 1) - 128*a**2*x*gamma(-m + 1) + 128*a*gamma(-m +
1)) + 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(-m + 1) - 128*a**3*x**2*gamma(-m + 1) - 128*a**
2*x*gamma(-m + 1) + 128*a*gamma(-m + 1)) + 2*a*e**m*m**2*x*x**m*gamma(-m)/(128*a
**4*x**3*gamma(-m + 1) - 128*a**3*x**2*gamma(-m + 1) - 128*a**2*x*gamma(-m + 1)
+ 128*a*gamma(-m + 1)) + 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(-m + 1) - 128*a**3*x**2*gamma(-m + 1) - 128*a*
*2*x*gamma(-m + 1) + 128*a*gamma(-m + 1)) - 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(-m + 1) - 128
*a**3*x**2*gamma(-m + 1) - 128*a**2*x*gamma(-m + 1) + 128*a*gamma(-m + 1)) - 10*
a*e**m*m*x*x**m*gamma(-m)/(128*a**4*x**3*gamma(-m + 1) - 128*a**3*x**2*gamma(-m
+ 1) - 128*a**2*x*gamma(-m + 1) + 128*a*gamma(-m + 1)) - 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(-m + 1) - 128*a
**3*x**2*gamma(-m + 1) - 128*a**2*x*gamma(-m + 1) + 128*a*gamma(-m + 1)) + 6*e**
m*m**2*x**m*lerchphi(1/(a*x), 1, m*exp_polar(I*pi))*gamma(-m)/(128*a**4*x**3*gam
ma(-m + 1) - 128*a**3*x**2*gamma(-m + 1) - 128*a**2*x*gamma(-m + 1) + 128*a*gamm
a(-m + 1)) - 2*e**m*m**2*x**m*lerchphi(exp_polar(I*pi)/(a*x), 1, m*exp_polar(I*p
i))*gamma(-m)/(128*a**4*x**3*gamma(-m + 1) - 128*a**3*x**2*gamma(-m + 1) - 128*a
**2*x*gamma(-m + 1) + 128*a*gamma(-m + 1)) - 3*e**m*m*x**m*lerchphi(1/(a*x), 1,
m*exp_polar(I*pi))*gamma(-m)/(128*a**4*x**3*gamma(-m + 1) - 128*a**3*x**2*gamma(
-m + 1) - 128*a**2*x*gamma(-m + 1) + 128*a*gamma(-m + 1)) + 3*e**m*m*x**m*lerchp
hi(exp_polar(I*pi)/(a*x), 1, m*exp_polar(I*pi))*gamma(-m)/(128*a**4*x**3*gamma(-
m + 1) - 128*a**3*x**2*gamma(-m + 1) - 128*a**2*x*gamma(-m + 1) + 128*a*gamma(-m
+ 1))
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int -\frac{\left (e x\right )^{m}}{8 \,{\left (a x + 1\right )}^{2}{\left (a x - 1\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/8*(e*x)^m/((a*x + 1)^2*(a*x - 1)^3),x, algorithm="giac")
[Out]
integrate(-1/8*(e*x)^m/((a*x + 1)^2*(a*x - 1)^3), x)