3.1.61 \(\int \frac {(e x)^m}{(2-2 a x)^4 (1+a x)^3} \, dx\) [61]
Optimal. Leaf size=86 \[ \frac {(e x)^{1+m} \, _2F_1\left (4,\frac {1+m}{2};\frac {3+m}{2};a^2 x^2\right )}{16 e (1+m)}+\frac {a (e x)^{2+m} \, _2F_1\left (4,\frac {2+m}{2};\frac {4+m}{2};a^2 x^2\right )}{16 e^2 (2+m)} \]
[Out]
1/16*(e*x)^(1+m)*hypergeom([4, 1/2+1/2*m],[3/2+1/2*m],a^2*x^2)/e/(1+m)+1/16*a*(e*x)^(2+m)*hypergeom([4, 1+1/2*
m],[2+1/2*m],a^2*x^2)/e^2/(2+m)
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Rubi [A]
time = 0.02, antiderivative size = 86, normalized size of antiderivative = 1.00, 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 = {83, 74, 371}
\begin {gather*} \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)} \end {gather*}
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 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*} \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*}
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Mathematica [A]
time = 0.09, size = 77, normalized size = 0.90 \begin {gather*} \frac {x (e x)^m \left (a (1+m) x \, _2F_1\left (4,1+\frac {m}{2};2+\frac {m}{2};a^2 x^2\right )+(2+m) \, _2F_1\left (4,\frac {1+m}{2};\frac {3+m}{2};a^2 x^2\right )\right )}{16 (1+m) (2+m)} \end {gather*}
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 = 0.05, size = 0, normalized size = 0.00 \[\int \frac {\left (e x \right )^{m}}{\left (-2 a x +2\right )^{4} \left (a x +1\right )^{3}}\, dx\]
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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
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((x*e)^m/((a*x + 1)^3*(a*x - 1)^4), x)
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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*(x*e)^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] Result contains complex when optimal does not.
time = 4.11, size = 5872, normalized size = 68.28 \begin {gather*} \text {Too large to display} \end {gather*}
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*gamm...
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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*(x*e)^m/((a*x + 1)^3*(a*x - 1)^4), x)
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (e\,x\right )}^m}{{\left (a\,x+1\right )}^3\,{\left (2\,a\,x-2\right )}^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((e*x)^m/((a*x + 1)^3*(2*a*x - 2)^4),x)
[Out]
int((e*x)^m/((a*x + 1)^3*(2*a*x - 2)^4), x)
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