3.1.67 \(\int \frac {(e x)^m}{(a+b x)^3 (a d-b d x)^4} \, dx\) [67]
Optimal. Leaf size=98 \[ \frac {(e x)^{1+m} \, _2F_1\left (4,\frac {1+m}{2};\frac {3+m}{2};\frac {b^2 x^2}{a^2}\right )}{a^7 d^4 e (1+m)}+\frac {b (e x)^{2+m} \, _2F_1\left (4,\frac {2+m}{2};\frac {4+m}{2};\frac {b^2 x^2}{a^2}\right )}{a^8 d^4 e^2 (2+m)} \]
[Out]
(e*x)^(1+m)*hypergeom([4, 1/2+1/2*m],[3/2+1/2*m],b^2*x^2/a^2)/a^7/d^4/e/(1+m)+b*(e*x)^(2+m)*hypergeom([4, 1+1/
2*m],[2+1/2*m],b^2*x^2/a^2)/a^8/d^4/e^2/(2+m)
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Rubi [A]
time = 0.03, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {83, 74, 371}
\begin {gather*} \frac {b (e x)^{m+2} \, _2F_1\left (4,\frac {m+2}{2};\frac {m+4}{2};\frac {b^2 x^2}{a^2}\right )}{a^8 d^4 e^2 (m+2)}+\frac {(e x)^{m+1} \, _2F_1\left (4,\frac {m+1}{2};\frac {m+3}{2};\frac {b^2 x^2}{a^2}\right )}{a^7 d^4 e (m+1)} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(e*x)^m/((a + b*x)^3*(a*d - b*d*x)^4),x]
[Out]
((e*x)^(1 + m)*Hypergeometric2F1[4, (1 + m)/2, (3 + m)/2, (b^2*x^2)/a^2])/(a^7*d^4*e*(1 + m)) + (b*(e*x)^(2 +
m)*Hypergeometric2F1[4, (2 + m)/2, (4 + m)/2, (b^2*x^2)/a^2])/(a^8*d^4*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}{(a+b x)^3 (a d-b d x)^4} \, dx &=a \int \frac {(e x)^m}{(a+b x)^4 (a d-b d x)^4} \, dx+\frac {b \int \frac {(e x)^{1+m}}{(a+b x)^4 (a d-b d x)^4} \, dx}{e}\\ &=a \int \frac {(e x)^m}{\left (a^2 d-b^2 d x^2\right )^4} \, dx+\frac {b \int \frac {(e x)^{1+m}}{\left (a^2 d-b^2 d x^2\right )^4} \, dx}{e}\\ &=\frac {(e x)^{1+m} \, _2F_1\left (4,\frac {1+m}{2};\frac {3+m}{2};\frac {b^2 x^2}{a^2}\right )}{a^7 d^4 e (1+m)}+\frac {b (e x)^{2+m} \, _2F_1\left (4,\frac {2+m}{2};\frac {4+m}{2};\frac {b^2 x^2}{a^2}\right )}{a^8 d^4 e^2 (2+m)}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 87, normalized size = 0.89 \begin {gather*} \frac {x (e x)^m \left (b (1+m) x \, _2F_1\left (4,1+\frac {m}{2};2+\frac {m}{2};\frac {b^2 x^2}{a^2}\right )+a (2+m) \, _2F_1\left (4,\frac {1+m}{2};\frac {3+m}{2};\frac {b^2 x^2}{a^2}\right )\right )}{a^8 d^4 (1+m) (2+m)} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(e*x)^m/((a + b*x)^3*(a*d - b*d*x)^4),x]
[Out]
(x*(e*x)^m*(b*(1 + m)*x*Hypergeometric2F1[4, 1 + m/2, 2 + m/2, (b^2*x^2)/a^2] + a*(2 + m)*Hypergeometric2F1[4,
(1 + m)/2, (3 + m)/2, (b^2*x^2)/a^2]))/(a^8*d^4*(1 + m)*(2 + m))
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Maple [F]
time = 0.07, size = 0, normalized size = 0.00 \[\int \frac {\left (e x \right )^{m}}{\left (b x +a \right )^{3} \left (-b d x +a d \right )^{4}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((e*x)^m/(b*x+a)^3/(-b*d*x+a*d)^4,x)
[Out]
int((e*x)^m/(b*x+a)^3/(-b*d*x+a*d)^4,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/(b*x+a)^3/(-b*d*x+a*d)^4,x, algorithm="maxima")
[Out]
integrate((x*e)^m/((b*d*x - a*d)^4*(b*x + a)^3), 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/(b*x+a)^3/(-b*d*x+a*d)^4,x, algorithm="fricas")
[Out]
integral((x*e)^m/(b^7*d^4*x^7 - a*b^6*d^4*x^6 - 3*a^2*b^5*d^4*x^5 + 3*a^3*b^4*d^4*x^4 + 3*a^4*b^3*d^4*x^3 - 3*
a^5*b^2*d^4*x^2 - a^6*b*d^4*x + a^7*d^4), x)
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Sympy [C] Result contains complex when optimal does not.
time = 5.25, size = 8284, normalized size = 84.53 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x)**m/(b*x+a)**3/(-b*d*x+a*d)**4,x)
[Out]
-2*a**5*e**m*m**4*x**m*lerchphi(a/(b*x), 1, m*exp_polar(I*pi))*gamma(-m)/(-96*a**11*b*d**4*gamma(1 - m) + 96*a
**10*b**2*d**4*x*gamma(1 - m) + 192*a**9*b**3*d**4*x**2*gamma(1 - m) - 192*a**8*b**4*d**4*x**3*gamma(1 - m) -
96*a**7*b**5*d**4*x**4*gamma(1 - m) + 96*a**6*b**6*d**4*x**5*gamma(1 - m)) + 15*a**5*e**m*m**3*x**m*lerchphi(a
/(b*x), 1, m*exp_polar(I*pi))*gamma(-m)/(-96*a**11*b*d**4*gamma(1 - m) + 96*a**10*b**2*d**4*x*gamma(1 - m) + 1
92*a**9*b**3*d**4*x**2*gamma(1 - m) - 192*a**8*b**4*d**4*x**3*gamma(1 - m) - 96*a**7*b**5*d**4*x**4*gamma(1 -
m) + 96*a**6*b**6*d**4*x**5*gamma(1 - m)) - 3*a**5*e**m*m**3*x**m*lerchphi(a*exp_polar(I*pi)/(b*x), 1, m*exp_p
olar(I*pi))*gamma(-m)/(-96*a**11*b*d**4*gamma(1 - m) + 96*a**10*b**2*d**4*x*gamma(1 - m) + 192*a**9*b**3*d**4*
x**2*gamma(1 - m) - 192*a**8*b**4*d**4*x**3*gamma(1 - m) - 96*a**7*b**5*d**4*x**4*gamma(1 - m) + 96*a**6*b**6*
d**4*x**5*gamma(1 - m)) - 31*a**5*e**m*m**2*x**m*lerchphi(a/(b*x), 1, m*exp_polar(I*pi))*gamma(-m)/(-96*a**11*
b*d**4*gamma(1 - m) + 96*a**10*b**2*d**4*x*gamma(1 - m) + 192*a**9*b**3*d**4*x**2*gamma(1 - m) - 192*a**8*b**4
*d**4*x**3*gamma(1 - m) - 96*a**7*b**5*d**4*x**4*gamma(1 - m) + 96*a**6*b**6*d**4*x**5*gamma(1 - m)) + 15*a**5
*e**m*m**2*x**m*lerchphi(a*exp_polar(I*pi)/(b*x), 1, m*exp_polar(I*pi))*gamma(-m)/(-96*a**11*b*d**4*gamma(1 -
m) + 96*a**10*b**2*d**4*x*gamma(1 - m) + 192*a**9*b**3*d**4*x**2*gamma(1 - m) - 192*a**8*b**4*d**4*x**3*gamma(
1 - m) - 96*a**7*b**5*d**4*x**4*gamma(1 - m) + 96*a**6*b**6*d**4*x**5*gamma(1 - m)) + 15*a**5*e**m*m*x**m*lerc
hphi(a/(b*x), 1, m*exp_polar(I*pi))*gamma(-m)/(-96*a**11*b*d**4*gamma(1 - m) + 96*a**10*b**2*d**4*x*gamma(1 -
m) + 192*a**9*b**3*d**4*x**2*gamma(1 - m) - 192*a**8*b**4*d**4*x**3*gamma(1 - m) - 96*a**7*b**5*d**4*x**4*gamm
a(1 - m) + 96*a**6*b**6*d**4*x**5*gamma(1 - m)) - 15*a**5*e**m*m*x**m*lerchphi(a*exp_polar(I*pi)/(b*x), 1, m*e
xp_polar(I*pi))*gamma(-m)/(-96*a**11*b*d**4*gamma(1 - m) + 96*a**10*b**2*d**4*x*gamma(1 - m) + 192*a**9*b**3*d
**4*x**2*gamma(1 - m) - 192*a**8*b**4*d**4*x**3*gamma(1 - m) - 96*a**7*b**5*d**4*x**4*gamma(1 - m) + 96*a**6*b
**6*d**4*x**5*gamma(1 - m)) + 2*a**4*b*e**m*m**4*x*x**m*lerchphi(a/(b*x), 1, m*exp_polar(I*pi))*gamma(-m)/(-96
*a**11*b*d**4*gamma(1 - m) + 96*a**10*b**2*d**4*x*gamma(1 - m) + 192*a**9*b**3*d**4*x**2*gamma(1 - m) - 192*a*
*8*b**4*d**4*x**3*gamma(1 - m) - 96*a**7*b**5*d**4*x**4*gamma(1 - m) + 96*a**6*b**6*d**4*x**5*gamma(1 - m)) -
15*a**4*b*e**m*m**3*x*x**m*lerchphi(a/(b*x), 1, m*exp_polar(I*pi))*gamma(-m)/(-96*a**11*b*d**4*gamma(1 - m) +
96*a**10*b**2*d**4*x*gamma(1 - m) + 192*a**9*b**3*d**4*x**2*gamma(1 - m) - 192*a**8*b**4*d**4*x**3*gamma(1 - m
) - 96*a**7*b**5*d**4*x**4*gamma(1 - m) + 96*a**6*b**6*d**4*x**5*gamma(1 - m)) + 3*a**4*b*e**m*m**3*x*x**m*ler
chphi(a*exp_polar(I*pi)/(b*x), 1, m*exp_polar(I*pi))*gamma(-m)/(-96*a**11*b*d**4*gamma(1 - m) + 96*a**10*b**2*
d**4*x*gamma(1 - m) + 192*a**9*b**3*d**4*x**2*gamma(1 - m) - 192*a**8*b**4*d**4*x**3*gamma(1 - m) - 96*a**7*b*
*5*d**4*x**4*gamma(1 - m) + 96*a**6*b**6*d**4*x**5*gamma(1 - m)) + 2*a**4*b*e**m*m**3*x*x**m*gamma(-m)/(-96*a*
*11*b*d**4*gamma(1 - m) + 96*a**10*b**2*d**4*x*gamma(1 - m) + 192*a**9*b**3*d**4*x**2*gamma(1 - m) - 192*a**8*
b**4*d**4*x**3*gamma(1 - m) - 96*a**7*b**5*d**4*x**4*gamma(1 - m) + 96*a**6*b**6*d**4*x**5*gamma(1 - m)) + 31*
a**4*b*e**m*m**2*x*x**m*lerchphi(a/(b*x), 1, m*exp_polar(I*pi))*gamma(-m)/(-96*a**11*b*d**4*gamma(1 - m) + 96*
a**10*b**2*d**4*x*gamma(1 - m) + 192*a**9*b**3*d**4*x**2*gamma(1 - m) - 192*a**8*b**4*d**4*x**3*gamma(1 - m) -
96*a**7*b**5*d**4*x**4*gamma(1 - m) + 96*a**6*b**6*d**4*x**5*gamma(1 - m)) - 15*a**4*b*e**m*m**2*x*x**m*lerch
phi(a*exp_polar(I*pi)/(b*x), 1, m*exp_polar(I*pi))*gamma(-m)/(-96*a**11*b*d**4*gamma(1 - m) + 96*a**10*b**2*d*
*4*x*gamma(1 - m) + 192*a**9*b**3*d**4*x**2*gamma(1 - m) - 192*a**8*b**4*d**4*x**3*gamma(1 - m) - 96*a**7*b**5
*d**4*x**4*gamma(1 - m) + 96*a**6*b**6*d**4*x**5*gamma(1 - m)) - 20*a**4*b*e**m*m**2*x*x**m*gamma(-m)/(-96*a**
11*b*d**4*gamma(1 - m) + 96*a**10*b**2*d**4*x*gamma(1 - m) + 192*a**9*b**3*d**4*x**2*gamma(1 - m) - 192*a**8*b
**4*d**4*x**3*gamma(1 - m) - 96*a**7*b**5*d**4*x**4*gamma(1 - m) + 96*a**6*b**6*d**4*x**5*gamma(1 - m)) - 15*a
**4*b*e**m*m*x*x**m*lerchphi(a/(b*x), 1, m*exp_polar(I*pi))*gamma(-m)/(-96*a**11*b*d**4*gamma(1 - m) + 96*a**1
0*b**2*d**4*x*gamma(1 - m) + 192*a**9*b**3*d**4*x**2*gamma(1 - m) - 192*a**8*b**4*d**4*x**3*gamma(1 - m) - 96*
a**7*b**5*d**4*x**4*gamma(1 - m) + 96*a**6*b**6*d**4*x**5*gamma(1 - m)) + 15*a**4*b*e**m*m*x*x**m*lerchphi(a*e
xp_polar(I*pi)/(b*x), 1, m*exp_polar(I*pi))*gamma(-m)/(-96*a**11*b*d**4*gamma(1 - m) + 96*a**10*b**2*d**4*x*ga
mma(1 - m) + 192*a**9*b**3*d**4*x**2*gamma(1 - m) - 192*a**8*b**4*d**4*x**3*gamma(1 - m) - 96*a**7*b**5*d**4*x
**4*gamma(1 - m) + 96*a**6*b**6*d**4*x**5*gamma(1 - m)) + 66*a**4*b*e**m*m*x*x**m*gamma(-m)/(-96*a**11*b*d**4*
gamma(1 - m) + 96*a**10*b**2*d**4*x*gamma(1 - m) + 192*a**9*b**3*d**4*x**2*gamma(1 - m) - 192*a**8*b**4*d**4*x
**3*gamma(1 - m) - 96*a**7*b**5*d**4*x**4*gamma...
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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/(b*x+a)^3/(-b*d*x+a*d)^4,x, algorithm="giac")
[Out]
integrate((x*e)^m/((b*d*x - a*d)^4*(b*x + a)^3), 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\,d-b\,d\,x\right )}^4\,{\left (a+b\,x\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((e*x)^m/((a*d - b*d*x)^4*(a + b*x)^3),x)
[Out]
int((e*x)^m/((a*d - b*d*x)^4*(a + b*x)^3), x)
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