3.66 \(\int \frac{(e x)^m}{(a+b x)^2 (a d-b d x)^3} \, dx\)
Optimal. Leaf size=98 \[ \frac{b (e x)^{m+2} \, _2F_1\left (3,\frac{m+2}{2};\frac{m+4}{2};\frac{b^2 x^2}{a^2}\right )}{a^6 d^3 e^2 (m+2)}+\frac{(e x)^{m+1} \, _2F_1\left (3,\frac{m+1}{2};\frac{m+3}{2};\frac{b^2 x^2}{a^2}\right )}{a^5 d^3 e (m+1)} \]
[Out]
((e*x)^(1 + m)*Hypergeometric2F1[3, (1 + m)/2, (3 + m)/2, (b^2*x^2)/a^2])/(a^5*d^3*e*(1 + m)) + (b*(e*x)^(2 +
m)*Hypergeometric2F1[3, (2 + m)/2, (4 + m)/2, (b^2*x^2)/a^2])/(a^6*d^3*e^2*(2 + m))
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Rubi [A] time = 0.0412521, antiderivative size = 98, normalized size of antiderivative = 1.,
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 =
{82, 73, 364} \[ \frac{b (e x)^{m+2} \, _2F_1\left (3,\frac{m+2}{2};\frac{m+4}{2};\frac{b^2 x^2}{a^2}\right )}{a^6 d^3 e^2 (m+2)}+\frac{(e x)^{m+1} \, _2F_1\left (3,\frac{m+1}{2};\frac{m+3}{2};\frac{b^2 x^2}{a^2}\right )}{a^5 d^3 e (m+1)} \]
Antiderivative was successfully verified.
[In]
Int[(e*x)^m/((a + b*x)^2*(a*d - b*d*x)^3),x]
[Out]
((e*x)^(1 + m)*Hypergeometric2F1[3, (1 + m)/2, (3 + m)/2, (b^2*x^2)/a^2])/(a^5*d^3*e*(1 + m)) + (b*(e*x)^(2 +
m)*Hypergeometric2F1[3, (2 + m)/2, (4 + m)/2, (b^2*x^2)/a^2])/(a^6*d^3*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}{(a+b x)^2 (a d-b d x)^3} \, dx &=a \int \frac{(e x)^m}{(a+b x)^3 (a d-b d x)^3} \, dx+\frac{b \int \frac{(e x)^{1+m}}{(a+b x)^3 (a d-b d x)^3} \, dx}{e}\\ &=a \int \frac{(e x)^m}{\left (a^2 d-b^2 d x^2\right )^3} \, dx+\frac{b \int \frac{(e x)^{1+m}}{\left (a^2 d-b^2 d x^2\right )^3} \, dx}{e}\\ &=\frac{(e x)^{1+m} \, _2F_1\left (3,\frac{1+m}{2};\frac{3+m}{2};\frac{b^2 x^2}{a^2}\right )}{a^5 d^3 e (1+m)}+\frac{b (e x)^{2+m} \, _2F_1\left (3,\frac{2+m}{2};\frac{4+m}{2};\frac{b^2 x^2}{a^2}\right )}{a^6 d^3 e^2 (2+m)}\\ \end{align*}
Mathematica [A] time = 0.0320356, size = 87, normalized size = 0.89 \[ \frac{x (e x)^m \left (b (m+1) x \, _2F_1\left (3,\frac{m}{2}+1;\frac{m}{2}+2;\frac{b^2 x^2}{a^2}\right )+a (m+2) \, _2F_1\left (3,\frac{m+1}{2};\frac{m+3}{2};\frac{b^2 x^2}{a^2}\right )\right )}{a^6 d^3 (m+1) (m+2)} \]
Antiderivative was successfully verified.
[In]
Integrate[(e*x)^m/((a + b*x)^2*(a*d - b*d*x)^3),x]
[Out]
(x*(e*x)^m*(b*(1 + m)*x*Hypergeometric2F1[3, 1 + m/2, 2 + m/2, (b^2*x^2)/a^2] + a*(2 + m)*Hypergeometric2F1[3,
(1 + m)/2, (3 + m)/2, (b^2*x^2)/a^2]))/(a^6*d^3*(1 + m)*(2 + m))
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Maple [F] time = 0.066, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex \right ) ^{m}}{ \left ( bx+a \right ) ^{2} \left ( -bdx+ad \right ) ^{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((e*x)^m/(b*x+a)^2/(-b*d*x+a*d)^3,x)
[Out]
int((e*x)^m/(b*x+a)^2/(-b*d*x+a*d)^3,x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{\left (e x\right )^{m}}{{\left (b d x - a d\right )}^{3}{\left (b x + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x)^m/(b*x+a)^2/(-b*d*x+a*d)^3,x, algorithm="maxima")
[Out]
-integrate((e*x)^m/((b*d*x - a*d)^3*(b*x + a)^2), x)
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\left (e x\right )^{m}}{b^{5} d^{3} x^{5} - a b^{4} d^{3} x^{4} - 2 \, a^{2} b^{3} d^{3} x^{3} + 2 \, a^{3} b^{2} d^{3} x^{2} + a^{4} b d^{3} x - a^{5} d^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x)^m/(b*x+a)^2/(-b*d*x+a*d)^3,x, algorithm="fricas")
[Out]
integral(-(e*x)^m/(b^5*d^3*x^5 - a*b^4*d^3*x^4 - 2*a^2*b^3*d^3*x^3 + 2*a^3*b^2*d^3*x^2 + a^4*b*d^3*x - a^5*d^3
), x)
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Sympy [C] time = 6.11648, size = 2717, normalized size = 27.72 \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/(b*x+a)**2/(-b*d*x+a*d)**3,x)
[Out]
-2*a**3*e**m*m**3*x**m*lerchphi(a/(b*x), 1, m*exp_polar(I*pi))*gamma(-m)/(16*a**7*b*d**3*gamma(1 - m) - 16*a**
6*b**2*d**3*x*gamma(1 - m) - 16*a**5*b**3*d**3*x**2*gamma(1 - m) + 16*a**4*b**4*d**3*x**3*gamma(1 - m)) + 6*a*
*3*e**m*m**2*x**m*lerchphi(a/(b*x), 1, m*exp_polar(I*pi))*gamma(-m)/(16*a**7*b*d**3*gamma(1 - m) - 16*a**6*b**
2*d**3*x*gamma(1 - m) - 16*a**5*b**3*d**3*x**2*gamma(1 - m) + 16*a**4*b**4*d**3*x**3*gamma(1 - m)) - 2*a**3*e*
*m*m**2*x**m*lerchphi(a*exp_polar(I*pi)/(b*x), 1, m*exp_polar(I*pi))*gamma(-m)/(16*a**7*b*d**3*gamma(1 - m) -
16*a**6*b**2*d**3*x*gamma(1 - m) - 16*a**5*b**3*d**3*x**2*gamma(1 - m) + 16*a**4*b**4*d**3*x**3*gamma(1 - m))
- 3*a**3*e**m*m*x**m*lerchphi(a/(b*x), 1, m*exp_polar(I*pi))*gamma(-m)/(16*a**7*b*d**3*gamma(1 - m) - 16*a**6*
b**2*d**3*x*gamma(1 - m) - 16*a**5*b**3*d**3*x**2*gamma(1 - m) + 16*a**4*b**4*d**3*x**3*gamma(1 - m)) + 3*a**3
*e**m*m*x**m*lerchphi(a*exp_polar(I*pi)/(b*x), 1, m*exp_polar(I*pi))*gamma(-m)/(16*a**7*b*d**3*gamma(1 - m) -
16*a**6*b**2*d**3*x*gamma(1 - m) - 16*a**5*b**3*d**3*x**2*gamma(1 - m) + 16*a**4*b**4*d**3*x**3*gamma(1 - m))
+ 2*a**2*b*e**m*m**3*x*x**m*lerchphi(a/(b*x), 1, m*exp_polar(I*pi))*gamma(-m)/(16*a**7*b*d**3*gamma(1 - m) - 1
6*a**6*b**2*d**3*x*gamma(1 - m) - 16*a**5*b**3*d**3*x**2*gamma(1 - m) + 16*a**4*b**4*d**3*x**3*gamma(1 - m)) -
6*a**2*b*e**m*m**2*x*x**m*lerchphi(a/(b*x), 1, m*exp_polar(I*pi))*gamma(-m)/(16*a**7*b*d**3*gamma(1 - m) - 16
*a**6*b**2*d**3*x*gamma(1 - m) - 16*a**5*b**3*d**3*x**2*gamma(1 - m) + 16*a**4*b**4*d**3*x**3*gamma(1 - m)) +
2*a**2*b*e**m*m**2*x*x**m*lerchphi(a*exp_polar(I*pi)/(b*x), 1, m*exp_polar(I*pi))*gamma(-m)/(16*a**7*b*d**3*ga
mma(1 - m) - 16*a**6*b**2*d**3*x*gamma(1 - m) - 16*a**5*b**3*d**3*x**2*gamma(1 - m) + 16*a**4*b**4*d**3*x**3*g
amma(1 - m)) + 2*a**2*b*e**m*m**2*x*x**m*gamma(-m)/(16*a**7*b*d**3*gamma(1 - m) - 16*a**6*b**2*d**3*x*gamma(1
- m) - 16*a**5*b**3*d**3*x**2*gamma(1 - m) + 16*a**4*b**4*d**3*x**3*gamma(1 - m)) + 3*a**2*b*e**m*m*x*x**m*ler
chphi(a/(b*x), 1, m*exp_polar(I*pi))*gamma(-m)/(16*a**7*b*d**3*gamma(1 - m) - 16*a**6*b**2*d**3*x*gamma(1 - m)
- 16*a**5*b**3*d**3*x**2*gamma(1 - m) + 16*a**4*b**4*d**3*x**3*gamma(1 - m)) - 3*a**2*b*e**m*m*x*x**m*lerchph
i(a*exp_polar(I*pi)/(b*x), 1, m*exp_polar(I*pi))*gamma(-m)/(16*a**7*b*d**3*gamma(1 - m) - 16*a**6*b**2*d**3*x*
gamma(1 - m) - 16*a**5*b**3*d**3*x**2*gamma(1 - m) + 16*a**4*b**4*d**3*x**3*gamma(1 - m)) - 10*a**2*b*e**m*m*x
*x**m*gamma(-m)/(16*a**7*b*d**3*gamma(1 - m) - 16*a**6*b**2*d**3*x*gamma(1 - m) - 16*a**5*b**3*d**3*x**2*gamma
(1 - m) + 16*a**4*b**4*d**3*x**3*gamma(1 - m)) + 2*a*b**2*e**m*m**3*x**2*x**m*lerchphi(a/(b*x), 1, m*exp_polar
(I*pi))*gamma(-m)/(16*a**7*b*d**3*gamma(1 - m) - 16*a**6*b**2*d**3*x*gamma(1 - m) - 16*a**5*b**3*d**3*x**2*gam
ma(1 - m) + 16*a**4*b**4*d**3*x**3*gamma(1 - m)) - 6*a*b**2*e**m*m**2*x**2*x**m*lerchphi(a/(b*x), 1, m*exp_pol
ar(I*pi))*gamma(-m)/(16*a**7*b*d**3*gamma(1 - m) - 16*a**6*b**2*d**3*x*gamma(1 - m) - 16*a**5*b**3*d**3*x**2*g
amma(1 - m) + 16*a**4*b**4*d**3*x**3*gamma(1 - m)) + 2*a*b**2*e**m*m**2*x**2*x**m*lerchphi(a*exp_polar(I*pi)/(
b*x), 1, m*exp_polar(I*pi))*gamma(-m)/(16*a**7*b*d**3*gamma(1 - m) - 16*a**6*b**2*d**3*x*gamma(1 - m) - 16*a**
5*b**3*d**3*x**2*gamma(1 - m) + 16*a**4*b**4*d**3*x**3*gamma(1 - m)) + 3*a*b**2*e**m*m*x**2*x**m*lerchphi(a/(b
*x), 1, m*exp_polar(I*pi))*gamma(-m)/(16*a**7*b*d**3*gamma(1 - m) - 16*a**6*b**2*d**3*x*gamma(1 - m) - 16*a**5
*b**3*d**3*x**2*gamma(1 - m) + 16*a**4*b**4*d**3*x**3*gamma(1 - m)) - 3*a*b**2*e**m*m*x**2*x**m*lerchphi(a*exp
_polar(I*pi)/(b*x), 1, m*exp_polar(I*pi))*gamma(-m)/(16*a**7*b*d**3*gamma(1 - m) - 16*a**6*b**2*d**3*x*gamma(1
- m) - 16*a**5*b**3*d**3*x**2*gamma(1 - m) + 16*a**4*b**4*d**3*x**3*gamma(1 - m)) + 2*a*b**2*e**m*m*x**2*x**m
*gamma(-m)/(16*a**7*b*d**3*gamma(1 - m) - 16*a**6*b**2*d**3*x*gamma(1 - m) - 16*a**5*b**3*d**3*x**2*gamma(1 -
m) + 16*a**4*b**4*d**3*x**3*gamma(1 - m)) - 2*b**3*e**m*m**3*x**3*x**m*lerchphi(a/(b*x), 1, m*exp_polar(I*pi))
*gamma(-m)/(16*a**7*b*d**3*gamma(1 - m) - 16*a**6*b**2*d**3*x*gamma(1 - m) - 16*a**5*b**3*d**3*x**2*gamma(1 -
m) + 16*a**4*b**4*d**3*x**3*gamma(1 - m)) + 6*b**3*e**m*m**2*x**3*x**m*lerchphi(a/(b*x), 1, m*exp_polar(I*pi))
*gamma(-m)/(16*a**7*b*d**3*gamma(1 - m) - 16*a**6*b**2*d**3*x*gamma(1 - m) - 16*a**5*b**3*d**3*x**2*gamma(1 -
m) + 16*a**4*b**4*d**3*x**3*gamma(1 - m)) - 2*b**3*e**m*m**2*x**3*x**m*lerchphi(a*exp_polar(I*pi)/(b*x), 1, m*
exp_polar(I*pi))*gamma(-m)/(16*a**7*b*d**3*gamma(1 - m) - 16*a**6*b**2*d**3*x*gamma(1 - m) - 16*a**5*b**3*d**3
*x**2*gamma(1 - m) + 16*a**4*b**4*d**3*x**3*gamma(1 - m)) - 2*b**3*e**m*m**2*x**3*x**m*gamma(-m)/(16*a**7*b*d*
*3*gamma(1 - m) - 16*a**6*b**2*d**3*x*gamma(1 - m) - 16*a**5*b**3*d**3*x**2*gamma(1 - m) + 16*a**4*b**4*d**3*x
**3*gamma(1 - m)) - 3*b**3*e**m*m*x**3*x**m*lerchphi(a/(b*x), 1, m*exp_polar(I*pi))*gamma(-m)/(16*a**7*b*d**3*
gamma(1 - m) - 16*a**6*b**2*d**3*x*gamma(1 - m) - 16*a**5*b**3*d**3*x**2*gamma(1 - m) + 16*a**4*b**4*d**3*x**3
*gamma(1 - m)) + 3*b**3*e**m*m*x**3*x**m*lerchphi(a*exp_polar(I*pi)/(b*x), 1, m*exp_polar(I*pi))*gamma(-m)/(16
*a**7*b*d**3*gamma(1 - m) - 16*a**6*b**2*d**3*x*gamma(1 - m) - 16*a**5*b**3*d**3*x**2*gamma(1 - m) + 16*a**4*b
**4*d**3*x**3*gamma(1 - m)) + 4*b**3*e**m*m*x**3*x**m*gamma(-m)/(16*a**7*b*d**3*gamma(1 - m) - 16*a**6*b**2*d*
*3*x*gamma(1 - m) - 16*a**5*b**3*d**3*x**2*gamma(1 - m) + 16*a**4*b**4*d**3*x**3*gamma(1 - m))
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{\left (e x\right )^{m}}{{\left (b d x - a d\right )}^{3}{\left (b x + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x)^m/(b*x+a)^2/(-b*d*x+a*d)^3,x, algorithm="giac")
[Out]
integrate(-(e*x)^m/((b*d*x - a*d)^3*(b*x + a)^2), x)