3.365 \(\int \frac{(e x)^m}{(a+b x)^2 (a d-b d x)^3} \, dx\)
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 = 0.181576, 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
\[ \frac{b (e x)^{m+2} \, _2F_1\left (3,\frac{m+2}{2};\frac{m+4}{2};\frac{b^2 x^2}{a^2}\right )}{a^6 d^3 e^2 (m+2)}+\frac{(e x)^{m+1} \, _2F_1\left (3,\frac{m+1}{2};\frac{m+3}{2};\frac{b^2 x^2}{a^2}\right )}{a^5 d^3 e (m+1)} \]
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))
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Rubi in Sympy [A] time = 25.3218, size = 80, normalized size = 0.82 \[ \frac{\left (e x\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 3, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{\frac{b^{2} x^{2}}{a^{2}}} \right )}}{a^{5} d^{3} e \left (m + 1\right )} + \frac{b \left (e x\right )^{m + 2}{{}_{2}F_{1}\left (\begin{matrix} 3, \frac{m}{2} + 1 \\ \frac{m}{2} + 2 \end{matrix}\middle |{\frac{b^{2} x^{2}}{a^{2}}} \right )}}{a^{6} d^{3} e^{2} \left (m + 2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x)**m/(b*x+a)**2/(-b*d*x+a*d)**3,x)
[Out]
(e*x)**(m + 1)*hyper((3, m/2 + 1/2), (m/2 + 3/2,), b**2*x**2/a**2)/(a**5*d**3*e*
(m + 1)) + b*(e*x)**(m + 2)*hyper((3, m/2 + 1), (m/2 + 2,), b**2*x**2/a**2)/(a**
6*d**3*e**2*(m + 2))
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Mathematica [C] time = 0.282687, size = 144, normalized size = 1.47 \[ \frac{a (m+2) x (e x)^m F_1\left (m+1;3,2;m+2;\frac{b x}{a},-\frac{b x}{a}\right )}{d^3 (m+1) (a-b x)^3 (a+b x)^2 \left (b x \left (3 F_1\left (m+2;4,2;m+3;\frac{b x}{a},-\frac{b x}{a}\right )-2 \, _2F_1\left (3,\frac{m}{2}+1;\frac{m}{2}+2;\frac{b^2 x^2}{a^2}\right )\right )+a (m+2) F_1\left (m+1;3,2;m+2;\frac{b x}{a},-\frac{b x}{a}\right )\right )} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(e*x)^m/((a + b*x)^2*(a*d - b*d*x)^3),x]
[Out]
(a*(2 + m)*x*(e*x)^m*AppellF1[1 + m, 3, 2, 2 + m, (b*x)/a, -((b*x)/a)])/(d^3*(1
+ m)*(a - b*x)^3*(a + b*x)^2*(a*(2 + m)*AppellF1[1 + m, 3, 2, 2 + m, (b*x)/a, -(
(b*x)/a)] + b*x*(3*AppellF1[2 + m, 4, 2, 3 + m, (b*x)/a, -((b*x)/a)] - 2*Hyperge
ometricPFQ[{3, 1 + m/2}, {2 + m/2}, (b^2*x^2)/a^2])))
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Maple [F] time = 0.099, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex \right ) ^{m}}{ \left ( bx+a \right ) ^{2} \left ( -bdx+ad \right ) ^{3}}}\, dx \]
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. \[ -\int \frac{\left (e x\right )^{m}}{{\left (b d x - a d\right )}^{3}{\left (b x + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(e*x)^m/((b*d*x - a*d)^3*(b*x + a)^2),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. \[{\rm integral}\left (-\frac{\left (e x\right )^{m}}{b^{5} d^{3} x^{5} - a b^{4} d^{3} x^{4} - 2 \, a^{2} b^{3} d^{3} x^{3} + 2 \, a^{3} b^{2} d^{3} x^{2} + a^{4} b d^{3} x - a^{5} d^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(e*x)^m/((b*d*x - a*d)^3*(b*x + a)^2),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 [A] time = 17.3325, size = 2717, normalized size = 27.72 \[ \text{result too large to display} \]
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(-m + 1) - 16*a**6*b**2*d**3*x*gamma(-m + 1) - 16*a**5*b**3*d**3*x
**2*gamma(-m + 1) + 16*a**4*b**4*d**3*x**3*gamma(-m + 1)) + 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(-m + 1
) - 16*a**6*b**2*d**3*x*gamma(-m + 1) - 16*a**5*b**3*d**3*x**2*gamma(-m + 1) + 1
6*a**4*b**4*d**3*x**3*gamma(-m + 1)) - 2*a**3*e**m*m**2*x**m*lerchphi(a*exp_pola
r(I*pi)/(b*x), 1, m*exp_polar(I*pi))*gamma(-m)/(16*a**7*b*d**3*gamma(-m + 1) - 1
6*a**6*b**2*d**3*x*gamma(-m + 1) - 16*a**5*b**3*d**3*x**2*gamma(-m + 1) + 16*a**
4*b**4*d**3*x**3*gamma(-m + 1)) - 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(-m + 1) - 16*a**6*b**2*d**3*x*gamma
(-m + 1) - 16*a**5*b**3*d**3*x**2*gamma(-m + 1) + 16*a**4*b**4*d**3*x**3*gamma(-
m + 1)) + 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(-m + 1) - 16*a**6*b**2*d**3*x*gamma(-m + 1)
- 16*a**5*b**3*d**3*x**2*gamma(-m + 1) + 16*a**4*b**4*d**3*x**3*gamma(-m + 1))
+ 2*a**2*b*e**m*m**3*x*x**m*lerchphi(a/(b*x), 1, m*exp_polar(I*pi))*gamma(-m)/(1
6*a**7*b*d**3*gamma(-m + 1) - 16*a**6*b**2*d**3*x*gamma(-m + 1) - 16*a**5*b**3*d
**3*x**2*gamma(-m + 1) + 16*a**4*b**4*d**3*x**3*gamma(-m + 1)) - 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*gam
ma(-m + 1) - 16*a**6*b**2*d**3*x*gamma(-m + 1) - 16*a**5*b**3*d**3*x**2*gamma(-m
+ 1) + 16*a**4*b**4*d**3*x**3*gamma(-m + 1)) + 2*a**2*b*e**m*m**2*x*x**m*lerchp
hi(a*exp_polar(I*pi)/(b*x), 1, m*exp_polar(I*pi))*gamma(-m)/(16*a**7*b*d**3*gamm
a(-m + 1) - 16*a**6*b**2*d**3*x*gamma(-m + 1) - 16*a**5*b**3*d**3*x**2*gamma(-m
+ 1) + 16*a**4*b**4*d**3*x**3*gamma(-m + 1)) + 2*a**2*b*e**m*m**2*x*x**m*gamma(-
m)/(16*a**7*b*d**3*gamma(-m + 1) - 16*a**6*b**2*d**3*x*gamma(-m + 1) - 16*a**5*b
**3*d**3*x**2*gamma(-m + 1) + 16*a**4*b**4*d**3*x**3*gamma(-m + 1)) + 3*a**2*b*e
**m*m*x*x**m*lerchphi(a/(b*x), 1, m*exp_polar(I*pi))*gamma(-m)/(16*a**7*b*d**3*g
amma(-m + 1) - 16*a**6*b**2*d**3*x*gamma(-m + 1) - 16*a**5*b**3*d**3*x**2*gamma(
-m + 1) + 16*a**4*b**4*d**3*x**3*gamma(-m + 1)) - 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
(-m + 1) - 16*a**6*b**2*d**3*x*gamma(-m + 1) - 16*a**5*b**3*d**3*x**2*gamma(-m +
1) + 16*a**4*b**4*d**3*x**3*gamma(-m + 1)) - 10*a**2*b*e**m*m*x*x**m*gamma(-m)/
(16*a**7*b*d**3*gamma(-m + 1) - 16*a**6*b**2*d**3*x*gamma(-m + 1) - 16*a**5*b**3
*d**3*x**2*gamma(-m + 1) + 16*a**4*b**4*d**3*x**3*gamma(-m + 1)) + 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(-m + 1) - 16*a**6*b**2*d**3*x*gamma(-m + 1) - 16*a**5*b**3*d**3*x**2*gam
ma(-m + 1) + 16*a**4*b**4*d**3*x**3*gamma(-m + 1)) - 6*a*b**2*e**m*m**2*x**2*x**
m*lerchphi(a/(b*x), 1, m*exp_polar(I*pi))*gamma(-m)/(16*a**7*b*d**3*gamma(-m + 1
) - 16*a**6*b**2*d**3*x*gamma(-m + 1) - 16*a**5*b**3*d**3*x**2*gamma(-m + 1) + 1
6*a**4*b**4*d**3*x**3*gamma(-m + 1)) + 2*a*b**2*e**m*m**2*x**2*x**m*lerchphi(a*e
xp_polar(I*pi)/(b*x), 1, m*exp_polar(I*pi))*gamma(-m)/(16*a**7*b*d**3*gamma(-m +
1) - 16*a**6*b**2*d**3*x*gamma(-m + 1) - 16*a**5*b**3*d**3*x**2*gamma(-m + 1) +
16*a**4*b**4*d**3*x**3*gamma(-m + 1)) + 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(-m + 1) - 16*a**6*b**
2*d**3*x*gamma(-m + 1) - 16*a**5*b**3*d**3*x**2*gamma(-m + 1) + 16*a**4*b**4*d**
3*x**3*gamma(-m + 1)) - 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(-m + 1) - 16*a**6*b**2
*d**3*x*gamma(-m + 1) - 16*a**5*b**3*d**3*x**2*gamma(-m + 1) + 16*a**4*b**4*d**3
*x**3*gamma(-m + 1)) + 2*a*b**2*e**m*m*x**2*x**m*gamma(-m)/(16*a**7*b*d**3*gamma
(-m + 1) - 16*a**6*b**2*d**3*x*gamma(-m + 1) - 16*a**5*b**3*d**3*x**2*gamma(-m +
1) + 16*a**4*b**4*d**3*x**3*gamma(-m + 1)) - 2*b**3*e**m*m**3*x**3*x**m*lerchph
i(a/(b*x), 1, m*exp_polar(I*pi))*gamma(-m)/(16*a**7*b*d**3*gamma(-m + 1) - 16*a*
*6*b**2*d**3*x*gamma(-m + 1) - 16*a**5*b**3*d**3*x**2*gamma(-m + 1) + 16*a**4*b*
*4*d**3*x**3*gamma(-m + 1)) + 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(-m + 1) - 16*a**6*b**2*d**3*x*g
amma(-m + 1) - 16*a**5*b**3*d**3*x**2*gamma(-m + 1) + 16*a**4*b**4*d**3*x**3*gam
ma(-m + 1)) - 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(-m + 1) - 16*a**6*b**2*d**3*x*g
amma(-m + 1) - 16*a**5*b**3*d**3*x**2*gamma(-m + 1) + 16*a**4*b**4*d**3*x**3*gam
ma(-m + 1)) - 2*b**3*e**m*m**2*x**3*x**m*gamma(-m)/(16*a**7*b*d**3*gamma(-m + 1)
- 16*a**6*b**2*d**3*x*gamma(-m + 1) - 16*a**5*b**3*d**3*x**2*gamma(-m + 1) + 16
*a**4*b**4*d**3*x**3*gamma(-m + 1)) - 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(-m + 1) - 16*a**6*b**2*d**
3*x*gamma(-m + 1) - 16*a**5*b**3*d**3*x**2*gamma(-m + 1) + 16*a**4*b**4*d**3*x**
3*gamma(-m + 1)) + 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(-m + 1) - 16*a**6*b**2*d**3*x
*gamma(-m + 1) - 16*a**5*b**3*d**3*x**2*gamma(-m + 1) + 16*a**4*b**4*d**3*x**3*g
amma(-m + 1)) + 4*b**3*e**m*m*x**3*x**m*gamma(-m)/(16*a**7*b*d**3*gamma(-m + 1)
- 16*a**6*b**2*d**3*x*gamma(-m + 1) - 16*a**5*b**3*d**3*x**2*gamma(-m + 1) + 16*
a**4*b**4*d**3*x**3*gamma(-m + 1))
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int -\frac{\left (e x\right )^{m}}{{\left (b d x - a d\right )}^{3}{\left (b x + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(e*x)^m/((b*d*x - a*d)^3*(b*x + a)^2),x, algorithm="giac")
[Out]
integrate(-(e*x)^m/((b*d*x - a*d)^3*(b*x + a)^2), x)