3.364 \(\int \frac{(e x)^m}{(a+b x) (a d-b d x)^2} \, dx\)

Optimal. Leaf size=98 \[ \frac{b (e x)^{m+2} \, _2F_1\left (2,\frac{m+2}{2};\frac{m+4}{2};\frac{b^2 x^2}{a^2}\right )}{a^4 d^2 e^2 (m+2)}+\frac{(e x)^{m+1} \, _2F_1\left (2,\frac{m+1}{2};\frac{m+3}{2};\frac{b^2 x^2}{a^2}\right )}{a^3 d^2 e (m+1)} \]

[Out]

((e*x)^(1 + m)*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, (b^2*x^2)/a^2])/(a^3*d
^2*e*(1 + m)) + (b*(e*x)^(2 + m)*Hypergeometric2F1[2, (2 + m)/2, (4 + m)/2, (b^2
*x^2)/a^2])/(a^4*d^2*e^2*(2 + m))

_______________________________________________________________________________________

Rubi [A]  time = 0.178545, 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 \[ \frac{b (e x)^{m+2} \, _2F_1\left (2,\frac{m+2}{2};\frac{m+4}{2};\frac{b^2 x^2}{a^2}\right )}{a^4 d^2 e^2 (m+2)}+\frac{(e x)^{m+1} \, _2F_1\left (2,\frac{m+1}{2};\frac{m+3}{2};\frac{b^2 x^2}{a^2}\right )}{a^3 d^2 e (m+1)} \]

Antiderivative was successfully verified.

[In]  Int[(e*x)^m/((a + b*x)*(a*d - b*d*x)^2),x]

[Out]

((e*x)^(1 + m)*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, (b^2*x^2)/a^2])/(a^3*d
^2*e*(1 + m)) + (b*(e*x)^(2 + m)*Hypergeometric2F1[2, (2 + m)/2, (4 + m)/2, (b^2
*x^2)/a^2])/(a^4*d^2*e^2*(2 + m))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 25.2905, size = 80, normalized size = 0.82 \[ \frac{\left (e x\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 2, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{\frac{b^{2} x^{2}}{a^{2}}} \right )}}{a^{3} d^{2} e \left (m + 1\right )} + \frac{b \left (e x\right )^{m + 2}{{}_{2}F_{1}\left (\begin{matrix} 2, \frac{m}{2} + 1 \\ \frac{m}{2} + 2 \end{matrix}\middle |{\frac{b^{2} x^{2}}{a^{2}}} \right )}}{a^{4} d^{2} e^{2} \left (m + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x)**m/(b*x+a)/(-b*d*x+a*d)**2,x)

[Out]

(e*x)**(m + 1)*hyper((2, m/2 + 1/2), (m/2 + 3/2,), b**2*x**2/a**2)/(a**3*d**2*e*
(m + 1)) + b*(e*x)**(m + 2)*hyper((2, m/2 + 1), (m/2 + 2,), b**2*x**2/a**2)/(a**
4*d**2*e**2*(m + 2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0774445, size = 67, normalized size = 0.68 \[ \frac{x (e x)^m \left (\, _2F_1\left (1,m+1;m+2;-\frac{b x}{a}\right )+\, _2F_1\left (1,m+1;m+2;\frac{b x}{a}\right )+2 \, _2F_1\left (2,m+1;m+2;\frac{b x}{a}\right )\right )}{4 a^3 d^2 (m+1)} \]

Antiderivative was successfully verified.

[In]  Integrate[(e*x)^m/((a + b*x)*(a*d - b*d*x)^2),x]

[Out]

(x*(e*x)^m*(Hypergeometric2F1[1, 1 + m, 2 + m, -((b*x)/a)] + Hypergeometric2F1[1
, 1 + m, 2 + m, (b*x)/a] + 2*Hypergeometric2F1[2, 1 + m, 2 + m, (b*x)/a]))/(4*a^
3*d^2*(1 + m))

_______________________________________________________________________________________

Maple [F]  time = 0.076, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex \right ) ^{m}}{ \left ( bx+a \right ) \left ( -bdx+ad \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x)^m/(b*x+a)/(-b*d*x+a*d)^2,x)

[Out]

int((e*x)^m/(b*x+a)/(-b*d*x+a*d)^2,x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (e x\right )^{m}}{{\left (b d x - a d\right )}^{2}{\left (b x + a\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x)^m/((b*d*x - a*d)^2*(b*x + a)),x, algorithm="maxima")

[Out]

integrate((e*x)^m/((b*d*x - a*d)^2*(b*x + a)), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\left (e x\right )^{m}}{b^{3} d^{2} x^{3} - a b^{2} d^{2} x^{2} - a^{2} b d^{2} x + a^{3} d^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x)^m/((b*d*x - a*d)^2*(b*x + a)),x, algorithm="fricas")

[Out]

integral((e*x)^m/(b^3*d^2*x^3 - a*b^2*d^2*x^2 - a^2*b*d^2*x + a^3*d^2), x)

_______________________________________________________________________________________

Sympy [A]  time = 9.51517, size = 440, normalized size = 4.49 \[ - \frac{2 a e^{m} m^{2} x^{m} \Phi \left (\frac{a}{b x}, 1, m e^{i \pi }\right ) \Gamma \left (- m\right )}{- 4 a^{3} b d^{2} \Gamma \left (- m + 1\right ) + 4 a^{2} b^{2} d^{2} x \Gamma \left (- m + 1\right )} + \frac{a e^{m} m x^{m} \Phi \left (\frac{a}{b x}, 1, m e^{i \pi }\right ) \Gamma \left (- m\right )}{- 4 a^{3} b d^{2} \Gamma \left (- m + 1\right ) + 4 a^{2} b^{2} d^{2} x \Gamma \left (- m + 1\right )} - \frac{a e^{m} m x^{m} \Phi \left (\frac{a e^{i \pi }}{b x}, 1, m e^{i \pi }\right ) \Gamma \left (- m\right )}{- 4 a^{3} b d^{2} \Gamma \left (- m + 1\right ) + 4 a^{2} b^{2} d^{2} x \Gamma \left (- m + 1\right )} + \frac{2 b e^{m} m^{2} x x^{m} \Phi \left (\frac{a}{b x}, 1, m e^{i \pi }\right ) \Gamma \left (- m\right )}{- 4 a^{3} b d^{2} \Gamma \left (- m + 1\right ) + 4 a^{2} b^{2} d^{2} x \Gamma \left (- m + 1\right )} - \frac{b e^{m} m x x^{m} \Phi \left (\frac{a}{b x}, 1, m e^{i \pi }\right ) \Gamma \left (- m\right )}{- 4 a^{3} b d^{2} \Gamma \left (- m + 1\right ) + 4 a^{2} b^{2} d^{2} x \Gamma \left (- m + 1\right )} + \frac{b e^{m} m x x^{m} \Phi \left (\frac{a e^{i \pi }}{b x}, 1, m e^{i \pi }\right ) \Gamma \left (- m\right )}{- 4 a^{3} b d^{2} \Gamma \left (- m + 1\right ) + 4 a^{2} b^{2} d^{2} x \Gamma \left (- m + 1\right )} + \frac{2 b e^{m} m x x^{m} \Gamma \left (- m\right )}{- 4 a^{3} b d^{2} \Gamma \left (- m + 1\right ) + 4 a^{2} b^{2} d^{2} x \Gamma \left (- m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x)**m/(b*x+a)/(-b*d*x+a*d)**2,x)

[Out]

-2*a*e**m*m**2*x**m*lerchphi(a/(b*x), 1, m*exp_polar(I*pi))*gamma(-m)/(-4*a**3*b
*d**2*gamma(-m + 1) + 4*a**2*b**2*d**2*x*gamma(-m + 1)) + a*e**m*m*x**m*lerchphi
(a/(b*x), 1, m*exp_polar(I*pi))*gamma(-m)/(-4*a**3*b*d**2*gamma(-m + 1) + 4*a**2
*b**2*d**2*x*gamma(-m + 1)) - a*e**m*m*x**m*lerchphi(a*exp_polar(I*pi)/(b*x), 1,
m*exp_polar(I*pi))*gamma(-m)/(-4*a**3*b*d**2*gamma(-m + 1) + 4*a**2*b**2*d**2*x
*gamma(-m + 1)) + 2*b*e**m*m**2*x*x**m*lerchphi(a/(b*x), 1, m*exp_polar(I*pi))*g
amma(-m)/(-4*a**3*b*d**2*gamma(-m + 1) + 4*a**2*b**2*d**2*x*gamma(-m + 1)) - b*e
**m*m*x*x**m*lerchphi(a/(b*x), 1, m*exp_polar(I*pi))*gamma(-m)/(-4*a**3*b*d**2*g
amma(-m + 1) + 4*a**2*b**2*d**2*x*gamma(-m + 1)) + b*e**m*m*x*x**m*lerchphi(a*ex
p_polar(I*pi)/(b*x), 1, m*exp_polar(I*pi))*gamma(-m)/(-4*a**3*b*d**2*gamma(-m +
1) + 4*a**2*b**2*d**2*x*gamma(-m + 1)) + 2*b*e**m*m*x*x**m*gamma(-m)/(-4*a**3*b*
d**2*gamma(-m + 1) + 4*a**2*b**2*d**2*x*gamma(-m + 1))

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (e x\right )^{m}}{{\left (b d x - a d\right )}^{2}{\left (b x + a\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x)^m/((b*d*x - a*d)^2*(b*x + a)),x, algorithm="giac")

[Out]

integrate((e*x)^m/((b*d*x - a*d)^2*(b*x + a)), x)