3.429 \(\int \left (a+\frac{b}{x}\right )^m \, dx\)

Optimal. Leaf size=40 \[ -\frac{b \left (a+\frac{b}{x}\right )^{m+1} \, _2F_1\left (2,m+1;m+2;\frac{b}{a x}+1\right )}{a^2 (m+1)} \]

[Out]

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

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Rubi [A]  time = 0.0292528, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -\frac{b \left (a+\frac{b}{x}\right )^{m+1} \, _2F_1\left (2,m+1;m+2;\frac{b}{a x}+1\right )}{a^2 (m+1)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^m,x]

[Out]

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

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Rubi in Sympy [A]  time = 2.04972, size = 29, normalized size = 0.72 \[ - \frac{b \left (a + \frac{b}{x}\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 2, m + 1 \\ m + 2 \end{matrix}\middle |{1 + \frac{b}{a x}} \right )}}{a^{2} \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x)**m,x)

[Out]

-b*(a + b/x)**(m + 1)*hyper((2, m + 1), (m + 2,), 1 + b/(a*x))/(a**2*(m + 1))

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Mathematica [A]  time = 0.0129052, size = 50, normalized size = 1.25 \[ -\frac{x \left (a+\frac{b}{x}\right )^m \left (\frac{a x}{b}+1\right )^{-m} \, _2F_1\left (1-m,-m;2-m;-\frac{a x}{b}\right )}{m-1} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x)^m,x]

[Out]

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

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Maple [F]  time = 0.018, size = 0, normalized size = 0. \[ \int \left ( a+{\frac{b}{x}} \right ) ^{m}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^m,x)

[Out]

int((a+b/x)^m,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^m,x, algorithm="maxima")

[Out]

integrate((a + b/x)^m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^m,x, algorithm="fricas")

[Out]

integral(((a*x + b)/x)^m, x)

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Sympy [A]  time = 4.79641, size = 34, normalized size = 0.85 \[ \frac{b^{m} x x^{- m} \Gamma \left (- m + 1\right ){{}_{2}F_{1}\left (\begin{matrix} - m, - m + 1 \\ - m + 2 \end{matrix}\middle |{\frac{a x e^{i \pi }}{b}} \right )}}{\Gamma \left (- m + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x)**m,x)

[Out]

b**m*x*x**(-m)*gamma(-m + 1)*hyper((-m, -m + 1), (-m + 2,), a*x*exp_polar(I*pi)/
b)/gamma(-m + 2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^m,x, algorithm="giac")

[Out]

integrate((a + b/x)^m, x)