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

Optimal. Leaf size=79 \[ \frac{d x^2 \left (a+\frac{b}{x}\right )^{m+1}}{2 a}-\frac{b \left (a+\frac{b}{x}\right )^{m+1} (2 a c-b d (1-m)) \, _2F_1\left (2,m+1;m+2;\frac{b}{a x}+1\right )}{2 a^3 (m+1)} \]

[Out]

(d*(a + b/x)^(1 + m)*x^2)/(2*a) - (b*(2*a*c - b*d*(1 - m))*(a + b/x)^(1 + m)*Hyp
ergeometric2F1[2, 1 + m, 2 + m, 1 + b/(a*x)])/(2*a^3*(1 + m))

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Rubi [A]  time = 0.105425, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{d x^2 \left (a+\frac{b}{x}\right )^{m+1}}{2 a}-\frac{b \left (a+\frac{b}{x}\right )^{m+1} (2 a c-b d (1-m)) \, _2F_1\left (2,m+1;m+2;\frac{b}{a x}+1\right )}{2 a^3 (m+1)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^m*(c + d*x),x]

[Out]

(d*(a + b/x)^(1 + m)*x^2)/(2*a) - (b*(2*a*c - b*d*(1 - m))*(a + b/x)^(1 + m)*Hyp
ergeometric2F1[2, 1 + m, 2 + m, 1 + b/(a*x)])/(2*a^3*(1 + m))

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Rubi in Sympy [A]  time = 6.47905, size = 58, normalized size = 0.73 \[ \frac{d x^{2} \left (a + \frac{b}{x}\right )^{m + 1}}{2 a} - \frac{b \left (a + \frac{b}{x}\right )^{m + 1} \left (2 a c - b d \left (- m + 1\right )\right ){{}_{2}F_{1}\left (\begin{matrix} 2, m + 1 \\ m + 2 \end{matrix}\middle |{1 + \frac{b}{a x}} \right )}}{2 a^{3} \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x)**m*(d*x+c),x)

[Out]

d*x**2*(a + b/x)**(m + 1)/(2*a) - b*(a + b/x)**(m + 1)*(2*a*c - b*d*(-m + 1))*hy
per((2, m + 1), (m + 2,), 1 + b/(a*x))/(2*a**3*(m + 1))

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Mathematica [A]  time = 0.048839, size = 88, normalized size = 1.11 \[ -\frac{x \left (a+\frac{b}{x}\right )^m \left (\frac{a x}{b}+1\right )^{-m} \left (c (m-2) \, _2F_1\left (1-m,-m;2-m;-\frac{a x}{b}\right )+d (m-1) x \, _2F_1\left (2-m,-m;3-m;-\frac{a x}{b}\right )\right )}{(m-2) (m-1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.028, size = 0, normalized size = 0. \[ \int \left ( a+{\frac{b}{x}} \right ) ^{m} \left ( dx+c \right ) \, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^m*(d*x+c),x)

[Out]

int((a+b/x)^m*(d*x+c),x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((d*x + c)*(a + b/x)^m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((d*x + c)*((a*x + b)/x)^m, x)

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Sympy [A]  time = 10.7185, size = 75, normalized size = 0.95 \[ \frac{b^{m} c x x^{- m} \Gamma \left (- m + 1\right ){{}_{2}F_{1}\left (\begin{matrix} - m, - m + 1 \\ - m + 2 \end{matrix}\middle |{\frac{a x e^{i \pi }}{b}} \right )}}{\Gamma \left (- m + 2\right )} + \frac{b^{m} d x^{2} x^{- m} \Gamma \left (- m + 2\right ){{}_{2}F_{1}\left (\begin{matrix} - m, - m + 2 \\ - m + 3 \end{matrix}\middle |{\frac{a x e^{i \pi }}{b}} \right )}}{\Gamma \left (- m + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x)**m*(d*x+c),x)

[Out]

b**m*c*x*x**(-m)*gamma(-m + 1)*hyper((-m, -m + 1), (-m + 2,), a*x*exp_polar(I*pi
)/b)/gamma(-m + 2) + b**m*d*x**2*x**(-m)*gamma(-m + 2)*hyper((-m, -m + 2), (-m +
3,), a*x*exp_polar(I*pi)/b)/gamma(-m + 3)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((d*x + c)*(a + b/x)^m, x)