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

Optimal. Leaf size=131 \[ -\frac{\left (a+\frac{b}{x}\right )^{n+1} (a c-b d n) \, _2F_1\left (1,n+1;n+2;\frac{b}{a x}+1\right )}{a^2 d^2 (n+1)}+\frac{c^2 \left (a+\frac{b}{x}\right )^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{c \left (a+\frac{b}{x}\right )}{a c-b d}\right )}{d^2 (n+1) (a c-b d)}+\frac{x \left (a+\frac{b}{x}\right )^{n+1}}{a d} \]

[Out]

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

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Rubi [A]  time = 0.278209, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{\left (a+\frac{b}{x}\right )^{n+1} (a c-b d n) \, _2F_1\left (1,n+1;n+2;\frac{b}{a x}+1\right )}{a^2 d^2 (n+1)}+\frac{c^2 \left (a+\frac{b}{x}\right )^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{c \left (a+\frac{b}{x}\right )}{a c-b d}\right )}{d^2 (n+1) (a c-b d)}+\frac{x \left (a+\frac{b}{x}\right )^{n+1}}{a d} \]

Antiderivative was successfully verified.

[In]  Int[((a + b/x)^n*x)/(c + d*x),x]

[Out]

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

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Rubi in Sympy [A]  time = 47.7277, size = 95, normalized size = 0.73 \[ \frac{c^{2} \left (a + \frac{b}{x}\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{c \left (a + \frac{b}{x}\right )}{a c - b d}} \right )}}{d^{2} \left (n + 1\right ) \left (a c - b d\right )} + \frac{x \left (a + \frac{b}{x}\right )^{n + 1}}{a d} - \frac{\left (a + \frac{b}{x}\right )^{n + 1} \left (a c - b d n\right ){{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{1 + \frac{b}{a x}} \right )}}{a^{2} d^{2} \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c**2*(a + b/x)**(n + 1)*hyper((1, n + 1), (n + 2,), c*(a + b/x)/(a*c - b*d))/(d*
*2*(n + 1)*(a*c - b*d)) + x*(a + b/x)**(n + 1)/(a*d) - (a + b/x)**(n + 1)*(a*c -
b*d*n)*hyper((1, n + 1), (n + 2,), 1 + b/(a*x))/(a**2*d**2*(n + 1))

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Mathematica [A]  time = 0.0946539, size = 0, normalized size = 0. \[ \int \frac{\left (a+\frac{b}{x}\right )^n x}{c+d x} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.05, size = 0, normalized size = 0. \[ \int{\frac{x}{dx+c} \left ( a+{\frac{b}{x}} \right ) ^{n}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^n*x/(d*x+c),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((a + b/x)^n*x/(d*x + c), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x)**n*x/(d*x+c),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((a + b/x)^n*x/(d*x + c), x)