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

Optimal. Leaf size=54 \[ \frac{\left (a+\frac{b}{x}\right )^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{c \left (a+\frac{b}{x}\right )}{a c-b d}\right )}{(n+1) (a c-b d)} \]

[Out]

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

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Rubi [A]  time = 0.106344, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{\left (a+\frac{b}{x}\right )^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{c \left (a+\frac{b}{x}\right )}{a c-b d}\right )}{(n+1) (a c-b d)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 14.1783, size = 36, normalized size = 0.67 \[ \frac{\left (a + \frac{b}{x}\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{c \left (a + \frac{b}{x}\right )}{a c - b d}} \right )}}{\left (n + 1\right ) \left (a c - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.064411, 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 = 0.056, size = 0, normalized size = 0. \[ \int{\frac{1}{x \left ( dx+c \right ) } \left ( a+{\frac{b}{x}} \right ) ^{n}}\, dx \]

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. \[ \int \frac{{\left (a + \frac{b}{x}\right )}^{n}}{{\left (d x + c\right )} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(((a*x + b)/x)^n/(d*x^2 + c*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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