3.289 \(\int \frac{\left (a+\frac{b}{x}\right )^n}{x^2 (c+d x)} \, dx\)

Optimal. Leaf size=84 \[ -\frac{d \left (a+\frac{b}{x}\right )^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{c \left (a+\frac{b}{x}\right )}{a c-b d}\right )}{c (n+1) (a c-b d)}-\frac{\left (a+\frac{b}{x}\right )^{n+1}}{b c (n+1)} \]

[Out]

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

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Rubi [A]  time = 0.153515, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{d \left (a+\frac{b}{x}\right )^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{c \left (a+\frac{b}{x}\right )}{a c-b d}\right )}{c (n+1) (a c-b d)}-\frac{\left (a+\frac{b}{x}\right )^{n+1}}{b c (n+1)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^n/(x^2*(c + d*x)),x]

[Out]

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

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Rubi in Sympy [A]  time = 19.7248, size = 56, normalized size = 0.67 \[ - \frac{d \left (a + \frac{b}{x}\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{c \left (a + \frac{b}{x}\right )}{a c - b d}} \right )}}{c \left (n + 1\right ) \left (a c - b d\right )} - \frac{\left (a + \frac{b}{x}\right )^{n + 1}}{b c \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x)**n/x**2/(d*x+c),x)

[Out]

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

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Mathematica [A]  time = 0.0780147, size = 0, normalized size = 0. \[ \int \frac{\left (a+\frac{b}{x}\right )^n}{x^2 (c+d x)} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[(a + b/x)^n/(x^2*(c + d*x)),x]

[Out]

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

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Maple [F]  time = 0.064, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{2} \left ( dx+c \right ) } \left ( a+{\frac{b}{x}} \right ) ^{n}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^n/x^2/(d*x+c),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(((a*x + b)/x)^n/(d*x^3 + c*x^2), 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**2/(d*x+c),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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