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

Optimal. Leaf size=101 \[ \frac{\left (a+\frac{b}{x}\right )^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{b}{a x}+1\right )}{a d (m+1)}-\frac{c \left (a+\frac{b}{x}\right )^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{c \left (a+\frac{b}{x}\right )}{a c-b d}\right )}{d (m+1) (a c-b d)} \]

[Out]

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

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Rubi [A]  time = 0.161163, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294 \[ \frac{\left (a+\frac{b}{x}\right )^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{b}{a x}+1\right )}{a d (m+1)}-\frac{c \left (a+\frac{b}{x}\right )^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{c \left (a+\frac{b}{x}\right )}{a c-b d}\right )}{d (m+1) (a c-b d)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^m/(c + d*x),x]

[Out]

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

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Rubi in Sympy [A]  time = 10.9484, size = 66, normalized size = 0.65 \[ - \frac{c \left (a + \frac{b}{x}\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{c \left (a + \frac{b}{x}\right )}{a c - b d}} \right )}}{d \left (m + 1\right ) \left (a c - b d\right )} + \frac{\left (a + \frac{b}{x}\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{1 + \frac{b}{a x}} \right )}}{a d \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0406183, size = 0, normalized size = 0. \[ \int \frac{\left (a+\frac{b}{x}\right )^m}{c+d x} \, dx \]

Verification is Not applicable to the result.

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

[Out]

Integrate[(a + b/x)^m/(c + d*x), x]

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Maple [F]  time = 0.056, size = 0, normalized size = 0. \[ \int{\frac{1}{dx+c} \left ( a+{\frac{b}{x}} \right ) ^{m}}\, dx \]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((a + b/x)^m/(d*x + c), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((a + b/x)^m/(d*x + c), x)