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

Optimal. Leaf size=56 \[ -\frac{b \left (a+\frac{b}{x}\right )^{m+1} \, _2F_1\left (2,m+1;m+2;\frac{c \left (a+\frac{b}{x}\right )}{a c-b d}\right )}{(m+1) (a c-b d)^2} \]

[Out]

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

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Rubi [A]  time = 0.0948967, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ -\frac{b \left (a+\frac{b}{x}\right )^{m+1} \, _2F_1\left (2,m+1;m+2;\frac{c \left (a+\frac{b}{x}\right )}{a c-b d}\right )}{(m+1) (a c-b d)^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^m/(c + d*x)^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 6.58278, size = 41, normalized size = 0.73 \[ - \frac{b \left (a + \frac{b}{x}\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 2, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{c \left (a + \frac{b}{x}\right )}{a c - b d}} \right )}}{\left (m + 1\right ) \left (a c - b d\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0440716, size = 0, normalized size = 0. \[ \int \frac{\left (a+\frac{b}{x}\right )^m}{(c+d x)^2} \, dx \]

Verification is Not applicable to the result.

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

[Out]

Integrate[(a + b/x)^m/(c + d*x)^2, x]

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Maple [F]  time = 0.059, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( dx+c \right ) ^{2}} \left ( a+{\frac{b}{x}} \right ) ^{m}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^m/(d*x+c)^2,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((a + b/x)^m/(d*x + c)^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((a + b/x)^m/(d*x + c)^2, x)