Optimal. Leaf size=56 \[ -\frac{b \left (a+\frac{b}{x}\right )^{m+1} \, _2F_1\left (2,m+1;m+2;\frac{c \left (a+\frac{b}{x}\right )}{a c-b d}\right )}{(m+1) (a c-b d)^2} \]
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Rubi [A] time = 0.0340559, 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, Rules used = {434, 444, 68} \[ -\frac{b \left (a+\frac{b}{x}\right )^{m+1} \, _2F_1\left (2,m+1;m+2;\frac{c \left (a+\frac{b}{x}\right )}{a c-b d}\right )}{(m+1) (a c-b d)^2} \]
Antiderivative was successfully verified.
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Rule 434
Rule 444
Rule 68
Rubi steps
\begin{align*} \int \frac{\left (a+\frac{b}{x}\right )^m}{(c+d x)^2} \, dx &=\int \frac{\left (a+\frac{b}{x}\right )^m}{\left (d+\frac{c}{x}\right )^2 x^2} \, dx\\ &=-\operatorname{Subst}\left (\int \frac{(a+b x)^m}{(d+c x)^2} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{b \left (a+\frac{b}{x}\right )^{1+m} \, _2F_1\left (2,1+m;2+m;\frac{c \left (a+\frac{b}{x}\right )}{a c-b d}\right )}{(a c-b d)^2 (1+m)}\\ \end{align*}
Mathematica [A] time = 0.0244326, size = 57, normalized size = 1.02 \[ -\frac{b \left (a+\frac{b}{x}\right )^{m+1} \, _2F_1\left (2,m+1;m+2;-\frac{c \left (a+\frac{b}{x}\right )}{b d-a c}\right )}{(m+1) (b d-a c)^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.037, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( dx+c \right ) ^{2}} \left ( a+{\frac{b}{x}} \right ) ^{m}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a + \frac{b}{x}\right )}^{m}}{{\left (d x + c\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\left (\frac{a x + b}{x}\right )^{m}}{d^{2} x^{2} + 2 \, c d x + c^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + \frac{b}{x}\right )^{m}}{\left (c + d x\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a + \frac{b}{x}\right )}^{m}}{{\left (d x + c\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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