Optimal. Leaf size=84 \[ -\frac{d \left (a+\frac{b}{x}\right )^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{c \left (a+\frac{b}{x}\right )}{a c-b d}\right )}{c (n+1) (a c-b d)}-\frac{\left (a+\frac{b}{x}\right )^{n+1}}{b c (n+1)} \]
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Rubi [A] time = 0.0555754, 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, Rules used = {514, 446, 80, 68} \[ -\frac{d \left (a+\frac{b}{x}\right )^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{c \left (a+\frac{b}{x}\right )}{a c-b d}\right )}{c (n+1) (a c-b d)}-\frac{\left (a+\frac{b}{x}\right )^{n+1}}{b c (n+1)} \]
Antiderivative was successfully verified.
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Rule 514
Rule 446
Rule 80
Rule 68
Rubi steps
\begin{align*} \int \frac{\left (a+\frac{b}{x}\right )^n}{x^2 (c+d x)} \, dx &=\int \frac{\left (a+\frac{b}{x}\right )^n}{\left (d+\frac{c}{x}\right ) x^3} \, dx\\ &=-\operatorname{Subst}\left (\int \frac{x (a+b x)^n}{d+c x} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{\left (a+\frac{b}{x}\right )^{1+n}}{b c (1+n)}+\frac{d \operatorname{Subst}\left (\int \frac{(a+b x)^n}{d+c x} \, dx,x,\frac{1}{x}\right )}{c}\\ &=-\frac{\left (a+\frac{b}{x}\right )^{1+n}}{b c (1+n)}-\frac{d \left (a+\frac{b}{x}\right )^{1+n} \, _2F_1\left (1,1+n;2+n;\frac{c \left (a+\frac{b}{x}\right )}{a c-b d}\right )}{c (a c-b d) (1+n)}\\ \end{align*}
Mathematica [A] time = 0.0234315, size = 77, normalized size = 0.92 \[ \frac{(a x+b) \left (a+\frac{b}{x}\right )^n \left (b d \, _2F_1\left (1,n+1;n+2;\frac{c \left (a+\frac{b}{x}\right )}{a c-b d}\right )+a c-b d\right )}{b c (n+1) x (b d-a c)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.511, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2} \left ( dx+c \right ) } \left ( a+{\frac{b}{x}} \right ) ^{n}}\, 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 )}^{n}}{{\left (d x + c\right )} x^{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 )^{n}}{d x^{3} + c x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 )}^{n}}{{\left (d x + c\right )} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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