Optimal. Leaf size=79 \[ \frac{d x^2 \left (a+\frac{b}{x}\right )^{m+1}}{2 a}-\frac{b \left (a+\frac{b}{x}\right )^{m+1} (2 a c-b d (1-m)) \, _2F_1\left (2,m+1;m+2;\frac{b}{a x}+1\right )}{2 a^3 (m+1)} \]
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Rubi [A] time = 0.0378231, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {434, 446, 78, 65} \[ \frac{d x^2 \left (a+\frac{b}{x}\right )^{m+1}}{2 a}-\frac{b \left (a+\frac{b}{x}\right )^{m+1} (2 a c-b d (1-m)) \, _2F_1\left (2,m+1;m+2;\frac{b}{a x}+1\right )}{2 a^3 (m+1)} \]
Antiderivative was successfully verified.
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Rule 434
Rule 446
Rule 78
Rule 65
Rubi steps
\begin{align*} \int \left (a+\frac{b}{x}\right )^m (c+d x) \, dx &=\int \left (a+\frac{b}{x}\right )^m \left (d+\frac{c}{x}\right ) x \, dx\\ &=-\operatorname{Subst}\left (\int \frac{(a+b x)^m (d+c x)}{x^3} \, dx,x,\frac{1}{x}\right )\\ &=\frac{d \left (a+\frac{b}{x}\right )^{1+m} x^2}{2 a}-\frac{(2 a c+b d (-1+m)) \operatorname{Subst}\left (\int \frac{(a+b x)^m}{x^2} \, dx,x,\frac{1}{x}\right )}{2 a}\\ &=\frac{d \left (a+\frac{b}{x}\right )^{1+m} x^2}{2 a}-\frac{b (2 a c-b d (1-m)) \left (a+\frac{b}{x}\right )^{1+m} \, _2F_1\left (2,1+m;2+m;1+\frac{b}{a x}\right )}{2 a^3 (1+m)}\\ \end{align*}
Mathematica [A] time = 0.0309613, size = 73, normalized size = 0.92 \[ \frac{(a x+b) \left (a+\frac{b}{x}\right )^m \left (a^2 d (m+1) x^2+b (-2 a c-b d (m-1)) \, _2F_1\left (2,m+1;m+2;\frac{b}{a x}+1\right )\right )}{2 a^3 (m+1) x} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.018, size = 0, normalized size = 0. \begin{align*} \int \left ( a+{\frac{b}{x}} \right ) ^{m} \left ( dx+c \right ) \, 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{\left (d x + c\right )}{\left (a + \frac{b}{x}\right )}^{m}\,{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 ({\left (d x + c\right )} \left (\frac{a x + b}{x}\right )^{m}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 3.42879, size = 75, normalized size = 0.95 \begin{align*} \frac{b^{m} c x x^{- m} \Gamma \left (1 - m\right ){{}_{2}F_{1}\left (\begin{matrix} - m, 1 - m \\ 2 - m \end{matrix}\middle |{\frac{a x e^{i \pi }}{b}} \right )}}{\Gamma \left (2 - m\right )} + \frac{b^{m} d x^{2} x^{- m} \Gamma \left (2 - m\right ){{}_{2}F_{1}\left (\begin{matrix} - m, 2 - m \\ 3 - m \end{matrix}\middle |{\frac{a x e^{i \pi }}{b}} \right )}}{\Gamma \left (3 - m\right )} \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{\left (d x + c\right )}{\left (a + \frac{b}{x}\right )}^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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