Optimal. Leaf size=40 \[ -\frac {b \left (a+\frac {b}{x}\right )^{1+m} \, _2F_1\left (2,1+m;2+m;1+\frac {b}{a x}\right )}{a^2 (1+m)} \]
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Rubi [A]
time = 0.01, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {248, 67}
\begin {gather*} -\frac {b \left (a+\frac {b}{x}\right )^{m+1} \, _2F_1\left (2,m+1;m+2;\frac {b}{a x}+1\right )}{a^2 (m+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 67
Rule 248
Rubi steps
\begin {align*} \int \left (a+\frac {b}{x}\right )^m \, dx &=-\text {Subst}\left (\int \frac {(a+b x)^m}{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;1+\frac {b}{a x}\right )}{a^2 (1+m)}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 50, normalized size = 1.25 \begin {gather*} -\frac {\left (a+\frac {b}{x}\right )^m x \left (1+\frac {a x}{b}\right )^{-m} \, _2F_1\left (1-m,-m;2-m;-\frac {a x}{b}\right )}{-1+m} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \left (a +\frac {b}{x}\right )^{m}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.65, size = 34, normalized size = 0.85 \begin {gather*} \frac {b^{m} 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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.13, size = 51, normalized size = 1.28 \begin {gather*} -\frac {x\,{\left (a+\frac {b}{x}\right )}^m\,{{}}_2{\mathrm {F}}_1\left (1-m,-m;\ 2-m;\ -\frac {a\,x}{b}\right )}{{\left (\frac {a\,x}{b}+1\right )}^m\,\left (m-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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