Optimal. Leaf size=64 \[ \frac {\left (a+\frac {b}{x}\right )^n \left (1+\frac {b}{a x}\right )^{-n} x^m F_1\left (-m;-n,1;1-m;-\frac {b}{a x},-\frac {c}{d x}\right )}{d m} \]
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Rubi [A]
time = 0.05, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {528, 511, 140,
138} \begin {gather*} \frac {x^m \left (a+\frac {b}{x}\right )^n \left (\frac {b}{a x}+1\right )^{-n} F_1\left (-m;-n,1;1-m;-\frac {b}{a x},-\frac {c}{d x}\right )}{d m} \end {gather*}
Antiderivative was successfully verified.
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Rule 138
Rule 140
Rule 511
Rule 528
Rubi steps
\begin {align*} \int \frac {\left (a+\frac {b}{x}\right )^n x^m}{c+d x} \, dx &=\int \frac {\left (a+\frac {b}{x}\right )^n x^{-1+m}}{d+\frac {c}{x}} \, dx\\ &=-\left (\left (\left (\frac {1}{x}\right )^m x^m\right ) \text {Subst}\left (\int \frac {x^{-1-m} (a+b x)^n}{d+c x} \, dx,x,\frac {1}{x}\right )\right )\\ &=-\left (\left (\left (a+\frac {b}{x}\right )^n \left (1+\frac {b}{a x}\right )^{-n} \left (\frac {1}{x}\right )^m x^m\right ) \text {Subst}\left (\int \frac {x^{-1-m} \left (1+\frac {b x}{a}\right )^n}{d+c x} \, dx,x,\frac {1}{x}\right )\right )\\ &=\frac {\left (a+\frac {b}{x}\right )^n \left (1+\frac {b}{a x}\right )^{-n} x^m F_1\left (-m;-n,1;1-m;-\frac {b}{a x},-\frac {c}{d x}\right )}{d m}\\ \end {align*}
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Mathematica [F]
time = 0.17, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+\frac {b}{x}\right )^n x^m}{c+d x} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {\left (a +\frac {b}{x}\right )^{n} x^{m}}{d x +c}\, dx\]
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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{m} \left (a + \frac {b}{x}\right )^{n}}{c + d x}\, dx \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 [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x^m\,{\left (a+\frac {b}{x}\right )}^n}{c+d\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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