Optimal. Leaf size=80 \[ \frac {x \left (a+\frac {b}{x}\right )^m \left (\frac {a x}{b}+1\right )^{-m} (c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} F_1\left (1-m;-m,-n;2-m;-\frac {a x}{b},-\frac {d x}{c}\right )}{1-m} \]
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Rubi [A] time = 0.06, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {435, 135, 133} \[ \frac {x \left (a+\frac {b}{x}\right )^m \left (\frac {a x}{b}+1\right )^{-m} (c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} F_1\left (1-m;-m,-n;2-m;-\frac {a x}{b},-\frac {d x}{c}\right )}{1-m} \]
Antiderivative was successfully verified.
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Rule 133
Rule 135
Rule 435
Rubi steps
\begin {align*} \int \left (a+\frac {b}{x}\right )^m (c+d x)^n \, dx &=\left (\left (a+\frac {b}{x}\right )^m x^m (b+a x)^{-m}\right ) \int x^{-m} (b+a x)^m (c+d x)^n \, dx\\ &=\left (\left (a+\frac {b}{x}\right )^m x^m \left (1+\frac {a x}{b}\right )^{-m}\right ) \int x^{-m} \left (1+\frac {a x}{b}\right )^m (c+d x)^n \, dx\\ &=\left (\left (a+\frac {b}{x}\right )^m x^m \left (1+\frac {a x}{b}\right )^{-m} (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n}\right ) \int x^{-m} \left (1+\frac {a x}{b}\right )^m \left (1+\frac {d x}{c}\right )^n \, dx\\ &=\frac {\left (a+\frac {b}{x}\right )^m x \left (1+\frac {a x}{b}\right )^{-m} (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n} F_1\left (1-m;-m,-n;2-m;-\frac {a x}{b},-\frac {d x}{c}\right )}{1-m}\\ \end {align*}
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Mathematica [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int \left (a+\frac {b}{x}\right )^m (c+d x)^n \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (d x + c\right )}^{n} \left (\frac {a x + b}{x}\right )^{m}, x\right ) \]
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 \[ \int {\left (d x + c\right )}^{n} {\left (a + \frac {b}{x}\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.11, size = 0, normalized size = 0.00 \[ \int \left (a +\frac {b}{x}\right )^{m} \left (d x +c \right )^{n}\, dx \]
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 \[ \int {\left (d x + c\right )}^{n} {\left (a + \frac {b}{x}\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a+\frac {b}{x}\right )}^m\,{\left (c+d\,x\right )}^n \,d x \]
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 \[ \int \left (a + \frac {b}{x}\right )^{m} \left (c + d x\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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