Optimal. Leaf size=39 \[ -\frac{(a+b x)^{n-1} (c+d x)^{1-n}}{(1-n) (b c-a d)} \]
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Rubi [A] time = 0.0041875, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {37} \[ -\frac{(a+b x)^{n-1} (c+d x)^{1-n}}{(1-n) (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 37
Rubi steps
\begin{align*} \int (a+b x)^{-2+n} (c+d x)^{-n} \, dx &=-\frac{(a+b x)^{-1+n} (c+d x)^{1-n}}{(b c-a d) (1-n)}\\ \end{align*}
Mathematica [A] time = 0.0120888, size = 36, normalized size = 0.92 \[ \frac{(a+b x)^{n-1} (c+d x)^{1-n}}{(n-1) (b c-a d)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 45, normalized size = 1.2 \begin{align*} -{\frac{ \left ( bx+a \right ) ^{-1+n} \left ( dx+c \right ) }{ \left ( adn-bcn-ad+bc \right ) \left ( dx+c \right ) ^{n}}} \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 (b x + a\right )}^{n - 2}}{{\left (d x + c\right )}^{n}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.82054, size = 127, normalized size = 3.26 \begin{align*} -\frac{{\left (b d x^{2} + a c +{\left (b c + a d\right )} x\right )}{\left (b x + a\right )}^{n - 2}}{{\left (b c - a d -{\left (b c - a d\right )} n\right )}{\left (d x + c\right )}^{n}} \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 (b x + a\right )}^{n - 2}}{{\left (d x + c\right )}^{n}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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