Optimal. Leaf size=62 \[ -\frac{4 b^2 (a-b x)^{1-n} (a+b x)^{n-1} \, _2F_1\left (3,1-n;2-n;\frac{a-b x}{a+b x}\right )}{a (1-n)} \]
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Rubi [A] time = 0.0175386, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {131} \[ -\frac{4 b^2 (a-b x)^{1-n} (a+b x)^{n-1} \, _2F_1\left (3,1-n;2-n;\frac{a-b x}{a+b x}\right )}{a (1-n)} \]
Antiderivative was successfully verified.
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Rule 131
Rubi steps
\begin{align*} \int \frac{(a-b x)^{-n} (a+b x)^{1+n}}{x^3} \, dx &=-\frac{4 b^2 (a-b x)^{1-n} (a+b x)^{-1+n} \, _2F_1\left (3,1-n;2-n;\frac{a-b x}{a+b x}\right )}{a (1-n)}\\ \end{align*}
Mathematica [A] time = 0.0255171, size = 62, normalized size = 1. \[ -\frac{4 b^2 (a-b x)^{1-n} (a+b x)^{n-1} \, _2F_1\left (3,1-n;2-n;\frac{a-b x}{a+b x}\right )}{a (1-n)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.059, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( bx+a \right ) ^{1+n}}{{x}^{3} \left ( -bx+a \right ) ^{n}}}\, 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 \frac{{\left (b x + a\right )}^{n + 1}}{{\left (-b x + a\right )}^{n} x^{3}}\,{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 (\frac{{\left (b x + a\right )}^{n + 1}}{{\left (-b x + a\right )}^{n} x^{3}}, x\right ) \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 + 1}}{{\left (-b x + a\right )}^{n} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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