Optimal. Leaf size=50 \[ -\frac{(b x)^{-2 m-1} \, _2F_1\left (\frac{1}{2} (-2 m-1),-m;\frac{1}{2} (1-2 m);a^2 x^2\right )}{b (2 m+1)} \]
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Rubi [A] time = 0.0184492, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {125, 364} \[ -\frac{(b x)^{-2 m-1} \, _2F_1\left (\frac{1}{2} (-2 m-1),-m;\frac{1}{2} (1-2 m);a^2 x^2\right )}{b (2 m+1)} \]
Antiderivative was successfully verified.
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Rule 125
Rule 364
Rubi steps
\begin{align*} \int (b x)^{-2-2 m} (1-a x)^m (1+a x)^m \, dx &=\int (b x)^{-2-2 m} \left (1-a^2 x^2\right )^m \, dx\\ &=-\frac{(b x)^{-1-2 m} \, _2F_1\left (\frac{1}{2} (-1-2 m),-m;\frac{1}{2} (1-2 m);a^2 x^2\right )}{b (1+2 m)}\\ \end{align*}
Mathematica [A] time = 0.0160963, size = 44, normalized size = 0.88 \[ -\frac{(b x)^{-2 m-1} \, _2F_1\left (-m-\frac{1}{2},-m;\frac{1}{2}-m;a^2 x^2\right )}{2 b m+b} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.113, size = 0, normalized size = 0. \begin{align*} \int \left ( bx \right ) ^{-2-2\,m} \left ( -ax+1 \right ) ^{m} \left ( ax+1 \right ) ^{m}\, 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{\left (a x + 1\right )}^{m}{\left (-a x + 1\right )}^{m} \left (b x\right )^{-2 \, m - 2}\,{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 ({\left (a x + 1\right )}^{m}{\left (-a x + 1\right )}^{m} \left (b x\right )^{-2 \, m - 2}, 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{\left (a x + 1\right )}^{m}{\left (-a x + 1\right )}^{m} \left (b x\right )^{-2 \, m - 2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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