Optimal. Leaf size=50 \[ -\frac {(b x)^{-1-2 m} \, _2F_1\left (\frac {1}{2} (-1-2 m),-m;\frac {1}{2} (1-2 m);a^2 x^2\right )}{b (1+2 m)} \]
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Rubi [A]
time = 0.01, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {126, 371}
\begin {gather*} -\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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 126
Rule 371
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*}
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Mathematica [A]
time = 0.09, size = 44, normalized size = 0.88 \begin {gather*} -\frac {(b x)^{-1-2 m} \, _2F_1\left (-\frac {1}{2}-m,-m;\frac {1}{2}-m;a^2 x^2\right )}{b+2 b m} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \left (b x \right )^{-2-2 m} \left (-a x +1\right )^{m} \left (a x +1\right )^{m}\, dx\]
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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 {{\left (1-a\,x\right )}^m\,{\left (a\,x+1\right )}^m}{{\left (b\,x\right )}^{2\,m+2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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