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 = 0.0640267, 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 \[ -\frac{(b x)^{-2 m-1} \, _2F_1\left (\frac{1}{2} (-2 m-1),-m;\frac{1}{2} (1-2 m);a^2 x^2\right )}{b (2 m+1)} \]
Antiderivative was successfully verified.
[In] Int[(b*x)^(-2 - 2*m)*(1 - a*x)^m*(1 + a*x)^m,x]
[Out]
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Rubi in Sympy [A] time = 8.11651, size = 36, normalized size = 0.72 \[ - \frac{\left (b x\right )^{- 2 m - 1}{{}_{2}F_{1}\left (\begin{matrix} - m, - m - \frac{1}{2} \\ - m + \frac{1}{2} \end{matrix}\middle |{a^{2} x^{2}} \right )}}{b \left (2 m + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x)**(-2-2*m)*(-a*x+1)**m*(a*x+1)**m,x)
[Out]
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Mathematica [A] time = 0.0484633, 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.
[In] Integrate[(b*x)^(-2 - 2*m)*(1 - a*x)^m*(1 + a*x)^m,x]
[Out]
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Maple [F] time = 0.178, size = 0, normalized size = 0. \[ \int \left ( bx \right ) ^{-2-2\,m} \left ( -ax+1 \right ) ^{m} \left ( ax+1 \right ) ^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x)^(-2-2*m)*(-a*x+1)^m*(a*x+1)^m,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (a x + 1\right )}^{m}{\left (-a x + 1\right )}^{m} \left (b x\right )^{-2 \, m - 2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a*x + 1)^m*(-a*x + 1)^m*(b*x)^(-2*m - 2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (a x + 1\right )}^{m}{\left (-a x + 1\right )}^{m} \left (b x\right )^{-2 \, m - 2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a*x + 1)^m*(-a*x + 1)^m*(b*x)^(-2*m - 2),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x)**(-2-2*m)*(-a*x+1)**m*(a*x+1)**m,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (a x + 1\right )}^{m}{\left (-a x + 1\right )}^{m} \left (b x\right )^{-2 \, m - 2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a*x + 1)^m*(-a*x + 1)^m*(b*x)^(-2*m - 2),x, algorithm="giac")
[Out]