3.987 \(\int (b x)^{-2-2 m} (1-a x)^m (1+a x)^m \, dx\)

Optimal. Leaf size=50 \[ -\frac{(b x)^{-2 m-1} \, _2F_1\left (\frac{1}{2} (-2 m-1),-m;\frac{1}{2} (1-2 m);a^2 x^2\right )}{b (2 m+1)} \]

[Out]

-(((b*x)^(-1 - 2*m)*Hypergeometric2F1[(-1 - 2*m)/2, -m, (1 - 2*m)/2, a^2*x^2])/(
b*(1 + 2*m)))

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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]

-(((b*x)^(-1 - 2*m)*Hypergeometric2F1[(-1 - 2*m)/2, -m, (1 - 2*m)/2, a^2*x^2])/(
b*(1 + 2*m)))

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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]

-(b*x)**(-2*m - 1)*hyper((-m, -m - 1/2), (-m + 1/2,), a**2*x**2)/(b*(2*m + 1))

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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]

-(((b*x)^(-1 - 2*m)*Hypergeometric2F1[-1/2 - m, -m, 1/2 - m, a^2*x^2])/(b + 2*b*
m))

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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]

int((b*x)^(-2-2*m)*(-a*x+1)^m*(a*x+1)^m,x)

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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]

integrate((a*x + 1)^m*(-a*x + 1)^m*(b*x)^(-2*m - 2), x)

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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]

integral((a*x + 1)^m*(-a*x + 1)^m*(b*x)^(-2*m - 2), x)

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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]

Timed 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]

integrate((a*x + 1)^m*(-a*x + 1)^m*(b*x)^(-2*m - 2), x)