\(\int (b x)^{-2-2 m} (1-a x)^m (1+a x)^m \, dx\) [999]

Optimal result

Integrand size = 25, antiderivative size = 50 \[ \int (b x)^{-2-2 m} (1-a x)^m (1+a x)^m \, dx=-\frac {(b x)^{-1-2 m} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-1-2 m),-m,\frac {1}{2} (1-2 m),a^2 x^2\right )}{b (1+2 m)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {126, 371} \[ \int (b x)^{-2-2 m} (1-a x)^m (1+a x)^m \, dx=-\frac {(b x)^{-2 m-1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-2 m-1),-m,\frac {1}{2} (1-2 m),a^2 x^2\right )}{b (2 m+1)} \]

[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)))

Rule 126

Int[((f_.)*(x_))^(p_.)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[(a*c + b*d*x^2)
^m*(f*x)^p, x] /; FreeQ[{a, b, c, d, f, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[n, m] && GtQ[a, 0] && GtQ[c,
0]

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps \begin{align*} \text {integral}& = \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] (verified)

Time = 0.09 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.88 \[ \int (b x)^{-2-2 m} (1-a x)^m (1+a x)^m \, dx=-\frac {(b x)^{-1-2 m} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2}-m,-m,\frac {1}{2}-m,a^2 x^2\right )}{b+2 b m} \]

[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))

Maple [F]

\[\int \left (b x \right )^{-2-2 m} \left (-a x +1\right )^{m} \left (a x +1\right )^{m}d x\]

[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)

Fricas [F]

\[ \int (b x)^{-2-2 m} (1-a x)^m (1+a x)^m \, dx=\int { {\left (a x + 1\right )}^{m} {\left (-a x + 1\right )}^{m} \left (b x\right )^{-2 \, m - 2} \,d x } \]

[In]

integrate((b*x)^(-2-2*m)*(-a*x+1)^m*(a*x+1)^m,x, algorithm="fricas")

[Out]

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 13.22 (sec) , antiderivative size = 138, normalized size of antiderivative = 2.76 \[ \int (b x)^{-2-2 m} (1-a x)^m (1+a x)^m \, dx=\frac {a^{2 m + 1} b^{- 2 m - 2} {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {m}{2} + 1, \frac {m}{2} + \frac {3}{2}, 1 & m + \frac {3}{2}, 1, \frac {3}{2} \\\frac {1}{2}, 1, \frac {m}{2} + 1, \frac {3}{2}, \frac {m}{2} + \frac {3}{2} & 0 \end {matrix} \middle | {\frac {e^{- 2 i \pi }}{a^{2} x^{2}}} \right )} e^{i \pi m}}{4 \pi \Gamma \left (- m\right )} - \frac {a^{2 m + 1} b^{- 2 m - 2} {G_{6, 6}^{2, 6}\left (\begin {matrix} m + \frac {1}{2}, m + 1, m + \frac {3}{2}, \frac {m}{2} + \frac {1}{2}, \frac {m}{2} + 1, 1 & \\\frac {m}{2} + \frac {1}{2}, \frac {m}{2} + 1 & m + \frac {1}{2}, m + 1, \frac {1}{2}, 0 \end {matrix} \middle | {\frac {1}{a^{2} x^{2}}} \right )}}{4 \pi \Gamma \left (- m\right )} \]

[In]

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

[Out]

a**(2*m + 1)*b**(-2*m - 2)*meijerg(((m/2 + 1, m/2 + 3/2, 1), (m + 3/2, 1, 3/2)), ((1/2, 1, m/2 + 1, 3/2, m/2 +
3/2), (0,)), exp_polar(-2*I*pi)/(a**2*x**2))*exp(I*pi*m)/(4*pi*gamma(-m)) - a**(2*m + 1)*b**(-2*m - 2)*meijer
g(((m + 1/2, m + 1, m + 3/2, m/2 + 1/2, m/2 + 1, 1), ()), ((m/2 + 1/2, m/2 + 1), (m + 1/2, m + 1, 1/2, 0)), 1/
(a**2*x**2))/(4*pi*gamma(-m))

Maxima [F]

\[ \int (b x)^{-2-2 m} (1-a x)^m (1+a x)^m \, dx=\int { {\left (a x + 1\right )}^{m} {\left (-a x + 1\right )}^{m} \left (b x\right )^{-2 \, m - 2} \,d x } \]

[In]

integrate((b*x)^(-2-2*m)*(-a*x+1)^m*(a*x+1)^m,x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int (b x)^{-2-2 m} (1-a x)^m (1+a x)^m \, dx=\int { {\left (a x + 1\right )}^{m} {\left (-a x + 1\right )}^{m} \left (b x\right )^{-2 \, m - 2} \,d x } \]

[In]

integrate((b*x)^(-2-2*m)*(-a*x+1)^m*(a*x+1)^m,x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int (b x)^{-2-2 m} (1-a x)^m (1+a x)^m \, dx=\int \frac {{\left (1-a\,x\right )}^m\,{\left (a\,x+1\right )}^m}{{\left (b\,x\right )}^{2\,m+2}} \,d x \]

[In]

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

[Out]

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