3.997 \(\int (1-x)^{-\frac{1}{2}+p} (c x)^{-2 (1+p)} (1+x)^{\frac{1}{2}+p} \, dx\)

Optimal. Leaf size=83 \[ -\frac{4^{p+1} (1-x)^{p+\frac{1}{2}} \left (\frac{x}{x+1}\right )^{2 (p+1)} (x+1)^{p+\frac{3}{2}} (c x)^{-2 (p+1)} \, _2F_1\left (p+\frac{1}{2},2 (p+1);p+\frac{3}{2};\frac{1-x}{x+1}\right )}{2 p+1} \]

[Out]

-((4^(1 + p)*(1 - x)^(1/2 + p)*(x/(1 + x))^(2*(1 + p))*(1 + x)^(3/2 + p)*Hypergeometric2F1[1/2 + p, 2*(1 + p),
3/2 + p, (1 - x)/(1 + x)])/((1 + 2*p)*(c*x)^(2*(1 + p))))

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Rubi [A]  time = 0.0208823, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.033, Rules used = {132} \[ -\frac{4^{p+1} (1-x)^{p+\frac{1}{2}} \left (\frac{x}{x+1}\right )^{2 (p+1)} (x+1)^{p+\frac{3}{2}} (c x)^{-2 (p+1)} \, _2F_1\left (p+\frac{1}{2},2 (p+1);p+\frac{3}{2};\frac{1-x}{x+1}\right )}{2 p+1} \]

Antiderivative was successfully verified.

[In]

Int[((1 - x)^(-1/2 + p)*(1 + x)^(1/2 + p))/(c*x)^(2*(1 + p)),x]

[Out]

-((4^(1 + p)*(1 - x)^(1/2 + p)*(x/(1 + x))^(2*(1 + p))*(1 + x)^(3/2 + p)*Hypergeometric2F1[1/2 + p, 2*(1 + p),
3/2 + p, (1 - x)/(1 + x)])/((1 + 2*p)*(c*x)^(2*(1 + p))))

Rule 132

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[((a + b*x
)^(m + 1)*(c + d*x)^n*(e + f*x)^(p + 1)*Hypergeometric2F1[m + 1, -n, m + 2, -(((d*e - c*f)*(a + b*x))/((b*c -
a*d)*(e + f*x)))])/(((b*e - a*f)*(m + 1))*(((b*e - a*f)*(c + d*x))/((b*c - a*d)*(e + f*x)))^n), x] /; FreeQ[{a
, b, c, d, e, f, m, n, p}, x] && EqQ[m + n + p + 2, 0] &&  !IntegerQ[n]

Rubi steps

\begin{align*} \int (1-x)^{-\frac{1}{2}+p} (c x)^{-2 (1+p)} (1+x)^{\frac{1}{2}+p} \, dx &=-\frac{4^{1+p} (1-x)^{\frac{1}{2}+p} (c x)^{-2 (1+p)} \left (\frac{x}{1+x}\right )^{2 (1+p)} (1+x)^{\frac{3}{2}+p} \, _2F_1\left (\frac{1}{2}+p,2 (1+p);\frac{3}{2}+p;\frac{1-x}{1+x}\right )}{1+2 p}\\ \end{align*}

Mathematica [A]  time = 0.0413498, size = 82, normalized size = 0.99 \[ -\frac{4^{p+1} (1-x)^{p+\frac{1}{2}} \left (\frac{x}{x+1}\right )^{2 p} (x+1)^{p-\frac{1}{2}} (c x)^{-2 p} \, _2F_1\left (p+\frac{1}{2},2 p+2;p+\frac{3}{2};\frac{1-x}{x+1}\right )}{c^2 (2 p+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[((1 - x)^(-1/2 + p)*(1 + x)^(1/2 + p))/(c*x)^(2*(1 + p)),x]

[Out]

-((4^(1 + p)*(1 - x)^(1/2 + p)*(x/(1 + x))^(2*p)*(1 + x)^(-1/2 + p)*Hypergeometric2F1[1/2 + p, 2 + 2*p, 3/2 +
p, (1 - x)/(1 + x)])/(c^2*(1 + 2*p)*(c*x)^(2*p)))

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Maple [F]  time = 0.125, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( cx \right ) ^{2+2\,p}} \left ( 1+x \right ) ^{{\frac{1}{2}}+p} \left ( 1-x \right ) ^{-{\frac{1}{2}}+p}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-x)^(-1/2+p)*(1+x)^(1/2+p)/((c*x)^(2+2*p)),x)

[Out]

int((1-x)^(-1/2+p)*(1+x)^(1/2+p)/((c*x)^(2+2*p)),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (c x\right )^{-2 \, p - 2}{\left (x + 1\right )}^{p + \frac{1}{2}}{\left (-x + 1\right )}^{p - \frac{1}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)^(-1/2+p)*(1+x)^(1/2+p)/((c*x)^(2+2*p)),x, algorithm="maxima")

[Out]

integrate((c*x)^(-2*p - 2)*(x + 1)^(p + 1/2)*(-x + 1)^(p - 1/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (x + 1\right )}^{p + \frac{1}{2}}{\left (-x + 1\right )}^{p - \frac{1}{2}}}{\left (c x\right )^{2 \, p + 2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)^(-1/2+p)*(1+x)^(1/2+p)/((c*x)^(2+2*p)),x, algorithm="fricas")

[Out]

integral((x + 1)^(p + 1/2)*(-x + 1)^(p - 1/2)/(c*x)^(2*p + 2), x)

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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.

[In]

integrate((1-x)**(-1/2+p)*(1+x)**(1/2+p)/((c*x)**(2+2*p)),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (x + 1\right )}^{p + \frac{1}{2}}{\left (-x + 1\right )}^{p - \frac{1}{2}}}{\left (c x\right )^{2 \, p + 2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)^(-1/2+p)*(1+x)^(1/2+p)/((c*x)^(2+2*p)),x, algorithm="giac")

[Out]

integrate((x + 1)^(p + 1/2)*(-x + 1)^(p - 1/2)/(c*x)^(2*p + 2), x)