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

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Rubi [A]  time = 0.02, antiderivative size = 83, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.04, 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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fricas [F]  time = 1.03, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (x + 1\right )}^{p + \frac {1}{2}} {\left (-x + 1\right )}^{p - \frac {1}{2}}}{\left (c x\right )^{2 \, p + 2}}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (x + 1\right )}^{p + \frac {1}{2}} {\left (-x + 1\right )}^{p - \frac {1}{2}}}{\left (c x\right )^{2 \, p + 2}}\,{d x} \]

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)

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maple [F]  time = 0.14, size = 0, normalized size = 0.00 \[ \int \left (c x \right )^{-2 p -2} \left (-x +1\right )^{p -\frac {1}{2}} \left (x +1\right )^{p +\frac {1}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c x\right )^{-2 \, p - 2} {\left (x + 1\right )}^{p + \frac {1}{2}} {\left (-x + 1\right )}^{p - \frac {1}{2}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (1-x\right )}^{p-\frac {1}{2}}\,{\left (x+1\right )}^{p+\frac {1}{2}}}{{\left (c\,x\right )}^{2\,p+2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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