3.1069 \(\int \frac {\sqrt {1+x}}{(1-x)^{3/2}} \, dx\)

Optimal. Leaf size=23 \[ \frac {2 \sqrt {x+1}}{\sqrt {1-x}}-\sin ^{-1}(x) \]

[Out]

-arcsin(x)+2*(1+x)^(1/2)/(1-x)^(1/2)

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Rubi [A]  time = 0.00, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {47, 41, 216} \[ \frac {2 \sqrt {x+1}}{\sqrt {1-x}}-\sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 + x]/(1 - x)^(3/2),x]

[Out]

(2*Sqrt[1 + x])/Sqrt[1 - x] - ArcSin[x]

Rule 41

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

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin {align*} \int \frac {\sqrt {1+x}}{(1-x)^{3/2}} \, dx &=\frac {2 \sqrt {1+x}}{\sqrt {1-x}}-\int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx\\ &=\frac {2 \sqrt {1+x}}{\sqrt {1-x}}-\int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=\frac {2 \sqrt {1+x}}{\sqrt {1-x}}-\sin ^{-1}(x)\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 36, normalized size = 1.57 \[ 2 \left (\frac {\sqrt {x+1}}{\sqrt {1-x}}+\sin ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {2}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 + x]/(1 - x)^(3/2),x]

[Out]

2*(Sqrt[1 + x]/Sqrt[1 - x] + ArcSin[Sqrt[1 - x]/Sqrt[2]])

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fricas [B]  time = 0.47, size = 48, normalized size = 2.09 \[ \frac {2 \, {\left ({\left (x - 1\right )} \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) + x - \sqrt {x + 1} \sqrt {-x + 1} - 1\right )}}{x - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)^(1/2)/(1-x)^(3/2),x, algorithm="fricas")

[Out]

2*((x - 1)*arctan((sqrt(x + 1)*sqrt(-x + 1) - 1)/x) + x - sqrt(x + 1)*sqrt(-x + 1) - 1)/(x - 1)

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giac [A]  time = 1.06, size = 33, normalized size = 1.43 \[ -\frac {2 \, \sqrt {x + 1} \sqrt {-x + 1}}{x - 1} - 2 \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)^(1/2)/(1-x)^(3/2),x, algorithm="giac")

[Out]

-2*sqrt(x + 1)*sqrt(-x + 1)/(x - 1) - 2*arcsin(1/2*sqrt(2)*sqrt(x + 1))

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maple [B]  time = 0.03, size = 64, normalized size = 2.78 \[ -\frac {\sqrt {\left (x +1\right ) \left (-x +1\right )}\, \arcsin \relax (x )}{\sqrt {x +1}\, \sqrt {-x +1}}+\frac {2 \sqrt {x +1}\, \sqrt {\left (x +1\right ) \left (-x +1\right )}}{\sqrt {-\left (x +1\right ) \left (x -1\right )}\, \sqrt {-x +1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x+1)^(1/2)/(-x+1)^(3/2),x)

[Out]

2*(x+1)^(1/2)/(-(x+1)*(-1+x))^(1/2)*((x+1)*(-x+1))^(1/2)/(-x+1)^(1/2)-((x+1)*(-x+1))^(1/2)/(x+1)^(1/2)/(-x+1)^
(1/2)*arcsin(x)

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maxima [A]  time = 2.95, size = 21, normalized size = 0.91 \[ -\frac {2 \, \sqrt {-x^{2} + 1}}{x - 1} - \arcsin \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)^(1/2)/(1-x)^(3/2),x, algorithm="maxima")

[Out]

-2*sqrt(-x^2 + 1)/(x - 1) - arcsin(x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {\sqrt {x+1}}{{\left (1-x\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + 1)^(1/2)/(1 - x)^(3/2),x)

[Out]

int((x + 1)^(1/2)/(1 - x)^(3/2), x)

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sympy [A]  time = 1.62, size = 71, normalized size = 3.09 \[ \begin {cases} 2 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} - \frac {2 i \sqrt {x + 1}}{\sqrt {x - 1}} & \text {for}\: \frac {\left |{x + 1}\right |}{2} > 1 \\- 2 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} + \frac {2 \sqrt {x + 1}}{\sqrt {1 - x}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)**(1/2)/(1-x)**(3/2),x)

[Out]

Piecewise((2*I*acosh(sqrt(2)*sqrt(x + 1)/2) - 2*I*sqrt(x + 1)/sqrt(x - 1), Abs(x + 1)/2 > 1), (-2*asin(sqrt(2)
*sqrt(x + 1)/2) + 2*sqrt(x + 1)/sqrt(1 - x), True))

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