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

Optimal. Leaf size=17 \[ \frac {\sqrt {x+1}}{\sqrt {1-x}} \]

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Rubi [A]  time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {37} \begin {gather*} \frac {\sqrt {x+1}}{\sqrt {1-x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((1 - x)^(3/2)*Sqrt[1 + x]),x]

[Out]

Sqrt[1 + x]/Sqrt[1 - x]

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} \frac {\sqrt {x+1}}{\sqrt {1-x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[1 + x]/Sqrt[1 - x]

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IntegrateAlgebraic [A]  time = 0.02, size = 17, normalized size = 1.00 \begin {gather*} \frac {\sqrt {x+1}}{\sqrt {1-x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[1 + x]/Sqrt[1 - x]

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fricas [A]  time = 1.15, size = 23, normalized size = 1.35 \begin {gather*} \frac {x - \sqrt {x + 1} \sqrt {-x + 1} - 1}{x - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.68, size = 19, normalized size = 1.12 \begin {gather*} -\frac {\sqrt {x + 1} \sqrt {-x + 1}}{x - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(x + 1)*sqrt(-x + 1)/(x - 1)

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maple [A]  time = 0.00, size = 14, normalized size = 0.82 \begin {gather*} \frac {\sqrt {x +1}}{\sqrt {-x +1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 2.98, size = 16, normalized size = 0.94 \begin {gather*} -\frac {\sqrt {-x^{2} + 1}}{x - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-x^2 + 1)/(x - 1)

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mupad [B]  time = 0.28, size = 13, normalized size = 0.76 \begin {gather*} \frac {\sqrt {x+1}}{\sqrt {1-x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x + 1)^(1/2)/(1 - x)^(1/2)

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sympy [A]  time = 0.94, size = 29, normalized size = 1.71 \begin {gather*} \begin {cases} \frac {1}{\sqrt {-1 + \frac {2}{x + 1}}} & \text {for}\: \frac {2}{\left |{x + 1}\right |} > 1 \\- \frac {i}{\sqrt {1 - \frac {2}{x + 1}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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