3.12.11 \(\int \frac {1}{(1-x)^{3/2} \sqrt {1+x}} \, dx\) [1111]

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

[Out]

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

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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 = 0.01, size = 17, normalized size = 1.00 \begin {gather*} \frac {\sqrt {1+x}}{\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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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in optimal.
time = 2.66, size = 48, normalized size = 2.82 \begin {gather*} \text {Piecewise}\left [\left \{\left \{-1+\frac {2}{1+x},\left \{\frac {1-x}{1+x}\text {!=}0\right \}\text {\&\&}\frac {1}{\text {Abs}\left [1+x\right ]}>\frac {1}{2}\right \}\right \},-\frac {I}{\sqrt {1-\frac {2}{1+x}}}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Piecewise[{{-1 + 2 / (1 + x), {(1 - x) / (1 + x) != 0} && 1 / Abs[1 + x] > 1 / 2}}, -I / Sqrt[1 - 2 / (1 + x)]
]

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Maple [A]
time = 0.13, size = 14, normalized size = 0.82

method	result	size
gosper	\(\frac {\sqrt {1+x}}{\sqrt {1-x}}\)	\(14\)
default	\(\frac {\sqrt {1+x}}{\sqrt {1-x}}\)	\(14\)
risch	\(\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \sqrt {1+x}}{\sqrt {1-x}\, \sqrt {-\left (1+x \right ) \left (-1+x \right )}}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.35, 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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Fricas [A]
time = 0.30, 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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Sympy [A]
time = 0.49, size = 31, normalized size = 1.82 \begin {gather*} \begin {cases} \frac {1}{\sqrt {-1 + \frac {2}{x + 1}}} & \text {for}\: \frac {1}{\left |{x + 1}\right |} > \frac {1}{2} \\- \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)), 1/Abs(x + 1) > 1/2), (-I/sqrt(1 - 2/(x + 1)), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 43 vs. \(2 (13) = 26\).
time = 0.00, size = 64, normalized size = 3.76 \begin {gather*} 2 \left (\frac {\sqrt {-x+1}}{2 \left (-2 \sqrt {x+1}+2 \sqrt {2}\right )}-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{8 \sqrt {-x+1}}\right ) \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]

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