3.9.31 \(\int \frac {\sqrt {1+x}}{(1-x)^{5/2}} \, dx\) [831]

Optimal. Leaf size=20 \[ \frac {(1+x)^{3/2}}{3 (1-x)^{3/2}} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 20, 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 {(x+1)^{3/2}}{3 (1-x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 + x]/(1 - x)^(5/2),x]

[Out]

(1 + x)^(3/2)/(3*(1 - x)^(3/2))

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 {\sqrt {1+x}}{(1-x)^{5/2}} \, dx &=\frac {(1+x)^{3/2}}{3 (1-x)^{3/2}}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 20, normalized size = 1.00 \begin {gather*} \frac {(1+x)^{3/2}}{3 (1-x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 + x]/(1 - x)^(5/2),x]

[Out]

(1 + x)^(3/2)/(3*(1 - x)^(3/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(29\) vs. \(2(14)=28\).
time = 0.08, size = 30, normalized size = 1.50

method	result	size
gosper	\(\frac {\left (1+x \right )^{\frac {3}{2}}}{3 \left (1-x \right )^{\frac {3}{2}}}\)	\(15\)
default	\(\frac {2 \sqrt {1+x}}{3 \left (1-x \right )^{\frac {3}{2}}}-\frac {\sqrt {1+x}}{3 \sqrt {1-x}}\)	\(30\)
risch	\(-\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \left (x^{2}+2 x +1\right )}{3 \sqrt {1-x}\, \sqrt {1+x}\, \left (-1+x \right ) \sqrt {-\left (1+x \right ) \left (-1+x \right )}}\)	\(49\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+x)^(1/2)/(1-x)^(5/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 38 vs. \(2 (14) = 28\).
time = 0.40, size = 38, normalized size = 1.90 \begin {gather*} \frac {2 \, \sqrt {-x^{2} + 1}}{3 \, {\left (x^{2} - 2 \, x + 1\right )}} + \frac {\sqrt {-x^{2} + 1}}{3 \, {\left (x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)^(1/2)/(1-x)^(5/2),x, algorithm="maxima")

[Out]

2/3*sqrt(-x^2 + 1)/(x^2 - 2*x + 1) + 1/3*sqrt(-x^2 + 1)/(x - 1)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 33 vs. \(2 (14) = 28\).
time = 1.02, size = 33, normalized size = 1.65 \begin {gather*} \frac {x^{2} + {\left (x + 1\right )}^{\frac {3}{2}} \sqrt {-x + 1} - 2 \, x + 1}{3 \, {\left (x^{2} - 2 \, x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)^(1/2)/(1-x)^(5/2),x, algorithm="fricas")

[Out]

1/3*(x^2 + (x + 1)^(3/2)*sqrt(-x + 1) - 2*x + 1)/(x^2 - 2*x + 1)

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Sympy [C] Result contains complex when optimal does not.
time = 0.88, size = 60, normalized size = 3.00 \begin {gather*} \begin {cases} \frac {i \left (x + 1\right )^{\frac {3}{2}}}{3 \sqrt {x - 1} \left (x + 1\right ) - 6 \sqrt {x - 1}} & \text {for}\: \left |{x + 1}\right | > 2 \\- \frac {\left (x + 1\right )^{\frac {3}{2}}}{3 \sqrt {1 - x} \left (x + 1\right ) - 6 \sqrt {1 - x}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)**(1/2)/(1-x)**(5/2),x)

[Out]

Piecewise((I*(x + 1)**(3/2)/(3*sqrt(x - 1)*(x + 1) - 6*sqrt(x - 1)), Abs(x + 1) > 2), (-(x + 1)**(3/2)/(3*sqrt
(1 - x)*(x + 1) - 6*sqrt(1 - x)), True))

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Giac [A]
time = 1.14, size = 19, normalized size = 0.95 \begin {gather*} \frac {{\left (x + 1\right )}^{\frac {3}{2}} \sqrt {-x + 1}}{3 \, {\left (x - 1\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)^(1/2)/(1-x)^(5/2),x, algorithm="giac")

[Out]

1/3*(x + 1)^(3/2)*sqrt(-x + 1)/(x - 1)^2

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Mupad [B]
time = 1.11, size = 34, normalized size = 1.70 \begin {gather*} \frac {\left (\frac {x\,\sqrt {x+1}}{3}+\frac {\sqrt {x+1}}{3}\right )\,\sqrt {1-x}}{x^2-2\,x+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + 1)^(1/2)/(1 - x)^(5/2),x)

[Out]

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

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