3.831 \(\int \frac {\sqrt {1+x}}{(1-x)^{5/2}} \, dx\)

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

[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} \[ \frac {(x+1)^{3/2}}{3 (1-x)^{3/2}} \]

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 \[ \frac {(x+1)^{3/2}}{3 (1-x)^{3/2}} \]

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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fricas [B]  time = 1.15, size = 33, normalized size = 1.65 \[ \frac {x^{2} + {\left (x + 1\right )}^{\frac {3}{2}} \sqrt {-x + 1} - 2 \, x + 1}{3 \, {\left (x^{2} - 2 \, x + 1\right )}} \]

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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giac [A]  time = 1.24, size = 19, normalized size = 0.95 \[ \frac {{\left (x + 1\right )}^{\frac {3}{2}} \sqrt {-x + 1}}{3 \, {\left (x - 1\right )}^{2}} \]

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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maple [A]  time = 0.00, size = 15, normalized size = 0.75 \[ \frac {\left (x +1\right )^{\frac {3}{2}}}{3 \left (-x +1\right )^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.51, size = 38, normalized size = 1.90 \[ \frac {2 \, \sqrt {-x^{2} + 1}}{3 \, {\left (x^{2} - 2 \, x + 1\right )}} + \frac {\sqrt {-x^{2} + 1}}{3 \, {\left (x - 1\right )}} \]

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

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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sympy [A]  time = 1.53, size = 61, normalized size = 3.05 \[ \begin {cases} \frac {i \left (x + 1\right )^{\frac {3}{2}}}{3 \sqrt {x - 1} \left (x + 1\right ) - 6 \sqrt {x - 1}} & \text {for}\: \frac {\left |{x + 1}\right |}{2} > 1 \\- \frac {\left (x + 1\right )^{\frac {3}{2}}}{3 \sqrt {1 - x} \left (x + 1\right ) - 6 \sqrt {1 - x}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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