3.1131 \(\int \frac{1}{\sqrt{1-x} (1+x)^{5/2}} \, dx\)

Optimal. Leaf size=41 \[ -\frac{\sqrt{1-x}}{3 \sqrt{x+1}}-\frac{\sqrt{1-x}}{3 (x+1)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0043043, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {45, 37} \[ -\frac{\sqrt{1-x}}{3 \sqrt{x+1}}-\frac{\sqrt{1-x}}{3 (x+1)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[1 - x]*(1 + x)^(5/2)),x]

[Out]

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

Rule 45

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] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

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

Mathematica [A]  time = 0.0111907, size = 23, normalized size = 0.56 \[ -\frac{\sqrt{1-x} (x+2)}{3 (x+1)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.001, size = 18, normalized size = 0.4 \begin{align*} -{\frac{2+x}{3}\sqrt{1-x} \left ( 1+x \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.49294, size = 51, normalized size = 1.24 \begin{align*} -\frac{\sqrt{-x^{2} + 1}}{3 \,{\left (x^{2} + 2 \, x + 1\right )}} - \frac{\sqrt{-x^{2} + 1}}{3 \,{\left (x + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.92189, size = 101, normalized size = 2.46 \begin{align*} -\frac{2 \, x^{2} +{\left (x + 2\right )} \sqrt{x + 1} \sqrt{-x + 1} + 4 \, x + 2}{3 \,{\left (x^{2} + 2 \, x + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(2*x^2 + (x + 2)*sqrt(x + 1)*sqrt(-x + 1) + 4*x + 2)/(x^2 + 2*x + 1)

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Sympy [A]  time = 4.23274, size = 65, normalized size = 1.59 \begin{align*} \begin{cases} - \frac{\sqrt{-1 + \frac{2}{x + 1}}}{3} - \frac{\sqrt{-1 + \frac{2}{x + 1}}}{3 \left (x + 1\right )} & \text{for}\: \frac{2}{\left |{x + 1}\right |} > 1 \\- \frac{i \sqrt{1 - \frac{2}{x + 1}}}{3} - \frac{i \sqrt{1 - \frac{2}{x + 1}}}{3 \left (x + 1\right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.07142, size = 120, normalized size = 2.93 \begin{align*} \frac{{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{3}}{48 \,{\left (x + 1\right )}^{\frac{3}{2}}} + \frac{3 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}}{16 \, \sqrt{x + 1}} - \frac{{\left (x + 1\right )}^{\frac{3}{2}}{\left (\frac{9 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{2}}{x + 1} + 1\right )}}{48 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/48*(sqrt(2) - sqrt(-x + 1))^3/(x + 1)^(3/2) + 3/16*(sqrt(2) - sqrt(-x + 1))/sqrt(x + 1) - 1/48*(x + 1)^(3/2)
*(9*(sqrt(2) - sqrt(-x + 1))^2/(x + 1) + 1)/(sqrt(2) - sqrt(-x + 1))^3