3.1122 \(\int \frac{1}{(1-x)^{5/2} (1+x)^{3/2}} \, dx\)

Optimal. Leaf size=42 \[ \frac{2 x}{3 \sqrt{1-x} \sqrt{x+1}}+\frac{1}{3 (1-x)^{3/2} \sqrt{x+1}} \]

[Out]

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

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Rubi [A]  time = 0.0045252, antiderivative size = 42, 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, 39} \[ \frac{2 x}{3 \sqrt{1-x} \sqrt{x+1}}+\frac{1}{3 (1-x)^{3/2} \sqrt{x+1}} \]

Antiderivative was successfully verified.

[In]

Int[1/((1 - x)^(5/2)*(1 + x)^(3/2)),x]

[Out]

1/(3*(1 - x)^(3/2)*Sqrt[1 + x]) + (2*x)/(3*Sqrt[1 - x]*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 39

Int[1/(((a_) + (b_.)*(x_))^(3/2)*((c_) + (d_.)*(x_))^(3/2)), x_Symbol] :> Simp[x/(a*c*Sqrt[a + b*x]*Sqrt[c + d
*x]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0]

Rubi steps

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

Mathematica [A]  time = 0.007004, size = 30, normalized size = 0.71 \[ \frac{2 x^2-2 x-1}{3 (x-1) \sqrt{1-x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((1 - x)^(5/2)*(1 + x)^(3/2)),x]

[Out]

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

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Maple [A]  time = 0.002, size = 25, normalized size = 0.6 \begin{align*} -{\frac{2\,{x}^{2}-2\,x-1}{3} \left ( 1-x \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{1+x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1-x)^(5/2)/(1+x)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.04189, size = 54, normalized size = 1.29 \begin{align*} \frac{2 \, x}{3 \, \sqrt{-x^{2} + 1}} - \frac{1}{3 \,{\left (\sqrt{-x^{2} + 1} x - \sqrt{-x^{2} + 1}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)^(5/2)/(1+x)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.74559, size = 122, normalized size = 2.9 \begin{align*} \frac{x^{3} - x^{2} -{\left (2 \, x^{2} - 2 \, x - 1\right )} \sqrt{x + 1} \sqrt{-x + 1} - x + 1}{3 \,{\left (x^{3} - x^{2} - x + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)^(5/2)/(1+x)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 17.5289, size = 158, normalized size = 3.76 \begin{align*} \begin{cases} - \frac{2 \sqrt{-1 + \frac{2}{x + 1}} \left (x + 1\right )^{2}}{- 12 x + 3 \left (x + 1\right )^{2}} + \frac{6 \sqrt{-1 + \frac{2}{x + 1}} \left (x + 1\right )}{- 12 x + 3 \left (x + 1\right )^{2}} - \frac{3 \sqrt{-1 + \frac{2}{x + 1}}}{- 12 x + 3 \left (x + 1\right )^{2}} & \text{for}\: \frac{2}{\left |{x + 1}\right |} > 1 \\- \frac{2 i \sqrt{1 - \frac{2}{x + 1}} \left (x + 1\right )^{2}}{- 12 x + 3 \left (x + 1\right )^{2}} + \frac{6 i \sqrt{1 - \frac{2}{x + 1}} \left (x + 1\right )}{- 12 x + 3 \left (x + 1\right )^{2}} - \frac{3 i \sqrt{1 - \frac{2}{x + 1}}}{- 12 x + 3 \left (x + 1\right )^{2}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)**(5/2)/(1+x)**(3/2),x)

[Out]

Piecewise((-2*sqrt(-1 + 2/(x + 1))*(x + 1)**2/(-12*x + 3*(x + 1)**2) + 6*sqrt(-1 + 2/(x + 1))*(x + 1)/(-12*x +
3*(x + 1)**2) - 3*sqrt(-1 + 2/(x + 1))/(-12*x + 3*(x + 1)**2), 2/Abs(x + 1) > 1), (-2*I*sqrt(1 - 2/(x + 1))*(
x + 1)**2/(-12*x + 3*(x + 1)**2) + 6*I*sqrt(1 - 2/(x + 1))*(x + 1)/(-12*x + 3*(x + 1)**2) - 3*I*sqrt(1 - 2/(x
+ 1))/(-12*x + 3*(x + 1)**2), True))

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Giac [B]  time = 1.07, size = 90, normalized size = 2.14 \begin{align*} \frac{\sqrt{2} - \sqrt{-x + 1}}{8 \, \sqrt{x + 1}} - \frac{{\left (5 \, x - 7\right )} \sqrt{x + 1} \sqrt{-x + 1}}{12 \,{\left (x - 1\right )}^{2}} - \frac{\sqrt{x + 1}}{8 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)^(5/2)/(1+x)^(3/2),x, algorithm="giac")

[Out]

1/8*(sqrt(2) - sqrt(-x + 1))/sqrt(x + 1) - 1/12*(5*x - 7)*sqrt(x + 1)*sqrt(-x + 1)/(x - 1)^2 - 1/8*sqrt(x + 1)
/(sqrt(2) - sqrt(-x + 1))