3.1132 \(\int \frac{1}{(1-x)^{3/2} (1+x)^{5/2}} \, dx\)

Optimal. Leaf size=58 \[ -\frac{2 \sqrt{1-x}}{3 \sqrt{x+1}}-\frac{2 \sqrt{1-x}}{3 (x+1)^{3/2}}+\frac{1}{(x+1)^{3/2} \sqrt{1-x}} \]

[Out]

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

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Rubi [A]  time = 0.0084024, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {45, 37} \[ -\frac{2 \sqrt{1-x}}{3 \sqrt{x+1}}-\frac{2 \sqrt{1-x}}{3 (x+1)^{3/2}}+\frac{1}{(x+1)^{3/2} \sqrt{1-x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0058078, size = 30, normalized size = 0.52 \[ \frac{2 x^2+2 x-1}{3 \sqrt{1-x} (x+1)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.004, size = 25, normalized size = 0.4 \begin{align*}{\frac{2\,{x}^{2}+2\,x-1}{3}{\frac{1}{\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)^(3/2)/(1+x)^(5/2),x)

[Out]

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

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Maxima [A]  time = 0.991325, size = 51, normalized size = 0.88 \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)^(3/2)/(1+x)^(5/2),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 = 1.90355, size = 123, normalized size = 2.12 \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)^(3/2)/(1+x)^(5/2),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 [A]  time = 17.4688, size = 165, normalized size = 2.84 \begin{align*} \begin{cases} - \frac{2 \sqrt{-1 + \frac{2}{x + 1}} \left (x + 1\right )^{2}}{- 6 x + 3 \left (x + 1\right )^{2} - 6} + \frac{2 \sqrt{-1 + \frac{2}{x + 1}} \left (x + 1\right )}{- 6 x + 3 \left (x + 1\right )^{2} - 6} + \frac{\sqrt{-1 + \frac{2}{x + 1}}}{- 6 x + 3 \left (x + 1\right )^{2} - 6} & \text{for}\: \frac{2}{\left |{x + 1}\right |} > 1 \\- \frac{2 i \sqrt{1 - \frac{2}{x + 1}} \left (x + 1\right )^{2}}{- 6 x + 3 \left (x + 1\right )^{2} - 6} + \frac{2 i \sqrt{1 - \frac{2}{x + 1}} \left (x + 1\right )}{- 6 x + 3 \left (x + 1\right )^{2} - 6} + \frac{i \sqrt{1 - \frac{2}{x + 1}}}{- 6 x + 3 \left (x + 1\right )^{2} - 6} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.08483, size = 146, normalized size = 2.52 \begin{align*} \frac{{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{3}}{96 \,{\left (x + 1\right )}^{\frac{3}{2}}} + \frac{7 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}}{32 \, \sqrt{x + 1}} - \frac{\sqrt{x + 1} \sqrt{-x + 1}}{4 \,{\left (x - 1\right )}} - \frac{{\left (x + 1\right )}^{\frac{3}{2}}{\left (\frac{21 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{2}}{x + 1} + 1\right )}}{96 \,{\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)^(3/2)/(1+x)^(5/2),x, algorithm="giac")

[Out]

1/96*(sqrt(2) - sqrt(-x + 1))^3/(x + 1)^(3/2) + 7/32*(sqrt(2) - sqrt(-x + 1))/sqrt(x + 1) - 1/4*sqrt(x + 1)*sq
rt(-x + 1)/(x - 1) - 1/96*(x + 1)^(3/2)*(21*(sqrt(2) - sqrt(-x + 1))^2/(x + 1) + 1)/(sqrt(2) - sqrt(-x + 1))^3