3.1112 \(\int \frac{1}{(1-x)^{5/2} \sqrt{1+x}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0043011, 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} \[ \frac{\sqrt{x+1}}{3 \sqrt{1-x}}+\frac{\sqrt{x+1}}{3 (1-x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

Mathematica [A]  time = 0.0063659, size = 23, normalized size = 0.56 \[ -\frac{(x-2) \sqrt{x+1}}{3 (1-x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.003, 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)^(5/2)/(1+x)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.49189, 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)^(5/2)/(1+x)^(1/2),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 = 1.72421, size = 100, normalized size = 2.44 \begin{align*} \frac{2 \, x^{2} - \sqrt{x + 1}{\left (x - 2\right )} \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)^(5/2)/(1+x)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 4.17845, size = 126, normalized size = 3.07 \begin{align*} \begin{cases} \frac{x + 1}{3 \sqrt{-1 + \frac{2}{x + 1}} \left (x + 1\right ) - 6 \sqrt{-1 + \frac{2}{x + 1}}} - \frac{3}{3 \sqrt{-1 + \frac{2}{x + 1}} \left (x + 1\right ) - 6 \sqrt{-1 + \frac{2}{x + 1}}} & \text{for}\: \frac{2}{\left |{x + 1}\right |} > 1 \\- \frac{i \left (x + 1\right )}{3 \sqrt{1 - \frac{2}{x + 1}} \left (x + 1\right ) - 6 \sqrt{1 - \frac{2}{x + 1}}} + \frac{3 i}{3 \sqrt{1 - \frac{2}{x + 1}} \left (x + 1\right ) - 6 \sqrt{1 - \frac{2}{x + 1}}} & \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)**(1/2),x)

[Out]

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

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Giac [A]  time = 1.06955, size = 30, normalized size = 0.73 \begin{align*} -\frac{\sqrt{x + 1}{\left (x - 2\right )} \sqrt{-x + 1}}{3 \,{\left (x - 1\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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