3.1071 \(\int \frac{\sqrt{1+x}}{(1-x)^{7/2}} \, dx\)

Optimal. Leaf size=41 \[ \frac{(x+1)^{3/2}}{15 (1-x)^{3/2}}+\frac{(x+1)^{3/2}}{5 (1-x)^{5/2}} \]

[Out]

(1 + x)^(3/2)/(5*(1 - x)^(5/2)) + (1 + x)^(3/2)/(15*(1 - x)^(3/2))

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Rubi [A]  time = 0.0042463, 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} \[ \frac{(x+1)^{3/2}}{15 (1-x)^{3/2}}+\frac{(x+1)^{3/2}}{5 (1-x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 + x]/(1 - x)^(7/2),x]

[Out]

(1 + x)^(3/2)/(5*(1 - x)^(5/2)) + (1 + x)^(3/2)/(15*(1 - x)^(3/2))

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{\sqrt{1+x}}{(1-x)^{7/2}} \, dx &=\frac{(1+x)^{3/2}}{5 (1-x)^{5/2}}+\frac{1}{5} \int \frac{\sqrt{1+x}}{(1-x)^{5/2}} \, dx\\ &=\frac{(1+x)^{3/2}}{5 (1-x)^{5/2}}+\frac{(1+x)^{3/2}}{15 (1-x)^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0081319, size = 23, normalized size = 0.56 \[ -\frac{(x-4) (x+1)^{3/2}}{15 (1-x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 + x]/(1 - x)^(7/2),x]

[Out]

-((-4 + x)*(1 + x)^(3/2))/(15*(1 - x)^(5/2))

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Maple [A]  time = 0.002, size = 18, normalized size = 0.4 \begin{align*} -{\frac{x-4}{15} \left ( 1+x \right ) ^{{\frac{3}{2}}} \left ( 1-x \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+x)^(1/2)/(1-x)^(7/2),x)

[Out]

-1/15*(1+x)^(3/2)*(x-4)/(1-x)^(5/2)

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Maxima [B]  time = 1.03062, size = 86, normalized size = 2.1 \begin{align*} -\frac{2 \, \sqrt{-x^{2} + 1}}{5 \,{\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} - \frac{\sqrt{-x^{2} + 1}}{15 \,{\left (x^{2} - 2 \, x + 1\right )}} + \frac{\sqrt{-x^{2} + 1}}{15 \,{\left (x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)^(1/2)/(1-x)^(7/2),x, algorithm="maxima")

[Out]

-2/5*sqrt(-x^2 + 1)/(x^3 - 3*x^2 + 3*x - 1) - 1/15*sqrt(-x^2 + 1)/(x^2 - 2*x + 1) + 1/15*sqrt(-x^2 + 1)/(x - 1
)

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Fricas [A]  time = 1.58183, size = 136, normalized size = 3.32 \begin{align*} \frac{4 \, x^{3} - 12 \, x^{2} +{\left (x^{2} - 3 \, x - 4\right )} \sqrt{x + 1} \sqrt{-x + 1} + 12 \, x - 4}{15 \,{\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)^(1/2)/(1-x)^(7/2),x, algorithm="fricas")

[Out]

1/15*(4*x^3 - 12*x^2 + (x^2 - 3*x - 4)*sqrt(x + 1)*sqrt(-x + 1) + 12*x - 4)/(x^3 - 3*x^2 + 3*x - 1)

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Sympy [B]  time = 23.1053, size = 173, normalized size = 4.22 \begin{align*} \begin{cases} \frac{i \left (x + 1\right )^{\frac{5}{2}}}{15 \sqrt{x - 1} \left (x + 1\right )^{2} - 60 \sqrt{x - 1} \left (x + 1\right ) + 60 \sqrt{x - 1}} - \frac{5 i \left (x + 1\right )^{\frac{3}{2}}}{15 \sqrt{x - 1} \left (x + 1\right )^{2} - 60 \sqrt{x - 1} \left (x + 1\right ) + 60 \sqrt{x - 1}} & \text{for}\: \frac{\left |{x + 1}\right |}{2} > 1 \\- \frac{\left (x + 1\right )^{\frac{5}{2}}}{15 \sqrt{1 - x} \left (x + 1\right )^{2} - 60 \sqrt{1 - x} \left (x + 1\right ) + 60 \sqrt{1 - x}} + \frac{5 \left (x + 1\right )^{\frac{3}{2}}}{15 \sqrt{1 - x} \left (x + 1\right )^{2} - 60 \sqrt{1 - x} \left (x + 1\right ) + 60 \sqrt{1 - x}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)**(1/2)/(1-x)**(7/2),x)

[Out]

Piecewise((I*(x + 1)**(5/2)/(15*sqrt(x - 1)*(x + 1)**2 - 60*sqrt(x - 1)*(x + 1) + 60*sqrt(x - 1)) - 5*I*(x + 1
)**(3/2)/(15*sqrt(x - 1)*(x + 1)**2 - 60*sqrt(x - 1)*(x + 1) + 60*sqrt(x - 1)), Abs(x + 1)/2 > 1), (-(x + 1)**
(5/2)/(15*sqrt(1 - x)*(x + 1)**2 - 60*sqrt(1 - x)*(x + 1) + 60*sqrt(1 - x)) + 5*(x + 1)**(3/2)/(15*sqrt(1 - x)
*(x + 1)**2 - 60*sqrt(1 - x)*(x + 1) + 60*sqrt(1 - x)), True))

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Giac [A]  time = 1.09774, size = 30, normalized size = 0.73 \begin{align*} \frac{{\left (x + 1\right )}^{\frac{3}{2}}{\left (x - 4\right )} \sqrt{-x + 1}}{15 \,{\left (x - 1\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)^(1/2)/(1-x)^(7/2),x, algorithm="giac")

[Out]

1/15*(x + 1)^(3/2)*(x - 4)*sqrt(-x + 1)/(x - 1)^3