3.1123 \(\int \frac{1}{(1-x)^{7/2} (1+x)^{3/2}} \, dx\)

Optimal. Leaf size=62 \[ \frac{2 x}{5 \sqrt{1-x} \sqrt{x+1}}+\frac{1}{5 (1-x)^{3/2} \sqrt{x+1}}+\frac{1}{5 (1-x)^{5/2} \sqrt{x+1}} \]

[Out]

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

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Rubi [A]  time = 0.008368, antiderivative size = 62, 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, 39} \[ \frac{2 x}{5 \sqrt{1-x} \sqrt{x+1}}+\frac{1}{5 (1-x)^{3/2} \sqrt{x+1}}+\frac{1}{5 (1-x)^{5/2} \sqrt{x+1}} \]

Antiderivative was successfully verified.

[In]

Int[1/((1 - x)^(7/2)*(1 + x)^(3/2)),x]

[Out]

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

Mathematica [A]  time = 0.0084147, size = 33, normalized size = 0.53 \[ \frac{2 x^3-4 x^2+x+2}{5 (x-1)^2 \sqrt{1-x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((1 - x)^(7/2)*(1 + x)^(3/2)),x]

[Out]

(2 + x - 4*x^2 + 2*x^3)/(5*(-1 + x)^2*Sqrt[1 - x^2])

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Maple [A]  time = 0.003, size = 28, normalized size = 0.5 \begin{align*}{\frac{2\,{x}^{3}-4\,{x}^{2}+x+2}{5} \left ( 1-x \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt{1+x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1-x)^(7/2)/(1+x)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.06284, size = 107, normalized size = 1.73 \begin{align*} \frac{2 \, x}{5 \, \sqrt{-x^{2} + 1}} + \frac{1}{5 \,{\left (\sqrt{-x^{2} + 1} x^{2} - 2 \, \sqrt{-x^{2} + 1} x + \sqrt{-x^{2} + 1}\right )}} - \frac{1}{5 \,{\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)^(7/2)/(1+x)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.94946, size = 143, normalized size = 2.31 \begin{align*} \frac{2 \, x^{4} - 4 \, x^{3} -{\left (2 \, x^{3} - 4 \, x^{2} + x + 2\right )} \sqrt{x + 1} \sqrt{-x + 1} + 4 \, x - 2}{5 \,{\left (x^{4} - 2 \, x^{3} + 2 \, x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)^(7/2)/(1+x)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 70.1819, size = 282, normalized size = 4.55 \begin{align*} \begin{cases} - \frac{2 \sqrt{-1 + \frac{2}{x + 1}} \left (x + 1\right )^{3}}{60 x + 5 \left (x + 1\right )^{3} - 30 \left (x + 1\right )^{2} + 20} + \frac{10 \sqrt{-1 + \frac{2}{x + 1}} \left (x + 1\right )^{2}}{60 x + 5 \left (x + 1\right )^{3} - 30 \left (x + 1\right )^{2} + 20} - \frac{15 \sqrt{-1 + \frac{2}{x + 1}} \left (x + 1\right )}{60 x + 5 \left (x + 1\right )^{3} - 30 \left (x + 1\right )^{2} + 20} + \frac{5 \sqrt{-1 + \frac{2}{x + 1}}}{60 x + 5 \left (x + 1\right )^{3} - 30 \left (x + 1\right )^{2} + 20} & \text{for}\: \frac{2}{\left |{x + 1}\right |} > 1 \\- \frac{2 i \sqrt{1 - \frac{2}{x + 1}} \left (x + 1\right )^{3}}{60 x + 5 \left (x + 1\right )^{3} - 30 \left (x + 1\right )^{2} + 20} + \frac{10 i \sqrt{1 - \frac{2}{x + 1}} \left (x + 1\right )^{2}}{60 x + 5 \left (x + 1\right )^{3} - 30 \left (x + 1\right )^{2} + 20} - \frac{15 i \sqrt{1 - \frac{2}{x + 1}} \left (x + 1\right )}{60 x + 5 \left (x + 1\right )^{3} - 30 \left (x + 1\right )^{2} + 20} + \frac{5 i \sqrt{1 - \frac{2}{x + 1}}}{60 x + 5 \left (x + 1\right )^{3} - 30 \left (x + 1\right )^{2} + 20} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)**(7/2)/(1+x)**(3/2),x)

[Out]

Piecewise((-2*sqrt(-1 + 2/(x + 1))*(x + 1)**3/(60*x + 5*(x + 1)**3 - 30*(x + 1)**2 + 20) + 10*sqrt(-1 + 2/(x +
1))*(x + 1)**2/(60*x + 5*(x + 1)**3 - 30*(x + 1)**2 + 20) - 15*sqrt(-1 + 2/(x + 1))*(x + 1)/(60*x + 5*(x + 1)
**3 - 30*(x + 1)**2 + 20) + 5*sqrt(-1 + 2/(x + 1))/(60*x + 5*(x + 1)**3 - 30*(x + 1)**2 + 20), 2/Abs(x + 1) >
1), (-2*I*sqrt(1 - 2/(x + 1))*(x + 1)**3/(60*x + 5*(x + 1)**3 - 30*(x + 1)**2 + 20) + 10*I*sqrt(1 - 2/(x + 1))
*(x + 1)**2/(60*x + 5*(x + 1)**3 - 30*(x + 1)**2 + 20) - 15*I*sqrt(1 - 2/(x + 1))*(x + 1)/(60*x + 5*(x + 1)**3
- 30*(x + 1)**2 + 20) + 5*I*sqrt(1 - 2/(x + 1))/(60*x + 5*(x + 1)**3 - 30*(x + 1)**2 + 20), True))

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Giac [A]  time = 1.0699, size = 99, normalized size = 1.6 \begin{align*} \frac{\sqrt{2} - \sqrt{-x + 1}}{16 \, \sqrt{x + 1}} - \frac{\sqrt{x + 1}}{16 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}} - \frac{{\left ({\left (11 \, x - 39\right )}{\left (x + 1\right )} + 60\right )} \sqrt{x + 1} \sqrt{-x + 1}}{40 \,{\left (x - 1\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)^(7/2)/(1+x)^(3/2),x, algorithm="giac")

[Out]

1/16*(sqrt(2) - sqrt(-x + 1))/sqrt(x + 1) - 1/16*sqrt(x + 1)/(sqrt(2) - sqrt(-x + 1)) - 1/40*((11*x - 39)*(x +
1) + 60)*sqrt(x + 1)*sqrt(-x + 1)/(x - 1)^3