3.1134 \(\int \frac{1}{(1-x)^{7/2} (1+x)^{5/2}} \, dx\)

Optimal. Leaf size=63 \[ \frac{8 x}{15 \sqrt{1-x} \sqrt{x+1}}+\frac{4 x}{15 (1-x)^{3/2} (x+1)^{3/2}}+\frac{1}{5 (1-x)^{5/2} (x+1)^{3/2}} \]

[Out]

1/(5*(1 - x)^(5/2)*(1 + x)^(3/2)) + (4*x)/(15*(1 - x)^(3/2)*(1 + x)^(3/2)) + (8*x)/(15*Sqrt[1 - x]*Sqrt[1 + x]
)

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Rubi [A]  time = 0.0081462, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {45, 40, 39} \[ \frac{8 x}{15 \sqrt{1-x} \sqrt{x+1}}+\frac{4 x}{15 (1-x)^{3/2} (x+1)^{3/2}}+\frac{1}{5 (1-x)^{5/2} (x+1)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/((1 - x)^(7/2)*(1 + x)^(5/2)),x]

[Out]

1/(5*(1 - x)^(5/2)*(1 + x)^(3/2)) + (4*x)/(15*(1 - x)^(3/2)*(1 + x)^(3/2)) + (8*x)/(15*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 40

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> -Simp[(x*(a + b*x)^(m + 1)*(c + d*x)^(m +
1))/(2*a*c*(m + 1)), x] + Dist[(2*m + 3)/(2*a*c*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(m + 1), x], x] /; F
reeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0] && ILtQ[m + 3/2, 0]

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)^{5/2}} \, dx &=\frac{1}{5 (1-x)^{5/2} (1+x)^{3/2}}+\frac{4}{5} \int \frac{1}{(1-x)^{5/2} (1+x)^{5/2}} \, dx\\ &=\frac{1}{5 (1-x)^{5/2} (1+x)^{3/2}}+\frac{4 x}{15 (1-x)^{3/2} (1+x)^{3/2}}+\frac{8}{15} \int \frac{1}{(1-x)^{3/2} (1+x)^{3/2}} \, dx\\ &=\frac{1}{5 (1-x)^{5/2} (1+x)^{3/2}}+\frac{4 x}{15 (1-x)^{3/2} (1+x)^{3/2}}+\frac{8 x}{15 \sqrt{1-x} \sqrt{1+x}}\\ \end{align*}

Mathematica [A]  time = 0.0111156, size = 40, normalized size = 0.63 \[ \frac{8 x^4-8 x^3-12 x^2+12 x+3}{15 (1-x)^{5/2} (x+1)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((1 - x)^(7/2)*(1 + x)^(5/2)),x]

[Out]

(3 + 12*x - 12*x^2 - 8*x^3 + 8*x^4)/(15*(1 - x)^(5/2)*(1 + x)^(3/2))

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Maple [A]  time = 0.003, size = 35, normalized size = 0.6 \begin{align*}{\frac{8\,{x}^{4}-8\,{x}^{3}-12\,{x}^{2}+12\,x+3}{15} \left ( 1-x \right ) ^{-{\frac{5}{2}}} \left ( 1+x \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1-x)^(7/2)/(1+x)^(5/2),x)

[Out]

1/15*(8*x^4-8*x^3-12*x^2+12*x+3)/(1+x)^(3/2)/(1-x)^(5/2)

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Maxima [A]  time = 0.999271, size = 70, normalized size = 1.11 \begin{align*} \frac{8 \, x}{15 \, \sqrt{-x^{2} + 1}} + \frac{4 \, x}{15 \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}}} - \frac{1}{5 \,{\left ({\left (-x^{2} + 1\right )}^{\frac{3}{2}} x -{\left (-x^{2} + 1\right )}^{\frac{3}{2}}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)^(7/2)/(1+x)^(5/2),x, algorithm="maxima")

[Out]

8/15*x/sqrt(-x^2 + 1) + 4/15*x/(-x^2 + 1)^(3/2) - 1/5/((-x^2 + 1)^(3/2)*x - (-x^2 + 1)^(3/2))

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Fricas [A]  time = 1.50386, size = 198, normalized size = 3.14 \begin{align*} \frac{3 \, x^{5} - 3 \, x^{4} - 6 \, x^{3} + 6 \, x^{2} -{\left (8 \, x^{4} - 8 \, x^{3} - 12 \, x^{2} + 12 \, x + 3\right )} \sqrt{x + 1} \sqrt{-x + 1} + 3 \, x - 3}{15 \,{\left (x^{5} - x^{4} - 2 \, x^{3} + 2 \, x^{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)^(5/2),x, algorithm="fricas")

[Out]

1/15*(3*x^5 - 3*x^4 - 6*x^3 + 6*x^2 - (8*x^4 - 8*x^3 - 12*x^2 + 12*x + 3)*sqrt(x + 1)*sqrt(-x + 1) + 3*x - 3)/
(x^5 - x^4 - 2*x^3 + 2*x^2 + x - 1)

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Sympy [B]  time = 149.631, size = 423, normalized size = 6.71 \begin{align*} \begin{cases} - \frac{8 \sqrt{-1 + \frac{2}{x + 1}} \left (x + 1\right )^{4}}{- 120 x + 15 \left (x + 1\right )^{4} - 90 \left (x + 1\right )^{3} + 180 \left (x + 1\right )^{2} - 120} + \frac{40 \sqrt{-1 + \frac{2}{x + 1}} \left (x + 1\right )^{3}}{- 120 x + 15 \left (x + 1\right )^{4} - 90 \left (x + 1\right )^{3} + 180 \left (x + 1\right )^{2} - 120} - \frac{60 \sqrt{-1 + \frac{2}{x + 1}} \left (x + 1\right )^{2}}{- 120 x + 15 \left (x + 1\right )^{4} - 90 \left (x + 1\right )^{3} + 180 \left (x + 1\right )^{2} - 120} + \frac{20 \sqrt{-1 + \frac{2}{x + 1}} \left (x + 1\right )}{- 120 x + 15 \left (x + 1\right )^{4} - 90 \left (x + 1\right )^{3} + 180 \left (x + 1\right )^{2} - 120} + \frac{5 \sqrt{-1 + \frac{2}{x + 1}}}{- 120 x + 15 \left (x + 1\right )^{4} - 90 \left (x + 1\right )^{3} + 180 \left (x + 1\right )^{2} - 120} & \text{for}\: \frac{2}{\left |{x + 1}\right |} > 1 \\- \frac{8 i \sqrt{1 - \frac{2}{x + 1}} \left (x + 1\right )^{4}}{- 120 x + 15 \left (x + 1\right )^{4} - 90 \left (x + 1\right )^{3} + 180 \left (x + 1\right )^{2} - 120} + \frac{40 i \sqrt{1 - \frac{2}{x + 1}} \left (x + 1\right )^{3}}{- 120 x + 15 \left (x + 1\right )^{4} - 90 \left (x + 1\right )^{3} + 180 \left (x + 1\right )^{2} - 120} - \frac{60 i \sqrt{1 - \frac{2}{x + 1}} \left (x + 1\right )^{2}}{- 120 x + 15 \left (x + 1\right )^{4} - 90 \left (x + 1\right )^{3} + 180 \left (x + 1\right )^{2} - 120} + \frac{20 i \sqrt{1 - \frac{2}{x + 1}} \left (x + 1\right )}{- 120 x + 15 \left (x + 1\right )^{4} - 90 \left (x + 1\right )^{3} + 180 \left (x + 1\right )^{2} - 120} + \frac{5 i \sqrt{1 - \frac{2}{x + 1}}}{- 120 x + 15 \left (x + 1\right )^{4} - 90 \left (x + 1\right )^{3} + 180 \left (x + 1\right )^{2} - 120} & \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)**(5/2),x)

[Out]

Piecewise((-8*sqrt(-1 + 2/(x + 1))*(x + 1)**4/(-120*x + 15*(x + 1)**4 - 90*(x + 1)**3 + 180*(x + 1)**2 - 120)
+ 40*sqrt(-1 + 2/(x + 1))*(x + 1)**3/(-120*x + 15*(x + 1)**4 - 90*(x + 1)**3 + 180*(x + 1)**2 - 120) - 60*sqrt
(-1 + 2/(x + 1))*(x + 1)**2/(-120*x + 15*(x + 1)**4 - 90*(x + 1)**3 + 180*(x + 1)**2 - 120) + 20*sqrt(-1 + 2/(
x + 1))*(x + 1)/(-120*x + 15*(x + 1)**4 - 90*(x + 1)**3 + 180*(x + 1)**2 - 120) + 5*sqrt(-1 + 2/(x + 1))/(-120
*x + 15*(x + 1)**4 - 90*(x + 1)**3 + 180*(x + 1)**2 - 120), 2/Abs(x + 1) > 1), (-8*I*sqrt(1 - 2/(x + 1))*(x +
1)**4/(-120*x + 15*(x + 1)**4 - 90*(x + 1)**3 + 180*(x + 1)**2 - 120) + 40*I*sqrt(1 - 2/(x + 1))*(x + 1)**3/(-
120*x + 15*(x + 1)**4 - 90*(x + 1)**3 + 180*(x + 1)**2 - 120) - 60*I*sqrt(1 - 2/(x + 1))*(x + 1)**2/(-120*x +
15*(x + 1)**4 - 90*(x + 1)**3 + 180*(x + 1)**2 - 120) + 20*I*sqrt(1 - 2/(x + 1))*(x + 1)/(-120*x + 15*(x + 1)*
*4 - 90*(x + 1)**3 + 180*(x + 1)**2 - 120) + 5*I*sqrt(1 - 2/(x + 1))/(-120*x + 15*(x + 1)**4 - 90*(x + 1)**3 +
180*(x + 1)**2 - 120), True))

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Giac [B]  time = 1.09646, size = 161, normalized size = 2.56 \begin{align*} \frac{{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{3}}{384 \,{\left (x + 1\right )}^{\frac{3}{2}}} + \frac{15 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}}{128 \, \sqrt{x + 1}} - \frac{{\left (x + 1\right )}^{\frac{3}{2}}{\left (\frac{45 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{2}}{x + 1} + 1\right )}}{384 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{3}} - \frac{{\left ({\left (73 \, x - 247\right )}{\left (x + 1\right )} + 360\right )} \sqrt{x + 1} \sqrt{-x + 1}}{240 \,{\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)^(5/2),x, algorithm="giac")

[Out]

1/384*(sqrt(2) - sqrt(-x + 1))^3/(x + 1)^(3/2) + 15/128*(sqrt(2) - sqrt(-x + 1))/sqrt(x + 1) - 1/384*(x + 1)^(
3/2)*(45*(sqrt(2) - sqrt(-x + 1))^2/(x + 1) + 1)/(sqrt(2) - sqrt(-x + 1))^3 - 1/240*((73*x - 247)*(x + 1) + 36
0)*sqrt(x + 1)*sqrt(-x + 1)/(x - 1)^3