3.1135 \(\int \frac {1}{(1-x)^{9/2} (1+x)^{5/2}} \, dx\)

Optimal. Leaf size=83 \[ \frac {8 x}{21 \sqrt {1-x} \sqrt {x+1}}+\frac {4 x}{21 (1-x)^{3/2} (x+1)^{3/2}}+\frac {1}{7 (1-x)^{5/2} (x+1)^{3/2}}+\frac {1}{7 (1-x)^{7/2} (x+1)^{3/2}} \]

[Out]

1/7/(1-x)^(7/2)/(1+x)^(3/2)+1/7/(1-x)^(5/2)/(1+x)^(3/2)+4/21*x/(1-x)^(3/2)/(1+x)^(3/2)+8/21*x/(1-x)^(1/2)/(1+x
)^(1/2)

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Rubi [A]  time = 0.01, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {45, 40, 39} \[ \frac {8 x}{21 \sqrt {1-x} \sqrt {x+1}}+\frac {4 x}{21 (1-x)^{3/2} (x+1)^{3/2}}+\frac {1}{7 (1-x)^{5/2} (x+1)^{3/2}}+\frac {1}{7 (1-x)^{7/2} (x+1)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/((1 - x)^(9/2)*(1 + x)^(5/2)),x]

[Out]

1/(7*(1 - x)^(7/2)*(1 + x)^(3/2)) + 1/(7*(1 - x)^(5/2)*(1 + x)^(3/2)) + (4*x)/(21*(1 - x)^(3/2)*(1 + x)^(3/2))
+ (8*x)/(21*Sqrt[1 - x]*Sqrt[1 + x])

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]

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 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])

Rubi steps

\begin {align*} \int \frac {1}{(1-x)^{9/2} (1+x)^{5/2}} \, dx &=\frac {1}{7 (1-x)^{7/2} (1+x)^{3/2}}+\frac {5}{7} \int \frac {1}{(1-x)^{7/2} (1+x)^{5/2}} \, dx\\ &=\frac {1}{7 (1-x)^{7/2} (1+x)^{3/2}}+\frac {1}{7 (1-x)^{5/2} (1+x)^{3/2}}+\frac {4}{7} \int \frac {1}{(1-x)^{5/2} (1+x)^{5/2}} \, dx\\ &=\frac {1}{7 (1-x)^{7/2} (1+x)^{3/2}}+\frac {1}{7 (1-x)^{5/2} (1+x)^{3/2}}+\frac {4 x}{21 (1-x)^{3/2} (1+x)^{3/2}}+\frac {8}{21} \int \frac {1}{(1-x)^{3/2} (1+x)^{3/2}} \, dx\\ &=\frac {1}{7 (1-x)^{7/2} (1+x)^{3/2}}+\frac {1}{7 (1-x)^{5/2} (1+x)^{3/2}}+\frac {4 x}{21 (1-x)^{3/2} (1+x)^{3/2}}+\frac {8 x}{21 \sqrt {1-x} \sqrt {1+x}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 45, normalized size = 0.54 \[ \frac {-8 x^5+16 x^4+4 x^3-24 x^2+9 x+6}{21 (1-x)^{7/2} (x+1)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((1 - x)^(9/2)*(1 + x)^(5/2)),x]

[Out]

(6 + 9*x - 24*x^2 + 4*x^3 + 16*x^4 - 8*x^5)/(21*(1 - x)^(7/2)*(1 + x)^(3/2))

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fricas [A]  time = 0.44, size = 101, normalized size = 1.22 \[ \frac {6 \, x^{6} - 12 \, x^{5} - 6 \, x^{4} + 24 \, x^{3} - 6 \, x^{2} - {\left (8 \, x^{5} - 16 \, x^{4} - 4 \, x^{3} + 24 \, x^{2} - 9 \, x - 6\right )} \sqrt {x + 1} \sqrt {-x + 1} - 12 \, x + 6}{21 \, {\left (x^{6} - 2 \, x^{5} - x^{4} + 4 \, x^{3} - x^{2} - 2 \, x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)^(9/2)/(1+x)^(5/2),x, algorithm="fricas")

[Out]

1/21*(6*x^6 - 12*x^5 - 6*x^4 + 24*x^3 - 6*x^2 - (8*x^5 - 16*x^4 - 4*x^3 + 24*x^2 - 9*x - 6)*sqrt(x + 1)*sqrt(-
x + 1) - 12*x + 6)/(x^6 - 2*x^5 - x^4 + 4*x^3 - x^2 - 2*x + 1)

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giac [B]  time = 0.69, size = 125, normalized size = 1.51 \[ \frac {{\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{3}}{768 \, {\left (x + 1\right )}^{\frac {3}{2}}} + \frac {19 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}}{256 \, \sqrt {x + 1}} - \frac {{\left (x + 1\right )}^{\frac {3}{2}} {\left (\frac {57 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{2}}{x + 1} + 1\right )}}{768 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{3}} - \frac {{\left ({\left ({\left (79 \, x - 432\right )} {\left (x + 1\right )} + 1120\right )} {\left (x + 1\right )} - 840\right )} \sqrt {x + 1} \sqrt {-x + 1}}{336 \, {\left (x - 1\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)^(9/2)/(1+x)^(5/2),x, algorithm="giac")

[Out]

1/768*(sqrt(2) - sqrt(-x + 1))^3/(x + 1)^(3/2) + 19/256*(sqrt(2) - sqrt(-x + 1))/sqrt(x + 1) - 1/768*(x + 1)^(
3/2)*(57*(sqrt(2) - sqrt(-x + 1))^2/(x + 1) + 1)/(sqrt(2) - sqrt(-x + 1))^3 - 1/336*(((79*x - 432)*(x + 1) + 1
120)*(x + 1) - 840)*sqrt(x + 1)*sqrt(-x + 1)/(x - 1)^4

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maple [A]  time = 0.00, size = 40, normalized size = 0.48 \[ -\frac {8 x^{5}-16 x^{4}-4 x^{3}+24 x^{2}-9 x -6}{21 \left (x +1\right )^{\frac {3}{2}} \left (-x +1\right )^{\frac {7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-x+1)^(9/2)/(x+1)^(5/2),x)

[Out]

-1/21*(8*x^5-16*x^4-4*x^3+24*x^2-9*x-6)/(x+1)^(3/2)/(-x+1)^(7/2)

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maxima [A]  time = 1.34, size = 91, normalized size = 1.10 \[ \frac {8 \, x}{21 \, \sqrt {-x^{2} + 1}} + \frac {4 \, x}{21 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}}} + \frac {1}{7 \, {\left ({\left (-x^{2} + 1\right )}^{\frac {3}{2}} x^{2} - 2 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} x + {\left (-x^{2} + 1\right )}^{\frac {3}{2}}\right )}} - \frac {1}{7 \, {\left ({\left (-x^{2} + 1\right )}^{\frac {3}{2}} x - {\left (-x^{2} + 1\right )}^{\frac {3}{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)^(9/2)/(1+x)^(5/2),x, algorithm="maxima")

[Out]

8/21*x/sqrt(-x^2 + 1) + 4/21*x/(-x^2 + 1)^(3/2) + 1/7/((-x^2 + 1)^(3/2)*x^2 - 2*(-x^2 + 1)^(3/2)*x + (-x^2 + 1
)^(3/2)) - 1/7/((-x^2 + 1)^(3/2)*x - (-x^2 + 1)^(3/2))

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mupad [B]  time = 0.41, size = 86, normalized size = 1.04 \[ \frac {9\,x\,\sqrt {1-x}+6\,\sqrt {1-x}-24\,x^2\,\sqrt {1-x}+4\,x^3\,\sqrt {1-x}+16\,x^4\,\sqrt {1-x}-8\,x^5\,\sqrt {1-x}}{\left (21\,x+21\right )\,{\left (x-1\right )}^4\,\sqrt {x+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((1 - x)^(9/2)*(x + 1)^(5/2)),x)

[Out]

(9*x*(1 - x)^(1/2) + 6*(1 - x)^(1/2) - 24*x^2*(1 - x)^(1/2) + 4*x^3*(1 - x)^(1/2) + 16*x^4*(1 - x)^(1/2) - 8*x
^5*(1 - x)^(1/2))/((21*x + 21)*(x - 1)^4*(x + 1)^(1/2))

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sympy [B]  time = 71.01, size = 592, normalized size = 7.13 \[ \begin {cases} \frac {8 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{5}}{- 336 x - 21 \left (x + 1\right )^{5} + 168 \left (x + 1\right )^{4} - 504 \left (x + 1\right )^{3} + 672 \left (x + 1\right )^{2} - 336} - \frac {56 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{4}}{- 336 x - 21 \left (x + 1\right )^{5} + 168 \left (x + 1\right )^{4} - 504 \left (x + 1\right )^{3} + 672 \left (x + 1\right )^{2} - 336} + \frac {140 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{3}}{- 336 x - 21 \left (x + 1\right )^{5} + 168 \left (x + 1\right )^{4} - 504 \left (x + 1\right )^{3} + 672 \left (x + 1\right )^{2} - 336} - \frac {140 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{2}}{- 336 x - 21 \left (x + 1\right )^{5} + 168 \left (x + 1\right )^{4} - 504 \left (x + 1\right )^{3} + 672 \left (x + 1\right )^{2} - 336} + \frac {35 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )}{- 336 x - 21 \left (x + 1\right )^{5} + 168 \left (x + 1\right )^{4} - 504 \left (x + 1\right )^{3} + 672 \left (x + 1\right )^{2} - 336} + \frac {7 \sqrt {-1 + \frac {2}{x + 1}}}{- 336 x - 21 \left (x + 1\right )^{5} + 168 \left (x + 1\right )^{4} - 504 \left (x + 1\right )^{3} + 672 \left (x + 1\right )^{2} - 336} & \text {for}\: \frac {2}{\left |{x + 1}\right |} > 1 \\\frac {8 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{5}}{- 336 x - 21 \left (x + 1\right )^{5} + 168 \left (x + 1\right )^{4} - 504 \left (x + 1\right )^{3} + 672 \left (x + 1\right )^{2} - 336} - \frac {56 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{4}}{- 336 x - 21 \left (x + 1\right )^{5} + 168 \left (x + 1\right )^{4} - 504 \left (x + 1\right )^{3} + 672 \left (x + 1\right )^{2} - 336} + \frac {140 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{3}}{- 336 x - 21 \left (x + 1\right )^{5} + 168 \left (x + 1\right )^{4} - 504 \left (x + 1\right )^{3} + 672 \left (x + 1\right )^{2} - 336} - \frac {140 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{2}}{- 336 x - 21 \left (x + 1\right )^{5} + 168 \left (x + 1\right )^{4} - 504 \left (x + 1\right )^{3} + 672 \left (x + 1\right )^{2} - 336} + \frac {35 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )}{- 336 x - 21 \left (x + 1\right )^{5} + 168 \left (x + 1\right )^{4} - 504 \left (x + 1\right )^{3} + 672 \left (x + 1\right )^{2} - 336} + \frac {7 i \sqrt {1 - \frac {2}{x + 1}}}{- 336 x - 21 \left (x + 1\right )^{5} + 168 \left (x + 1\right )^{4} - 504 \left (x + 1\right )^{3} + 672 \left (x + 1\right )^{2} - 336} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)**(9/2)/(1+x)**(5/2),x)

[Out]

Piecewise((8*sqrt(-1 + 2/(x + 1))*(x + 1)**5/(-336*x - 21*(x + 1)**5 + 168*(x + 1)**4 - 504*(x + 1)**3 + 672*(
x + 1)**2 - 336) - 56*sqrt(-1 + 2/(x + 1))*(x + 1)**4/(-336*x - 21*(x + 1)**5 + 168*(x + 1)**4 - 504*(x + 1)**
3 + 672*(x + 1)**2 - 336) + 140*sqrt(-1 + 2/(x + 1))*(x + 1)**3/(-336*x - 21*(x + 1)**5 + 168*(x + 1)**4 - 504
*(x + 1)**3 + 672*(x + 1)**2 - 336) - 140*sqrt(-1 + 2/(x + 1))*(x + 1)**2/(-336*x - 21*(x + 1)**5 + 168*(x + 1
)**4 - 504*(x + 1)**3 + 672*(x + 1)**2 - 336) + 35*sqrt(-1 + 2/(x + 1))*(x + 1)/(-336*x - 21*(x + 1)**5 + 168*
(x + 1)**4 - 504*(x + 1)**3 + 672*(x + 1)**2 - 336) + 7*sqrt(-1 + 2/(x + 1))/(-336*x - 21*(x + 1)**5 + 168*(x
+ 1)**4 - 504*(x + 1)**3 + 672*(x + 1)**2 - 336), 2/Abs(x + 1) > 1), (8*I*sqrt(1 - 2/(x + 1))*(x + 1)**5/(-336
*x - 21*(x + 1)**5 + 168*(x + 1)**4 - 504*(x + 1)**3 + 672*(x + 1)**2 - 336) - 56*I*sqrt(1 - 2/(x + 1))*(x + 1
)**4/(-336*x - 21*(x + 1)**5 + 168*(x + 1)**4 - 504*(x + 1)**3 + 672*(x + 1)**2 - 336) + 140*I*sqrt(1 - 2/(x +
1))*(x + 1)**3/(-336*x - 21*(x + 1)**5 + 168*(x + 1)**4 - 504*(x + 1)**3 + 672*(x + 1)**2 - 336) - 140*I*sqrt
(1 - 2/(x + 1))*(x + 1)**2/(-336*x - 21*(x + 1)**5 + 168*(x + 1)**4 - 504*(x + 1)**3 + 672*(x + 1)**2 - 336) +
35*I*sqrt(1 - 2/(x + 1))*(x + 1)/(-336*x - 21*(x + 1)**5 + 168*(x + 1)**4 - 504*(x + 1)**3 + 672*(x + 1)**2 -
336) + 7*I*sqrt(1 - 2/(x + 1))/(-336*x - 21*(x + 1)**5 + 168*(x + 1)**4 - 504*(x + 1)**3 + 672*(x + 1)**2 - 3
36), True))

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