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

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Rubi [A]  time = 0.0137007, antiderivative size = 83, normalized size of antiderivative = 1., 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 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)^{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*}

Mathematica [A]  time = 0.0133994, 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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Maple [A]  time = 0.001, size = 40, normalized size = 0.5 \begin{align*} -{\frac{8\,{x}^{5}-16\,{x}^{4}-4\,{x}^{3}+24\,{x}^{2}-9\,x-6}{21} \left ( 1-x \right ) ^{-{\frac{7}{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)^(9/2)/(1+x)^(5/2),x)

[Out]

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

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Maxima [A]  time = 0.994069, size = 123, normalized size = 1.48 \begin{align*} \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 )}} \end{align*}

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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Fricas [A]  time = 1.55858, size = 235, normalized size = 2.83 \begin{align*} \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 )}} \end{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 1.08924, size = 169, normalized size = 2.04 \begin{align*} \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}} \end{align*}

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