3.1124 \(\int \frac{1}{(1-x)^{9/2} (1+x)^{3/2}} \, dx\)

Optimal. Leaf size=82 \[ \frac{8 x}{35 \sqrt{1-x} \sqrt{x+1}}+\frac{4}{35 (1-x)^{3/2} \sqrt{x+1}}+\frac{4}{35 (1-x)^{5/2} \sqrt{x+1}}+\frac{1}{7 (1-x)^{7/2} \sqrt{x+1}} \]

[Out]

1/(7*(1 - x)^(7/2)*Sqrt[1 + x]) + 4/(35*(1 - x)^(5/2)*Sqrt[1 + x]) + 4/(35*(1 - x)^(3/2)*Sqrt[1 + x]) + (8*x)/
(35*Sqrt[1 - x]*Sqrt[1 + x])

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Rubi [A]  time = 0.0134444, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {45, 39} \[ \frac{8 x}{35 \sqrt{1-x} \sqrt{x+1}}+\frac{4}{35 (1-x)^{3/2} \sqrt{x+1}}+\frac{4}{35 (1-x)^{5/2} \sqrt{x+1}}+\frac{1}{7 (1-x)^{7/2} \sqrt{x+1}} \]

Antiderivative was successfully verified.

[In]

Int[1/((1 - x)^(9/2)*(1 + x)^(3/2)),x]

[Out]

1/(7*(1 - x)^(7/2)*Sqrt[1 + x]) + 4/(35*(1 - x)^(5/2)*Sqrt[1 + x]) + 4/(35*(1 - x)^(3/2)*Sqrt[1 + x]) + (8*x)/
(35*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)^{9/2} (1+x)^{3/2}} \, dx &=\frac{1}{7 (1-x)^{7/2} \sqrt{1+x}}+\frac{4}{7} \int \frac{1}{(1-x)^{7/2} (1+x)^{3/2}} \, dx\\ &=\frac{1}{7 (1-x)^{7/2} \sqrt{1+x}}+\frac{4}{35 (1-x)^{5/2} \sqrt{1+x}}+\frac{12}{35} \int \frac{1}{(1-x)^{5/2} (1+x)^{3/2}} \, dx\\ &=\frac{1}{7 (1-x)^{7/2} \sqrt{1+x}}+\frac{4}{35 (1-x)^{5/2} \sqrt{1+x}}+\frac{4}{35 (1-x)^{3/2} \sqrt{1+x}}+\frac{8}{35} \int \frac{1}{(1-x)^{3/2} (1+x)^{3/2}} \, dx\\ &=\frac{1}{7 (1-x)^{7/2} \sqrt{1+x}}+\frac{4}{35 (1-x)^{5/2} \sqrt{1+x}}+\frac{4}{35 (1-x)^{3/2} \sqrt{1+x}}+\frac{8 x}{35 \sqrt{1-x} \sqrt{1+x}}\\ \end{align*}

Mathematica [A]  time = 0.0096104, size = 40, normalized size = 0.49 \[ \frac{8 x^4-24 x^3+20 x^2+4 x-13}{35 (x-1)^3 \sqrt{1-x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((1 - x)^(9/2)*(1 + x)^(3/2)),x]

[Out]

(-13 + 4*x + 20*x^2 - 24*x^3 + 8*x^4)/(35*(-1 + x)^3*Sqrt[1 - x^2])

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Maple [A]  time = 0.003, size = 35, normalized size = 0.4 \begin{align*} -{\frac{8\,{x}^{4}-24\,{x}^{3}+20\,{x}^{2}+4\,x-13}{35} \left ( 1-x \right ) ^{-{\frac{7}{2}}}{\frac{1}{\sqrt{1+x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1-x)^(9/2)/(1+x)^(3/2),x)

[Out]

-1/35*(8*x^4-24*x^3+20*x^2+4*x-13)/(1+x)^(1/2)/(1-x)^(7/2)

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Maxima [B]  time = 1.00894, size = 181, normalized size = 2.21 \begin{align*} \frac{8 \, x}{35 \, \sqrt{-x^{2} + 1}} - \frac{1}{7 \,{\left (\sqrt{-x^{2} + 1} x^{3} - 3 \, \sqrt{-x^{2} + 1} x^{2} + 3 \, \sqrt{-x^{2} + 1} x - \sqrt{-x^{2} + 1}\right )}} + \frac{4}{35 \,{\left (\sqrt{-x^{2} + 1} x^{2} - 2 \, \sqrt{-x^{2} + 1} x + \sqrt{-x^{2} + 1}\right )}} - \frac{4}{35 \,{\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)^(9/2)/(1+x)^(3/2),x, algorithm="maxima")

[Out]

8/35*x/sqrt(-x^2 + 1) - 1/7/(sqrt(-x^2 + 1)*x^3 - 3*sqrt(-x^2 + 1)*x^2 + 3*sqrt(-x^2 + 1)*x - sqrt(-x^2 + 1))
+ 4/35/(sqrt(-x^2 + 1)*x^2 - 2*sqrt(-x^2 + 1)*x + sqrt(-x^2 + 1)) - 4/35/(sqrt(-x^2 + 1)*x - sqrt(-x^2 + 1))

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Fricas [A]  time = 1.80086, size = 213, normalized size = 2.6 \begin{align*} \frac{13 \, x^{5} - 39 \, x^{4} + 26 \, x^{3} + 26 \, x^{2} -{\left (8 \, x^{4} - 24 \, x^{3} + 20 \, x^{2} + 4 \, x - 13\right )} \sqrt{x + 1} \sqrt{-x + 1} - 39 \, x + 13}{35 \,{\left (x^{5} - 3 \, x^{4} + 2 \, x^{3} + 2 \, x^{2} - 3 \, x + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)^(9/2)/(1+x)^(3/2),x, algorithm="fricas")

[Out]

1/35*(13*x^5 - 39*x^4 + 26*x^3 + 26*x^2 - (8*x^4 - 24*x^3 + 20*x^2 + 4*x - 13)*sqrt(x + 1)*sqrt(-x + 1) - 39*x
+ 13)/(x^5 - 3*x^4 + 2*x^3 + 2*x^2 - 3*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)**(3/2),x)

[Out]

Timed out

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Giac [A]  time = 1.08629, size = 107, normalized size = 1.3 \begin{align*} \frac{\sqrt{2} - \sqrt{-x + 1}}{32 \, \sqrt{x + 1}} - \frac{\sqrt{x + 1}}{32 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}} - \frac{{\left ({\left ({\left (93 \, x - 523\right )}{\left (x + 1\right )} + 1400\right )}{\left (x + 1\right )} - 1120\right )} \sqrt{x + 1} \sqrt{-x + 1}}{560 \,{\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)^(3/2),x, algorithm="giac")

[Out]

1/32*(sqrt(2) - sqrt(-x + 1))/sqrt(x + 1) - 1/32*sqrt(x + 1)/(sqrt(2) - sqrt(-x + 1)) - 1/560*(((93*x - 523)*(
x + 1) + 1400)*(x + 1) - 1120)*sqrt(x + 1)*sqrt(-x + 1)/(x - 1)^4