3.1125 \(\int \frac{1}{(1-x)^{11/2} (1+x)^{3/2}} \, dx\)

Optimal. Leaf size=102 \[ \frac{8 x}{63 \sqrt{1-x} \sqrt{x+1}}+\frac{4}{63 (1-x)^{3/2} \sqrt{x+1}}+\frac{4}{63 (1-x)^{5/2} \sqrt{x+1}}+\frac{5}{63 (1-x)^{7/2} \sqrt{x+1}}+\frac{1}{9 (1-x)^{9/2} \sqrt{x+1}} \]

[Out]

1/(9*(1 - x)^(9/2)*Sqrt[1 + x]) + 5/(63*(1 - x)^(7/2)*Sqrt[1 + x]) + 4/(63*(1 - x)^(5/2)*Sqrt[1 + x]) + 4/(63*
(1 - x)^(3/2)*Sqrt[1 + x]) + (8*x)/(63*Sqrt[1 - x]*Sqrt[1 + x])

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Rubi [A]  time = 0.0203065, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {45, 39} \[ \frac{8 x}{63 \sqrt{1-x} \sqrt{x+1}}+\frac{4}{63 (1-x)^{3/2} \sqrt{x+1}}+\frac{4}{63 (1-x)^{5/2} \sqrt{x+1}}+\frac{5}{63 (1-x)^{7/2} \sqrt{x+1}}+\frac{1}{9 (1-x)^{9/2} \sqrt{x+1}} \]

Antiderivative was successfully verified.

[In]

Int[1/((1 - x)^(11/2)*(1 + x)^(3/2)),x]

[Out]

1/(9*(1 - x)^(9/2)*Sqrt[1 + x]) + 5/(63*(1 - x)^(7/2)*Sqrt[1 + x]) + 4/(63*(1 - x)^(5/2)*Sqrt[1 + x]) + 4/(63*
(1 - x)^(3/2)*Sqrt[1 + x]) + (8*x)/(63*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)^{11/2} (1+x)^{3/2}} \, dx &=\frac{1}{9 (1-x)^{9/2} \sqrt{1+x}}+\frac{5}{9} \int \frac{1}{(1-x)^{9/2} (1+x)^{3/2}} \, dx\\ &=\frac{1}{9 (1-x)^{9/2} \sqrt{1+x}}+\frac{5}{63 (1-x)^{7/2} \sqrt{1+x}}+\frac{20}{63} \int \frac{1}{(1-x)^{7/2} (1+x)^{3/2}} \, dx\\ &=\frac{1}{9 (1-x)^{9/2} \sqrt{1+x}}+\frac{5}{63 (1-x)^{7/2} \sqrt{1+x}}+\frac{4}{63 (1-x)^{5/2} \sqrt{1+x}}+\frac{4}{21} \int \frac{1}{(1-x)^{5/2} (1+x)^{3/2}} \, dx\\ &=\frac{1}{9 (1-x)^{9/2} \sqrt{1+x}}+\frac{5}{63 (1-x)^{7/2} \sqrt{1+x}}+\frac{4}{63 (1-x)^{5/2} \sqrt{1+x}}+\frac{4}{63 (1-x)^{3/2} \sqrt{1+x}}+\frac{8}{63} \int \frac{1}{(1-x)^{3/2} (1+x)^{3/2}} \, dx\\ &=\frac{1}{9 (1-x)^{9/2} \sqrt{1+x}}+\frac{5}{63 (1-x)^{7/2} \sqrt{1+x}}+\frac{4}{63 (1-x)^{5/2} \sqrt{1+x}}+\frac{4}{63 (1-x)^{3/2} \sqrt{1+x}}+\frac{8 x}{63 \sqrt{1-x} \sqrt{1+x}}\\ \end{align*}

Mathematica [A]  time = 0.0115801, size = 45, normalized size = 0.44 \[ \frac{8 x^5-32 x^4+44 x^3-16 x^2-17 x+20}{63 (x-1)^4 \sqrt{1-x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((1 - x)^(11/2)*(1 + x)^(3/2)),x]

[Out]

(20 - 17*x - 16*x^2 + 44*x^3 - 32*x^4 + 8*x^5)/(63*(-1 + x)^4*Sqrt[1 - x^2])

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Maple [A]  time = 0.003, size = 40, normalized size = 0.4 \begin{align*}{\frac{8\,{x}^{5}-32\,{x}^{4}+44\,{x}^{3}-16\,{x}^{2}-17\,x+20}{63} \left ( 1-x \right ) ^{-{\frac{9}{2}}}{\frac{1}{\sqrt{1+x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1-x)^(11/2)/(1+x)^(3/2),x)

[Out]

1/63*(8*x^5-32*x^4+44*x^3-16*x^2-17*x+20)/(1+x)^(1/2)/(1-x)^(9/2)

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Maxima [B]  time = 1.03223, size = 271, normalized size = 2.66 \begin{align*} \frac{8 \, x}{63 \, \sqrt{-x^{2} + 1}} + \frac{1}{9 \,{\left (\sqrt{-x^{2} + 1} x^{4} - 4 \, \sqrt{-x^{2} + 1} x^{3} + 6 \, \sqrt{-x^{2} + 1} x^{2} - 4 \, \sqrt{-x^{2} + 1} x + \sqrt{-x^{2} + 1}\right )}} - \frac{5}{63 \,{\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}{63 \,{\left (\sqrt{-x^{2} + 1} x^{2} - 2 \, \sqrt{-x^{2} + 1} x + \sqrt{-x^{2} + 1}\right )}} - \frac{4}{63 \,{\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)^(11/2)/(1+x)^(3/2),x, algorithm="maxima")

[Out]

8/63*x/sqrt(-x^2 + 1) + 1/9/(sqrt(-x^2 + 1)*x^4 - 4*sqrt(-x^2 + 1)*x^3 + 6*sqrt(-x^2 + 1)*x^2 - 4*sqrt(-x^2 +
1)*x + sqrt(-x^2 + 1)) - 5/63/(sqrt(-x^2 + 1)*x^3 - 3*sqrt(-x^2 + 1)*x^2 + 3*sqrt(-x^2 + 1)*x - sqrt(-x^2 + 1)
) + 4/63/(sqrt(-x^2 + 1)*x^2 - 2*sqrt(-x^2 + 1)*x + sqrt(-x^2 + 1)) - 4/63/(sqrt(-x^2 + 1)*x - sqrt(-x^2 + 1))

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Fricas [A]  time = 1.7967, size = 230, normalized size = 2.25 \begin{align*} \frac{20 \, x^{6} - 80 \, x^{5} + 100 \, x^{4} - 100 \, x^{2} -{\left (8 \, x^{5} - 32 \, x^{4} + 44 \, x^{3} - 16 \, x^{2} - 17 \, x + 20\right )} \sqrt{x + 1} \sqrt{-x + 1} + 80 \, x - 20}{63 \,{\left (x^{6} - 4 \, x^{5} + 5 \, x^{4} - 5 \, x^{2} + 4 \, x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)^(11/2)/(1+x)^(3/2),x, algorithm="fricas")

[Out]

1/63*(20*x^6 - 80*x^5 + 100*x^4 - 100*x^2 - (8*x^5 - 32*x^4 + 44*x^3 - 16*x^2 - 17*x + 20)*sqrt(x + 1)*sqrt(-x
+ 1) + 80*x - 20)/(x^6 - 4*x^5 + 5*x^4 - 5*x^2 + 4*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)**(11/2)/(1+x)**(3/2),x)

[Out]

Timed out

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Giac [A]  time = 1.06085, size = 115, normalized size = 1.13 \begin{align*} \frac{\sqrt{2} - \sqrt{-x + 1}}{64 \, \sqrt{x + 1}} - \frac{\sqrt{x + 1}}{64 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}} - \frac{{\left ({\left ({\left ({\left (193 \, x - 1481\right )}{\left (x + 1\right )} + 5544\right )}{\left (x + 1\right )} - 8400\right )}{\left (x + 1\right )} + 5040\right )} \sqrt{x + 1} \sqrt{-x + 1}}{2016 \,{\left (x - 1\right )}^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)^(11/2)/(1+x)^(3/2),x, algorithm="giac")

[Out]

1/64*(sqrt(2) - sqrt(-x + 1))/sqrt(x + 1) - 1/64*sqrt(x + 1)/(sqrt(2) - sqrt(-x + 1)) - 1/2016*((((193*x - 148
1)*(x + 1) + 5544)*(x + 1) - 8400)*(x + 1) + 5040)*sqrt(x + 1)*sqrt(-x + 1)/(x - 1)^5