3.1136 \(\int \frac{1}{(1-x)^{11/2} (1+x)^{5/2}} \, dx\)

Optimal. Leaf size=103 \[ \frac{16 x}{63 \sqrt{1-x} \sqrt{x+1}}+\frac{8 x}{63 (1-x)^{3/2} (x+1)^{3/2}}+\frac{2}{21 (1-x)^{5/2} (x+1)^{3/2}}+\frac{2}{21 (1-x)^{7/2} (x+1)^{3/2}}+\frac{1}{9 (1-x)^{9/2} (x+1)^{3/2}} \]

[Out]

1/(9*(1 - x)^(9/2)*(1 + x)^(3/2)) + 2/(21*(1 - x)^(7/2)*(1 + x)^(3/2)) + 2/(21*(1 - x)^(5/2)*(1 + x)^(3/2)) +
(8*x)/(63*(1 - x)^(3/2)*(1 + x)^(3/2)) + (16*x)/(63*Sqrt[1 - x]*Sqrt[1 + x])

________________________________________________________________________________________

Rubi [A]  time = 0.0193206, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {45, 40, 39} \[ \frac{16 x}{63 \sqrt{1-x} \sqrt{x+1}}+\frac{8 x}{63 (1-x)^{3/2} (x+1)^{3/2}}+\frac{2}{21 (1-x)^{5/2} (x+1)^{3/2}}+\frac{2}{21 (1-x)^{7/2} (x+1)^{3/2}}+\frac{1}{9 (1-x)^{9/2} (x+1)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/((1 - x)^(11/2)*(1 + x)^(5/2)),x]

[Out]

1/(9*(1 - x)^(9/2)*(1 + x)^(3/2)) + 2/(21*(1 - x)^(7/2)*(1 + x)^(3/2)) + 2/(21*(1 - x)^(5/2)*(1 + x)^(3/2)) +
(8*x)/(63*(1 - x)^(3/2)*(1 + x)^(3/2)) + (16*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 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)^{11/2} (1+x)^{5/2}} \, dx &=\frac{1}{9 (1-x)^{9/2} (1+x)^{3/2}}+\frac{2}{3} \int \frac{1}{(1-x)^{9/2} (1+x)^{5/2}} \, dx\\ &=\frac{1}{9 (1-x)^{9/2} (1+x)^{3/2}}+\frac{2}{21 (1-x)^{7/2} (1+x)^{3/2}}+\frac{10}{21} \int \frac{1}{(1-x)^{7/2} (1+x)^{5/2}} \, dx\\ &=\frac{1}{9 (1-x)^{9/2} (1+x)^{3/2}}+\frac{2}{21 (1-x)^{7/2} (1+x)^{3/2}}+\frac{2}{21 (1-x)^{5/2} (1+x)^{3/2}}+\frac{8}{21} \int \frac{1}{(1-x)^{5/2} (1+x)^{5/2}} \, dx\\ &=\frac{1}{9 (1-x)^{9/2} (1+x)^{3/2}}+\frac{2}{21 (1-x)^{7/2} (1+x)^{3/2}}+\frac{2}{21 (1-x)^{5/2} (1+x)^{3/2}}+\frac{8 x}{63 (1-x)^{3/2} (1+x)^{3/2}}+\frac{16}{63} \int \frac{1}{(1-x)^{3/2} (1+x)^{3/2}} \, dx\\ &=\frac{1}{9 (1-x)^{9/2} (1+x)^{3/2}}+\frac{2}{21 (1-x)^{7/2} (1+x)^{3/2}}+\frac{2}{21 (1-x)^{5/2} (1+x)^{3/2}}+\frac{8 x}{63 (1-x)^{3/2} (1+x)^{3/2}}+\frac{16 x}{63 \sqrt{1-x} \sqrt{1+x}}\\ \end{align*}

Mathematica [A]  time = 0.0127024, size = 50, normalized size = 0.49 \[ \frac{16 x^6-48 x^5+24 x^4+56 x^3-66 x^2+6 x+19}{63 (1-x)^{9/2} (x+1)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((1 - x)^(11/2)*(1 + x)^(5/2)),x]

[Out]

(19 + 6*x - 66*x^2 + 56*x^3 + 24*x^4 - 48*x^5 + 16*x^6)/(63*(1 - x)^(9/2)*(1 + x)^(3/2))

________________________________________________________________________________________

Maple [A]  time = 0.002, size = 45, normalized size = 0.4 \begin{align*}{\frac{16\,{x}^{6}-48\,{x}^{5}+24\,{x}^{4}+56\,{x}^{3}-66\,{x}^{2}+6\,x+19}{63} \left ( 1-x \right ) ^{-{\frac{9}{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)^(11/2)/(1+x)^(5/2),x)

[Out]

1/63*(16*x^6-48*x^5+24*x^4+56*x^3-66*x^2+6*x+19)/(1+x)^(3/2)/(1-x)^(9/2)

________________________________________________________________________________________

Maxima [A]  time = 1.00592, size = 197, normalized size = 1.91 \begin{align*} \frac{16 \, x}{63 \, \sqrt{-x^{2} + 1}} + \frac{8 \, x}{63 \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}}} - \frac{1}{9 \,{\left ({\left (-x^{2} + 1\right )}^{\frac{3}{2}} x^{3} - 3 \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}} x^{2} + 3 \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}} x -{\left (-x^{2} + 1\right )}^{\frac{3}{2}}\right )}} + \frac{2}{21 \,{\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{2}{21 \,{\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)^(11/2)/(1+x)^(5/2),x, algorithm="maxima")

[Out]

16/63*x/sqrt(-x^2 + 1) + 8/63*x/(-x^2 + 1)^(3/2) - 1/9/((-x^2 + 1)^(3/2)*x^3 - 3*(-x^2 + 1)^(3/2)*x^2 + 3*(-x^
2 + 1)^(3/2)*x - (-x^2 + 1)^(3/2)) + 2/21/((-x^2 + 1)^(3/2)*x^2 - 2*(-x^2 + 1)^(3/2)*x + (-x^2 + 1)^(3/2)) - 2
/21/((-x^2 + 1)^(3/2)*x - (-x^2 + 1)^(3/2))

________________________________________________________________________________________

Fricas [A]  time = 1.53236, size = 279, normalized size = 2.71 \begin{align*} \frac{19 \, x^{7} - 57 \, x^{6} + 19 \, x^{5} + 95 \, x^{4} - 95 \, x^{3} - 19 \, x^{2} -{\left (16 \, x^{6} - 48 \, x^{5} + 24 \, x^{4} + 56 \, x^{3} - 66 \, x^{2} + 6 \, x + 19\right )} \sqrt{x + 1} \sqrt{-x + 1} + 57 \, x - 19}{63 \,{\left (x^{7} - 3 \, x^{6} + x^{5} + 5 \, x^{4} - 5 \, x^{3} - x^{2} + 3 \, x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)^(11/2)/(1+x)^(5/2),x, algorithm="fricas")

[Out]

1/63*(19*x^7 - 57*x^6 + 19*x^5 + 95*x^4 - 95*x^3 - 19*x^2 - (16*x^6 - 48*x^5 + 24*x^4 + 56*x^3 - 66*x^2 + 6*x
+ 19)*sqrt(x + 1)*sqrt(-x + 1) + 57*x - 19)/(x^7 - 3*x^6 + x^5 + 5*x^4 - 5*x^3 - x^2 + 3*x - 1)

________________________________________________________________________________________

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)**(5/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.08194, size = 177, normalized size = 1.72 \begin{align*} \frac{{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{3}}{1536 \,{\left (x + 1\right )}^{\frac{3}{2}}} + \frac{23 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}}{512 \, \sqrt{x + 1}} - \frac{{\left (x + 1\right )}^{\frac{3}{2}}{\left (\frac{69 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{2}}{x + 1} + 1\right )}}{1536 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{3}} - \frac{{\left ({\left ({\left ({\left (667 \, x - 5021\right )}{\left (x + 1\right )} + 18396\right )}{\left (x + 1\right )} - 26880\right )}{\left (x + 1\right )} + 15120\right )} \sqrt{x + 1} \sqrt{-x + 1}}{4032 \,{\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)^(5/2),x, algorithm="giac")

[Out]

1/1536*(sqrt(2) - sqrt(-x + 1))^3/(x + 1)^(3/2) + 23/512*(sqrt(2) - sqrt(-x + 1))/sqrt(x + 1) - 1/1536*(x + 1)
^(3/2)*(69*(sqrt(2) - sqrt(-x + 1))^2/(x + 1) + 1)/(sqrt(2) - sqrt(-x + 1))^3 - 1/4032*((((667*x - 5021)*(x +
1) + 18396)*(x + 1) - 26880)*(x + 1) + 15120)*sqrt(x + 1)*sqrt(-x + 1)/(x - 1)^5