3.1074 \(\int \frac{\sqrt{1+x}}{(1-x)^{13/2}} \, dx\)

Optimal. Leaf size=101 \[ \frac{8 (x+1)^{3/2}}{3465 (1-x)^{3/2}}+\frac{8 (x+1)^{3/2}}{1155 (1-x)^{5/2}}+\frac{4 (x+1)^{3/2}}{231 (1-x)^{7/2}}+\frac{4 (x+1)^{3/2}}{99 (1-x)^{9/2}}+\frac{(x+1)^{3/2}}{11 (1-x)^{11/2}} \]

[Out]

(1 + x)^(3/2)/(11*(1 - x)^(11/2)) + (4*(1 + x)^(3/2))/(99*(1 - x)^(9/2)) + (4*(1 + x)^(3/2))/(231*(1 - x)^(7/2
)) + (8*(1 + x)^(3/2))/(1155*(1 - x)^(5/2)) + (8*(1 + x)^(3/2))/(3465*(1 - x)^(3/2))

________________________________________________________________________________________

Rubi [A]  time = 0.0186567, antiderivative size = 101, 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, 37} \[ \frac{8 (x+1)^{3/2}}{3465 (1-x)^{3/2}}+\frac{8 (x+1)^{3/2}}{1155 (1-x)^{5/2}}+\frac{4 (x+1)^{3/2}}{231 (1-x)^{7/2}}+\frac{4 (x+1)^{3/2}}{99 (1-x)^{9/2}}+\frac{(x+1)^{3/2}}{11 (1-x)^{11/2}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 + x]/(1 - x)^(13/2),x]

[Out]

(1 + x)^(3/2)/(11*(1 - x)^(11/2)) + (4*(1 + x)^(3/2))/(99*(1 - x)^(9/2)) + (4*(1 + x)^(3/2))/(231*(1 - x)^(7/2
)) + (8*(1 + x)^(3/2))/(1155*(1 - x)^(5/2)) + (8*(1 + x)^(3/2))/(3465*(1 - x)^(3/2))

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 37

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] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{\sqrt{1+x}}{(1-x)^{13/2}} \, dx &=\frac{(1+x)^{3/2}}{11 (1-x)^{11/2}}+\frac{4}{11} \int \frac{\sqrt{1+x}}{(1-x)^{11/2}} \, dx\\ &=\frac{(1+x)^{3/2}}{11 (1-x)^{11/2}}+\frac{4 (1+x)^{3/2}}{99 (1-x)^{9/2}}+\frac{4}{33} \int \frac{\sqrt{1+x}}{(1-x)^{9/2}} \, dx\\ &=\frac{(1+x)^{3/2}}{11 (1-x)^{11/2}}+\frac{4 (1+x)^{3/2}}{99 (1-x)^{9/2}}+\frac{4 (1+x)^{3/2}}{231 (1-x)^{7/2}}+\frac{8}{231} \int \frac{\sqrt{1+x}}{(1-x)^{7/2}} \, dx\\ &=\frac{(1+x)^{3/2}}{11 (1-x)^{11/2}}+\frac{4 (1+x)^{3/2}}{99 (1-x)^{9/2}}+\frac{4 (1+x)^{3/2}}{231 (1-x)^{7/2}}+\frac{8 (1+x)^{3/2}}{1155 (1-x)^{5/2}}+\frac{8 \int \frac{\sqrt{1+x}}{(1-x)^{5/2}} \, dx}{1155}\\ &=\frac{(1+x)^{3/2}}{11 (1-x)^{11/2}}+\frac{4 (1+x)^{3/2}}{99 (1-x)^{9/2}}+\frac{4 (1+x)^{3/2}}{231 (1-x)^{7/2}}+\frac{8 (1+x)^{3/2}}{1155 (1-x)^{5/2}}+\frac{8 (1+x)^{3/2}}{3465 (1-x)^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.017413, size = 40, normalized size = 0.4 \[ \frac{(x+1)^{3/2} \left (8 x^4-56 x^3+180 x^2-364 x+547\right )}{3465 (1-x)^{11/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 + x]/(1 - x)^(13/2),x]

[Out]

((1 + x)^(3/2)*(547 - 364*x + 180*x^2 - 56*x^3 + 8*x^4))/(3465*(1 - x)^(11/2))

________________________________________________________________________________________

Maple [A]  time = 0.002, size = 35, normalized size = 0.4 \begin{align*}{\frac{8\,{x}^{4}-56\,{x}^{3}+180\,{x}^{2}-364\,x+547}{3465} \left ( 1+x \right ) ^{{\frac{3}{2}}} \left ( 1-x \right ) ^{-{\frac{11}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+x)^(1/2)/(1-x)^(13/2),x)

[Out]

1/3465*(1+x)^(3/2)*(8*x^4-56*x^3+180*x^2-364*x+547)/(1-x)^(11/2)

________________________________________________________________________________________

Maxima [B]  time = 1.04257, size = 232, normalized size = 2.3 \begin{align*} \frac{2 \, \sqrt{-x^{2} + 1}}{11 \,{\left (x^{6} - 6 \, x^{5} + 15 \, x^{4} - 20 \, x^{3} + 15 \, x^{2} - 6 \, x + 1\right )}} + \frac{\sqrt{-x^{2} + 1}}{99 \,{\left (x^{5} - 5 \, x^{4} + 10 \, x^{3} - 10 \, x^{2} + 5 \, x - 1\right )}} - \frac{4 \, \sqrt{-x^{2} + 1}}{693 \,{\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} + \frac{4 \, \sqrt{-x^{2} + 1}}{1155 \,{\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} - \frac{8 \, \sqrt{-x^{2} + 1}}{3465 \,{\left (x^{2} - 2 \, x + 1\right )}} + \frac{8 \, \sqrt{-x^{2} + 1}}{3465 \,{\left (x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)^(1/2)/(1-x)^(13/2),x, algorithm="maxima")

[Out]

2/11*sqrt(-x^2 + 1)/(x^6 - 6*x^5 + 15*x^4 - 20*x^3 + 15*x^2 - 6*x + 1) + 1/99*sqrt(-x^2 + 1)/(x^5 - 5*x^4 + 10
*x^3 - 10*x^2 + 5*x - 1) - 4/693*sqrt(-x^2 + 1)/(x^4 - 4*x^3 + 6*x^2 - 4*x + 1) + 4/1155*sqrt(-x^2 + 1)/(x^3 -
3*x^2 + 3*x - 1) - 8/3465*sqrt(-x^2 + 1)/(x^2 - 2*x + 1) + 8/3465*sqrt(-x^2 + 1)/(x - 1)

________________________________________________________________________________________

Fricas [A]  time = 1.53397, size = 279, normalized size = 2.76 \begin{align*} \frac{547 \, x^{6} - 3282 \, x^{5} + 8205 \, x^{4} - 10940 \, x^{3} + 8205 \, x^{2} +{\left (8 \, x^{5} - 48 \, x^{4} + 124 \, x^{3} - 184 \, x^{2} + 183 \, x + 547\right )} \sqrt{x + 1} \sqrt{-x + 1} - 3282 \, x + 547}{3465 \,{\left (x^{6} - 6 \, x^{5} + 15 \, x^{4} - 20 \, x^{3} + 15 \, x^{2} - 6 \, x + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)^(1/2)/(1-x)^(13/2),x, algorithm="fricas")

[Out]

1/3465*(547*x^6 - 3282*x^5 + 8205*x^4 - 10940*x^3 + 8205*x^2 + (8*x^5 - 48*x^4 + 124*x^3 - 184*x^2 + 183*x + 5
47)*sqrt(x + 1)*sqrt(-x + 1) - 3282*x + 547)/(x^6 - 6*x^5 + 15*x^4 - 20*x^3 + 15*x^2 - 6*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+x)**(1/2)/(1-x)**(13/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.15099, size = 57, normalized size = 0.56 \begin{align*} \frac{{\left (4 \,{\left ({\left (2 \,{\left (x + 1\right )}{\left (x - 10\right )} + 99\right )}{\left (x + 1\right )} - 231\right )}{\left (x + 1\right )} + 1155\right )}{\left (x + 1\right )}^{\frac{3}{2}} \sqrt{-x + 1}}{3465 \,{\left (x - 1\right )}^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)^(1/2)/(1-x)^(13/2),x, algorithm="giac")

[Out]

1/3465*(4*((2*(x + 1)*(x - 10) + 99)*(x + 1) - 231)*(x + 1) + 1155)*(x + 1)^(3/2)*sqrt(-x + 1)/(x - 1)^6