3.1073 \(\int \frac{\sqrt{1+x}}{(1-x)^{11/2}} \, dx\)

Optimal. Leaf size=81 \[ \frac{2 (x+1)^{3/2}}{315 (1-x)^{3/2}}+\frac{2 (x+1)^{3/2}}{105 (1-x)^{5/2}}+\frac{(x+1)^{3/2}}{21 (1-x)^{7/2}}+\frac{(x+1)^{3/2}}{9 (1-x)^{9/2}} \]

[Out]

(1 + x)^(3/2)/(9*(1 - x)^(9/2)) + (1 + x)^(3/2)/(21*(1 - x)^(7/2)) + (2*(1 + x)^(3/2))/(105*(1 - x)^(5/2)) + (
2*(1 + x)^(3/2))/(315*(1 - x)^(3/2))

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Rubi [A]  time = 0.0130847, antiderivative size = 81, 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, 37} \[ \frac{2 (x+1)^{3/2}}{315 (1-x)^{3/2}}+\frac{2 (x+1)^{3/2}}{105 (1-x)^{5/2}}+\frac{(x+1)^{3/2}}{21 (1-x)^{7/2}}+\frac{(x+1)^{3/2}}{9 (1-x)^{9/2}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 + x]/(1 - x)^(11/2),x]

[Out]

(1 + x)^(3/2)/(9*(1 - x)^(9/2)) + (1 + x)^(3/2)/(21*(1 - x)^(7/2)) + (2*(1 + x)^(3/2))/(105*(1 - x)^(5/2)) + (
2*(1 + x)^(3/2))/(315*(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)^{11/2}} \, dx &=\frac{(1+x)^{3/2}}{9 (1-x)^{9/2}}+\frac{1}{3} \int \frac{\sqrt{1+x}}{(1-x)^{9/2}} \, dx\\ &=\frac{(1+x)^{3/2}}{9 (1-x)^{9/2}}+\frac{(1+x)^{3/2}}{21 (1-x)^{7/2}}+\frac{2}{21} \int \frac{\sqrt{1+x}}{(1-x)^{7/2}} \, dx\\ &=\frac{(1+x)^{3/2}}{9 (1-x)^{9/2}}+\frac{(1+x)^{3/2}}{21 (1-x)^{7/2}}+\frac{2 (1+x)^{3/2}}{105 (1-x)^{5/2}}+\frac{2}{105} \int \frac{\sqrt{1+x}}{(1-x)^{5/2}} \, dx\\ &=\frac{(1+x)^{3/2}}{9 (1-x)^{9/2}}+\frac{(1+x)^{3/2}}{21 (1-x)^{7/2}}+\frac{2 (1+x)^{3/2}}{105 (1-x)^{5/2}}+\frac{2 (1+x)^{3/2}}{315 (1-x)^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0135358, size = 35, normalized size = 0.43 \[ \frac{(x+1)^{3/2} \left (-2 x^3+12 x^2-33 x+58\right )}{315 (1-x)^{9/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((1 + x)^(3/2)*(58 - 33*x + 12*x^2 - 2*x^3))/(315*(1 - x)^(9/2))

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Maple [A]  time = 0.002, size = 30, normalized size = 0.4 \begin{align*} -{\frac{2\,{x}^{3}-12\,{x}^{2}+33\,x-58}{315} \left ( 1+x \right ) ^{{\frac{3}{2}}} \left ( 1-x \right ) ^{-{\frac{9}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/315*(1+x)^(3/2)*(2*x^3-12*x^2+33*x-58)/(1-x)^(9/2)

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Maxima [B]  time = 1.03188, size = 177, normalized size = 2.19 \begin{align*} -\frac{2 \, \sqrt{-x^{2} + 1}}{9 \,{\left (x^{5} - 5 \, x^{4} + 10 \, x^{3} - 10 \, x^{2} + 5 \, x - 1\right )}} - \frac{\sqrt{-x^{2} + 1}}{63 \,{\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} + \frac{\sqrt{-x^{2} + 1}}{105 \,{\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} - \frac{2 \, \sqrt{-x^{2} + 1}}{315 \,{\left (x^{2} - 2 \, x + 1\right )}} + \frac{2 \, \sqrt{-x^{2} + 1}}{315 \,{\left (x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/9*sqrt(-x^2 + 1)/(x^5 - 5*x^4 + 10*x^3 - 10*x^2 + 5*x - 1) - 1/63*sqrt(-x^2 + 1)/(x^4 - 4*x^3 + 6*x^2 - 4*x
+ 1) + 1/105*sqrt(-x^2 + 1)/(x^3 - 3*x^2 + 3*x - 1) - 2/315*sqrt(-x^2 + 1)/(x^2 - 2*x + 1) + 2/315*sqrt(-x^2
+ 1)/(x - 1)

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Fricas [A]  time = 1.54972, size = 224, normalized size = 2.77 \begin{align*} \frac{58 \, x^{5} - 290 \, x^{4} + 580 \, x^{3} - 580 \, x^{2} +{\left (2 \, x^{4} - 10 \, x^{3} + 21 \, x^{2} - 25 \, x - 58\right )} \sqrt{x + 1} \sqrt{-x + 1} + 290 \, x - 58}{315 \,{\left (x^{5} - 5 \, x^{4} + 10 \, x^{3} - 10 \, x^{2} + 5 \, x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/315*(58*x^5 - 290*x^4 + 580*x^3 - 580*x^2 + (2*x^4 - 10*x^3 + 21*x^2 - 25*x - 58)*sqrt(x + 1)*sqrt(-x + 1) +
290*x - 58)/(x^5 - 5*x^4 + 10*x^3 - 10*x^2 + 5*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+x)**(1/2)/(1-x)**(11/2),x)

[Out]

Timed out

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Giac [A]  time = 1.08433, size = 47, normalized size = 0.58 \begin{align*} \frac{{\left ({\left (2 \,{\left (x + 1\right )}{\left (x - 8\right )} + 63\right )}{\left (x + 1\right )} - 105\right )}{\left (x + 1\right )}^{\frac{3}{2}} \sqrt{-x + 1}}{315 \,{\left (x - 1\right )}^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/315*((2*(x + 1)*(x - 8) + 63)*(x + 1) - 105)*(x + 1)^(3/2)*sqrt(-x + 1)/(x - 1)^5