3.1072 \(\int \frac{\sqrt{1+x}}{(1-x)^{9/2}} \, dx\)

Optimal. Leaf size=61 \[ \frac{2 (x+1)^{3/2}}{105 (1-x)^{3/2}}+\frac{2 (x+1)^{3/2}}{35 (1-x)^{5/2}}+\frac{(x+1)^{3/2}}{7 (1-x)^{7/2}} \]

[Out]

(1 + x)^(3/2)/(7*(1 - x)^(7/2)) + (2*(1 + x)^(3/2))/(35*(1 - x)^(5/2)) + (2*(1 + x)^(3/2))/(105*(1 - x)^(3/2))

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Rubi [A]  time = 0.0082057, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {45, 37} \[ \frac{2 (x+1)^{3/2}}{105 (1-x)^{3/2}}+\frac{2 (x+1)^{3/2}}{35 (1-x)^{5/2}}+\frac{(x+1)^{3/2}}{7 (1-x)^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 + x]/(1 - x)^(9/2),x]

[Out]

(1 + x)^(3/2)/(7*(1 - x)^(7/2)) + (2*(1 + x)^(3/2))/(35*(1 - x)^(5/2)) + (2*(1 + x)^(3/2))/(105*(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)^{9/2}} \, dx &=\frac{(1+x)^{3/2}}{7 (1-x)^{7/2}}+\frac{2}{7} \int \frac{\sqrt{1+x}}{(1-x)^{7/2}} \, dx\\ &=\frac{(1+x)^{3/2}}{7 (1-x)^{7/2}}+\frac{2 (1+x)^{3/2}}{35 (1-x)^{5/2}}+\frac{2}{35} \int \frac{\sqrt{1+x}}{(1-x)^{5/2}} \, dx\\ &=\frac{(1+x)^{3/2}}{7 (1-x)^{7/2}}+\frac{2 (1+x)^{3/2}}{35 (1-x)^{5/2}}+\frac{2 (1+x)^{3/2}}{105 (1-x)^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0114946, size = 30, normalized size = 0.49 \[ \frac{(x+1)^{3/2} \left (2 x^2-10 x+23\right )}{105 (1-x)^{7/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((1 + x)^(3/2)*(23 - 10*x + 2*x^2))/(105*(1 - x)^(7/2))

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Maple [A]  time = 0.004, size = 25, normalized size = 0.4 \begin{align*}{\frac{2\,{x}^{2}-10\,x+23}{105} \left ( 1+x \right ) ^{{\frac{3}{2}}} \left ( 1-x \right ) ^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/105*(1+x)^(3/2)*(2*x^2-10*x+23)/(1-x)^(7/2)

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Maxima [B]  time = 1.03258, size = 128, normalized size = 2.1 \begin{align*} \frac{2 \, \sqrt{-x^{2} + 1}}{7 \,{\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} + \frac{\sqrt{-x^{2} + 1}}{35 \,{\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} - \frac{2 \, \sqrt{-x^{2} + 1}}{105 \,{\left (x^{2} - 2 \, x + 1\right )}} + \frac{2 \, \sqrt{-x^{2} + 1}}{105 \,{\left (x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/7*sqrt(-x^2 + 1)/(x^4 - 4*x^3 + 6*x^2 - 4*x + 1) + 1/35*sqrt(-x^2 + 1)/(x^3 - 3*x^2 + 3*x - 1) - 2/105*sqrt(
-x^2 + 1)/(x^2 - 2*x + 1) + 2/105*sqrt(-x^2 + 1)/(x - 1)

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Fricas [A]  time = 1.52901, size = 181, normalized size = 2.97 \begin{align*} \frac{23 \, x^{4} - 92 \, x^{3} + 138 \, x^{2} +{\left (2 \, x^{3} - 8 \, x^{2} + 13 \, x + 23\right )} \sqrt{x + 1} \sqrt{-x + 1} - 92 \, x + 23}{105 \,{\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/105*(23*x^4 - 92*x^3 + 138*x^2 + (2*x^3 - 8*x^2 + 13*x + 23)*sqrt(x + 1)*sqrt(-x + 1) - 92*x + 23)/(x^4 - 4*
x^3 + 6*x^2 - 4*x + 1)

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Sympy [B]  time = 87.1587, size = 568, normalized size = 9.31 \begin{align*} \begin{cases} \frac{2 i \left (x + 1\right )^{\frac{9}{2}}}{105 \sqrt{x - 1} \left (x + 1\right )^{4} - 840 \sqrt{x - 1} \left (x + 1\right )^{3} + 2520 \sqrt{x - 1} \left (x + 1\right )^{2} - 3360 \sqrt{x - 1} \left (x + 1\right ) + 1680 \sqrt{x - 1}} - \frac{18 i \left (x + 1\right )^{\frac{7}{2}}}{105 \sqrt{x - 1} \left (x + 1\right )^{4} - 840 \sqrt{x - 1} \left (x + 1\right )^{3} + 2520 \sqrt{x - 1} \left (x + 1\right )^{2} - 3360 \sqrt{x - 1} \left (x + 1\right ) + 1680 \sqrt{x - 1}} + \frac{63 i \left (x + 1\right )^{\frac{5}{2}}}{105 \sqrt{x - 1} \left (x + 1\right )^{4} - 840 \sqrt{x - 1} \left (x + 1\right )^{3} + 2520 \sqrt{x - 1} \left (x + 1\right )^{2} - 3360 \sqrt{x - 1} \left (x + 1\right ) + 1680 \sqrt{x - 1}} - \frac{70 i \left (x + 1\right )^{\frac{3}{2}}}{105 \sqrt{x - 1} \left (x + 1\right )^{4} - 840 \sqrt{x - 1} \left (x + 1\right )^{3} + 2520 \sqrt{x - 1} \left (x + 1\right )^{2} - 3360 \sqrt{x - 1} \left (x + 1\right ) + 1680 \sqrt{x - 1}} & \text{for}\: \frac{\left |{x + 1}\right |}{2} > 1 \\- \frac{2 \left (x + 1\right )^{\frac{9}{2}}}{105 \sqrt{1 - x} \left (x + 1\right )^{4} - 840 \sqrt{1 - x} \left (x + 1\right )^{3} + 2520 \sqrt{1 - x} \left (x + 1\right )^{2} - 3360 \sqrt{1 - x} \left (x + 1\right ) + 1680 \sqrt{1 - x}} + \frac{18 \left (x + 1\right )^{\frac{7}{2}}}{105 \sqrt{1 - x} \left (x + 1\right )^{4} - 840 \sqrt{1 - x} \left (x + 1\right )^{3} + 2520 \sqrt{1 - x} \left (x + 1\right )^{2} - 3360 \sqrt{1 - x} \left (x + 1\right ) + 1680 \sqrt{1 - x}} - \frac{63 \left (x + 1\right )^{\frac{5}{2}}}{105 \sqrt{1 - x} \left (x + 1\right )^{4} - 840 \sqrt{1 - x} \left (x + 1\right )^{3} + 2520 \sqrt{1 - x} \left (x + 1\right )^{2} - 3360 \sqrt{1 - x} \left (x + 1\right ) + 1680 \sqrt{1 - x}} + \frac{70 \left (x + 1\right )^{\frac{3}{2}}}{105 \sqrt{1 - x} \left (x + 1\right )^{4} - 840 \sqrt{1 - x} \left (x + 1\right )^{3} + 2520 \sqrt{1 - x} \left (x + 1\right )^{2} - 3360 \sqrt{1 - x} \left (x + 1\right ) + 1680 \sqrt{1 - x}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)**(1/2)/(1-x)**(9/2),x)

[Out]

Piecewise((2*I*(x + 1)**(9/2)/(105*sqrt(x - 1)*(x + 1)**4 - 840*sqrt(x - 1)*(x + 1)**3 + 2520*sqrt(x - 1)*(x +
1)**2 - 3360*sqrt(x - 1)*(x + 1) + 1680*sqrt(x - 1)) - 18*I*(x + 1)**(7/2)/(105*sqrt(x - 1)*(x + 1)**4 - 840*
sqrt(x - 1)*(x + 1)**3 + 2520*sqrt(x - 1)*(x + 1)**2 - 3360*sqrt(x - 1)*(x + 1) + 1680*sqrt(x - 1)) + 63*I*(x
+ 1)**(5/2)/(105*sqrt(x - 1)*(x + 1)**4 - 840*sqrt(x - 1)*(x + 1)**3 + 2520*sqrt(x - 1)*(x + 1)**2 - 3360*sqrt
(x - 1)*(x + 1) + 1680*sqrt(x - 1)) - 70*I*(x + 1)**(3/2)/(105*sqrt(x - 1)*(x + 1)**4 - 840*sqrt(x - 1)*(x + 1
)**3 + 2520*sqrt(x - 1)*(x + 1)**2 - 3360*sqrt(x - 1)*(x + 1) + 1680*sqrt(x - 1)), Abs(x + 1)/2 > 1), (-2*(x +
1)**(9/2)/(105*sqrt(1 - x)*(x + 1)**4 - 840*sqrt(1 - x)*(x + 1)**3 + 2520*sqrt(1 - x)*(x + 1)**2 - 3360*sqrt(
1 - x)*(x + 1) + 1680*sqrt(1 - x)) + 18*(x + 1)**(7/2)/(105*sqrt(1 - x)*(x + 1)**4 - 840*sqrt(1 - x)*(x + 1)**
3 + 2520*sqrt(1 - x)*(x + 1)**2 - 3360*sqrt(1 - x)*(x + 1) + 1680*sqrt(1 - x)) - 63*(x + 1)**(5/2)/(105*sqrt(1
- x)*(x + 1)**4 - 840*sqrt(1 - x)*(x + 1)**3 + 2520*sqrt(1 - x)*(x + 1)**2 - 3360*sqrt(1 - x)*(x + 1) + 1680*
sqrt(1 - x)) + 70*(x + 1)**(3/2)/(105*sqrt(1 - x)*(x + 1)**4 - 840*sqrt(1 - x)*(x + 1)**3 + 2520*sqrt(1 - x)*(
x + 1)**2 - 3360*sqrt(1 - x)*(x + 1) + 1680*sqrt(1 - x)), True))

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Giac [A]  time = 1.07231, size = 39, normalized size = 0.64 \begin{align*} \frac{{\left (2 \,{\left (x + 1\right )}{\left (x - 6\right )} + 35\right )}{\left (x + 1\right )}^{\frac{3}{2}} \sqrt{-x + 1}}{105 \,{\left (x - 1\right )}^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/105*(2*(x + 1)*(x - 6) + 35)*(x + 1)^(3/2)*sqrt(-x + 1)/(x - 1)^4