3.1114 \(\int \frac{1}{(1-x)^{9/2} \sqrt{1+x}} \, dx\)

Optimal. Leaf size=81 \[ \frac{2 \sqrt{x+1}}{35 \sqrt{1-x}}+\frac{2 \sqrt{x+1}}{35 (1-x)^{3/2}}+\frac{3 \sqrt{x+1}}{35 (1-x)^{5/2}}+\frac{\sqrt{x+1}}{7 (1-x)^{7/2}} \]

[Out]

Sqrt[1 + x]/(7*(1 - x)^(7/2)) + (3*Sqrt[1 + x])/(35*(1 - x)^(5/2)) + (2*Sqrt[1 + x])/(35*(1 - x)^(3/2)) + (2*S
qrt[1 + x])/(35*Sqrt[1 - x])

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Rubi [A]  time = 0.0138147, 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} \[ \frac{2 \sqrt{x+1}}{35 \sqrt{1-x}}+\frac{2 \sqrt{x+1}}{35 (1-x)^{3/2}}+\frac{3 \sqrt{x+1}}{35 (1-x)^{5/2}}+\frac{\sqrt{x+1}}{7 (1-x)^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/((1 - x)^(9/2)*Sqrt[1 + x]),x]

[Out]

Sqrt[1 + x]/(7*(1 - x)^(7/2)) + (3*Sqrt[1 + x])/(35*(1 - x)^(5/2)) + (2*Sqrt[1 + x])/(35*(1 - x)^(3/2)) + (2*S
qrt[1 + x])/(35*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 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{1}{(1-x)^{9/2} \sqrt{1+x}} \, dx &=\frac{\sqrt{1+x}}{7 (1-x)^{7/2}}+\frac{3}{7} \int \frac{1}{(1-x)^{7/2} \sqrt{1+x}} \, dx\\ &=\frac{\sqrt{1+x}}{7 (1-x)^{7/2}}+\frac{3 \sqrt{1+x}}{35 (1-x)^{5/2}}+\frac{6}{35} \int \frac{1}{(1-x)^{5/2} \sqrt{1+x}} \, dx\\ &=\frac{\sqrt{1+x}}{7 (1-x)^{7/2}}+\frac{3 \sqrt{1+x}}{35 (1-x)^{5/2}}+\frac{2 \sqrt{1+x}}{35 (1-x)^{3/2}}+\frac{2}{35} \int \frac{1}{(1-x)^{3/2} \sqrt{1+x}} \, dx\\ &=\frac{\sqrt{1+x}}{7 (1-x)^{7/2}}+\frac{3 \sqrt{1+x}}{35 (1-x)^{5/2}}+\frac{2 \sqrt{1+x}}{35 (1-x)^{3/2}}+\frac{2 \sqrt{1+x}}{35 \sqrt{1-x}}\\ \end{align*}

Mathematica [A]  time = 0.010828, size = 35, normalized size = 0.43 \[ \frac{\sqrt{x+1} \left (-2 x^3+8 x^2-13 x+12\right )}{35 (1-x)^{7/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 + x]*(12 - 13*x + 8*x^2 - 2*x^3))/(35*(1 - x)^(7/2))

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Maple [A]  time = 0.003, size = 30, normalized size = 0.4 \begin{align*} -{\frac{2\,{x}^{3}-8\,{x}^{2}+13\,x-12}{35}\sqrt{1+x} \left ( 1-x \right ) ^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/35*(1+x)^(1/2)*(2*x^3-8*x^2+13*x-12)/(1-x)^(7/2)

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Maxima [A]  time = 1.49483, size = 128, normalized size = 1.58 \begin{align*} \frac{\sqrt{-x^{2} + 1}}{7 \,{\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} - \frac{3 \, \sqrt{-x^{2} + 1}}{35 \,{\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} + \frac{2 \, \sqrt{-x^{2} + 1}}{35 \,{\left (x^{2} - 2 \, x + 1\right )}} - \frac{2 \, \sqrt{-x^{2} + 1}}{35 \,{\left (x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.83633, size = 178, normalized size = 2.2 \begin{align*} \frac{12 \, x^{4} - 48 \, x^{3} + 72 \, x^{2} -{\left (2 \, x^{3} - 8 \, x^{2} + 13 \, x - 12\right )} \sqrt{x + 1} \sqrt{-x + 1} - 48 \, x + 12}{35 \,{\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/(1-x)^(9/2)/(1+x)^(1/2),x, algorithm="fricas")

[Out]

1/35*(12*x^4 - 48*x^3 + 72*x^2 - (2*x^3 - 8*x^2 + 13*x - 12)*sqrt(x + 1)*sqrt(-x + 1) - 48*x + 12)/(x^4 - 4*x^
3 + 6*x^2 - 4*x + 1)

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Sympy [B]  time = 126.932, size = 541, normalized size = 6.68 \begin{align*} \begin{cases} \frac{2 \left (x + 1\right )^{3}}{35 \sqrt{-1 + \frac{2}{x + 1}} \left (x + 1\right )^{3} - 210 \sqrt{-1 + \frac{2}{x + 1}} \left (x + 1\right )^{2} + 420 \sqrt{-1 + \frac{2}{x + 1}} \left (x + 1\right ) - 280 \sqrt{-1 + \frac{2}{x + 1}}} - \frac{14 \left (x + 1\right )^{2}}{35 \sqrt{-1 + \frac{2}{x + 1}} \left (x + 1\right )^{3} - 210 \sqrt{-1 + \frac{2}{x + 1}} \left (x + 1\right )^{2} + 420 \sqrt{-1 + \frac{2}{x + 1}} \left (x + 1\right ) - 280 \sqrt{-1 + \frac{2}{x + 1}}} + \frac{35 \left (x + 1\right )}{35 \sqrt{-1 + \frac{2}{x + 1}} \left (x + 1\right )^{3} - 210 \sqrt{-1 + \frac{2}{x + 1}} \left (x + 1\right )^{2} + 420 \sqrt{-1 + \frac{2}{x + 1}} \left (x + 1\right ) - 280 \sqrt{-1 + \frac{2}{x + 1}}} - \frac{35}{35 \sqrt{-1 + \frac{2}{x + 1}} \left (x + 1\right )^{3} - 210 \sqrt{-1 + \frac{2}{x + 1}} \left (x + 1\right )^{2} + 420 \sqrt{-1 + \frac{2}{x + 1}} \left (x + 1\right ) - 280 \sqrt{-1 + \frac{2}{x + 1}}} & \text{for}\: \frac{2}{\left |{x + 1}\right |} > 1 \\- \frac{2 i \left (x + 1\right )^{3}}{35 \sqrt{1 - \frac{2}{x + 1}} \left (x + 1\right )^{3} - 210 \sqrt{1 - \frac{2}{x + 1}} \left (x + 1\right )^{2} + 420 \sqrt{1 - \frac{2}{x + 1}} \left (x + 1\right ) - 280 \sqrt{1 - \frac{2}{x + 1}}} + \frac{14 i \left (x + 1\right )^{2}}{35 \sqrt{1 - \frac{2}{x + 1}} \left (x + 1\right )^{3} - 210 \sqrt{1 - \frac{2}{x + 1}} \left (x + 1\right )^{2} + 420 \sqrt{1 - \frac{2}{x + 1}} \left (x + 1\right ) - 280 \sqrt{1 - \frac{2}{x + 1}}} - \frac{35 i \left (x + 1\right )}{35 \sqrt{1 - \frac{2}{x + 1}} \left (x + 1\right )^{3} - 210 \sqrt{1 - \frac{2}{x + 1}} \left (x + 1\right )^{2} + 420 \sqrt{1 - \frac{2}{x + 1}} \left (x + 1\right ) - 280 \sqrt{1 - \frac{2}{x + 1}}} + \frac{35 i}{35 \sqrt{1 - \frac{2}{x + 1}} \left (x + 1\right )^{3} - 210 \sqrt{1 - \frac{2}{x + 1}} \left (x + 1\right )^{2} + 420 \sqrt{1 - \frac{2}{x + 1}} \left (x + 1\right ) - 280 \sqrt{1 - \frac{2}{x + 1}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((2*(x + 1)**3/(35*sqrt(-1 + 2/(x + 1))*(x + 1)**3 - 210*sqrt(-1 + 2/(x + 1))*(x + 1)**2 + 420*sqrt(-
1 + 2/(x + 1))*(x + 1) - 280*sqrt(-1 + 2/(x + 1))) - 14*(x + 1)**2/(35*sqrt(-1 + 2/(x + 1))*(x + 1)**3 - 210*s
qrt(-1 + 2/(x + 1))*(x + 1)**2 + 420*sqrt(-1 + 2/(x + 1))*(x + 1) - 280*sqrt(-1 + 2/(x + 1))) + 35*(x + 1)/(35
*sqrt(-1 + 2/(x + 1))*(x + 1)**3 - 210*sqrt(-1 + 2/(x + 1))*(x + 1)**2 + 420*sqrt(-1 + 2/(x + 1))*(x + 1) - 28
0*sqrt(-1 + 2/(x + 1))) - 35/(35*sqrt(-1 + 2/(x + 1))*(x + 1)**3 - 210*sqrt(-1 + 2/(x + 1))*(x + 1)**2 + 420*s
qrt(-1 + 2/(x + 1))*(x + 1) - 280*sqrt(-1 + 2/(x + 1))), 2/Abs(x + 1) > 1), (-2*I*(x + 1)**3/(35*sqrt(1 - 2/(x
+ 1))*(x + 1)**3 - 210*sqrt(1 - 2/(x + 1))*(x + 1)**2 + 420*sqrt(1 - 2/(x + 1))*(x + 1) - 280*sqrt(1 - 2/(x +
1))) + 14*I*(x + 1)**2/(35*sqrt(1 - 2/(x + 1))*(x + 1)**3 - 210*sqrt(1 - 2/(x + 1))*(x + 1)**2 + 420*sqrt(1 -
2/(x + 1))*(x + 1) - 280*sqrt(1 - 2/(x + 1))) - 35*I*(x + 1)/(35*sqrt(1 - 2/(x + 1))*(x + 1)**3 - 210*sqrt(1
- 2/(x + 1))*(x + 1)**2 + 420*sqrt(1 - 2/(x + 1))*(x + 1) - 280*sqrt(1 - 2/(x + 1))) + 35*I/(35*sqrt(1 - 2/(x
+ 1))*(x + 1)**3 - 210*sqrt(1 - 2/(x + 1))*(x + 1)**2 + 420*sqrt(1 - 2/(x + 1))*(x + 1) - 280*sqrt(1 - 2/(x +
1))), True))

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Giac [A]  time = 1.08594, size = 47, normalized size = 0.58 \begin{align*} -\frac{{\left ({\left (2 \,{\left (x + 1\right )}{\left (x - 6\right )} + 35\right )}{\left (x + 1\right )} - 35\right )} \sqrt{x + 1} \sqrt{-x + 1}}{35 \,{\left (x - 1\right )}^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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