3.1084 \(\int \frac{(1+x)^{3/2}}{(1-x)^{9/2}} \, dx\)

Optimal. Leaf size=41 \[ \frac{(x+1)^{5/2}}{35 (1-x)^{5/2}}+\frac{(x+1)^{5/2}}{7 (1-x)^{7/2}} \]

[Out]

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

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Rubi [A]  time = 0.0041896, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {45, 37} \[ \frac{(x+1)^{5/2}}{35 (1-x)^{5/2}}+\frac{(x+1)^{5/2}}{7 (1-x)^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[(1 + x)^(3/2)/(1 - x)^(9/2),x]

[Out]

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

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

Mathematica [A]  time = 0.0107669, size = 23, normalized size = 0.56 \[ -\frac{(x-6) (x+1)^{5/2}}{35 (1-x)^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + x)^(3/2)/(1 - x)^(9/2),x]

[Out]

-((-6 + x)*(1 + x)^(5/2))/(35*(1 - x)^(7/2))

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Maple [A]  time = 0.002, size = 18, normalized size = 0.4 \begin{align*} -{\frac{x-6}{35} \left ( 1+x \right ) ^{{\frac{5}{2}}} \left ( 1-x \right ) ^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+x)^(3/2)/(1-x)^(9/2),x)

[Out]

-1/35*(1+x)^(5/2)*(x-6)/(1-x)^(7/2)

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Maxima [B]  time = 1.02331, size = 177, normalized size = 4.32 \begin{align*} -\frac{{\left (-x^{2} + 1\right )}^{\frac{3}{2}}}{2 \,{\left (x^{5} - 5 \, x^{4} + 10 \, x^{3} - 10 \, x^{2} + 5 \, x - 1\right )}} - \frac{3 \, \sqrt{-x^{2} + 1}}{7 \,{\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} - \frac{3 \, \sqrt{-x^{2} + 1}}{70 \,{\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} + \frac{\sqrt{-x^{2} + 1}}{35 \,{\left (x^{2} - 2 \, x + 1\right )}} - \frac{\sqrt{-x^{2} + 1}}{35 \,{\left (x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)^(3/2)/(1-x)^(9/2),x, algorithm="maxima")

[Out]

-1/2*(-x^2 + 1)^(3/2)/(x^5 - 5*x^4 + 10*x^3 - 10*x^2 + 5*x - 1) - 3/7*sqrt(-x^2 + 1)/(x^4 - 4*x^3 + 6*x^2 - 4*
x + 1) - 3/70*sqrt(-x^2 + 1)/(x^3 - 3*x^2 + 3*x - 1) + 1/35*sqrt(-x^2 + 1)/(x^2 - 2*x + 1) - 1/35*sqrt(-x^2 +
1)/(x - 1)

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Fricas [B]  time = 1.67713, size = 171, normalized size = 4.17 \begin{align*} \frac{6 \, x^{4} - 24 \, x^{3} + 36 \, x^{2} -{\left (x^{3} - 4 \, x^{2} - 11 \, x - 6\right )} \sqrt{x + 1} \sqrt{-x + 1} - 24 \, x + 6}{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+x)^(3/2)/(1-x)^(9/2),x, algorithm="fricas")

[Out]

1/35*(6*x^4 - 24*x^3 + 36*x^2 - (x^3 - 4*x^2 - 11*x - 6)*sqrt(x + 1)*sqrt(-x + 1) - 24*x + 6)/(x^4 - 4*x^3 + 6
*x^2 - 4*x + 1)

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Sympy [B]  time = 86.9206, size = 228, normalized size = 5.56 \begin{align*} \begin{cases} - \frac{i \left (x + 1\right )^{\frac{7}{2}}}{35 \sqrt{x - 1} \left (x + 1\right )^{3} - 210 \sqrt{x - 1} \left (x + 1\right )^{2} + 420 \sqrt{x - 1} \left (x + 1\right ) - 280 \sqrt{x - 1}} + \frac{7 i \left (x + 1\right )^{\frac{5}{2}}}{35 \sqrt{x - 1} \left (x + 1\right )^{3} - 210 \sqrt{x - 1} \left (x + 1\right )^{2} + 420 \sqrt{x - 1} \left (x + 1\right ) - 280 \sqrt{x - 1}} & \text{for}\: \frac{\left |{x + 1}\right |}{2} > 1 \\\frac{\left (x + 1\right )^{\frac{7}{2}}}{35 \sqrt{1 - x} \left (x + 1\right )^{3} - 210 \sqrt{1 - x} \left (x + 1\right )^{2} + 420 \sqrt{1 - x} \left (x + 1\right ) - 280 \sqrt{1 - x}} - \frac{7 \left (x + 1\right )^{\frac{5}{2}}}{35 \sqrt{1 - x} \left (x + 1\right )^{3} - 210 \sqrt{1 - x} \left (x + 1\right )^{2} + 420 \sqrt{1 - x} \left (x + 1\right ) - 280 \sqrt{1 - x}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)**(3/2)/(1-x)**(9/2),x)

[Out]

Piecewise((-I*(x + 1)**(7/2)/(35*sqrt(x - 1)*(x + 1)**3 - 210*sqrt(x - 1)*(x + 1)**2 + 420*sqrt(x - 1)*(x + 1)
- 280*sqrt(x - 1)) + 7*I*(x + 1)**(5/2)/(35*sqrt(x - 1)*(x + 1)**3 - 210*sqrt(x - 1)*(x + 1)**2 + 420*sqrt(x
- 1)*(x + 1) - 280*sqrt(x - 1)), Abs(x + 1)/2 > 1), ((x + 1)**(7/2)/(35*sqrt(1 - x)*(x + 1)**3 - 210*sqrt(1 -
x)*(x + 1)**2 + 420*sqrt(1 - x)*(x + 1) - 280*sqrt(1 - x)) - 7*(x + 1)**(5/2)/(35*sqrt(1 - x)*(x + 1)**3 - 210
*sqrt(1 - x)*(x + 1)**2 + 420*sqrt(1 - x)*(x + 1) - 280*sqrt(1 - x)), True))

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Giac [A]  time = 1.08455, size = 30, normalized size = 0.73 \begin{align*} -\frac{{\left (x + 1\right )}^{\frac{5}{2}}{\left (x - 6\right )} \sqrt{-x + 1}}{35 \,{\left (x - 1\right )}^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)^(3/2)/(1-x)^(9/2),x, algorithm="giac")

[Out]

-1/35*(x + 1)^(5/2)*(x - 6)*sqrt(-x + 1)/(x - 1)^4