3.1098 \(\int \frac{(1+x)^{5/2}}{(1-x)^{9/2}} \, dx\)

Optimal. Leaf size=20 \[ \frac{(x+1)^{7/2}}{7 (1-x)^{7/2}} \]

[Out]

(1 + x)^(7/2)/(7*(1 - x)^(7/2))

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Rubi [A]  time = 0.0017708, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {37} \[ \frac{(x+1)^{7/2}}{7 (1-x)^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[(1 + x)^(5/2)/(1 - x)^(9/2),x]

[Out]

(1 + x)^(7/2)/(7*(1 - x)^(7/2))

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

Mathematica [A]  time = 0.007489, size = 20, normalized size = 1. \[ \frac{(x+1)^{7/2}}{7 (1-x)^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + x)^(5/2)/(1 - x)^(9/2),x]

[Out]

(1 + x)^(7/2)/(7*(1 - x)^(7/2))

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Maple [A]  time = 0.001, size = 15, normalized size = 0.8 \begin{align*}{\frac{1}{7} \left ( 1+x \right ) ^{{\frac{7}{2}}} \left ( 1-x \right ) ^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+x)^(5/2)/(1-x)^(9/2),x)

[Out]

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

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Maxima [B]  time = 1.02651, size = 231, normalized size = 11.55 \begin{align*} \frac{{\left (-x^{2} + 1\right )}^{\frac{5}{2}}}{x^{6} - 6 \, x^{5} + 15 \, x^{4} - 20 \, x^{3} + 15 \, x^{2} - 6 \, x + 1} + \frac{5 \,{\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{15 \, \sqrt{-x^{2} + 1}}{7 \,{\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} + \frac{3 \, \sqrt{-x^{2} + 1}}{14 \,{\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} - \frac{\sqrt{-x^{2} + 1}}{7 \,{\left (x^{2} - 2 \, x + 1\right )}} + \frac{\sqrt{-x^{2} + 1}}{7 \,{\left (x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)^(5/2)/(1-x)^(9/2),x, algorithm="maxima")

[Out]

(-x^2 + 1)^(5/2)/(x^6 - 6*x^5 + 15*x^4 - 20*x^3 + 15*x^2 - 6*x + 1) + 5/2*(-x^2 + 1)^(3/2)/(x^5 - 5*x^4 + 10*x
^3 - 10*x^2 + 5*x - 1) + 15/7*sqrt(-x^2 + 1)/(x^4 - 4*x^3 + 6*x^2 - 4*x + 1) + 3/14*sqrt(-x^2 + 1)/(x^3 - 3*x^
2 + 3*x - 1) - 1/7*sqrt(-x^2 + 1)/(x^2 - 2*x + 1) + 1/7*sqrt(-x^2 + 1)/(x - 1)

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Fricas [B]  time = 1.5855, size = 162, normalized size = 8.1 \begin{align*} \frac{x^{4} - 4 \, x^{3} + 6 \, x^{2} +{\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )} \sqrt{x + 1} \sqrt{-x + 1} - 4 \, x + 1}{7 \,{\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)^(5/2)/(1-x)^(9/2),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 86.5537, size = 116, normalized size = 5.8 \begin{align*} \begin{cases} \frac{i \left (x + 1\right )^{\frac{7}{2}}}{7 \sqrt{x - 1} \left (x + 1\right )^{3} - 42 \sqrt{x - 1} \left (x + 1\right )^{2} + 84 \sqrt{x - 1} \left (x + 1\right ) - 56 \sqrt{x - 1}} & \text{for}\: \frac{\left |{x + 1}\right |}{2} > 1 \\- \frac{\left (x + 1\right )^{\frac{7}{2}}}{7 \sqrt{1 - x} \left (x + 1\right )^{3} - 42 \sqrt{1 - x} \left (x + 1\right )^{2} + 84 \sqrt{1 - x} \left (x + 1\right ) - 56 \sqrt{1 - x}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)**(5/2)/(1-x)**(9/2),x)

[Out]

Piecewise((I*(x + 1)**(7/2)/(7*sqrt(x - 1)*(x + 1)**3 - 42*sqrt(x - 1)*(x + 1)**2 + 84*sqrt(x - 1)*(x + 1) - 5
6*sqrt(x - 1)), Abs(x + 1)/2 > 1), (-(x + 1)**(7/2)/(7*sqrt(1 - x)*(x + 1)**3 - 42*sqrt(1 - x)*(x + 1)**2 + 84
*sqrt(1 - x)*(x + 1) - 56*sqrt(1 - x)), True))

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Giac [A]  time = 1.08752, size = 26, normalized size = 1.3 \begin{align*} \frac{{\left (x + 1\right )}^{\frac{7}{2}} \sqrt{-x + 1}}{7 \,{\left (x - 1\right )}^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)^(5/2)/(1-x)^(9/2),x, algorithm="giac")

[Out]

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