3.1099 \(\int \frac{(1+x)^{5/2}}{(1-x)^{11/2}} \, dx\)

Optimal. Leaf size=41 \[ \frac{(x+1)^{7/2}}{63 (1-x)^{7/2}}+\frac{(x+1)^{7/2}}{9 (1-x)^{9/2}} \]

[Out]

(1 + x)^(7/2)/(9*(1 - x)^(9/2)) + (1 + x)^(7/2)/(63*(1 - x)^(7/2))

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Rubi [A]  time = 0.0043577, 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} \[ \frac{(x+1)^{7/2}}{63 (1-x)^{7/2}}+\frac{(x+1)^{7/2}}{9 (1-x)^{9/2}} \]

Antiderivative was successfully verified.

[In]

Int[(1 + x)^(5/2)/(1 - x)^(11/2),x]

[Out]

(1 + x)^(7/2)/(9*(1 - x)^(9/2)) + (1 + x)^(7/2)/(63*(1 - x)^(7/2))

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)^{5/2}}{(1-x)^{11/2}} \, dx &=\frac{(1+x)^{7/2}}{9 (1-x)^{9/2}}+\frac{1}{9} \int \frac{(1+x)^{5/2}}{(1-x)^{9/2}} \, dx\\ &=\frac{(1+x)^{7/2}}{9 (1-x)^{9/2}}+\frac{(1+x)^{7/2}}{63 (1-x)^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.0148399, size = 23, normalized size = 0.56 \[ -\frac{(x-8) (x+1)^{7/2}}{63 (1-x)^{9/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + x)^(5/2)/(1 - x)^(11/2),x]

[Out]

-((-8 + x)*(1 + x)^(7/2))/(63*(1 - x)^(9/2))

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Maple [A]  time = 0.002, size = 18, normalized size = 0.4 \begin{align*} -{\frac{x-8}{63} \left ( 1+x \right ) ^{{\frac{7}{2}}} \left ( 1-x \right ) ^{-{\frac{9}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+x)^(5/2)/(1-x)^(11/2),x)

[Out]

-1/63*(1+x)^(7/2)*(x-8)/(1-x)^(9/2)

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Maxima [B]  time = 1.02933, size = 294, normalized size = 7.17 \begin{align*} -\frac{{\left (-x^{2} + 1\right )}^{\frac{5}{2}}}{2 \,{\left (x^{7} - 7 \, x^{6} + 21 \, x^{5} - 35 \, x^{4} + 35 \, x^{3} - 21 \, x^{2} + 7 \, x - 1\right )}} - \frac{5 \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}}}{6 \,{\left (x^{6} - 6 \, x^{5} + 15 \, x^{4} - 20 \, x^{3} + 15 \, x^{2} - 6 \, x + 1\right )}} - \frac{5 \, \sqrt{-x^{2} + 1}}{9 \,{\left (x^{5} - 5 \, x^{4} + 10 \, x^{3} - 10 \, x^{2} + 5 \, x - 1\right )}} - \frac{5 \, \sqrt{-x^{2} + 1}}{126 \,{\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} + \frac{\sqrt{-x^{2} + 1}}{42 \,{\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} - \frac{\sqrt{-x^{2} + 1}}{63 \,{\left (x^{2} - 2 \, x + 1\right )}} + \frac{\sqrt{-x^{2} + 1}}{63 \,{\left (x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)^(5/2)/(1-x)^(11/2),x, algorithm="maxima")

[Out]

-1/2*(-x^2 + 1)^(5/2)/(x^7 - 7*x^6 + 21*x^5 - 35*x^4 + 35*x^3 - 21*x^2 + 7*x - 1) - 5/6*(-x^2 + 1)^(3/2)/(x^6
- 6*x^5 + 15*x^4 - 20*x^3 + 15*x^2 - 6*x + 1) - 5/9*sqrt(-x^2 + 1)/(x^5 - 5*x^4 + 10*x^3 - 10*x^2 + 5*x - 1) -
5/126*sqrt(-x^2 + 1)/(x^4 - 4*x^3 + 6*x^2 - 4*x + 1) + 1/42*sqrt(-x^2 + 1)/(x^3 - 3*x^2 + 3*x - 1) - 1/63*sqr
t(-x^2 + 1)/(x^2 - 2*x + 1) + 1/63*sqrt(-x^2 + 1)/(x - 1)

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Fricas [B]  time = 1.67573, size = 209, normalized size = 5.1 \begin{align*} \frac{8 \, x^{5} - 40 \, x^{4} + 80 \, x^{3} - 80 \, x^{2} +{\left (x^{4} - 5 \, x^{3} - 21 \, x^{2} - 23 \, x - 8\right )} \sqrt{x + 1} \sqrt{-x + 1} + 40 \, x - 8}{63 \,{\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)^(5/2)/(1-x)^(11/2),x, algorithm="fricas")

[Out]

1/63*(8*x^5 - 40*x^4 + 80*x^3 - 80*x^2 + (x^4 - 5*x^3 - 21*x^2 - 23*x - 8)*sqrt(x + 1)*sqrt(-x + 1) + 40*x - 8
)/(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)**(5/2)/(1-x)**(11/2),x)

[Out]

Timed out

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Giac [A]  time = 1.09704, size = 30, normalized size = 0.73 \begin{align*} \frac{{\left (x + 1\right )}^{\frac{7}{2}}{\left (x - 8\right )} \sqrt{-x + 1}}{63 \,{\left (x - 1\right )}^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)^(5/2)/(1-x)^(11/2),x, algorithm="giac")

[Out]

1/63*(x + 1)^(7/2)*(x - 8)*sqrt(-x + 1)/(x - 1)^5