3.1125 \(\int \frac {1}{(1-x)^{11/2} (1+x)^{3/2}} \, dx\)

Optimal. Leaf size=102 \[ \frac {8 x}{63 \sqrt {1-x} \sqrt {x+1}}+\frac {4}{63 (1-x)^{3/2} \sqrt {x+1}}+\frac {4}{63 (1-x)^{5/2} \sqrt {x+1}}+\frac {5}{63 (1-x)^{7/2} \sqrt {x+1}}+\frac {1}{9 (1-x)^{9/2} \sqrt {x+1}} \]

[Out]

1/9/(1-x)^(9/2)/(1+x)^(1/2)+5/63/(1-x)^(7/2)/(1+x)^(1/2)+4/63/(1-x)^(5/2)/(1+x)^(1/2)+4/63/(1-x)^(3/2)/(1+x)^(
1/2)+8/63*x/(1-x)^(1/2)/(1+x)^(1/2)

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Rubi [A]  time = 0.02, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {45, 39} \[ \frac {8 x}{63 \sqrt {1-x} \sqrt {x+1}}+\frac {4}{63 (1-x)^{3/2} \sqrt {x+1}}+\frac {4}{63 (1-x)^{5/2} \sqrt {x+1}}+\frac {5}{63 (1-x)^{7/2} \sqrt {x+1}}+\frac {1}{9 (1-x)^{9/2} \sqrt {x+1}} \]

Antiderivative was successfully verified.

[In]

Int[1/((1 - x)^(11/2)*(1 + x)^(3/2)),x]

[Out]

1/(9*(1 - x)^(9/2)*Sqrt[1 + x]) + 5/(63*(1 - x)^(7/2)*Sqrt[1 + x]) + 4/(63*(1 - x)^(5/2)*Sqrt[1 + x]) + 4/(63*
(1 - x)^(3/2)*Sqrt[1 + x]) + (8*x)/(63*Sqrt[1 - x]*Sqrt[1 + x])

Rule 39

Int[1/(((a_) + (b_.)*(x_))^(3/2)*((c_) + (d_.)*(x_))^(3/2)), x_Symbol] :> Simp[x/(a*c*Sqrt[a + b*x]*Sqrt[c + d
*x]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0]

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])

Rubi steps

\begin {align*} \int \frac {1}{(1-x)^{11/2} (1+x)^{3/2}} \, dx &=\frac {1}{9 (1-x)^{9/2} \sqrt {1+x}}+\frac {5}{9} \int \frac {1}{(1-x)^{9/2} (1+x)^{3/2}} \, dx\\ &=\frac {1}{9 (1-x)^{9/2} \sqrt {1+x}}+\frac {5}{63 (1-x)^{7/2} \sqrt {1+x}}+\frac {20}{63} \int \frac {1}{(1-x)^{7/2} (1+x)^{3/2}} \, dx\\ &=\frac {1}{9 (1-x)^{9/2} \sqrt {1+x}}+\frac {5}{63 (1-x)^{7/2} \sqrt {1+x}}+\frac {4}{63 (1-x)^{5/2} \sqrt {1+x}}+\frac {4}{21} \int \frac {1}{(1-x)^{5/2} (1+x)^{3/2}} \, dx\\ &=\frac {1}{9 (1-x)^{9/2} \sqrt {1+x}}+\frac {5}{63 (1-x)^{7/2} \sqrt {1+x}}+\frac {4}{63 (1-x)^{5/2} \sqrt {1+x}}+\frac {4}{63 (1-x)^{3/2} \sqrt {1+x}}+\frac {8}{63} \int \frac {1}{(1-x)^{3/2} (1+x)^{3/2}} \, dx\\ &=\frac {1}{9 (1-x)^{9/2} \sqrt {1+x}}+\frac {5}{63 (1-x)^{7/2} \sqrt {1+x}}+\frac {4}{63 (1-x)^{5/2} \sqrt {1+x}}+\frac {4}{63 (1-x)^{3/2} \sqrt {1+x}}+\frac {8 x}{63 \sqrt {1-x} \sqrt {1+x}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 45, normalized size = 0.44 \[ \frac {8 x^5-32 x^4+44 x^3-16 x^2-17 x+20}{63 (x-1)^4 \sqrt {1-x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((1 - x)^(11/2)*(1 + x)^(3/2)),x]

[Out]

(20 - 17*x - 16*x^2 + 44*x^3 - 32*x^4 + 8*x^5)/(63*(-1 + x)^4*Sqrt[1 - x^2])

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fricas [A]  time = 0.48, size = 91, normalized size = 0.89 \[ \frac {20 \, x^{6} - 80 \, x^{5} + 100 \, x^{4} - 100 \, x^{2} - {\left (8 \, x^{5} - 32 \, x^{4} + 44 \, x^{3} - 16 \, x^{2} - 17 \, x + 20\right )} \sqrt {x + 1} \sqrt {-x + 1} + 80 \, x - 20}{63 \, {\left (x^{6} - 4 \, x^{5} + 5 \, x^{4} - 5 \, x^{2} + 4 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)^(11/2)/(1+x)^(3/2),x, algorithm="fricas")

[Out]

1/63*(20*x^6 - 80*x^5 + 100*x^4 - 100*x^2 - (8*x^5 - 32*x^4 + 44*x^3 - 16*x^2 - 17*x + 20)*sqrt(x + 1)*sqrt(-x
+ 1) + 80*x - 20)/(x^6 - 4*x^5 + 5*x^4 - 5*x^2 + 4*x - 1)

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giac [A]  time = 0.68, size = 85, normalized size = 0.83 \[ \frac {\sqrt {2} - \sqrt {-x + 1}}{64 \, \sqrt {x + 1}} - \frac {\sqrt {x + 1}}{64 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}} - \frac {{\left ({\left ({\left ({\left (193 \, x - 1481\right )} {\left (x + 1\right )} + 5544\right )} {\left (x + 1\right )} - 8400\right )} {\left (x + 1\right )} + 5040\right )} \sqrt {x + 1} \sqrt {-x + 1}}{2016 \, {\left (x - 1\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)^(11/2)/(1+x)^(3/2),x, algorithm="giac")

[Out]

1/64*(sqrt(2) - sqrt(-x + 1))/sqrt(x + 1) - 1/64*sqrt(x + 1)/(sqrt(2) - sqrt(-x + 1)) - 1/2016*((((193*x - 148
1)*(x + 1) + 5544)*(x + 1) - 8400)*(x + 1) + 5040)*sqrt(x + 1)*sqrt(-x + 1)/(x - 1)^5

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maple [A]  time = 0.00, size = 40, normalized size = 0.39 \[ \frac {8 x^{5}-32 x^{4}+44 x^{3}-16 x^{2}-17 x +20}{63 \sqrt {x +1}\, \left (-x +1\right )^{\frac {9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-x+1)^(11/2)/(x+1)^(3/2),x)

[Out]

1/63*(8*x^5-32*x^4+44*x^3-16*x^2-17*x+20)/(x+1)^(1/2)/(-x+1)^(9/2)

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maxima [B]  time = 1.37, size = 201, normalized size = 1.97 \[ \frac {8 \, x}{63 \, \sqrt {-x^{2} + 1}} + \frac {1}{9 \, {\left (\sqrt {-x^{2} + 1} x^{4} - 4 \, \sqrt {-x^{2} + 1} x^{3} + 6 \, \sqrt {-x^{2} + 1} x^{2} - 4 \, \sqrt {-x^{2} + 1} x + \sqrt {-x^{2} + 1}\right )}} - \frac {5}{63 \, {\left (\sqrt {-x^{2} + 1} x^{3} - 3 \, \sqrt {-x^{2} + 1} x^{2} + 3 \, \sqrt {-x^{2} + 1} x - \sqrt {-x^{2} + 1}\right )}} + \frac {4}{63 \, {\left (\sqrt {-x^{2} + 1} x^{2} - 2 \, \sqrt {-x^{2} + 1} x + \sqrt {-x^{2} + 1}\right )}} - \frac {4}{63 \, {\left (\sqrt {-x^{2} + 1} x - \sqrt {-x^{2} + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)^(11/2)/(1+x)^(3/2),x, algorithm="maxima")

[Out]

8/63*x/sqrt(-x^2 + 1) + 1/9/(sqrt(-x^2 + 1)*x^4 - 4*sqrt(-x^2 + 1)*x^3 + 6*sqrt(-x^2 + 1)*x^2 - 4*sqrt(-x^2 +
1)*x + sqrt(-x^2 + 1)) - 5/63/(sqrt(-x^2 + 1)*x^3 - 3*sqrt(-x^2 + 1)*x^2 + 3*sqrt(-x^2 + 1)*x - sqrt(-x^2 + 1)
) + 4/63/(sqrt(-x^2 + 1)*x^2 - 2*sqrt(-x^2 + 1)*x + sqrt(-x^2 + 1)) - 4/63/(sqrt(-x^2 + 1)*x - sqrt(-x^2 + 1))

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mupad [B]  time = 0.36, size = 80, normalized size = 0.78 \[ \frac {17\,x\,\sqrt {1-x}-20\,\sqrt {1-x}+16\,x^2\,\sqrt {1-x}-44\,x^3\,\sqrt {1-x}+32\,x^4\,\sqrt {1-x}-8\,x^5\,\sqrt {1-x}}{63\,{\left (x-1\right )}^5\,\sqrt {x+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((1 - x)^(11/2)*(x + 1)^(3/2)),x)

[Out]

(17*x*(1 - x)^(1/2) - 20*(1 - x)^(1/2) + 16*x^2*(1 - x)^(1/2) - 44*x^3*(1 - x)^(1/2) + 32*x^4*(1 - x)^(1/2) -
8*x^5*(1 - x)^(1/2))/(63*(x - 1)^5*(x + 1)^(1/2))

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sympy [B]  time = 113.61, size = 592, normalized size = 5.80 \[ \begin {cases} \frac {8 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{5}}{- 5040 x - 63 \left (x + 1\right )^{5} + 630 \left (x + 1\right )^{4} - 2520 \left (x + 1\right )^{3} + 5040 \left (x + 1\right )^{2} - 3024} - \frac {72 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{4}}{- 5040 x - 63 \left (x + 1\right )^{5} + 630 \left (x + 1\right )^{4} - 2520 \left (x + 1\right )^{3} + 5040 \left (x + 1\right )^{2} - 3024} + \frac {252 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{3}}{- 5040 x - 63 \left (x + 1\right )^{5} + 630 \left (x + 1\right )^{4} - 2520 \left (x + 1\right )^{3} + 5040 \left (x + 1\right )^{2} - 3024} - \frac {420 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{2}}{- 5040 x - 63 \left (x + 1\right )^{5} + 630 \left (x + 1\right )^{4} - 2520 \left (x + 1\right )^{3} + 5040 \left (x + 1\right )^{2} - 3024} + \frac {315 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )}{- 5040 x - 63 \left (x + 1\right )^{5} + 630 \left (x + 1\right )^{4} - 2520 \left (x + 1\right )^{3} + 5040 \left (x + 1\right )^{2} - 3024} - \frac {63 \sqrt {-1 + \frac {2}{x + 1}}}{- 5040 x - 63 \left (x + 1\right )^{5} + 630 \left (x + 1\right )^{4} - 2520 \left (x + 1\right )^{3} + 5040 \left (x + 1\right )^{2} - 3024} & \text {for}\: \frac {2}{\left |{x + 1}\right |} > 1 \\\frac {8 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{5}}{- 5040 x - 63 \left (x + 1\right )^{5} + 630 \left (x + 1\right )^{4} - 2520 \left (x + 1\right )^{3} + 5040 \left (x + 1\right )^{2} - 3024} - \frac {72 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{4}}{- 5040 x - 63 \left (x + 1\right )^{5} + 630 \left (x + 1\right )^{4} - 2520 \left (x + 1\right )^{3} + 5040 \left (x + 1\right )^{2} - 3024} + \frac {252 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{3}}{- 5040 x - 63 \left (x + 1\right )^{5} + 630 \left (x + 1\right )^{4} - 2520 \left (x + 1\right )^{3} + 5040 \left (x + 1\right )^{2} - 3024} - \frac {420 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{2}}{- 5040 x - 63 \left (x + 1\right )^{5} + 630 \left (x + 1\right )^{4} - 2520 \left (x + 1\right )^{3} + 5040 \left (x + 1\right )^{2} - 3024} + \frac {315 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )}{- 5040 x - 63 \left (x + 1\right )^{5} + 630 \left (x + 1\right )^{4} - 2520 \left (x + 1\right )^{3} + 5040 \left (x + 1\right )^{2} - 3024} - \frac {63 i \sqrt {1 - \frac {2}{x + 1}}}{- 5040 x - 63 \left (x + 1\right )^{5} + 630 \left (x + 1\right )^{4} - 2520 \left (x + 1\right )^{3} + 5040 \left (x + 1\right )^{2} - 3024} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)**(11/2)/(1+x)**(3/2),x)

[Out]

Piecewise((8*sqrt(-1 + 2/(x + 1))*(x + 1)**5/(-5040*x - 63*(x + 1)**5 + 630*(x + 1)**4 - 2520*(x + 1)**3 + 504
0*(x + 1)**2 - 3024) - 72*sqrt(-1 + 2/(x + 1))*(x + 1)**4/(-5040*x - 63*(x + 1)**5 + 630*(x + 1)**4 - 2520*(x
+ 1)**3 + 5040*(x + 1)**2 - 3024) + 252*sqrt(-1 + 2/(x + 1))*(x + 1)**3/(-5040*x - 63*(x + 1)**5 + 630*(x + 1)
**4 - 2520*(x + 1)**3 + 5040*(x + 1)**2 - 3024) - 420*sqrt(-1 + 2/(x + 1))*(x + 1)**2/(-5040*x - 63*(x + 1)**5
+ 630*(x + 1)**4 - 2520*(x + 1)**3 + 5040*(x + 1)**2 - 3024) + 315*sqrt(-1 + 2/(x + 1))*(x + 1)/(-5040*x - 63
*(x + 1)**5 + 630*(x + 1)**4 - 2520*(x + 1)**3 + 5040*(x + 1)**2 - 3024) - 63*sqrt(-1 + 2/(x + 1))/(-5040*x -
63*(x + 1)**5 + 630*(x + 1)**4 - 2520*(x + 1)**3 + 5040*(x + 1)**2 - 3024), 2/Abs(x + 1) > 1), (8*I*sqrt(1 - 2
/(x + 1))*(x + 1)**5/(-5040*x - 63*(x + 1)**5 + 630*(x + 1)**4 - 2520*(x + 1)**3 + 5040*(x + 1)**2 - 3024) - 7
2*I*sqrt(1 - 2/(x + 1))*(x + 1)**4/(-5040*x - 63*(x + 1)**5 + 630*(x + 1)**4 - 2520*(x + 1)**3 + 5040*(x + 1)*
*2 - 3024) + 252*I*sqrt(1 - 2/(x + 1))*(x + 1)**3/(-5040*x - 63*(x + 1)**5 + 630*(x + 1)**4 - 2520*(x + 1)**3
+ 5040*(x + 1)**2 - 3024) - 420*I*sqrt(1 - 2/(x + 1))*(x + 1)**2/(-5040*x - 63*(x + 1)**5 + 630*(x + 1)**4 - 2
520*(x + 1)**3 + 5040*(x + 1)**2 - 3024) + 315*I*sqrt(1 - 2/(x + 1))*(x + 1)/(-5040*x - 63*(x + 1)**5 + 630*(x
+ 1)**4 - 2520*(x + 1)**3 + 5040*(x + 1)**2 - 3024) - 63*I*sqrt(1 - 2/(x + 1))/(-5040*x - 63*(x + 1)**5 + 630
*(x + 1)**4 - 2520*(x + 1)**3 + 5040*(x + 1)**2 - 3024), True))

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