3.12.25 \(\int \frac {1}{(1-x)^{11/2} (1+x)^{3/2}} \, dx\) [1125]

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

[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 = 0.01, 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 = {47, 39} \begin {gather*} \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}} \end {gather*}

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 47

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 = 0.07, size = 45, normalized size = 0.44 \begin {gather*} \frac {20-17 x-16 x^2+44 x^3-32 x^4+8 x^5}{63 (-1+x)^4 \sqrt {1-x^2}} \end {gather*}

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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Maple [A]
time = 0.14, size = 86, normalized size = 0.84

method	result	size
gosper	\(\frac {8 x^{5}-32 x^{4}+44 x^{3}-16 x^{2}-17 x +20}{63 \sqrt {1+x}\, \left (1-x \right )^{\frac {9}{2}}}\)	\(40\)
risch	\(\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \left (8 x^{5}-32 x^{4}+44 x^{3}-16 x^{2}-17 x +20\right )}{63 \sqrt {1-x}\, \sqrt {1+x}\, \left (-1+x \right )^{4} \sqrt {-\left (1+x \right ) \left (-1+x \right )}}\)	\(66\)
default	\(\frac {1}{9 \left (1-x \right )^{\frac {9}{2}} \sqrt {1+x}}+\frac {5}{63 \left (1-x \right )^{\frac {7}{2}} \sqrt {1+x}}+\frac {4}{63 \left (1-x \right )^{\frac {5}{2}} \sqrt {1+x}}+\frac {4}{63 \left (1-x \right )^{\frac {3}{2}} \sqrt {1+x}}+\frac {8}{63 \sqrt {1-x}\, \sqrt {1+x}}-\frac {8 \sqrt {1-x}}{63 \sqrt {1+x}}\)	\(86\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 201 vs. \(2 (72) = 144\).
time = 0.28, size = 201, normalized size = 1.97 \begin {gather*} \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 )}} \end {gather*}

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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Fricas [A]
time = 1.02, size = 91, normalized size = 0.89 \begin {gather*} \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 )}} \end {gather*}

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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Sympy [C] Result contains complex when optimal does not.
time = 82.95, size = 593, normalized size = 5.81 \begin {gather*} \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 {1}{\left |{x + 1}\right |} > \frac {1}{2} \\- \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} \end {gather*}

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), 1/Abs(x + 1) > 1/2), (-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) + 72*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 - 504
0*(x + 1)**2 + 3024) + 420*I*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*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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Giac [A]
time = 1.24, size = 85, normalized size = 0.83 \begin {gather*} \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}} \end {gather*}

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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Mupad [B]
time = 0.36, size = 80, normalized size = 0.78 \begin {gather*} \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}} \end {gather*}

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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