3.12.31 \(\int \frac {1}{\sqrt {1-x} (1+x)^{5/2}} \, dx\) [1131]

Optimal. Leaf size=41 \[ -\frac {\sqrt {1-x}}{3 (1+x)^{3/2}}-\frac {\sqrt {1-x}}{3 \sqrt {1+x}} \]

[Out]

-1/3*(1-x)^(1/2)/(1+x)^(3/2)-1/3*(1-x)^(1/2)/(1+x)^(1/2)

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Rubi [A]
time = 0.00, antiderivative size = 41, normalized size of antiderivative = 1.00, 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 = {47, 37} \begin {gather*} -\frac {\sqrt {1-x}}{3 \sqrt {x+1}}-\frac {\sqrt {1-x}}{3 (x+1)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[1 - x]*(1 + x)^(5/2)),x]

[Out]

-1/3*Sqrt[1 - x]/(1 + x)^(3/2) - Sqrt[1 - x]/(3*Sqrt[1 + x])

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]

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

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Mathematica [A]
time = 0.04, size = 23, normalized size = 0.56 \begin {gather*} -\frac {\sqrt {1-x} (2+x)}{3 (1+x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*(Sqrt[1 - x]*(2 + x))/(1 + x)^(3/2)

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in optimal.
time = 3.42, size = 68, normalized size = 1.66 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {\left (-2-x\right ) \sqrt {\frac {1-x}{1+x}}}{3 \left (1+x\right )},\frac {1}{\text {Abs}\left [1+x\right ]}>\frac {1}{2}\right \}\right \},-\frac {I \sqrt {1-\frac {2}{1+x}}}{3 \left (1+x\right )}-\frac {I \sqrt {1-\frac {2}{1+x}}}{3}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[1/((1 - x)^(1/2)*(1 + x)^(5/2)),x]')

[Out]

Piecewise[{{(-2 - x) Sqrt[(1 - x) / (1 + x)] / (3 (1 + x)), 1 / Abs[1 + x] > 1 / 2}}, -I Sqrt[1 - 2 / (1 + x)]
 / (3 (1 + x)) - I Sqrt[1 - 2 / (1 + x)] / 3]

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Maple [A]
time = 0.16, size = 30, normalized size = 0.73

method result size
gosper \(-\frac {\left (2+x \right ) \sqrt {1-x}}{3 \left (1+x \right )^{\frac {3}{2}}}\) \(18\)
default \(-\frac {\sqrt {1-x}}{3 \left (1+x \right )^{\frac {3}{2}}}-\frac {\sqrt {1-x}}{3 \sqrt {1+x}}\) \(30\)
risch \(\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \left (x^{2}+x -2\right )}{3 \sqrt {1-x}\, \left (1+x \right )^{\frac {3}{2}} \sqrt {-\left (1+x \right ) \left (-1+x \right )}}\) \(42\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1-x)^(1/2)/(1+x)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-1/3*(1-x)^(1/2)/(1+x)^(3/2)-1/3*(1-x)^(1/2)/(1+x)^(1/2)

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Maxima [A]
time = 0.34, size = 38, normalized size = 0.93 \begin {gather*} -\frac {\sqrt {-x^{2} + 1}}{3 \, {\left (x^{2} + 2 \, x + 1\right )}} - \frac {\sqrt {-x^{2} + 1}}{3 \, {\left (x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.30, size = 38, normalized size = 0.93 \begin {gather*} -\frac {2 \, x^{2} + {\left (x + 2\right )} \sqrt {x + 1} \sqrt {-x + 1} + 4 \, x + 2}{3 \, {\left (x^{2} + 2 \, x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 1.37, size = 66, normalized size = 1.61 \begin {gather*} \begin {cases} - \frac {\sqrt {-1 + \frac {2}{x + 1}}}{3} - \frac {\sqrt {-1 + \frac {2}{x + 1}}}{3 \left (x + 1\right )} & \text {for}\: \frac {1}{\left |{x + 1}\right |} > \frac {1}{2} \\- \frac {i \sqrt {1 - \frac {2}{x + 1}}}{3} - \frac {i \sqrt {1 - \frac {2}{x + 1}}}{3 \left (x + 1\right )} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-sqrt(-1 + 2/(x + 1))/3 - sqrt(-1 + 2/(x + 1))/(3*(x + 1)), 1/Abs(x + 1) > 1/2), (-I*sqrt(1 - 2/(x
+ 1))/3 - I*sqrt(1 - 2/(x + 1))/(3*(x + 1)), True))

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Giac [A]
time = 0.00, size = 47, normalized size = 1.15 \begin {gather*} -\frac {4 \left (\frac 1{4}-\frac {1}{12} \sqrt {-x+1} \sqrt {-x+1}\right ) \sqrt {-x+1} \sqrt {x+1}}{\left (x+1\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)^(1/2)/(1+x)^(5/2),x)

[Out]

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

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Mupad [B]
time = 0.31, size = 33, normalized size = 0.80 \begin {gather*} -\frac {x\,\sqrt {1-x}+2\,\sqrt {1-x}}{\left (3\,x+3\right )\,\sqrt {x+1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x*(1 - x)^(1/2) + 2*(1 - x)^(1/2))/((3*x + 3)*(x + 1)^(1/2))

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