3.12.33 \(\int \frac {1}{(1-x)^{5/2} (1+x)^{5/2}} \, dx\) [1133]

Optimal. Leaf size=43 \[ \frac {x}{3 (1-x)^{3/2} (1+x)^{3/2}}+\frac {2 x}{3 \sqrt {1-x} \sqrt {1+x}} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 43, 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 = {40, 39} \begin {gather*} \frac {2 x}{3 \sqrt {1-x} \sqrt {x+1}}+\frac {x}{3 (1-x)^{3/2} (x+1)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x/(3*(1 - x)^(3/2)*(1 + x)^(3/2)) + (2*x)/(3*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 40

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

Rubi steps

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

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Mathematica [A]
time = 0.04, size = 24, normalized size = 0.56 \begin {gather*} \frac {3 x-2 x^3}{3 \left (1-x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*x - 2*x^3)/(3*(1 - x^2)^(3/2))

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in optimal.
time = 8.56, size = 199, normalized size = 4.63 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {x \left (3-2 x^2\right ) \sqrt {\frac {1-x}{1+x}}}{3 \left (1-x-x^2+x^3\right )},\frac {1}{\text {Abs}\left [1+x\right ]}>\frac {1}{2}\right \}\right \},\frac {-3 I \left (1+x\right ) \sqrt {1-\frac {2}{1+x}}}{12+12 x-12 \left (1+x\right )^2+3 \left (1+x\right )^3}-\frac {2 I \left (1+x\right )^3 \sqrt {1-\frac {2}{1+x}}}{12+12 x-12 \left (1+x\right )^2+3 \left (1+x\right )^3}-\frac {I \sqrt {1-\frac {2}{1+x}}}{12+12 x-12 \left (1+x\right )^2+3 \left (1+x\right )^3}+\frac {I 6 \left (1+x\right )^2 \sqrt {1-\frac {2}{1+x}}}{12+12 x-12 \left (1+x\right )^2+3 \left (1+x\right )^3}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Piecewise[{{x (3 - 2 x ^ 2) Sqrt[(1 - x) / (1 + x)] / (3 (1 - x - x ^ 2 + x ^ 3)), 1 / Abs[1 + x] > 1 / 2}}, -
3 I (1 + x) Sqrt[1 - 2 / (1 + x)] / (12 + 12 x - 12 (1 + x) ^ 2 + 3 (1 + x) ^ 3) - 2 I (1 + x) ^ 3 Sqrt[1 - 2
/ (1 + x)] / (12 + 12 x - 12 (1 + x) ^ 2 + 3 (1 + x) ^ 3) - I Sqrt[1 - 2 / (1 + x)] / (12 + 12 x - 12 (1 + x)
^ 2 + 3 (1 + x) ^ 3) + I 6 (1 + x) ^ 2 Sqrt[1 - 2 / (1 + x)] / (12 + 12 x - 12 (1 + x) ^ 2 + 3 (1 + x) ^ 3)]

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Maple [A]
time = 0.15, size = 57, normalized size = 1.33

method	result	size
gosper	\(-\frac {x \left (2 x^{2}-3\right )}{3 \left (1+x \right )^{\frac {3}{2}} \left (1-x \right )^{\frac {3}{2}}}\)	\(23\)
default	\(\frac {1}{3 \left (1-x \right )^{\frac {3}{2}} \left (1+x \right )^{\frac {3}{2}}}+\frac {1}{\sqrt {1-x}\, \left (1+x \right )^{\frac {3}{2}}}-\frac {2 \sqrt {1-x}}{3 \left (1+x \right )^{\frac {3}{2}}}-\frac {2 \sqrt {1-x}}{3 \sqrt {1+x}}\)	\(57\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.27, size = 25, normalized size = 0.58 \begin {gather*} \frac {2 \, x}{3 \, \sqrt {-x^{2} + 1}} + \frac {x}{3 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.29, size = 35, normalized size = 0.81 \begin {gather*} -\frac {{\left (2 \, x^{3} - 3 \, x\right )} \sqrt {x + 1} \sqrt {-x + 1}}{3 \, {\left (x^{4} - 2 \, x^{2} + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C] Result contains complex when optimal does not.
time = 6.71, size = 280, normalized size = 6.51 \begin {gather*} \begin {cases} - \frac {2 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{3}}{12 x + 3 \left (x + 1\right )^{3} - 12 \left (x + 1\right )^{2} + 12} + \frac {6 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{2}}{12 x + 3 \left (x + 1\right )^{3} - 12 \left (x + 1\right )^{2} + 12} - \frac {3 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )}{12 x + 3 \left (x + 1\right )^{3} - 12 \left (x + 1\right )^{2} + 12} - \frac {\sqrt {-1 + \frac {2}{x + 1}}}{12 x + 3 \left (x + 1\right )^{3} - 12 \left (x + 1\right )^{2} + 12} & \text {for}\: \frac {1}{\left |{x + 1}\right |} > \frac {1}{2} \\- \frac {2 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{3}}{12 x + 3 \left (x + 1\right )^{3} - 12 \left (x + 1\right )^{2} + 12} + \frac {6 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{2}}{12 x + 3 \left (x + 1\right )^{3} - 12 \left (x + 1\right )^{2} + 12} - \frac {3 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )}{12 x + 3 \left (x + 1\right )^{3} - 12 \left (x + 1\right )^{2} + 12} - \frac {i \sqrt {1 - \frac {2}{x + 1}}}{12 x + 3 \left (x + 1\right )^{3} - 12 \left (x + 1\right )^{2} + 12} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-2*sqrt(-1 + 2/(x + 1))*(x + 1)**3/(12*x + 3*(x + 1)**3 - 12*(x + 1)**2 + 12) + 6*sqrt(-1 + 2/(x +
1))*(x + 1)**2/(12*x + 3*(x + 1)**3 - 12*(x + 1)**2 + 12) - 3*sqrt(-1 + 2/(x + 1))*(x + 1)/(12*x + 3*(x + 1)**
3 - 12*(x + 1)**2 + 12) - sqrt(-1 + 2/(x + 1))/(12*x + 3*(x + 1)**3 - 12*(x + 1)**2 + 12), 1/Abs(x + 1) > 1/2)
, (-2*I*sqrt(1 - 2/(x + 1))*(x + 1)**3/(12*x + 3*(x + 1)**3 - 12*(x + 1)**2 + 12) + 6*I*sqrt(1 - 2/(x + 1))*(x
+ 1)**2/(12*x + 3*(x + 1)**3 - 12*(x + 1)**2 + 12) - 3*I*sqrt(1 - 2/(x + 1))*(x + 1)/(12*x + 3*(x + 1)**3 - 1
2*(x + 1)**2 + 12) - I*sqrt(1 - 2/(x + 1))/(12*x + 3*(x + 1)**3 - 12*(x + 1)**2 + 12), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 106 vs. \(2 (31) = 62\).
time = 0.01, size = 197, normalized size = 4.58 \begin {gather*} -2 \left (\frac {-\frac {16384}{3} \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{3}+\frac {90112 \left (-2 \sqrt {x+1}+2 \sqrt {2}\right )}{\sqrt {-x+1}}}{2097152}+\frac {33 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{2}+1}{384 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{3}}+\frac {2 \left (\frac {3}{16}-\frac {1}{12} \sqrt {-x+1} \sqrt {-x+1}\right ) \sqrt {-x+1} \sqrt {x+1}}{\left (x+1\right )^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/192*(sqrt(2) - sqrt(x + 1))^3/(-x + 1)^(3/2) - 11/64*(sqrt(2) - sqrt(x + 1))/sqrt(-x + 1) - 1/12*(4*x + 5)*
sqrt(-x + 1)/(x + 1)^(3/2) - 1/192*(-x + 1)^(3/2)*(33*(sqrt(2) - sqrt(x + 1))^2/(x - 1) - 1)/(sqrt(2) - sqrt(x
+ 1))^3

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Mupad [B]
time = 0.37, size = 41, normalized size = 0.95 \begin {gather*} \frac {3\,x\,\sqrt {1-x}-2\,x^3\,\sqrt {1-x}}{\left (3\,x+3\right )\,{\left (x-1\right )}^2\,\sqrt {x+1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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