3.11.91 \(\int (1-x)^{5/2} (1+x)^{5/2} \, dx\) [1091]

Optimal. Leaf size=70 \[ \frac {5}{16} \sqrt {1-x} x \sqrt {1+x}+\frac {5}{24} (1-x)^{3/2} x (1+x)^{3/2}+\frac {1}{6} (1-x)^{5/2} x (1+x)^{5/2}+\frac {5}{16} \sin ^{-1}(x) \]

[Out]

5/24*(1-x)^(3/2)*x*(1+x)^(3/2)+1/6*(1-x)^(5/2)*x*(1+x)^(5/2)+5/16*arcsin(x)+5/16*x*(1-x)^(1/2)*(1+x)^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {38, 41, 222} \begin {gather*} \frac {5 \text {ArcSin}(x)}{16}+\frac {1}{6} (1-x)^{5/2} x (x+1)^{5/2}+\frac {5}{24} (1-x)^{3/2} x (x+1)^{3/2}+\frac {5}{16} \sqrt {1-x} x \sqrt {x+1} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - x)^(5/2)*(1 + x)^(5/2),x]

[Out]

(5*Sqrt[1 - x]*x*Sqrt[1 + x])/16 + (5*(1 - x)^(3/2)*x*(1 + x)^(3/2))/24 + ((1 - x)^(5/2)*x*(1 + x)^(5/2))/6 +
(5*ArcSin[x])/16

Rule 38

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[x*(a + b*x)^m*((c + d*x)^m/(2*m + 1))
, x] + Dist[2*a*c*(m/(2*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] && IGtQ[m + 1/2, 0]

Rule 41

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[(a*c + b*d*x^2)^m, x] /; FreeQ[{a, b
, c, d, m}, x] && EqQ[b*c + a*d, 0] && (IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))

Rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin {align*} \int (1-x)^{5/2} (1+x)^{5/2} \, dx &=\frac {1}{6} (1-x)^{5/2} x (1+x)^{5/2}+\frac {5}{6} \int (1-x)^{3/2} (1+x)^{3/2} \, dx\\ &=\frac {5}{24} (1-x)^{3/2} x (1+x)^{3/2}+\frac {1}{6} (1-x)^{5/2} x (1+x)^{5/2}+\frac {5}{8} \int \sqrt {1-x} \sqrt {1+x} \, dx\\ &=\frac {5}{16} \sqrt {1-x} x \sqrt {1+x}+\frac {5}{24} (1-x)^{3/2} x (1+x)^{3/2}+\frac {1}{6} (1-x)^{5/2} x (1+x)^{5/2}+\frac {5}{16} \int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx\\ &=\frac {5}{16} \sqrt {1-x} x \sqrt {1+x}+\frac {5}{24} (1-x)^{3/2} x (1+x)^{3/2}+\frac {1}{6} (1-x)^{5/2} x (1+x)^{5/2}+\frac {5}{16} \int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=\frac {5}{16} \sqrt {1-x} x \sqrt {1+x}+\frac {5}{24} (1-x)^{3/2} x (1+x)^{3/2}+\frac {1}{6} (1-x)^{5/2} x (1+x)^{5/2}+\frac {5}{16} \sin ^{-1}(x)\\ \end {align*}

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Mathematica [A]
time = 0.09, size = 51, normalized size = 0.73 \begin {gather*} \frac {1}{48} x \sqrt {1-x^2} \left (33-26 x^2+8 x^4\right )-\frac {5}{8} \tan ^{-1}\left (\frac {\sqrt {1-x^2}}{1+x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*Sqrt[1 - x^2]*(33 - 26*x^2 + 8*x^4))/48 - (5*ArcTan[Sqrt[1 - x^2]/(1 + x)])/8

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(112\) vs. \(2(50)=100\).
time = 0.15, size = 113, normalized size = 1.61

method	result	size
risch	\(-\frac {x \left (8 x^{4}-26 x^{2}+33\right ) \sqrt {1+x}\, \left (-1+x \right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{48 \sqrt {-\left (1+x \right ) \left (-1+x \right )}\, \sqrt {1-x}}+\frac {5 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{16 \sqrt {1+x}\, \sqrt {1-x}}\)	\(80\)
default	\(\frac {\left (1-x \right )^{\frac {5}{2}} \left (1+x \right )^{\frac {7}{2}}}{6}+\frac {\left (1-x \right )^{\frac {3}{2}} \left (1+x \right )^{\frac {7}{2}}}{6}+\frac {\sqrt {1-x}\, \left (1+x \right )^{\frac {7}{2}}}{8}-\frac {\sqrt {1-x}\, \left (1+x \right )^{\frac {5}{2}}}{24}-\frac {5 \sqrt {1-x}\, \left (1+x \right )^{\frac {3}{2}}}{48}-\frac {5 \sqrt {1-x}\, \sqrt {1+x}}{16}+\frac {5 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{16 \sqrt {1+x}\, \sqrt {1-x}}\)	\(113\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(1-x)^(5/2)*(1+x)^(7/2)+1/6*(1-x)^(3/2)*(1+x)^(7/2)+1/8*(1-x)^(1/2)*(1+x)^(7/2)-1/24*(1-x)^(1/2)*(1+x)^(5/
2)-5/48*(1-x)^(1/2)*(1+x)^(3/2)-5/16*(1-x)^(1/2)*(1+x)^(1/2)+5/16*((1+x)*(1-x))^(1/2)/(1+x)^(1/2)/(1-x)^(1/2)*
arcsin(x)

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Maxima [A]
time = 0.50, size = 41, normalized size = 0.59 \begin {gather*} \frac {1}{6} \, {\left (-x^{2} + 1\right )}^{\frac {5}{2}} x + \frac {5}{24} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} x + \frac {5}{16} \, \sqrt {-x^{2} + 1} x + \frac {5}{16} \, \arcsin \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(-x^2 + 1)^(5/2)*x + 5/24*(-x^2 + 1)^(3/2)*x + 5/16*sqrt(-x^2 + 1)*x + 5/16*arcsin(x)

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Fricas [A]
time = 0.72, size = 51, normalized size = 0.73 \begin {gather*} \frac {1}{48} \, {\left (8 \, x^{5} - 26 \, x^{3} + 33 \, x\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {5}{8} \, \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/48*(8*x^5 - 26*x^3 + 33*x)*sqrt(x + 1)*sqrt(-x + 1) - 5/8*arctan((sqrt(x + 1)*sqrt(-x + 1) - 1)/x)

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Sympy [C] Result contains complex when optimal does not.
time = 186.20, size = 284, normalized size = 4.06 \begin {gather*} \begin {cases} - \frac {5 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{8} + \frac {i \left (x + 1\right )^{\frac {13}{2}}}{6 \sqrt {x - 1}} - \frac {7 i \left (x + 1\right )^{\frac {11}{2}}}{6 \sqrt {x - 1}} + \frac {67 i \left (x + 1\right )^{\frac {9}{2}}}{24 \sqrt {x - 1}} - \frac {55 i \left (x + 1\right )^{\frac {7}{2}}}{24 \sqrt {x - 1}} - \frac {i \left (x + 1\right )^{\frac {5}{2}}}{48 \sqrt {x - 1}} - \frac {5 i \left (x + 1\right )^{\frac {3}{2}}}{48 \sqrt {x - 1}} + \frac {5 i \sqrt {x + 1}}{8 \sqrt {x - 1}} & \text {for}\: \left |{x + 1}\right | > 2 \\\frac {5 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{8} - \frac {\left (x + 1\right )^{\frac {13}{2}}}{6 \sqrt {1 - x}} + \frac {7 \left (x + 1\right )^{\frac {11}{2}}}{6 \sqrt {1 - x}} - \frac {67 \left (x + 1\right )^{\frac {9}{2}}}{24 \sqrt {1 - x}} + \frac {55 \left (x + 1\right )^{\frac {7}{2}}}{24 \sqrt {1 - x}} + \frac {\left (x + 1\right )^{\frac {5}{2}}}{48 \sqrt {1 - x}} + \frac {5 \left (x + 1\right )^{\frac {3}{2}}}{48 \sqrt {1 - x}} - \frac {5 \sqrt {x + 1}}{8 \sqrt {1 - x}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-5*I*acosh(sqrt(2)*sqrt(x + 1)/2)/8 + I*(x + 1)**(13/2)/(6*sqrt(x - 1)) - 7*I*(x + 1)**(11/2)/(6*sq
rt(x - 1)) + 67*I*(x + 1)**(9/2)/(24*sqrt(x - 1)) - 55*I*(x + 1)**(7/2)/(24*sqrt(x - 1)) - I*(x + 1)**(5/2)/(4
8*sqrt(x - 1)) - 5*I*(x + 1)**(3/2)/(48*sqrt(x - 1)) + 5*I*sqrt(x + 1)/(8*sqrt(x - 1)), Abs(x + 1) > 2), (5*as
in(sqrt(2)*sqrt(x + 1)/2)/8 - (x + 1)**(13/2)/(6*sqrt(1 - x)) + 7*(x + 1)**(11/2)/(6*sqrt(1 - x)) - 67*(x + 1)
**(9/2)/(24*sqrt(1 - x)) + 55*(x + 1)**(7/2)/(24*sqrt(1 - x)) + (x + 1)**(5/2)/(48*sqrt(1 - x)) + 5*(x + 1)**(
3/2)/(48*sqrt(1 - x)) - 5*sqrt(x + 1)/(8*sqrt(1 - x)), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 185 vs. \(2 (50) = 100\).
time = 1.69, size = 185, normalized size = 2.64 \begin {gather*} \frac {1}{240} \, {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, x - 26\right )} {\left (x + 1\right )} + 321\right )} {\left (x + 1\right )} - 451\right )} {\left (x + 1\right )} + 745\right )} {\left (x + 1\right )} - 405\right )} \sqrt {x + 1} \sqrt {-x + 1} + \frac {1}{120} \, {\left ({\left (2 \, {\left (3 \, {\left (4 \, x - 17\right )} {\left (x + 1\right )} + 133\right )} {\left (x + 1\right )} - 295\right )} {\left (x + 1\right )} + 195\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {1}{12} \, {\left ({\left (2 \, {\left (3 \, x - 10\right )} {\left (x + 1\right )} + 43\right )} {\left (x + 1\right )} - 39\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {1}{3} \, {\left ({\left (2 \, x - 5\right )} {\left (x + 1\right )} + 9\right )} \sqrt {x + 1} \sqrt {-x + 1} + \frac {1}{2} \, \sqrt {x + 1} {\left (x - 2\right )} \sqrt {-x + 1} + \sqrt {x + 1} \sqrt {-x + 1} + \frac {5}{8} \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/240*((2*((4*(5*x - 26)*(x + 1) + 321)*(x + 1) - 451)*(x + 1) + 745)*(x + 1) - 405)*sqrt(x + 1)*sqrt(-x + 1)
+ 1/120*((2*(3*(4*x - 17)*(x + 1) + 133)*(x + 1) - 295)*(x + 1) + 195)*sqrt(x + 1)*sqrt(-x + 1) - 1/12*((2*(3*
x - 10)*(x + 1) + 43)*(x + 1) - 39)*sqrt(x + 1)*sqrt(-x + 1) - 1/3*((2*x - 5)*(x + 1) + 9)*sqrt(x + 1)*sqrt(-x
+ 1) + 1/2*sqrt(x + 1)*(x - 2)*sqrt(-x + 1) + sqrt(x + 1)*sqrt(-x + 1) + 5/8*arcsin(1/2*sqrt(2)*sqrt(x + 1))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (1-x\right )}^{5/2}\,{\left (x+1\right )}^{5/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - x)^(5/2)*(x + 1)^(5/2),x)

[Out]

int((1 - x)^(5/2)*(x + 1)^(5/2), x)

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