3.1091 \(\int (1-x)^{5/2} (1+x)^{5/2} \, dx\)

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

[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

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Rubi [A]  time = 0.0107479, antiderivative size = 70, normalized size of antiderivative = 1., 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, 216} \[ \frac{1}{6} (1-x)^{5/2} x (x+1)^{5/2}+\frac{5}{24} (1-x)^{3/2} x (x+1)^{3/2}+\frac{5}{16} \sqrt{1-x} x \sqrt{x+1}+\frac{5}{16} \sin ^{-1}(x) \]

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 216

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*}

Mathematica [A]  time = 0.0153676, size = 34, normalized size = 0.49 \[ \frac{1}{48} \left (x \sqrt{1-x^2} \left (8 x^4-26 x^2+33\right )+15 \sin ^{-1}(x)\right ) \]

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) + 15*ArcSin[x])/48

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Maple [B]  time = 0.003, size = 113, normalized size = 1.6 \begin{align*}{\frac{1}{6} \left ( 1-x \right ) ^{{\frac{5}{2}}} \left ( 1+x \right ) ^{{\frac{7}{2}}}}+{\frac{1}{6} \left ( 1-x \right ) ^{{\frac{3}{2}}} \left ( 1+x \right ) ^{{\frac{7}{2}}}}+{\frac{1}{8}\sqrt{1-x} \left ( 1+x \right ) ^{{\frac{7}{2}}}}-{\frac{1}{24}\sqrt{1-x} \left ( 1+x \right ) ^{{\frac{5}{2}}}}-{\frac{5}{48}\sqrt{1-x} \left ( 1+x \right ) ^{{\frac{3}{2}}}}-{\frac{5}{16}\sqrt{1-x}\sqrt{1+x}}+{\frac{5\,\arcsin \left ( x \right ) }{16}\sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) }{\frac{1}{\sqrt{1-x}}}{\frac{1}{\sqrt{1+x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 1.4925, size = 55, normalized size = 0.79 \begin{align*} \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{align*}

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 = 1.50185, size = 138, normalized size = 1.97 \begin{align*} \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{align*}

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 [B]  time = 79.3469, size = 286, normalized size = 4.09 \begin{align*} \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}\: \frac{\left |{x + 1}\right |}{2} > 1 \\\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{align*}

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 > 1), (5*
asin(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]  time = 1.12763, size = 138, normalized size = 1.97 \begin{align*} \frac{1}{48} \,{\left ({\left (2 \,{\left ({\left (4 \,{\left (x + 1\right )}{\left (x - 4\right )} + 39\right )}{\left (x + 1\right )} - 37\right )}{\left (x + 1\right )} + 31\right )}{\left (x + 1\right )} - 3\right )} \sqrt{x + 1} \sqrt{-x + 1} - \frac{1}{4} \,{\left ({\left (2 \,{\left (x + 1\right )}{\left (x - 2\right )} + 5\right )}{\left (x + 1\right )} - 1\right )} \sqrt{x + 1} \sqrt{-x + 1} + \frac{1}{2} \, \sqrt{x + 1} x \sqrt{-x + 1} + \frac{5}{8} \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{x + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/48*((2*((4*(x + 1)*(x - 4) + 39)*(x + 1) - 37)*(x + 1) + 31)*(x + 1) - 3)*sqrt(x + 1)*sqrt(-x + 1) - 1/4*((2
*(x + 1)*(x - 2) + 5)*(x + 1) - 1)*sqrt(x + 1)*sqrt(-x + 1) + 1/2*sqrt(x + 1)*x*sqrt(-x + 1) + 5/8*arcsin(1/2*
sqrt(2)*sqrt(x + 1))