3.11.9 \(\int (1-x)^{3/2} (1+x)^{3/2} \, dx\)

Optimal. Leaf size=49 \[ \frac {1}{4} (1-x)^{3/2} x (x+1)^{3/2}+\frac {3}{8} \sqrt {1-x} x \sqrt {x+1}+\frac {3}{8} \sin ^{-1}(x) \]

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Rubi [A]  time = 0.01, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {38, 41, 216} \begin {gather*} \frac {1}{4} (1-x)^{3/2} x (x+1)^{3/2}+\frac {3}{8} \sqrt {1-x} x \sqrt {x+1}+\frac {3}{8} \sin ^{-1}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - x)^(3/2)*(1 + x)^(3/2),x]

[Out]

(3*Sqrt[1 - x]*x*Sqrt[1 + x])/8 + ((1 - x)^(3/2)*x*(1 + x)^(3/2))/4 + (3*ArcSin[x])/8

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)^{3/2} (1+x)^{3/2} \, dx &=\frac {1}{4} (1-x)^{3/2} x (1+x)^{3/2}+\frac {3}{4} \int \sqrt {1-x} \sqrt {1+x} \, dx\\ &=\frac {3}{8} \sqrt {1-x} x \sqrt {1+x}+\frac {1}{4} (1-x)^{3/2} x (1+x)^{3/2}+\frac {3}{8} \int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx\\ &=\frac {3}{8} \sqrt {1-x} x \sqrt {1+x}+\frac {1}{4} (1-x)^{3/2} x (1+x)^{3/2}+\frac {3}{8} \int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=\frac {3}{8} \sqrt {1-x} x \sqrt {1+x}+\frac {1}{4} (1-x)^{3/2} x (1+x)^{3/2}+\frac {3}{8} \sin ^{-1}(x)\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 29, normalized size = 0.59 \begin {gather*} \frac {1}{8} \left (x \sqrt {1-x^2} \left (5-2 x^2\right )+3 \sin ^{-1}(x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(5 - 2*x^2)*Sqrt[1 - x^2] + 3*ArcSin[x])/8

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IntegrateAlgebraic [B]  time = 0.10, size = 115, normalized size = 2.35 \begin {gather*} \frac {-\frac {3 (1-x)^{7/2}}{(x+1)^{7/2}}-\frac {11 (1-x)^{5/2}}{(x+1)^{5/2}}+\frac {11 (1-x)^{3/2}}{(x+1)^{3/2}}+\frac {3 \sqrt {1-x}}{\sqrt {x+1}}}{4 \left (\frac {1-x}{x+1}+1\right )^4}-\frac {3}{4} \tan ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {x+1}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-3*(1 - x)^(7/2))/(1 + x)^(7/2) - (11*(1 - x)^(5/2))/(1 + x)^(5/2) + (11*(1 - x)^(3/2))/(1 + x)^(3/2) + (3*S
qrt[1 - x])/Sqrt[1 + x])/(4*(1 + (1 - x)/(1 + x))^4) - (3*ArcTan[Sqrt[1 - x]/Sqrt[1 + x]])/4

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fricas [A]  time = 1.19, size = 46, normalized size = 0.94 \begin {gather*} -\frac {1}{8} \, {\left (2 \, x^{3} - 5 \, x\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {3}{4} \, \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)^(3/2)*(1+x)^(3/2),x, algorithm="fricas")

[Out]

-1/8*(2*x^3 - 5*x)*sqrt(x + 1)*sqrt(-x + 1) - 3/4*arctan((sqrt(x + 1)*sqrt(-x + 1) - 1)/x)

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giac [B]  time = 1.12, size = 101, normalized size = 2.06 \begin {gather*} -\frac {1}{24} \, {\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}{6} \, {\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 {3}{4} \, \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)^(3/2)*(1+x)^(3/2),x, algorithm="giac")

[Out]

-1/24*((2*(3*x - 10)*(x + 1) + 43)*(x + 1) - 39)*sqrt(x + 1)*sqrt(-x + 1) - 1/6*((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) + 3/4*arcsin(1/2*sqrt(2)*
sqrt(x + 1))

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maple [B]  time = 0.00, size = 85, normalized size = 1.73 \begin {gather*} \frac {3 \sqrt {\left (x +1\right ) \left (-x +1\right )}\, \arcsin \relax (x )}{8 \sqrt {x +1}\, \sqrt {-x +1}}+\frac {\left (-x +1\right )^{\frac {3}{2}} \left (x +1\right )^{\frac {5}{2}}}{4}+\frac {\sqrt {-x +1}\, \left (x +1\right )^{\frac {5}{2}}}{4}-\frac {\sqrt {-x +1}\, \left (x +1\right )^{\frac {3}{2}}}{8}-\frac {3 \sqrt {-x +1}\, \sqrt {x +1}}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(-x+1)^(3/2)*(x+1)^(5/2)+1/4*(-x+1)^(1/2)*(x+1)^(5/2)-1/8*(-x+1)^(1/2)*(x+1)^(3/2)-3/8*(-x+1)^(1/2)*(x+1)^
(1/2)+3/8*((x+1)*(-x+1))^(1/2)/(x+1)^(1/2)/(-x+1)^(1/2)*arcsin(x)

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maxima [A]  time = 2.91, size = 29, normalized size = 0.59 \begin {gather*} \frac {1}{4} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} x + \frac {3}{8} \, \sqrt {-x^{2} + 1} x + \frac {3}{8} \, \arcsin \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)^(3/2)*(1+x)^(3/2),x, algorithm="maxima")

[Out]

1/4*(-x^2 + 1)^(3/2)*x + 3/8*sqrt(-x^2 + 1)*x + 3/8*arcsin(x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int {\left (1-x\right )}^{3/2}\,{\left (x+1\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - x)^(3/2)*(x + 1)^(3/2),x)

[Out]

int((1 - x)^(3/2)*(x + 1)^(3/2), x)

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sympy [B]  time = 7.46, size = 214, normalized size = 4.37 \begin {gather*} \begin {cases} - \frac {3 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{4} - \frac {i \left (x + 1\right )^{\frac {9}{2}}}{4 \sqrt {x - 1}} + \frac {5 i \left (x + 1\right )^{\frac {7}{2}}}{4 \sqrt {x - 1}} - \frac {13 i \left (x + 1\right )^{\frac {5}{2}}}{8 \sqrt {x - 1}} - \frac {i \left (x + 1\right )^{\frac {3}{2}}}{8 \sqrt {x - 1}} + \frac {3 i \sqrt {x + 1}}{4 \sqrt {x - 1}} & \text {for}\: \frac {\left |{x + 1}\right |}{2} > 1 \\\frac {3 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{4} + \frac {\left (x + 1\right )^{\frac {9}{2}}}{4 \sqrt {1 - x}} - \frac {5 \left (x + 1\right )^{\frac {7}{2}}}{4 \sqrt {1 - x}} + \frac {13 \left (x + 1\right )^{\frac {5}{2}}}{8 \sqrt {1 - x}} + \frac {\left (x + 1\right )^{\frac {3}{2}}}{8 \sqrt {1 - x}} - \frac {3 \sqrt {x + 1}}{4 \sqrt {1 - x}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)**(3/2)*(1+x)**(3/2),x)

[Out]

Piecewise((-3*I*acosh(sqrt(2)*sqrt(x + 1)/2)/4 - I*(x + 1)**(9/2)/(4*sqrt(x - 1)) + 5*I*(x + 1)**(7/2)/(4*sqrt
(x - 1)) - 13*I*(x + 1)**(5/2)/(8*sqrt(x - 1)) - I*(x + 1)**(3/2)/(8*sqrt(x - 1)) + 3*I*sqrt(x + 1)/(4*sqrt(x
- 1)), Abs(x + 1)/2 > 1), (3*asin(sqrt(2)*sqrt(x + 1)/2)/4 + (x + 1)**(9/2)/(4*sqrt(1 - x)) - 5*(x + 1)**(7/2)
/(4*sqrt(1 - x)) + 13*(x + 1)**(5/2)/(8*sqrt(1 - x)) + (x + 1)**(3/2)/(8*sqrt(1 - x)) - 3*sqrt(x + 1)/(4*sqrt(
1 - x)), True))

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