3.1077 \(\int (1-x)^{5/2} (1+x)^{3/2} \, dx\)

Optimal. Leaf size=69 \[ \frac{1}{5} (1-x)^{5/2} (x+1)^{5/2}+\frac{1}{4} (1-x)^{3/2} x (x+1)^{3/2}+\frac{3}{8} \sqrt{1-x} x \sqrt{x+1}+\frac{3}{8} \sin ^{-1}(x) \]

[Out]

(3*Sqrt[1 - x]*x*Sqrt[1 + x])/8 + ((1 - x)^(3/2)*x*(1 + x)^(3/2))/4 + ((1 - x)^(5/2)*(1 + x)^(5/2))/5 + (3*Arc
Sin[x])/8

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Rubi [A]  time = 0.0081353, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {49, 38, 41, 216} \[ \frac{1}{5} (1-x)^{5/2} (x+1)^{5/2}+\frac{1}{4} (1-x)^{3/2} x (x+1)^{3/2}+\frac{3}{8} \sqrt{1-x} x \sqrt{x+1}+\frac{3}{8} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[(1 - x)^(5/2)*(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 + ((1 - x)^(5/2)*(1 + x)^(5/2))/5 + (3*Arc
Sin[x])/8

Rule 49

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

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)^{3/2} \, dx &=\frac{1}{5} (1-x)^{5/2} (1+x)^{5/2}+\int (1-x)^{3/2} (1+x)^{3/2} \, dx\\ &=\frac{1}{4} (1-x)^{3/2} x (1+x)^{3/2}+\frac{1}{5} (1-x)^{5/2} (1+x)^{5/2}+\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{1}{5} (1-x)^{5/2} (1+x)^{5/2}+\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{1}{5} (1-x)^{5/2} (1+x)^{5/2}+\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{1}{5} (1-x)^{5/2} (1+x)^{5/2}+\frac{3}{8} \sin ^{-1}(x)\\ \end{align*}

Mathematica [A]  time = 0.0444378, size = 55, normalized size = 0.8 \[ \frac{1}{40} \left (\sqrt{1-x^2} \left (8 x^4-10 x^3-16 x^2+25 x+8\right )-30 \sin ^{-1}\left (\frac{\sqrt{1-x}}{\sqrt{2}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - x)^(5/2)*(1 + x)^(3/2),x]

[Out]

(Sqrt[1 - x^2]*(8 + 25*x - 16*x^2 - 10*x^3 + 8*x^4) - 30*ArcSin[Sqrt[1 - x]/Sqrt[2]])/40

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Maple [A]  time = 0.005, size = 99, normalized size = 1.4 \begin{align*}{\frac{1}{5} \left ( 1-x \right ) ^{{\frac{5}{2}}} \left ( 1+x \right ) ^{{\frac{5}{2}}}}+{\frac{1}{4} \left ( 1-x \right ) ^{{\frac{3}{2}}} \left ( 1+x \right ) ^{{\frac{5}{2}}}}+{\frac{1}{4}\sqrt{1-x} \left ( 1+x \right ) ^{{\frac{5}{2}}}}-{\frac{1}{8}\sqrt{1-x} \left ( 1+x \right ) ^{{\frac{3}{2}}}}-{\frac{3}{8}\sqrt{1-x}\sqrt{1+x}}+{\frac{3\,\arcsin \left ( x \right ) }{8}\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)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.57968, size = 54, normalized size = 0.78 \begin{align*} \frac{1}{5} \,{\left (-x^{2} + 1\right )}^{\frac{5}{2}} + \frac{1}{4} \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}} x + \frac{3}{8} \, \sqrt{-x^{2} + 1} x + \frac{3}{8} \, \arcsin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)^(5/2)*(1+x)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.48994, size = 155, normalized size = 2.25 \begin{align*} \frac{1}{40} \,{\left (8 \, x^{4} - 10 \, x^{3} - 16 \, x^{2} + 25 \, x + 8\right )} \sqrt{x + 1} \sqrt{-x + 1} - \frac{3}{4} \, \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)^(3/2),x, algorithm="fricas")

[Out]

1/40*(8*x^4 - 10*x^3 - 16*x^2 + 25*x + 8)*sqrt(x + 1)*sqrt(-x + 1) - 3/4*arctan((sqrt(x + 1)*sqrt(-x + 1) - 1)
/x)

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Sympy [B]  time = 37.3425, size = 250, normalized size = 3.62 \begin{align*} \begin{cases} - \frac{3 i \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )}}{4} + \frac{i \left (x + 1\right )^{\frac{11}{2}}}{5 \sqrt{x - 1}} - \frac{29 i \left (x + 1\right )^{\frac{9}{2}}}{20 \sqrt{x - 1}} + \frac{73 i \left (x + 1\right )^{\frac{7}{2}}}{20 \sqrt{x - 1}} - \frac{129 i \left (x + 1\right )^{\frac{5}{2}}}{40 \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{11}{2}}}{5 \sqrt{1 - x}} + \frac{29 \left (x + 1\right )^{\frac{9}{2}}}{20 \sqrt{1 - x}} - \frac{73 \left (x + 1\right )^{\frac{7}{2}}}{20 \sqrt{1 - x}} + \frac{129 \left (x + 1\right )^{\frac{5}{2}}}{40 \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{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)**(5/2)*(1+x)**(3/2),x)

[Out]

Piecewise((-3*I*acosh(sqrt(2)*sqrt(x + 1)/2)/4 + I*(x + 1)**(11/2)/(5*sqrt(x - 1)) - 29*I*(x + 1)**(9/2)/(20*s
qrt(x - 1)) + 73*I*(x + 1)**(7/2)/(20*sqrt(x - 1)) - 129*I*(x + 1)**(5/2)/(40*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)**(11/2)/(5*sqrt(1 - x)) + 29*(x + 1)**(9/2)/(20*sqrt(1 - x)) - 73*(x + 1)**(7/2)/(20*sqrt(1 - x)) + 129*(x
+ 1)**(5/2)/(40*sqrt(1 - x)) + (x + 1)**(3/2)/(8*sqrt(1 - x)) - 3*sqrt(x + 1)/(4*sqrt(1 - x)), True))

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Giac [B]  time = 1.09546, size = 143, normalized size = 2.07 \begin{align*} \frac{1}{15} \,{\left ({\left (3 \,{\left (x + 1\right )}{\left (x - 3\right )} + 17\right )}{\left (x + 1\right )} - 10\right )}{\left (x + 1\right )}^{\frac{3}{2}} \sqrt{-x + 1} - \frac{1}{3} \,{\left (x + 1\right )}^{\frac{3}{2}}{\left (x - 1\right )} \sqrt{-x + 1} - \frac{1}{8} \,{\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{3}{4} \, \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)^(3/2),x, algorithm="giac")

[Out]

1/15*((3*(x + 1)*(x - 3) + 17)*(x + 1) - 10)*(x + 1)^(3/2)*sqrt(-x + 1) - 1/3*(x + 1)^(3/2)*(x - 1)*sqrt(-x +
1) - 1/8*((2*(x + 1)*(x - 2) + 5)*(x + 1) - 1)*sqrt(x + 1)*sqrt(-x + 1) + 1/2*sqrt(x + 1)*x*sqrt(-x + 1) + 3/4
*arcsin(1/2*sqrt(2)*sqrt(x + 1))