3.1065 \(\int (1-x)^{5/2} \sqrt{1+x} \, dx\)

Optimal. Leaf size=68 \[ \frac{1}{4} (x+1)^{3/2} (1-x)^{5/2}+\frac{5}{12} (x+1)^{3/2} (1-x)^{3/2}+\frac{5}{8} x \sqrt{x+1} \sqrt{1-x}+\frac{5}{8} \sin ^{-1}(x) \]

[Out]

(5*Sqrt[1 - x]*x*Sqrt[1 + x])/8 + (5*(1 - x)^(3/2)*(1 + x)^(3/2))/12 + ((1 - x)^(5/2)*(1 + x)^(3/2))/4 + (5*Ar
cSin[x])/8

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Rubi [A]  time = 0.010737, antiderivative size = 68, 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} \[ \frac{1}{4} (x+1)^{3/2} (1-x)^{5/2}+\frac{5}{12} (x+1)^{3/2} (1-x)^{3/2}+\frac{5}{8} x \sqrt{x+1} \sqrt{1-x}+\frac{5}{8} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[(1 - x)^(5/2)*Sqrt[1 + x],x]

[Out]

(5*Sqrt[1 - x]*x*Sqrt[1 + x])/8 + (5*(1 - x)^(3/2)*(1 + x)^(3/2))/12 + ((1 - x)^(5/2)*(1 + x)^(3/2))/4 + (5*Ar
cSin[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} \sqrt{1+x} \, dx &=\frac{1}{4} (1-x)^{5/2} (1+x)^{3/2}+\frac{5}{4} \int (1-x)^{3/2} \sqrt{1+x} \, dx\\ &=\frac{5}{12} (1-x)^{3/2} (1+x)^{3/2}+\frac{1}{4} (1-x)^{5/2} (1+x)^{3/2}+\frac{5}{4} \int \sqrt{1-x} \sqrt{1+x} \, dx\\ &=\frac{5}{8} \sqrt{1-x} x \sqrt{1+x}+\frac{5}{12} (1-x)^{3/2} (1+x)^{3/2}+\frac{1}{4} (1-x)^{5/2} (1+x)^{3/2}+\frac{5}{8} \int \frac{1}{\sqrt{1-x} \sqrt{1+x}} \, dx\\ &=\frac{5}{8} \sqrt{1-x} x \sqrt{1+x}+\frac{5}{12} (1-x)^{3/2} (1+x)^{3/2}+\frac{1}{4} (1-x)^{5/2} (1+x)^{3/2}+\frac{5}{8} \int \frac{1}{\sqrt{1-x^2}} \, dx\\ &=\frac{5}{8} \sqrt{1-x} x \sqrt{1+x}+\frac{5}{12} (1-x)^{3/2} (1+x)^{3/2}+\frac{1}{4} (1-x)^{5/2} (1+x)^{3/2}+\frac{5}{8} \sin ^{-1}(x)\\ \end{align*}

Mathematica [A]  time = 0.0403838, size = 50, normalized size = 0.74 \[ \frac{1}{24} \left (\sqrt{1-x^2} \left (6 x^3-16 x^2+9 x+16\right )-30 \sin ^{-1}\left (\frac{\sqrt{1-x}}{\sqrt{2}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - x^2]*(16 + 9*x - 16*x^2 + 6*x^3) - 30*ArcSin[Sqrt[1 - x]/Sqrt[2]])/24

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Maple [A]  time = 0.005, size = 85, normalized size = 1.3 \begin{align*}{\frac{1}{4} \left ( 1-x \right ) ^{{\frac{5}{2}}} \left ( 1+x \right ) ^{{\frac{3}{2}}}}+{\frac{5}{12} \left ( 1-x \right ) ^{{\frac{3}{2}}} \left ( 1+x \right ) ^{{\frac{3}{2}}}}+{\frac{5}{8}\sqrt{1-x} \left ( 1+x \right ) ^{{\frac{3}{2}}}}-{\frac{5}{8}\sqrt{1-x}\sqrt{1+x}}+{\frac{5\,\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)^(1/2),x)

[Out]

1/4*(1-x)^(5/2)*(1+x)^(3/2)+5/12*(1-x)^(3/2)*(1+x)^(3/2)+5/8*(1-x)^(1/2)*(1+x)^(3/2)-5/8*(1-x)^(1/2)*(1+x)^(1/
2)+5/8*((1+x)*(1-x))^(1/2)/(1+x)^(1/2)/(1-x)^(1/2)*arcsin(x)

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Maxima [A]  time = 1.54095, size = 54, normalized size = 0.79 \begin{align*} -\frac{1}{4} \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}} x + \frac{2}{3} \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}} + \frac{5}{8} \, \sqrt{-x^{2} + 1} x + \frac{5}{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)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.58123, size = 143, normalized size = 2.1 \begin{align*} \frac{1}{24} \,{\left (6 \, x^{3} - 16 \, x^{2} + 9 \, x + 16\right )} \sqrt{x + 1} \sqrt{-x + 1} - \frac{5}{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)^(1/2),x, algorithm="fricas")

[Out]

1/24*(6*x^3 - 16*x^2 + 9*x + 16)*sqrt(x + 1)*sqrt(-x + 1) - 5/4*arctan((sqrt(x + 1)*sqrt(-x + 1) - 1)/x)

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Sympy [A]  time = 15.673, size = 218, normalized size = 3.21 \begin{align*} \begin{cases} - \frac{5 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{23 i \left (x + 1\right )^{\frac{7}{2}}}{12 \sqrt{x - 1}} + \frac{127 i \left (x + 1\right )^{\frac{5}{2}}}{24 \sqrt{x - 1}} - \frac{133 i \left (x + 1\right )^{\frac{3}{2}}}{24 \sqrt{x - 1}} + \frac{5 i \sqrt{x + 1}}{4 \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 )}}{4} - \frac{\left (x + 1\right )^{\frac{9}{2}}}{4 \sqrt{1 - x}} + \frac{23 \left (x + 1\right )^{\frac{7}{2}}}{12 \sqrt{1 - x}} - \frac{127 \left (x + 1\right )^{\frac{5}{2}}}{24 \sqrt{1 - x}} + \frac{133 \left (x + 1\right )^{\frac{3}{2}}}{24 \sqrt{1 - x}} - \frac{5 \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)**(1/2),x)

[Out]

Piecewise((-5*I*acosh(sqrt(2)*sqrt(x + 1)/2)/4 + I*(x + 1)**(9/2)/(4*sqrt(x - 1)) - 23*I*(x + 1)**(7/2)/(12*sq
rt(x - 1)) + 127*I*(x + 1)**(5/2)/(24*sqrt(x - 1)) - 133*I*(x + 1)**(3/2)/(24*sqrt(x - 1)) + 5*I*sqrt(x + 1)/(
4*sqrt(x - 1)), Abs(x + 1)/2 > 1), (5*asin(sqrt(2)*sqrt(x + 1)/2)/4 - (x + 1)**(9/2)/(4*sqrt(1 - x)) + 23*(x +
1)**(7/2)/(12*sqrt(1 - x)) - 127*(x + 1)**(5/2)/(24*sqrt(1 - x)) + 133*(x + 1)**(3/2)/(24*sqrt(1 - x)) - 5*sq
rt(x + 1)/(4*sqrt(1 - x)), True))

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Giac [A]  time = 1.0887, size = 103, normalized size = 1.51 \begin{align*} -\frac{2}{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{5}{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)^(1/2),x, algorithm="giac")

[Out]

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