3.1107 \(\int \frac{(1-x)^{5/2}}{\sqrt{1+x}} \, dx\)

Optimal. Leaf size=67 \[ \frac{1}{3} \sqrt{x+1} (1-x)^{5/2}+\frac{5}{6} \sqrt{x+1} (1-x)^{3/2}+\frac{5}{2} \sqrt{x+1} \sqrt{1-x}+\frac{5}{2} \sin ^{-1}(x) \]

[Out]

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

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Rubi [A]  time = 0.0106446, antiderivative size = 67, 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 = {50, 41, 216} \[ \frac{1}{3} \sqrt{x+1} (1-x)^{5/2}+\frac{5}{6} \sqrt{x+1} (1-x)^{3/2}+\frac{5}{2} \sqrt{x+1} \sqrt{1-x}+\frac{5}{2} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[(1 - x)^(5/2)/Sqrt[1 + x],x]

[Out]

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

Rule 50

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[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

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

Mathematica [A]  time = 0.0256524, size = 54, normalized size = 0.81 \[ \frac{\sqrt{x+1} \left (-2 x^3+11 x^2-31 x+22\right )}{6 \sqrt{1-x}}-5 \sin ^{-1}\left (\frac{\sqrt{1-x}}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 + x]*(22 - 31*x + 11*x^2 - 2*x^3))/(6*Sqrt[1 - x]) - 5*ArcSin[Sqrt[1 - x]/Sqrt[2]]

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Maple [A]  time = 0.005, size = 71, normalized size = 1.1 \begin{align*}{\frac{1}{3} \left ( 1-x \right ) ^{{\frac{5}{2}}}\sqrt{1+x}}+{\frac{5}{6} \left ( 1-x \right ) ^{{\frac{3}{2}}}\sqrt{1+x}}+{\frac{5}{2}\sqrt{1-x}\sqrt{1+x}}+{\frac{5\,\arcsin \left ( x \right ) }{2}\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]

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

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Maxima [A]  time = 1.50438, size = 57, normalized size = 0.85 \begin{align*} \frac{1}{3} \, \sqrt{-x^{2} + 1} x^{2} - \frac{3}{2} \, \sqrt{-x^{2} + 1} x + \frac{11}{3} \, \sqrt{-x^{2} + 1} + \frac{5}{2} \, \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]

1/3*sqrt(-x^2 + 1)*x^2 - 3/2*sqrt(-x^2 + 1)*x + 11/3*sqrt(-x^2 + 1) + 5/2*arcsin(x)

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Fricas [A]  time = 1.54617, size = 127, normalized size = 1.9 \begin{align*} \frac{1}{6} \,{\left (2 \, x^{2} - 9 \, x + 22\right )} \sqrt{x + 1} \sqrt{-x + 1} - 5 \, \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]

1/6*(2*x^2 - 9*x + 22)*sqrt(x + 1)*sqrt(-x + 1) - 5*arctan((sqrt(x + 1)*sqrt(-x + 1) - 1)/x)

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Sympy [A]  time = 10.0033, size = 175, normalized size = 2.61 \begin{align*} \begin{cases} - 5 i \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )} + \frac{i \left (x + 1\right )^{\frac{7}{2}}}{3 \sqrt{x - 1}} - \frac{17 i \left (x + 1\right )^{\frac{5}{2}}}{6 \sqrt{x - 1}} + \frac{59 i \left (x + 1\right )^{\frac{3}{2}}}{6 \sqrt{x - 1}} - \frac{11 i \sqrt{x + 1}}{\sqrt{x - 1}} & \text{for}\: \frac{\left |{x + 1}\right |}{2} > 1 \\5 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )} - \frac{\left (x + 1\right )^{\frac{7}{2}}}{3 \sqrt{1 - x}} + \frac{17 \left (x + 1\right )^{\frac{5}{2}}}{6 \sqrt{1 - x}} - \frac{59 \left (x + 1\right )^{\frac{3}{2}}}{6 \sqrt{1 - x}} + \frac{11 \sqrt{x + 1}}{\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) + I*(x + 1)**(7/2)/(3*sqrt(x - 1)) - 17*I*(x + 1)**(5/2)/(6*sqrt(
x - 1)) + 59*I*(x + 1)**(3/2)/(6*sqrt(x - 1)) - 11*I*sqrt(x + 1)/sqrt(x - 1), Abs(x + 1)/2 > 1), (5*asin(sqrt(
2)*sqrt(x + 1)/2) - (x + 1)**(7/2)/(3*sqrt(1 - x)) + 17*(x + 1)**(5/2)/(6*sqrt(1 - x)) - 59*(x + 1)**(3/2)/(6*
sqrt(1 - x)) + 11*sqrt(x + 1)/sqrt(1 - x), True))

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Giac [A]  time = 1.07833, size = 93, normalized size = 1.39 \begin{align*} \frac{1}{6} \,{\left ({\left (2 \, x - 5\right )}{\left (x + 1\right )} + 9\right )} \sqrt{x + 1} \sqrt{-x + 1} - \sqrt{x + 1}{\left (x - 2\right )} \sqrt{-x + 1} + \sqrt{x + 1} \sqrt{-x + 1} + 5 \, \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]

1/6*((2*x - 5)*(x + 1) + 9)*sqrt(x + 1)*sqrt(-x + 1) - sqrt(x + 1)*(x - 2)*sqrt(-x + 1) + sqrt(x + 1)*sqrt(-x
+ 1) + 5*arcsin(1/2*sqrt(2)*sqrt(x + 1))