3.1094 \(\int \frac{(1+x)^{5/2}}{\sqrt{1-x}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0103963, 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} \[ -\frac{1}{3} \sqrt{1-x} (x+1)^{5/2}-\frac{5}{6} \sqrt{1-x} (x+1)^{3/2}-\frac{5}{2} \sqrt{1-x} \sqrt{x+1}+\frac{5}{2} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[(1 + x)^(5/2)/Sqrt[1 - x],x]

[Out]

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

Mathematica [A]  time = 0.0220195, size = 44, normalized size = 0.66 \[ -\frac{1}{6} \sqrt{1-x^2} \left (2 x^2+9 x+22\right )-5 \sin ^{-1}\left (\frac{\sqrt{1-x}}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[1 - x^2]*(22 + 9*x + 2*x^2))/6 - 5*ArcSin[Sqrt[1 - x]/Sqrt[2]]

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Maple [A]  time = 0.004, size = 71, normalized size = 1.1 \begin{align*} -{\frac{1}{3}\sqrt{1-x} \left ( 1+x \right ) ^{{\frac{5}{2}}}}-{\frac{5}{6}\sqrt{1-x} \left ( 1+x \right ) ^{{\frac{3}{2}}}}-{\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)^(1/2)*(1+x)^(5/2)-5/6*(1-x)^(1/2)*(1+x)^(3/2)-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 = 1.55981, 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 = 1.63123, size = 128, normalized size = 1.91 \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 = 11.5651, size = 172, normalized size = 2.57 \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{i \left (x + 1\right )^{\frac{5}{2}}}{6 \sqrt{x - 1}} - \frac{5 i \left (x + 1\right )^{\frac{3}{2}}}{6 \sqrt{x - 1}} + \frac{5 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{\left (x + 1\right )^{\frac{5}{2}}}{6 \sqrt{1 - x}} + \frac{5 \left (x + 1\right )^{\frac{3}{2}}}{6 \sqrt{1 - x}} - \frac{5 \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)) - I*(x + 1)**(5/2)/(6*sqrt(x -
1)) - 5*I*(x + 1)**(3/2)/(6*sqrt(x - 1)) + 5*I*sqrt(x + 1)/sqrt(x - 1), Abs(x + 1)/2 > 1), (5*asin(sqrt(2)*sq
rt(x + 1)/2) + (x + 1)**(7/2)/(3*sqrt(1 - x)) + (x + 1)**(5/2)/(6*sqrt(1 - x)) + 5*(x + 1)**(3/2)/(6*sqrt(1 -
x)) - 5*sqrt(x + 1)/sqrt(1 - x), True))

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Giac [A]  time = 1.0604, size = 53, normalized size = 0.79 \begin{align*} -\frac{1}{6} \,{\left ({\left (2 \, x + 7\right )}{\left (x + 1\right )} + 15\right )} \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 + 7)*(x + 1) + 15)*sqrt(x + 1)*sqrt(-x + 1) + 5*arcsin(1/2*sqrt(2)*sqrt(x + 1))