3.12.7 \(\int \frac {(1-x)^{5/2}}{\sqrt {1+x}} \, dx\) [1107]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 67, normalized size of antiderivative = 1.00, 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 = {52, 41, 222} \begin {gather*} \frac {5 \text {ArcSin}(x)}{2}+\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} \end {gather*}

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 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 52

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 222

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*}

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Mathematica [A]
time = 0.06, size = 56, normalized size = 0.84 \begin {gather*} \frac {\sqrt {1+x} \left (22-31 x+11 x^2-2 x^3\right )}{6 \sqrt {1-x}}+5 \tan ^{-1}\left (\frac {\sqrt {1+x}}{\sqrt {1-x}}\right ) \end {gather*}

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*ArcTan[Sqrt[1 + x]/Sqrt[1 - x]]

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Maple [A]
time = 0.16, size = 71, normalized size = 1.06

method	result	size
default	\(\frac {\left (1-x \right )^{\frac {5}{2}} \sqrt {1+x}}{3}+\frac {5 \left (1-x \right )^{\frac {3}{2}} \sqrt {1+x}}{6}+\frac {5 \sqrt {1-x}\, \sqrt {1+x}}{2}+\frac {5 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{2 \sqrt {1+x}\, \sqrt {1-x}}\)	\(71\)
risch	\(-\frac {\left (2 x^{2}-9 x +22\right ) \sqrt {1+x}\, \left (-1+x \right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{6 \sqrt {-\left (1+x \right ) \left (-1+x \right )}\, \sqrt {1-x}}+\frac {5 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{2 \sqrt {1+x}\, \sqrt {1-x}}\)	\(77\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-x)^(5/2)/(1+x)^(1/2),x,method=_RETURNVERBOSE)

[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 = 0.50, size = 42, normalized size = 0.63 \begin {gather*} \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 {gather*}

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 = 0.92, size = 47, normalized size = 0.70 \begin {gather*} \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 {gather*}

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 [C] Result contains complex when optimal does not.
time = 4.11, size = 173, normalized size = 2.58 \begin {gather*} \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}\: \left |{x + 1}\right | > 2 \\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 {gather*}

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), (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*sq
rt(1 - x)) + 11*sqrt(x + 1)/sqrt(1 - x), True))

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Giac [A]
time = 1.50, size = 69, normalized size = 1.03 \begin {gather*} \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 {gather*}

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))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (1-x\right )}^{5/2}}{\sqrt {x+1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - x)^(5/2)/(x + 1)^(1/2),x)

[Out]

int((1 - x)^(5/2)/(x + 1)^(1/2), x)

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