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

Optimal. Leaf size=48 \[ \frac {1}{3} (1-x)^{3/2} (x+1)^{3/2}+\frac {1}{2} \sqrt {1-x} x \sqrt {x+1}+\frac {1}{2} \sin ^{-1}(x) \]

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Rubi [A]  time = 0.01, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {49, 38, 41, 216} \begin {gather*} \frac {1}{3} (1-x)^{3/2} (x+1)^{3/2}+\frac {1}{2} \sqrt {1-x} x \sqrt {x+1}+\frac {1}{2} \sin ^{-1}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - x)^(3/2)*Sqrt[1 + x],x]

[Out]

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

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

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Mathematica [A]  time = 0.04, size = 44, normalized size = 0.92 \begin {gather*} \frac {1}{6} \left (-2 x^2+3 x+2\right ) \sqrt {1-x^2}-\sin ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.07, size = 82, normalized size = 1.71 \begin {gather*} \frac {\sqrt {x+1} \left (\frac {3 (x+1)^2}{(1-x)^2}+\frac {8 (x+1)}{1-x}-3\right )}{3 \sqrt {1-x} \left (\frac {x+1}{1-x}+1\right )^3}+\tan ^{-1}\left (\frac {\sqrt {x+1}}{\sqrt {1-x}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(1 - x)^(3/2)*Sqrt[1 + x],x]

[Out]

(Sqrt[1 + x]*(-3 + (8*(1 + x))/(1 - x) + (3*(1 + x)^2)/(1 - x)^2))/(3*Sqrt[1 - x]*(1 + (1 + x)/(1 - x))^3) + A
rcTan[Sqrt[1 + x]/Sqrt[1 - x]]

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fricas [A]  time = 1.36, size = 47, normalized size = 0.98 \begin {gather*} -\frac {1}{6} \, {\left (2 \, x^{2} - 3 \, x - 2\right )} \sqrt {x + 1} \sqrt {-x + 1} - \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)^(3/2)*(1+x)^(1/2),x, algorithm="fricas")

[Out]

-1/6*(2*x^2 - 3*x - 2)*sqrt(x + 1)*sqrt(-x + 1) - arctan((sqrt(x + 1)*sqrt(-x + 1) - 1)/x)

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giac [A]  time = 1.02, size = 50, normalized size = 1.04 \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} \sqrt {-x + 1} + \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)^(3/2)*(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)*sqrt(-x + 1) + arcsin(1/2*sqrt(2)*sqrt(x +
1))

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maple [B]  time = 0.00, size = 71, normalized size = 1.48 \begin {gather*} \frac {\sqrt {\left (x +1\right ) \left (-x +1\right )}\, \arcsin \relax (x )}{2 \sqrt {x +1}\, \sqrt {-x +1}}+\frac {\left (-x +1\right )^{\frac {3}{2}} \left (x +1\right )^{\frac {3}{2}}}{3}+\frac {\sqrt {-x +1}\, \left (x +1\right )^{\frac {3}{2}}}{2}-\frac {\sqrt {-x +1}\, \sqrt {x +1}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x+1)^(3/2)*(x+1)^(1/2),x)

[Out]

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

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maxima [A]  time = 2.90, size = 28, normalized size = 0.58 \begin {gather*} \frac {1}{3} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} + \frac {1}{2} \, \sqrt {-x^{2} + 1} x + \frac {1}{2} \, \arcsin \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int {\left (1-x\right )}^{3/2}\,\sqrt {x+1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - x)^(3/2)*(x + 1)^(1/2),x)

[Out]

int((1 - x)^(3/2)*(x + 1)^(1/2), x)

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sympy [B]  time = 4.49, size = 168, normalized size = 3.50 \begin {gather*} \begin {cases} - 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 {11 i \left (x + 1\right )^{\frac {5}{2}}}{6 \sqrt {x - 1}} - \frac {17 i \left (x + 1\right )^{\frac {3}{2}}}{6 \sqrt {x - 1}} + \frac {i \sqrt {x + 1}}{\sqrt {x - 1}} & \text {for}\: \frac {\left |{x + 1}\right |}{2} > 1 \\\operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} + \frac {\left (x + 1\right )^{\frac {7}{2}}}{3 \sqrt {1 - x}} - \frac {11 \left (x + 1\right )^{\frac {5}{2}}}{6 \sqrt {1 - x}} + \frac {17 \left (x + 1\right )^{\frac {3}{2}}}{6 \sqrt {1 - x}} - \frac {\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)**(3/2)*(1+x)**(1/2),x)

[Out]

Piecewise((-I*acosh(sqrt(2)*sqrt(x + 1)/2) - I*(x + 1)**(7/2)/(3*sqrt(x - 1)) + 11*I*(x + 1)**(5/2)/(6*sqrt(x
- 1)) - 17*I*(x + 1)**(3/2)/(6*sqrt(x - 1)) + I*sqrt(x + 1)/sqrt(x - 1), Abs(x + 1)/2 > 1), (asin(sqrt(2)*sqrt
(x + 1)/2) + (x + 1)**(7/2)/(3*sqrt(1 - x)) - 11*(x + 1)**(5/2)/(6*sqrt(1 - x)) + 17*(x + 1)**(3/2)/(6*sqrt(1
- x)) - sqrt(x + 1)/sqrt(1 - x), True))

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