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

Optimal. Leaf size=41 \[ -\frac {2 (1-x)^{3/2}}{\sqrt {x+1}}-3 \sqrt {x+1} \sqrt {1-x}-3 \sin ^{-1}(x) \]

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Rubi [A]  time = 0.01, antiderivative size = 41, 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 = {47, 50, 41, 216} \begin {gather*} -\frac {2 (1-x)^{3/2}}{\sqrt {x+1}}-3 \sqrt {x+1} \sqrt {1-x}-3 \sin ^{-1}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - x)^(3/2)/(1 + x)^(3/2),x]

[Out]

(-2*(1 - x)^(3/2))/Sqrt[1 + x] - 3*Sqrt[1 - x]*Sqrt[1 + x] - 3*ArcSin[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 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

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

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Mathematica [C]  time = 0.01, size = 37, normalized size = 0.90 \begin {gather*} -\frac {(1-x)^{5/2} \, _2F_1\left (\frac {3}{2},\frac {5}{2};\frac {7}{2};\frac {1-x}{2}\right )}{5 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/5*((1 - x)^(5/2)*Hypergeometric2F1[3/2, 5/2, 7/2, (1 - x)/2])/Sqrt[2]

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IntegrateAlgebraic [C]  time = 0.13, size = 49, normalized size = 1.20 \begin {gather*} \frac {(-x-5) \sqrt {1-x}}{\sqrt {x+1}}-6 i \log \left (\sqrt {1-x}-i \sqrt {x+1}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-5 - x)*Sqrt[1 - x])/Sqrt[1 + x] - (6*I)*Log[Sqrt[1 - x] - I*Sqrt[1 + x]]

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fricas [A]  time = 1.39, size = 53, normalized size = 1.29 \begin {gather*} -\frac {{\left (x + 5\right )} \sqrt {x + 1} \sqrt {-x + 1} - 6 \, {\left (x + 1\right )} \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) + 5 \, x + 5}{x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((x + 5)*sqrt(x + 1)*sqrt(-x + 1) - 6*(x + 1)*arctan((sqrt(x + 1)*sqrt(-x + 1) - 1)/x) + 5*x + 5)/(x + 1)

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giac [B]  time = 0.73, size = 70, normalized size = 1.71 \begin {gather*} -\sqrt {x + 1} \sqrt {-x + 1} + \frac {2 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}}{\sqrt {x + 1}} - \frac {2 \, \sqrt {x + 1}}{\sqrt {2} - \sqrt {-x + 1}} - 6 \, \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)^(3/2),x, algorithm="giac")

[Out]

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

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maple [B]  time = 0.02, size = 71, normalized size = 1.73 \begin {gather*} -\frac {3 \sqrt {\left (x +1\right ) \left (-x +1\right )}\, \arcsin \relax (x )}{\sqrt {x +1}\, \sqrt {-x +1}}+\frac {\left (x^{2}+4 x -5\right ) \sqrt {\left (x +1\right ) \left (-x +1\right )}}{\sqrt {-\left (x +1\right ) \left (x -1\right )}\, \sqrt {-x +1}\, \sqrt {x +1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 2.86, size = 41, normalized size = 1.00 \begin {gather*} \frac {{\left (-x^{2} + 1\right )}^{\frac {3}{2}}}{x^{2} + 2 \, x + 1} - \frac {6 \, \sqrt {-x^{2} + 1}}{x + 1} - 3 \, \arcsin \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - x)^(3/2)/(x + 1)^(3/2),x)

[Out]

int((1 - x)^(3/2)/(x + 1)^(3/2), x)

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sympy [A]  time = 2.48, size = 133, normalized size = 3.24 \begin {gather*} \begin {cases} 6 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} - \frac {i \left (x + 1\right )^{\frac {3}{2}}}{\sqrt {x - 1}} - \frac {2 i \sqrt {x + 1}}{\sqrt {x - 1}} + \frac {8 i}{\sqrt {x - 1} \sqrt {x + 1}} & \text {for}\: \frac {\left |{x + 1}\right |}{2} > 1 \\- 6 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} + \frac {\left (x + 1\right )^{\frac {3}{2}}}{\sqrt {1 - x}} + \frac {2 \sqrt {x + 1}}{\sqrt {1 - x}} - \frac {8}{\sqrt {1 - x} \sqrt {x + 1}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)**(3/2)/(1+x)**(3/2),x)

[Out]

Piecewise((6*I*acosh(sqrt(2)*sqrt(x + 1)/2) - I*(x + 1)**(3/2)/sqrt(x - 1) - 2*I*sqrt(x + 1)/sqrt(x - 1) + 8*I
/(sqrt(x - 1)*sqrt(x + 1)), Abs(x + 1)/2 > 1), (-6*asin(sqrt(2)*sqrt(x + 1)/2) + (x + 1)**(3/2)/sqrt(1 - x) +
2*sqrt(x + 1)/sqrt(1 - x) - 8/(sqrt(1 - x)*sqrt(x + 1)), True))

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