3.11.13 \(\int \frac {(1+x)^{3/2}}{(1-x)^{5/2}} \, dx\)

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

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {47, 41, 216} \begin {gather*} \frac {2 (x+1)^{3/2}}{3 (1-x)^{3/2}}-\frac {2 \sqrt {x+1}}{\sqrt {1-x}}+\sin ^{-1}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + x)^(3/2)/(1 - x)^(5/2),x]

[Out]

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

________________________________________________________________________________________

Mathematica [C]  time = 0.01, size = 37, normalized size = 0.90 \begin {gather*} \frac {4 \sqrt {2} \, _2F_1\left (-\frac {3}{2},-\frac {3}{2};-\frac {1}{2};\frac {1-x}{2}\right )}{3 (1-x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + x)^(3/2)/(1 - x)^(5/2),x]

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.06, size = 55, normalized size = 1.34 \begin {gather*} -\frac {2 \left (\frac {3 (1-x)}{x+1}-1\right ) (x+1)^{3/2}}{3 (1-x)^{3/2}}-2 \tan ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {x+1}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(1 + x)^(3/2)/(1 - x)^(5/2),x]

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 1.31, size = 71, normalized size = 1.73 \begin {gather*} -\frac {2 \, {\left (2 \, x^{2} - 2 \, {\left (2 \, x - 1\right )} \sqrt {x + 1} \sqrt {-x + 1} + 3 \, {\left (x^{2} - 2 \, x + 1\right )} \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) - 4 \, x + 2\right )}}{3 \, {\left (x^{2} - 2 \, x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)^(3/2)/(1-x)^(5/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [A]  time = 1.02, size = 38, normalized size = 0.93 \begin {gather*} \frac {4 \, {\left (2 \, x - 1\right )} \sqrt {x + 1} \sqrt {-x + 1}}{3 \, {\left (x - 1\right )}^{2}} + 2 \, \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)^(5/2),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [B]  time = 0.02, size = 76, normalized size = 1.85 \begin {gather*} \frac {\sqrt {\left (x +1\right ) \left (-x +1\right )}\, \arcsin \relax (x )}{\sqrt {x +1}\, \sqrt {-x +1}}-\frac {4 \left (2 x^{2}+x -1\right ) \sqrt {\left (x +1\right ) \left (-x +1\right )}}{3 \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)^(5/2),x)

[Out]

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

________________________________________________________________________________________

maxima [B]  time = 2.97, size = 66, normalized size = 1.61 \begin {gather*} -\frac {{\left (-x^{2} + 1\right )}^{\frac {3}{2}}}{3 \, {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} + \frac {2 \, \sqrt {-x^{2} + 1}}{3 \, {\left (x^{2} - 2 \, x + 1\right )}} + \frac {7 \, \sqrt {-x^{2} + 1}}{3 \, {\left (x - 1\right )}} + \arcsin \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)^(3/2)/(1-x)^(5/2),x, algorithm="maxima")

[Out]

-1/3*(-x^2 + 1)^(3/2)/(x^3 - 3*x^2 + 3*x - 1) + 2/3*sqrt(-x^2 + 1)/(x^2 - 2*x + 1) + 7/3*sqrt(-x^2 + 1)/(x - 1
) + arcsin(x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\left (x+1\right )}^{3/2}}{{\left (1-x\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + 1)^(3/2)/(1 - x)^(5/2),x)

[Out]

int((x + 1)^(3/2)/(1 - x)^(5/2), x)

________________________________________________________________________________________

sympy [B]  time = 3.70, size = 500, normalized size = 12.20 \begin {gather*} \begin {cases} \frac {6 i \sqrt {x - 1} \left (x + 1\right )^{\frac {15}{2}} \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{- 3 \sqrt {x - 1} \left (x + 1\right )^{\frac {15}{2}} + 6 \sqrt {x - 1} \left (x + 1\right )^{\frac {13}{2}}} - \frac {3 \pi \sqrt {x - 1} \left (x + 1\right )^{\frac {15}{2}}}{- 3 \sqrt {x - 1} \left (x + 1\right )^{\frac {15}{2}} + 6 \sqrt {x - 1} \left (x + 1\right )^{\frac {13}{2}}} - \frac {12 i \sqrt {x - 1} \left (x + 1\right )^{\frac {13}{2}} \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{- 3 \sqrt {x - 1} \left (x + 1\right )^{\frac {15}{2}} + 6 \sqrt {x - 1} \left (x + 1\right )^{\frac {13}{2}}} + \frac {6 \pi \sqrt {x - 1} \left (x + 1\right )^{\frac {13}{2}}}{- 3 \sqrt {x - 1} \left (x + 1\right )^{\frac {15}{2}} + 6 \sqrt {x - 1} \left (x + 1\right )^{\frac {13}{2}}} - \frac {8 i \left (x + 1\right )^{8}}{- 3 \sqrt {x - 1} \left (x + 1\right )^{\frac {15}{2}} + 6 \sqrt {x - 1} \left (x + 1\right )^{\frac {13}{2}}} + \frac {12 i \left (x + 1\right )^{7}}{- 3 \sqrt {x - 1} \left (x + 1\right )^{\frac {15}{2}} + 6 \sqrt {x - 1} \left (x + 1\right )^{\frac {13}{2}}} & \text {for}\: \frac {\left |{x + 1}\right |}{2} > 1 \\\frac {6 \sqrt {1 - x} \left (x + 1\right )^{\frac {15}{2}} \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{3 \sqrt {1 - x} \left (x + 1\right )^{\frac {15}{2}} - 6 \sqrt {1 - x} \left (x + 1\right )^{\frac {13}{2}}} - \frac {12 \sqrt {1 - x} \left (x + 1\right )^{\frac {13}{2}} \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{3 \sqrt {1 - x} \left (x + 1\right )^{\frac {15}{2}} - 6 \sqrt {1 - x} \left (x + 1\right )^{\frac {13}{2}}} - \frac {8 \left (x + 1\right )^{8}}{3 \sqrt {1 - x} \left (x + 1\right )^{\frac {15}{2}} - 6 \sqrt {1 - x} \left (x + 1\right )^{\frac {13}{2}}} + \frac {12 \left (x + 1\right )^{7}}{3 \sqrt {1 - x} \left (x + 1\right )^{\frac {15}{2}} - 6 \sqrt {1 - x} \left (x + 1\right )^{\frac {13}{2}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)**(3/2)/(1-x)**(5/2),x)

[Out]

Piecewise((6*I*sqrt(x - 1)*(x + 1)**(15/2)*acosh(sqrt(2)*sqrt(x + 1)/2)/(-3*sqrt(x - 1)*(x + 1)**(15/2) + 6*sq
rt(x - 1)*(x + 1)**(13/2)) - 3*pi*sqrt(x - 1)*(x + 1)**(15/2)/(-3*sqrt(x - 1)*(x + 1)**(15/2) + 6*sqrt(x - 1)*
(x + 1)**(13/2)) - 12*I*sqrt(x - 1)*(x + 1)**(13/2)*acosh(sqrt(2)*sqrt(x + 1)/2)/(-3*sqrt(x - 1)*(x + 1)**(15/
2) + 6*sqrt(x - 1)*(x + 1)**(13/2)) + 6*pi*sqrt(x - 1)*(x + 1)**(13/2)/(-3*sqrt(x - 1)*(x + 1)**(15/2) + 6*sqr
t(x - 1)*(x + 1)**(13/2)) - 8*I*(x + 1)**8/(-3*sqrt(x - 1)*(x + 1)**(15/2) + 6*sqrt(x - 1)*(x + 1)**(13/2)) +
12*I*(x + 1)**7/(-3*sqrt(x - 1)*(x + 1)**(15/2) + 6*sqrt(x - 1)*(x + 1)**(13/2)), Abs(x + 1)/2 > 1), (6*sqrt(1
- x)*(x + 1)**(15/2)*asin(sqrt(2)*sqrt(x + 1)/2)/(3*sqrt(1 - x)*(x + 1)**(15/2) - 6*sqrt(1 - x)*(x + 1)**(13/
2)) - 12*sqrt(1 - x)*(x + 1)**(13/2)*asin(sqrt(2)*sqrt(x + 1)/2)/(3*sqrt(1 - x)*(x + 1)**(15/2) - 6*sqrt(1 - x
)*(x + 1)**(13/2)) - 8*(x + 1)**8/(3*sqrt(1 - x)*(x + 1)**(15/2) - 6*sqrt(1 - x)*(x + 1)**(13/2)) + 12*(x + 1)
**7/(3*sqrt(1 - x)*(x + 1)**(15/2) - 6*sqrt(1 - x)*(x + 1)**(13/2)), True))

________________________________________________________________________________________