3.1082 \(\int \frac{(1+x)^{3/2}}{(1-x)^{5/2}} \, dx\)

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0050945, antiderivative size = 41, normalized size of antiderivative = 1., 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} \[ \frac{2 (x+1)^{3/2}}{3 (1-x)^{3/2}}-\frac{2 \sqrt{x+1}}{\sqrt{1-x}}+\sin ^{-1}(x) \]

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 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 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)^{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 = 0.0063796, size = 37, normalized size = 0.9 \[ \frac{4 \sqrt{2} \, _2F_1\left (-\frac{3}{2},-\frac{3}{2};-\frac{1}{2};\frac{1-x}{2}\right )}{3 (1-x)^{3/2}} \]

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))

________________________________________________________________________________________

Maple [B]  time = 0.016, size = 76, normalized size = 1.9 \begin{align*} -{\frac{8\,{x}^{2}+4\,x-4}{-3+3\,x}\sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) }{\frac{1}{\sqrt{- \left ( 1+x \right ) \left ( -1+x \right ) }}}{\frac{1}{\sqrt{1-x}}}{\frac{1}{\sqrt{1+x}}}}+{\arcsin \left ( x \right ) \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)^(3/2)/(1-x)^(5/2),x)

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 1.51751, size = 89, normalized size = 2.17 \begin{align*} -\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 \left (x\right ) \end{align*}

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)

________________________________________________________________________________________

Fricas [B]  time = 1.46875, size = 189, normalized size = 4.61 \begin{align*} -\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{align*}

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)

________________________________________________________________________________________

Sympy [B]  time = 5.58419, size = 500, normalized size = 12.2 \begin{align*} \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{align*}

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*sqrt(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))

________________________________________________________________________________________

Giac [A]  time = 1.07073, size = 51, normalized size = 1.24 \begin{align*} \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{align*}

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))