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

Optimal. Leaf size=69 \[ -\frac{\sqrt{1-x} (x+1)^{3/2}}{2 x^2}-\frac{3 \sqrt{1-x} \sqrt{x+1}}{2 x}-\frac{3}{2} \tanh ^{-1}\left (\sqrt{1-x} \sqrt{x+1}\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0100055, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {94, 92, 206} \[ -\frac{\sqrt{1-x} (x+1)^{3/2}}{2 x^2}-\frac{3 \sqrt{1-x} \sqrt{x+1}}{2 x}-\frac{3}{2} \tanh ^{-1}\left (\sqrt{1-x} \sqrt{x+1}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 94

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a + b
*x)^(m + 1)*(c + d*x)^n*(e + f*x)^(p + 1))/((m + 1)*(b*e - a*f)), x] - Dist[(n*(d*e - c*f))/((m + 1)*(b*e - a*
f)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[
m + n + p + 2, 0] && GtQ[n, 0] &&  !(SumSimplerQ[p, 1] &&  !SumSimplerQ[m, 1])

Rule 92

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))), x_Symbol] :> Dist[b*f, Subst[I
nt[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sqrt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] &&
EqQ[2*b*d*e - f*(b*c + a*d), 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{(1+x)^{3/2}}{\sqrt{1-x} x^3} \, dx &=-\frac{\sqrt{1-x} (1+x)^{3/2}}{2 x^2}+\frac{3}{2} \int \frac{\sqrt{1+x}}{\sqrt{1-x} x^2} \, dx\\ &=-\frac{3 \sqrt{1-x} \sqrt{1+x}}{2 x}-\frac{\sqrt{1-x} (1+x)^{3/2}}{2 x^2}+\frac{3}{2} \int \frac{1}{\sqrt{1-x} x \sqrt{1+x}} \, dx\\ &=-\frac{3 \sqrt{1-x} \sqrt{1+x}}{2 x}-\frac{\sqrt{1-x} (1+x)^{3/2}}{2 x^2}-\frac{3}{2} \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt{1-x} \sqrt{1+x}\right )\\ &=-\frac{3 \sqrt{1-x} \sqrt{1+x}}{2 x}-\frac{\sqrt{1-x} (1+x)^{3/2}}{2 x^2}-\frac{3}{2} \tanh ^{-1}\left (\sqrt{1-x} \sqrt{1+x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0136508, size = 59, normalized size = 0.86 \[ \frac{4 x^3+x^2-3 \sqrt{1-x^2} x^2 \tanh ^{-1}\left (\sqrt{1-x^2}\right )-4 x-1}{2 x^2 \sqrt{1-x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-1 - 4*x + x^2 + 4*x^3 - 3*x^2*Sqrt[1 - x^2]*ArcTanh[Sqrt[1 - x^2]])/(2*x^2*Sqrt[1 - x^2])

________________________________________________________________________________________

Maple [A]  time = 0.01, size = 64, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,{x}^{2}}\sqrt{1-x}\sqrt{1+x} \left ( 3\,{\it Artanh} \left ({\frac{1}{\sqrt{-{x}^{2}+1}}} \right ){x}^{2}+4\,x\sqrt{-{x}^{2}+1}+\sqrt{-{x}^{2}+1} \right ){\frac{1}{\sqrt{-{x}^{2}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.71664, size = 73, normalized size = 1.06 \begin{align*} -\frac{2 \, \sqrt{-x^{2} + 1}}{x} - \frac{\sqrt{-x^{2} + 1}}{2 \, x^{2}} - \frac{3}{2} \, \log \left (\frac{2 \, \sqrt{-x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.66511, size = 124, normalized size = 1.8 \begin{align*} \frac{3 \, x^{2} \log \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) -{\left (4 \, x + 1\right )} \sqrt{x + 1} \sqrt{-x + 1}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError