\(\int \frac {(1+x)^{3/2}}{\sqrt {1-x} x^3} \, dx\) [730]

Optimal result

Integrand size = 20, antiderivative size = 69 \[ \int \frac {(1+x)^{3/2}}{\sqrt {1-x} x^3} \, dx=-\frac {3 \sqrt {1-x} \sqrt {1+x}}{2 x}-\frac {\sqrt {1-x} (1+x)^{3/2}}{2 x^2}-\frac {3}{2} \text {arctanh}\left (\sqrt {1-x} \sqrt {1+x}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {96, 94, 212} \[ \int \frac {(1+x)^{3/2}}{\sqrt {1-x} x^3} \, dx=-\frac {3}{2} \text {arctanh}\left (\sqrt {1-x} \sqrt {x+1}\right )-\frac {\sqrt {1-x} (x+1)^{3/2}}{2 x^2}-\frac {3 \sqrt {1-x} \sqrt {x+1}}{2 x} \]

[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[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 96

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[m, 1] ||  !SumSimplerQ[p, 1]) && NeQ[m, -1]

Rule 212

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

Rubi steps \begin{align*} \text {integral}& = -\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} \text {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] (verified)

Time = 0.19 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.78 \[ \int \frac {(1+x)^{3/2}}{\sqrt {1-x} x^3} \, dx=-\frac {\sqrt {1-x} \left (1+5 x+4 x^2\right )}{2 x^2 \sqrt {1+x}}-3 \text {arctanh}\left (\frac {\sqrt {1-x}}{\sqrt {1+x}}\right ) \]

[In]

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

[Out]

-1/2*(Sqrt[1 - x]*(1 + 5*x + 4*x^2))/(x^2*Sqrt[1 + x]) - 3*ArcTanh[Sqrt[1 - x]/Sqrt[1 + x]]

Maple [A] (verified)

Time = 0.57 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.93

method	result	size
default	\(-\frac {\sqrt {1+x}\, \sqrt {1-x}\, \left (3 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-x^{2}+1}}\right ) x^{2}+4 x \sqrt {-x^{2}+1}+\sqrt {-x^{2}+1}\right )}{2 x^{2} \sqrt {-x^{2}+1}}\)	\(64\)
risch	\(\frac {\left (-1+x \right ) \sqrt {1+x}\, \left (1+4 x \right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{2 x^{2} \sqrt {-\left (-1+x \right ) \left (1+x \right )}\, \sqrt {1-x}}-\frac {3 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-x^{2}+1}}\right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{2 \sqrt {1-x}\, \sqrt {1+x}}\)	\(83\)

[In]

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

[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)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.72 \[ \int \frac {(1+x)^{3/2}}{\sqrt {1-x} x^3} \, dx=\frac {3 \, x^{2} \log \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) - {\left (4 \, x + 1\right )} \sqrt {x + 1} \sqrt {-x + 1}}{2 \, x^{2}} \]

[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]

\[ \int \frac {(1+x)^{3/2}}{\sqrt {1-x} x^3} \, dx=\int \frac {\left (x + 1\right )^{\frac {3}{2}}}{x^{3} \sqrt {1 - x}}\, dx \]

[In]

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

[Out]

Integral((x + 1)**(3/2)/(x**3*sqrt(1 - x)), x)

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.78 \[ \int \frac {(1+x)^{3/2}}{\sqrt {1-x} x^3} \, dx=-\frac {2 \, \sqrt {-x^{2} + 1}}{x} - \frac {\sqrt {-x^{2} + 1}}{2 \, x^{2}} - \frac {3}{2} \, \log \left (\frac {2 \, \sqrt {-x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) \]

[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))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 234 vs. \(2 (51) = 102\).

Time = 0.34 (sec) , antiderivative size = 234, normalized size of antiderivative = 3.39 \[ \int \frac {(1+x)^{3/2}}{\sqrt {1-x} x^3} \, dx=-\frac {2 \, {\left (3 \, {\left (\frac {\sqrt {2} - \sqrt {-x + 1}}{\sqrt {x + 1}} - \frac {\sqrt {x + 1}}{\sqrt {2} - \sqrt {-x + 1}}\right )}^{3} - \frac {20 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}}{\sqrt {x + 1}} + \frac {20 \, \sqrt {x + 1}}{\sqrt {2} - \sqrt {-x + 1}}\right )}}{{\left ({\left (\frac {\sqrt {2} - \sqrt {-x + 1}}{\sqrt {x + 1}} - \frac {\sqrt {x + 1}}{\sqrt {2} - \sqrt {-x + 1}}\right )}^{2} - 4\right )}^{2}} - \frac {3}{2} \, \log \left ({\left | -\frac {\sqrt {2} - \sqrt {-x + 1}}{\sqrt {x + 1}} + \frac {\sqrt {x + 1}}{\sqrt {2} - \sqrt {-x + 1}} + 2 \right |}\right ) + \frac {3}{2} \, \log \left ({\left | -\frac {\sqrt {2} - \sqrt {-x + 1}}{\sqrt {x + 1}} + \frac {\sqrt {x + 1}}{\sqrt {2} - \sqrt {-x + 1}} - 2 \right |}\right ) \]

[In]

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

[Out]

-2*(3*((sqrt(2) - sqrt(-x + 1))/sqrt(x + 1) - sqrt(x + 1)/(sqrt(2) - sqrt(-x + 1)))^3 - 20*(sqrt(2) - sqrt(-x
+ 1))/sqrt(x + 1) + 20*sqrt(x + 1)/(sqrt(2) - sqrt(-x + 1)))/(((sqrt(2) - sqrt(-x + 1))/sqrt(x + 1) - sqrt(x +
1)/(sqrt(2) - sqrt(-x + 1)))^2 - 4)^2 - 3/2*log(abs(-(sqrt(2) - sqrt(-x + 1))/sqrt(x + 1) + sqrt(x + 1)/(sqrt
(2) - sqrt(-x + 1)) + 2)) + 3/2*log(abs(-(sqrt(2) - sqrt(-x + 1))/sqrt(x + 1) + sqrt(x + 1)/(sqrt(2) - sqrt(-x
+ 1)) - 2))

Mupad [F(-1)]

Timed out. \[ \int \frac {(1+x)^{3/2}}{\sqrt {1-x} x^3} \, dx=\int \frac {{\left (x+1\right )}^{3/2}}{x^3\,\sqrt {1-x}} \,d x \]

[In]

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

[Out]

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