\(\int \frac {\sqrt {1+x}}{(1-x)^{5/2} x^2} \, dx\) [834]

Optimal result

Integrand size = 20, antiderivative size = 87 \[ \int \frac {\sqrt {1+x}}{(1-x)^{5/2} x^2} \, dx=\frac {14 \sqrt {1+x}}{3 \sqrt {1-x}}+\frac {2 \sqrt {1+x}}{3 (1-x)^{3/2} x}-\frac {5 \sqrt {1+x}}{3 \sqrt {1-x} x}-3 \text {arctanh}\left (\sqrt {1-x} \sqrt {1+x}\right ) \]

[Out]

-3*arctanh((1-x)^(1/2)*(1+x)^(1/2))+2/3*(1+x)^(1/2)/(1-x)^(3/2)/x+14/3*(1+x)^(1/2)/(1-x)^(1/2)-5/3*(1+x)^(1/2)
/x/(1-x)^(1/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {101, 156, 157, 12, 94, 212} \[ \int \frac {\sqrt {1+x}}{(1-x)^{5/2} x^2} \, dx=-3 \text {arctanh}\left (\sqrt {1-x} \sqrt {x+1}\right )+\frac {14 \sqrt {x+1}}{3 \sqrt {1-x}}-\frac {5 \sqrt {x+1}}{3 \sqrt {1-x} x}+\frac {2 \sqrt {x+1}}{3 (1-x)^{3/2} x} \]

[In]

Int[Sqrt[1 + x]/((1 - x)^(5/2)*x^2),x]

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 101

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[1/((m + 1)*(b*e - a*f)), Int[(a +
b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p
] || IntegersQ[p, m + n])

Rule 156

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f
))), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]

Rule 157

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f
))), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n, 2*p]

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 {2 \sqrt {1+x}}{3 (1-x)^{3/2} x}-\frac {2}{3} \int \frac {-\frac {5}{2}-2 x}{(1-x)^{3/2} x^2 \sqrt {1+x}} \, dx \\ & = \frac {2 \sqrt {1+x}}{3 (1-x)^{3/2} x}-\frac {5 \sqrt {1+x}}{3 \sqrt {1-x} x}+\frac {2}{3} \int \frac {\frac {9}{2}+\frac {5 x}{2}}{(1-x)^{3/2} x \sqrt {1+x}} \, dx \\ & = \frac {14 \sqrt {1+x}}{3 \sqrt {1-x}}+\frac {2 \sqrt {1+x}}{3 (1-x)^{3/2} x}-\frac {5 \sqrt {1+x}}{3 \sqrt {1-x} x}-\frac {2}{3} \int -\frac {9}{2 \sqrt {1-x} x \sqrt {1+x}} \, dx \\ & = \frac {14 \sqrt {1+x}}{3 \sqrt {1-x}}+\frac {2 \sqrt {1+x}}{3 (1-x)^{3/2} x}-\frac {5 \sqrt {1+x}}{3 \sqrt {1-x} x}+3 \int \frac {1}{\sqrt {1-x} x \sqrt {1+x}} \, dx \\ & = \frac {14 \sqrt {1+x}}{3 \sqrt {1-x}}+\frac {2 \sqrt {1+x}}{3 (1-x)^{3/2} x}-\frac {5 \sqrt {1+x}}{3 \sqrt {1-x} x}-3 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {1-x} \sqrt {1+x}\right ) \\ & = \frac {14 \sqrt {1+x}}{3 \sqrt {1-x}}+\frac {2 \sqrt {1+x}}{3 (1-x)^{3/2} x}-\frac {5 \sqrt {1+x}}{3 \sqrt {1-x} x}-3 \tanh ^{-1}\left (\sqrt {1-x} \sqrt {1+x}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.62 \[ \int \frac {\sqrt {1+x}}{(1-x)^{5/2} x^2} \, dx=-\frac {\sqrt {1+x} \left (3-19 x+14 x^2\right )}{3 (1-x)^{3/2} x}-6 \text {arctanh}\left (\frac {\sqrt {1+x}}{\sqrt {1-x}}\right ) \]

[In]

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

[Out]

-1/3*(Sqrt[1 + x]*(3 - 19*x + 14*x^2))/((1 - x)^(3/2)*x) - 6*ArcTanh[Sqrt[1 + x]/Sqrt[1 - x]]

Maple [A] (verified)

Time = 1.78 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.09

method	result	size
risch	\(\frac {\left (14 x^{3}-5 x^{2}-16 x +3\right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{3 x \sqrt {-\left (-1+x \right ) \left (1+x \right )}\, \left (-1+x \right ) \sqrt {1-x}\, \sqrt {1+x}}-\frac {3 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-x^{2}+1}}\right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{\sqrt {1-x}\, \sqrt {1+x}}\)	\(95\)
default	\(-\frac {\left (9 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-x^{2}+1}}\right ) x^{3}-18 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-x^{2}+1}}\right ) x^{2}+14 x^{2} \sqrt {-x^{2}+1}+9 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-x^{2}+1}}\right ) x -19 x \sqrt {-x^{2}+1}+3 \sqrt {-x^{2}+1}\right ) \sqrt {1-x}\, \sqrt {1+x}}{3 x \left (-1+x \right )^{2} \sqrt {-x^{2}+1}}\)	\(113\)

[In]

int((1+x)^(1/2)/(1-x)^(5/2)/x^2,x,method=_RETURNVERBOSE)

[Out]

1/3*(14*x^3-5*x^2-16*x+3)/x/(-(-1+x)*(1+x))^(1/2)/(-1+x)*((1+x)*(1-x))^(1/2)/(1-x)^(1/2)/(1+x)^(1/2)-3*arctanh
(1/(-x^2+1)^(1/2))*((1+x)*(1-x))^(1/2)/(1-x)^(1/2)/(1+x)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.97 \[ \int \frac {\sqrt {1+x}}{(1-x)^{5/2} x^2} \, dx=\frac {13 \, x^{3} - 26 \, x^{2} - {\left (14 \, x^{2} - 19 \, x + 3\right )} \sqrt {x + 1} \sqrt {-x + 1} + 9 \, {\left (x^{3} - 2 \, x^{2} + x\right )} \log \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) + 13 \, x}{3 \, {\left (x^{3} - 2 \, x^{2} + x\right )}} \]

[In]

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

[Out]

1/3*(13*x^3 - 26*x^2 - (14*x^2 - 19*x + 3)*sqrt(x + 1)*sqrt(-x + 1) + 9*(x^3 - 2*x^2 + x)*log((sqrt(x + 1)*sqr
t(-x + 1) - 1)/x) + 13*x)/(x^3 - 2*x^2 + x)

Sympy [F]

\[ \int \frac {\sqrt {1+x}}{(1-x)^{5/2} x^2} \, dx=\int \frac {\sqrt {x + 1}}{x^{2} \left (1 - x\right )^{\frac {5}{2}}}\, dx \]

[In]

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

[Out]

Integral(sqrt(x + 1)/(x**2*(1 - x)**(5/2)), x)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.99 \[ \int \frac {\sqrt {1+x}}{(1-x)^{5/2} x^2} \, dx=\frac {14 \, x}{3 \, \sqrt {-x^{2} + 1}} + \frac {3}{\sqrt {-x^{2} + 1}} + \frac {7 \, x}{3 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}}} + \frac {4}{3 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}}} - \frac {1}{{\left (-x^{2} + 1\right )}^{\frac {3}{2}} x} - 3 \, \log \left (\frac {2 \, \sqrt {-x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) \]

[In]

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

[Out]

14/3*x/sqrt(-x^2 + 1) + 3/sqrt(-x^2 + 1) + 7/3*x/(-x^2 + 1)^(3/2) + 4/3/(-x^2 + 1)^(3/2) - 1/((-x^2 + 1)^(3/2)
*x) - 3*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. 211 vs. \(2 (65) = 130\).

Time = 0.36 (sec) , antiderivative size = 211, normalized size of antiderivative = 2.43 \[ \int \frac {\sqrt {1+x}}{(1-x)^{5/2} x^2} \, dx=-\frac {{\left (11 \, x - 13\right )} \sqrt {x + 1} \sqrt {-x + 1}}{3 \, {\left (x - 1\right )}^{2}} - \frac {4 \, {\left (\frac {\sqrt {2} - \sqrt {-x + 1}}{\sqrt {x + 1}} - \frac {\sqrt {x + 1}}{\sqrt {2} - \sqrt {-x + 1}}\right )}}{{\left (\frac {\sqrt {2} - \sqrt {-x + 1}}{\sqrt {x + 1}} - \frac {\sqrt {x + 1}}{\sqrt {2} - \sqrt {-x + 1}}\right )}^{2} - 4} - 3 \, \log \left ({\left | -\frac {\sqrt {2} - \sqrt {-x + 1}}{\sqrt {x + 1}} + \frac {\sqrt {x + 1}}{\sqrt {2} - \sqrt {-x + 1}} + 2 \right |}\right ) + 3 \, \log \left ({\left | -\frac {\sqrt {2} - \sqrt {-x + 1}}{\sqrt {x + 1}} + \frac {\sqrt {x + 1}}{\sqrt {2} - \sqrt {-x + 1}} - 2 \right |}\right ) \]

[In]

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

[Out]

-1/3*(11*x - 13)*sqrt(x + 1)*sqrt(-x + 1)/(x - 1)^2 - 4*((sqrt(2) - sqrt(-x + 1))/sqrt(x + 1) - sqrt(x + 1)/(s
qrt(2) - sqrt(-x + 1)))/(((sqrt(2) - sqrt(-x + 1))/sqrt(x + 1) - sqrt(x + 1)/(sqrt(2) - sqrt(-x + 1)))^2 - 4)
- 3*log(abs(-(sqrt(2) - sqrt(-x + 1))/sqrt(x + 1) + sqrt(x + 1)/(sqrt(2) - sqrt(-x + 1)) + 2)) + 3*log(abs(-(s
qrt(2) - sqrt(-x + 1))/sqrt(x + 1) + sqrt(x + 1)/(sqrt(2) - sqrt(-x + 1)) - 2))

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {1+x}}{(1-x)^{5/2} x^2} \, dx=\int \frac {\sqrt {x+1}}{x^2\,{\left (1-x\right )}^{5/2}} \,d x \]

[In]

int((x + 1)^(1/2)/(x^2*(1 - x)^(5/2)),x)

[Out]

int((x + 1)^(1/2)/(x^2*(1 - x)^(5/2)), x)