Integrand size = 20, antiderivative size = 38 \[ \int \frac {(1+x)^{3/2}}{\sqrt {1-x} x^2} \, dx=-\frac {\sqrt {1-x} \sqrt {1+x}}{x}+\arcsin (x)-2 \text {arctanh}\left (\sqrt {1-x^2}\right ) \] Output:
-(1-x)^(1/2)*(1+x)^(1/2)/x+arcsin(x)-2*arctanh((-x^2+1)^(1/2))
Time = 0.08 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.55 \[ \int \frac {(1+x)^{3/2}}{\sqrt {1-x} x^2} \, dx=-\frac {\sqrt {1-x^2}+2 x \arctan \left (\frac {\sqrt {1-x}}{\sqrt {1+x}}\right )+4 x \text {arctanh}\left (\frac {\sqrt {1-x}}{\sqrt {1+x}}\right )}{x} \] Input:
Integrate[(1 + x)^(3/2)/(Sqrt[1 - x]*x^2),x]
Output:
-((Sqrt[1 - x^2] + 2*x*ArcTan[Sqrt[1 - x]/Sqrt[1 + x]] + 4*x*ArcTanh[Sqrt[ 1 - x]/Sqrt[1 + x]])/x)
Time = 0.16 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.16, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {109, 25, 175, 39, 103, 219, 223}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {(x+1)^{3/2}}{\sqrt {1-x} x^2} \, dx\) |
\(\Big \downarrow \) 109 |
\(\displaystyle -\int -\frac {x+2}{\sqrt {1-x} x \sqrt {x+1}}dx-\frac {\sqrt {1-x} \sqrt {x+1}}{x}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \int \frac {x+2}{\sqrt {1-x} x \sqrt {x+1}}dx-\frac {\sqrt {1-x} \sqrt {x+1}}{x}\) |
\(\Big \downarrow \) 175 |
\(\displaystyle \int \frac {1}{\sqrt {1-x} \sqrt {x+1}}dx+2 \int \frac {1}{\sqrt {1-x} x \sqrt {x+1}}dx-\frac {\sqrt {1-x} \sqrt {x+1}}{x}\) |
\(\Big \downarrow \) 39 |
\(\displaystyle \int \frac {1}{\sqrt {1-x^2}}dx+2 \int \frac {1}{\sqrt {1-x} x \sqrt {x+1}}dx-\frac {\sqrt {1-x} \sqrt {x+1}}{x}\) |
\(\Big \downarrow \) 103 |
\(\displaystyle \int \frac {1}{\sqrt {1-x^2}}dx-2 \int \frac {1}{1-(1-x) (x+1)}d\left (\sqrt {1-x} \sqrt {x+1}\right )-\frac {\sqrt {1-x} \sqrt {x+1}}{x}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \int \frac {1}{\sqrt {1-x^2}}dx-2 \text {arctanh}\left (\sqrt {1-x} \sqrt {x+1}\right )-\frac {\sqrt {1-x} \sqrt {x+1}}{x}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle \arcsin (x)-2 \text {arctanh}\left (\sqrt {1-x} \sqrt {x+1}\right )-\frac {\sqrt {1-x} \sqrt {x+1}}{x}\) |
Input:
Int[(1 + x)^(3/2)/(Sqrt[1 - x]*x^2),x]
Output:
-((Sqrt[1 - x]*Sqrt[1 + x])/x) + ArcSin[x] - 2*ArcTanh[Sqrt[1 - x]*Sqrt[1 + x]]
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]))
Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_ ))), x_] :> Simp[b*f Subst[Int[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sq rt[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]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(b*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f *x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_ )))/((a_.) + (b_.)*(x_)), x_] :> Simp[h/b Int[(c + d*x)^n*(e + f*x)^p, x] , x] + Simp[(b*g - a*h)/b Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]
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] && (Gt Q[a, 0] || LtQ[b, 0])
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]
Time = 0.16 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.45
method | result | size |
default | \(\frac {\sqrt {1+x}\, \sqrt {1-x}\, \left (\arcsin \left (x \right ) x -2 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-x^{2}+1}}\right ) x -\sqrt {-x^{2}+1}\right )}{x \sqrt {-x^{2}+1}}\) | \(55\) |
risch | \(\frac {\sqrt {1+x}\, \left (-1+x \right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{x \sqrt {-\left (1+x \right ) \left (-1+x \right )}\, \sqrt {1-x}}+\frac {\left (\arcsin \left (x \right )-2 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-x^{2}+1}}\right )\right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{\sqrt {1-x}\, \sqrt {1+x}}\) | \(81\) |
Input:
int((1+x)^(3/2)/(1-x)^(1/2)/x^2,x,method=_RETURNVERBOSE)
Output:
(1+x)^(1/2)*(1-x)^(1/2)*(arcsin(x)*x-2*arctanh(1/(-x^2+1)^(1/2))*x-(-x^2+1 )^(1/2))/x/(-x^2+1)^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (32) = 64\).
Time = 0.08 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.71 \[ \int \frac {(1+x)^{3/2}}{\sqrt {1-x} x^2} \, dx=-\frac {2 \, x \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) - 2 \, x \log \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) + \sqrt {x + 1} \sqrt {-x + 1}}{x} \] Input:
integrate((1+x)^(3/2)/(1-x)^(1/2)/x^2,x, algorithm="fricas")
Output:
-(2*x*arctan((sqrt(x + 1)*sqrt(-x + 1) - 1)/x) - 2*x*log((sqrt(x + 1)*sqrt (-x + 1) - 1)/x) + sqrt(x + 1)*sqrt(-x + 1))/x
\[ \int \frac {(1+x)^{3/2}}{\sqrt {1-x} x^2} \, dx=\int \frac {\left (x + 1\right )^{\frac {3}{2}}}{x^{2} \sqrt {1 - x}}\, dx \] Input:
integrate((1+x)**(3/2)/(1-x)**(1/2)/x**2,x)
Output:
Integral((x + 1)**(3/2)/(x**2*sqrt(1 - x)), x)
Time = 0.12 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.11 \[ \int \frac {(1+x)^{3/2}}{\sqrt {1-x} x^2} \, dx=-\frac {\sqrt {-x^{2} + 1}}{x} + \arcsin \left (x\right ) - 2 \, \log \left (\frac {2 \, \sqrt {-x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) \] Input:
integrate((1+x)^(3/2)/(1-x)^(1/2)/x^2,x, algorithm="maxima")
Output:
-sqrt(-x^2 + 1)/x + arcsin(x) - 2*log(2*sqrt(-x^2 + 1)/abs(x) + 2/abs(x))
Leaf count of result is larger than twice the leaf count of optimal. 236 vs. \(2 (32) = 64\).
Time = 0.17 (sec) , antiderivative size = 236, normalized size of antiderivative = 6.21 \[ \int \frac {(1+x)^{3/2}}{\sqrt {1-x} x^2} \, dx=\pi - \frac {4 \, {\left (\frac {\sqrt {2} - \sqrt {-x + 1}}{\sqrt {x + 1}} - \frac {\sqrt {x + 1}}{\sqrt {2} - \sqrt {-x + 1}}\right )}}{{\left (\frac {\sqrt {2} - \sqrt {-x + 1}}{\sqrt {x + 1}} - \frac {\sqrt {x + 1}}{\sqrt {2} - \sqrt {-x + 1}}\right )}^{2} - 4} + 2 \, \arctan \left (\frac {\sqrt {x + 1} {\left (\frac {{\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{2}}{x + 1} - 1\right )}}{2 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}}\right ) - 2 \, \log \left ({\left | -\frac {\sqrt {2} - \sqrt {-x + 1}}{\sqrt {x + 1}} + \frac {\sqrt {x + 1}}{\sqrt {2} - \sqrt {-x + 1}} + 2 \right |}\right ) + 2 \, \log \left ({\left | -\frac {\sqrt {2} - \sqrt {-x + 1}}{\sqrt {x + 1}} + \frac {\sqrt {x + 1}}{\sqrt {2} - \sqrt {-x + 1}} - 2 \right |}\right ) \] Input:
integrate((1+x)^(3/2)/(1-x)^(1/2)/x^2,x, algorithm="giac")
Output:
pi - 4*((sqrt(2) - sqrt(-x + 1))/sqrt(x + 1) - 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*arctan(1/2*sqrt(x + 1)*((sqrt(2) - sqrt(-x + 1) )^2/(x + 1) - 1)/(sqrt(2) - sqrt(-x + 1))) - 2*log(abs(-(sqrt(2) - sqrt(-x + 1))/sqrt(x + 1) + sqrt(x + 1)/(sqrt(2) - sqrt(-x + 1)) + 2)) + 2*log(ab s(-(sqrt(2) - sqrt(-x + 1))/sqrt(x + 1) + sqrt(x + 1)/(sqrt(2) - sqrt(-x + 1)) - 2))
Timed out. \[ \int \frac {(1+x)^{3/2}}{\sqrt {1-x} x^2} \, dx=\int \frac {{\left (x+1\right )}^{3/2}}{x^2\,\sqrt {1-x}} \,d x \] Input:
int((x + 1)^(3/2)/(x^2*(1 - x)^(1/2)),x)
Output:
int((x + 1)^(3/2)/(x^2*(1 - x)^(1/2)), x)
Time = 0.19 (sec) , antiderivative size = 128, normalized size of antiderivative = 3.37 \[ \int \frac {(1+x)^{3/2}}{\sqrt {1-x} x^2} \, dx=\frac {-2 \mathit {asin} \left (\frac {\sqrt {1-x}}{\sqrt {2}}\right ) x -\sqrt {x +1}\, \sqrt {1-x}-2 \,\mathrm {log}\left (-\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {1-x}}{\sqrt {2}}\right )}{2}\right )-1\right ) x +2 \,\mathrm {log}\left (-\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {1-x}}{\sqrt {2}}\right )}{2}\right )+1\right ) x -2 \,\mathrm {log}\left (\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {1-x}}{\sqrt {2}}\right )}{2}\right )-1\right ) x +2 \,\mathrm {log}\left (\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {1-x}}{\sqrt {2}}\right )}{2}\right )+1\right ) x}{x} \] Input:
int((1+x)^(3/2)/(1-x)^(1/2)/x^2,x)
Output:
( - 2*asin(sqrt( - x + 1)/sqrt(2))*x - sqrt(x + 1)*sqrt( - x + 1) - 2*log( - sqrt(2) + tan(asin(sqrt( - x + 1)/sqrt(2))/2) - 1)*x + 2*log( - sqrt(2) + tan(asin(sqrt( - x + 1)/sqrt(2))/2) + 1)*x - 2*log(sqrt(2) + tan(asin(s qrt( - x + 1)/sqrt(2))/2) - 1)*x + 2*log(sqrt(2) + tan(asin(sqrt( - x + 1) /sqrt(2))/2) + 1)*x)/x