Integrand size = 20, antiderivative size = 58 \[ \int \frac {(1+x)^{3/2}}{\sqrt {1-x} x^3} \, dx=-\frac {\sqrt {1-x} (1+x)^{3/2}}{2 x^2}-\frac {3 \sqrt {1-x^2}}{2 x}-\frac {3}{2} \text {arctanh}\left (\sqrt {1-x^2}\right ) \] Output:
-1/2*(1-x)^(1/2)*(1+x)^(3/2)/x^2-3/2*(-x^2+1)^(1/2)/x-3/2*arctanh((-x^2+1) ^(1/2))
Time = 0.11 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.93 \[ \int \frac {(1+x)^{3/2}}{\sqrt {1-x} x^3} \, dx=-\frac {\sqrt {1-x} \left (1+5 x+4 x^2\right )}{2 x^2 \sqrt {1+x}}-3 \text {arctanh}\left (\frac {\sqrt {1-x}}{\sqrt {1+x}}\right ) \] Input:
Integrate[(1 + x)^(3/2)/(Sqrt[1 - x]*x^3),x]
Output:
-1/2*(Sqrt[1 - x]*(1 + 5*x + 4*x^2))/(x^2*Sqrt[1 + x]) - 3*ArcTanh[Sqrt[1 - x]/Sqrt[1 + x]]
Time = 0.16 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.21, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {105, 105, 103, 219}
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.
\(\displaystyle \int \frac {(x+1)^{3/2}}{\sqrt {1-x} x^3} \, dx\) |
\(\Big \downarrow \) 105 |
\(\displaystyle \frac {3}{2} \int \frac {\sqrt {x+1}}{\sqrt {1-x} x^2}dx-\frac {\sqrt {1-x} (x+1)^{3/2}}{2 x^2}\) |
\(\Big \downarrow \) 105 |
\(\displaystyle \frac {3}{2} \left (\int \frac {1}{\sqrt {1-x} x \sqrt {x+1}}dx-\frac {\sqrt {1-x} \sqrt {x+1}}{x}\right )-\frac {\sqrt {1-x} (x+1)^{3/2}}{2 x^2}\) |
\(\Big \downarrow \) 103 |
\(\displaystyle \frac {3}{2} \left (-\int \frac {1}{1-(1-x) (x+1)}d\left (\sqrt {1-x} \sqrt {x+1}\right )-\frac {\sqrt {1-x} \sqrt {x+1}}{x}\right )-\frac {\sqrt {1-x} (x+1)^{3/2}}{2 x^2}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {3}{2} \left (-\text {arctanh}\left (\sqrt {1-x} \sqrt {x+1}\right )-\frac {\sqrt {1-x} \sqrt {x+1}}{x}\right )-\frac {\sqrt {1-x} (x+1)^{3/2}}{2 x^2}\) |
Input:
Int[(1 + x)^(3/2)/(Sqrt[1 - x]*x^3),x]
Output:
-1/2*(Sqrt[1 - x]*(1 + x)^(3/2))/x^2 + (3*(-((Sqrt[1 - x]*Sqrt[1 + x])/x) - ArcTanh[Sqrt[1 - x]*Sqrt[1 + x]]))/2
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[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Simp[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]
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])
Time = 0.16 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.10
method | result | size |
default | \(-\frac {\sqrt {1+x}\, \sqrt {1-x}\, \left (3 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-x^{2}+1}}\right ) x^{2}+4 x \sqrt {-x^{2}+1}+\sqrt {-x^{2}+1}\right )}{2 \sqrt {-x^{2}+1}\, x^{2}}\) | \(64\) |
risch | \(\frac {\sqrt {1+x}\, \left (-1+x \right ) \left (1+4 x \right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{2 x^{2} \sqrt {-\left (1+x \right ) \left (-1+x \right )}\, \sqrt {1-x}}-\frac {3 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-x^{2}+1}}\right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{2 \sqrt {1-x}\, \sqrt {1+x}}\) | \(83\) |
Input:
int((1+x)^(3/2)/(1-x)^(1/2)/x^3,x,method=_RETURNVERBOSE)
Output:
-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+1)^(1/2)/x^2
Time = 0.08 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.86 \[ \int \frac {(1+x)^{3/2}}{\sqrt {1-x} x^3} \, dx=\frac {3 \, x^{2} \log \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) - {\left (4 \, x + 1\right )} \sqrt {x + 1} \sqrt {-x + 1}}{2 \, x^{2}} \] Input:
integrate((1+x)^(3/2)/(1-x)^(1/2)/x^3,x, algorithm="fricas")
Output:
1/2*(3*x^2*log((sqrt(x + 1)*sqrt(-x + 1) - 1)/x) - (4*x + 1)*sqrt(x + 1)*s qrt(-x + 1))/x^2
\[ \int \frac {(1+x)^{3/2}}{\sqrt {1-x} x^3} \, dx=\int \frac {\left (x + 1\right )^{\frac {3}{2}}}{x^{3} \sqrt {1 - x}}\, dx \] Input:
integrate((1+x)**(3/2)/(1-x)**(1/2)/x**3,x)
Output:
Integral((x + 1)**(3/2)/(x**3*sqrt(1 - x)), x)
Time = 0.11 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.93 \[ \int \frac {(1+x)^{3/2}}{\sqrt {1-x} x^3} \, dx=-\frac {2 \, \sqrt {-x^{2} + 1}}{x} - \frac {\sqrt {-x^{2} + 1}}{2 \, x^{2}} - \frac {3}{2} \, \log \left (\frac {2 \, \sqrt {-x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) \] Input:
integrate((1+x)^(3/2)/(1-x)^(1/2)/x^3,x, algorithm="maxima")
Output:
-2*sqrt(-x^2 + 1)/x - 1/2*sqrt(-x^2 + 1)/x^2 - 3/2*log(2*sqrt(-x^2 + 1)/ab s(x) + 2/abs(x))
Leaf count of result is larger than twice the leaf count of optimal. 234 vs. \(2 (44) = 88\).
Time = 0.18 (sec) , antiderivative size = 234, normalized size of antiderivative = 4.03 \[ \int \frac {(1+x)^{3/2}}{\sqrt {1-x} x^3} \, dx=-\frac {2 \, {\left (3 \, {\left (\frac {\sqrt {2} - \sqrt {-x + 1}}{\sqrt {x + 1}} - \frac {\sqrt {x + 1}}{\sqrt {2} - \sqrt {-x + 1}}\right )}^{3} - \frac {20 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}}{\sqrt {x + 1}} + \frac {20 \, \sqrt {x + 1}}{\sqrt {2} - \sqrt {-x + 1}}\right )}}{{\left ({\left (\frac {\sqrt {2} - \sqrt {-x + 1}}{\sqrt {x + 1}} - \frac {\sqrt {x + 1}}{\sqrt {2} - \sqrt {-x + 1}}\right )}^{2} - 4\right )}^{2}} - \frac {3}{2} \, \log \left ({\left | -\frac {\sqrt {2} - \sqrt {-x + 1}}{\sqrt {x + 1}} + \frac {\sqrt {x + 1}}{\sqrt {2} - \sqrt {-x + 1}} + 2 \right |}\right ) + \frac {3}{2} \, \log \left ({\left | -\frac {\sqrt {2} - \sqrt {-x + 1}}{\sqrt {x + 1}} + \frac {\sqrt {x + 1}}{\sqrt {2} - \sqrt {-x + 1}} - 2 \right |}\right ) \] Input:
integrate((1+x)^(3/2)/(1-x)^(1/2)/x^3,x, algorithm="giac")
Output:
-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)/(sq rt(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))
Timed out. \[ \int \frac {(1+x)^{3/2}}{\sqrt {1-x} x^3} \, dx=\int \frac {{\left (x+1\right )}^{3/2}}{x^3\,\sqrt {1-x}} \,d x \] Input:
int((x + 1)^(3/2)/(x^3*(1 - x)^(1/2)),x)
Output:
int((x + 1)^(3/2)/(x^3*(1 - x)^(1/2)), x)
Time = 0.20 (sec) , antiderivative size = 135, normalized size of antiderivative = 2.33 \[ \int \frac {(1+x)^{3/2}}{\sqrt {1-x} x^3} \, dx=\frac {-4 \sqrt {x +1}\, \sqrt {1-x}\, x -\sqrt {x +1}\, \sqrt {1-x}-3 \,\mathrm {log}\left (-\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {1-x}}{\sqrt {2}}\right )}{2}\right )-1\right ) x^{2}+3 \,\mathrm {log}\left (-\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {1-x}}{\sqrt {2}}\right )}{2}\right )+1\right ) x^{2}-3 \,\mathrm {log}\left (\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {1-x}}{\sqrt {2}}\right )}{2}\right )-1\right ) x^{2}+3 \,\mathrm {log}\left (\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {1-x}}{\sqrt {2}}\right )}{2}\right )+1\right ) x^{2}}{2 x^{2}} \] Input:
int((1+x)^(3/2)/(1-x)^(1/2)/x^3,x)
Output:
( - 4*sqrt(x + 1)*sqrt( - x + 1)*x - sqrt(x + 1)*sqrt( - x + 1) - 3*log( - sqrt(2) + tan(asin(sqrt( - x + 1)/sqrt(2))/2) - 1)*x**2 + 3*log( - sqrt(2 ) + tan(asin(sqrt( - x + 1)/sqrt(2))/2) + 1)*x**2 - 3*log(sqrt(2) + tan(as in(sqrt( - x + 1)/sqrt(2))/2) - 1)*x**2 + 3*log(sqrt(2) + tan(asin(sqrt( - x + 1)/sqrt(2))/2) + 1)*x**2)/(2*x**2)