Integrand size = 20, antiderivative size = 101 \[ \int \frac {\sqrt {1+x}}{(1-x)^{5/2} x^3} \, dx=\frac {19 \sqrt {1+x}}{6 (1-x)^{3/2}}+\frac {26 \sqrt {1+x}}{3 \sqrt {1-x}}-\frac {\sqrt {1+x}}{2 (1-x)^{3/2} x^2}-\frac {2 \sqrt {1+x}}{(1-x)^{3/2} x}-\frac {11}{2} \text {arctanh}\left (\sqrt {1-x^2}\right ) \] Output:
19/6*(1+x)^(1/2)/(1-x)^(3/2)+26/3*(1+x)^(1/2)/(1-x)^(1/2)-1/2*(1+x)^(1/2)/ (1-x)^(3/2)/x^2-2*(1+x)^(1/2)/(1-x)^(3/2)/x-11/2*arctanh((-x^2+1)^(1/2))
Time = 0.13 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.58 \[ \int \frac {\sqrt {1+x}}{(1-x)^{5/2} x^3} \, dx=-\frac {\sqrt {1+x} \left (3+12 x-71 x^2+52 x^3\right )}{6 (1-x)^{3/2} x^2}-11 \text {arctanh}\left (\frac {\sqrt {1+x}}{\sqrt {1-x}}\right ) \] Input:
Integrate[Sqrt[1 + x]/((1 - x)^(5/2)*x^3),x]
Output:
-1/6*(Sqrt[1 + x]*(3 + 12*x - 71*x^2 + 52*x^3))/((1 - x)^(3/2)*x^2) - 11*A rcTanh[Sqrt[1 + x]/Sqrt[1 - x]]
Time = 0.21 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.15, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {110, 27, 168, 25, 168, 25, 169, 27, 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.
| \(\displaystyle \int \frac {\sqrt {x+1}}{(1-x)^{5/2} x^3} \, dx\) |
| \(\Big \downarrow \) 110 |
| \(\displaystyle \frac {2 \sqrt {x+1}}{3 (1-x)^{3/2} x^2}-\frac {2}{3} \int -\frac {6 x+7}{2 (1-x)^{3/2} x^3 \sqrt {x+1}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \int \frac {6 x+7}{(1-x)^{3/2} x^3 \sqrt {x+1}}dx+\frac {2 \sqrt {x+1}}{3 (1-x)^{3/2} x^2}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{3} \left (-\frac {1}{2} \int -\frac {14 x+19}{(1-x)^{3/2} x^2 \sqrt {x+1}}dx-\frac {7 \sqrt {x+1}}{2 \sqrt {1-x} x^2}\right )+\frac {2 \sqrt {x+1}}{3 (1-x)^{3/2} x^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \int \frac {14 x+19}{(1-x)^{3/2} x^2 \sqrt {x+1}}dx-\frac {7 \sqrt {x+1}}{2 \sqrt {1-x} x^2}\right )+\frac {2 \sqrt {x+1}}{3 (1-x)^{3/2} x^2}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (-\int -\frac {19 x+33}{(1-x)^{3/2} x \sqrt {x+1}}dx-\frac {19 \sqrt {x+1}}{\sqrt {1-x} x}\right )-\frac {7 \sqrt {x+1}}{2 \sqrt {1-x} x^2}\right )+\frac {2 \sqrt {x+1}}{3 (1-x)^{3/2} x^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (\int \frac {19 x+33}{(1-x)^{3/2} x \sqrt {x+1}}dx-\frac {19 \sqrt {x+1}}{\sqrt {1-x} x}\right )-\frac {7 \sqrt {x+1}}{2 \sqrt {1-x} x^2}\right )+\frac {2 \sqrt {x+1}}{3 (1-x)^{3/2} x^2}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (-\int -\frac {33}{\sqrt {1-x} x \sqrt {x+1}}dx+\frac {52 \sqrt {x+1}}{\sqrt {1-x}}-\frac {19 \sqrt {x+1}}{\sqrt {1-x} x}\right )-\frac {7 \sqrt {x+1}}{2 \sqrt {1-x} x^2}\right )+\frac {2 \sqrt {x+1}}{3 (1-x)^{3/2} x^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (33 \int \frac {1}{\sqrt {1-x} x \sqrt {x+1}}dx+\frac {52 \sqrt {x+1}}{\sqrt {1-x}}-\frac {19 \sqrt {x+1}}{\sqrt {1-x} x}\right )-\frac {7 \sqrt {x+1}}{2 \sqrt {1-x} x^2}\right )+\frac {2 \sqrt {x+1}}{3 (1-x)^{3/2} x^2}\) |
| \(\Big \downarrow \) 103 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (-33 \int \frac {1}{1-(1-x) (x+1)}d\left (\sqrt {1-x} \sqrt {x+1}\right )+\frac {52 \sqrt {x+1}}{\sqrt {1-x}}-\frac {19 \sqrt {x+1}}{\sqrt {1-x} x}\right )-\frac {7 \sqrt {x+1}}{2 \sqrt {1-x} x^2}\right )+\frac {2 \sqrt {x+1}}{3 (1-x)^{3/2} x^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (-33 \text {arctanh}\left (\sqrt {1-x} \sqrt {x+1}\right )+\frac {52 \sqrt {x+1}}{\sqrt {1-x}}-\frac {19 \sqrt {x+1}}{\sqrt {1-x} x}\right )-\frac {7 \sqrt {x+1}}{2 \sqrt {1-x} x^2}\right )+\frac {2 \sqrt {x+1}}{3 (1-x)^{3/2} x^2}\) |
Input:
Int[Sqrt[1 + x]/((1 - x)^(5/2)*x^3),x]
Output:
(2*Sqrt[1 + x])/(3*(1 - x)^(3/2)*x^2) + ((-7*Sqrt[1 + x])/(2*Sqrt[1 - x]*x ^2) + ((52*Sqrt[1 + x])/Sqrt[1 - x] - (19*Sqrt[1 + x])/(Sqrt[1 - x]*x) - 3 3*ArcTanh[Sqrt[1 - x]*Sqrt[1 + x]])/2)/3
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[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] && Gt Q[n, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> 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] + S imp[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]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> 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] + S imp[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]
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 = 100, normalized size of antiderivative = 0.99
| method | result | size |
| risch | \(\frac {\left (52 x^{4}-19 x^{3}-59 x^{2}+15 x +3\right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{6 \left (-1+x \right ) \sqrt {-\left (1+x \right ) \left (-1+x \right )}\, x^{2} \sqrt {1-x}\, \sqrt {1+x}}-\frac {11 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-x^{2}+1}}\right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{2 \sqrt {1-x}\, \sqrt {1+x}}\) | \(100\) |
| default | \(-\frac {\left (33 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-x^{2}+1}}\right ) x^{4}-66 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-x^{2}+1}}\right ) x^{3}+52 x^{3} \sqrt {-x^{2}+1}+33 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-x^{2}+1}}\right ) x^{2}-71 \sqrt {-x^{2}+1}\, x^{2}+12 x \sqrt {-x^{2}+1}+3 \sqrt {-x^{2}+1}\right ) \sqrt {1+x}}{6 \left (1-x \right )^{\frac {3}{2}} \sqrt {-x^{2}+1}\, x^{2}}\) | \(124\) |
Input:
int((1+x)^(1/2)/(1-x)^(5/2)/x^3,x,method=_RETURNVERBOSE)
Output:
1/6*(52*x^4-19*x^3-59*x^2+15*x+3)/(-1+x)/(-(1+x)*(-1+x))^(1/2)/x^2*((1+x)* (1-x))^(1/2)/(1-x)^(1/2)/(1+x)^(1/2)-11/2*arctanh(1/(-x^2+1)^(1/2))*((1+x) *(1-x))^(1/2)/(1-x)^(1/2)/(1+x)^(1/2)
Time = 0.08 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt {1+x}}{(1-x)^{5/2} x^3} \, dx=\frac {38 \, x^{4} - 76 \, x^{3} + 38 \, x^{2} - {\left (52 \, x^{3} - 71 \, x^{2} + 12 \, x + 3\right )} \sqrt {x + 1} \sqrt {-x + 1} + 33 \, {\left (x^{4} - 2 \, x^{3} + x^{2}\right )} \log \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right )}{6 \, {\left (x^{4} - 2 \, x^{3} + x^{2}\right )}} \] Input:
integrate((1+x)^(1/2)/(1-x)^(5/2)/x^3,x, algorithm="fricas")
Output:
1/6*(38*x^4 - 76*x^3 + 38*x^2 - (52*x^3 - 71*x^2 + 12*x + 3)*sqrt(x + 1)*s qrt(-x + 1) + 33*(x^4 - 2*x^3 + x^2)*log((sqrt(x + 1)*sqrt(-x + 1) - 1)/x) )/(x^4 - 2*x^3 + x^2)
\[ \int \frac {\sqrt {1+x}}{(1-x)^{5/2} x^3} \, dx=\int \frac {\sqrt {x + 1}}{x^{3} \left (1 - x\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((1+x)**(1/2)/(1-x)**(5/2)/x**3,x)
Output:
Integral(sqrt(x + 1)/(x**3*(1 - x)**(5/2)), x)
Time = 0.05 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.99 \[ \int \frac {\sqrt {1+x}}{(1-x)^{5/2} x^3} \, dx=\frac {26 \, x}{3 \, \sqrt {-x^{2} + 1}} + \frac {11}{2 \, \sqrt {-x^{2} + 1}} + \frac {13 \, x}{3 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}}} + \frac {11}{6 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}}} - \frac {3}{{\left (-x^{2} + 1\right )}^{\frac {3}{2}} x} - \frac {1}{2 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} x^{2}} - \frac {11}{2} \, \log \left (\frac {2 \, \sqrt {-x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) \] Input:
integrate((1+x)^(1/2)/(1-x)^(5/2)/x^3,x, algorithm="maxima")
Output:
26/3*x/sqrt(-x^2 + 1) + 11/2/sqrt(-x^2 + 1) + 13/3*x/(-x^2 + 1)^(3/2) + 11 /6/(-x^2 + 1)^(3/2) - 3/((-x^2 + 1)^(3/2)*x) - 1/2/((-x^2 + 1)^(3/2)*x^2) - 11/2*log(2*sqrt(-x^2 + 1)/abs(x) + 2/abs(x))
Leaf count of result is larger than twice the leaf count of optimal. 258 vs. \(2 (75) = 150\).
Time = 0.21 (sec) , antiderivative size = 258, normalized size of antiderivative = 2.55 \[ \int \frac {\sqrt {1+x}}{(1-x)^{5/2} x^3} \, dx=-\frac {{\left (17 \, x - 19\right )} \sqrt {x + 1} \sqrt {-x + 1}}{3 \, {\left (x - 1\right )}^{2}} - \frac {2 \, {\left (5 \, {\left (\frac {\sqrt {2} - \sqrt {-x + 1}}{\sqrt {x + 1}} - \frac {\sqrt {x + 1}}{\sqrt {2} - \sqrt {-x + 1}}\right )}^{3} - \frac {28 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}}{\sqrt {x + 1}} + \frac {28 \, \sqrt {x + 1}}{\sqrt {2} - \sqrt {-x + 1}}\right )}}{{\left ({\left (\frac {\sqrt {2} - \sqrt {-x + 1}}{\sqrt {x + 1}} - \frac {\sqrt {x + 1}}{\sqrt {2} - \sqrt {-x + 1}}\right )}^{2} - 4\right )}^{2}} - \frac {11}{2} \, \log \left ({\left | -\frac {\sqrt {2} - \sqrt {-x + 1}}{\sqrt {x + 1}} + \frac {\sqrt {x + 1}}{\sqrt {2} - \sqrt {-x + 1}} + 2 \right |}\right ) + \frac {11}{2} \, \log \left ({\left | -\frac {\sqrt {2} - \sqrt {-x + 1}}{\sqrt {x + 1}} + \frac {\sqrt {x + 1}}{\sqrt {2} - \sqrt {-x + 1}} - 2 \right |}\right ) \] Input:
integrate((1+x)^(1/2)/(1-x)^(5/2)/x^3,x, algorithm="giac")
Output:
-1/3*(17*x - 19)*sqrt(x + 1)*sqrt(-x + 1)/(x - 1)^2 - 2*(5*((sqrt(2) - sqr t(-x + 1))/sqrt(x + 1) - sqrt(x + 1)/(sqrt(2) - sqrt(-x + 1)))^3 - 28*(sqr t(2) - sqrt(-x + 1))/sqrt(x + 1) + 28*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 - 11/2*log(abs(-(sqrt(2) - sqrt(-x + 1))/sqrt(x + 1) + sqr t(x + 1)/(sqrt(2) - sqrt(-x + 1)) + 2)) + 11/2*log(abs(-(sqrt(2) - sqrt(-x + 1))/sqrt(x + 1) + sqrt(x + 1)/(sqrt(2) - sqrt(-x + 1)) - 2))
Timed out. \[ \int \frac {\sqrt {1+x}}{(1-x)^{5/2} x^3} \, dx=\int \frac {\sqrt {x+1}}{x^3\,{\left (1-x\right )}^{5/2}} \,d x \] Input:
int((x + 1)^(1/2)/(x^3*(1 - x)^(5/2)),x)
Output:
int((x + 1)^(1/2)/(x^3*(1 - x)^(5/2)), x)
Time = 0.19 (sec) , antiderivative size = 306, normalized size of antiderivative = 3.03 \[ \int \frac {\sqrt {1+x}}{(1-x)^{5/2} x^3} \, dx=\frac {-33 \sqrt {1-x}\, \mathrm {log}\left (-\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {1-x}}{\sqrt {2}}\right )}{2}\right )-1\right ) x^{3}+33 \sqrt {1-x}\, \mathrm {log}\left (-\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {1-x}}{\sqrt {2}}\right )}{2}\right )-1\right ) x^{2}+33 \sqrt {1-x}\, \mathrm {log}\left (-\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {1-x}}{\sqrt {2}}\right )}{2}\right )+1\right ) x^{3}-33 \sqrt {1-x}\, \mathrm {log}\left (-\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {1-x}}{\sqrt {2}}\right )}{2}\right )+1\right ) x^{2}-33 \sqrt {1-x}\, \mathrm {log}\left (\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {1-x}}{\sqrt {2}}\right )}{2}\right )-1\right ) x^{3}+33 \sqrt {1-x}\, \mathrm {log}\left (\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {1-x}}{\sqrt {2}}\right )}{2}\right )-1\right ) x^{2}+33 \sqrt {1-x}\, \mathrm {log}\left (\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {1-x}}{\sqrt {2}}\right )}{2}\right )+1\right ) x^{3}-33 \sqrt {1-x}\, \mathrm {log}\left (\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {1-x}}{\sqrt {2}}\right )}{2}\right )+1\right ) x^{2}+52 \sqrt {x +1}\, x^{3}-71 \sqrt {x +1}\, x^{2}+12 \sqrt {x +1}\, x +3 \sqrt {x +1}}{6 \sqrt {1-x}\, x^{2} \left (x -1\right )} \] Input:
int((1+x)^(1/2)/(1-x)^(5/2)/x^3,x)
Output:
( - 33*sqrt( - x + 1)*log( - sqrt(2) + tan(asin(sqrt( - x + 1)/sqrt(2))/2) - 1)*x**3 + 33*sqrt( - x + 1)*log( - sqrt(2) + tan(asin(sqrt( - x + 1)/sq rt(2))/2) - 1)*x**2 + 33*sqrt( - x + 1)*log( - sqrt(2) + tan(asin(sqrt( - x + 1)/sqrt(2))/2) + 1)*x**3 - 33*sqrt( - x + 1)*log( - sqrt(2) + tan(asin (sqrt( - x + 1)/sqrt(2))/2) + 1)*x**2 - 33*sqrt( - x + 1)*log(sqrt(2) + ta n(asin(sqrt( - x + 1)/sqrt(2))/2) - 1)*x**3 + 33*sqrt( - x + 1)*log(sqrt(2 ) + tan(asin(sqrt( - x + 1)/sqrt(2))/2) - 1)*x**2 + 33*sqrt( - x + 1)*log( sqrt(2) + tan(asin(sqrt( - x + 1)/sqrt(2))/2) + 1)*x**3 - 33*sqrt( - x + 1 )*log(sqrt(2) + tan(asin(sqrt( - x + 1)/sqrt(2))/2) + 1)*x**2 + 52*sqrt(x + 1)*x**3 - 71*sqrt(x + 1)*x**2 + 12*sqrt(x + 1)*x + 3*sqrt(x + 1))/(6*sqr t( - x + 1)*x**2*(x - 1))