Integrand size = 13, antiderivative size = 106 \[ \int \frac {\sqrt {x}}{\left (1+x^2\right )^3} \, dx=\frac {x^{3/2}}{4 \left (1+x^2\right )^2}+\frac {5 x^{3/2}}{16 \left (1+x^2\right )}-\frac {5 \arctan \left (1-\sqrt {2} \sqrt {x}\right )}{32 \sqrt {2}}+\frac {5 \arctan \left (1+\sqrt {2} \sqrt {x}\right )}{32 \sqrt {2}}-\frac {5 \text {arctanh}\left (\frac {\sqrt {2} \sqrt {x}}{1+x}\right )}{32 \sqrt {2}} \] Output:
1/4*x^(3/2)/(x^2+1)^2+5*x^(3/2)/(16*x^2+16)+5/64*arctan(-1+2^(1/2)*x^(1/2) )*2^(1/2)+5/64*arctan(1+2^(1/2)*x^(1/2))*2^(1/2)-5/64*arctanh(2^(1/2)*x^(1 /2)/(1+x))*2^(1/2)
Time = 0.09 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.68 \[ \int \frac {\sqrt {x}}{\left (1+x^2\right )^3} \, dx=\frac {1}{64} \left (\frac {4 x^{3/2} \left (9+5 x^2\right )}{\left (1+x^2\right )^2}+5 \sqrt {2} \arctan \left (\frac {-1+x}{\sqrt {2} \sqrt {x}}\right )-5 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {x}}{1+x}\right )\right ) \] Input:
Integrate[Sqrt[x]/(1 + x^2)^3,x]
Output:
((4*x^(3/2)*(9 + 5*x^2))/(1 + x^2)^2 + 5*Sqrt[2]*ArcTan[(-1 + x)/(Sqrt[2]* Sqrt[x])] - 5*Sqrt[2]*ArcTanh[(Sqrt[2]*Sqrt[x])/(1 + x)])/64
Time = 0.30 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.36, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.846, Rules used = {253, 253, 266, 826, 1476, 1082, 217, 1479, 25, 27, 1103}
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.
| \(\displaystyle \int \frac {\sqrt {x}}{\left (x^2+1\right )^3} \, dx\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {5}{8} \int \frac {\sqrt {x}}{\left (x^2+1\right )^2}dx+\frac {x^{3/2}}{4 \left (x^2+1\right )^2}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {5}{8} \left (\frac {1}{4} \int \frac {\sqrt {x}}{x^2+1}dx+\frac {x^{3/2}}{2 \left (x^2+1\right )}\right )+\frac {x^{3/2}}{4 \left (x^2+1\right )^2}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {5}{8} \left (\frac {1}{2} \int \frac {x}{x^2+1}d\sqrt {x}+\frac {x^{3/2}}{2 \left (x^2+1\right )}\right )+\frac {x^{3/2}}{4 \left (x^2+1\right )^2}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {5}{8} \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {x+1}{x^2+1}d\sqrt {x}-\frac {1}{2} \int \frac {1-x}{x^2+1}d\sqrt {x}\right )+\frac {x^{3/2}}{2 \left (x^2+1\right )}\right )+\frac {x^{3/2}}{4 \left (x^2+1\right )^2}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {5}{8} \left (\frac {1}{2} \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{x-\sqrt {2} \sqrt {x}+1}d\sqrt {x}+\frac {1}{2} \int \frac {1}{x+\sqrt {2} \sqrt {x}+1}d\sqrt {x}\right )-\frac {1}{2} \int \frac {1-x}{x^2+1}d\sqrt {x}\right )+\frac {x^{3/2}}{2 \left (x^2+1\right )}\right )+\frac {x^{3/2}}{4 \left (x^2+1\right )^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {5}{8} \left (\frac {1}{2} \left (\frac {1}{2} \left (\frac {\int \frac {1}{-x-1}d\left (1-\sqrt {2} \sqrt {x}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-x-1}d\left (\sqrt {2} \sqrt {x}+1\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-x}{x^2+1}d\sqrt {x}\right )+\frac {x^{3/2}}{2 \left (x^2+1\right )}\right )+\frac {x^{3/2}}{4 \left (x^2+1\right )^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {5}{8} \left (\frac {1}{2} \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {x}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {x}\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-x}{x^2+1}d\sqrt {x}\right )+\frac {x^{3/2}}{2 \left (x^2+1\right )}\right )+\frac {x^{3/2}}{4 \left (x^2+1\right )^2}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {5}{8} \left (\frac {1}{2} \left (\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2}-2 \sqrt {x}}{x-\sqrt {2} \sqrt {x}+1}d\sqrt {x}}{2 \sqrt {2}}+\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {x}+1\right )}{x+\sqrt {2} \sqrt {x}+1}d\sqrt {x}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {x}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {x}\right )}{\sqrt {2}}\right )\right )+\frac {x^{3/2}}{2 \left (x^2+1\right )}\right )+\frac {x^{3/2}}{4 \left (x^2+1\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {5}{8} \left (\frac {1}{2} \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-2 \sqrt {x}}{x-\sqrt {2} \sqrt {x}+1}d\sqrt {x}}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {x}+1\right )}{x+\sqrt {2} \sqrt {x}+1}d\sqrt {x}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {x}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {x}\right )}{\sqrt {2}}\right )\right )+\frac {x^{3/2}}{2 \left (x^2+1\right )}\right )+\frac {x^{3/2}}{4 \left (x^2+1\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{8} \left (\frac {1}{2} \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-2 \sqrt {x}}{x-\sqrt {2} \sqrt {x}+1}d\sqrt {x}}{2 \sqrt {2}}-\frac {1}{2} \int \frac {\sqrt {2} \sqrt {x}+1}{x+\sqrt {2} \sqrt {x}+1}d\sqrt {x}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {x}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {x}\right )}{\sqrt {2}}\right )\right )+\frac {x^{3/2}}{2 \left (x^2+1\right )}\right )+\frac {x^{3/2}}{4 \left (x^2+1\right )^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {5}{8} \left (\frac {1}{2} \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {x}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {x}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (x-\sqrt {2} \sqrt {x}+1\right )}{2 \sqrt {2}}-\frac {\log \left (x+\sqrt {2} \sqrt {x}+1\right )}{2 \sqrt {2}}\right )\right )+\frac {x^{3/2}}{2 \left (x^2+1\right )}\right )+\frac {x^{3/2}}{4 \left (x^2+1\right )^2}\) |
Input:
Int[Sqrt[x]/(1 + x^2)^3,x]
Output:
x^(3/2)/(4*(1 + x^2)^2) + (5*(x^(3/2)/(2*(1 + x^2)) + ((-(ArcTan[1 - Sqrt[ 2]*Sqrt[x]]/Sqrt[2]) + ArcTan[1 + Sqrt[2]*Sqrt[x]]/Sqrt[2])/2 + (Log[1 - S qrt[2]*Sqrt[x] + x]/(2*Sqrt[2]) - Log[1 + Sqrt[2]*Sqrt[x] + x]/(2*Sqrt[2]) )/2)/2))/8
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(c*x )^(m + 1))*((a + b*x^2)^(p + 1)/(2*a*c*(p + 1))), x] + Simp[(m + 2*p + 3)/( 2*a*(p + 1)) Int[(c*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, m }, x] && LtQ[p, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s) Int[(r + s*x^2)/(a + b*x^ 4), x], x] - Simp[1/(2*s) Int[(r - s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{ a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b]]))
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Time = 0.50 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.72
| method | result | size |
| risch | \(\frac {x^{\frac {3}{2}} \left (5 x^{2}+9\right )}{16 \left (x^{2}+1\right )^{2}}+\frac {5 \sqrt {2}\, \left (\ln \left (\frac {x -\sqrt {2}\, \sqrt {x}+1}{x +\sqrt {2}\, \sqrt {x}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {x}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {x}\right )\right )}{128}\) | \(76\) |
| derivativedivides | \(\frac {x^{\frac {3}{2}}}{4 \left (x^{2}+1\right )^{2}}+\frac {5 x^{\frac {3}{2}}}{16 \left (x^{2}+1\right )}+\frac {5 \sqrt {2}\, \left (\ln \left (\frac {x -\sqrt {2}\, \sqrt {x}+1}{x +\sqrt {2}\, \sqrt {x}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {x}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {x}\right )\right )}{128}\) | \(81\) |
| default | \(\frac {x^{\frac {3}{2}}}{4 \left (x^{2}+1\right )^{2}}+\frac {5 x^{\frac {3}{2}}}{16 \left (x^{2}+1\right )}+\frac {5 \sqrt {2}\, \left (\ln \left (\frac {x -\sqrt {2}\, \sqrt {x}+1}{x +\sqrt {2}\, \sqrt {x}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {x}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {x}\right )\right )}{128}\) | \(81\) |
| meijerg | \(\frac {x^{\frac {3}{2}} \left (15 x^{2}+27\right )}{48 \left (x^{2}+1\right )^{2}}+\frac {5 x^{\frac {3}{2}} \sqrt {2}\, \ln \left (1-\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}+\sqrt {x^{2}}\right )}{128 \left (x^{2}\right )^{\frac {3}{4}}}+\frac {5 x^{\frac {3}{2}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}{2-\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}\right )}{64 \left (x^{2}\right )^{\frac {3}{4}}}-\frac {5 x^{\frac {3}{2}} \sqrt {2}\, \ln \left (1+\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}+\sqrt {x^{2}}\right )}{128 \left (x^{2}\right )^{\frac {3}{4}}}+\frac {5 x^{\frac {3}{2}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}{2+\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}\right )}{64 \left (x^{2}\right )^{\frac {3}{4}}}\) | \(158\) |
| trager | \(\frac {x^{\frac {3}{2}} \left (5 x^{2}+9\right )}{16 \left (x^{2}+1\right )^{2}}+\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{5} x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{5}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )+4 \sqrt {x}}{\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2}-x -1}\right )}{64}+\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{5} x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{5}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x +\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x +\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )+4 \sqrt {x}}{\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2}+x +1}\right )}{64}\) | \(201\) |
Input:
int(x^(1/2)/(x^2+1)^3,x,method=_RETURNVERBOSE)
Output:
1/16*x^(3/2)*(5*x^2+9)/(x^2+1)^2+5/128*2^(1/2)*(ln((x-2^(1/2)*x^(1/2)+1)/( x+2^(1/2)*x^(1/2)+1))+2*arctan(1+2^(1/2)*x^(1/2))+2*arctan(-1+2^(1/2)*x^(1 /2)))
Time = 0.07 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.25 \[ \int \frac {\sqrt {x}}{\left (1+x^2\right )^3} \, dx=\frac {10 \, \sqrt {2} {\left (x^{4} + 2 \, x^{2} + 1\right )} \arctan \left (\sqrt {2} \sqrt {x} + 1\right ) + 10 \, \sqrt {2} {\left (x^{4} + 2 \, x^{2} + 1\right )} \arctan \left (\sqrt {2} \sqrt {x} - 1\right ) - 5 \, \sqrt {2} {\left (x^{4} + 2 \, x^{2} + 1\right )} \log \left (\sqrt {2} \sqrt {x} + x + 1\right ) + 5 \, \sqrt {2} {\left (x^{4} + 2 \, x^{2} + 1\right )} \log \left (-\sqrt {2} \sqrt {x} + x + 1\right ) + 8 \, {\left (5 \, x^{3} + 9 \, x\right )} \sqrt {x}}{128 \, {\left (x^{4} + 2 \, x^{2} + 1\right )}} \] Input:
integrate(x^(1/2)/(x^2+1)^3,x, algorithm="fricas")
Output:
1/128*(10*sqrt(2)*(x^4 + 2*x^2 + 1)*arctan(sqrt(2)*sqrt(x) + 1) + 10*sqrt( 2)*(x^4 + 2*x^2 + 1)*arctan(sqrt(2)*sqrt(x) - 1) - 5*sqrt(2)*(x^4 + 2*x^2 + 1)*log(sqrt(2)*sqrt(x) + x + 1) + 5*sqrt(2)*(x^4 + 2*x^2 + 1)*log(-sqrt( 2)*sqrt(x) + x + 1) + 8*(5*x^3 + 9*x)*sqrt(x))/(x^4 + 2*x^2 + 1)
Leaf count of result is larger than twice the leaf count of optimal. 481 vs. \(2 (94) = 188\).
Time = 1.04 (sec) , antiderivative size = 481, normalized size of antiderivative = 4.54 \[ \int \frac {\sqrt {x}}{\left (1+x^2\right )^3} \, dx=\frac {40 x^{\frac {7}{2}}}{128 x^{4} + 256 x^{2} + 128} + \frac {72 x^{\frac {3}{2}}}{128 x^{4} + 256 x^{2} + 128} + \frac {5 \sqrt {2} x^{4} \log {\left (- 4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{128 x^{4} + 256 x^{2} + 128} - \frac {5 \sqrt {2} x^{4} \log {\left (4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac {10 \sqrt {2} x^{4} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} - 1 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac {10 \sqrt {2} x^{4} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} + 1 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac {10 \sqrt {2} x^{2} \log {\left (- 4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{128 x^{4} + 256 x^{2} + 128} - \frac {10 \sqrt {2} x^{2} \log {\left (4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac {20 \sqrt {2} x^{2} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} - 1 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac {20 \sqrt {2} x^{2} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} + 1 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac {5 \sqrt {2} \log {\left (- 4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{128 x^{4} + 256 x^{2} + 128} - \frac {5 \sqrt {2} \log {\left (4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac {10 \sqrt {2} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} - 1 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac {10 \sqrt {2} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} + 1 \right )}}{128 x^{4} + 256 x^{2} + 128} \] Input:
integrate(x**(1/2)/(x**2+1)**3,x)
Output:
40*x**(7/2)/(128*x**4 + 256*x**2 + 128) + 72*x**(3/2)/(128*x**4 + 256*x**2 + 128) + 5*sqrt(2)*x**4*log(-4*sqrt(2)*sqrt(x) + 4*x + 4)/(128*x**4 + 256 *x**2 + 128) - 5*sqrt(2)*x**4*log(4*sqrt(2)*sqrt(x) + 4*x + 4)/(128*x**4 + 256*x**2 + 128) + 10*sqrt(2)*x**4*atan(sqrt(2)*sqrt(x) - 1)/(128*x**4 + 2 56*x**2 + 128) + 10*sqrt(2)*x**4*atan(sqrt(2)*sqrt(x) + 1)/(128*x**4 + 256 *x**2 + 128) + 10*sqrt(2)*x**2*log(-4*sqrt(2)*sqrt(x) + 4*x + 4)/(128*x**4 + 256*x**2 + 128) - 10*sqrt(2)*x**2*log(4*sqrt(2)*sqrt(x) + 4*x + 4)/(128 *x**4 + 256*x**2 + 128) + 20*sqrt(2)*x**2*atan(sqrt(2)*sqrt(x) - 1)/(128*x **4 + 256*x**2 + 128) + 20*sqrt(2)*x**2*atan(sqrt(2)*sqrt(x) + 1)/(128*x** 4 + 256*x**2 + 128) + 5*sqrt(2)*log(-4*sqrt(2)*sqrt(x) + 4*x + 4)/(128*x** 4 + 256*x**2 + 128) - 5*sqrt(2)*log(4*sqrt(2)*sqrt(x) + 4*x + 4)/(128*x**4 + 256*x**2 + 128) + 10*sqrt(2)*atan(sqrt(2)*sqrt(x) - 1)/(128*x**4 + 256* x**2 + 128) + 10*sqrt(2)*atan(sqrt(2)*sqrt(x) + 1)/(128*x**4 + 256*x**2 + 128)
Time = 0.11 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt {x}}{\left (1+x^2\right )^3} \, dx=\frac {5}{64} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {x}\right )}\right ) + \frac {5}{64} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {x}\right )}\right ) - \frac {5}{128} \, \sqrt {2} \log \left (\sqrt {2} \sqrt {x} + x + 1\right ) + \frac {5}{128} \, \sqrt {2} \log \left (-\sqrt {2} \sqrt {x} + x + 1\right ) + \frac {5 \, x^{\frac {7}{2}} + 9 \, x^{\frac {3}{2}}}{16 \, {\left (x^{4} + 2 \, x^{2} + 1\right )}} \] Input:
integrate(x^(1/2)/(x^2+1)^3,x, algorithm="maxima")
Output:
5/64*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(x))) + 5/64*sqrt(2)*arct an(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(x))) - 5/128*sqrt(2)*log(sqrt(2)*sqrt(x) + x + 1) + 5/128*sqrt(2)*log(-sqrt(2)*sqrt(x) + x + 1) + 1/16*(5*x^(7/2) + 9*x^(3/2))/(x^4 + 2*x^2 + 1)
Time = 0.12 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.89 \[ \int \frac {\sqrt {x}}{\left (1+x^2\right )^3} \, dx=\frac {5}{64} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {x}\right )}\right ) + \frac {5}{64} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {x}\right )}\right ) - \frac {5}{128} \, \sqrt {2} \log \left (\sqrt {2} \sqrt {x} + x + 1\right ) + \frac {5}{128} \, \sqrt {2} \log \left (-\sqrt {2} \sqrt {x} + x + 1\right ) + \frac {5 \, x^{\frac {7}{2}} + 9 \, x^{\frac {3}{2}}}{16 \, {\left (x^{2} + 1\right )}^{2}} \] Input:
integrate(x^(1/2)/(x^2+1)^3,x, algorithm="giac")
Output:
5/64*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(x))) + 5/64*sqrt(2)*arct an(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(x))) - 5/128*sqrt(2)*log(sqrt(2)*sqrt(x) + x + 1) + 5/128*sqrt(2)*log(-sqrt(2)*sqrt(x) + x + 1) + 1/16*(5*x^(7/2) + 9*x^(3/2))/(x^2 + 1)^2
Time = 0.04 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.58 \[ \int \frac {\sqrt {x}}{\left (1+x^2\right )^3} \, dx=\frac {\frac {9\,x^{3/2}}{16}+\frac {5\,x^{7/2}}{16}}{x^4+2\,x^2+1}+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,\sqrt {x}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {5}{64}-\frac {5}{64}{}\mathrm {i}\right )+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,\sqrt {x}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {5}{64}+\frac {5}{64}{}\mathrm {i}\right ) \] Input:
int(x^(1/2)/(x^2 + 1)^3,x)
Output:
2^(1/2)*atan(2^(1/2)*x^(1/2)*(1/2 - 1i/2))*(5/64 - 5i/64) + 2^(1/2)*atan(2 ^(1/2)*x^(1/2)*(1/2 + 1i/2))*(5/64 + 5i/64) + ((9*x^(3/2))/16 + (5*x^(7/2) )/16)/(2*x^2 + x^4 + 1)
Time = 0.21 (sec) , antiderivative size = 241, normalized size of antiderivative = 2.27 \[ \int \frac {\sqrt {x}}{\left (1+x^2\right )^3} \, dx=\frac {10 \sqrt {2}\, \mathit {atan} \left (\frac {2 \sqrt {x}-\sqrt {2}}{\sqrt {2}}\right ) x^{4}+20 \sqrt {2}\, \mathit {atan} \left (\frac {2 \sqrt {x}-\sqrt {2}}{\sqrt {2}}\right ) x^{2}+10 \sqrt {2}\, \mathit {atan} \left (\frac {2 \sqrt {x}-\sqrt {2}}{\sqrt {2}}\right )+10 \sqrt {2}\, \mathit {atan} \left (\frac {2 \sqrt {x}+\sqrt {2}}{\sqrt {2}}\right ) x^{4}+20 \sqrt {2}\, \mathit {atan} \left (\frac {2 \sqrt {x}+\sqrt {2}}{\sqrt {2}}\right ) x^{2}+10 \sqrt {2}\, \mathit {atan} \left (\frac {2 \sqrt {x}+\sqrt {2}}{\sqrt {2}}\right )+40 \sqrt {x}\, x^{3}+72 \sqrt {x}\, x +5 \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, \sqrt {2}+x +1\right ) x^{4}+10 \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, \sqrt {2}+x +1\right ) x^{2}+5 \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, \sqrt {2}+x +1\right )-5 \sqrt {2}\, \mathrm {log}\left (\sqrt {x}\, \sqrt {2}+x +1\right ) x^{4}-10 \sqrt {2}\, \mathrm {log}\left (\sqrt {x}\, \sqrt {2}+x +1\right ) x^{2}-5 \sqrt {2}\, \mathrm {log}\left (\sqrt {x}\, \sqrt {2}+x +1\right )}{128 x^{4}+256 x^{2}+128} \] Input:
int(x^(1/2)/(x^2+1)^3,x)
Output:
(10*sqrt(2)*atan((2*sqrt(x) - sqrt(2))/sqrt(2))*x**4 + 20*sqrt(2)*atan((2* sqrt(x) - sqrt(2))/sqrt(2))*x**2 + 10*sqrt(2)*atan((2*sqrt(x) - sqrt(2))/s qrt(2)) + 10*sqrt(2)*atan((2*sqrt(x) + sqrt(2))/sqrt(2))*x**4 + 20*sqrt(2) *atan((2*sqrt(x) + sqrt(2))/sqrt(2))*x**2 + 10*sqrt(2)*atan((2*sqrt(x) + s qrt(2))/sqrt(2)) + 40*sqrt(x)*x**3 + 72*sqrt(x)*x + 5*sqrt(2)*log( - sqrt( x)*sqrt(2) + x + 1)*x**4 + 10*sqrt(2)*log( - sqrt(x)*sqrt(2) + x + 1)*x**2 + 5*sqrt(2)*log( - sqrt(x)*sqrt(2) + x + 1) - 5*sqrt(2)*log(sqrt(x)*sqrt( 2) + x + 1)*x**4 - 10*sqrt(2)*log(sqrt(x)*sqrt(2) + x + 1)*x**2 - 5*sqrt(2 )*log(sqrt(x)*sqrt(2) + x + 1))/(128*(x**4 + 2*x**2 + 1))