Integrand size = 13, antiderivative size = 106 \[ \int \frac {x^{3/2}}{\left (1+x^2\right )^3} \, dx=-\frac {\sqrt {x}}{4 \left (1+x^2\right )^2}+\frac {\sqrt {x}}{16 \left (1+x^2\right )}-\frac {3 \arctan \left (1-\sqrt {2} \sqrt {x}\right )}{32 \sqrt {2}}+\frac {3 \arctan \left (1+\sqrt {2} \sqrt {x}\right )}{32 \sqrt {2}}+\frac {3 \text {arctanh}\left (\frac {\sqrt {2} \sqrt {x}}{1+x}\right )}{32 \sqrt {2}} \] Output:
-1/4*x^(1/2)/(x^2+1)^2+x^(1/2)/(16*x^2+16)+3/64*arctan(-1+2^(1/2)*x^(1/2)) *2^(1/2)+3/64*arctan(1+2^(1/2)*x^(1/2))*2^(1/2)+3/64*arctanh(2^(1/2)*x^(1/ 2)/(1+x))*2^(1/2)
Time = 0.14 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.66 \[ \int \frac {x^{3/2}}{\left (1+x^2\right )^3} \, dx=\frac {1}{64} \left (\frac {4 \sqrt {x} \left (-3+x^2\right )}{\left (1+x^2\right )^2}+3 \sqrt {2} \arctan \left (\frac {-1+x}{\sqrt {2} \sqrt {x}}\right )+3 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {x}}{1+x}\right )\right ) \] Input:
Integrate[x^(3/2)/(1 + x^2)^3,x]
Output:
((4*Sqrt[x]*(-3 + x^2))/(1 + x^2)^2 + 3*Sqrt[2]*ArcTan[(-1 + x)/(Sqrt[2]*S qrt[x])] + 3*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 = {252, 253, 266, 755, 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.
| \(\displaystyle \int \frac {x^{3/2}}{\left (x^2+1\right )^3} \, dx\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {1}{8} \int \frac {1}{\sqrt {x} \left (x^2+1\right )^2}dx-\frac {\sqrt {x}}{4 \left (x^2+1\right )^2}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {1}{8} \left (\frac {3}{4} \int \frac {1}{\sqrt {x} \left (x^2+1\right )}dx+\frac {\sqrt {x}}{2 \left (x^2+1\right )}\right )-\frac {\sqrt {x}}{4 \left (x^2+1\right )^2}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {1}{8} \left (\frac {3}{2} \int \frac {1}{x^2+1}d\sqrt {x}+\frac {\sqrt {x}}{2 \left (x^2+1\right )}\right )-\frac {\sqrt {x}}{4 \left (x^2+1\right )^2}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {1}{8} \left (\frac {3}{2} \left (\frac {1}{2} \int \frac {1-x}{x^2+1}d\sqrt {x}+\frac {1}{2} \int \frac {x+1}{x^2+1}d\sqrt {x}\right )+\frac {\sqrt {x}}{2 \left (x^2+1\right )}\right )-\frac {\sqrt {x}}{4 \left (x^2+1\right )^2}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {1}{8} \left (\frac {3}{2} \left (\frac {1}{2} \int \frac {1-x}{x^2+1}d\sqrt {x}+\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{x-\sqrt {2} \sqrt {x}+1}d\sqrt {x}+\frac {1}{2} \int \frac {1}{x+\sqrt {2} \sqrt {x}+1}d\sqrt {x}\right )\right )+\frac {\sqrt {x}}{2 \left (x^2+1\right )}\right )-\frac {\sqrt {x}}{4 \left (x^2+1\right )^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {1}{8} \left (\frac {3}{2} \left (\frac {1}{2} \int \frac {1-x}{x^2+1}d\sqrt {x}+\frac {1}{2} \left (\frac {\int \frac {1}{-x-1}d\left (1-\sqrt {2} \sqrt {x}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-x-1}d\left (\sqrt {2} \sqrt {x}+1\right )}{\sqrt {2}}\right )\right )+\frac {\sqrt {x}}{2 \left (x^2+1\right )}\right )-\frac {\sqrt {x}}{4 \left (x^2+1\right )^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{8} \left (\frac {3}{2} \left (\frac {1}{2} \int \frac {1-x}{x^2+1}d\sqrt {x}+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {x}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {x}\right )}{\sqrt {2}}\right )\right )+\frac {\sqrt {x}}{2 \left (x^2+1\right )}\right )-\frac {\sqrt {x}}{4 \left (x^2+1\right )^2}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {1}{8} \left (\frac {3}{2} \left (\frac {1}{2} \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {x}}{x-\sqrt {2} \sqrt {x}+1}d\sqrt {x}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {x}+1\right )}{x+\sqrt {2} \sqrt {x}+1}d\sqrt {x}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {x}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {x}\right )}{\sqrt {2}}\right )\right )+\frac {\sqrt {x}}{2 \left (x^2+1\right )}\right )-\frac {\sqrt {x}}{4 \left (x^2+1\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{8} \left (\frac {3}{2} \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {x}}{x-\sqrt {2} \sqrt {x}+1}d\sqrt {x}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {x}+1\right )}{x+\sqrt {2} \sqrt {x}+1}d\sqrt {x}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {x}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {x}\right )}{\sqrt {2}}\right )\right )+\frac {\sqrt {x}}{2 \left (x^2+1\right )}\right )-\frac {\sqrt {x}}{4 \left (x^2+1\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \left (\frac {3}{2} \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {x}}{x-\sqrt {2} \sqrt {x}+1}d\sqrt {x}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {x}+1}{x+\sqrt {2} \sqrt {x}+1}d\sqrt {x}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {x}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {x}\right )}{\sqrt {2}}\right )\right )+\frac {\sqrt {x}}{2 \left (x^2+1\right )}\right )-\frac {\sqrt {x}}{4 \left (x^2+1\right )^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{8} \left (\frac {3}{2} \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {x}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {x}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (x+\sqrt {2} \sqrt {x}+1\right )}{2 \sqrt {2}}-\frac {\log \left (x-\sqrt {2} \sqrt {x}+1\right )}{2 \sqrt {2}}\right )\right )+\frac {\sqrt {x}}{2 \left (x^2+1\right )}\right )-\frac {\sqrt {x}}{4 \left (x^2+1\right )^2}\) |
Input:
Int[x^(3/2)/(1 + x^2)^3,x]
Output:
-1/4*Sqrt[x]/(1 + x^2)^2 + (Sqrt[x]/(2*(1 + x^2)) + (3*((-(ArcTan[1 - Sqrt [2]*Sqrt[x]]/Sqrt[2]) + ArcTan[1 + Sqrt[2]*Sqrt[x]]/Sqrt[2])/2 + (-1/2*Log [1 - Sqrt[2]*Sqrt[x] + x]/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*(c*x )^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* (p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c }, x] && LtQ[p, -1] && GtQ[m, 1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi alQ[a, b, c, 2, m, p, x]
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[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] ], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r) Int[(r - s*x^2)/(a + b*x^4) , x], x] + Simp[1/(2*r) 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 = 0.53 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.70
| method | result | size |
| risch | \(\frac {\left (x^{2}-3\right ) \sqrt {x}}{16 \left (x^{2}+1\right )^{2}}+\frac {3 \sqrt {2}\, \left (\ln \left (\frac {x +\sqrt {2}\, \sqrt {x}+1}{x -\sqrt {2}\, \sqrt {x}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {x}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {x}\right )\right )}{128}\) | \(74\) |
| derivativedivides | \(\frac {\frac {x^{\frac {5}{2}}}{16}-\frac {3 \sqrt {x}}{16}}{\left (x^{2}+1\right )^{2}}+\frac {3 \sqrt {2}\, \left (\ln \left (\frac {x +\sqrt {2}\, \sqrt {x}+1}{x -\sqrt {2}\, \sqrt {x}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {x}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {x}\right )\right )}{128}\) | \(77\) |
| default | \(\frac {\frac {x^{\frac {5}{2}}}{16}-\frac {3 \sqrt {x}}{16}}{\left (x^{2}+1\right )^{2}}+\frac {3 \sqrt {2}\, \left (\ln \left (\frac {x +\sqrt {2}\, \sqrt {x}+1}{x -\sqrt {2}\, \sqrt {x}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {x}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {x}\right )\right )}{128}\) | \(77\) |
| meijerg | \(-\frac {\sqrt {x}\, \left (-5 x^{2}+15\right )}{80 \left (x^{2}+1\right )^{2}}+\frac {3 \sqrt {x}\, \left (-\frac {\sqrt {2}\, \ln \left (1-\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}+\sqrt {x^{2}}\right )}{2 \left (x^{2}\right )^{\frac {1}{4}}}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}{2-\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}\right )}{\left (x^{2}\right )^{\frac {1}{4}}}+\frac {\sqrt {2}\, \ln \left (1+\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}+\sqrt {x^{2}}\right )}{2 \left (x^{2}\right )^{\frac {1}{4}}}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}{2+\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}\right )}{\left (x^{2}\right )^{\frac {1}{4}}}\right )}{64}\) | \(150\) |
| trager | \(\frac {\left (x^{2}-3\right ) \sqrt {x}}{16 \left (x^{2}+1\right )^{2}}+\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{5} x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{5}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x +\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x +\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )+4 \sqrt {x}}{\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2}+x +1}\right )}{64}+\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{5} x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{5}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )+4 \sqrt {x}}{\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2}-x -1}\right )}{64}\) | \(199\) |
Input:
int(x^(3/2)/(x^2+1)^3,x,method=_RETURNVERBOSE)
Output:
1/16*(x^2-3)/(x^2+1)^2*x^(1/2)+3/128*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 = 128, normalized size of antiderivative = 1.21 \[ \int \frac {x^{3/2}}{\left (1+x^2\right )^3} \, dx=\frac {6 \, \sqrt {2} {\left (x^{4} + 2 \, x^{2} + 1\right )} \arctan \left (\sqrt {2} \sqrt {x} + 1\right ) + 6 \, \sqrt {2} {\left (x^{4} + 2 \, x^{2} + 1\right )} \arctan \left (\sqrt {2} \sqrt {x} - 1\right ) + 3 \, \sqrt {2} {\left (x^{4} + 2 \, x^{2} + 1\right )} \log \left (\sqrt {2} \sqrt {x} + x + 1\right ) - 3 \, \sqrt {2} {\left (x^{4} + 2 \, x^{2} + 1\right )} \log \left (-\sqrt {2} \sqrt {x} + x + 1\right ) + 8 \, {\left (x^{2} - 3\right )} \sqrt {x}}{128 \, {\left (x^{4} + 2 \, x^{2} + 1\right )}} \] Input:
integrate(x^(3/2)/(x^2+1)^3,x, algorithm="fricas")
Output:
1/128*(6*sqrt(2)*(x^4 + 2*x^2 + 1)*arctan(sqrt(2)*sqrt(x) + 1) + 6*sqrt(2) *(x^4 + 2*x^2 + 1)*arctan(sqrt(2)*sqrt(x) - 1) + 3*sqrt(2)*(x^4 + 2*x^2 + 1)*log(sqrt(2)*sqrt(x) + x + 1) - 3*sqrt(2)*(x^4 + 2*x^2 + 1)*log(-sqrt(2) *sqrt(x) + x + 1) + 8*(x^2 - 3)*sqrt(x))/(x^4 + 2*x^2 + 1)
Leaf count of result is larger than twice the leaf count of optimal. 481 vs. \(2 (92) = 184\).
Time = 1.30 (sec) , antiderivative size = 481, normalized size of antiderivative = 4.54 \[ \int \frac {x^{3/2}}{\left (1+x^2\right )^3} \, dx=\frac {8 x^{\frac {5}{2}}}{128 x^{4} + 256 x^{2} + 128} - \frac {24 \sqrt {x}}{128 x^{4} + 256 x^{2} + 128} - \frac {3 \sqrt {2} x^{4} \log {\left (- 4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac {3 \sqrt {2} x^{4} \log {\left (4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac {6 \sqrt {2} x^{4} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} - 1 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac {6 \sqrt {2} x^{4} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} + 1 \right )}}{128 x^{4} + 256 x^{2} + 128} - \frac {6 \sqrt {2} x^{2} \log {\left (- 4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac {6 \sqrt {2} x^{2} \log {\left (4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac {12 \sqrt {2} x^{2} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} - 1 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac {12 \sqrt {2} x^{2} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} + 1 \right )}}{128 x^{4} + 256 x^{2} + 128} - \frac {3 \sqrt {2} \log {\left (- 4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac {3 \sqrt {2} \log {\left (4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac {6 \sqrt {2} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} - 1 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac {6 \sqrt {2} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} + 1 \right )}}{128 x^{4} + 256 x^{2} + 128} \] Input:
integrate(x**(3/2)/(x**2+1)**3,x)
Output:
8*x**(5/2)/(128*x**4 + 256*x**2 + 128) - 24*sqrt(x)/(128*x**4 + 256*x**2 + 128) - 3*sqrt(2)*x**4*log(-4*sqrt(2)*sqrt(x) + 4*x + 4)/(128*x**4 + 256*x **2 + 128) + 3*sqrt(2)*x**4*log(4*sqrt(2)*sqrt(x) + 4*x + 4)/(128*x**4 + 2 56*x**2 + 128) + 6*sqrt(2)*x**4*atan(sqrt(2)*sqrt(x) - 1)/(128*x**4 + 256* x**2 + 128) + 6*sqrt(2)*x**4*atan(sqrt(2)*sqrt(x) + 1)/(128*x**4 + 256*x** 2 + 128) - 6*sqrt(2)*x**2*log(-4*sqrt(2)*sqrt(x) + 4*x + 4)/(128*x**4 + 25 6*x**2 + 128) + 6*sqrt(2)*x**2*log(4*sqrt(2)*sqrt(x) + 4*x + 4)/(128*x**4 + 256*x**2 + 128) + 12*sqrt(2)*x**2*atan(sqrt(2)*sqrt(x) - 1)/(128*x**4 + 256*x**2 + 128) + 12*sqrt(2)*x**2*atan(sqrt(2)*sqrt(x) + 1)/(128*x**4 + 25 6*x**2 + 128) - 3*sqrt(2)*log(-4*sqrt(2)*sqrt(x) + 4*x + 4)/(128*x**4 + 25 6*x**2 + 128) + 3*sqrt(2)*log(4*sqrt(2)*sqrt(x) + 4*x + 4)/(128*x**4 + 256 *x**2 + 128) + 6*sqrt(2)*atan(sqrt(2)*sqrt(x) - 1)/(128*x**4 + 256*x**2 + 128) + 6*sqrt(2)*atan(sqrt(2)*sqrt(x) + 1)/(128*x**4 + 256*x**2 + 128)
Time = 0.11 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.92 \[ \int \frac {x^{3/2}}{\left (1+x^2\right )^3} \, dx=\frac {3}{64} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {x}\right )}\right ) + \frac {3}{64} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {x}\right )}\right ) + \frac {3}{128} \, \sqrt {2} \log \left (\sqrt {2} \sqrt {x} + x + 1\right ) - \frac {3}{128} \, \sqrt {2} \log \left (-\sqrt {2} \sqrt {x} + x + 1\right ) + \frac {x^{\frac {5}{2}} - 3 \, \sqrt {x}}{16 \, {\left (x^{4} + 2 \, x^{2} + 1\right )}} \] Input:
integrate(x^(3/2)/(x^2+1)^3,x, algorithm="maxima")
Output:
3/64*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(x))) + 3/64*sqrt(2)*arct an(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(x))) + 3/128*sqrt(2)*log(sqrt(2)*sqrt(x) + x + 1) - 3/128*sqrt(2)*log(-sqrt(2)*sqrt(x) + x + 1) + 1/16*(x^(5/2) - 3*sqrt(x))/(x^4 + 2*x^2 + 1)
Time = 0.12 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.87 \[ \int \frac {x^{3/2}}{\left (1+x^2\right )^3} \, dx=\frac {3}{64} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {x}\right )}\right ) + \frac {3}{64} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {x}\right )}\right ) + \frac {3}{128} \, \sqrt {2} \log \left (\sqrt {2} \sqrt {x} + x + 1\right ) - \frac {3}{128} \, \sqrt {2} \log \left (-\sqrt {2} \sqrt {x} + x + 1\right ) + \frac {x^{\frac {5}{2}} - 3 \, \sqrt {x}}{16 \, {\left (x^{2} + 1\right )}^{2}} \] Input:
integrate(x^(3/2)/(x^2+1)^3,x, algorithm="giac")
Output:
3/64*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(x))) + 3/64*sqrt(2)*arct an(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(x))) + 3/128*sqrt(2)*log(sqrt(2)*sqrt(x) + x + 1) - 3/128*sqrt(2)*log(-sqrt(2)*sqrt(x) + x + 1) + 1/16*(x^(5/2) - 3*sqrt(x))/(x^2 + 1)^2
Time = 0.05 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.58 \[ \int \frac {x^{3/2}}{\left (1+x^2\right )^3} \, dx=-\frac {\frac {3\,\sqrt {x}}{16}-\frac {x^{5/2}}{16}}{x^4+2\,x^2+1}+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,\sqrt {x}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {3}{64}+\frac {3}{64}{}\mathrm {i}\right )+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,\sqrt {x}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {3}{64}-\frac {3}{64}{}\mathrm {i}\right ) \] Input:
int(x^(3/2)/(x^2 + 1)^3,x)
Output:
2^(1/2)*atan(2^(1/2)*x^(1/2)*(1/2 - 1i/2))*(3/64 + 3i/64) + 2^(1/2)*atan(2 ^(1/2)*x^(1/2)*(1/2 + 1i/2))*(3/64 - 3i/64) - ((3*x^(1/2))/16 - x^(5/2)/16 )/(2*x^2 + x^4 + 1)
Time = 0.25 (sec) , antiderivative size = 240, normalized size of antiderivative = 2.26 \[ \int \frac {x^{3/2}}{\left (1+x^2\right )^3} \, dx=\frac {6 \sqrt {2}\, \mathit {atan} \left (\frac {2 \sqrt {x}-\sqrt {2}}{\sqrt {2}}\right ) x^{4}+12 \sqrt {2}\, \mathit {atan} \left (\frac {2 \sqrt {x}-\sqrt {2}}{\sqrt {2}}\right ) x^{2}+6 \sqrt {2}\, \mathit {atan} \left (\frac {2 \sqrt {x}-\sqrt {2}}{\sqrt {2}}\right )+6 \sqrt {2}\, \mathit {atan} \left (\frac {2 \sqrt {x}+\sqrt {2}}{\sqrt {2}}\right ) x^{4}+12 \sqrt {2}\, \mathit {atan} \left (\frac {2 \sqrt {x}+\sqrt {2}}{\sqrt {2}}\right ) x^{2}+6 \sqrt {2}\, \mathit {atan} \left (\frac {2 \sqrt {x}+\sqrt {2}}{\sqrt {2}}\right )+8 \sqrt {x}\, x^{2}-24 \sqrt {x}-3 \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, \sqrt {2}+x +1\right ) x^{4}-6 \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, \sqrt {2}+x +1\right ) x^{2}-3 \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, \sqrt {2}+x +1\right )+3 \sqrt {2}\, \mathrm {log}\left (\sqrt {x}\, \sqrt {2}+x +1\right ) x^{4}+6 \sqrt {2}\, \mathrm {log}\left (\sqrt {x}\, \sqrt {2}+x +1\right ) x^{2}+3 \sqrt {2}\, \mathrm {log}\left (\sqrt {x}\, \sqrt {2}+x +1\right )}{128 x^{4}+256 x^{2}+128} \] Input:
int(x^(3/2)/(x^2+1)^3,x)
Output:
(6*sqrt(2)*atan((2*sqrt(x) - sqrt(2))/sqrt(2))*x**4 + 12*sqrt(2)*atan((2*s qrt(x) - sqrt(2))/sqrt(2))*x**2 + 6*sqrt(2)*atan((2*sqrt(x) - sqrt(2))/sqr t(2)) + 6*sqrt(2)*atan((2*sqrt(x) + sqrt(2))/sqrt(2))*x**4 + 12*sqrt(2)*at an((2*sqrt(x) + sqrt(2))/sqrt(2))*x**2 + 6*sqrt(2)*atan((2*sqrt(x) + sqrt( 2))/sqrt(2)) + 8*sqrt(x)*x**2 - 24*sqrt(x) - 3*sqrt(2)*log( - sqrt(x)*sqrt (2) + x + 1)*x**4 - 6*sqrt(2)*log( - sqrt(x)*sqrt(2) + x + 1)*x**2 - 3*sqr t(2)*log( - sqrt(x)*sqrt(2) + x + 1) + 3*sqrt(2)*log(sqrt(x)*sqrt(2) + x + 1)*x**4 + 6*sqrt(2)*log(sqrt(x)*sqrt(2) + x + 1)*x**2 + 3*sqrt(2)*log(sqr t(x)*sqrt(2) + x + 1))/(128*(x**4 + 2*x**2 + 1))