Optimal. Leaf size=129 \[ \frac {\sqrt {x}}{16 \left (x^2+1\right )}-\frac {\sqrt {x}}{4 \left (x^2+1\right )^2}-\frac {3 \log \left (x-\sqrt {2} \sqrt {x}+1\right )}{64 \sqrt {2}}+\frac {3 \log \left (x+\sqrt {2} \sqrt {x}+1\right )}{64 \sqrt {2}}-\frac {3 \tan ^{-1}\left (1-\sqrt {2} \sqrt {x}\right )}{32 \sqrt {2}}+\frac {3 \tan ^{-1}\left (\sqrt {2} \sqrt {x}+1\right )}{32 \sqrt {2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.692, Rules used = {288, 290, 329, 211, 1165, 628, 1162, 617, 204} \[ \frac {\sqrt {x}}{16 \left (x^2+1\right )}-\frac {\sqrt {x}}{4 \left (x^2+1\right )^2}-\frac {3 \log \left (x-\sqrt {2} \sqrt {x}+1\right )}{64 \sqrt {2}}+\frac {3 \log \left (x+\sqrt {2} \sqrt {x}+1\right )}{64 \sqrt {2}}-\frac {3 \tan ^{-1}\left (1-\sqrt {2} \sqrt {x}\right )}{32 \sqrt {2}}+\frac {3 \tan ^{-1}\left (\sqrt {2} \sqrt {x}+1\right )}{32 \sqrt {2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 204
Rule 211
Rule 288
Rule 290
Rule 329
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {x^{3/2}}{\left (1+x^2\right )^3} \, dx &=-\frac {\sqrt {x}}{4 \left (1+x^2\right )^2}+\frac {1}{8} \int \frac {1}{\sqrt {x} \left (1+x^2\right )^2} \, dx\\ &=-\frac {\sqrt {x}}{4 \left (1+x^2\right )^2}+\frac {\sqrt {x}}{16 \left (1+x^2\right )}+\frac {3}{32} \int \frac {1}{\sqrt {x} \left (1+x^2\right )} \, dx\\ &=-\frac {\sqrt {x}}{4 \left (1+x^2\right )^2}+\frac {\sqrt {x}}{16 \left (1+x^2\right )}+\frac {3}{16} \operatorname {Subst}\left (\int \frac {1}{1+x^4} \, dx,x,\sqrt {x}\right )\\ &=-\frac {\sqrt {x}}{4 \left (1+x^2\right )^2}+\frac {\sqrt {x}}{16 \left (1+x^2\right )}+\frac {3}{32} \operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {x}\right )+\frac {3}{32} \operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {x}\right )\\ &=-\frac {\sqrt {x}}{4 \left (1+x^2\right )^2}+\frac {\sqrt {x}}{16 \left (1+x^2\right )}+\frac {3}{64} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {x}\right )+\frac {3}{64} \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {x}\right )-\frac {3 \operatorname {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {x}\right )}{64 \sqrt {2}}-\frac {3 \operatorname {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {x}\right )}{64 \sqrt {2}}\\ &=-\frac {\sqrt {x}}{4 \left (1+x^2\right )^2}+\frac {\sqrt {x}}{16 \left (1+x^2\right )}-\frac {3 \log \left (1-\sqrt {2} \sqrt {x}+x\right )}{64 \sqrt {2}}+\frac {3 \log \left (1+\sqrt {2} \sqrt {x}+x\right )}{64 \sqrt {2}}+\frac {3 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {x}\right )}{32 \sqrt {2}}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {x}\right )}{32 \sqrt {2}}\\ &=-\frac {\sqrt {x}}{4 \left (1+x^2\right )^2}+\frac {\sqrt {x}}{16 \left (1+x^2\right )}-\frac {3 \tan ^{-1}\left (1-\sqrt {2} \sqrt {x}\right )}{32 \sqrt {2}}+\frac {3 \tan ^{-1}\left (1+\sqrt {2} \sqrt {x}\right )}{32 \sqrt {2}}-\frac {3 \log \left (1-\sqrt {2} \sqrt {x}+x\right )}{64 \sqrt {2}}+\frac {3 \log \left (1+\sqrt {2} \sqrt {x}+x\right )}{64 \sqrt {2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 121, normalized size = 0.94 \[ \frac {1}{128} \left (\frac {8 \sqrt {x}}{x^2+1}-\frac {32 \sqrt {x}}{\left (x^2+1\right )^2}-3 \sqrt {2} \log \left (x-\sqrt {2} \sqrt {x}+1\right )+3 \sqrt {2} \log \left (x+\sqrt {2} \sqrt {x}+1\right )-6 \sqrt {2} \tan ^{-1}\left (1-\sqrt {2} \sqrt {x}\right )+6 \sqrt {2} \tan ^{-1}\left (\sqrt {2} \sqrt {x}+1\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.66, size = 171, normalized size = 1.33 \[ -\frac {12 \, \sqrt {2} {\left (x^{4} + 2 \, x^{2} + 1\right )} \arctan \left (\sqrt {2} \sqrt {\sqrt {2} \sqrt {x} + x + 1} - \sqrt {2} \sqrt {x} - 1\right ) + 12 \, \sqrt {2} {\left (x^{4} + 2 \, x^{2} + 1\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {-4 \, \sqrt {2} \sqrt {x} + 4 \, x + 4} - \sqrt {2} \sqrt {x} + 1\right ) - 3 \, \sqrt {2} {\left (x^{4} + 2 \, x^{2} + 1\right )} \log \left (4 \, \sqrt {2} \sqrt {x} + 4 \, x + 4\right ) + 3 \, \sqrt {2} {\left (x^{4} + 2 \, x^{2} + 1\right )} \log \left (-4 \, \sqrt {2} \sqrt {x} + 4 \, x + 4\right ) - 8 \, {\left (x^{2} - 3\right )} \sqrt {x}}{128 \, {\left (x^{4} + 2 \, x^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.64, size = 92, normalized size = 0.71 \[ \frac {3}{64} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {x}\right )}\right ) + \frac {3}{64} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {x}\right )}\right ) + \frac {3}{128} \, \sqrt {2} \log \left (\sqrt {2} \sqrt {x} + x + 1\right ) - \frac {3}{128} \, \sqrt {2} \log \left (-\sqrt {2} \sqrt {x} + x + 1\right ) + \frac {x^{\frac {5}{2}} - 3 \, \sqrt {x}}{16 \, {\left (x^{2} + 1\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 82, normalized size = 0.64 \[ \frac {3 \sqrt {2}\, \arctan \left (\sqrt {2}\, \sqrt {x}-1\right )}{64}+\frac {3 \sqrt {2}\, \arctan \left (\sqrt {2}\, \sqrt {x}+1\right )}{64}+\frac {3 \sqrt {2}\, \ln \left (\frac {x +\sqrt {2}\, \sqrt {x}+1}{x -\sqrt {2}\, \sqrt {x}+1}\right )}{128}+\frac {\frac {x^{\frac {5}{2}}}{16}-\frac {3 \sqrt {x}}{16}}{\left (x^{2}+1\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 2.97, size = 97, normalized size = 0.75 \[ \frac {3}{64} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {x}\right )}\right ) + \frac {3}{64} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {x}\right )}\right ) + \frac {3}{128} \, \sqrt {2} \log \left (\sqrt {2} \sqrt {x} + x + 1\right ) - \frac {3}{128} \, \sqrt {2} \log \left (-\sqrt {2} \sqrt {x} + x + 1\right ) + \frac {x^{\frac {5}{2}} - 3 \, \sqrt {x}}{16 \, {\left (x^{4} + 2 \, x^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.69, size = 62, normalized size = 0.48 \[ -\frac {\frac {3\,\sqrt {x}}{16}-\frac {x^{5/2}}{16}}{x^4+2\,x^2+1}+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,\sqrt {x}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {3}{64}+\frac {3}{64}{}\mathrm {i}\right )+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,\sqrt {x}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {3}{64}-\frac {3}{64}{}\mathrm {i}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 9.81, size = 481, normalized size = 3.73 \[ \frac {8 x^{\frac {5}{2}}}{128 x^{4} + 256 x^{2} + 128} - \frac {24 \sqrt {x}}{128 x^{4} + 256 x^{2} + 128} - \frac {3 \sqrt {2} x^{4} \log {\left (- 4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac {3 \sqrt {2} x^{4} \log {\left (4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac {6 \sqrt {2} x^{4} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} - 1 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac {6 \sqrt {2} x^{4} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} + 1 \right )}}{128 x^{4} + 256 x^{2} + 128} - \frac {6 \sqrt {2} x^{2} \log {\left (- 4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac {6 \sqrt {2} x^{2} \log {\left (4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac {12 \sqrt {2} x^{2} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} - 1 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac {12 \sqrt {2} x^{2} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} + 1 \right )}}{128 x^{4} + 256 x^{2} + 128} - \frac {3 \sqrt {2} \log {\left (- 4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac {3 \sqrt {2} \log {\left (4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac {6 \sqrt {2} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} - 1 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac {6 \sqrt {2} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} + 1 \right )}}{128 x^{4} + 256 x^{2} + 128} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________