Optimal. Leaf size=129 \[ \frac {5 x^{3/2}}{16 \left (x^2+1\right )}+\frac {x^{3/2}}{4 \left (x^2+1\right )^2}+\frac {5 \log \left (x-\sqrt {2} \sqrt {x}+1\right )}{64 \sqrt {2}}-\frac {5 \log \left (x+\sqrt {2} \sqrt {x}+1\right )}{64 \sqrt {2}}-\frac {5 \tan ^{-1}\left (1-\sqrt {2} \sqrt {x}\right )}{32 \sqrt {2}}+\frac {5 \tan ^{-1}\left (\sqrt {2} \sqrt {x}+1\right )}{32 \sqrt {2}} \]
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Rubi [A] time = 0.07, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {290, 329, 297, 1162, 617, 204, 1165, 628} \[ \frac {5 x^{3/2}}{16 \left (x^2+1\right )}+\frac {x^{3/2}}{4 \left (x^2+1\right )^2}+\frac {5 \log \left (x-\sqrt {2} \sqrt {x}+1\right )}{64 \sqrt {2}}-\frac {5 \log \left (x+\sqrt {2} \sqrt {x}+1\right )}{64 \sqrt {2}}-\frac {5 \tan ^{-1}\left (1-\sqrt {2} \sqrt {x}\right )}{32 \sqrt {2}}+\frac {5 \tan ^{-1}\left (\sqrt {2} \sqrt {x}+1\right )}{32 \sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 290
Rule 297
Rule 329
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {\sqrt {x}}{\left (1+x^2\right )^3} \, dx &=\frac {x^{3/2}}{4 \left (1+x^2\right )^2}+\frac {5}{8} \int \frac {\sqrt {x}}{\left (1+x^2\right )^2} \, 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}{32} \int \frac {\sqrt {x}}{1+x^2} \, 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}{16} \operatorname {Subst}\left (\int \frac {x^2}{1+x^4} \, dx,x,\sqrt {x}\right )\\ &=\frac {x^{3/2}}{4 \left (1+x^2\right )^2}+\frac {5 x^{3/2}}{16 \left (1+x^2\right )}-\frac {5}{32} \operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {x}\right )+\frac {5}{32} \operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {x}\right )\\ &=\frac {x^{3/2}}{4 \left (1+x^2\right )^2}+\frac {5 x^{3/2}}{16 \left (1+x^2\right )}+\frac {5}{64} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {x}\right )+\frac {5}{64} \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {x}\right )+\frac {5 \operatorname {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {x}\right )}{64 \sqrt {2}}+\frac {5 \operatorname {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {x}\right )}{64 \sqrt {2}}\\ &=\frac {x^{3/2}}{4 \left (1+x^2\right )^2}+\frac {5 x^{3/2}}{16 \left (1+x^2\right )}+\frac {5 \log \left (1-\sqrt {2} \sqrt {x}+x\right )}{64 \sqrt {2}}-\frac {5 \log \left (1+\sqrt {2} \sqrt {x}+x\right )}{64 \sqrt {2}}+\frac {5 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {x}\right )}{32 \sqrt {2}}-\frac {5 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {x}\right )}{32 \sqrt {2}}\\ &=\frac {x^{3/2}}{4 \left (1+x^2\right )^2}+\frac {5 x^{3/2}}{16 \left (1+x^2\right )}-\frac {5 \tan ^{-1}\left (1-\sqrt {2} \sqrt {x}\right )}{32 \sqrt {2}}+\frac {5 \tan ^{-1}\left (1+\sqrt {2} \sqrt {x}\right )}{32 \sqrt {2}}+\frac {5 \log \left (1-\sqrt {2} \sqrt {x}+x\right )}{64 \sqrt {2}}-\frac {5 \log \left (1+\sqrt {2} \sqrt {x}+x\right )}{64 \sqrt {2}}\\ \end {align*}
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Mathematica [C] time = 0.00, size = 22, normalized size = 0.17 \[ \frac {2}{3} x^{3/2} \, _2F_1\left (\frac {3}{4},3;\frac {7}{4};-x^2\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 175, normalized size = 1.36 \[ -\frac {20 \, \sqrt {2} {\left (x^{4} + 2 \, x^{2} + 1\right )} \arctan \left (\sqrt {2} \sqrt {\sqrt {2} \sqrt {x} + x + 1} - \sqrt {2} \sqrt {x} - 1\right ) + 20 \, \sqrt {2} {\left (x^{4} + 2 \, x^{2} + 1\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {-4 \, \sqrt {2} \sqrt {x} + 4 \, x + 4} - \sqrt {2} \sqrt {x} + 1\right ) + 5 \, \sqrt {2} {\left (x^{4} + 2 \, x^{2} + 1\right )} \log \left (4 \, \sqrt {2} \sqrt {x} + 4 \, x + 4\right ) - 5 \, \sqrt {2} {\left (x^{4} + 2 \, x^{2} + 1\right )} \log \left (-4 \, \sqrt {2} \sqrt {x} + 4 \, x + 4\right ) - 8 \, {\left (5 \, x^{3} + 9 \, x\right )} \sqrt {x}}{128 \, {\left (x^{4} + 2 \, x^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.59, size = 94, normalized size = 0.73 \[ \frac {5}{64} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {x}\right )}\right ) + \frac {5}{64} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {x}\right )}\right ) - \frac {5}{128} \, \sqrt {2} \log \left (\sqrt {2} \sqrt {x} + x + 1\right ) + \frac {5}{128} \, \sqrt {2} \log \left (-\sqrt {2} \sqrt {x} + x + 1\right ) + \frac {5 \, x^{\frac {7}{2}} + 9 \, x^{\frac {3}{2}}}{16 \, {\left (x^{2} + 1\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 86, normalized size = 0.67 \[ \frac {x^{\frac {3}{2}}}{4 \left (x^{2}+1\right )^{2}}+\frac {5 x^{\frac {3}{2}}}{16 \left (x^{2}+1\right )}+\frac {5 \sqrt {2}\, \arctan \left (\sqrt {2}\, \sqrt {x}-1\right )}{64}+\frac {5 \sqrt {2}\, \arctan \left (\sqrt {2}\, \sqrt {x}+1\right )}{64}+\frac {5 \sqrt {2}\, \ln \left (\frac {x -\sqrt {2}\, \sqrt {x}+1}{x +\sqrt {2}\, \sqrt {x}+1}\right )}{128} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.02, size = 99, normalized size = 0.77 \[ \frac {5}{64} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {x}\right )}\right ) + \frac {5}{64} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {x}\right )}\right ) - \frac {5}{128} \, \sqrt {2} \log \left (\sqrt {2} \sqrt {x} + x + 1\right ) + \frac {5}{128} \, \sqrt {2} \log \left (-\sqrt {2} \sqrt {x} + x + 1\right ) + \frac {5 \, x^{\frac {7}{2}} + 9 \, x^{\frac {3}{2}}}{16 \, {\left (x^{4} + 2 \, x^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 61, normalized size = 0.47 \[ \frac {\frac {9\,x^{3/2}}{16}+\frac {5\,x^{7/2}}{16}}{x^4+2\,x^2+1}+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,\sqrt {x}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {5}{64}-\frac {5}{64}{}\mathrm {i}\right )+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,\sqrt {x}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {5}{64}+\frac {5}{64}{}\mathrm {i}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 6.38, size = 481, normalized size = 3.73 \[ \frac {40 x^{\frac {7}{2}}}{128 x^{4} + 256 x^{2} + 128} + \frac {72 x^{\frac {3}{2}}}{128 x^{4} + 256 x^{2} + 128} + \frac {5 \sqrt {2} x^{4} \log {\left (- 4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{128 x^{4} + 256 x^{2} + 128} - \frac {5 \sqrt {2} x^{4} \log {\left (4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac {10 \sqrt {2} x^{4} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} - 1 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac {10 \sqrt {2} x^{4} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} + 1 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac {10 \sqrt {2} x^{2} \log {\left (- 4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{128 x^{4} + 256 x^{2} + 128} - \frac {10 \sqrt {2} x^{2} \log {\left (4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac {20 \sqrt {2} x^{2} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} - 1 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac {20 \sqrt {2} x^{2} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} + 1 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac {5 \sqrt {2} \log {\left (- 4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{128 x^{4} + 256 x^{2} + 128} - \frac {5 \sqrt {2} \log {\left (4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac {10 \sqrt {2} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} - 1 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac {10 \sqrt {2} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} + 1 \right )}}{128 x^{4} + 256 x^{2} + 128} \]
Verification of antiderivative is not currently implemented for this CAS.
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