Optimal. Leaf size=129 \[ -\frac {x^{3/2}}{4 \left (1+x^2\right )^2}+\frac {3 x^{3/2}}{16 \left (1+x^2\right )}-\frac {3 \tan ^{-1}\left (1-\sqrt {2} \sqrt {x}\right )}{32 \sqrt {2}}+\frac {3 \tan ^{-1}\left (1+\sqrt {2} \sqrt {x}\right )}{32 \sqrt {2}}+\frac {3 \log \left (1-\sqrt {2} \sqrt {x}+x\right )}{64 \sqrt {2}}-\frac {3 \log \left (1+\sqrt {2} \sqrt {x}+x\right )}{64 \sqrt {2}} \]
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Rubi [A]
time = 0.05, 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 = {294, 296, 335,
303, 1176, 631, 210, 1179, 642} \begin {gather*} -\frac {3 \text {ArcTan}\left (1-\sqrt {2} \sqrt {x}\right )}{32 \sqrt {2}}+\frac {3 \text {ArcTan}\left (\sqrt {2} \sqrt {x}+1\right )}{32 \sqrt {2}}+\frac {3 x^{3/2}}{16 \left (x^2+1\right )}-\frac {x^{3/2}}{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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 294
Rule 296
Rule 303
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps
\begin {align*} \int \frac {x^{5/2}}{\left (1+x^2\right )^3} \, dx &=-\frac {x^{3/2}}{4 \left (1+x^2\right )^2}+\frac {3}{8} \int \frac {\sqrt {x}}{\left (1+x^2\right )^2} \, dx\\ &=-\frac {x^{3/2}}{4 \left (1+x^2\right )^2}+\frac {3 x^{3/2}}{16 \left (1+x^2\right )}+\frac {3}{32} \int \frac {\sqrt {x}}{1+x^2} \, dx\\ &=-\frac {x^{3/2}}{4 \left (1+x^2\right )^2}+\frac {3 x^{3/2}}{16 \left (1+x^2\right )}+\frac {3}{16} \text {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 {3 x^{3/2}}{16 \left (1+x^2\right )}-\frac {3}{32} \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {x}\right )+\frac {3}{32} \text {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 {3 x^{3/2}}{16 \left (1+x^2\right )}+\frac {3}{64} \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {x}\right )+\frac {3}{64} \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {x}\right )+\frac {3 \text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {x}\right )}{64 \sqrt {2}}+\frac {3 \text {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 {3 x^{3/2}}{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 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {x}\right )}{32 \sqrt {2}}-\frac {3 \text {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 {3 x^{3/2}}{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*}
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Mathematica [A]
time = 0.17, size = 72, normalized size = 0.56 \begin {gather*} \frac {1}{64} \left (\frac {4 x^{3/2} \left (-1+3 x^2\right )}{\left (1+x^2\right )^2}+3 \sqrt {2} \tan ^{-1}\left (\frac {-1+x}{\sqrt {2} \sqrt {x}}\right )-3 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {x}}{1+x}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.45, size = 77, normalized size = 0.60
method | result | size |
derivativedivides | \(\frac {\frac {3 x^{\frac {7}{2}}}{16}-\frac {x^{\frac {3}{2}}}{16}}{\left (x^{2}+1\right )^{2}}+\frac {3 \sqrt {2}\, \left (\ln \left (\frac {1+x -\sqrt {2}\, \sqrt {x}}{1+x +\sqrt {2}\, \sqrt {x}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {x}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {x}\right )\right )}{128}\) | \(77\) |
default | \(\frac {\frac {3 x^{\frac {7}{2}}}{16}-\frac {x^{\frac {3}{2}}}{16}}{\left (x^{2}+1\right )^{2}}+\frac {3 \sqrt {2}\, \left (\ln \left (\frac {1+x -\sqrt {2}\, \sqrt {x}}{1+x +\sqrt {2}\, \sqrt {x}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {x}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {x}\right )\right )}{128}\) | \(77\) |
risch | \(\frac {x^{\frac {3}{2}} \left (3 x^{2}-1\right )}{16 \left (x^{2}+1\right )^{2}}+\frac {3 \arctan \left (1+\sqrt {2}\, \sqrt {x}\right ) \sqrt {2}}{64}+\frac {3 \arctan \left (-1+\sqrt {2}\, \sqrt {x}\right ) \sqrt {2}}{64}+\frac {3 \sqrt {2}\, \ln \left (\frac {1+x -\sqrt {2}\, \sqrt {x}}{1+x +\sqrt {2}\, \sqrt {x}}\right )}{128}\) | \(81\) |
meijerg | \(-\frac {x^{\frac {3}{2}} \left (-21 x^{2}+7\right )}{112 \left (x^{2}+1\right )^{2}}+\frac {3 x^{\frac {3}{2}} \left (\frac {\sqrt {2}\, \ln \left (1-\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}+\sqrt {x^{2}}\right )}{2 \left (x^{2}\right )^{\frac {3}{4}}}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}{2-\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}\right )}{\left (x^{2}\right )^{\frac {3}{4}}}-\frac {\sqrt {2}\, \ln \left (1+\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}+\sqrt {x^{2}}\right )}{2 \left (x^{2}\right )^{\frac {3}{4}}}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}{2+\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}\right )}{\left (x^{2}\right )^{\frac {3}{4}}}\right )}{64}\) | \(150\) |
trager | \(\frac {x^{\frac {3}{2}} \left (3 x^{2}-1\right )}{16 \left (x^{2}+1\right )^{2}}-\frac {3 \RootOf \left (\textit {\_Z}^{4}+1\right ) \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{4}+1\right )^{5} x +\RootOf \left (\textit {\_Z}^{4}+1\right )^{5}-2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3}+\RootOf \left (\textit {\_Z}^{4}+1\right ) x +4 \sqrt {x}+\RootOf \left (\textit {\_Z}^{4}+1\right )}{\RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x -\RootOf \left (\textit {\_Z}^{4}+1\right )^{2}-x -1}\right )}{64}-\frac {3 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{4}+1\right )^{5} x +\RootOf \left (\textit {\_Z}^{4}+1\right )^{5}+2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x -\RootOf \left (\textit {\_Z}^{4}+1\right ) x +4 \sqrt {x}-\RootOf \left (\textit {\_Z}^{4}+1\right )}{\RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x -\RootOf \left (\textit {\_Z}^{4}+1\right )^{2}+x +1}\right )}{64}\) | \(199\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 99, normalized size = 0.77 \begin {gather*} \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 {3 \, x^{\frac {7}{2}} - x^{\frac {3}{2}}}{16 \, {\left (x^{4} + 2 \, x^{2} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.29, size = 175, normalized size = 1.36 \begin {gather*} -\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 (3 \, x^{3} - x\right )} \sqrt {x}}{128 \, {\left (x^{4} + 2 \, x^{2} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 481 vs.
\(2 (117) = 234\).
time = 2.45, size = 481, normalized size = 3.73 \begin {gather*} \frac {24 x^{\frac {7}{2}}}{128 x^{4} + 256 x^{2} + 128} - \frac {8 x^{\frac {3}{2}}}{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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.70, size = 94, normalized size = 0.73 \begin {gather*} \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 {3 \, x^{\frac {7}{2}} - x^{\frac {3}{2}}}{16 \, {\left (x^{2} + 1\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 62, normalized size = 0.48 \begin {gather*} -\frac {\frac {x^{3/2}}{16}-\frac {3\,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 {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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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