Optimal. Leaf size=122 \[ -\frac {7}{6 x^{3/2}}+\frac {1}{2 x^{3/2} \left (1+x^2\right )}+\frac {7 \tan ^{-1}\left (1-\sqrt {2} \sqrt {x}\right )}{4 \sqrt {2}}-\frac {7 \tan ^{-1}\left (1+\sqrt {2} \sqrt {x}\right )}{4 \sqrt {2}}+\frac {7 \log \left (1-\sqrt {2} \sqrt {x}+x\right )}{8 \sqrt {2}}-\frac {7 \log \left (1+\sqrt {2} \sqrt {x}+x\right )}{8 \sqrt {2}} \]
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Rubi [A]
time = 0.04, antiderivative size = 122, 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 = {296, 331, 335,
217, 1179, 642, 1176, 631, 210} \begin {gather*} \frac {7 \text {ArcTan}\left (1-\sqrt {2} \sqrt {x}\right )}{4 \sqrt {2}}-\frac {7 \text {ArcTan}\left (\sqrt {2} \sqrt {x}+1\right )}{4 \sqrt {2}}-\frac {7}{6 x^{3/2}}+\frac {1}{2 x^{3/2} \left (x^2+1\right )}+\frac {7 \log \left (x-\sqrt {2} \sqrt {x}+1\right )}{8 \sqrt {2}}-\frac {7 \log \left (x+\sqrt {2} \sqrt {x}+1\right )}{8 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 217
Rule 296
Rule 331
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps
\begin {align*} \int \frac {1}{x^{5/2} \left (1+x^2\right )^2} \, dx &=\frac {1}{2 x^{3/2} \left (1+x^2\right )}+\frac {7}{4} \int \frac {1}{x^{5/2} \left (1+x^2\right )} \, dx\\ &=-\frac {7}{6 x^{3/2}}+\frac {1}{2 x^{3/2} \left (1+x^2\right )}-\frac {7}{4} \int \frac {1}{\sqrt {x} \left (1+x^2\right )} \, dx\\ &=-\frac {7}{6 x^{3/2}}+\frac {1}{2 x^{3/2} \left (1+x^2\right )}-\frac {7}{2} \text {Subst}\left (\int \frac {1}{1+x^4} \, dx,x,\sqrt {x}\right )\\ &=-\frac {7}{6 x^{3/2}}+\frac {1}{2 x^{3/2} \left (1+x^2\right )}-\frac {7}{4} \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {x}\right )-\frac {7}{4} \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {x}\right )\\ &=-\frac {7}{6 x^{3/2}}+\frac {1}{2 x^{3/2} \left (1+x^2\right )}-\frac {7}{8} \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {x}\right )-\frac {7}{8} \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {x}\right )+\frac {7 \text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {x}\right )}{8 \sqrt {2}}+\frac {7 \text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {x}\right )}{8 \sqrt {2}}\\ &=-\frac {7}{6 x^{3/2}}+\frac {1}{2 x^{3/2} \left (1+x^2\right )}+\frac {7 \log \left (1-\sqrt {2} \sqrt {x}+x\right )}{8 \sqrt {2}}-\frac {7 \log \left (1+\sqrt {2} \sqrt {x}+x\right )}{8 \sqrt {2}}-\frac {7 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {x}\right )}{4 \sqrt {2}}+\frac {7 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {x}\right )}{4 \sqrt {2}}\\ &=-\frac {7}{6 x^{3/2}}+\frac {1}{2 x^{3/2} \left (1+x^2\right )}+\frac {7 \tan ^{-1}\left (1-\sqrt {2} \sqrt {x}\right )}{4 \sqrt {2}}-\frac {7 \tan ^{-1}\left (1+\sqrt {2} \sqrt {x}\right )}{4 \sqrt {2}}+\frac {7 \log \left (1-\sqrt {2} \sqrt {x}+x\right )}{8 \sqrt {2}}-\frac {7 \log \left (1+\sqrt {2} \sqrt {x}+x\right )}{8 \sqrt {2}}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 72, normalized size = 0.59 \begin {gather*} \frac {1}{24} \left (-\frac {4 \left (4+7 x^2\right )}{x^{3/2} \left (1+x^2\right )}-21 \sqrt {2} \tan ^{-1}\left (\frac {-1+x}{\sqrt {2} \sqrt {x}}\right )-21 \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 = 0.47, size = 74, normalized size = 0.61
method | result | size |
derivativedivides | \(-\frac {\sqrt {x}}{2 \left (x^{2}+1\right )}-\frac {7 \sqrt {2}\, \left (\ln \left (\frac {1+x +\sqrt {2}\, \sqrt {x}}{1+x -\sqrt {2}\, \sqrt {x}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {x}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {x}\right )\right )}{16}-\frac {2}{3 x^{\frac {3}{2}}}\) | \(74\) |
default | \(-\frac {\sqrt {x}}{2 \left (x^{2}+1\right )}-\frac {7 \sqrt {2}\, \left (\ln \left (\frac {1+x +\sqrt {2}\, \sqrt {x}}{1+x -\sqrt {2}\, \sqrt {x}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {x}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {x}\right )\right )}{16}-\frac {2}{3 x^{\frac {3}{2}}}\) | \(74\) |
risch | \(-\frac {7 x^{2}+4}{6 \left (x^{2}+1\right ) x^{\frac {3}{2}}}-\frac {7 \arctan \left (1+\sqrt {2}\, \sqrt {x}\right ) \sqrt {2}}{8}-\frac {7 \arctan \left (-1+\sqrt {2}\, \sqrt {x}\right ) \sqrt {2}}{8}-\frac {7 \sqrt {2}\, \ln \left (\frac {1+x +\sqrt {2}\, \sqrt {x}}{1+x -\sqrt {2}\, \sqrt {x}}\right )}{16}\) | \(81\) |
meijerg | \(-\frac {2 \left (7 x^{2}+4\right )}{3 x^{\frac {3}{2}} \left (4 x^{2}+4\right )}-\frac {7 \sqrt {x}\, \left (-\frac {\sqrt {2}\, \ln \left (1-\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}+\sqrt {x^{2}}\right )}{2 \left (x^{2}\right )^{\frac {1}{4}}}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}{2-\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}\right )}{\left (x^{2}\right )^{\frac {1}{4}}}+\frac {\sqrt {2}\, \ln \left (1+\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}+\sqrt {x^{2}}\right )}{2 \left (x^{2}\right )^{\frac {1}{4}}}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}{2+\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}\right )}{\left (x^{2}\right )^{\frac {1}{4}}}\right )}{8}\) | \(152\) |
trager | \(-\frac {7 x^{2}+4}{6 \left (x^{2}+1\right ) x^{\frac {3}{2}}}+\frac {7 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{4}+1\right )^{5} x +\RootOf \left (\textit {\_Z}^{4}+1\right )^{5}-2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3}+\RootOf \left (\textit {\_Z}^{4}+1\right ) x +4 \sqrt {x}+\RootOf \left (\textit {\_Z}^{4}+1\right )}{\RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x -\RootOf \left (\textit {\_Z}^{4}+1\right )^{2}-x -1}\right )}{8}-\frac {7 \RootOf \left (\textit {\_Z}^{4}+1\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{4}+1\right )^{5} x -\RootOf \left (\textit {\_Z}^{4}+1\right )^{5}-2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x +\RootOf \left (\textit {\_Z}^{4}+1\right ) x +4 \sqrt {x}+\RootOf \left (\textit {\_Z}^{4}+1\right )}{\RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x -\RootOf \left (\textit {\_Z}^{4}+1\right )^{2}+x +1}\right )}{8}\) | \(197\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.62, size = 92, normalized size = 0.75 \begin {gather*} -\frac {7}{8} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {x}\right )}\right ) - \frac {7}{8} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {x}\right )}\right ) - \frac {7}{16} \, \sqrt {2} \log \left (\sqrt {2} \sqrt {x} + x + 1\right ) + \frac {7}{16} \, \sqrt {2} \log \left (-\sqrt {2} \sqrt {x} + x + 1\right ) - \frac {7 \, x^{2} + 4}{6 \, {\left (x^{\frac {7}{2}} + x^{\frac {3}{2}}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.06, size = 158, normalized size = 1.30 \begin {gather*} \frac {84 \, \sqrt {2} {\left (x^{4} + x^{2}\right )} \arctan \left (\sqrt {2} \sqrt {\sqrt {2} \sqrt {x} + x + 1} - \sqrt {2} \sqrt {x} - 1\right ) + 84 \, \sqrt {2} {\left (x^{4} + x^{2}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {-4 \, \sqrt {2} \sqrt {x} + 4 \, x + 4} - \sqrt {2} \sqrt {x} + 1\right ) - 21 \, \sqrt {2} {\left (x^{4} + x^{2}\right )} \log \left (4 \, \sqrt {2} \sqrt {x} + 4 \, x + 4\right ) + 21 \, \sqrt {2} {\left (x^{4} + x^{2}\right )} \log \left (-4 \, \sqrt {2} \sqrt {x} + 4 \, x + 4\right ) - 8 \, {\left (7 \, x^{2} + 4\right )} \sqrt {x}}{48 \, {\left (x^{4} + x^{2}\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. 366 vs.
\(2 (112) = 224\).
time = 1.39, size = 366, normalized size = 3.00 \begin {gather*} \frac {21 \sqrt {2} x^{\frac {7}{2}} \log {\left (- 4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{48 x^{\frac {7}{2}} + 48 x^{\frac {3}{2}}} - \frac {21 \sqrt {2} x^{\frac {7}{2}} \log {\left (4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{48 x^{\frac {7}{2}} + 48 x^{\frac {3}{2}}} - \frac {42 \sqrt {2} x^{\frac {7}{2}} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} - 1 \right )}}{48 x^{\frac {7}{2}} + 48 x^{\frac {3}{2}}} - \frac {42 \sqrt {2} x^{\frac {7}{2}} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} + 1 \right )}}{48 x^{\frac {7}{2}} + 48 x^{\frac {3}{2}}} + \frac {21 \sqrt {2} x^{\frac {3}{2}} \log {\left (- 4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{48 x^{\frac {7}{2}} + 48 x^{\frac {3}{2}}} - \frac {21 \sqrt {2} x^{\frac {3}{2}} \log {\left (4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{48 x^{\frac {7}{2}} + 48 x^{\frac {3}{2}}} - \frac {42 \sqrt {2} x^{\frac {3}{2}} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} - 1 \right )}}{48 x^{\frac {7}{2}} + 48 x^{\frac {3}{2}}} - \frac {42 \sqrt {2} x^{\frac {3}{2}} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} + 1 \right )}}{48 x^{\frac {7}{2}} + 48 x^{\frac {3}{2}}} - \frac {56 x^{2}}{48 x^{\frac {7}{2}} + 48 x^{\frac {3}{2}}} - \frac {32}{48 x^{\frac {7}{2}} + 48 x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.02, size = 91, normalized size = 0.75 \begin {gather*} -\frac {7}{8} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {x}\right )}\right ) - \frac {7}{8} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {x}\right )}\right ) - \frac {7}{16} \, \sqrt {2} \log \left (\sqrt {2} \sqrt {x} + x + 1\right ) + \frac {7}{16} \, \sqrt {2} \log \left (-\sqrt {2} \sqrt {x} + x + 1\right ) - \frac {\sqrt {x}}{2 \, {\left (x^{2} + 1\right )}} - \frac {2}{3 \, x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.08, size = 55, normalized size = 0.45 \begin {gather*} -\frac {\frac {7\,x^2}{6}+\frac {2}{3}}{x^{3/2}+x^{7/2}}+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,\sqrt {x}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {7}{8}-\frac {7}{8}{}\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 {7}{8}+\frac {7}{8}{}\mathrm {i}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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