Optimal. Leaf size=131 \[ -\frac {9}{10 x^{5/2}}+\frac {1}{2 x^{5/2} \left (x^2+1\right )}+\frac {9}{2 \sqrt {x}}+\frac {9 \log \left (x-\sqrt {2} \sqrt {x}+1\right )}{8 \sqrt {2}}-\frac {9 \log \left (x+\sqrt {2} \sqrt {x}+1\right )}{8 \sqrt {2}}-\frac {9 \tan ^{-1}\left (1-\sqrt {2} \sqrt {x}\right )}{4 \sqrt {2}}+\frac {9 \tan ^{-1}\left (\sqrt {2} \sqrt {x}+1\right )}{4 \sqrt {2}} \]
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Rubi [A] time = 0.07, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.692, Rules used = {290, 325, 329, 297, 1162, 617, 204, 1165, 628} \[ \frac {1}{2 x^{5/2} \left (x^2+1\right )}-\frac {9}{10 x^{5/2}}+\frac {9}{2 \sqrt {x}}+\frac {9 \log \left (x-\sqrt {2} \sqrt {x}+1\right )}{8 \sqrt {2}}-\frac {9 \log \left (x+\sqrt {2} \sqrt {x}+1\right )}{8 \sqrt {2}}-\frac {9 \tan ^{-1}\left (1-\sqrt {2} \sqrt {x}\right )}{4 \sqrt {2}}+\frac {9 \tan ^{-1}\left (\sqrt {2} \sqrt {x}+1\right )}{4 \sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 290
Rule 297
Rule 325
Rule 329
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {1}{x^{7/2} \left (1+x^2\right )^2} \, dx &=\frac {1}{2 x^{5/2} \left (1+x^2\right )}+\frac {9}{4} \int \frac {1}{x^{7/2} \left (1+x^2\right )} \, dx\\ &=-\frac {9}{10 x^{5/2}}+\frac {1}{2 x^{5/2} \left (1+x^2\right )}-\frac {9}{4} \int \frac {1}{x^{3/2} \left (1+x^2\right )} \, dx\\ &=-\frac {9}{10 x^{5/2}}+\frac {9}{2 \sqrt {x}}+\frac {1}{2 x^{5/2} \left (1+x^2\right )}+\frac {9}{4} \int \frac {\sqrt {x}}{1+x^2} \, dx\\ &=-\frac {9}{10 x^{5/2}}+\frac {9}{2 \sqrt {x}}+\frac {1}{2 x^{5/2} \left (1+x^2\right )}+\frac {9}{2} \operatorname {Subst}\left (\int \frac {x^2}{1+x^4} \, dx,x,\sqrt {x}\right )\\ &=-\frac {9}{10 x^{5/2}}+\frac {9}{2 \sqrt {x}}+\frac {1}{2 x^{5/2} \left (1+x^2\right )}-\frac {9}{4} \operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {x}\right )+\frac {9}{4} \operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {x}\right )\\ &=-\frac {9}{10 x^{5/2}}+\frac {9}{2 \sqrt {x}}+\frac {1}{2 x^{5/2} \left (1+x^2\right )}+\frac {9}{8} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {x}\right )+\frac {9}{8} \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {x}\right )+\frac {9 \operatorname {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {x}\right )}{8 \sqrt {2}}+\frac {9 \operatorname {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {x}\right )}{8 \sqrt {2}}\\ &=-\frac {9}{10 x^{5/2}}+\frac {9}{2 \sqrt {x}}+\frac {1}{2 x^{5/2} \left (1+x^2\right )}+\frac {9 \log \left (1-\sqrt {2} \sqrt {x}+x\right )}{8 \sqrt {2}}-\frac {9 \log \left (1+\sqrt {2} \sqrt {x}+x\right )}{8 \sqrt {2}}+\frac {9 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {x}\right )}{4 \sqrt {2}}-\frac {9 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {x}\right )}{4 \sqrt {2}}\\ &=-\frac {9}{10 x^{5/2}}+\frac {9}{2 \sqrt {x}}+\frac {1}{2 x^{5/2} \left (1+x^2\right )}-\frac {9 \tan ^{-1}\left (1-\sqrt {2} \sqrt {x}\right )}{4 \sqrt {2}}+\frac {9 \tan ^{-1}\left (1+\sqrt {2} \sqrt {x}\right )}{4 \sqrt {2}}+\frac {9 \log \left (1-\sqrt {2} \sqrt {x}+x\right )}{8 \sqrt {2}}-\frac {9 \log \left (1+\sqrt {2} \sqrt {x}+x\right )}{8 \sqrt {2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 22, normalized size = 0.17 \[ -\frac {2 \, _2F_1\left (-\frac {5}{4},2;-\frac {1}{4};-x^2\right )}{5 x^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.03, size = 163, normalized size = 1.24 \[ -\frac {180 \, \sqrt {2} {\left (x^{5} + x^{3}\right )} \arctan \left (\sqrt {2} \sqrt {\sqrt {2} \sqrt {x} + x + 1} - \sqrt {2} \sqrt {x} - 1\right ) + 180 \, \sqrt {2} {\left (x^{5} + x^{3}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {-4 \, \sqrt {2} \sqrt {x} + 4 \, x + 4} - \sqrt {2} \sqrt {x} + 1\right ) + 45 \, \sqrt {2} {\left (x^{5} + x^{3}\right )} \log \left (4 \, \sqrt {2} \sqrt {x} + 4 \, x + 4\right ) - 45 \, \sqrt {2} {\left (x^{5} + x^{3}\right )} \log \left (-4 \, \sqrt {2} \sqrt {x} + 4 \, x + 4\right ) - 8 \, {\left (45 \, x^{4} + 36 \, x^{2} - 4\right )} \sqrt {x}}{80 \, {\left (x^{5} + x^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.64, size = 98, normalized size = 0.75 \[ \frac {9}{8} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {x}\right )}\right ) + \frac {9}{8} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {x}\right )}\right ) - \frac {9}{16} \, \sqrt {2} \log \left (\sqrt {2} \sqrt {x} + x + 1\right ) + \frac {9}{16} \, \sqrt {2} \log \left (-\sqrt {2} \sqrt {x} + x + 1\right ) + \frac {x^{\frac {3}{2}}}{2 \, {\left (x^{2} + 1\right )}} + \frac {2 \, {\left (10 \, x^{2} - 1\right )}}{5 \, x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 84, normalized size = 0.64 \[ \frac {x^{\frac {3}{2}}}{2 x^{2}+2}+\frac {9 \sqrt {2}\, \arctan \left (\sqrt {2}\, \sqrt {x}-1\right )}{8}+\frac {9 \sqrt {2}\, \arctan \left (\sqrt {2}\, \sqrt {x}+1\right )}{8}+\frac {9 \sqrt {2}\, \ln \left (\frac {x -\sqrt {2}\, \sqrt {x}+1}{x +\sqrt {2}\, \sqrt {x}+1}\right )}{16}+\frac {4}{\sqrt {x}}-\frac {2}{5 x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.04, size = 97, normalized size = 0.74 \[ \frac {9}{8} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {x}\right )}\right ) + \frac {9}{8} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {x}\right )}\right ) - \frac {9}{16} \, \sqrt {2} \log \left (\sqrt {2} \sqrt {x} + x + 1\right ) + \frac {9}{16} \, \sqrt {2} \log \left (-\sqrt {2} \sqrt {x} + x + 1\right ) + \frac {45 \, x^{4} + 36 \, x^{2} - 4}{10 \, {\left (x^{\frac {9}{2}} + x^{\frac {5}{2}}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 59, normalized size = 0.45 \[ \frac {\frac {9\,x^4}{2}+\frac {18\,x^2}{5}-\frac {2}{5}}{x^{5/2}+x^{9/2}}+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,\sqrt {x}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {9}{8}-\frac {9}{8}{}\mathrm {i}\right )+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,\sqrt {x}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {9}{8}+\frac {9}{8}{}\mathrm {i}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 18.73, size = 384, normalized size = 2.93 \[ \frac {45 \sqrt {2} x^{\frac {9}{2}} \log {\left (- 4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{80 x^{\frac {9}{2}} + 80 x^{\frac {5}{2}}} - \frac {45 \sqrt {2} x^{\frac {9}{2}} \log {\left (4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{80 x^{\frac {9}{2}} + 80 x^{\frac {5}{2}}} + \frac {90 \sqrt {2} x^{\frac {9}{2}} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} - 1 \right )}}{80 x^{\frac {9}{2}} + 80 x^{\frac {5}{2}}} + \frac {90 \sqrt {2} x^{\frac {9}{2}} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} + 1 \right )}}{80 x^{\frac {9}{2}} + 80 x^{\frac {5}{2}}} + \frac {45 \sqrt {2} x^{\frac {5}{2}} \log {\left (- 4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{80 x^{\frac {9}{2}} + 80 x^{\frac {5}{2}}} - \frac {45 \sqrt {2} x^{\frac {5}{2}} \log {\left (4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{80 x^{\frac {9}{2}} + 80 x^{\frac {5}{2}}} + \frac {90 \sqrt {2} x^{\frac {5}{2}} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} - 1 \right )}}{80 x^{\frac {9}{2}} + 80 x^{\frac {5}{2}}} + \frac {90 \sqrt {2} x^{\frac {5}{2}} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} + 1 \right )}}{80 x^{\frac {9}{2}} + 80 x^{\frac {5}{2}}} + \frac {360 x^{4}}{80 x^{\frac {9}{2}} + 80 x^{\frac {5}{2}}} + \frac {288 x^{2}}{80 x^{\frac {9}{2}} + 80 x^{\frac {5}{2}}} - \frac {32}{80 x^{\frac {9}{2}} + 80 x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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