Optimal. Leaf size=147 \[ -\frac {117}{80 x^{5/2}}+\frac {13}{16 x^{5/2} \left (x^2+1\right )}+\frac {1}{4 x^{5/2} \left (x^2+1\right )^2}+\frac {117}{16 \sqrt {x}}+\frac {117 \log \left (x-\sqrt {2} \sqrt {x}+1\right )}{64 \sqrt {2}}-\frac {117 \log \left (x+\sqrt {2} \sqrt {x}+1\right )}{64 \sqrt {2}}-\frac {117 \tan ^{-1}\left (1-\sqrt {2} \sqrt {x}\right )}{32 \sqrt {2}}+\frac {117 \tan ^{-1}\left (\sqrt {2} \sqrt {x}+1\right )}{32 \sqrt {2}} \]
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Rubi [A] time = 0.07, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 14, 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} \[ \frac {13}{16 x^{5/2} \left (x^2+1\right )}+\frac {1}{4 x^{5/2} \left (x^2+1\right )^2}-\frac {117}{80 x^{5/2}}+\frac {117}{16 \sqrt {x}}+\frac {117 \log \left (x-\sqrt {2} \sqrt {x}+1\right )}{64 \sqrt {2}}-\frac {117 \log \left (x+\sqrt {2} \sqrt {x}+1\right )}{64 \sqrt {2}}-\frac {117 \tan ^{-1}\left (1-\sqrt {2} \sqrt {x}\right )}{32 \sqrt {2}}+\frac {117 \tan ^{-1}\left (\sqrt {2} \sqrt {x}+1\right )}{32 \sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 290
Rule 297
Rule 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 )^3} \, dx &=\frac {1}{4 x^{5/2} \left (1+x^2\right )^2}+\frac {13}{8} \int \frac {1}{x^{7/2} \left (1+x^2\right )^2} \, dx\\ &=\frac {1}{4 x^{5/2} \left (1+x^2\right )^2}+\frac {13}{16 x^{5/2} \left (1+x^2\right )}+\frac {117}{32} \int \frac {1}{x^{7/2} \left (1+x^2\right )} \, dx\\ &=-\frac {117}{80 x^{5/2}}+\frac {1}{4 x^{5/2} \left (1+x^2\right )^2}+\frac {13}{16 x^{5/2} \left (1+x^2\right )}-\frac {117}{32} \int \frac {1}{x^{3/2} \left (1+x^2\right )} \, dx\\ &=-\frac {117}{80 x^{5/2}}+\frac {117}{16 \sqrt {x}}+\frac {1}{4 x^{5/2} \left (1+x^2\right )^2}+\frac {13}{16 x^{5/2} \left (1+x^2\right )}+\frac {117}{32} \int \frac {\sqrt {x}}{1+x^2} \, dx\\ &=-\frac {117}{80 x^{5/2}}+\frac {117}{16 \sqrt {x}}+\frac {1}{4 x^{5/2} \left (1+x^2\right )^2}+\frac {13}{16 x^{5/2} \left (1+x^2\right )}+\frac {117}{16} \operatorname {Subst}\left (\int \frac {x^2}{1+x^4} \, dx,x,\sqrt {x}\right )\\ &=-\frac {117}{80 x^{5/2}}+\frac {117}{16 \sqrt {x}}+\frac {1}{4 x^{5/2} \left (1+x^2\right )^2}+\frac {13}{16 x^{5/2} \left (1+x^2\right )}-\frac {117}{32} \operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {x}\right )+\frac {117}{32} \operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {x}\right )\\ &=-\frac {117}{80 x^{5/2}}+\frac {117}{16 \sqrt {x}}+\frac {1}{4 x^{5/2} \left (1+x^2\right )^2}+\frac {13}{16 x^{5/2} \left (1+x^2\right )}+\frac {117}{64} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {x}\right )+\frac {117}{64} \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {x}\right )+\frac {117 \operatorname {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {x}\right )}{64 \sqrt {2}}+\frac {117 \operatorname {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {x}\right )}{64 \sqrt {2}}\\ &=-\frac {117}{80 x^{5/2}}+\frac {117}{16 \sqrt {x}}+\frac {1}{4 x^{5/2} \left (1+x^2\right )^2}+\frac {13}{16 x^{5/2} \left (1+x^2\right )}+\frac {117 \log \left (1-\sqrt {2} \sqrt {x}+x\right )}{64 \sqrt {2}}-\frac {117 \log \left (1+\sqrt {2} \sqrt {x}+x\right )}{64 \sqrt {2}}+\frac {117 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {x}\right )}{32 \sqrt {2}}-\frac {117 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {x}\right )}{32 \sqrt {2}}\\ &=-\frac {117}{80 x^{5/2}}+\frac {117}{16 \sqrt {x}}+\frac {1}{4 x^{5/2} \left (1+x^2\right )^2}+\frac {13}{16 x^{5/2} \left (1+x^2\right )}-\frac {117 \tan ^{-1}\left (1-\sqrt {2} \sqrt {x}\right )}{32 \sqrt {2}}+\frac {117 \tan ^{-1}\left (1+\sqrt {2} \sqrt {x}\right )}{32 \sqrt {2}}+\frac {117 \log \left (1-\sqrt {2} \sqrt {x}+x\right )}{64 \sqrt {2}}-\frac {117 \log \left (1+\sqrt {2} \sqrt {x}+x\right )}{64 \sqrt {2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 22, normalized size = 0.15 \[ -\frac {2 \, _2F_1\left (-\frac {5}{4},3;-\frac {1}{4};-x^2\right )}{5 x^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 193, normalized size = 1.31 \[ -\frac {2340 \, \sqrt {2} {\left (x^{7} + 2 \, x^{5} + x^{3}\right )} \arctan \left (\sqrt {2} \sqrt {\sqrt {2} \sqrt {x} + x + 1} - \sqrt {2} \sqrt {x} - 1\right ) + 2340 \, \sqrt {2} {\left (x^{7} + 2 \, x^{5} + x^{3}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {-4 \, \sqrt {2} \sqrt {x} + 4 \, x + 4} - \sqrt {2} \sqrt {x} + 1\right ) + 585 \, \sqrt {2} {\left (x^{7} + 2 \, x^{5} + x^{3}\right )} \log \left (4 \, \sqrt {2} \sqrt {x} + 4 \, x + 4\right ) - 585 \, \sqrt {2} {\left (x^{7} + 2 \, x^{5} + x^{3}\right )} \log \left (-4 \, \sqrt {2} \sqrt {x} + 4 \, x + 4\right ) - 8 \, {\left (585 \, x^{6} + 1053 \, x^{4} + 416 \, x^{2} - 32\right )} \sqrt {x}}{640 \, {\left (x^{7} + 2 \, x^{5} + x^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.64, size = 106, normalized size = 0.72 \[ \frac {117}{64} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {x}\right )}\right ) + \frac {117}{64} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {x}\right )}\right ) - \frac {117}{128} \, \sqrt {2} \log \left (\sqrt {2} \sqrt {x} + x + 1\right ) + \frac {117}{128} \, \sqrt {2} \log \left (-\sqrt {2} \sqrt {x} + x + 1\right ) + \frac {21 \, x^{\frac {7}{2}} + 25 \, x^{\frac {3}{2}}}{16 \, {\left (x^{2} + 1\right )}^{2}} + \frac {2 \, {\left (15 \, x^{2} - 1\right )}}{5 \, x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 92, normalized size = 0.63 \[ \frac {117 \sqrt {2}\, \arctan \left (\sqrt {2}\, \sqrt {x}-1\right )}{64}+\frac {117 \sqrt {2}\, \arctan \left (\sqrt {2}\, \sqrt {x}+1\right )}{64}+\frac {117 \sqrt {2}\, \ln \left (\frac {x -\sqrt {2}\, \sqrt {x}+1}{x +\sqrt {2}\, \sqrt {x}+1}\right )}{128}+\frac {6}{\sqrt {x}}-\frac {2}{5 x^{\frac {5}{2}}}+\frac {\frac {21 x^{\frac {7}{2}}}{16}+\frac {25 x^{\frac {3}{2}}}{16}}{\left (x^{2}+1\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.95, size = 107, normalized size = 0.73 \[ \frac {117}{64} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {x}\right )}\right ) + \frac {117}{64} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {x}\right )}\right ) - \frac {117}{128} \, \sqrt {2} \log \left (\sqrt {2} \sqrt {x} + x + 1\right ) + \frac {117}{128} \, \sqrt {2} \log \left (-\sqrt {2} \sqrt {x} + x + 1\right ) + \frac {585 \, x^{6} + 1053 \, x^{4} + 416 \, x^{2} - 32}{80 \, {\left (x^{\frac {13}{2}} + 2 \, x^{\frac {9}{2}} + x^{\frac {5}{2}}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 69, normalized size = 0.47 \[ \frac {\frac {117\,x^6}{16}+\frac {1053\,x^4}{80}+\frac {26\,x^2}{5}-\frac {2}{5}}{x^{5/2}+2\,x^{9/2}+x^{13/2}}+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,\sqrt {x}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {117}{64}-\frac {117}{64}{}\mathrm {i}\right )+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,\sqrt {x}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {117}{64}+\frac {117}{64}{}\mathrm {i}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 51.16, size = 678, normalized size = 4.61 \[ \frac {585 \sqrt {2} x^{\frac {13}{2}} \log {\left (- 4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{640 x^{\frac {13}{2}} + 1280 x^{\frac {9}{2}} + 640 x^{\frac {5}{2}}} - \frac {585 \sqrt {2} x^{\frac {13}{2}} \log {\left (4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{640 x^{\frac {13}{2}} + 1280 x^{\frac {9}{2}} + 640 x^{\frac {5}{2}}} + \frac {1170 \sqrt {2} x^{\frac {13}{2}} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} - 1 \right )}}{640 x^{\frac {13}{2}} + 1280 x^{\frac {9}{2}} + 640 x^{\frac {5}{2}}} + \frac {1170 \sqrt {2} x^{\frac {13}{2}} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} + 1 \right )}}{640 x^{\frac {13}{2}} + 1280 x^{\frac {9}{2}} + 640 x^{\frac {5}{2}}} + \frac {1170 \sqrt {2} x^{\frac {9}{2}} \log {\left (- 4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{640 x^{\frac {13}{2}} + 1280 x^{\frac {9}{2}} + 640 x^{\frac {5}{2}}} - \frac {1170 \sqrt {2} x^{\frac {9}{2}} \log {\left (4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{640 x^{\frac {13}{2}} + 1280 x^{\frac {9}{2}} + 640 x^{\frac {5}{2}}} + \frac {2340 \sqrt {2} x^{\frac {9}{2}} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} - 1 \right )}}{640 x^{\frac {13}{2}} + 1280 x^{\frac {9}{2}} + 640 x^{\frac {5}{2}}} + \frac {2340 \sqrt {2} x^{\frac {9}{2}} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} + 1 \right )}}{640 x^{\frac {13}{2}} + 1280 x^{\frac {9}{2}} + 640 x^{\frac {5}{2}}} + \frac {585 \sqrt {2} x^{\frac {5}{2}} \log {\left (- 4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{640 x^{\frac {13}{2}} + 1280 x^{\frac {9}{2}} + 640 x^{\frac {5}{2}}} - \frac {585 \sqrt {2} x^{\frac {5}{2}} \log {\left (4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{640 x^{\frac {13}{2}} + 1280 x^{\frac {9}{2}} + 640 x^{\frac {5}{2}}} + \frac {1170 \sqrt {2} x^{\frac {5}{2}} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} - 1 \right )}}{640 x^{\frac {13}{2}} + 1280 x^{\frac {9}{2}} + 640 x^{\frac {5}{2}}} + \frac {1170 \sqrt {2} x^{\frac {5}{2}} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} + 1 \right )}}{640 x^{\frac {13}{2}} + 1280 x^{\frac {9}{2}} + 640 x^{\frac {5}{2}}} + \frac {4680 x^{6}}{640 x^{\frac {13}{2}} + 1280 x^{\frac {9}{2}} + 640 x^{\frac {5}{2}}} + \frac {8424 x^{4}}{640 x^{\frac {13}{2}} + 1280 x^{\frac {9}{2}} + 640 x^{\frac {5}{2}}} + \frac {3328 x^{2}}{640 x^{\frac {13}{2}} + 1280 x^{\frac {9}{2}} + 640 x^{\frac {5}{2}}} - \frac {256}{640 x^{\frac {13}{2}} + 1280 x^{\frac {9}{2}} + 640 x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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