Optimal. Leaf size=122 \[ \frac {1}{2 \sqrt {x} \left (x^2+1\right )}-\frac {5}{2 \sqrt {x}}-\frac {5 \log \left (x-\sqrt {2} \sqrt {x}+1\right )}{8 \sqrt {2}}+\frac {5 \log \left (x+\sqrt {2} \sqrt {x}+1\right )}{8 \sqrt {2}}+\frac {5 \tan ^{-1}\left (1-\sqrt {2} \sqrt {x}\right )}{4 \sqrt {2}}-\frac {5 \tan ^{-1}\left (\sqrt {2} \sqrt {x}+1\right )}{4 \sqrt {2}} \]
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Rubi [A] time = 0.06, 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 = {290, 325, 329, 297, 1162, 617, 204, 1165, 628} \[ \frac {1}{2 \sqrt {x} \left (x^2+1\right )}-\frac {5}{2 \sqrt {x}}-\frac {5 \log \left (x-\sqrt {2} \sqrt {x}+1\right )}{8 \sqrt {2}}+\frac {5 \log \left (x+\sqrt {2} \sqrt {x}+1\right )}{8 \sqrt {2}}+\frac {5 \tan ^{-1}\left (1-\sqrt {2} \sqrt {x}\right )}{4 \sqrt {2}}-\frac {5 \tan ^{-1}\left (\sqrt {2} \sqrt {x}+1\right )}{4 \sqrt {2}} \]
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^{3/2} \left (1+x^2\right )^2} \, dx &=\frac {1}{2 \sqrt {x} \left (1+x^2\right )}+\frac {5}{4} \int \frac {1}{x^{3/2} \left (1+x^2\right )} \, dx\\ &=-\frac {5}{2 \sqrt {x}}+\frac {1}{2 \sqrt {x} \left (1+x^2\right )}-\frac {5}{4} \int \frac {\sqrt {x}}{1+x^2} \, dx\\ &=-\frac {5}{2 \sqrt {x}}+\frac {1}{2 \sqrt {x} \left (1+x^2\right )}-\frac {5}{2} \operatorname {Subst}\left (\int \frac {x^2}{1+x^4} \, dx,x,\sqrt {x}\right )\\ &=-\frac {5}{2 \sqrt {x}}+\frac {1}{2 \sqrt {x} \left (1+x^2\right )}+\frac {5}{4} \operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {x}\right )-\frac {5}{4} \operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {x}\right )\\ &=-\frac {5}{2 \sqrt {x}}+\frac {1}{2 \sqrt {x} \left (1+x^2\right )}-\frac {5}{8} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {x}\right )-\frac {5}{8} \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 )}{8 \sqrt {2}}-\frac {5 \operatorname {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {x}\right )}{8 \sqrt {2}}\\ &=-\frac {5}{2 \sqrt {x}}+\frac {1}{2 \sqrt {x} \left (1+x^2\right )}-\frac {5 \log \left (1-\sqrt {2} \sqrt {x}+x\right )}{8 \sqrt {2}}+\frac {5 \log \left (1+\sqrt {2} \sqrt {x}+x\right )}{8 \sqrt {2}}-\frac {5 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {x}\right )}{4 \sqrt {2}}+\frac {5 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {x}\right )}{4 \sqrt {2}}\\ &=-\frac {5}{2 \sqrt {x}}+\frac {1}{2 \sqrt {x} \left (1+x^2\right )}+\frac {5 \tan ^{-1}\left (1-\sqrt {2} \sqrt {x}\right )}{4 \sqrt {2}}-\frac {5 \tan ^{-1}\left (1+\sqrt {2} \sqrt {x}\right )}{4 \sqrt {2}}-\frac {5 \log \left (1-\sqrt {2} \sqrt {x}+x\right )}{8 \sqrt {2}}+\frac {5 \log \left (1+\sqrt {2} \sqrt {x}+x\right )}{8 \sqrt {2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 20, normalized size = 0.16 \[ -\frac {2 \, _2F_1\left (-\frac {1}{4},2;\frac {3}{4};-x^2\right )}{\sqrt {x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 148, normalized size = 1.21 \[ \frac {20 \, \sqrt {2} {\left (x^{3} + x\right )} \arctan \left (\sqrt {2} \sqrt {\sqrt {2} \sqrt {x} + x + 1} - \sqrt {2} \sqrt {x} - 1\right ) + 20 \, \sqrt {2} {\left (x^{3} + x\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {-4 \, \sqrt {2} \sqrt {x} + 4 \, x + 4} - \sqrt {2} \sqrt {x} + 1\right ) + 5 \, \sqrt {2} {\left (x^{3} + x\right )} \log \left (4 \, \sqrt {2} \sqrt {x} + 4 \, x + 4\right ) - 5 \, \sqrt {2} {\left (x^{3} + x\right )} \log \left (-4 \, \sqrt {2} \sqrt {x} + 4 \, x + 4\right ) - 8 \, {\left (5 \, x^{2} + 4\right )} \sqrt {x}}{16 \, {\left (x^{3} + x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.64, size = 92, normalized size = 0.75 \[ -\frac {5}{8} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {x}\right )}\right ) - \frac {5}{8} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {x}\right )}\right ) + \frac {5}{16} \, \sqrt {2} \log \left (\sqrt {2} \sqrt {x} + x + 1\right ) - \frac {5}{16} \, \sqrt {2} \log \left (-\sqrt {2} \sqrt {x} + x + 1\right ) - \frac {5 \, x^{2} + 4}{2 \, {\left (x^{\frac {5}{2}} + \sqrt {x}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 79, normalized size = 0.65 \[ -\frac {x^{\frac {3}{2}}}{2 \left (x^{2}+1\right )}-\frac {5 \sqrt {2}\, \arctan \left (\sqrt {2}\, \sqrt {x}-1\right )}{8}-\frac {5 \sqrt {2}\, \arctan \left (\sqrt {2}\, \sqrt {x}+1\right )}{8}-\frac {5 \sqrt {2}\, \ln \left (\frac {x -\sqrt {2}\, \sqrt {x}+1}{x +\sqrt {2}\, \sqrt {x}+1}\right )}{16}-\frac {2}{\sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.04, size = 92, normalized size = 0.75 \[ -\frac {5}{8} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {x}\right )}\right ) - \frac {5}{8} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {x}\right )}\right ) + \frac {5}{16} \, \sqrt {2} \log \left (\sqrt {2} \sqrt {x} + x + 1\right ) - \frac {5}{16} \, \sqrt {2} \log \left (-\sqrt {2} \sqrt {x} + x + 1\right ) - \frac {5 \, x^{2} + 4}{2 \, {\left (x^{\frac {5}{2}} + \sqrt {x}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.68, size = 55, normalized size = 0.45 \[ -\frac {\frac {5\,x^2}{2}+2}{\sqrt {x}+x^{5/2}}+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,\sqrt {x}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {5}{8}+\frac {5}{8}{}\mathrm {i}\right )+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,\sqrt {x}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {5}{8}-\frac {5}{8}{}\mathrm {i}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 4.56, size = 366, normalized size = 3.00 \[ - \frac {5 \sqrt {2} x^{\frac {5}{2}} \log {\left (- 4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{16 x^{\frac {5}{2}} + 16 \sqrt {x}} + \frac {5 \sqrt {2} x^{\frac {5}{2}} \log {\left (4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{16 x^{\frac {5}{2}} + 16 \sqrt {x}} - \frac {10 \sqrt {2} x^{\frac {5}{2}} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} - 1 \right )}}{16 x^{\frac {5}{2}} + 16 \sqrt {x}} - \frac {10 \sqrt {2} x^{\frac {5}{2}} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} + 1 \right )}}{16 x^{\frac {5}{2}} + 16 \sqrt {x}} - \frac {5 \sqrt {2} \sqrt {x} \log {\left (- 4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{16 x^{\frac {5}{2}} + 16 \sqrt {x}} + \frac {5 \sqrt {2} \sqrt {x} \log {\left (4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{16 x^{\frac {5}{2}} + 16 \sqrt {x}} - \frac {10 \sqrt {2} \sqrt {x} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} - 1 \right )}}{16 x^{\frac {5}{2}} + 16 \sqrt {x}} - \frac {10 \sqrt {2} \sqrt {x} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} + 1 \right )}}{16 x^{\frac {5}{2}} + 16 \sqrt {x}} - \frac {40 x^{2}}{16 x^{\frac {5}{2}} + 16 \sqrt {x}} - \frac {32}{16 x^{\frac {5}{2}} + 16 \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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