Optimal. Leaf size=122 \[ \frac{1}{2 x^{3/2} \left (x^2+1\right )}-\frac{7}{6 x^{3/2}}+\frac{7 \log \left (x-\sqrt{2} \sqrt{x}+1\right )}{8 \sqrt{2}}-\frac{7 \log \left (x+\sqrt{2} \sqrt{x}+1\right )}{8 \sqrt{2}}+\frac{7 \tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )}{4 \sqrt{2}}-\frac{7 \tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )}{4 \sqrt{2}} \]
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Rubi [A] time = 0.0608723, antiderivative size = 122, normalized size of antiderivative = 1., 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, 211, 1165, 628, 1162, 617, 204} \[ \frac{1}{2 x^{3/2} \left (x^2+1\right )}-\frac{7}{6 x^{3/2}}+\frac{7 \log \left (x-\sqrt{2} \sqrt{x}+1\right )}{8 \sqrt{2}}-\frac{7 \log \left (x+\sqrt{2} \sqrt{x}+1\right )}{8 \sqrt{2}}+\frac{7 \tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )}{4 \sqrt{2}}-\frac{7 \tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )}{4 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 290
Rule 325
Rule 329
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
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} \operatorname{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} \operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\sqrt{x}\right )-\frac{7}{4} \operatorname{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} \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\sqrt{x}\right )-\frac{7}{8} \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\sqrt{x}\right )+\frac{7 \operatorname{Subst}\left (\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx,x,\sqrt{x}\right )}{8 \sqrt{2}}+\frac{7 \operatorname{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 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} \sqrt{x}\right )}{4 \sqrt{2}}+\frac{7 \operatorname{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*}
Mathematica [C] time = 0.0051349, size = 22, normalized size = 0.18 \[ -\frac{2 \, _2F_1\left (-\frac{3}{4},2;\frac{1}{4};-x^2\right )}{3 x^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 79, normalized size = 0.7 \begin{align*} -{\frac{2}{3}{x}^{-{\frac{3}{2}}}}-{\frac{1}{2\,{x}^{2}+2}\sqrt{x}}-{\frac{7\,\sqrt{2}}{8}\arctan \left ( 1+\sqrt{2}\sqrt{x} \right ) }-{\frac{7\,\sqrt{2}}{8}\arctan \left ( -1+\sqrt{2}\sqrt{x} \right ) }-{\frac{7\,\sqrt{2}}{16}\ln \left ({ \left ( 1+x+\sqrt{2}\sqrt{x} \right ) \left ( 1+x-\sqrt{2}\sqrt{x} \right ) ^{-1}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 3.28638, size = 124, normalized size = 1.02 \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.34961, size = 478, normalized size = 3.92 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 13.6437, size = 366, normalized size = 3. \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.65837, size = 123, normalized size = 1.01 \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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