Optimal. Leaf size=138 \[ \frac{9}{16 \sqrt{x} \left (x^2+1\right )}+\frac{1}{4 \sqrt{x} \left (x^2+1\right )^2}-\frac{45}{16 \sqrt{x}}-\frac{45 \log \left (x-\sqrt{2} \sqrt{x}+1\right )}{64 \sqrt{2}}+\frac{45 \log \left (x+\sqrt{2} \sqrt{x}+1\right )}{64 \sqrt{2}}+\frac{45 \tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )}{32 \sqrt{2}}-\frac{45 \tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )}{32 \sqrt{2}} \]
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Rubi [A] time = 0.0703358, antiderivative size = 138, normalized size of antiderivative = 1., 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} \[ \frac{9}{16 \sqrt{x} \left (x^2+1\right )}+\frac{1}{4 \sqrt{x} \left (x^2+1\right )^2}-\frac{45}{16 \sqrt{x}}-\frac{45 \log \left (x-\sqrt{2} \sqrt{x}+1\right )}{64 \sqrt{2}}+\frac{45 \log \left (x+\sqrt{2} \sqrt{x}+1\right )}{64 \sqrt{2}}+\frac{45 \tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )}{32 \sqrt{2}}-\frac{45 \tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )}{32 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 290
Rule 325
Rule 329
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{x^{3/2} \left (1+x^2\right )^3} \, dx &=\frac{1}{4 \sqrt{x} \left (1+x^2\right )^2}+\frac{9}{8} \int \frac{1}{x^{3/2} \left (1+x^2\right )^2} \, dx\\ &=\frac{1}{4 \sqrt{x} \left (1+x^2\right )^2}+\frac{9}{16 \sqrt{x} \left (1+x^2\right )}+\frac{45}{32} \int \frac{1}{x^{3/2} \left (1+x^2\right )} \, dx\\ &=-\frac{45}{16 \sqrt{x}}+\frac{1}{4 \sqrt{x} \left (1+x^2\right )^2}+\frac{9}{16 \sqrt{x} \left (1+x^2\right )}-\frac{45}{32} \int \frac{\sqrt{x}}{1+x^2} \, dx\\ &=-\frac{45}{16 \sqrt{x}}+\frac{1}{4 \sqrt{x} \left (1+x^2\right )^2}+\frac{9}{16 \sqrt{x} \left (1+x^2\right )}-\frac{45}{16} \operatorname{Subst}\left (\int \frac{x^2}{1+x^4} \, dx,x,\sqrt{x}\right )\\ &=-\frac{45}{16 \sqrt{x}}+\frac{1}{4 \sqrt{x} \left (1+x^2\right )^2}+\frac{9}{16 \sqrt{x} \left (1+x^2\right )}+\frac{45}{32} \operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\sqrt{x}\right )-\frac{45}{32} \operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\sqrt{x}\right )\\ &=-\frac{45}{16 \sqrt{x}}+\frac{1}{4 \sqrt{x} \left (1+x^2\right )^2}+\frac{9}{16 \sqrt{x} \left (1+x^2\right )}-\frac{45}{64} \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\sqrt{x}\right )-\frac{45}{64} \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\sqrt{x}\right )-\frac{45 \operatorname{Subst}\left (\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx,x,\sqrt{x}\right )}{64 \sqrt{2}}-\frac{45 \operatorname{Subst}\left (\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx,x,\sqrt{x}\right )}{64 \sqrt{2}}\\ &=-\frac{45}{16 \sqrt{x}}+\frac{1}{4 \sqrt{x} \left (1+x^2\right )^2}+\frac{9}{16 \sqrt{x} \left (1+x^2\right )}-\frac{45 \log \left (1-\sqrt{2} \sqrt{x}+x\right )}{64 \sqrt{2}}+\frac{45 \log \left (1+\sqrt{2} \sqrt{x}+x\right )}{64 \sqrt{2}}-\frac{45 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} \sqrt{x}\right )}{32 \sqrt{2}}+\frac{45 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt{2} \sqrt{x}\right )}{32 \sqrt{2}}\\ &=-\frac{45}{16 \sqrt{x}}+\frac{1}{4 \sqrt{x} \left (1+x^2\right )^2}+\frac{9}{16 \sqrt{x} \left (1+x^2\right )}+\frac{45 \tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )}{32 \sqrt{2}}-\frac{45 \tan ^{-1}\left (1+\sqrt{2} \sqrt{x}\right )}{32 \sqrt{2}}-\frac{45 \log \left (1-\sqrt{2} \sqrt{x}+x\right )}{64 \sqrt{2}}+\frac{45 \log \left (1+\sqrt{2} \sqrt{x}+x\right )}{64 \sqrt{2}}\\ \end{align*}
Mathematica [C] time = 0.0046598, size = 20, normalized size = 0.14 \[ -\frac{2 \, _2F_1\left (-\frac{1}{4},3;\frac{3}{4};-x^2\right )}{\sqrt{x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 87, normalized size = 0.6 \begin{align*} -2\,{\frac{1}{\sqrt{x}}}-2\,{\frac{1}{ \left ({x}^{2}+1 \right ) ^{2}} \left ({\frac{13\,{x}^{7/2}}{32}}+{\frac{17\,{x}^{3/2}}{32}} \right ) }-{\frac{45\,\sqrt{2}}{64}\arctan \left ( 1+\sqrt{2}\sqrt{x} \right ) }-{\frac{45\,\sqrt{2}}{64}\arctan \left ( -1+\sqrt{2}\sqrt{x} \right ) }-{\frac{45\,\sqrt{2}}{128}\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 = 4.07082, size = 138, normalized size = 1. \begin{align*} -\frac{45}{64} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{x}\right )}\right ) - \frac{45}{64} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \sqrt{x}\right )}\right ) + \frac{45}{128} \, \sqrt{2} \log \left (\sqrt{2} \sqrt{x} + x + 1\right ) - \frac{45}{128} \, \sqrt{2} \log \left (-\sqrt{2} \sqrt{x} + x + 1\right ) - \frac{45 \, x^{4} + 81 \, x^{2} + 32}{16 \,{\left (x^{\frac{9}{2}} + 2 \, x^{\frac{5}{2}} + \sqrt{x}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.33975, size = 537, normalized size = 3.89 \begin{align*} \frac{180 \, \sqrt{2}{\left (x^{5} + 2 \, x^{3} + x\right )} \arctan \left (\sqrt{2} \sqrt{\sqrt{2} \sqrt{x} + x + 1} - \sqrt{2} \sqrt{x} - 1\right ) + 180 \, \sqrt{2}{\left (x^{5} + 2 \, 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 ) + 45 \, \sqrt{2}{\left (x^{5} + 2 \, x^{3} + x\right )} \log \left (4 \, \sqrt{2} \sqrt{x} + 4 \, x + 4\right ) - 45 \, \sqrt{2}{\left (x^{5} + 2 \, x^{3} + x\right )} \log \left (-4 \, \sqrt{2} \sqrt{x} + 4 \, x + 4\right ) - 8 \,{\left (45 \, x^{4} + 81 \, x^{2} + 32\right )} \sqrt{x}}{128 \,{\left (x^{5} + 2 \, x^{3} + x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 22.5495, size = 653, normalized size = 4.73 \begin{align*} - \frac{45 \sqrt{2} x^{\frac{9}{2}} \log{\left (- 4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{128 x^{\frac{9}{2}} + 256 x^{\frac{5}{2}} + 128 \sqrt{x}} + \frac{45 \sqrt{2} x^{\frac{9}{2}} \log{\left (4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{128 x^{\frac{9}{2}} + 256 x^{\frac{5}{2}} + 128 \sqrt{x}} - \frac{90 \sqrt{2} x^{\frac{9}{2}} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} - 1 \right )}}{128 x^{\frac{9}{2}} + 256 x^{\frac{5}{2}} + 128 \sqrt{x}} - \frac{90 \sqrt{2} x^{\frac{9}{2}} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} + 1 \right )}}{128 x^{\frac{9}{2}} + 256 x^{\frac{5}{2}} + 128 \sqrt{x}} - \frac{90 \sqrt{2} x^{\frac{5}{2}} \log{\left (- 4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{128 x^{\frac{9}{2}} + 256 x^{\frac{5}{2}} + 128 \sqrt{x}} + \frac{90 \sqrt{2} x^{\frac{5}{2}} \log{\left (4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{128 x^{\frac{9}{2}} + 256 x^{\frac{5}{2}} + 128 \sqrt{x}} - \frac{180 \sqrt{2} x^{\frac{5}{2}} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} - 1 \right )}}{128 x^{\frac{9}{2}} + 256 x^{\frac{5}{2}} + 128 \sqrt{x}} - \frac{180 \sqrt{2} x^{\frac{5}{2}} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} + 1 \right )}}{128 x^{\frac{9}{2}} + 256 x^{\frac{5}{2}} + 128 \sqrt{x}} - \frac{45 \sqrt{2} \sqrt{x} \log{\left (- 4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{128 x^{\frac{9}{2}} + 256 x^{\frac{5}{2}} + 128 \sqrt{x}} + \frac{45 \sqrt{2} \sqrt{x} \log{\left (4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{128 x^{\frac{9}{2}} + 256 x^{\frac{5}{2}} + 128 \sqrt{x}} - \frac{90 \sqrt{2} \sqrt{x} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} - 1 \right )}}{128 x^{\frac{9}{2}} + 256 x^{\frac{5}{2}} + 128 \sqrt{x}} - \frac{90 \sqrt{2} \sqrt{x} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} + 1 \right )}}{128 x^{\frac{9}{2}} + 256 x^{\frac{5}{2}} + 128 \sqrt{x}} - \frac{360 x^{4}}{128 x^{\frac{9}{2}} + 256 x^{\frac{5}{2}} + 128 \sqrt{x}} - \frac{648 x^{2}}{128 x^{\frac{9}{2}} + 256 x^{\frac{5}{2}} + 128 \sqrt{x}} - \frac{256}{128 x^{\frac{9}{2}} + 256 x^{\frac{5}{2}} + 128 \sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.474, size = 134, normalized size = 0.97 \begin{align*} -\frac{45}{64} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{x}\right )}\right ) - \frac{45}{64} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \sqrt{x}\right )}\right ) + \frac{45}{128} \, \sqrt{2} \log \left (\sqrt{2} \sqrt{x} + x + 1\right ) - \frac{45}{128} \, \sqrt{2} \log \left (-\sqrt{2} \sqrt{x} + x + 1\right ) - \frac{2}{\sqrt{x}} - \frac{13 \, x^{\frac{7}{2}} + 17 \, x^{\frac{3}{2}}}{16 \,{\left (x^{2} + 1\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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