Optimal. Leaf size=67 \[ \frac{1}{20} \left (5 x^2+4\right ) \left (x^4+5\right )^{5/2}-\frac{5}{16} x^2 \left (x^4+5\right )^{3/2}-\frac{75}{32} x^2 \sqrt{x^4+5}-\frac{375}{32} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right ) \]
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Rubi [A] time = 0.039551, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1252, 780, 195, 215} \[ \frac{1}{20} \left (5 x^2+4\right ) \left (x^4+5\right )^{5/2}-\frac{5}{16} x^2 \left (x^4+5\right )^{3/2}-\frac{75}{32} x^2 \sqrt{x^4+5}-\frac{375}{32} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right ) \]
Antiderivative was successfully verified.
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Rule 1252
Rule 780
Rule 195
Rule 215
Rubi steps
\begin{align*} \int x^3 \left (2+3 x^2\right ) \left (5+x^4\right )^{3/2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x (2+3 x) \left (5+x^2\right )^{3/2} \, dx,x,x^2\right )\\ &=\frac{1}{20} \left (4+5 x^2\right ) \left (5+x^4\right )^{5/2}-\frac{5}{4} \operatorname{Subst}\left (\int \left (5+x^2\right )^{3/2} \, dx,x,x^2\right )\\ &=-\frac{5}{16} x^2 \left (5+x^4\right )^{3/2}+\frac{1}{20} \left (4+5 x^2\right ) \left (5+x^4\right )^{5/2}-\frac{75}{16} \operatorname{Subst}\left (\int \sqrt{5+x^2} \, dx,x,x^2\right )\\ &=-\frac{75}{32} x^2 \sqrt{5+x^4}-\frac{5}{16} x^2 \left (5+x^4\right )^{3/2}+\frac{1}{20} \left (4+5 x^2\right ) \left (5+x^4\right )^{5/2}-\frac{375}{32} \operatorname{Subst}\left (\int \frac{1}{\sqrt{5+x^2}} \, dx,x,x^2\right )\\ &=-\frac{75}{32} x^2 \sqrt{5+x^4}-\frac{5}{16} x^2 \left (5+x^4\right )^{3/2}+\frac{1}{20} \left (4+5 x^2\right ) \left (5+x^4\right )^{5/2}-\frac{375}{32} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )\\ \end{align*}
Mathematica [A] time = 0.0414293, size = 54, normalized size = 0.81 \[ \frac{1}{160} \left (\sqrt{x^4+5} \left (40 x^{10}+32 x^8+350 x^6+320 x^4+375 x^2+800\right )-1875 \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 58, normalized size = 0.9 \begin{align*}{\frac{{x}^{10}}{4}\sqrt{{x}^{4}+5}}+{\frac{35\,{x}^{6}}{16}\sqrt{{x}^{4}+5}}+{\frac{75\,{x}^{2}}{32}\sqrt{{x}^{4}+5}}-{\frac{375}{32}{\it Arcsinh} \left ({\frac{{x}^{2}\sqrt{5}}{5}} \right ) }+{\frac{1}{5} \left ({x}^{4}+5 \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.41751, size = 159, normalized size = 2.37 \begin{align*} \frac{1}{5} \,{\left (x^{4} + 5\right )}^{\frac{5}{2}} - \frac{125 \,{\left (\frac{3 \, \sqrt{x^{4} + 5}}{x^{2}} - \frac{8 \,{\left (x^{4} + 5\right )}^{\frac{3}{2}}}{x^{6}} - \frac{3 \,{\left (x^{4} + 5\right )}^{\frac{5}{2}}}{x^{10}}\right )}}{32 \,{\left (\frac{3 \,{\left (x^{4} + 5\right )}}{x^{4}} - \frac{3 \,{\left (x^{4} + 5\right )}^{2}}{x^{8}} + \frac{{\left (x^{4} + 5\right )}^{3}}{x^{12}} - 1\right )}} - \frac{375}{64} \, \log \left (\frac{\sqrt{x^{4} + 5}}{x^{2}} + 1\right ) + \frac{375}{64} \, \log \left (\frac{\sqrt{x^{4} + 5}}{x^{2}} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.5947, size = 150, normalized size = 2.24 \begin{align*} \frac{1}{160} \,{\left (40 \, x^{10} + 32 \, x^{8} + 350 \, x^{6} + 320 \, x^{4} + 375 \, x^{2} + 800\right )} \sqrt{x^{4} + 5} + \frac{375}{32} \, \log \left (-x^{2} + \sqrt{x^{4} + 5}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 10.5095, size = 124, normalized size = 1.85 \begin{align*} \frac{x^{14}}{4 \sqrt{x^{4} + 5}} + \frac{55 x^{10}}{16 \sqrt{x^{4} + 5}} + \frac{x^{8} \sqrt{x^{4} + 5}}{5} + \frac{425 x^{6}}{32 \sqrt{x^{4} + 5}} + \frac{x^{4} \sqrt{x^{4} + 5}}{3} + \frac{375 x^{2}}{32 \sqrt{x^{4} + 5}} + \frac{5 \left (x^{4} + 5\right )^{\frac{3}{2}}}{3} - \frac{10 \sqrt{x^{4} + 5}}{3} - \frac{375 \operatorname{asinh}{\left (\frac{\sqrt{5} x^{2}}{5} \right )}}{32} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15825, size = 80, normalized size = 1.19 \begin{align*} \frac{1}{160} \, \sqrt{x^{4} + 5}{\left ({\left (2 \,{\left ({\left (4 \,{\left (5 \, x^{2} + 4\right )} x^{2} + 175\right )} x^{2} + 160\right )} x^{2} + 375\right )} x^{2} + 800\right )} + \frac{375}{32} \, \log \left (-x^{2} + \sqrt{x^{4} + 5}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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