Optimal. Leaf size=58 \[ -\frac {45}{4} \sinh ^{-1}\left (\frac {x^2}{\sqrt {5}}\right )-\frac {\left (3 x^2+2\right ) x^4}{2 \sqrt {x^4+5}}+\frac {1}{4} \left (9 x^2+8\right ) \sqrt {x^4+5} \]
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Rubi [A] time = 0.05, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1252, 819, 780, 215} \[ -\frac {\left (3 x^2+2\right ) x^4}{2 \sqrt {x^4+5}}+\frac {1}{4} \left (9 x^2+8\right ) \sqrt {x^4+5}-\frac {45}{4} \sinh ^{-1}\left (\frac {x^2}{\sqrt {5}}\right ) \]
Antiderivative was successfully verified.
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Rule 215
Rule 780
Rule 819
Rule 1252
Rubi steps
\begin {align*} \int \frac {x^7 \left (2+3 x^2\right )}{\left (5+x^4\right )^{3/2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^3 (2+3 x)}{\left (5+x^2\right )^{3/2}} \, dx,x,x^2\right )\\ &=-\frac {x^4 \left (2+3 x^2\right )}{2 \sqrt {5+x^4}}+\frac {1}{10} \operatorname {Subst}\left (\int \frac {x (20+45 x)}{\sqrt {5+x^2}} \, dx,x,x^2\right )\\ &=-\frac {x^4 \left (2+3 x^2\right )}{2 \sqrt {5+x^4}}+\frac {1}{4} \left (8+9 x^2\right ) \sqrt {5+x^4}-\frac {45}{4} \operatorname {Subst}\left (\int \frac {1}{\sqrt {5+x^2}} \, dx,x,x^2\right )\\ &=-\frac {x^4 \left (2+3 x^2\right )}{2 \sqrt {5+x^4}}+\frac {1}{4} \left (8+9 x^2\right ) \sqrt {5+x^4}-\frac {45}{4} \sinh ^{-1}\left (\frac {x^2}{\sqrt {5}}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 51, normalized size = 0.88 \[ \frac {3 x^6+4 x^4+45 x^2-45 \sqrt {x^4+5} \sinh ^{-1}\left (\frac {x^2}{\sqrt {5}}\right )+40}{4 \sqrt {x^4+5}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 62, normalized size = 1.07 \[ \frac {30 \, x^{4} + 45 \, {\left (x^{4} + 5\right )} \log \left (-x^{2} + \sqrt {x^{4} + 5}\right ) + {\left (3 \, x^{6} + 4 \, x^{4} + 45 \, x^{2} + 40\right )} \sqrt {x^{4} + 5} + 150}{4 \, {\left (x^{4} + 5\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 45, normalized size = 0.78 \[ \frac {{\left ({\left (3 \, x^{2} + 4\right )} x^{2} + 45\right )} x^{2} + 40}{4 \, \sqrt {x^{4} + 5}} + \frac {45}{4} \, \log \left (-x^{2} + \sqrt {x^{4} + 5}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 50, normalized size = 0.86 \[ \frac {3 x^{6}}{4 \sqrt {x^{4}+5}}+\frac {45 x^{2}}{4 \sqrt {x^{4}+5}}-\frac {45 \arcsinh \left (\frac {\sqrt {5}\, x^{2}}{5}\right )}{4}+\frac {x^{4}+10}{\sqrt {x^{4}+5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.21, size = 89, normalized size = 1.53 \[ \sqrt {x^{4} + 5} - \frac {15 \, {\left (\frac {3 \, {\left (x^{4} + 5\right )}}{x^{4}} - 2\right )}}{4 \, {\left (\frac {\sqrt {x^{4} + 5}}{x^{2}} - \frac {{\left (x^{4} + 5\right )}^{\frac {3}{2}}}{x^{6}}\right )}} + \frac {5}{\sqrt {x^{4} + 5}} - \frac {45}{8} \, \log \left (\frac {\sqrt {x^{4} + 5}}{x^{2}} + 1\right ) + \frac {45}{8} \, \log \left (\frac {\sqrt {x^{4} + 5}}{x^{2}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.11, size = 97, normalized size = 1.67 \[ \sqrt {x^4+5}\,\left (\frac {3\,x^2}{4}+1\right )-\frac {45\,\mathrm {asinh}\left (\frac {\sqrt {5}\,x^2}{5}\right )}{4}+\frac {\sqrt {5}\,\left (10+\sqrt {5}\,15{}\mathrm {i}\right )\,\sqrt {x^4+5}\,1{}\mathrm {i}}{20\,\left (-x^2+\sqrt {5}\,1{}\mathrm {i}\right )}-\frac {\sqrt {5}\,\left (-10+\sqrt {5}\,15{}\mathrm {i}\right )\,\sqrt {x^4+5}\,1{}\mathrm {i}}{20\,\left (x^2+\sqrt {5}\,1{}\mathrm {i}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 14.28, size = 66, normalized size = 1.14 \[ \frac {3 x^{6}}{4 \sqrt {x^{4} + 5}} + \frac {x^{4}}{\sqrt {x^{4} + 5}} + \frac {45 x^{2}}{4 \sqrt {x^{4} + 5}} - \frac {45 \operatorname {asinh}{\left (\frac {\sqrt {5} x^{2}}{5} \right )}}{4} + \frac {10}{\sqrt {x^{4} + 5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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