Optimal. Leaf size=47 \[ \frac {1}{4} x^4 \sqrt {a+\frac {b}{x^4}}+\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^4}}}{\sqrt {a}}\right )}{4 \sqrt {a}} \]
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Rubi [A] time = 0.03, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {266, 47, 63, 208} \[ \frac {1}{4} x^4 \sqrt {a+\frac {b}{x^4}}+\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^4}}}{\sqrt {a}}\right )}{4 \sqrt {a}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 208
Rule 266
Rubi steps
\begin {align*} \int \sqrt {a+\frac {b}{x^4}} x^3 \, dx &=-\left (\frac {1}{4} \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x^2} \, dx,x,\frac {1}{x^4}\right )\right )\\ &=\frac {1}{4} \sqrt {a+\frac {b}{x^4}} x^4-\frac {1}{8} b \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x^4}\right )\\ &=\frac {1}{4} \sqrt {a+\frac {b}{x^4}} x^4-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x^4}}\right )\\ &=\frac {1}{4} \sqrt {a+\frac {b}{x^4}} x^4+\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^4}}}{\sqrt {a}}\right )}{4 \sqrt {a}}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 62, normalized size = 1.32 \[ \frac {1}{4} x^2 \sqrt {a+\frac {b}{x^4}} \left (\frac {\sqrt {b} \sinh ^{-1}\left (\frac {\sqrt {a} x^2}{\sqrt {b}}\right )}{\sqrt {a} \sqrt {\frac {a x^4}{b}+1}}+x^2\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 127, normalized size = 2.70 \[ \left [\frac {2 \, a x^{4} \sqrt {\frac {a x^{4} + b}{x^{4}}} + \sqrt {a} b \log \left (-2 \, a x^{4} - 2 \, \sqrt {a} x^{4} \sqrt {\frac {a x^{4} + b}{x^{4}}} - b\right )}{8 \, a}, \frac {a x^{4} \sqrt {\frac {a x^{4} + b}{x^{4}}} - \sqrt {-a} b \arctan \left (\frac {\sqrt {-a} x^{4} \sqrt {\frac {a x^{4} + b}{x^{4}}}}{a x^{4} + b}\right )}{4 \, a}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 41, normalized size = 0.87 \[ \frac {1}{4} \, \sqrt {a x^{4} + b} x^{2} - \frac {b \log \left ({\left | -\sqrt {a} x^{2} + \sqrt {a x^{4} + b} \right |}\right )}{4 \, \sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 68, normalized size = 1.45 \[ \frac {\sqrt {\frac {a \,x^{4}+b}{x^{4}}}\, \left (\sqrt {a \,x^{4}+b}\, \sqrt {a}\, x^{2}+b \ln \left (\sqrt {a}\, x^{2}+\sqrt {a \,x^{4}+b}\right )\right ) x^{2}}{4 \sqrt {a \,x^{4}+b}\, \sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.95, size = 53, normalized size = 1.13 \[ \frac {1}{4} \, \sqrt {a + \frac {b}{x^{4}}} x^{4} - \frac {b \log \left (\frac {\sqrt {a + \frac {b}{x^{4}}} - \sqrt {a}}{\sqrt {a + \frac {b}{x^{4}}} + \sqrt {a}}\right )}{8 \, \sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.57, size = 35, normalized size = 0.74 \[ \frac {x^4\,\sqrt {a+\frac {b}{x^4}}}{4}+\frac {b\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x^4}}}{\sqrt {a}}\right )}{4\,\sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.50, size = 44, normalized size = 0.94 \[ \frac {\sqrt {b} x^{2} \sqrt {\frac {a x^{4}}{b} + 1}}{4} + \frac {b \operatorname {asinh}{\left (\frac {\sqrt {a} x^{2}}{\sqrt {b}} \right )}}{4 \sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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