Optimal. Leaf size=46 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^4}}}{\sqrt {a}}\right )}{2 a^{3/2}}-\frac {1}{2 a \sqrt {a+\frac {b}{x^4}}} \]
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Rubi [A] time = 0.03, antiderivative size = 46, 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, 51, 63, 208} \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^4}}}{\sqrt {a}}\right )}{2 a^{3/2}}-\frac {1}{2 a \sqrt {a+\frac {b}{x^4}}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{\left (a+\frac {b}{x^4}\right )^{3/2} x} \, dx &=-\left (\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{x (a+b x)^{3/2}} \, dx,x,\frac {1}{x^4}\right )\right )\\ &=-\frac {1}{2 a \sqrt {a+\frac {b}{x^4}}}-\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x^4}\right )}{4 a}\\ &=-\frac {1}{2 a \sqrt {a+\frac {b}{x^4}}}-\frac {\operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x^4}}\right )}{2 a b}\\ &=-\frac {1}{2 a \sqrt {a+\frac {b}{x^4}}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^4}}}{\sqrt {a}}\right )}{2 a^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 69, normalized size = 1.50 \[ \frac {\sqrt {b} \sqrt {\frac {a x^4}{b}+1} \sinh ^{-1}\left (\frac {\sqrt {a} x^2}{\sqrt {b}}\right )-\sqrt {a} x^2}{2 a^{3/2} x^2 \sqrt {a+\frac {b}{x^4}}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.82, size = 163, normalized size = 3.54 \[ \left [-\frac {2 \, a x^{4} \sqrt {\frac {a x^{4} + b}{x^{4}}} - {\left (a x^{4} + b\right )} \sqrt {a} \log \left (-2 \, a x^{4} - 2 \, \sqrt {a} x^{4} \sqrt {\frac {a x^{4} + b}{x^{4}}} - b\right )}{4 \, {\left (a^{3} x^{4} + a^{2} b\right )}}, -\frac {a x^{4} \sqrt {\frac {a x^{4} + b}{x^{4}}} + {\left (a x^{4} + b\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} x^{4} \sqrt {\frac {a x^{4} + b}{x^{4}}}}{a x^{4} + b}\right )}{2 \, {\left (a^{3} x^{4} + a^{2} b\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 51, normalized size = 1.11 \[ -\frac {x^{2}}{2 \, \sqrt {a x^{4} + b} a} - \frac {\log \left ({\left | -\sqrt {a} x^{2} + \sqrt {a x^{4} + b} \right |}\right )}{2 \, a^{\frac {3}{2}}} + \frac {\log \left ({\left | b \right |}\right )}{4 \, a^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 67, normalized size = 1.46 \[ -\frac {\left (a \,x^{4}+b \right ) \left (a^{\frac {3}{2}} x^{2}-\sqrt {a \,x^{4}+b}\, a \ln \left (\sqrt {a}\, x^{2}+\sqrt {a \,x^{4}+b}\right )\right )}{2 \left (\frac {a \,x^{4}+b}{x^{4}}\right )^{\frac {3}{2}} a^{\frac {5}{2}} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.93, size = 52, normalized size = 1.13 \[ -\frac {\log \left (\frac {\sqrt {a + \frac {b}{x^{4}}} - \sqrt {a}}{\sqrt {a + \frac {b}{x^{4}}} + \sqrt {a}}\right )}{4 \, a^{\frac {3}{2}}} - \frac {1}{2 \, \sqrt {a + \frac {b}{x^{4}}} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.36, size = 34, normalized size = 0.74 \[ \frac {\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x^4}}}{\sqrt {a}}\right )}{2\,a^{3/2}}-\frac {1}{2\,a\,\sqrt {a+\frac {b}{x^4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.33, size = 187, normalized size = 4.07 \[ - \frac {2 a^{3} x^{4} \sqrt {1 + \frac {b}{a x^{4}}}}{4 a^{\frac {9}{2}} x^{4} + 4 a^{\frac {7}{2}} b} - \frac {a^{3} x^{4} \log {\left (\frac {b}{a x^{4}} \right )}}{4 a^{\frac {9}{2}} x^{4} + 4 a^{\frac {7}{2}} b} + \frac {2 a^{3} x^{4} \log {\left (\sqrt {1 + \frac {b}{a x^{4}}} + 1 \right )}}{4 a^{\frac {9}{2}} x^{4} + 4 a^{\frac {7}{2}} b} - \frac {a^{2} b \log {\left (\frac {b}{a x^{4}} \right )}}{4 a^{\frac {9}{2}} x^{4} + 4 a^{\frac {7}{2}} b} + \frac {2 a^{2} b \log {\left (\sqrt {1 + \frac {b}{a x^{4}}} + 1 \right )}}{4 a^{\frac {9}{2}} x^{4} + 4 a^{\frac {7}{2}} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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