Optimal. Leaf size=59 \[ \frac {1}{2} a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^4}}}{\sqrt {a}}\right )-\frac {1}{2} a \sqrt {a+\frac {b}{x^4}}-\frac {1}{6} \left (a+\frac {b}{x^4}\right )^{3/2} \]
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Rubi [A] time = 0.03, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {266, 50, 63, 208} \[ \frac {1}{2} a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^4}}}{\sqrt {a}}\right )-\frac {1}{2} a \sqrt {a+\frac {b}{x^4}}-\frac {1}{6} \left (a+\frac {b}{x^4}\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 208
Rule 266
Rubi steps
\begin {align*} \int \frac {\left (a+\frac {b}{x^4}\right )^{3/2}}{x} \, dx &=-\left (\frac {1}{4} \operatorname {Subst}\left (\int \frac {(a+b x)^{3/2}}{x} \, dx,x,\frac {1}{x^4}\right )\right )\\ &=-\frac {1}{6} \left (a+\frac {b}{x^4}\right )^{3/2}-\frac {1}{4} a \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x} \, dx,x,\frac {1}{x^4}\right )\\ &=-\frac {1}{2} a \sqrt {a+\frac {b}{x^4}}-\frac {1}{6} \left (a+\frac {b}{x^4}\right )^{3/2}-\frac {1}{4} a^2 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x^4}\right )\\ &=-\frac {1}{2} a \sqrt {a+\frac {b}{x^4}}-\frac {1}{6} \left (a+\frac {b}{x^4}\right )^{3/2}-\frac {a^2 \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x^4}}\right )}{2 b}\\ &=-\frac {1}{2} a \sqrt {a+\frac {b}{x^4}}-\frac {1}{6} \left (a+\frac {b}{x^4}\right )^{3/2}+\frac {1}{2} a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^4}}}{\sqrt {a}}\right )\\ \end {align*}
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Mathematica [C] time = 0.02, size = 52, normalized size = 0.88 \[ -\frac {b \sqrt {a+\frac {b}{x^4}} \, _2F_1\left (-\frac {3}{2},-\frac {3}{2};-\frac {1}{2};-\frac {a x^4}{b}\right )}{6 x^4 \sqrt {\frac {a x^4}{b}+1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 141, normalized size = 2.39 \[ \left [\frac {3 \, a^{\frac {3}{2}} x^{4} \log \left (-2 \, a x^{4} - 2 \, \sqrt {a} x^{4} \sqrt {\frac {a x^{4} + b}{x^{4}}} - b\right ) - 2 \, {\left (4 \, a x^{4} + b\right )} \sqrt {\frac {a x^{4} + b}{x^{4}}}}{12 \, x^{4}}, -\frac {3 \, \sqrt {-a} a x^{4} \arctan \left (\frac {\sqrt {-a} x^{4} \sqrt {\frac {a x^{4} + b}{x^{4}}}}{a x^{4} + b}\right ) + {\left (4 \, a x^{4} + b\right )} \sqrt {\frac {a x^{4} + b}{x^{4}}}}{6 \, x^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.25, size = 122, normalized size = 2.07 \[ -\frac {1}{4} \, a^{\frac {3}{2}} \log \left ({\left (\sqrt {a} x^{2} - \sqrt {a x^{4} + b}\right )}^{2}\right ) + \frac {2 \, {\left (3 \, {\left (\sqrt {a} x^{2} - \sqrt {a x^{4} + b}\right )}^{4} a^{\frac {3}{2}} b - 3 \, {\left (\sqrt {a} x^{2} - \sqrt {a x^{4} + b}\right )}^{2} a^{\frac {3}{2}} b^{2} + 2 \, a^{\frac {3}{2}} b^{3}\right )}}{3 \, {\left ({\left (\sqrt {a} x^{2} - \sqrt {a x^{4} + b}\right )}^{2} - b\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 78, normalized size = 1.32 \[ -\frac {\left (\frac {a \,x^{4}+b}{x^{4}}\right )^{\frac {3}{2}} \left (-3 a^{\frac {3}{2}} x^{6} \ln \left (\sqrt {a}\, x^{2}+\sqrt {a \,x^{4}+b}\right )+4 \sqrt {a \,x^{4}+b}\, a \,x^{4}+\sqrt {a \,x^{4}+b}\, b \right )}{6 \left (a \,x^{4}+b \right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.95, size = 61, normalized size = 1.03 \[ -\frac {1}{4} \, a^{\frac {3}{2}} \log \left (\frac {\sqrt {a + \frac {b}{x^{4}}} - \sqrt {a}}{\sqrt {a + \frac {b}{x^{4}}} + \sqrt {a}}\right ) - \frac {1}{6} \, {\left (a + \frac {b}{x^{4}}\right )}^{\frac {3}{2}} - \frac {1}{2} \, \sqrt {a + \frac {b}{x^{4}}} a \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.50, size = 43, normalized size = 0.73 \[ \frac {a^{3/2}\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x^4}}}{\sqrt {a}}\right )}{2}-\frac {a\,\sqrt {a+\frac {b}{x^4}}}{2}-\frac {{\left (a+\frac {b}{x^4}\right )}^{3/2}}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.48, size = 80, normalized size = 1.36 \[ - \frac {2 a^{\frac {3}{2}} \sqrt {1 + \frac {b}{a x^{4}}}}{3} - \frac {a^{\frac {3}{2}} \log {\left (\frac {b}{a x^{4}} \right )}}{4} + \frac {a^{\frac {3}{2}} \log {\left (\sqrt {1 + \frac {b}{a x^{4}}} + 1 \right )}}{2} - \frac {\sqrt {a} b \sqrt {1 + \frac {b}{a x^{4}}}}{6 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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