Optimal. Leaf size=80 \[ -\frac {2 \sqrt [4]{a x^4+b x^2}}{x}-\sqrt [4]{a} \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^2}}\right )+\sqrt [4]{a} \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^2}}\right ) \]
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Rubi [A] time = 0.15, antiderivative size = 142, normalized size of antiderivative = 1.78, number of steps used = 7, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {2020, 2032, 329, 331, 298, 203, 206} \begin {gather*} -\frac {2 \sqrt [4]{a x^4+b x^2}}{x}-\frac {\sqrt [4]{a} x^{3/2} \left (a x^2+b\right )^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{\left (a x^4+b x^2\right )^{3/4}}+\frac {\sqrt [4]{a} x^{3/2} \left (a x^2+b\right )^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{\left (a x^4+b x^2\right )^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 298
Rule 329
Rule 331
Rule 2020
Rule 2032
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{b x^2+a x^4}}{x^2} \, dx &=-\frac {2 \sqrt [4]{b x^2+a x^4}}{x}+a \int \frac {x^2}{\left (b x^2+a x^4\right )^{3/4}} \, dx\\ &=-\frac {2 \sqrt [4]{b x^2+a x^4}}{x}+\frac {\left (a x^{3/2} \left (b+a x^2\right )^{3/4}\right ) \int \frac {\sqrt {x}}{\left (b+a x^2\right )^{3/4}} \, dx}{\left (b x^2+a x^4\right )^{3/4}}\\ &=-\frac {2 \sqrt [4]{b x^2+a x^4}}{x}+\frac {\left (2 a x^{3/2} \left (b+a x^2\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{\left (b x^2+a x^4\right )^{3/4}}\\ &=-\frac {2 \sqrt [4]{b x^2+a x^4}}{x}+\frac {\left (2 a x^{3/2} \left (b+a x^2\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\left (b x^2+a x^4\right )^{3/4}}\\ &=-\frac {2 \sqrt [4]{b x^2+a x^4}}{x}+\frac {\left (\sqrt {a} x^{3/2} \left (b+a x^2\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\left (b x^2+a x^4\right )^{3/4}}-\frac {\left (\sqrt {a} x^{3/2} \left (b+a x^2\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\left (b x^2+a x^4\right )^{3/4}}\\ &=-\frac {2 \sqrt [4]{b x^2+a x^4}}{x}-\frac {\sqrt [4]{a} x^{3/2} \left (b+a x^2\right )^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\left (b x^2+a x^4\right )^{3/4}}+\frac {\sqrt [4]{a} x^{3/2} \left (b+a x^2\right )^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\left (b x^2+a x^4\right )^{3/4}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 53, normalized size = 0.66 \begin {gather*} -\frac {2 \sqrt [4]{x^2 \left (a x^2+b\right )} \, _2F_1\left (-\frac {1}{4},-\frac {1}{4};\frac {3}{4};-\frac {a x^2}{b}\right )}{x \sqrt [4]{\frac {a x^2}{b}+1}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.23, size = 80, normalized size = 1.00 \begin {gather*} -\frac {2 \sqrt [4]{a x^4+b x^2}}{x}-\sqrt [4]{a} \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^2}}\right )+\sqrt [4]{a} \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.37, size = 185, normalized size = 2.31 \begin {gather*} \frac {1}{2} \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right ) + \frac {1}{2} \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right ) + \frac {1}{4} \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} \log \left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x^{2}}}\right ) - \frac {1}{4} \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} \log \left (-\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x^{2}}}\right ) - 2 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.10, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a \,x^{4}+b \,x^{2}\right )^{\frac {1}{4}}}{x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a x^{4} + b x^{2}\right )}^{\frac {1}{4}}}{x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.98, size = 44, normalized size = 0.55 \begin {gather*} -\frac {2\,{\left (a\,x^4+b\,x^2\right )}^{1/4}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},-\frac {1}{4};\ \frac {3}{4};\ -\frac {a\,x^2}{b}\right )}{x\,{\left (\frac {a\,x^2}{b}+1\right )}^{1/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [4]{x^{2} \left (a x^{2} + b\right )}}{x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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