Optimal. Leaf size=94 \[ a^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^2}}\right )+a^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^2}}\right )+\frac {4 \left (a x^4+b x^2\right )^{3/4} \left (4 a x^2-3 b\right )}{21 b x^5} \]
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Rubi [A] time = 0.29, antiderivative size = 169, normalized size of antiderivative = 1.80, number of steps used = 10, number of rules used = 9, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {2052, 2011, 329, 240, 212, 206, 203, 2016, 2014} \begin {gather*} \frac {a^{3/4} \sqrt {x} \sqrt [4]{a x^2+b} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{\sqrt [4]{a x^4+b x^2}}+\frac {a^{3/4} \sqrt {x} \sqrt [4]{a x^2+b} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{\sqrt [4]{a x^4+b x^2}}-\frac {4 \left (a x^4+b x^2\right )^{3/4}}{7 x^5}+\frac {16 a \left (a x^4+b x^2\right )^{3/4}}{21 b x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 212
Rule 240
Rule 329
Rule 2011
Rule 2014
Rule 2016
Rule 2052
Rubi steps
\begin {align*} \int \frac {2 b+a x^4}{x^4 \sqrt [4]{b x^2+a x^4}} \, dx &=\int \left (\frac {a}{\sqrt [4]{b x^2+a x^4}}+\frac {2 b}{x^4 \sqrt [4]{b x^2+a x^4}}\right ) \, dx\\ &=a \int \frac {1}{\sqrt [4]{b x^2+a x^4}} \, dx+(2 b) \int \frac {1}{x^4 \sqrt [4]{b x^2+a x^4}} \, dx\\ &=-\frac {4 \left (b x^2+a x^4\right )^{3/4}}{7 x^5}-\frac {1}{7} (8 a) \int \frac {1}{x^2 \sqrt [4]{b x^2+a x^4}} \, dx+\frac {\left (a \sqrt {x} \sqrt [4]{b+a x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt [4]{b+a x^2}} \, dx}{\sqrt [4]{b x^2+a x^4}}\\ &=-\frac {4 \left (b x^2+a x^4\right )^{3/4}}{7 x^5}+\frac {16 a \left (b x^2+a x^4\right )^{3/4}}{21 b x^3}+\frac {\left (2 a \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^4}}\\ &=-\frac {4 \left (b x^2+a x^4\right )^{3/4}}{7 x^5}+\frac {16 a \left (b x^2+a x^4\right )^{3/4}}{21 b x^3}+\frac {\left (2 a \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{1-a x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt [4]{b x^2+a x^4}}\\ &=-\frac {4 \left (b x^2+a x^4\right )^{3/4}}{7 x^5}+\frac {16 a \left (b x^2+a x^4\right )^{3/4}}{21 b x^3}+\frac {\left (a \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt [4]{b x^2+a x^4}}+\frac {\left (a \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt [4]{b x^2+a x^4}}\\ &=-\frac {4 \left (b x^2+a x^4\right )^{3/4}}{7 x^5}+\frac {16 a \left (b x^2+a x^4\right )^{3/4}}{21 b x^3}+\frac {a^{3/4} \sqrt {x} \sqrt [4]{b+a x^2} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt [4]{b x^2+a x^4}}+\frac {a^{3/4} \sqrt {x} \sqrt [4]{b+a x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt [4]{b x^2+a x^4}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 142, normalized size = 1.51 \begin {gather*} \frac {21 a^{3/4} b x^{7/2} \sqrt [4]{a x^2+b} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )+21 a^{3/4} b x^{7/2} \sqrt [4]{a x^2+b} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )+4 \left (4 a^2 x^4+a b x^2-3 b^2\right )}{21 b x^3 \sqrt [4]{x^2 \left (a x^2+b\right )}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.49, size = 94, normalized size = 1.00 \begin {gather*} \frac {4 \left (-3 b+4 a x^2\right ) \left (b x^2+a x^4\right )^{3/4}}{21 b x^5}+a^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^2+a x^4}}\right )+a^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^2+a x^4}}\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 = 0.23, size = 209, normalized size = 2.22 \begin {gather*} \frac {1}{2} \, \sqrt {2} \left (-a\right )^{\frac {3}{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 {3}{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 {3}{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 {3}{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 {4 \, {\left (3 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {7}{4}} b^{6} - 7 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {3}{4}} a b^{6}\right )}}{21 \, b^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {a \,x^{4}+2 b}{x^{4} \left (a \,x^{4}+b \,x^{2}\right )^{\frac {1}{4}}}\, dx\]
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 {a x^{4} + 2 \, b}{{\left (a x^{4} + b x^{2}\right )}^{\frac {1}{4}} x^{4}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.05, size = 75, normalized size = 0.80 \begin {gather*} \frac {2\,a\,x\,{\left (\frac {a\,x^2}{b}+1\right )}^{1/4}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {1}{4};\ \frac {5}{4};\ -\frac {a\,x^2}{b}\right )}{{\left (a\,x^4+b\,x^2\right )}^{1/4}}-\frac {4\,{\left (a\,x^4+b\,x^2\right )}^{3/4}\,\left (3\,b-4\,a\,x^2\right )}{21\,b\,x^5} \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 {a x^{4} + 2 b}{x^{4} \sqrt [4]{x^{2} \left (a x^{2} + b\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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