Optimal. Leaf size=99 \[ \frac {1}{4} \text {RootSum}\left [\text {$\#$1}^{16}-4 \text {$\#$1}^{12} a+6 \text {$\#$1}^8 a^2+\text {$\#$1}^8 a b-4 \text {$\#$1}^4 a^3-2 \text {$\#$1}^4 a^2 b+a^4+a^3 b-2 b^3\& ,\frac {\log \left (\sqrt [4]{a x^4+b x^2}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}}\& \right ] \]
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Rubi [B] time = 3.34, antiderivative size = 1161, normalized size of antiderivative = 11.73, number of steps used = 22, number of rules used = 8, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2056, 6715, 6728, 1429, 377, 212, 208, 205} \begin {gather*} -\frac {\sqrt [8]{a-\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{a x^2+b} \tan ^{-1}\left (\frac {\sqrt [4]{a \sqrt {a-\sqrt {a^2+8 b}}-2 b} \sqrt {x}}{\sqrt [8]{a-\sqrt {a^2+8 b}} \sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a \sqrt {a-\sqrt {a^2+8 b}}-2 b} \sqrt [4]{a x^4+b x^2}}-\frac {\sqrt [8]{a-\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{a x^2+b} \tan ^{-1}\left (\frac {\sqrt [4]{\sqrt {a-\sqrt {a^2+8 b}} a+2 b} \sqrt {x}}{\sqrt [8]{a-\sqrt {a^2+8 b}} \sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{\sqrt {a-\sqrt {a^2+8 b}} a+2 b} \sqrt [4]{a x^4+b x^2}}-\frac {\sqrt [8]{a+\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{a x^2+b} \tan ^{-1}\left (\frac {\sqrt [4]{a \sqrt {a+\sqrt {a^2+8 b}}-2 b} \sqrt {x}}{\sqrt [8]{a+\sqrt {a^2+8 b}} \sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a \sqrt {a+\sqrt {a^2+8 b}}-2 b} \sqrt [4]{a x^4+b x^2}}-\frac {\sqrt [8]{a+\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{a x^2+b} \tan ^{-1}\left (\frac {\sqrt [4]{\sqrt {a+\sqrt {a^2+8 b}} a+2 b} \sqrt {x}}{\sqrt [8]{a+\sqrt {a^2+8 b}} \sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{\sqrt {a+\sqrt {a^2+8 b}} a+2 b} \sqrt [4]{a x^4+b x^2}}-\frac {\sqrt [8]{a-\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{a x^2+b} \tanh ^{-1}\left (\frac {\sqrt [4]{a \sqrt {a-\sqrt {a^2+8 b}}-2 b} \sqrt {x}}{\sqrt [8]{a-\sqrt {a^2+8 b}} \sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a \sqrt {a-\sqrt {a^2+8 b}}-2 b} \sqrt [4]{a x^4+b x^2}}-\frac {\sqrt [8]{a-\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{a x^2+b} \tanh ^{-1}\left (\frac {\sqrt [4]{\sqrt {a-\sqrt {a^2+8 b}} a+2 b} \sqrt {x}}{\sqrt [8]{a-\sqrt {a^2+8 b}} \sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{\sqrt {a-\sqrt {a^2+8 b}} a+2 b} \sqrt [4]{a x^4+b x^2}}-\frac {\sqrt [8]{a+\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{a x^2+b} \tanh ^{-1}\left (\frac {\sqrt [4]{a \sqrt {a+\sqrt {a^2+8 b}}-2 b} \sqrt {x}}{\sqrt [8]{a+\sqrt {a^2+8 b}} \sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a \sqrt {a+\sqrt {a^2+8 b}}-2 b} \sqrt [4]{a x^4+b x^2}}-\frac {\sqrt [8]{a+\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{a x^2+b} \tanh ^{-1}\left (\frac {\sqrt [4]{\sqrt {a+\sqrt {a^2+8 b}} a+2 b} \sqrt {x}}{\sqrt [8]{a+\sqrt {a^2+8 b}} \sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{\sqrt {a+\sqrt {a^2+8 b}} a+2 b} \sqrt [4]{a x^4+b x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 208
Rule 212
Rule 377
Rule 1429
Rule 2056
Rule 6715
Rule 6728
Rubi steps
\begin {align*} \int \frac {2 b+a x^4}{\sqrt [4]{b x^2+a x^4} \left (-b-a x^4+2 x^8\right )} \, dx &=\frac {\left (\sqrt {x} \sqrt [4]{b+a x^2}\right ) \int \frac {2 b+a x^4}{\sqrt {x} \sqrt [4]{b+a x^2} \left (-b-a x^4+2 x^8\right )} \, dx}{\sqrt [4]{b x^2+a x^4}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {2 b+a x^8}{\sqrt [4]{b+a x^4} \left (-b-a x^8+2 x^{16}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^4}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {a+\sqrt {a^2+8 b}}{\sqrt [4]{b+a x^4} \left (-a-\sqrt {a^2+8 b}+4 x^8\right )}+\frac {a-\sqrt {a^2+8 b}}{\sqrt [4]{b+a x^4} \left (-a+\sqrt {a^2+8 b}+4 x^8\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^4}}\\ &=\frac {\left (2 \left (a-\sqrt {a^2+8 b}\right ) \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{b+a x^4} \left (-a+\sqrt {a^2+8 b}+4 x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^4}}+\frac {\left (2 \left (a+\sqrt {a^2+8 b}\right ) \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{b+a x^4} \left (-a-\sqrt {a^2+8 b}+4 x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^4}}\\ &=-\frac {\left (2 \sqrt {a-\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (2 \sqrt {a-\sqrt {a^2+8 b}}-4 x^4\right ) \sqrt [4]{b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^4}}-\frac {\left (2 \sqrt {a-\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (2 \sqrt {a-\sqrt {a^2+8 b}}+4 x^4\right ) \sqrt [4]{b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^4}}-\frac {\left (2 \sqrt {a+\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (2 \sqrt {a+\sqrt {a^2+8 b}}-4 x^4\right ) \sqrt [4]{b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^4}}-\frac {\left (2 \sqrt {a+\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (2 \sqrt {a+\sqrt {a^2+8 b}}+4 x^4\right ) \sqrt [4]{b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^4}}\\ &=-\frac {\left (2 \sqrt {a-\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{2 \sqrt {a-\sqrt {a^2+8 b}}-\left (-4 b+2 a \sqrt {a-\sqrt {a^2+8 b}}\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt [4]{b x^2+a x^4}}-\frac {\left (2 \sqrt {a-\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{2 \sqrt {a-\sqrt {a^2+8 b}}-\left (4 b+2 a \sqrt {a-\sqrt {a^2+8 b}}\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt [4]{b x^2+a x^4}}-\frac {\left (2 \sqrt {a+\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{2 \sqrt {a+\sqrt {a^2+8 b}}-\left (-4 b+2 a \sqrt {a+\sqrt {a^2+8 b}}\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt [4]{b x^2+a x^4}}-\frac {\left (2 \sqrt {a+\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{2 \sqrt {a+\sqrt {a^2+8 b}}-\left (4 b+2 a \sqrt {a+\sqrt {a^2+8 b}}\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt [4]{b x^2+a x^4}}\\ &=-\frac {\left (\sqrt [4]{a-\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{a-\sqrt {a^2+8 b}}-\sqrt {-2 b+a \sqrt {a-\sqrt {a^2+8 b}}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt [4]{b x^2+a x^4}}-\frac {\left (\sqrt [4]{a-\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{a-\sqrt {a^2+8 b}}+\sqrt {-2 b+a \sqrt {a-\sqrt {a^2+8 b}}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt [4]{b x^2+a x^4}}-\frac {\left (\sqrt [4]{a-\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{a-\sqrt {a^2+8 b}}-\sqrt {2 b+a \sqrt {a-\sqrt {a^2+8 b}}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt [4]{b x^2+a x^4}}-\frac {\left (\sqrt [4]{a-\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{a-\sqrt {a^2+8 b}}+\sqrt {2 b+a \sqrt {a-\sqrt {a^2+8 b}}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt [4]{b x^2+a x^4}}-\frac {\left (\sqrt [4]{a+\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{a+\sqrt {a^2+8 b}}-\sqrt {-2 b+a \sqrt {a+\sqrt {a^2+8 b}}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt [4]{b x^2+a x^4}}-\frac {\left (\sqrt [4]{a+\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{a+\sqrt {a^2+8 b}}+\sqrt {-2 b+a \sqrt {a+\sqrt {a^2+8 b}}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt [4]{b x^2+a x^4}}-\frac {\left (\sqrt [4]{a+\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{a+\sqrt {a^2+8 b}}-\sqrt {2 b+a \sqrt {a+\sqrt {a^2+8 b}}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt [4]{b x^2+a x^4}}-\frac {\left (\sqrt [4]{a+\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{a+\sqrt {a^2+8 b}}+\sqrt {2 b+a \sqrt {a+\sqrt {a^2+8 b}}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt [4]{b x^2+a x^4}}\\ &=-\frac {\sqrt [8]{a-\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2} \tan ^{-1}\left (\frac {\sqrt [4]{-2 b+a \sqrt {a-\sqrt {a^2+8 b}}} \sqrt {x}}{\sqrt [8]{a-\sqrt {a^2+8 b}} \sqrt [4]{b+a x^2}}\right )}{2 \sqrt [4]{-2 b+a \sqrt {a-\sqrt {a^2+8 b}}} \sqrt [4]{b x^2+a x^4}}-\frac {\sqrt [8]{a-\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2} \tan ^{-1}\left (\frac {\sqrt [4]{2 b+a \sqrt {a-\sqrt {a^2+8 b}}} \sqrt {x}}{\sqrt [8]{a-\sqrt {a^2+8 b}} \sqrt [4]{b+a x^2}}\right )}{2 \sqrt [4]{2 b+a \sqrt {a-\sqrt {a^2+8 b}}} \sqrt [4]{b x^2+a x^4}}-\frac {\sqrt [8]{a+\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2} \tan ^{-1}\left (\frac {\sqrt [4]{-2 b+a \sqrt {a+\sqrt {a^2+8 b}}} \sqrt {x}}{\sqrt [8]{a+\sqrt {a^2+8 b}} \sqrt [4]{b+a x^2}}\right )}{2 \sqrt [4]{-2 b+a \sqrt {a+\sqrt {a^2+8 b}}} \sqrt [4]{b x^2+a x^4}}-\frac {\sqrt [8]{a+\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2} \tan ^{-1}\left (\frac {\sqrt [4]{2 b+a \sqrt {a+\sqrt {a^2+8 b}}} \sqrt {x}}{\sqrt [8]{a+\sqrt {a^2+8 b}} \sqrt [4]{b+a x^2}}\right )}{2 \sqrt [4]{2 b+a \sqrt {a+\sqrt {a^2+8 b}}} \sqrt [4]{b x^2+a x^4}}-\frac {\sqrt [8]{a-\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{-2 b+a \sqrt {a-\sqrt {a^2+8 b}}} \sqrt {x}}{\sqrt [8]{a-\sqrt {a^2+8 b}} \sqrt [4]{b+a x^2}}\right )}{2 \sqrt [4]{-2 b+a \sqrt {a-\sqrt {a^2+8 b}}} \sqrt [4]{b x^2+a x^4}}-\frac {\sqrt [8]{a-\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{2 b+a \sqrt {a-\sqrt {a^2+8 b}}} \sqrt {x}}{\sqrt [8]{a-\sqrt {a^2+8 b}} \sqrt [4]{b+a x^2}}\right )}{2 \sqrt [4]{2 b+a \sqrt {a-\sqrt {a^2+8 b}}} \sqrt [4]{b x^2+a x^4}}-\frac {\sqrt [8]{a+\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{-2 b+a \sqrt {a+\sqrt {a^2+8 b}}} \sqrt {x}}{\sqrt [8]{a+\sqrt {a^2+8 b}} \sqrt [4]{b+a x^2}}\right )}{2 \sqrt [4]{-2 b+a \sqrt {a+\sqrt {a^2+8 b}}} \sqrt [4]{b x^2+a x^4}}-\frac {\sqrt [8]{a+\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{2 b+a \sqrt {a+\sqrt {a^2+8 b}}} \sqrt {x}}{\sqrt [8]{a+\sqrt {a^2+8 b}} \sqrt [4]{b+a x^2}}\right )}{2 \sqrt [4]{2 b+a \sqrt {a+\sqrt {a^2+8 b}}} \sqrt [4]{b x^2+a x^4}}\\ \end {align*}
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Mathematica [B] time = 3.07, size = 782, normalized size = 7.90 \begin {gather*} \frac {x \sqrt [4]{a+\frac {b}{x^2}} \left (-\left (a^2 (2 a+b) \text {RootSum}\left [\text {$\#$1}^4-4 \text {$\#$1}^3 a+6 \text {$\#$1}^2 a^2+\text {$\#$1}^2 a b-4 \text {$\#$1} a^3-2 \text {$\#$1} a^2 b+a^4+a^3 b-2 b^3\&,\frac {\frac {\log \left (\sqrt [4]{\text {$\#$1}}-\sqrt [4]{a+\frac {b}{x^2}}\right )}{\sqrt [4]{\text {$\#$1}}}-\frac {\log \left (\sqrt [4]{\text {$\#$1}}+\sqrt [4]{a+\frac {b}{x^2}}\right )}{\sqrt [4]{\text {$\#$1}}}+\frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{a+\frac {b}{x^2}}}{\sqrt [4]{\text {$\#$1}}}\right )}{\sqrt [4]{\text {$\#$1}}}}{2 \text {$\#$1}^3-6 \text {$\#$1}^2 a+6 \text {$\#$1} a^2+\text {$\#$1} a b-2 a^3-a^2 b}\&\right ]\right )+a (6 a+b) \text {RootSum}\left [\text {$\#$1}^4-4 \text {$\#$1}^3 a+6 \text {$\#$1}^2 a^2+\text {$\#$1}^2 a b-4 \text {$\#$1} a^3-2 \text {$\#$1} a^2 b+a^4+a^3 b-2 b^3\&,\frac {\text {$\#$1}^{3/4} \log \left (\sqrt [4]{\text {$\#$1}}-\sqrt [4]{a+\frac {b}{x^2}}\right )-\text {$\#$1}^{3/4} \log \left (\sqrt [4]{\text {$\#$1}}+\sqrt [4]{a+\frac {b}{x^2}}\right )+2 \text {$\#$1}^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a+\frac {b}{x^2}}}{\sqrt [4]{\text {$\#$1}}}\right )}{2 \text {$\#$1}^3-6 \text {$\#$1}^2 a+6 \text {$\#$1} a^2+\text {$\#$1} a b-2 a^3-a^2 b}\&\right ]-6 a \text {RootSum}\left [\text {$\#$1}^4-4 \text {$\#$1}^3 a+6 \text {$\#$1}^2 a^2+\text {$\#$1}^2 a b-4 \text {$\#$1} a^3-2 \text {$\#$1} a^2 b+a^4+a^3 b-2 b^3\&,\frac {\text {$\#$1}^{7/4} \log \left (\sqrt [4]{\text {$\#$1}}-\sqrt [4]{a+\frac {b}{x^2}}\right )-\text {$\#$1}^{7/4} \log \left (\sqrt [4]{\text {$\#$1}}+\sqrt [4]{a+\frac {b}{x^2}}\right )+2 \text {$\#$1}^{7/4} \tan ^{-1}\left (\frac {\sqrt [4]{a+\frac {b}{x^2}}}{\sqrt [4]{\text {$\#$1}}}\right )}{2 \text {$\#$1}^3-6 \text {$\#$1}^2 a+6 \text {$\#$1} a^2+\text {$\#$1} a b-2 a^3-a^2 b}\&\right ]+2 \text {RootSum}\left [\text {$\#$1}^4-4 \text {$\#$1}^3 a+6 \text {$\#$1}^2 a^2+\text {$\#$1}^2 a b-4 \text {$\#$1} a^3-2 \text {$\#$1} a^2 b+a^4+a^3 b-2 b^3\&,\frac {\text {$\#$1}^{11/4} \log \left (\sqrt [4]{\text {$\#$1}}-\sqrt [4]{a+\frac {b}{x^2}}\right )-\text {$\#$1}^{11/4} \log \left (\sqrt [4]{\text {$\#$1}}+\sqrt [4]{a+\frac {b}{x^2}}\right )+2 \text {$\#$1}^{11/4} \tan ^{-1}\left (\frac {\sqrt [4]{a+\frac {b}{x^2}}}{\sqrt [4]{\text {$\#$1}}}\right )}{2 \text {$\#$1}^3-6 \text {$\#$1}^2 a+6 \text {$\#$1} a^2+\text {$\#$1} a b-2 a^3-a^2 b}\&\right ]\right )}{4 \sqrt [4]{x^2 \left (a x^2+b\right )}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 11.80, size = 99, normalized size = 1.00 \begin {gather*} \frac {1}{4} \text {RootSum}\left [a^4+a^3 b-2 b^3-4 a^3 \text {$\#$1}^4-2 a^2 b \text {$\#$1}^4+6 a^2 \text {$\#$1}^8+a b \text {$\#$1}^8-4 a \text {$\#$1}^{12}+\text {$\#$1}^{16}\&,\frac {-\log (x)+\log \left (\sqrt [4]{b x^2+a x^4}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\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 [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a x^{4} + 2 \, b}{{\left (2 \, x^{8} - a x^{4} - b\right )} {\left (a x^{4} + b x^{2}\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {a \,x^{4}+2 b}{\left (a \,x^{4}+b \,x^{2}\right )^{\frac {1}{4}} \left (2 x^{8}-a \,x^{4}-b \right )}\, dx\]
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 (2 \, x^{8} - a x^{4} - b\right )} {\left (a x^{4} + b x^{2}\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {a\,x^4+2\,b}{{\left (a\,x^4+b\,x^2\right )}^{1/4}\,\left (-2\,x^8+a\,x^4+b\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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