Optimal. Leaf size=139 \[ \frac {\text {ArcTan}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^2+a x^4}}\right )}{\sqrt [4]{a}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^2+a x^4}}\right )}{\sqrt [4]{a}}+\frac {1}{4} \text {RootSum}\left [a^4+a b^3-4 a^3 \text {$\#$1}^4+6 a^2 \text {$\#$1}^8-4 a \text {$\#$1}^{12}+\text {$\#$1}^{16}\& ,\frac {-\log (x)+\log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right )}{\text {$\#$1}}\& \right ] \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(1123\) vs. \(2(139)=278\).
time = 1.70, antiderivative size = 1123, normalized size of antiderivative = 8.08, number of steps
used = 28, number of rules used = 11, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.314, Rules used = {2081, 6847,
6857, 246, 218, 212, 209, 1443, 385, 214, 211} \begin {gather*} \frac {\sqrt {x} \sqrt [4]{a x^2-b} \text {ArcTan}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2-b}}\right )}{\sqrt [4]{a} \sqrt [4]{a x^4-b x^2}}-\frac {\sqrt {x} \sqrt [4]{a x^2-b} \text {ArcTan}\left (\frac {\sqrt [4]{a-\sqrt [4]{-a} b^{3/4}} \sqrt {x}}{\sqrt [4]{a x^2-b}}\right )}{2 \sqrt [4]{a-\sqrt [4]{-a} b^{3/4}} \sqrt [4]{a x^4-b x^2}}-\frac {\sqrt {x} \sqrt [4]{a x^2-b} \text {ArcTan}\left (\frac {\sqrt [4]{a+\sqrt [4]{-a} b^{3/4}} \sqrt {x}}{\sqrt [4]{a x^2-b}}\right )}{2 \sqrt [4]{a+\sqrt [4]{-a} b^{3/4}} \sqrt [4]{a x^4-b x^2}}-\frac {\sqrt [8]{-\sqrt {-a}} \sqrt {x} \sqrt [4]{a x^2-b} \text {ArcTan}\left (\frac {\sqrt [4]{\sqrt {-\sqrt {-a}} a-\sqrt {-a} b^{3/4}} \sqrt {x}}{\sqrt [8]{-\sqrt {-a}} \sqrt [4]{a x^2-b}}\right )}{2 \sqrt [4]{\sqrt {-\sqrt {-a}} a-\sqrt {-a} b^{3/4}} \sqrt [4]{a x^4-b x^2}}-\frac {\sqrt [8]{-\sqrt {-a}} \sqrt {x} \sqrt [4]{a x^2-b} \text {ArcTan}\left (\frac {\sqrt [4]{\sqrt {-\sqrt {-a}} a+\sqrt {-a} b^{3/4}} \sqrt {x}}{\sqrt [8]{-\sqrt {-a}} \sqrt [4]{a x^2-b}}\right )}{2 \sqrt [4]{\sqrt {-\sqrt {-a}} a+\sqrt {-a} b^{3/4}} \sqrt [4]{a x^4-b x^2}}+\frac {\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} \sqrt [4]{a x^4-b x^2}}-\frac {\sqrt {x} \sqrt [4]{a x^2-b} \tanh ^{-1}\left (\frac {\sqrt [4]{a-\sqrt [4]{-a} b^{3/4}} \sqrt {x}}{\sqrt [4]{a x^2-b}}\right )}{2 \sqrt [4]{a-\sqrt [4]{-a} b^{3/4}} \sqrt [4]{a x^4-b x^2}}-\frac {\sqrt {x} \sqrt [4]{a x^2-b} \tanh ^{-1}\left (\frac {\sqrt [4]{a+\sqrt [4]{-a} b^{3/4}} \sqrt {x}}{\sqrt [4]{a x^2-b}}\right )}{2 \sqrt [4]{a+\sqrt [4]{-a} b^{3/4}} \sqrt [4]{a x^4-b x^2}}-\frac {\sqrt [8]{-\sqrt {-a}} \sqrt {x} \sqrt [4]{a x^2-b} \tanh ^{-1}\left (\frac {\sqrt [4]{\sqrt {-\sqrt {-a}} a-\sqrt {-a} b^{3/4}} \sqrt {x}}{\sqrt [8]{-\sqrt {-a}} \sqrt [4]{a x^2-b}}\right )}{2 \sqrt [4]{\sqrt {-\sqrt {-a}} a-\sqrt {-a} b^{3/4}} \sqrt [4]{a x^4-b x^2}}-\frac {\sqrt [8]{-\sqrt {-a}} \sqrt {x} \sqrt [4]{a x^2-b} \tanh ^{-1}\left (\frac {\sqrt [4]{\sqrt {-\sqrt {-a}} a+\sqrt {-a} b^{3/4}} \sqrt {x}}{\sqrt [8]{-\sqrt {-a}} \sqrt [4]{a x^2-b}}\right )}{2 \sqrt [4]{\sqrt {-\sqrt {-a}} a+\sqrt {-a} b^{3/4}} \sqrt [4]{a x^4-b x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 211
Rule 212
Rule 214
Rule 218
Rule 246
Rule 385
Rule 1443
Rule 2081
Rule 6847
Rule 6857
Rubi steps
\begin {align*} \int \frac {-b+a x^8}{\sqrt [4]{-b x^2+a x^4} \left (b+a x^8\right )} \, dx &=\frac {\left (\sqrt {x} \sqrt [4]{-b+a x^2}\right ) \int \frac {-b+a x^8}{\sqrt {x} \sqrt [4]{-b+a x^2} \left (b+a x^8\right )} \, dx}{\sqrt [4]{-b x^2+a x^4}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {-b+a x^{16}}{\sqrt [4]{-b+a x^4} \left (b+a 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 ) \text {Subst}\left (\int \left (\frac {1}{\sqrt [4]{-b+a x^4}}-\frac {2 b}{\sqrt [4]{-b+a x^4} \left (b+a x^{16}\right )}\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 ) \text {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 {\left (4 b \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-b+a x^4} \left (b+a 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 ) \text {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 {\left (4 b \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{2 \sqrt {b} \sqrt [4]{-b+a x^4} \left (\sqrt {b}-\sqrt {-a} x^8\right )}+\frac {1}{2 \sqrt {b} \sqrt [4]{-b+a x^4} \left (\sqrt {b}+\sqrt {-a} x^8\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-b x^2+a x^4}}\\ &=\frac {\left (\sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {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 (\sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {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 (2 \sqrt {b} \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-b+a x^4} \left (\sqrt {b}-\sqrt {-a} x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-b x^2+a x^4}}-\frac {\left (2 \sqrt {b} \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-b+a x^4} \left (\sqrt {b}+\sqrt {-a} x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-b x^2+a x^4}}\\ &=\frac {\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]{a} \sqrt [4]{-b x^2+a x^4}}+\frac {\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]{a} \sqrt [4]{-b x^2+a x^4}}-\frac {\left (\sqrt {-\sqrt {-a}} \sqrt [4]{b} \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {-\sqrt {-a}} \sqrt [4]{b}-\sqrt {-a} x^4\right ) \sqrt [4]{-b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-b x^2+a x^4}}-\frac {\left (\sqrt {-\sqrt {-a}} \sqrt [4]{b} \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {-\sqrt {-a}} \sqrt [4]{b}+\sqrt {-a} x^4\right ) \sqrt [4]{-b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-b x^2+a x^4}}-\frac {\left (\sqrt [4]{-a} \sqrt [4]{b} \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt [4]{-a} \sqrt [4]{b}-\sqrt {-a} x^4\right ) \sqrt [4]{-b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-b x^2+a x^4}}-\frac {\left (\sqrt [4]{-a} \sqrt [4]{b} \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt [4]{-a} \sqrt [4]{b}+\sqrt {-a} x^4\right ) \sqrt [4]{-b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-b x^2+a x^4}}\\ &=\frac {\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]{a} \sqrt [4]{-b x^2+a x^4}}+\frac {\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]{a} \sqrt [4]{-b x^2+a x^4}}-\frac {\left (\sqrt {-\sqrt {-a}} \sqrt [4]{b} \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-\sqrt {-a}} \sqrt [4]{b}-\left (\sqrt {-\sqrt {-a}} a \sqrt [4]{b}-\sqrt {-a} 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 {-\sqrt {-a}} \sqrt [4]{b} \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-\sqrt {-a}} \sqrt [4]{b}-\left (\sqrt {-\sqrt {-a}} a \sqrt [4]{b}+\sqrt {-a} 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 [4]{b} \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-a} \sqrt [4]{b}-\left (\sqrt [4]{-a} a \sqrt [4]{b}-\sqrt {-a} 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 [4]{b} \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-a} \sqrt [4]{b}-\left (\sqrt [4]{-a} a \sqrt [4]{b}+\sqrt {-a} 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 {\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]{a} \sqrt [4]{-b x^2+a x^4}}+\frac {\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]{a} \sqrt [4]{-b x^2+a x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {a-\sqrt [4]{-a} b^{3/4}} 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 {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {a-\sqrt [4]{-a} b^{3/4}} 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 {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {a+\sqrt [4]{-a} b^{3/4}} 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 {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {a+\sqrt [4]{-a} b^{3/4}} 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]{-\sqrt {-a}} \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-\sqrt {-a}}-\sqrt {\sqrt {-\sqrt {-a}} a-\sqrt {-a} b^{3/4}} 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]{-\sqrt {-a}} \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-\sqrt {-a}}+\sqrt {\sqrt {-\sqrt {-a}} a-\sqrt {-a} b^{3/4}} 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]{-\sqrt {-a}} \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-\sqrt {-a}}-\sqrt {\sqrt {-\sqrt {-a}} a+\sqrt {-a} b^{3/4}} 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]{-\sqrt {-a}} \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-\sqrt {-a}}+\sqrt {\sqrt {-\sqrt {-a}} a+\sqrt {-a} b^{3/4}} 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 {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]{a} \sqrt [4]{-b x^2+a x^4}}-\frac {\sqrt {x} \sqrt [4]{-b+a x^2} \tan ^{-1}\left (\frac {\sqrt [4]{a-\sqrt [4]{-a} b^{3/4}} \sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{2 \sqrt [4]{a-\sqrt [4]{-a} b^{3/4}} \sqrt [4]{-b x^2+a x^4}}-\frac {\sqrt {x} \sqrt [4]{-b+a x^2} \tan ^{-1}\left (\frac {\sqrt [4]{a+\sqrt [4]{-a} b^{3/4}} \sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{2 \sqrt [4]{a+\sqrt [4]{-a} b^{3/4}} \sqrt [4]{-b x^2+a x^4}}-\frac {\sqrt [8]{-\sqrt {-a}} \sqrt {x} \sqrt [4]{-b+a x^2} \tan ^{-1}\left (\frac {\sqrt [4]{\sqrt {-\sqrt {-a}} a-\sqrt {-a} b^{3/4}} \sqrt {x}}{\sqrt [8]{-\sqrt {-a}} \sqrt [4]{-b+a x^2}}\right )}{2 \sqrt [4]{\sqrt {-\sqrt {-a}} a-\sqrt {-a} b^{3/4}} \sqrt [4]{-b x^2+a x^4}}-\frac {\sqrt [8]{-\sqrt {-a}} \sqrt {x} \sqrt [4]{-b+a x^2} \tan ^{-1}\left (\frac {\sqrt [4]{\sqrt {-\sqrt {-a}} a+\sqrt {-a} b^{3/4}} \sqrt {x}}{\sqrt [8]{-\sqrt {-a}} \sqrt [4]{-b+a x^2}}\right )}{2 \sqrt [4]{\sqrt {-\sqrt {-a}} a+\sqrt {-a} b^{3/4}} \sqrt [4]{-b x^2+a x^4}}+\frac {\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]{a} \sqrt [4]{-b x^2+a x^4}}-\frac {\sqrt {x} \sqrt [4]{-b+a x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{a-\sqrt [4]{-a} b^{3/4}} \sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{2 \sqrt [4]{a-\sqrt [4]{-a} b^{3/4}} \sqrt [4]{-b x^2+a x^4}}-\frac {\sqrt {x} \sqrt [4]{-b+a x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{a+\sqrt [4]{-a} b^{3/4}} \sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{2 \sqrt [4]{a+\sqrt [4]{-a} b^{3/4}} \sqrt [4]{-b x^2+a x^4}}-\frac {\sqrt [8]{-\sqrt {-a}} \sqrt {x} \sqrt [4]{-b+a x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{\sqrt {-\sqrt {-a}} a-\sqrt {-a} b^{3/4}} \sqrt {x}}{\sqrt [8]{-\sqrt {-a}} \sqrt [4]{-b+a x^2}}\right )}{2 \sqrt [4]{\sqrt {-\sqrt {-a}} a-\sqrt {-a} b^{3/4}} \sqrt [4]{-b x^2+a x^4}}-\frac {\sqrt [8]{-\sqrt {-a}} \sqrt {x} \sqrt [4]{-b+a x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{\sqrt {-\sqrt {-a}} a+\sqrt {-a} b^{3/4}} \sqrt {x}}{\sqrt [8]{-\sqrt {-a}} \sqrt [4]{-b+a x^2}}\right )}{2 \sqrt [4]{\sqrt {-\sqrt {-a}} a+\sqrt {-a} b^{3/4}} \sqrt [4]{-b x^2+a x^4}}\\ \end {align*}
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Mathematica [A]
time = 10.50, size = 213, normalized size = 1.53 \begin {gather*} \frac {\sqrt [4]{-a+\frac {b}{x^2}} x \left (2 \sqrt {2} \left (-\text {ArcTan}\left (\frac {-\sqrt {a}+\sqrt {-a+\frac {b}{x^2}}}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{-a+\frac {b}{x^2}}}\right )+\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{-a+\frac {b}{x^2}}}{\sqrt {a}+\sqrt {-a+\frac {b}{x^2}}}\right )\right )+\sqrt [4]{a} \text {RootSum}\left [a^4+a b^3+4 a^3 \text {$\#$1}^4+6 a^2 \text {$\#$1}^8+4 a \text {$\#$1}^{12}+\text {$\#$1}^{16}\&,\frac {\log \left (\sqrt [4]{-a+\frac {b}{x^2}}-\text {$\#$1}\right )}{\text {$\#$1}}\&\right ]\right )}{4 \sqrt [4]{a} \sqrt [4]{-b x^2+a x^4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {a \,x^{8}-b}{\left (a \,x^{4}-b \,x^{2}\right )^{\frac {1}{4}} \left (a \,x^{8}+b \right )}\, dx\]
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a x^{8} - b}{\sqrt [4]{x^{2} \left (a x^{2} - b\right )} \left (a x^{8} + b\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 {b-a\,x^8}{\left (a\,x^8+b\right )\,{\left (a\,x^4-b\,x^2\right )}^{1/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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