Optimal. Leaf size=81 \[ -\frac {\text {RootSum}\left [\text {$\#$1}^{16}-4 \text {$\#$1}^{12} a+6 \text {$\#$1}^8 a^2-4 \text {$\#$1}^4 a^3+a^4+a b^3\& ,\frac {\log \left (\sqrt [4]{a x^4-b x^2}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}}\& \right ]}{8 b} \]
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Rubi [B] time = 2.66, antiderivative size = 1017, normalized size of antiderivative = 12.56, number of steps used = 22, number of rules used = 10, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2056, 6715, 6725, 1429, 377, 212, 206, 203, 208, 205} \begin {gather*} \frac {\sqrt {x} \sqrt [4]{a x^2-b} \tan ^{-1}\left (\frac {\sqrt [4]{a-\sqrt [4]{-a} b^{3/4}} \sqrt {x}}{\sqrt [4]{a x^2-b}}\right )}{4 \sqrt [4]{a-\sqrt [4]{-a} b^{3/4}} b \sqrt [4]{a x^4-b x^2}}+\frac {\sqrt {x} \sqrt [4]{a x^2-b} \tan ^{-1}\left (\frac {\sqrt [4]{a+\sqrt [4]{-a} b^{3/4}} \sqrt {x}}{\sqrt [4]{a x^2-b}}\right )}{4 \sqrt [4]{a+\sqrt [4]{-a} b^{3/4}} b \sqrt [4]{a x^4-b x^2}}+\frac {\sqrt [8]{-\sqrt {-a}} \sqrt {x} \sqrt [4]{a x^2-b} \tan ^{-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 )}{4 \sqrt [4]{\sqrt {-\sqrt {-a}} a-\sqrt {-a} b^{3/4}} b \sqrt [4]{a x^4-b x^2}}+\frac {\sqrt [8]{-\sqrt {-a}} \sqrt {x} \sqrt [4]{a x^2-b} \tan ^{-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 )}{4 \sqrt [4]{\sqrt {-\sqrt {-a}} a+\sqrt {-a} b^{3/4}} b \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 )}{4 \sqrt [4]{a-\sqrt [4]{-a} b^{3/4}} b \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 )}{4 \sqrt [4]{a+\sqrt [4]{-a} b^{3/4}} b \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 )}{4 \sqrt [4]{\sqrt {-\sqrt {-a}} a-\sqrt {-a} b^{3/4}} b \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 )}{4 \sqrt [4]{\sqrt {-\sqrt {-a}} a+\sqrt {-a} b^{3/4}} b \sqrt [4]{a x^4-b x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 205
Rule 206
Rule 208
Rule 212
Rule 377
Rule 1429
Rule 2056
Rule 6715
Rule 6725
Rubi steps
\begin {align*} \int \frac {1}{\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 {1}{\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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 {b} \sqrt [4]{-b x^2+a x^4}}+\frac {\left (\sqrt {x} \sqrt [4]{-b+a x^2}\right ) \operatorname {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 {b} \sqrt [4]{-b x^2+a x^4}}\\ &=\frac {\left (\sqrt {-\sqrt {-a}} \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \operatorname {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 )}{2 b^{3/4} \sqrt [4]{-b x^2+a x^4}}+\frac {\left (\sqrt {-\sqrt {-a}} \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \operatorname {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 )}{2 b^{3/4} \sqrt [4]{-b x^2+a x^4}}+\frac {\left (\sqrt [4]{-a} \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \operatorname {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 )}{2 b^{3/4} \sqrt [4]{-b x^2+a x^4}}+\frac {\left (\sqrt [4]{-a} \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \operatorname {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 )}{2 b^{3/4} \sqrt [4]{-b x^2+a x^4}}\\ &=\frac {\left (\sqrt {-\sqrt {-a}} \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \operatorname {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 )}{2 b^{3/4} \sqrt [4]{-b x^2+a x^4}}+\frac {\left (\sqrt {-\sqrt {-a}} \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \operatorname {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 )}{2 b^{3/4} \sqrt [4]{-b x^2+a x^4}}+\frac {\left (\sqrt [4]{-a} \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \operatorname {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 )}{2 b^{3/4} \sqrt [4]{-b x^2+a x^4}}+\frac {\left (\sqrt [4]{-a} \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \operatorname {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 )}{2 b^{3/4} \sqrt [4]{-b x^2+a x^4}}\\ &=\frac {\left (\sqrt {x} \sqrt [4]{-b+a x^2}\right ) \operatorname {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 )}{4 b \sqrt [4]{-b x^2+a x^4}}+\frac {\left (\sqrt {x} \sqrt [4]{-b+a x^2}\right ) \operatorname {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 )}{4 b \sqrt [4]{-b x^2+a x^4}}+\frac {\left (\sqrt {x} \sqrt [4]{-b+a x^2}\right ) \operatorname {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 )}{4 b \sqrt [4]{-b x^2+a x^4}}+\frac {\left (\sqrt {x} \sqrt [4]{-b+a x^2}\right ) \operatorname {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 )}{4 b \sqrt [4]{-b x^2+a x^4}}+\frac {\left (\sqrt [4]{-\sqrt {-a}} \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \operatorname {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 )}{4 b \sqrt [4]{-b x^2+a x^4}}+\frac {\left (\sqrt [4]{-\sqrt {-a}} \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \operatorname {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 )}{4 b \sqrt [4]{-b x^2+a x^4}}+\frac {\left (\sqrt [4]{-\sqrt {-a}} \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \operatorname {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 )}{4 b \sqrt [4]{-b x^2+a x^4}}+\frac {\left (\sqrt [4]{-\sqrt {-a}} \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \operatorname {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 )}{4 b \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 )}{4 \sqrt [4]{a-\sqrt [4]{-a} b^{3/4}} b \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 )}{4 \sqrt [4]{a+\sqrt [4]{-a} b^{3/4}} b \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 )}{4 \sqrt [4]{\sqrt {-\sqrt {-a}} a-\sqrt {-a} b^{3/4}} b \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 )}{4 \sqrt [4]{\sqrt {-\sqrt {-a}} a+\sqrt {-a} b^{3/4}} b \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 )}{4 \sqrt [4]{a-\sqrt [4]{-a} b^{3/4}} b \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 )}{4 \sqrt [4]{a+\sqrt [4]{-a} b^{3/4}} b \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 )}{4 \sqrt [4]{\sqrt {-\sqrt {-a}} a-\sqrt {-a} b^{3/4}} b \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 )}{4 \sqrt [4]{\sqrt {-\sqrt {-a}} a+\sqrt {-a} b^{3/4}} b \sqrt [4]{-b x^2+a x^4}}\\ \end {align*}
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Mathematica [B] time = 2.81, size = 662, normalized size = 8.17 \begin {gather*} -\frac {x \sqrt [4]{\frac {b}{x^2}-a} \left (a^3 \text {RootSum}\left [\text {$\#$1}^4+4 \text {$\#$1}^3 a+6 \text {$\#$1}^2 a^2+4 \text {$\#$1} a^3+a^4+a b^3\&,\frac {\frac {\log \left (\sqrt [4]{\text {$\#$1}}-\sqrt [4]{\frac {b}{x^2}-a}\right )}{\sqrt [4]{\text {$\#$1}}}-\frac {\log \left (\sqrt [4]{\text {$\#$1}}+\sqrt [4]{\frac {b}{x^2}-a}\right )}{\sqrt [4]{\text {$\#$1}}}+\frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{\frac {b}{x^2}-a}}{\sqrt [4]{\text {$\#$1}}}\right )}{\sqrt [4]{\text {$\#$1}}}}{\text {$\#$1}^3+3 \text {$\#$1}^2 a+3 \text {$\#$1} a^2+a^3}\&\right ]+3 a^2 \text {RootSum}\left [\text {$\#$1}^4+4 \text {$\#$1}^3 a+6 \text {$\#$1}^2 a^2+4 \text {$\#$1} a^3+a^4+a b^3\&,\frac {\text {$\#$1}^{3/4} \log \left (\sqrt [4]{\text {$\#$1}}-\sqrt [4]{\frac {b}{x^2}-a}\right )-\text {$\#$1}^{3/4} \log \left (\sqrt [4]{\text {$\#$1}}+\sqrt [4]{\frac {b}{x^2}-a}\right )+2 \text {$\#$1}^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {b}{x^2}-a}}{\sqrt [4]{\text {$\#$1}}}\right )}{\text {$\#$1}^3+3 \text {$\#$1}^2 a+3 \text {$\#$1} a^2+a^3}\&\right ]+3 a \text {RootSum}\left [\text {$\#$1}^4+4 \text {$\#$1}^3 a+6 \text {$\#$1}^2 a^2+4 \text {$\#$1} a^3+a^4+a b^3\&,\frac {\text {$\#$1}^{7/4} \log \left (\sqrt [4]{\text {$\#$1}}-\sqrt [4]{\frac {b}{x^2}-a}\right )-\text {$\#$1}^{7/4} \log \left (\sqrt [4]{\text {$\#$1}}+\sqrt [4]{\frac {b}{x^2}-a}\right )+2 \text {$\#$1}^{7/4} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {b}{x^2}-a}}{\sqrt [4]{\text {$\#$1}}}\right )}{\text {$\#$1}^3+3 \text {$\#$1}^2 a+3 \text {$\#$1} a^2+a^3}\&\right ]+\text {RootSum}\left [\text {$\#$1}^4+4 \text {$\#$1}^3 a+6 \text {$\#$1}^2 a^2+4 \text {$\#$1} a^3+a^4+a b^3\&,\frac {\text {$\#$1}^{11/4} \log \left (\sqrt [4]{\text {$\#$1}}-\sqrt [4]{\frac {b}{x^2}-a}\right )-\text {$\#$1}^{11/4} \log \left (\sqrt [4]{\text {$\#$1}}+\sqrt [4]{\frac {b}{x^2}-a}\right )+2 \text {$\#$1}^{11/4} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {b}{x^2}-a}}{\sqrt [4]{\text {$\#$1}}}\right )}{\text {$\#$1}^3+3 \text {$\#$1}^2 a+3 \text {$\#$1} a^2+a^3}\&\right ]\right )}{8 b \sqrt [4]{a x^4-b x^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 2.34, size = 81, normalized size = 1.00 \begin {gather*} -\frac {\text {RootSum}\left [a^4+a b^3-4 a^3 \text {$\#$1}^4+6 a^2 \text {$\#$1}^8-4 a \text {$\#$1}^{12}+\text {$\#$1}^{16}\&,\frac {-\log (x)+\log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{8 b} \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(-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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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a \,x^{4}-b \,x^{2}\right )^{\frac {1}{4}} \left (a \,x^{8}+b \right )}\, dx\]
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*} \int \frac {1}{{\left (a x^{8} + 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 {1}{\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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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\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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