Optimal. Leaf size=53 \[ -\frac {1}{4} \text {RootSum}\left [-\text {$\#$1}^8+\text {$\#$1}^4 a+2 b\& ,\frac {\log \left (\sqrt [4]{a x^4-b}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}}\& \right ] \]
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Rubi [B] time = 0.89, antiderivative size = 421, normalized size of antiderivative = 7.94, number of steps used = 10, number of rules used = 5, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.128, Rules used = {6728, 377, 212, 208, 205} \begin {gather*} \frac {\sqrt [4]{a-\sqrt {a^2+8 b}} \tan ^{-1}\left (\frac {x \sqrt [4]{-a \sqrt {a^2+8 b}+a^2+4 b}}{\sqrt [4]{a-\sqrt {a^2+8 b}} \sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{-a \sqrt {a^2+8 b}+a^2+4 b}}+\frac {\sqrt [4]{\sqrt {a^2+8 b}+a} \tan ^{-1}\left (\frac {x \sqrt [4]{a \sqrt {a^2+8 b}+a^2+4 b}}{\sqrt [4]{\sqrt {a^2+8 b}+a} \sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{a \sqrt {a^2+8 b}+a^2+4 b}}+\frac {\sqrt [4]{a-\sqrt {a^2+8 b}} \tanh ^{-1}\left (\frac {x \sqrt [4]{-a \sqrt {a^2+8 b}+a^2+4 b}}{\sqrt [4]{a-\sqrt {a^2+8 b}} \sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{-a \sqrt {a^2+8 b}+a^2+4 b}}+\frac {\sqrt [4]{\sqrt {a^2+8 b}+a} \tanh ^{-1}\left (\frac {x \sqrt [4]{a \sqrt {a^2+8 b}+a^2+4 b}}{\sqrt [4]{\sqrt {a^2+8 b}+a} \sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{a \sqrt {a^2+8 b}+a^2+4 b}} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 208
Rule 212
Rule 377
Rule 6728
Rubi steps
\begin {align*} \int \frac {-2 b+a x^4}{\sqrt [4]{-b+a x^4} \left (-b+a x^4+2 x^8\right )} \, dx &=\int \left (\frac {a-\sqrt {a^2+8 b}}{\left (a-\sqrt {a^2+8 b}+4 x^4\right ) \sqrt [4]{-b+a x^4}}+\frac {a+\sqrt {a^2+8 b}}{\left (a+\sqrt {a^2+8 b}+4 x^4\right ) \sqrt [4]{-b+a x^4}}\right ) \, dx\\ &=\left (a-\sqrt {a^2+8 b}\right ) \int \frac {1}{\left (a-\sqrt {a^2+8 b}+4 x^4\right ) \sqrt [4]{-b+a x^4}} \, dx+\left (a+\sqrt {a^2+8 b}\right ) \int \frac {1}{\left (a+\sqrt {a^2+8 b}+4 x^4\right ) \sqrt [4]{-b+a x^4}} \, dx\\ &=\left (a-\sqrt {a^2+8 b}\right ) \operatorname {Subst}\left (\int \frac {1}{a-\sqrt {a^2+8 b}-\left (4 b+a \left (a-\sqrt {a^2+8 b}\right )\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )+\left (a+\sqrt {a^2+8 b}\right ) \operatorname {Subst}\left (\int \frac {1}{a+\sqrt {a^2+8 b}-\left (4 b+a \left (a+\sqrt {a^2+8 b}\right )\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )\\ &=\frac {1}{2} \sqrt {a-\sqrt {a^2+8 b}} \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-\sqrt {a^2+8 b}}-\sqrt {a^2+4 b-a \sqrt {a^2+8 b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )+\frac {1}{2} \sqrt {a-\sqrt {a^2+8 b}} \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-\sqrt {a^2+8 b}}+\sqrt {a^2+4 b-a \sqrt {a^2+8 b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )+\frac {1}{2} \sqrt {a+\sqrt {a^2+8 b}} \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+\sqrt {a^2+8 b}}-\sqrt {a^2+4 b+a \sqrt {a^2+8 b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )+\frac {1}{2} \sqrt {a+\sqrt {a^2+8 b}} \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+\sqrt {a^2+8 b}}+\sqrt {a^2+4 b+a \sqrt {a^2+8 b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )\\ &=\frac {\sqrt [4]{a-\sqrt {a^2+8 b}} \tan ^{-1}\left (\frac {\sqrt [4]{a^2+4 b-a \sqrt {a^2+8 b}} x}{\sqrt [4]{a-\sqrt {a^2+8 b}} \sqrt [4]{-b+a x^4}}\right )}{2 \sqrt [4]{a^2+4 b-a \sqrt {a^2+8 b}}}+\frac {\sqrt [4]{a+\sqrt {a^2+8 b}} \tan ^{-1}\left (\frac {\sqrt [4]{a^2+4 b+a \sqrt {a^2+8 b}} x}{\sqrt [4]{a+\sqrt {a^2+8 b}} \sqrt [4]{-b+a x^4}}\right )}{2 \sqrt [4]{a^2+4 b+a \sqrt {a^2+8 b}}}+\frac {\sqrt [4]{a-\sqrt {a^2+8 b}} \tanh ^{-1}\left (\frac {\sqrt [4]{a^2+4 b-a \sqrt {a^2+8 b}} x}{\sqrt [4]{a-\sqrt {a^2+8 b}} \sqrt [4]{-b+a x^4}}\right )}{2 \sqrt [4]{a^2+4 b-a \sqrt {a^2+8 b}}}+\frac {\sqrt [4]{a+\sqrt {a^2+8 b}} \tanh ^{-1}\left (\frac {\sqrt [4]{a^2+4 b+a \sqrt {a^2+8 b}} x}{\sqrt [4]{a+\sqrt {a^2+8 b}} \sqrt [4]{-b+a x^4}}\right )}{2 \sqrt [4]{a^2+4 b+a \sqrt {a^2+8 b}}}\\ \end {align*}
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Mathematica [B] time = 0.67, size = 413, normalized size = 7.79 \begin {gather*} \frac {1}{2} \left (\frac {\sqrt [4]{a-\sqrt {a^2+8 b}} \tan ^{-1}\left (\frac {x \sqrt [4]{-a \sqrt {a^2+8 b}+a^2+4 b}}{\sqrt [4]{a-\sqrt {a^2+8 b}} \sqrt [4]{a x^4-b}}\right )}{\sqrt [4]{-a \sqrt {a^2+8 b}+a^2+4 b}}+\frac {\sqrt [4]{\sqrt {a^2+8 b}+a} \tan ^{-1}\left (\frac {x \sqrt [4]{a \sqrt {a^2+8 b}+a^2+4 b}}{\sqrt [4]{\sqrt {a^2+8 b}+a} \sqrt [4]{a x^4-b}}\right )}{\sqrt [4]{a \sqrt {a^2+8 b}+a^2+4 b}}+\frac {\sqrt [4]{a-\sqrt {a^2+8 b}} \tanh ^{-1}\left (\frac {x \sqrt [4]{-a \sqrt {a^2+8 b}+a^2+4 b}}{\sqrt [4]{a-\sqrt {a^2+8 b}} \sqrt [4]{a x^4-b}}\right )}{\sqrt [4]{-a \sqrt {a^2+8 b}+a^2+4 b}}+\frac {\sqrt [4]{\sqrt {a^2+8 b}+a} \tanh ^{-1}\left (\frac {x \sqrt [4]{a \sqrt {a^2+8 b}+a^2+4 b}}{\sqrt [4]{\sqrt {a^2+8 b}+a} \sqrt [4]{a x^4-b}}\right )}{\sqrt [4]{a \sqrt {a^2+8 b}+a^2+4 b}}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.70, size = 53, normalized size = 1.00 \begin {gather*} -\frac {1}{4} \text {RootSum}\left [2 b+a \text {$\#$1}^4-\text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right )}{\text {$\#$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\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 \[\int \frac {a \,x^{4}-2 b}{\left (a \,x^{4}-b \right )^{\frac {1}{4}} \left (2 x^{8}+a \,x^{4}-b \right )}\, dx\]
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\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 = -0.02 \begin {gather*} \int -\frac {2\,b-a\,x^4}{{\left (a\,x^4-b\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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