Optimal. Leaf size=103 \[ \frac {1}{4} \text {RootSum}\left [2 a^2-b-3 a \text {$\#$1}^4+\text {$\#$1}^8\& ,\frac {-a \log (x)+a \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right )-\log (x) \text {$\#$1}^4+\log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{3 a \text {$\#$1}-2 \text {$\#$1}^5}\& \right ] \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(499\) vs. \(2(103)=206\).
time = 0.86, antiderivative size = 499, normalized size of antiderivative = 4.84, 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 = {6860, 385,
218, 214, 211} \begin {gather*} -\frac {\left (a-\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \text {ArcTan}\left (\frac {x \sqrt [4]{-a \sqrt {a^2+4 b}+a^2-2 b}}{\sqrt [4]{a-\sqrt {a^2+4 b}} \sqrt [4]{a x^4-b}}\right )}{\left (a-\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{-a \sqrt {a^2+4 b}+a^2-2 b}}-\frac {\left (\frac {a^2-b}{\sqrt {a^2+4 b}}+a\right ) \text {ArcTan}\left (\frac {x \sqrt [4]{a \sqrt {a^2+4 b}+a^2-2 b}}{\sqrt [4]{\sqrt {a^2+4 b}+a} \sqrt [4]{a x^4-b}}\right )}{\left (\sqrt {a^2+4 b}+a\right )^{3/4} \sqrt [4]{a \sqrt {a^2+4 b}+a^2-2 b}}-\frac {\left (a-\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \tanh ^{-1}\left (\frac {x \sqrt [4]{-a \sqrt {a^2+4 b}+a^2-2 b}}{\sqrt [4]{a-\sqrt {a^2+4 b}} \sqrt [4]{a x^4-b}}\right )}{\left (a-\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{-a \sqrt {a^2+4 b}+a^2-2 b}}-\frac {\left (\frac {a^2-b}{\sqrt {a^2+4 b}}+a\right ) \tanh ^{-1}\left (\frac {x \sqrt [4]{a \sqrt {a^2+4 b}+a^2-2 b}}{\sqrt [4]{\sqrt {a^2+4 b}+a} \sqrt [4]{a x^4-b}}\right )}{\left (\sqrt {a^2+4 b}+a\right )^{3/4} \sqrt [4]{a \sqrt {a^2+4 b}+a^2-2 b}} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 214
Rule 218
Rule 385
Rule 6860
Rubi steps
\begin {align*} \int \frac {-b+2 a x^4}{\sqrt [4]{-b+a x^4} \left (-b-a x^4+x^8\right )} \, dx &=\int \left (\frac {2 a+\frac {2 \left (a^2-b\right )}{\sqrt {a^2+4 b}}}{\left (-a-\sqrt {a^2+4 b}+2 x^4\right ) \sqrt [4]{-b+a x^4}}+\frac {2 a-\frac {2 \left (a^2-b\right )}{\sqrt {a^2+4 b}}}{\left (-a+\sqrt {a^2+4 b}+2 x^4\right ) \sqrt [4]{-b+a x^4}}\right ) \, dx\\ &=\left (2 \left (a-\frac {a^2-b}{\sqrt {a^2+4 b}}\right )\right ) \int \frac {1}{\left (-a+\sqrt {a^2+4 b}+2 x^4\right ) \sqrt [4]{-b+a x^4}} \, dx+\left (2 \left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right )\right ) \int \frac {1}{\left (-a-\sqrt {a^2+4 b}+2 x^4\right ) \sqrt [4]{-b+a x^4}} \, dx\\ &=\left (2 \left (a-\frac {a^2-b}{\sqrt {a^2+4 b}}\right )\right ) \text {Subst}\left (\int \frac {1}{-a+\sqrt {a^2+4 b}-\left (2 b+a \left (-a+\sqrt {a^2+4 b}\right )\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )+\left (2 \left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right )\right ) \text {Subst}\left (\int \frac {1}{-a-\sqrt {a^2+4 b}-\left (2 b+a \left (-a-\sqrt {a^2+4 b}\right )\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )\\ &=-\frac {\left (a-\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-\sqrt {a^2+4 b}}-\sqrt {a^2-2 b-a \sqrt {a^2+4 b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{\sqrt {a-\sqrt {a^2+4 b}}}-\frac {\left (a-\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-\sqrt {a^2+4 b}}+\sqrt {a^2-2 b-a \sqrt {a^2+4 b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{\sqrt {a-\sqrt {a^2+4 b}}}-\frac {\left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+\sqrt {a^2+4 b}}-\sqrt {a^2-2 b+a \sqrt {a^2+4 b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{\sqrt {a+\sqrt {a^2+4 b}}}-\frac {\left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+\sqrt {a^2+4 b}}+\sqrt {a^2-2 b+a \sqrt {a^2+4 b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{\sqrt {a+\sqrt {a^2+4 b}}}\\ &=-\frac {\left (a-\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a^2-2 b-a \sqrt {a^2+4 b}} x}{\sqrt [4]{a-\sqrt {a^2+4 b}} \sqrt [4]{-b+a x^4}}\right )}{\left (a-\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{a^2-2 b-a \sqrt {a^2+4 b}}}-\frac {\left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a^2-2 b+a \sqrt {a^2+4 b}} x}{\sqrt [4]{a+\sqrt {a^2+4 b}} \sqrt [4]{-b+a x^4}}\right )}{\left (a+\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{a^2-2 b+a \sqrt {a^2+4 b}}}-\frac {\left (a-\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a^2-2 b-a \sqrt {a^2+4 b}} x}{\sqrt [4]{a-\sqrt {a^2+4 b}} \sqrt [4]{-b+a x^4}}\right )}{\left (a-\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{a^2-2 b-a \sqrt {a^2+4 b}}}-\frac {\left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a^2-2 b+a \sqrt {a^2+4 b}} x}{\sqrt [4]{a+\sqrt {a^2+4 b}} \sqrt [4]{-b+a x^4}}\right )}{\left (a+\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{a^2-2 b+a \sqrt {a^2+4 b}}}\\ \end {align*}
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Mathematica [A]
time = 0.59, size = 103, normalized size = 1.00 \begin {gather*} \frac {1}{4} \text {RootSum}\left [2 a^2-b-3 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {a \log (x)-a \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-3 a \text {$\#$1}+2 \text {$\#$1}^5}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {2 a \,x^{4}-b}{\left (a \,x^{4}-b \right )^{\frac {1}{4}} \left (x^{8}-a \,x^{4}-b \right )}\, dx\]
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*} \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 {2 a x^{4} - b}{\sqrt [4]{a x^{4} - b} \left (- a x^{4} - b + x^{8}\right )}\, dx \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*} \text {could not integrate} \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-2\,a\,x^4}{{\left (a\,x^4-b\right )}^{1/4}\,\left (-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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