Optimal. Leaf size=50 \[ \frac {1}{8} \text {RootSum}\left [b+2 a \text {$\#$1}^4-2 \text {$\#$1}^8\& ,\frac {-\log (x)+\log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right )}{\text {$\#$1}}\& \right ] \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(397\) vs. \(2(50)=100\).
time = 0.60, antiderivative size = 397, normalized size of antiderivative = 7.94, number of steps
used = 10, number of rules used = 5, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.139, Rules used = {6860, 385,
218, 214, 211} \begin {gather*} -\frac {\sqrt [4]{a-\sqrt {a^2+2 b}} \text {ArcTan}\left (\frac {x \sqrt [4]{-a \sqrt {a^2+2 b}+a^2+b}}{\sqrt [4]{a-\sqrt {a^2+2 b}} \sqrt [4]{a x^4+b}}\right )}{4 \sqrt [4]{-a \sqrt {a^2+2 b}+a^2+b}}-\frac {\sqrt [4]{\sqrt {a^2+2 b}+a} \text {ArcTan}\left (\frac {x \sqrt [4]{a \sqrt {a^2+2 b}+a^2+b}}{\sqrt [4]{\sqrt {a^2+2 b}+a} \sqrt [4]{a x^4+b}}\right )}{4 \sqrt [4]{a \sqrt {a^2+2 b}+a^2+b}}-\frac {\sqrt [4]{a-\sqrt {a^2+2 b}} \tanh ^{-1}\left (\frac {x \sqrt [4]{-a \sqrt {a^2+2 b}+a^2+b}}{\sqrt [4]{a-\sqrt {a^2+2 b}} \sqrt [4]{a x^4+b}}\right )}{4 \sqrt [4]{-a \sqrt {a^2+2 b}+a^2+b}}-\frac {\sqrt [4]{\sqrt {a^2+2 b}+a} \tanh ^{-1}\left (\frac {x \sqrt [4]{a \sqrt {a^2+2 b}+a^2+b}}{\sqrt [4]{\sqrt {a^2+2 b}+a} \sqrt [4]{a x^4+b}}\right )}{4 \sqrt [4]{a \sqrt {a^2+2 b}+a^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 {2 b+a x^4}{\sqrt [4]{b+a x^4} \left (-2 b-2 a x^4+x^8\right )} \, dx &=\int \left (\frac {a+\sqrt {a^2+2 b}}{\left (-2 a-2 \sqrt {a^2+2 b}+2 x^4\right ) \sqrt [4]{b+a x^4}}+\frac {a-\sqrt {a^2+2 b}}{\left (-2 a+2 \sqrt {a^2+2 b}+2 x^4\right ) \sqrt [4]{b+a x^4}}\right ) \, dx\\ &=\left (a-\sqrt {a^2+2 b}\right ) \int \frac {1}{\left (-2 a+2 \sqrt {a^2+2 b}+2 x^4\right ) \sqrt [4]{b+a x^4}} \, dx+\left (a+\sqrt {a^2+2 b}\right ) \int \frac {1}{\left (-2 a-2 \sqrt {a^2+2 b}+2 x^4\right ) \sqrt [4]{b+a x^4}} \, dx\\ &=\left (a-\sqrt {a^2+2 b}\right ) \text {Subst}\left (\int \frac {1}{-2 a+2 \sqrt {a^2+2 b}-\left (-2 b+a \left (-2 a+2 \sqrt {a^2+2 b}\right )\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )+\left (a+\sqrt {a^2+2 b}\right ) \text {Subst}\left (\int \frac {1}{-2 a-2 \sqrt {a^2+2 b}-\left (-2 b+a \left (-2 a-2 \sqrt {a^2+2 b}\right )\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )\\ &=-\left (\frac {1}{4} \sqrt {a-\sqrt {a^2+2 b}} \text {Subst}\left (\int \frac {1}{\sqrt {a-\sqrt {a^2+2 b}}-\sqrt {a^2+b-a \sqrt {a^2+2 b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )\right )-\frac {1}{4} \sqrt {a-\sqrt {a^2+2 b}} \text {Subst}\left (\int \frac {1}{\sqrt {a-\sqrt {a^2+2 b}}+\sqrt {a^2+b-a \sqrt {a^2+2 b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )-\frac {1}{4} \sqrt {a+\sqrt {a^2+2 b}} \text {Subst}\left (\int \frac {1}{\sqrt {a+\sqrt {a^2+2 b}}-\sqrt {a^2+b+a \sqrt {a^2+2 b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )-\frac {1}{4} \sqrt {a+\sqrt {a^2+2 b}} \text {Subst}\left (\int \frac {1}{\sqrt {a+\sqrt {a^2+2 b}}+\sqrt {a^2+b+a \sqrt {a^2+2 b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )\\ &=-\frac {\sqrt [4]{a-\sqrt {a^2+2 b}} \tan ^{-1}\left (\frac {\sqrt [4]{a^2+b-a \sqrt {a^2+2 b}} x}{\sqrt [4]{a-\sqrt {a^2+2 b}} \sqrt [4]{b+a x^4}}\right )}{4 \sqrt [4]{a^2+b-a \sqrt {a^2+2 b}}}-\frac {\sqrt [4]{a+\sqrt {a^2+2 b}} \tan ^{-1}\left (\frac {\sqrt [4]{a^2+b+a \sqrt {a^2+2 b}} x}{\sqrt [4]{a+\sqrt {a^2+2 b}} \sqrt [4]{b+a x^4}}\right )}{4 \sqrt [4]{a^2+b+a \sqrt {a^2+2 b}}}-\frac {\sqrt [4]{a-\sqrt {a^2+2 b}} \tanh ^{-1}\left (\frac {\sqrt [4]{a^2+b-a \sqrt {a^2+2 b}} x}{\sqrt [4]{a-\sqrt {a^2+2 b}} \sqrt [4]{b+a x^4}}\right )}{4 \sqrt [4]{a^2+b-a \sqrt {a^2+2 b}}}-\frac {\sqrt [4]{a+\sqrt {a^2+2 b}} \tanh ^{-1}\left (\frac {\sqrt [4]{a^2+b+a \sqrt {a^2+2 b}} x}{\sqrt [4]{a+\sqrt {a^2+2 b}} \sqrt [4]{b+a x^4}}\right )}{4 \sqrt [4]{a^2+b+a \sqrt {a^2+2 b}}}\\ \end {align*}
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Mathematica [A]
time = 0.45, size = 50, normalized size = 1.00 \begin {gather*} \frac {1}{8} \text {RootSum}\left [b+2 a \text {$\#$1}^4-2 \text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.17, size = 0, normalized size = 0.00 \[\int \frac {a \,x^{4}+2 b}{\left (a \,x^{4}+b \right )^{\frac {1}{4}} \left (x^{8}-2 a \,x^{4}-2 b \right )}\, dx\]
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(-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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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 = -0.02 \begin {gather*} \int -\frac {a\,x^4+2\,b}{{\left (a\,x^4+b\right )}^{1/4}\,\left (-x^8+2\,a\,x^4+2\,b\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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