Optimal. Leaf size=99 \[ -\frac {1}{4} \text {RootSum}\left [2 \text {$\#$1}^8-3 \text {$\#$1}^4 a+a^2-b\& ,\frac {\text {$\#$1}^4 \log \left (\sqrt [4]{a x^4+b}-\text {$\#$1} x\right )+\text {$\#$1}^4 (-\log (x))+a \log \left (\sqrt [4]{a x^4+b}-\text {$\#$1} x\right )-a \log (x)}{3 \text {$\#$1} a-4 \text {$\#$1}^5}\& \right ] \]
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Rubi [B] time = 1.48, antiderivative size = 483, normalized size of antiderivative = 4.88, number of steps used = 10, number of rules used = 5, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6728, 377, 212, 208, 205} \begin {gather*} -\frac {\left (a-\frac {a^2+b}{\sqrt {a^2+8 b}}\right ) \tan ^{-1}\left (\frac {x \sqrt [4]{-a \sqrt {a^2+8 b}+a^2+2 b}}{\sqrt [4]{a-\sqrt {a^2+8 b}} \sqrt [4]{a x^4+b}}\right )}{\left (a-\sqrt {a^2+8 b}\right )^{3/4} \sqrt [4]{-a \sqrt {a^2+8 b}+a^2+2 b}}-\frac {\left (\frac {a^2+b}{\sqrt {a^2+8 b}}+a\right ) \tan ^{-1}\left (\frac {x \sqrt [4]{a \sqrt {a^2+8 b}+a^2+2 b}}{\sqrt [4]{\sqrt {a^2+8 b}+a} \sqrt [4]{a x^4+b}}\right )}{\left (\sqrt {a^2+8 b}+a\right )^{3/4} \sqrt [4]{a \sqrt {a^2+8 b}+a^2+2 b}}-\frac {\left (a-\frac {a^2+b}{\sqrt {a^2+8 b}}\right ) \tanh ^{-1}\left (\frac {x \sqrt [4]{-a \sqrt {a^2+8 b}+a^2+2 b}}{\sqrt [4]{a-\sqrt {a^2+8 b}} \sqrt [4]{a x^4+b}}\right )}{\left (a-\sqrt {a^2+8 b}\right )^{3/4} \sqrt [4]{-a \sqrt {a^2+8 b}+a^2+2 b}}-\frac {\left (\frac {a^2+b}{\sqrt {a^2+8 b}}+a\right ) \tanh ^{-1}\left (\frac {x \sqrt [4]{a \sqrt {a^2+8 b}+a^2+2 b}}{\sqrt [4]{\sqrt {a^2+8 b}+a} \sqrt [4]{a x^4+b}}\right )}{\left (\sqrt {a^2+8 b}+a\right )^{3/4} \sqrt [4]{a \sqrt {a^2+8 b}+a^2+2 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 {b+2 a x^4}{\sqrt [4]{b+a x^4} \left (-2 b-a x^4+x^8\right )} \, dx &=\int \left (\frac {2 a+\frac {2 \left (a^2+b\right )}{\sqrt {a^2+8 b}}}{\left (-a-\sqrt {a^2+8 b}+2 x^4\right ) \sqrt [4]{b+a x^4}}+\frac {2 a-\frac {2 \left (a^2+b\right )}{\sqrt {a^2+8 b}}}{\left (-a+\sqrt {a^2+8 b}+2 x^4\right ) \sqrt [4]{b+a x^4}}\right ) \, dx\\ &=\left (2 \left (a-\frac {a^2+b}{\sqrt {a^2+8 b}}\right )\right ) \int \frac {1}{\left (-a+\sqrt {a^2+8 b}+2 x^4\right ) \sqrt [4]{b+a x^4}} \, dx+\left (2 \left (a+\frac {a^2+b}{\sqrt {a^2+8 b}}\right )\right ) \int \frac {1}{\left (-a-\sqrt {a^2+8 b}+2 x^4\right ) \sqrt [4]{b+a x^4}} \, dx\\ &=\left (2 \left (a-\frac {a^2+b}{\sqrt {a^2+8 b}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-a+\sqrt {a^2+8 b}-\left (-2 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 (2 \left (a+\frac {a^2+b}{\sqrt {a^2+8 b}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-a-\sqrt {a^2+8 b}-\left (-2 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 {\left (a-\frac {a^2+b}{\sqrt {a^2+8 b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-\sqrt {a^2+8 b}}-\sqrt {a^2+2 b-a \sqrt {a^2+8 b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt {a-\sqrt {a^2+8 b}}}-\frac {\left (a-\frac {a^2+b}{\sqrt {a^2+8 b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-\sqrt {a^2+8 b}}+\sqrt {a^2+2 b-a \sqrt {a^2+8 b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt {a-\sqrt {a^2+8 b}}}-\frac {\left (a+\frac {a^2+b}{\sqrt {a^2+8 b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+\sqrt {a^2+8 b}}-\sqrt {a^2+2 b+a \sqrt {a^2+8 b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt {a+\sqrt {a^2+8 b}}}-\frac {\left (a+\frac {a^2+b}{\sqrt {a^2+8 b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+\sqrt {a^2+8 b}}+\sqrt {a^2+2 b+a \sqrt {a^2+8 b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt {a+\sqrt {a^2+8 b}}}\\ &=-\frac {\left (a-\frac {a^2+b}{\sqrt {a^2+8 b}}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a^2+2 b-a \sqrt {a^2+8 b}} x}{\sqrt [4]{a-\sqrt {a^2+8 b}} \sqrt [4]{b+a x^4}}\right )}{\left (a-\sqrt {a^2+8 b}\right )^{3/4} \sqrt [4]{a^2+2 b-a \sqrt {a^2+8 b}}}-\frac {\left (a+\frac {a^2+b}{\sqrt {a^2+8 b}}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a^2+2 b+a \sqrt {a^2+8 b}} x}{\sqrt [4]{a+\sqrt {a^2+8 b}} \sqrt [4]{b+a x^4}}\right )}{\left (a+\sqrt {a^2+8 b}\right )^{3/4} \sqrt [4]{a^2+2 b+a \sqrt {a^2+8 b}}}-\frac {\left (a-\frac {a^2+b}{\sqrt {a^2+8 b}}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a^2+2 b-a \sqrt {a^2+8 b}} x}{\sqrt [4]{a-\sqrt {a^2+8 b}} \sqrt [4]{b+a x^4}}\right )}{\left (a-\sqrt {a^2+8 b}\right )^{3/4} \sqrt [4]{a^2+2 b-a \sqrt {a^2+8 b}}}-\frac {\left (a+\frac {a^2+b}{\sqrt {a^2+8 b}}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a^2+2 b+a \sqrt {a^2+8 b}} x}{\sqrt [4]{a+\sqrt {a^2+8 b}} \sqrt [4]{b+a x^4}}\right )}{\left (a+\sqrt {a^2+8 b}\right )^{3/4} \sqrt [4]{a^2+2 b+a \sqrt {a^2+8 b}}}\\ \end {align*}
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Mathematica [B] time = 0.89, size = 503, normalized size = 5.08 \begin {gather*} -\frac {\left (a-\frac {a^2+b}{\sqrt {a^2+8 b}}\right ) \tan ^{-1}\left (\frac {x \sqrt [4]{-a \sqrt {a^2+8 b}+a^2+2 b}}{\sqrt [4]{a-\sqrt {a^2+8 b}} \sqrt [4]{a x^4+b}}\right )}{\left (a-\sqrt {a^2+8 b}\right )^{3/4} \sqrt [4]{-a \sqrt {a^2+8 b}+a^2+2 b}}-\frac {\left (a \sqrt {a^2+8 b}+a^2+b\right ) \tan ^{-1}\left (\frac {x \sqrt [4]{a \sqrt {a^2+8 b}+a^2+2 b}}{\sqrt [4]{\sqrt {a^2+8 b}+a} \sqrt [4]{a x^4+b}}\right )}{\sqrt {a^2+8 b} \left (\sqrt {a^2+8 b}+a\right )^{3/4} \sqrt [4]{a \sqrt {a^2+8 b}+a^2+2 b}}-\frac {\left (a-\frac {a^2+b}{\sqrt {a^2+8 b}}\right ) \tanh ^{-1}\left (\frac {x \sqrt [4]{-a \sqrt {a^2+8 b}+a^2+2 b}}{\sqrt [4]{a-\sqrt {a^2+8 b}} \sqrt [4]{a x^4+b}}\right )}{\left (a-\sqrt {a^2+8 b}\right )^{3/4} \sqrt [4]{-a \sqrt {a^2+8 b}+a^2+2 b}}-\frac {\left (a \sqrt {a^2+8 b}+a^2+b\right ) \tanh ^{-1}\left (\frac {x \sqrt [4]{a \sqrt {a^2+8 b}+a^2+2 b}}{\sqrt [4]{\sqrt {a^2+8 b}+a} \sqrt [4]{a x^4+b}}\right )}{\sqrt {a^2+8 b} \left (\sqrt {a^2+8 b}+a\right )^{3/4} \sqrt [4]{a \sqrt {a^2+8 b}+a^2+2 b}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.94, size = 99, normalized size = 1.00 \begin {gather*} -\frac {1}{4} \text {RootSum}\left [a^2-b-3 a \text {$\#$1}^4+2 \text {$\#$1}^8\&,\frac {a \log (x)-a \log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-3 a \text {$\#$1}+4 \text {$\#$1}^5}\&\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 {2 \, a x^{4} + b}{{\left (x^{8} - a x^{4} - 2 \, 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 \[\int \frac {2 a \,x^{4}+b}{\left (a \,x^{4}+b \right )^{\frac {1}{4}} \left (x^{8}-a \,x^{4}-2 b \right )}\, dx\]
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*} \int \frac {2 \, a x^{4} + b}{{\left (x^{8} - a x^{4} - 2 \, 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 = -0.01 \begin {gather*} \int -\frac {2\,a\,x^4+b}{{\left (a\,x^4+b\right )}^{1/4}\,\left (-x^8+a\,x^4+2\,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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