Optimal. Leaf size=117 \[ \frac {\text {ArcTan}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt [4]{a}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt [4]{a}}+\frac {1}{4} \text {RootSum}\left [b+a \text {$\#$1}^4-\text {$\#$1}^8\& ,\frac {-\log (x) \text {$\#$1}^3+\log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-a+2 \text {$\#$1}^4}\& \right ] \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(507\) vs. \(2(117)=234\).
time = 0.46, antiderivative size = 507, normalized size of antiderivative = 4.33, number of steps
used = 25, number of rules used = 10, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {6860, 246,
218, 212, 209, 1442, 399, 385, 214, 211} \begin {gather*} \frac {\left (-a \sqrt {a^2+4 b}+a^2+2 b\right )^{3/4} \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 )}{2 \sqrt {a^2+4 b} \left (a-\sqrt {a^2+4 b}\right )^{3/4}}-\frac {\left (a \sqrt {a^2+4 b}+a^2+2 b\right )^{3/4} \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 )}{2 \sqrt {a^2+4 b} \left (\sqrt {a^2+4 b}+a\right )^{3/4}}+\frac {\left (-a \sqrt {a^2+4 b}+a^2+2 b\right )^{3/4} \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 )}{2 \sqrt {a^2+4 b} \left (a-\sqrt {a^2+4 b}\right )^{3/4}}-\frac {\left (a \sqrt {a^2+4 b}+a^2+2 b\right )^{3/4} \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 )}{2 \sqrt {a^2+4 b} \left (\sqrt {a^2+4 b}+a\right )^{3/4}}+\frac {\text {ArcTan}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{\sqrt [4]{a}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{\sqrt [4]{a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 211
Rule 212
Rule 214
Rule 218
Rule 246
Rule 385
Rule 399
Rule 1442
Rule 6860
Rubi steps
\begin {align*} \int \frac {-b-a x^4+2 x^8}{\sqrt [4]{b+a x^4} \left (-b-a x^4+x^8\right )} \, dx &=\int \left (\frac {2}{\sqrt [4]{b+a x^4}}+\frac {\left (b+a x^4\right )^{3/4}}{-b-a x^4+x^8}\right ) \, dx\\ &=2 \int \frac {1}{\sqrt [4]{b+a x^4}} \, dx+\int \frac {\left (b+a x^4\right )^{3/4}}{-b-a x^4+x^8} \, dx\\ &=2 \text {Subst}\left (\int \frac {1}{1-a x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )+\frac {2 \int \frac {\left (b+a x^4\right )^{3/4}}{-a-\sqrt {a^2+4 b}+2 x^4} \, dx}{\sqrt {a^2+4 b}}-\frac {2 \int \frac {\left (b+a x^4\right )^{3/4}}{-a+\sqrt {a^2+4 b}+2 x^4} \, dx}{\sqrt {a^2+4 b}}\\ &=\frac {\left (a^2+2 b+a \sqrt {a^2+4 b}\right ) \int \frac {1}{\left (-a-\sqrt {a^2+4 b}+2 x^4\right ) \sqrt [4]{b+a x^4}} \, dx}{\sqrt {a^2+4 b}}+\frac {\left (-2 b+a \left (-a+\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}{\sqrt {a^2+4 b}}+\text {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )+\text {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt [4]{a}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt [4]{a}}+\frac {\left (a^2+2 b+a \sqrt {a^2+4 b}\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 )}{\sqrt {a^2+4 b}}+\frac {\left (-2 b+a \left (-a+\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 )}{\sqrt {a^2+4 b}}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt [4]{a}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt [4]{a}}-\frac {\left (a^2+2 b+a \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 )}{2 \sqrt {a^2+4 b} \sqrt {a+\sqrt {a^2+4 b}}}-\frac {\left (a^2+2 b+a \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 )}{2 \sqrt {a^2+4 b} \sqrt {a+\sqrt {a^2+4 b}}}-\frac {\left (-2 b+a \left (-a+\sqrt {a^2+4 b}\right )\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 )}{2 \sqrt {a^2+4 b} \sqrt {a-\sqrt {a^2+4 b}}}-\frac {\left (-2 b+a \left (-a+\sqrt {a^2+4 b}\right )\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 )}{2 \sqrt {a^2+4 b} \sqrt {a-\sqrt {a^2+4 b}}}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt [4]{a}}+\frac {\left (a^2+2 b-a \sqrt {a^2+4 b}\right )^{3/4} \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 )}{2 \sqrt {a^2+4 b} \left (a-\sqrt {a^2+4 b}\right )^{3/4}}-\frac {\left (a^2+2 b+a \sqrt {a^2+4 b}\right )^{3/4} \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 )}{2 \sqrt {a^2+4 b} \left (a+\sqrt {a^2+4 b}\right )^{3/4}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt [4]{a}}+\frac {\left (a^2+2 b-a \sqrt {a^2+4 b}\right )^{3/4} \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 )}{2 \sqrt {a^2+4 b} \left (a-\sqrt {a^2+4 b}\right )^{3/4}}-\frac {\left (a^2+2 b+a \sqrt {a^2+4 b}\right )^{3/4} \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 )}{2 \sqrt {a^2+4 b} \left (a+\sqrt {a^2+4 b}\right )^{3/4}}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 112, normalized size = 0.96 \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )+\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt [4]{a}}+\frac {1}{4} \text {RootSum}\left [b+a \text {$\#$1}^4-\text {$\#$1}^8\&,\frac {-\log (x) \text {$\#$1}^3+\log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-a+2 \text {$\#$1}^4}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {2 x^{8}-a \,x^{4}-b}{\left (a \,x^{4}+b \right )^{\frac {1}{4}} \left (x^{8}-a \,x^{4}-b \right )}\, dx\]
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 [C] Result contains higher order function than in optimal. Order 3 vs. order
1.
time = 0.67, size = 3119, normalized size = 26.66 \begin {gather*} \text {Too large to display} \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 {- a x^{4} - b + 2 x^{8}}{\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 {-2\,x^8+a\,x^4+b}{{\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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