Optimal. Leaf size=105 \[ \frac {\text {ArcTan}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt [4]{a}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt [4]{a}}+\frac {1}{8} \text {RootSum}\left [a^2-a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\& ,\frac {-\log (x)+\log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right )}{\text {$\#$1}}\& \right ] \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(291\) vs. \(2(105)=210\).
time = 0.40, antiderivative size = 291, normalized size of antiderivative = 2.77, number of steps
used = 15, number of rules used = 7, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {6857, 246,
218, 212, 209, 1443, 385} \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{\sqrt [4]{a}}-\frac {\text {ArcTan}\left (\frac {\sqrt [8]{a} x \sqrt [4]{\sqrt {a}-\sqrt {b}}}{\sqrt [4]{a x^4+b}}\right )}{4 \sqrt [8]{a} \sqrt [4]{\sqrt {a}-\sqrt {b}}}-\frac {\text {ArcTan}\left (\frac {\sqrt [8]{a} x \sqrt [4]{\sqrt {a}+\sqrt {b}}}{\sqrt [4]{a x^4+b}}\right )}{4 \sqrt [8]{a} \sqrt [4]{\sqrt {a}+\sqrt {b}}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{\sqrt [4]{a}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [8]{a} x \sqrt [4]{\sqrt {a}-\sqrt {b}}}{\sqrt [4]{a x^4+b}}\right )}{4 \sqrt [8]{a} \sqrt [4]{\sqrt {a}-\sqrt {b}}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [8]{a} x \sqrt [4]{\sqrt {a}+\sqrt {b}}}{\sqrt [4]{a x^4+b}}\right )}{4 \sqrt [8]{a} \sqrt [4]{\sqrt {a}+\sqrt {b}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 212
Rule 218
Rule 246
Rule 385
Rule 1443
Rule 6857
Rubi steps
\begin {align*} \int \frac {-b+2 a x^8}{\sqrt [4]{b+a x^4} \left (-b+a x^8\right )} \, dx &=\int \left (\frac {2}{\sqrt [4]{b+a x^4}}+\frac {b}{\sqrt [4]{b+a x^4} \left (-b+a x^8\right )}\right ) \, dx\\ &=2 \int \frac {1}{\sqrt [4]{b+a x^4}} \, dx+b \int \frac {1}{\sqrt [4]{b+a x^4} \left (-b+a x^8\right )} \, dx\\ &=2 \text {Subst}\left (\int \frac {1}{1-a x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )-\frac {1}{2} \left (\sqrt {a} \sqrt {b}\right ) \int \frac {1}{\left (\sqrt {a} \sqrt {b}-a x^4\right ) \sqrt [4]{b+a x^4}} \, dx-\frac {1}{2} \left (\sqrt {a} \sqrt {b}\right ) \int \frac {1}{\left (\sqrt {a} \sqrt {b}+a x^4\right ) \sqrt [4]{b+a x^4}} \, dx\\ &=-\left (\frac {1}{2} \left (\sqrt {a} \sqrt {b}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a} \sqrt {b}-\left (a^{3/2} \sqrt {b}-a b\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )\right )-\frac {1}{2} \left (\sqrt {a} \sqrt {b}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a} \sqrt {b}-\left (a^{3/2} \sqrt {b}+a b\right ) x^4} \, 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 )+\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 {1}{4} \text {Subst}\left (\int \frac {1}{1-\sqrt [4]{a} \sqrt {\sqrt {a}-\sqrt {b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )-\frac {1}{4} \text {Subst}\left (\int \frac {1}{1+\sqrt [4]{a} \sqrt {\sqrt {a}-\sqrt {b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )-\frac {1}{4} \text {Subst}\left (\int \frac {1}{1-\sqrt [4]{a} \sqrt {\sqrt {a}+\sqrt {b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )-\frac {1}{4} \text {Subst}\left (\int \frac {1}{1+\sqrt [4]{a} \sqrt {\sqrt {a}+\sqrt {b}} 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 {\tan ^{-1}\left (\frac {\sqrt [8]{a} \sqrt [4]{\sqrt {a}-\sqrt {b}} x}{\sqrt [4]{b+a x^4}}\right )}{4 \sqrt [8]{a} \sqrt [4]{\sqrt {a}-\sqrt {b}}}-\frac {\tan ^{-1}\left (\frac {\sqrt [8]{a} \sqrt [4]{\sqrt {a}+\sqrt {b}} x}{\sqrt [4]{b+a x^4}}\right )}{4 \sqrt [8]{a} \sqrt [4]{\sqrt {a}+\sqrt {b}}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt [4]{a}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [8]{a} \sqrt [4]{\sqrt {a}-\sqrt {b}} x}{\sqrt [4]{b+a x^4}}\right )}{4 \sqrt [8]{a} \sqrt [4]{\sqrt {a}-\sqrt {b}}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [8]{a} \sqrt [4]{\sqrt {a}+\sqrt {b}} x}{\sqrt [4]{b+a x^4}}\right )}{4 \sqrt [8]{a} \sqrt [4]{\sqrt {a}+\sqrt {b}}}\\ \end {align*}
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Mathematica [A]
time = 0.83, size = 100, normalized size = 0.95 \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}{8} \text {RootSum}\left [a^2-a b-2 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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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {2 a \,x^{8}-b}{\left (a \,x^{4}+b \right )^{\frac {1}{4}} \left (a \,x^{8}-b \right )}\, dx\]
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 = 0.42, size = 1077, normalized size = 10.26 \begin {gather*} -\frac {1}{2} \, \left (-\frac {{\left (a - b\right )} \sqrt {\frac {b}{a^{3} - 2 \, a^{2} b + a b^{2}}} - 1}{a - b}\right )^{\frac {1}{4}} \arctan \left (\frac {{\left (x \sqrt {\frac {{\left ({\left (a^{2} - a b\right )} x^{2} \sqrt {\frac {b}{a^{3} - 2 \, a^{2} b + a b^{2}}} + a x^{2}\right )} \sqrt {-\frac {{\left (a - b\right )} \sqrt {\frac {b}{a^{3} - 2 \, a^{2} b + a b^{2}}} - 1}{a - b}} + \sqrt {a x^{4} + b}}{x^{2}}} - {\left (a x^{4} + b\right )}^{\frac {1}{4}}\right )} \left (-\frac {{\left (a - b\right )} \sqrt {\frac {b}{a^{3} - 2 \, a^{2} b + a b^{2}}} - 1}{a - b}\right )^{\frac {1}{4}}}{x}\right ) - \frac {1}{2} \, \left (\frac {{\left (a - b\right )} \sqrt {\frac {b}{a^{3} - 2 \, a^{2} b + a b^{2}}} + 1}{a - b}\right )^{\frac {1}{4}} \arctan \left (\frac {x \left (\frac {{\left (a - b\right )} \sqrt {\frac {b}{a^{3} - 2 \, a^{2} b + a b^{2}}} + 1}{a - b}\right )^{\frac {1}{4}} \sqrt {-\frac {{\left ({\left (a^{2} - a b\right )} x^{2} \sqrt {\frac {b}{a^{3} - 2 \, a^{2} b + a b^{2}}} - a x^{2}\right )} \sqrt {\frac {{\left (a - b\right )} \sqrt {\frac {b}{a^{3} - 2 \, a^{2} b + a b^{2}}} + 1}{a - b}} - \sqrt {a x^{4} + b}}{x^{2}}} - {\left (a x^{4} + b\right )}^{\frac {1}{4}} \left (\frac {{\left (a - b\right )} \sqrt {\frac {b}{a^{3} - 2 \, a^{2} b + a b^{2}}} + 1}{a - b}\right )^{\frac {1}{4}}}{x}\right ) + \frac {1}{8} \, \left (\frac {{\left (a - b\right )} \sqrt {\frac {b}{a^{3} - 2 \, a^{2} b + a b^{2}}} + 1}{a - b}\right )^{\frac {1}{4}} \log \left (\frac {{\left ({\left (a^{2} - a b\right )} x \sqrt {\frac {b}{a^{3} - 2 \, a^{2} b + a b^{2}}} - a x\right )} \left (\frac {{\left (a - b\right )} \sqrt {\frac {b}{a^{3} - 2 \, a^{2} b + a b^{2}}} + 1}{a - b}\right )^{\frac {3}{4}} + {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{8} \, \left (\frac {{\left (a - b\right )} \sqrt {\frac {b}{a^{3} - 2 \, a^{2} b + a b^{2}}} + 1}{a - b}\right )^{\frac {1}{4}} \log \left (-\frac {{\left ({\left (a^{2} - a b\right )} x \sqrt {\frac {b}{a^{3} - 2 \, a^{2} b + a b^{2}}} - a x\right )} \left (\frac {{\left (a - b\right )} \sqrt {\frac {b}{a^{3} - 2 \, a^{2} b + a b^{2}}} + 1}{a - b}\right )^{\frac {3}{4}} - {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{8} \, \left (-\frac {{\left (a - b\right )} \sqrt {\frac {b}{a^{3} - 2 \, a^{2} b + a b^{2}}} - 1}{a - b}\right )^{\frac {1}{4}} \log \left (\frac {{\left ({\left (a^{2} - a b\right )} x \sqrt {\frac {b}{a^{3} - 2 \, a^{2} b + a b^{2}}} + a x\right )} \left (-\frac {{\left (a - b\right )} \sqrt {\frac {b}{a^{3} - 2 \, a^{2} b + a b^{2}}} - 1}{a - b}\right )^{\frac {3}{4}} + {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{8} \, \left (-\frac {{\left (a - b\right )} \sqrt {\frac {b}{a^{3} - 2 \, a^{2} b + a b^{2}}} - 1}{a - b}\right )^{\frac {1}{4}} \log \left (-\frac {{\left ({\left (a^{2} - a b\right )} x \sqrt {\frac {b}{a^{3} - 2 \, a^{2} b + a b^{2}}} + a x\right )} \left (-\frac {{\left (a - b\right )} \sqrt {\frac {b}{a^{3} - 2 \, a^{2} b + a b^{2}}} - 1}{a - b}\right )^{\frac {3}{4}} - {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}\right ) + \frac {2 \, \arctan \left (\frac {\frac {x \sqrt {\frac {\sqrt {a} x^{2} + \sqrt {a x^{4} + b}}{x^{2}}}}{a^{\frac {1}{4}}} - \frac {{\left (a x^{4} + b\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}}}}{x}\right )}{a^{\frac {1}{4}}} + \frac {\log \left (\frac {a^{\frac {1}{4}} x + {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}\right )}{2 \, a^{\frac {1}{4}}} - \frac {\log \left (-\frac {a^{\frac {1}{4}} x - {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}\right )}{2 \, a^{\frac {1}{4}}} \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^{8} - b}{\sqrt [4]{a x^{4} + b} \left (a x^{8} - b\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^8}{{\left (a\,x^4+b\right )}^{1/4}\,\left (b-a\,x^8\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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