Optimal. Leaf size=116 \[ \frac {\text {RootSum}\left [\text {$\#$1}^8-2 \text {$\#$1}^4 a-\text {$\#$1}^4+a^2-a b+a\& ,\frac {\text {$\#$1}^4 \log \left (\sqrt [4]{a x^4-b}-\text {$\#$1} x\right )+\text {$\#$1}^4 (-\log (x))-a \log \left (\sqrt [4]{a x^4-b}-\text {$\#$1} x\right )+a \log (x)}{2 \text {$\#$1}^5-2 \text {$\#$1} a-\text {$\#$1}}\& \right ]}{4 b} \]
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Rubi [B] time = 0.65, antiderivative size = 447, normalized size of antiderivative = 3.85, number of steps used = 9, number of rules used = 5, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1428, 377, 212, 208, 205} \begin {gather*} \frac {a^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a} x \sqrt [4]{-\sqrt {4 a b+1}-2 b+1}}{\sqrt [4]{1-\sqrt {4 a b+1}} \sqrt [4]{a x^4-b}}\right )}{\sqrt {4 a b+1} \left (1-\sqrt {4 a b+1}\right )^{3/4} \sqrt [4]{-\sqrt {4 a b+1}-2 b+1}}-\frac {a^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a} x \sqrt [4]{\sqrt {4 a b+1}-2 b+1}}{\sqrt [4]{\sqrt {4 a b+1}+1} \sqrt [4]{a x^4-b}}\right )}{\sqrt {4 a b+1} \left (\sqrt {4 a b+1}+1\right )^{3/4} \sqrt [4]{\sqrt {4 a b+1}-2 b+1}}+\frac {a^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} x \sqrt [4]{-\sqrt {4 a b+1}-2 b+1}}{\sqrt [4]{1-\sqrt {4 a b+1}} \sqrt [4]{a x^4-b}}\right )}{\sqrt {4 a b+1} \left (1-\sqrt {4 a b+1}\right )^{3/4} \sqrt [4]{-\sqrt {4 a b+1}-2 b+1}}-\frac {a^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} x \sqrt [4]{\sqrt {4 a b+1}-2 b+1}}{\sqrt [4]{\sqrt {4 a b+1}+1} \sqrt [4]{a x^4-b}}\right )}{\sqrt {4 a b+1} \left (\sqrt {4 a b+1}+1\right )^{3/4} \sqrt [4]{\sqrt {4 a b+1}-2 b+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 208
Rule 212
Rule 377
Rule 1428
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [4]{-b+a x^4} \left (-b-x^4+a x^8\right )} \, dx &=\frac {(2 a) \int \frac {1}{\sqrt [4]{-b+a x^4} \left (-1-\sqrt {1+4 a b}+2 a x^4\right )} \, dx}{\sqrt {1+4 a b}}-\frac {(2 a) \int \frac {1}{\sqrt [4]{-b+a x^4} \left (-1+\sqrt {1+4 a b}+2 a x^4\right )} \, dx}{\sqrt {1+4 a b}}\\ &=\frac {(2 a) \operatorname {Subst}\left (\int \frac {1}{-1-\sqrt {1+4 a b}-\left (2 a b+a \left (-1-\sqrt {1+4 a b}\right )\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{\sqrt {1+4 a b}}-\frac {(2 a) \operatorname {Subst}\left (\int \frac {1}{-1+\sqrt {1+4 a b}-\left (2 a b+a \left (-1+\sqrt {1+4 a b}\right )\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{\sqrt {1+4 a b}}\\ &=\frac {a \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\sqrt {1+4 a b}}-\sqrt {a} \sqrt {1-2 b-\sqrt {1+4 a b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{\sqrt {1+4 a b} \sqrt {1-\sqrt {1+4 a b}}}+\frac {a \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\sqrt {1+4 a b}}+\sqrt {a} \sqrt {1-2 b-\sqrt {1+4 a b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{\sqrt {1+4 a b} \sqrt {1-\sqrt {1+4 a b}}}-\frac {a \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\sqrt {1+4 a b}}-\sqrt {a} \sqrt {1-2 b+\sqrt {1+4 a b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{\sqrt {1+4 a b} \sqrt {1+\sqrt {1+4 a b}}}-\frac {a \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\sqrt {1+4 a b}}+\sqrt {a} \sqrt {1-2 b+\sqrt {1+4 a b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{\sqrt {1+4 a b} \sqrt {1+\sqrt {1+4 a b}}}\\ &=\frac {a^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{1-2 b-\sqrt {1+4 a b}} x}{\sqrt [4]{1-\sqrt {1+4 a b}} \sqrt [4]{-b+a x^4}}\right )}{\sqrt {1+4 a b} \left (1-\sqrt {1+4 a b}\right )^{3/4} \sqrt [4]{1-2 b-\sqrt {1+4 a b}}}-\frac {a^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{1-2 b+\sqrt {1+4 a b}} x}{\sqrt [4]{1+\sqrt {1+4 a b}} \sqrt [4]{-b+a x^4}}\right )}{\sqrt {1+4 a b} \left (1+\sqrt {1+4 a b}\right )^{3/4} \sqrt [4]{1-2 b+\sqrt {1+4 a b}}}+\frac {a^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{1-2 b-\sqrt {1+4 a b}} x}{\sqrt [4]{1-\sqrt {1+4 a b}} \sqrt [4]{-b+a x^4}}\right )}{\sqrt {1+4 a b} \left (1-\sqrt {1+4 a b}\right )^{3/4} \sqrt [4]{1-2 b-\sqrt {1+4 a b}}}-\frac {a^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{1-2 b+\sqrt {1+4 a b}} x}{\sqrt [4]{1+\sqrt {1+4 a b}} \sqrt [4]{-b+a x^4}}\right )}{\sqrt {1+4 a b} \left (1+\sqrt {1+4 a b}\right )^{3/4} \sqrt [4]{1-2 b+\sqrt {1+4 a b}}}\\ \end {align*}
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Mathematica [B] time = 0.15, size = 404, normalized size = 3.48 \begin {gather*} -\frac {a^{3/4} \left (-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x \sqrt [4]{-\sqrt {4 a b+1}-2 b+1}}{\sqrt [4]{1-\sqrt {4 a b+1}} \sqrt [4]{a x^4-b}}\right )}{\left (1-\sqrt {4 a b+1}\right )^{3/4} \sqrt [4]{-\sqrt {4 a b+1}-2 b+1}}+\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x \sqrt [4]{\sqrt {4 a b+1}-2 b+1}}{\sqrt [4]{\sqrt {4 a b+1}+1} \sqrt [4]{a x^4-b}}\right )}{\left (\sqrt {4 a b+1}+1\right )^{3/4} \sqrt [4]{\sqrt {4 a b+1}-2 b+1}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x \sqrt [4]{-\sqrt {4 a b+1}-2 b+1}}{\sqrt [4]{1-\sqrt {4 a b+1}} \sqrt [4]{a x^4-b}}\right )}{\left (1-\sqrt {4 a b+1}\right )^{3/4} \sqrt [4]{-\sqrt {4 a b+1}-2 b+1}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x \sqrt [4]{\sqrt {4 a b+1}-2 b+1}}{\sqrt [4]{\sqrt {4 a b+1}+1} \sqrt [4]{a x^4-b}}\right )}{\left (\sqrt {4 a b+1}+1\right )^{3/4} \sqrt [4]{\sqrt {4 a b+1}-2 b+1}}\right )}{\sqrt {4 a b+1}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.00, size = 116, normalized size = 1.00 \begin {gather*} -\frac {\text {RootSum}\left [a+a^2-a b-\text {$\#$1}^4-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-a \log (x)+a \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}-2 a \text {$\#$1}+2 \text {$\#$1}^5}\&\right ]}{4 b} \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 {1}{{\left (a x^{8} - x^{4} - 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 = 0.00, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a \,x^{4}-b \right )^{\frac {1}{4}} \left (a \,x^{8}-x^{4}-b \right )}\, dx\]
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*} \int \frac {1}{{\left (a x^{8} - x^{4} - 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 {1}{{\left (a\,x^4-b\right )}^{1/4}\,\left (-a\,x^8+x^4+b\right )} \,d x \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 {1}{\sqrt [4]{a x^{4} - b} \left (a x^{8} - b - x^{4}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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