Optimal. Leaf size=101 \[ \frac {1}{8} \text {RootSum}\left [\text {$\#$1}^8-2 \text {$\#$1}^4 a+a^2-a b\& ,\frac {\text {$\#$1}^4 \log \left (\sqrt [4]{a x^4-b}-\text {$\#$1} x\right )+\text {$\#$1}^4 (-\log (x))+a \log \left (\sqrt [4]{a x^4-b}-\text {$\#$1} x\right )-a \log (x)}{\text {$\#$1} a-\text {$\#$1}^5}\& \right ] \]
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Rubi [B] time = 0.65, antiderivative size = 325, normalized size of antiderivative = 3.22, 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 = {6725, 377, 212, 206, 203} \begin {gather*} -\frac {\left (2 \sqrt {a}-\sqrt {b}\right ) \tan ^{-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 {b} \sqrt [4]{\sqrt {a}-\sqrt {b}}}+\frac {\left (2 \sqrt {a}+\sqrt {b}\right ) \tan ^{-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 {b} \sqrt [4]{\sqrt {a}+\sqrt {b}}}-\frac {\left (2 \sqrt {a}-\sqrt {b}\right ) \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 {b} \sqrt [4]{\sqrt {a}-\sqrt {b}}}+\frac {\left (2 \sqrt {a}+\sqrt {b}\right ) \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 {b} \sqrt [4]{\sqrt {a}+\sqrt {b}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 212
Rule 377
Rule 6725
Rubi steps
\begin {align*} \int \frac {-b+2 a x^4}{\sqrt [4]{-b+a x^4} \left (-b+a x^8\right )} \, dx &=\int \left (-\frac {2 a \sqrt {b}-\sqrt {a} b}{2 \sqrt {a} \sqrt {b} \left (\sqrt {b}-\sqrt {a} x^4\right ) \sqrt [4]{-b+a x^4}}+\frac {2 a \sqrt {b}+\sqrt {a} b}{2 \sqrt {a} \sqrt {b} \left (\sqrt {b}+\sqrt {a} x^4\right ) \sqrt [4]{-b+a x^4}}\right ) \, dx\\ &=-\left (\frac {1}{2} \left (2 \sqrt {a}-\sqrt {b}\right ) \int \frac {1}{\left (\sqrt {b}-\sqrt {a} x^4\right ) \sqrt [4]{-b+a x^4}} \, dx\right )+\frac {1}{2} \left (2 \sqrt {a}+\sqrt {b}\right ) \int \frac {1}{\left (\sqrt {b}+\sqrt {a} x^4\right ) \sqrt [4]{-b+a x^4}} \, dx\\ &=-\left (\frac {1}{2} \left (2 \sqrt {a}-\sqrt {b}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}-\left (a \sqrt {b}-\sqrt {a} b\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )\right )+\frac {1}{2} \left (2 \sqrt {a}+\sqrt {b}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}-\left (a \sqrt {b}+\sqrt {a} b\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )\\ &=-\left (\frac {1}{4} \left (-1+\frac {2 \sqrt {a}}{\sqrt {b}}\right ) \operatorname {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 )\right )-\frac {1}{4} \left (-1+\frac {2 \sqrt {a}}{\sqrt {b}}\right ) \operatorname {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} \left (1+\frac {2 \sqrt {a}}{\sqrt {b}}\right ) \operatorname {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} \left (1+\frac {2 \sqrt {a}}{\sqrt {b}}\right ) \operatorname {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 {\left (1-\frac {2 \sqrt {a}}{\sqrt {b}}\right ) \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 {\left (1+\frac {2 \sqrt {a}}{\sqrt {b}}\right ) \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 {\left (1-\frac {2 \sqrt {a}}{\sqrt {b}}\right ) \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 {\left (1+\frac {2 \sqrt {a}}{\sqrt {b}}\right ) \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 [F] time = 0.59, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-b+2 a x^4}{\sqrt [4]{-b+a x^4} \left (-b+a x^8\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 1.50, size = 100, normalized size = 0.99 \begin {gather*} \frac {1}{8} \text {RootSum}\left [a^2-a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {a \log (x)-a \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-a \text {$\#$1}+\text {$\#$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 (a x^{8} - 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 = 0.04, size = 0, normalized size = 0.00 \[\int \frac {2 a \,x^{4}-b}{\left (a \,x^{4}-b \right )^{\frac {1}{4}} \left (a \,x^{8}-b \right )}\, dx\]
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*} \int \frac {2 \, a x^{4} - b}{{\left (a x^{8} - 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 {b-2\,a\,x^4}{{\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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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 a x^{4} - 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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