Optimal. Leaf size=65 \[ \frac {1}{8} \text {RootSum}\left [\text {$\#$1}^8-2 \text {$\#$1}^4 a+a^2-a b\& ,\frac {\log \left (\sqrt [4]{a x^4-b}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1} a-\text {$\#$1}^5}\& \right ] \]
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Rubi [B] time = 0.26, antiderivative size = 269, normalized size of antiderivative = 4.14, number of steps used = 10, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {1529, 377, 212, 206, 203} \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt [8]{a} x \sqrt [4]{\sqrt {a}-\sqrt {b}}}{\sqrt [4]{a x^4-b}}\right )}{4 a^{5/8} \sqrt {b} \sqrt [4]{\sqrt {a}-\sqrt {b}}}+\frac {\tan ^{-1}\left (\frac {\sqrt [8]{a} x \sqrt [4]{\sqrt {a}+\sqrt {b}}}{\sqrt [4]{a x^4-b}}\right )}{4 a^{5/8} \sqrt {b} \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 a^{5/8} \sqrt {b} \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 a^{5/8} \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 1529
Rubi steps
\begin {align*} \int \frac {x^4}{\sqrt [4]{-b+a x^4} \left (-b+a x^8\right )} \, dx &=\int \left (-\frac {1}{2 \sqrt {a} \left (\sqrt {b}-\sqrt {a} x^4\right ) \sqrt [4]{-b+a x^4}}+\frac {1}{2 \sqrt {a} \left (\sqrt {b}+\sqrt {a} x^4\right ) \sqrt [4]{-b+a x^4}}\right ) \, dx\\ &=-\frac {\int \frac {1}{\left (\sqrt {b}-\sqrt {a} x^4\right ) \sqrt [4]{-b+a x^4}} \, dx}{2 \sqrt {a}}+\frac {\int \frac {1}{\left (\sqrt {b}+\sqrt {a} x^4\right ) \sqrt [4]{-b+a x^4}} \, dx}{2 \sqrt {a}}\\ &=-\frac {\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 )}{2 \sqrt {a}}+\frac {\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 )}{2 \sqrt {a}}\\ &=-\frac {\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 )}{4 \sqrt {a} \sqrt {b}}-\frac {\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 )}{4 \sqrt {a} \sqrt {b}}+\frac {\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 )}{4 \sqrt {a} \sqrt {b}}+\frac {\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 )}{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 a^{5/8} \sqrt [4]{\sqrt {a}-\sqrt {b}} \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 a^{5/8} \sqrt [4]{\sqrt {a}+\sqrt {b}} \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 a^{5/8} \sqrt [4]{\sqrt {a}-\sqrt {b}} \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 a^{5/8} \sqrt [4]{\sqrt {a}+\sqrt {b}} \sqrt {b}}\\ \end {align*}
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Mathematica [B] time = 0.20, size = 265, normalized size = 4.08 \begin {gather*} \frac {-\sqrt [4]{\sqrt {a}+\sqrt {b}} \tan ^{-1}\left (\frac {\sqrt [8]{a} x \sqrt [4]{\sqrt {a}-\sqrt {b}}}{\sqrt [4]{a x^4-b}}\right )+\sqrt [4]{\sqrt {a}-\sqrt {b}} \tan ^{-1}\left (\frac {\sqrt [8]{a} x \sqrt [4]{\sqrt {a}+\sqrt {b}}}{\sqrt [4]{a x^4-b}}\right )-\sqrt [4]{\sqrt {a}+\sqrt {b}} \tanh ^{-1}\left (\frac {\sqrt [8]{a} x \sqrt [4]{\sqrt {a}-\sqrt {b}}}{\sqrt [4]{a x^4-b}}\right )+\sqrt [4]{\sqrt {a}-\sqrt {b}} \tanh ^{-1}\left (\frac {\sqrt [8]{a} x \sqrt [4]{\sqrt {a}+\sqrt {b}}}{\sqrt [4]{a x^4-b}}\right )}{4 a^{5/8} \sqrt {b} \sqrt [4]{\sqrt {a}-\sqrt {b}} \sqrt [4]{\sqrt {a}+\sqrt {b}}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.50, size = 64, normalized size = 0.98 \begin {gather*} \frac {1}{8} \text {RootSum}\left [a^2-a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {\log (x)-\log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right )}{-a \text {$\#$1}+\text {$\#$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 {x^{4}}{{\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 = 0.03, size = 0, normalized size = 0.00 \[\int \frac {x^{4}}{\left (a \,x^{4}-b \right )^{\frac {1}{4}} \left (a \,x^{8}-b \right )}\, dx\]
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 {x^{4}}{{\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 = -0.02 \begin {gather*} -\int \frac {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 {x^{4}}{\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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