Optimal. Leaf size=171 \[ \frac {\text {RootSum}\left [\text {$\#$1}^8-2 \text {$\#$1}^4 a+a^2-a b\& ,\frac {\text {$\#$1}^4 \log \left (\sqrt [4]{a x^4+b x^2}-\text {$\#$1} x\right )+\text {$\#$1}^4 (-\log (x))-2 a \log \left (\sqrt [4]{a x^4+b x^2}-\text {$\#$1} x\right )+2 a \log (x)}{\text {$\#$1} a-\text {$\#$1}^5}\& \right ]}{32 a (a-b)}+\frac {\left (a x^4+b x^2\right )^{3/4} \left (b-a x^2\right )}{4 a b x (a-b) \left (a x^4-b\right )} \]
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Rubi [F] time = 0.23, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {x^4}{\left (-b+a x^4\right )^2 \sqrt [4]{b x^2+a x^4}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {x^4}{\left (-b+a x^4\right )^2 \sqrt [4]{b x^2+a x^4}} \, dx &=\frac {\left (\sqrt {x} \sqrt [4]{b+a x^2}\right ) \int \frac {x^{7/2}}{\sqrt [4]{b+a x^2} \left (-b+a x^4\right )^2} \, dx}{\sqrt [4]{b x^2+a x^4}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {x^8}{\sqrt [4]{b+a x^4} \left (-b+a x^8\right )^2} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^4}}\\ \end {align*}
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Mathematica [B] time = 1.36, size = 445, normalized size = 2.60 \begin {gather*} \frac {x \left (\frac {4 \sqrt [8]{a} \left (a^2 x^4-b^2\right )}{a x^4-b}+\frac {\sqrt {b} \sqrt [4]{a+\frac {b}{x^2}} \left (-\sqrt {a} \sqrt [4]{\sqrt {a}-\sqrt {b}} \tan ^{-1}\left (\frac {\sqrt [4]{a+\frac {b}{x^2}}}{\sqrt [8]{a} \sqrt [4]{\sqrt {a}+\sqrt {b}}}\right )+\sqrt {b} \sqrt [4]{\sqrt {a}-\sqrt {b}} \tan ^{-1}\left (\frac {\sqrt [4]{a+\frac {b}{x^2}}}{\sqrt [8]{a} \sqrt [4]{\sqrt {a}+\sqrt {b}}}\right )+\sqrt {b} \sqrt [4]{\sqrt {a}+\sqrt {b}} \tan ^{-1}\left (\frac {\sqrt [4]{a+\frac {b}{x^2}}}{\sqrt [8]{a} \sqrt [4]{\sqrt {a}-\sqrt {b}}}\right )+\sqrt {a} \sqrt [4]{\sqrt {a}+\sqrt {b}} \tan ^{-1}\left (\frac {\sqrt [4]{a+\frac {b}{x^2}}}{\sqrt [8]{a} \sqrt [4]{\sqrt {a}-\sqrt {b}}}\right )+\left (\sqrt {a}-\sqrt {b}\right )^{5/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a+\frac {b}{x^2}}}{\sqrt [8]{a} \sqrt [4]{\sqrt {a}+\sqrt {b}}}\right )-\left (\sqrt {a}+\sqrt {b}\right )^{5/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a+\frac {b}{x^2}}}{\sqrt [8]{a} \sqrt [4]{\sqrt {a}-\sqrt {b}}}\right )\right )}{\sqrt [4]{\sqrt {a}-\sqrt {b}} \sqrt [4]{\sqrt {a}+\sqrt {b}}}\right )}{16 a^{9/8} b (b-a) \sqrt [4]{x^2 \left (a x^2+b\right )}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.00, size = 170, normalized size = 0.99 \begin {gather*} \frac {\left (b-a x^2\right ) \left (b x^2+a x^4\right )^{3/4}}{4 a (a-b) b x \left (-b+a x^4\right )}+\frac {\text {RootSum}\left [a^2-a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-2 a \log (x)+2 a \log \left (\sqrt [4]{b x^2+a x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-a \text {$\#$1}+\text {$\#$1}^5}\&\right ]}{32 a (a-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 {x^{4}}{{\left (a x^{4} + b x^{2}\right )}^{\frac {1}{4}} {\left (a x^{4} - b\right )}^{2}}\,{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 \[\int \frac {x^{4}}{\left (a \,x^{4}-b \right )^{2} \left (a \,x^{4}+b \,x^{2}\right )^{\frac {1}{4}}}\, dx\]
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^{4} + b x^{2}\right )}^{\frac {1}{4}} {\left (a x^{4} - b\right )}^{2}}\,{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 {x^4}{{\left (b-a\,x^4\right )}^2\,{\left (a\,x^4+b\,x^2\right )}^{1/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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