Optimal. Leaf size=251 \[ \frac {\left (-a x^2-b\right ) \left (a x^4-b x^2\right )^{3/4}}{4 b^2 x (b-a) \left (b-a x^4\right )}-\frac {\text {RootSum}\left [\text {$\#$1}^8-2 \text {$\#$1}^4 a+a^2-a b\& ,\frac {-8 \text {$\#$1}^4 a \log \left (\sqrt [4]{a x^4-b x^2}-\text {$\#$1} x\right )+7 \text {$\#$1}^4 b \log \left (\sqrt [4]{a x^4-b x^2}-\text {$\#$1} x\right )+8 \text {$\#$1}^4 a \log (x)-7 \text {$\#$1}^4 b \log (x)+8 a^2 \log \left (\sqrt [4]{a x^4-b x^2}-\text {$\#$1} x\right )-6 a b \log \left (\sqrt [4]{a x^4-b x^2}-\text {$\#$1} x\right )-8 a^2 \log (x)+6 a b \log (x)}{\text {$\#$1} a-\text {$\#$1}^5}\& \right ]}{32 b^2 (a-b)} \]
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Rubi [F] time = 0.11, 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 {1}{\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 {1}{\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 {1}{\sqrt {x} \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 {1}{\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 = 1.80, size = 644, normalized size = 2.57 \begin {gather*} \frac {\left (a x^2-b\right ) \left (\frac {7 b \sqrt [4]{\sqrt {b}-\sqrt {a}} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {b}{x^2}-a}}{\sqrt [8]{a} \sqrt [4]{-\sqrt {a}-\sqrt {b}}}\right )+\sqrt {a} \sqrt {b} \sqrt [4]{\sqrt {b}-\sqrt {a}} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {b}{x^2}-a}}{\sqrt [8]{a} \sqrt [4]{-\sqrt {a}-\sqrt {b}}}\right )-8 a \sqrt [4]{\sqrt {b}-\sqrt {a}} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {b}{x^2}-a}}{\sqrt [8]{a} \sqrt [4]{-\sqrt {a}-\sqrt {b}}}\right )+7 b \sqrt [4]{-\sqrt {a}-\sqrt {b}} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {b}{x^2}-a}}{\sqrt [8]{a} \sqrt [4]{\sqrt {b}-\sqrt {a}}}\right )-\sqrt {a} \sqrt {b} \sqrt [4]{-\sqrt {a}-\sqrt {b}} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {b}{x^2}-a}}{\sqrt [8]{a} \sqrt [4]{\sqrt {b}-\sqrt {a}}}\right )-8 a \sqrt [4]{-\sqrt {a}-\sqrt {b}} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {b}{x^2}-a}}{\sqrt [8]{a} \sqrt [4]{\sqrt {b}-\sqrt {a}}}\right )+\sqrt [4]{\sqrt {b}-\sqrt {a}} \left (-\sqrt {a} \sqrt {b}+8 a-7 b\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {b}{x^2}-a}}{\sqrt [8]{a} \sqrt [4]{-\sqrt {a}-\sqrt {b}}}\right )+\sqrt [4]{-\sqrt {a}-\sqrt {b}} \left (\sqrt {a} \sqrt {b}+8 a-7 b\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {b}{x^2}-a}}{\sqrt [8]{a} \sqrt [4]{\sqrt {b}-\sqrt {a}}}\right )}{\sqrt [8]{a} \sqrt [4]{-\sqrt {a}-\sqrt {b}} \sqrt [4]{\sqrt {b}-\sqrt {a}} \left (\frac {b}{x^2}-a\right )^{3/4}}+\frac {4 x^2 \left (a x^2+b\right )}{a x^4-b}\right )}{16 b^2 x (b-a) \sqrt [4]{a x^4-b x^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.00, size = 233, normalized size = 0.93 \begin {gather*} \frac {\left (-b-a x^2\right ) \left (-b x^2+a x^4\right )^{3/4}}{4 b^2 (-a+b) x \left (b-a x^4\right )}-\frac {\text {RootSum}\left [a^2-a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{4 b^2}+\frac {\text {RootSum}\left [a^2-a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-2 a \log (x)+2 a \log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-a \text {$\#$1}+\text {$\#$1}^5}\&\right ]}{32 (a-b) 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^{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 \[\int \frac {1}{\left (a \,x^{4}-b \right )^{2} \left (a \,x^{4}-b \,x^{2}\right )^{\frac {1}{4}}}\, dx\]
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 {1}{{\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 = -0.00 \begin {gather*} \int \frac {1}{{\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] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [4]{x^{2} \left (a x^{2} - b\right )} \left (a x^{4} - b\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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