Optimal. Leaf size=65 \[ \frac {\text {RootSum}\left [2 \text {$\#$1}^8-4 \text {$\#$1}^4 a+2 a^2-a b\& ,\frac {\log \left (\sqrt [4]{a x^4+b x^2}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}}\& \right ]}{8 b} \]
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Rubi [B] time = 0.41, antiderivative size = 481, normalized size of antiderivative = 7.40, number of steps used = 11, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2056, 1270, 1429, 377, 212, 206, 203} \begin {gather*} -\frac {\sqrt {x} \sqrt [4]{a x^2+b} \tan ^{-1}\left (\frac {\sqrt [8]{a} \sqrt {x} \sqrt [4]{\sqrt {2} \sqrt {a}-\sqrt {b}}}{\sqrt [8]{2} \sqrt [4]{a x^2+b}}\right )}{2\ 2^{7/8} \sqrt [8]{a} b \sqrt [4]{\sqrt {2} \sqrt {a}-\sqrt {b}} \sqrt [4]{a x^4+b x^2}}-\frac {\sqrt {x} \sqrt [4]{a x^2+b} \tan ^{-1}\left (\frac {\sqrt [8]{a} \sqrt {x} \sqrt [4]{\sqrt {2} \sqrt {a}+\sqrt {b}}}{\sqrt [8]{2} \sqrt [4]{a x^2+b}}\right )}{2\ 2^{7/8} \sqrt [8]{a} b \sqrt [4]{\sqrt {2} \sqrt {a}+\sqrt {b}} \sqrt [4]{a x^4+b x^2}}-\frac {\sqrt {x} \sqrt [4]{a x^2+b} \tanh ^{-1}\left (\frac {\sqrt [8]{a} \sqrt {x} \sqrt [4]{\sqrt {2} \sqrt {a}-\sqrt {b}}}{\sqrt [8]{2} \sqrt [4]{a x^2+b}}\right )}{2\ 2^{7/8} \sqrt [8]{a} b \sqrt [4]{\sqrt {2} \sqrt {a}-\sqrt {b}} \sqrt [4]{a x^4+b x^2}}-\frac {\sqrt {x} \sqrt [4]{a x^2+b} \tanh ^{-1}\left (\frac {\sqrt [8]{a} \sqrt {x} \sqrt [4]{\sqrt {2} \sqrt {a}+\sqrt {b}}}{\sqrt [8]{2} \sqrt [4]{a x^2+b}}\right )}{2\ 2^{7/8} \sqrt [8]{a} b \sqrt [4]{\sqrt {2} \sqrt {a}+\sqrt {b}} \sqrt [4]{a x^4+b x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 212
Rule 377
Rule 1270
Rule 1429
Rule 2056
Rubi steps
\begin {align*} \int \frac {1}{\left (-2 b+a x^4\right ) \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 (-2 b+a x^4\right )} \, 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 (-2 b+a x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^4}}\\ &=-\frac {\left (\sqrt {a} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt {2} \sqrt {a} \sqrt {b}-a x^4\right ) \sqrt [4]{b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {2} \sqrt {b} \sqrt [4]{b x^2+a x^4}}-\frac {\left (\sqrt {a} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt {2} \sqrt {a} \sqrt {b}+a x^4\right ) \sqrt [4]{b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {2} \sqrt {b} \sqrt [4]{b x^2+a x^4}}\\ &=-\frac {\left (\sqrt {a} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2} \sqrt {a} \sqrt {b}-\left (\sqrt {2} a^{3/2} \sqrt {b}-a b\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt {2} \sqrt {b} \sqrt [4]{b x^2+a x^4}}-\frac {\left (\sqrt {a} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2} \sqrt {a} \sqrt {b}-\left (\sqrt {2} a^{3/2} \sqrt {b}+a b\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt {2} \sqrt {b} \sqrt [4]{b x^2+a x^4}}\\ &=-\frac {\left (\sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{2}-\sqrt [4]{a} \sqrt {\sqrt {2} \sqrt {a}-\sqrt {b}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2\ 2^{3/4} b \sqrt [4]{b x^2+a x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{2}+\sqrt [4]{a} \sqrt {\sqrt {2} \sqrt {a}-\sqrt {b}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2\ 2^{3/4} b \sqrt [4]{b x^2+a x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{2}-\sqrt [4]{a} \sqrt {\sqrt {2} \sqrt {a}+\sqrt {b}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2\ 2^{3/4} b \sqrt [4]{b x^2+a x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{2}+\sqrt [4]{a} \sqrt {\sqrt {2} \sqrt {a}+\sqrt {b}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2\ 2^{3/4} b \sqrt [4]{b x^2+a x^4}}\\ &=-\frac {\sqrt {x} \sqrt [4]{b+a x^2} \tan ^{-1}\left (\frac {\sqrt [8]{a} \sqrt [4]{\sqrt {2} \sqrt {a}-\sqrt {b}} \sqrt {x}}{\sqrt [8]{2} \sqrt [4]{b+a x^2}}\right )}{2\ 2^{7/8} \sqrt [8]{a} \sqrt [4]{\sqrt {2} \sqrt {a}-\sqrt {b}} b \sqrt [4]{b x^2+a x^4}}-\frac {\sqrt {x} \sqrt [4]{b+a x^2} \tan ^{-1}\left (\frac {\sqrt [8]{a} \sqrt [4]{\sqrt {2} \sqrt {a}+\sqrt {b}} \sqrt {x}}{\sqrt [8]{2} \sqrt [4]{b+a x^2}}\right )}{2\ 2^{7/8} \sqrt [8]{a} \sqrt [4]{\sqrt {2} \sqrt {a}+\sqrt {b}} b \sqrt [4]{b x^2+a x^4}}-\frac {\sqrt {x} \sqrt [4]{b+a x^2} \tanh ^{-1}\left (\frac {\sqrt [8]{a} \sqrt [4]{\sqrt {2} \sqrt {a}-\sqrt {b}} \sqrt {x}}{\sqrt [8]{2} \sqrt [4]{b+a x^2}}\right )}{2\ 2^{7/8} \sqrt [8]{a} \sqrt [4]{\sqrt {2} \sqrt {a}-\sqrt {b}} b \sqrt [4]{b x^2+a x^4}}-\frac {\sqrt {x} \sqrt [4]{b+a x^2} \tanh ^{-1}\left (\frac {\sqrt [8]{a} \sqrt [4]{\sqrt {2} \sqrt {a}+\sqrt {b}} \sqrt {x}}{\sqrt [8]{2} \sqrt [4]{b+a x^2}}\right )}{2\ 2^{7/8} \sqrt [8]{a} \sqrt [4]{\sqrt {2} \sqrt {a}+\sqrt {b}} b \sqrt [4]{b x^2+a x^4}}\\ \end {align*}
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Mathematica [B] time = 0.40, size = 350, normalized size = 5.38 \begin {gather*} -\frac {x \sqrt [4]{a+\frac {b}{x^2}} \left (-\sqrt [4]{2 \sqrt {a}+\sqrt {2} \sqrt {b}} \tan ^{-1}\left (\frac {\sqrt [4]{a+\frac {b}{x^2}}}{\sqrt [8]{a} \sqrt [4]{\sqrt {a}-\frac {\sqrt {b}}{\sqrt {2}}}}\right )-\sqrt [4]{2 \sqrt {a}-\sqrt {2} \sqrt {b}} \tan ^{-1}\left (\frac {\sqrt [4]{a+\frac {b}{x^2}}}{\sqrt [8]{a} \sqrt [4]{\sqrt {a}+\frac {\sqrt {b}}{\sqrt {2}}}}\right )+\sqrt [4]{2 \sqrt {a}+\sqrt {2} \sqrt {b}} \tanh ^{-1}\left (\frac {\sqrt [4]{a+\frac {b}{x^2}}}{\sqrt [8]{a} \sqrt [4]{\sqrt {a}-\frac {\sqrt {b}}{\sqrt {2}}}}\right )+\sqrt [4]{2 \sqrt {a}-\sqrt {2} \sqrt {b}} \tanh ^{-1}\left (\frac {\sqrt [4]{a+\frac {b}{x^2}}}{\sqrt [8]{a} \sqrt [4]{\sqrt {a}+\frac {\sqrt {b}}{\sqrt {2}}}}\right )\right )}{2\ 2^{3/4} \sqrt [8]{a} b \sqrt [4]{2 \sqrt {a}-\sqrt {2} \sqrt {b}} \sqrt [4]{2 \sqrt {a}+\sqrt {2} \sqrt {b}} \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 = 65, normalized size = 1.00 \begin {gather*} \frac {\text {RootSum}\left [2 a^2-a b-4 a \text {$\#$1}^4+2 \text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt [4]{b x^2+a x^4}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{8 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} - 2 \, b\right )}}\,{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 \[\int \frac {1}{\left (a \,x^{4}-2 b \right ) \left (a \,x^{4}+b \,x^{2}\right )^{\frac {1}{4}}}\, dx\]
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} - 2 \, b\right )}}\,{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 {1}{{\left (a\,x^4+b\,x^2\right )}^{1/4}\,\left (2\,b-a\,x^4\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]{x^{2} \left (a x^{2} + b\right )} \left (a x^{4} - 2 b\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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