Optimal. Leaf size=67 \[ -\frac {1}{4} \text {RootSum}\left [a^2-a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\& ,\frac {-\log (x)+\log \left (\sqrt [4]{b x^2+a x^4}-x \text {$\#$1}\right )}{a \text {$\#$1}-\text {$\#$1}^5}\& \right ] \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(401\) vs. \(2(67)=134\).
time = 0.37, antiderivative size = 401, normalized size of antiderivative = 5.99, number of steps
used = 12, number of rules used = 7, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {2081, 1284,
1543, 385, 218, 212, 209} \begin {gather*} \frac {\sqrt {x} \sqrt [4]{a x^2+b} \text {ArcTan}\left (\frac {\sqrt [8]{a} \sqrt {x} \sqrt [4]{\sqrt {a}-\sqrt {b}}}{\sqrt [4]{a x^2+b}}\right )}{2 a^{5/8} \sqrt {b} \sqrt [4]{\sqrt {a}-\sqrt {b}} \sqrt [4]{a x^4+b x^2}}-\frac {\sqrt {x} \sqrt [4]{a x^2+b} \text {ArcTan}\left (\frac {\sqrt [8]{a} \sqrt {x} \sqrt [4]{\sqrt {a}+\sqrt {b}}}{\sqrt [4]{a x^2+b}}\right )}{2 a^{5/8} \sqrt {b} \sqrt [4]{\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 {a}-\sqrt {b}}}{\sqrt [4]{a x^2+b}}\right )}{2 a^{5/8} \sqrt {b} \sqrt [4]{\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 {a}+\sqrt {b}}}{\sqrt [4]{a x^2+b}}\right )}{2 a^{5/8} \sqrt {b} \sqrt [4]{\sqrt {a}+\sqrt {b}} \sqrt [4]{a x^4+b x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 212
Rule 218
Rule 385
Rule 1284
Rule 1543
Rule 2081
Rubi steps
\begin {align*} \int \frac {x^2}{\left (-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 {x^{3/2}}{\sqrt [4]{b+a x^2} \left (-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 ) \text {Subst}\left (\int \frac {x^4}{\sqrt [4]{b+a x^4} \left (-b+a x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^4}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\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,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^4}}\\ &=-\frac {\left (\sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {b}-\sqrt {a} x^4\right ) \sqrt [4]{b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {a} \sqrt [4]{b x^2+a x^4}}+\frac {\left (\sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {b}+\sqrt {a} x^4\right ) \sqrt [4]{b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {a} \sqrt [4]{b x^2+a x^4}}\\ &=\frac {\left (\sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b}-\left (a \sqrt {b}-\sqrt {a} b\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt {a} \sqrt [4]{b x^2+a x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b}-\left (a \sqrt {b}+\sqrt {a} b\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt {a} \sqrt [4]{b x^2+a x^4}}\\ &=\frac {\left (\sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt [4]{a} \sqrt {\sqrt {a}-\sqrt {b}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt {a} \sqrt {b} \sqrt [4]{b x^2+a x^4}}+\frac {\left (\sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt [4]{a} \sqrt {\sqrt {a}-\sqrt {b}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt {a} \sqrt {b} \sqrt [4]{b x^2+a x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt [4]{a} \sqrt {\sqrt {a}+\sqrt {b}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt {a} \sqrt {b} \sqrt [4]{b x^2+a x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt [4]{a} \sqrt {\sqrt {a}+\sqrt {b}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt {a} \sqrt {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 {a}-\sqrt {b}} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 a^{5/8} \sqrt [4]{\sqrt {a}-\sqrt {b}} \sqrt {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 {a}+\sqrt {b}} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 a^{5/8} \sqrt [4]{\sqrt {a}+\sqrt {b}} \sqrt {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 {a}-\sqrt {b}} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 a^{5/8} \sqrt [4]{\sqrt {a}-\sqrt {b}} \sqrt {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 {a}+\sqrt {b}} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 a^{5/8} \sqrt [4]{\sqrt {a}+\sqrt {b}} \sqrt {b} \sqrt [4]{b x^2+a x^4}}\\ \end {align*}
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Mathematica [A]
time = 0.38, size = 102, normalized size = 1.52 \begin {gather*} -\frac {\sqrt {x} \sqrt [4]{b+a x^2} \text {RootSum}\left [a^2-a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log \left (\sqrt {x}\right )+\log \left (\sqrt [4]{b+a x^2}-\sqrt {x} \text {$\#$1}\right )}{a \text {$\#$1}-\text {$\#$1}^5}\&\right ]}{4 \sqrt [4]{x^2 \left (b+a x^2\right )}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.34, size = 0, normalized size = 0.00 \[\int \frac {x^{2}}{\left (a \,x^{4}-b \right ) \left (a \,x^{4}+b \,x^{2}\right )^{\frac {1}{4}}}\, dx\]
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{\sqrt [4]{x^{2} \left (a x^{2} + b\right )} \left (a x^{4} - b\right )}\, dx \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*} \text {could not integrate} \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^2}{\left (b-a\,x^4\right )\,{\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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