Optimal. Leaf size=55 \[ -\frac {\text {RootSum}\left [2 a^2-2 a \text {$\#$1}^4+\text {$\#$1}^8\& ,\frac {-\log (x)+\log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right )}{\text {$\#$1}^3}\& \right ]}{8 b^2} \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(325\) vs. \(2(55)=110\).
time = 0.53, antiderivative size = 325, normalized size of antiderivative = 5.91, number of steps
used = 10, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1543, 508,
304, 211} \begin {gather*} -\frac {\left (-a^2\right )^{3/8} \text {ArcTan}\left (\frac {\sqrt [4]{a} \sqrt [4]{\sqrt {-a^2}-a} x}{\sqrt [8]{-a^2} \sqrt [4]{a x^4+b}}\right )}{4 a^{3/4} \left (\sqrt {-a^2}-a\right )^{3/4} b^2}+\frac {\sqrt [4]{a} \text {ArcTan}\left (\frac {\left (-a^2\right )^{3/8} \sqrt [4]{\sqrt {-a^2}-a} x}{a^{3/4} \sqrt [4]{a x^4+b}}\right )}{4 \sqrt [8]{-a^2} \left (\sqrt {-a^2}-a\right )^{3/4} b^2}-\frac {\left (-a^2\right )^{3/8} \text {ArcTan}\left (\frac {\sqrt [4]{a} \sqrt [4]{\sqrt {-a^2}+a} x}{\sqrt [8]{-a^2} \sqrt [4]{a x^4+b}}\right )}{4 a^{3/4} \left (\sqrt {-a^2}+a\right )^{3/4} b^2}+\frac {\sqrt [4]{a} \text {ArcTan}\left (\frac {\left (-a^2\right )^{3/8} \sqrt [4]{\sqrt {-a^2}+a} x}{a^{3/4} \sqrt [4]{a x^4+b}}\right )}{4 \sqrt [8]{-a^2} \left (\sqrt {-a^2}+a\right )^{3/4} b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 304
Rule 508
Rule 1543
Rubi steps
\begin {align*} \int \frac {x^2}{\left (b+a x^4\right )^{3/4} \left (b^2+a^2 x^8\right )} \, dx &=\int \left (-\frac {a^2 x^2}{2 \sqrt {-a^2} b \left (b+a x^4\right )^{3/4} \left (\sqrt {-a^2} b-a^2 x^4\right )}-\frac {a^2 x^2}{2 \sqrt {-a^2} b \left (b+a x^4\right )^{3/4} \left (\sqrt {-a^2} b+a^2 x^4\right )}\right ) \, dx\\ &=\frac {\sqrt {-a^2} \int \frac {x^2}{\left (b+a x^4\right )^{3/4} \left (\sqrt {-a^2} b-a^2 x^4\right )} \, dx}{2 b}+\frac {\sqrt {-a^2} \int \frac {x^2}{\left (b+a x^4\right )^{3/4} \left (\sqrt {-a^2} b+a^2 x^4\right )} \, dx}{2 b}\\ &=\frac {\sqrt {-a^2} \text {Subst}\left (\int \frac {x^2}{\sqrt {-a^2} b-\left (-a^2 b+a \sqrt {-a^2} b\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{2 b}+\frac {\sqrt {-a^2} \text {Subst}\left (\int \frac {x^2}{\sqrt {-a^2} b-\left (a^2 b+a \sqrt {-a^2} b\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{2 b}\\ &=\frac {\sqrt {-a^2} \text {Subst}\left (\int \frac {1}{\sqrt [4]{-a^2}-\sqrt {a} \sqrt {-a+\sqrt {-a^2}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{4 \sqrt {a} \sqrt {-a+\sqrt {-a^2}} b^2}-\frac {\sqrt {-a^2} \text {Subst}\left (\int \frac {1}{\sqrt [4]{-a^2}+\sqrt {a} \sqrt {-a+\sqrt {-a^2}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{4 \sqrt {a} \sqrt {-a+\sqrt {-a^2}} b^2}+\frac {\sqrt {-a^2} \text {Subst}\left (\int \frac {1}{\sqrt [4]{-a^2}-\sqrt {a} \sqrt {a+\sqrt {-a^2}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{4 \sqrt {a} \sqrt {a+\sqrt {-a^2}} b^2}-\frac {\sqrt {-a^2} \text {Subst}\left (\int \frac {1}{\sqrt [4]{-a^2}+\sqrt {a} \sqrt {a+\sqrt {-a^2}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{4 \sqrt {a} \sqrt {a+\sqrt {-a^2}} b^2}\\ &=-\frac {\left (-a^2\right )^{3/8} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{-a+\sqrt {-a^2}} x}{\sqrt [8]{-a^2} \sqrt [4]{b+a x^4}}\right )}{4 a^{3/4} \left (-a+\sqrt {-a^2}\right )^{3/4} b^2}+\frac {\sqrt [4]{a} \tan ^{-1}\left (\frac {\left (-a^2\right )^{3/8} \sqrt [4]{-a+\sqrt {-a^2}} x}{a^{3/4} \sqrt [4]{b+a x^4}}\right )}{4 \sqrt [8]{-a^2} \left (-a+\sqrt {-a^2}\right )^{3/4} b^2}-\frac {\left (-a^2\right )^{3/8} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{a+\sqrt {-a^2}} x}{\sqrt [8]{-a^2} \sqrt [4]{b+a x^4}}\right )}{4 a^{3/4} \left (a+\sqrt {-a^2}\right )^{3/4} b^2}+\frac {\sqrt [4]{a} \tan ^{-1}\left (\frac {\left (-a^2\right )^{3/8} \sqrt [4]{a+\sqrt {-a^2}} x}{a^{3/4} \sqrt [4]{b+a x^4}}\right )}{4 \sqrt [8]{-a^2} \left (a+\sqrt {-a^2}\right )^{3/4} b^2}\\ \end {align*}
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Mathematica [A]
time = 0.30, size = 55, normalized size = 1.00 \begin {gather*} -\frac {\text {RootSum}\left [2 a^2-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right )}{\text {$\#$1}^3}\&\right ]}{8 b^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.32, size = 0, normalized size = 0.00 \[\int \frac {x^{2}}{\left (a \,x^{4}+b \right )^{\frac {3}{4}} \left (a^{2} x^{8}+b^{2}\right )}\, dx\]
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*} \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}}{\left (a x^{4} + b\right )^{\frac {3}{4}} \left (a^{2} x^{8} + b^{2}\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 = -0.02 \begin {gather*} \int \frac {x^2}{\left (a^2\,x^8+b^2\right )\,{\left (a\,x^4+b\right )}^{3/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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