Optimal. Leaf size=83 \[ -\frac {\text {RootSum}\left [a^5+b^5-4 a^4 \text {$\#$1}^4+6 a^3 \text {$\#$1}^8-4 a^2 \text {$\#$1}^{12}+a \text {$\#$1}^{16}\& ,\frac {-\log (x)+\log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right )}{\text {$\#$1}}\& \right ]}{8 a} \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(993\) vs. \(2(83)=166\).
time = 1.30, antiderivative size = 993, normalized size of antiderivative = 11.96, number of steps
used = 22, number of rules used = 8, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2081, 6847,
6857, 1443, 385, 218, 214, 211} \begin {gather*} -\frac {\sqrt {x} \sqrt [4]{a x^2-b} \text {ArcTan}\left (\frac {\sqrt [4]{\sqrt {-\sqrt {-a}} a-b^{5/4}} \sqrt {x}}{\sqrt [8]{-\sqrt {-a}} \sqrt [4]{a x^2-b}}\right )}{4 \left (-\sqrt {-a}\right )^{15/8} \sqrt [4]{\sqrt {-\sqrt {-a}} a-b^{5/4}} \sqrt [4]{a x^4-b x^2}}-\frac {\sqrt {x} \sqrt [4]{a x^2-b} \text {ArcTan}\left (\frac {\sqrt [4]{\sqrt [4]{-a} a-b^{5/4}} \sqrt {x}}{\sqrt [16]{-a} \sqrt [4]{a x^2-b}}\right )}{4 (-a)^{15/16} \sqrt [4]{\sqrt [4]{-a} a-b^{5/4}} \sqrt [4]{a x^4-b x^2}}-\frac {\sqrt {x} \sqrt [4]{a x^2-b} \text {ArcTan}\left (\frac {\sqrt [4]{b^{5/4}+\sqrt {-\sqrt {-a}} a} \sqrt {x}}{\sqrt [8]{-\sqrt {-a}} \sqrt [4]{a x^2-b}}\right )}{4 \left (-\sqrt {-a}\right )^{15/8} \sqrt [4]{b^{5/4}+\sqrt {-\sqrt {-a}} a} \sqrt [4]{a x^4-b x^2}}-\frac {\sqrt {x} \sqrt [4]{a x^2-b} \text {ArcTan}\left (\frac {\sqrt [4]{b^{5/4}+\sqrt [4]{-a} a} \sqrt {x}}{\sqrt [16]{-a} \sqrt [4]{a x^2-b}}\right )}{4 (-a)^{15/16} \sqrt [4]{b^{5/4}+\sqrt [4]{-a} a} \sqrt [4]{a x^4-b x^2}}-\frac {\sqrt {x} \sqrt [4]{a x^2-b} \tanh ^{-1}\left (\frac {\sqrt [4]{\sqrt {-\sqrt {-a}} a-b^{5/4}} \sqrt {x}}{\sqrt [8]{-\sqrt {-a}} \sqrt [4]{a x^2-b}}\right )}{4 \left (-\sqrt {-a}\right )^{15/8} \sqrt [4]{\sqrt {-\sqrt {-a}} a-b^{5/4}} \sqrt [4]{a x^4-b x^2}}-\frac {\sqrt {x} \sqrt [4]{a x^2-b} \tanh ^{-1}\left (\frac {\sqrt [4]{\sqrt [4]{-a} a-b^{5/4}} \sqrt {x}}{\sqrt [16]{-a} \sqrt [4]{a x^2-b}}\right )}{4 (-a)^{15/16} \sqrt [4]{\sqrt [4]{-a} a-b^{5/4}} \sqrt [4]{a x^4-b x^2}}-\frac {\sqrt {x} \sqrt [4]{a x^2-b} \tanh ^{-1}\left (\frac {\sqrt [4]{b^{5/4}+\sqrt {-\sqrt {-a}} a} \sqrt {x}}{\sqrt [8]{-\sqrt {-a}} \sqrt [4]{a x^2-b}}\right )}{4 \left (-\sqrt {-a}\right )^{15/8} \sqrt [4]{b^{5/4}+\sqrt {-\sqrt {-a}} a} \sqrt [4]{a x^4-b x^2}}-\frac {\sqrt {x} \sqrt [4]{a x^2-b} \tanh ^{-1}\left (\frac {\sqrt [4]{b^{5/4}+\sqrt [4]{-a} a} \sqrt {x}}{\sqrt [16]{-a} \sqrt [4]{a x^2-b}}\right )}{4 (-a)^{15/16} \sqrt [4]{b^{5/4}+\sqrt [4]{-a} a} \sqrt [4]{a x^4-b x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 214
Rule 218
Rule 385
Rule 1443
Rule 2081
Rule 6847
Rule 6857
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [4]{-b x^2+a x^4} \left (a+b x^8\right )} \, dx &=\frac {\left (\sqrt {x} \sqrt [4]{-b+a x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt [4]{-b+a x^2} \left (a+b x^8\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 {1}{\sqrt [4]{-b+a x^4} \left (a+b x^{16}\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 {\sqrt {-a}}{2 a \sqrt [4]{-b+a x^4} \left (\sqrt {-a}-\sqrt {b} x^8\right )}+\frac {\sqrt {-a}}{2 a \sqrt [4]{-b+a x^4} \left (\sqrt {-a}+\sqrt {b} x^8\right )}\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}{\sqrt [4]{-b+a x^4} \left (\sqrt {-a}-\sqrt {b} x^8\right )} \, 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 [4]{-b+a x^4} \left (\sqrt {-a}+\sqrt {b} x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {-a} \sqrt [4]{-b x^2+a x^4}}\\ &=-\frac {\left (\sqrt [4]{b} \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-b+a x^4} \left (\sqrt [4]{-a} \sqrt [4]{b}-\sqrt {b} x^4\right )} \, dx,x,\sqrt {x}\right )}{2 (-a)^{3/4} \sqrt [4]{-b x^2+a x^4}}-\frac {\left (\sqrt [4]{b} \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-b+a x^4} \left (\sqrt [4]{-a} \sqrt [4]{b}+\sqrt {b} x^4\right )} \, dx,x,\sqrt {x}\right )}{2 (-a)^{3/4} \sqrt [4]{-b x^2+a x^4}}+\frac {\left (\sqrt [4]{b} \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-b+a x^4} \left (\sqrt {-\sqrt {-a}} \sqrt [4]{b}-\sqrt {b} x^4\right )} \, dx,x,\sqrt {x}\right )}{2 \sqrt {-\sqrt {-a}} \sqrt {-a} \sqrt [4]{-b x^2+a x^4}}+\frac {\left (\sqrt [4]{b} \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-b+a x^4} \left (\sqrt {-\sqrt {-a}} \sqrt [4]{b}+\sqrt {b} x^4\right )} \, dx,x,\sqrt {x}\right )}{2 \sqrt {-\sqrt {-a}} \sqrt {-a} \sqrt [4]{-b x^2+a x^4}}\\ &=-\frac {\left (\sqrt [4]{b} \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-a} \sqrt [4]{b}-\left (\sqrt [4]{-a} a \sqrt [4]{b}-b^{3/2}\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{2 (-a)^{3/4} \sqrt [4]{-b x^2+a x^4}}-\frac {\left (\sqrt [4]{b} \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-a} \sqrt [4]{b}-\left (\sqrt [4]{-a} a \sqrt [4]{b}+b^{3/2}\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{2 (-a)^{3/4} \sqrt [4]{-b x^2+a x^4}}+\frac {\left (\sqrt [4]{b} \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-\sqrt {-a}} \sqrt [4]{b}-\left (\sqrt {-\sqrt {-a}} a \sqrt [4]{b}-b^{3/2}\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{2 \sqrt {-\sqrt {-a}} \sqrt {-a} \sqrt [4]{-b x^2+a x^4}}+\frac {\left (\sqrt [4]{b} \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-\sqrt {-a}} \sqrt [4]{b}-\left (\sqrt {-\sqrt {-a}} a \sqrt [4]{b}+b^{3/2}\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{2 \sqrt {-\sqrt {-a}} \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 [8]{-a}-\sqrt {\sqrt [4]{-a} a-b^{5/4}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{4 (-a)^{7/8} \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 [8]{-a}+\sqrt {\sqrt [4]{-a} a-b^{5/4}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{4 (-a)^{7/8} \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 [8]{-a}-\sqrt {\sqrt [4]{-a} a+b^{5/4}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{4 (-a)^{7/8} \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 [8]{-a}+\sqrt {\sqrt [4]{-a} a+b^{5/4}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{4 (-a)^{7/8} \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 [4]{-\sqrt {-a}}-\sqrt {\sqrt {-\sqrt {-a}} a-b^{5/4}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{4 \left (-\sqrt {-a}\right )^{3/4} \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 [4]{-\sqrt {-a}}+\sqrt {\sqrt {-\sqrt {-a}} a-b^{5/4}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{4 \left (-\sqrt {-a}\right )^{3/4} \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 [4]{-\sqrt {-a}}-\sqrt {\sqrt {-\sqrt {-a}} a+b^{5/4}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{4 \left (-\sqrt {-a}\right )^{3/4} \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 [4]{-\sqrt {-a}}+\sqrt {\sqrt {-\sqrt {-a}} a+b^{5/4}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{4 \left (-\sqrt {-a}\right )^{3/4} \sqrt {-a} \sqrt [4]{-b x^2+a x^4}}\\ &=-\frac {\sqrt {x} \sqrt [4]{-b+a x^2} \tan ^{-1}\left (\frac {\sqrt [4]{\sqrt {-\sqrt {-a}} a-b^{5/4}} \sqrt {x}}{\sqrt [8]{-\sqrt {-a}} \sqrt [4]{-b+a x^2}}\right )}{4 \left (-\sqrt {-a}\right )^{15/8} \sqrt [4]{\sqrt {-\sqrt {-a}} a-b^{5/4}} \sqrt [4]{-b x^2+a x^4}}-\frac {\sqrt {x} \sqrt [4]{-b+a x^2} \tan ^{-1}\left (\frac {\sqrt [4]{\sqrt [4]{-a} a-b^{5/4}} \sqrt {x}}{\sqrt [16]{-a} \sqrt [4]{-b+a x^2}}\right )}{4 (-a)^{15/16} \sqrt [4]{\sqrt [4]{-a} a-b^{5/4}} \sqrt [4]{-b x^2+a x^4}}-\frac {\sqrt {x} \sqrt [4]{-b+a x^2} \tan ^{-1}\left (\frac {\sqrt [4]{\sqrt {-\sqrt {-a}} a+b^{5/4}} \sqrt {x}}{\sqrt [8]{-\sqrt {-a}} \sqrt [4]{-b+a x^2}}\right )}{4 \left (-\sqrt {-a}\right )^{15/8} \sqrt [4]{\sqrt {-\sqrt {-a}} a+b^{5/4}} \sqrt [4]{-b x^2+a x^4}}-\frac {\sqrt {x} \sqrt [4]{-b+a x^2} \tan ^{-1}\left (\frac {\sqrt [4]{\sqrt [4]{-a} a+b^{5/4}} \sqrt {x}}{\sqrt [16]{-a} \sqrt [4]{-b+a x^2}}\right )}{4 (-a)^{15/16} \sqrt [4]{\sqrt [4]{-a} a+b^{5/4}} \sqrt [4]{-b x^2+a x^4}}-\frac {\sqrt {x} \sqrt [4]{-b+a x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{\sqrt {-\sqrt {-a}} a-b^{5/4}} \sqrt {x}}{\sqrt [8]{-\sqrt {-a}} \sqrt [4]{-b+a x^2}}\right )}{4 \left (-\sqrt {-a}\right )^{15/8} \sqrt [4]{\sqrt {-\sqrt {-a}} a-b^{5/4}} \sqrt [4]{-b x^2+a x^4}}-\frac {\sqrt {x} \sqrt [4]{-b+a x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{\sqrt [4]{-a} a-b^{5/4}} \sqrt {x}}{\sqrt [16]{-a} \sqrt [4]{-b+a x^2}}\right )}{4 (-a)^{15/16} \sqrt [4]{\sqrt [4]{-a} a-b^{5/4}} \sqrt [4]{-b x^2+a x^4}}-\frac {\sqrt {x} \sqrt [4]{-b+a x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{\sqrt {-\sqrt {-a}} a+b^{5/4}} \sqrt {x}}{\sqrt [8]{-\sqrt {-a}} \sqrt [4]{-b+a x^2}}\right )}{4 \left (-\sqrt {-a}\right )^{15/8} \sqrt [4]{\sqrt {-\sqrt {-a}} a+b^{5/4}} \sqrt [4]{-b x^2+a x^4}}-\frac {\sqrt {x} \sqrt [4]{-b+a x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{\sqrt [4]{-a} a+b^{5/4}} \sqrt {x}}{\sqrt [16]{-a} \sqrt [4]{-b+a x^2}}\right )}{4 (-a)^{15/16} \sqrt [4]{\sqrt [4]{-a} a+b^{5/4}} \sqrt [4]{-b x^2+a x^4}}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 122, normalized size = 1.47 \begin {gather*} -\frac {\sqrt {x} \sqrt [4]{-b+a x^2} \text {RootSum}\left [a^5+b^5-4 a^4 \text {$\#$1}^4+6 a^3 \text {$\#$1}^8-4 a^2 \text {$\#$1}^{12}+a \text {$\#$1}^{16}\&,\frac {-\log \left (\sqrt {x}\right )+\log \left (\sqrt [4]{-b+a x^2}-\sqrt {x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{8 a \sqrt [4]{-b x^2+a x^4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a \,x^{4}-b \,x^{2}\right )^{\frac {1}{4}} \left (b \,x^{8}+a \right )}\, dx\]
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 {1}{\sqrt [4]{x^{2} \left (a x^{2} - b\right )} \left (a + b x^{8}\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 {1}{\left (b\,x^8+a\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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