Optimal. Leaf size=40 \[ -\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} x^4}{\sqrt {a}}\right )}{4 a^{3/2}}-\frac {1}{4 a x^4} \]
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Rubi [A] time = 0.02, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {275, 325, 205} \[ -\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} x^4}{\sqrt {a}}\right )}{4 a^{3/2}}-\frac {1}{4 a x^4} \]
Antiderivative was successfully verified.
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Rule 205
Rule 275
Rule 325
Rubi steps
\begin {align*} \int \frac {1}{x^5 \left (a+b x^8\right )} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{x^2 \left (a+b x^2\right )} \, dx,x,x^4\right )\\ &=-\frac {1}{4 a x^4}-\frac {b \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,x^4\right )}{4 a}\\ &=-\frac {1}{4 a x^4}-\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} x^4}{\sqrt {a}}\right )}{4 a^{3/2}}\\ \end {align*}
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Mathematica [B] time = 0.05, size = 164, normalized size = 4.10 \[ \frac {\sqrt {b} x^4 \tan ^{-1}\left (\frac {\sqrt [8]{b} x \sec \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}-\tan \left (\frac {\pi }{8}\right )\right )-\sqrt {b} x^4 \tan ^{-1}\left (\frac {\sqrt [8]{b} x \sec \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}+\tan \left (\frac {\pi }{8}\right )\right )+\sqrt {b} x^4 \tan ^{-1}\left (\cot \left (\frac {\pi }{8}\right )-\frac {\sqrt [8]{b} x \csc \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}\right )+\sqrt {b} x^4 \tan ^{-1}\left (\frac {\sqrt [8]{b} x \csc \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}+\cot \left (\frac {\pi }{8}\right )\right )-\sqrt {a}}{4 a^{3/2} x^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 94, normalized size = 2.35 \[ \left [\frac {x^{4} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{8} - 2 \, a x^{4} \sqrt {-\frac {b}{a}} - a}{b x^{8} + a}\right ) - 2}{8 \, a x^{4}}, \frac {x^{4} \sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {\frac {b}{a}}}{b x^{4}}\right ) - 1}{4 \, a x^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 31, normalized size = 0.78 \[ -\frac {b \arctan \left (\frac {b x^{4}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} a} - \frac {1}{4 \, a x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 32, normalized size = 0.80 \[ -\frac {b \arctan \left (\frac {b \,x^{4}}{\sqrt {a b}}\right )}{4 \sqrt {a b}\, a}-\frac {1}{4 a \,x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.21, size = 31, normalized size = 0.78 \[ -\frac {b \arctan \left (\frac {b x^{4}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} a} - \frac {1}{4 \, a x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 28, normalized size = 0.70 \[ -\frac {1}{4\,a\,x^4}-\frac {\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x^4}{\sqrt {a}}\right )}{4\,a^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.69, size = 71, normalized size = 1.78 \[ \frac {\sqrt {- \frac {b}{a^{3}}} \log {\left (- \frac {a^{2} \sqrt {- \frac {b}{a^{3}}}}{b} + x^{4} \right )}}{8} - \frac {\sqrt {- \frac {b}{a^{3}}} \log {\left (\frac {a^{2} \sqrt {- \frac {b}{a^{3}}}}{b} + x^{4} \right )}}{8} - \frac {1}{4 a x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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