Optimal. Leaf size=29 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b} x^4}{\sqrt {a}}\right )}{4 \sqrt {a} \sqrt {b}} \]
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Rubi [A] time = 0.01, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {275, 205} \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b} x^4}{\sqrt {a}}\right )}{4 \sqrt {a} \sqrt {b}} \]
Antiderivative was successfully verified.
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Rule 205
Rule 275
Rubi steps
\begin {align*} \int \frac {x^3}{a+b x^8} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,x^4\right )\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {b} x^4}{\sqrt {a}}\right )}{4 \sqrt {a} \sqrt {b}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 29, normalized size = 1.00 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b} x^4}{\sqrt {a}}\right )}{4 \sqrt {a} \sqrt {b}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 72, normalized size = 2.48 \[ \left [-\frac {\sqrt {-a b} \log \left (\frac {b x^{8} - 2 \, \sqrt {-a b} x^{4} - a}{b x^{8} + a}\right )}{8 \, a b}, -\frac {\sqrt {a b} \arctan \left (\frac {\sqrt {a b}}{b x^{4}}\right )}{4 \, a b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 18, normalized size = 0.62 \[ \frac {\arctan \left (\frac {b x^{4}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 19, normalized size = 0.66 \[ \frac {\arctan \left (\frac {b \,x^{4}}{\sqrt {a b}}\right )}{4 \sqrt {a b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.34, size = 18, normalized size = 0.62 \[ \frac {\arctan \left (\frac {b x^{4}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 19, normalized size = 0.66 \[ \frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x^4}{\sqrt {a}}\right )}{4\,\sqrt {a}\,\sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.42, size = 56, normalized size = 1.93 \[ - \frac {\sqrt {- \frac {1}{a b}} \log {\left (- a \sqrt {- \frac {1}{a b}} + x^{4} \right )}}{8} + \frac {\sqrt {- \frac {1}{a b}} \log {\left (a \sqrt {- \frac {1}{a b}} + x^{4} \right )}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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