Optimal. Leaf size=40 \[ \frac {x^3}{3 b}-\frac {\sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} x^3}{\sqrt {a}}\right )}{3 b^{3/2}} \]
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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, 321, 205} \[ \frac {x^3}{3 b}-\frac {\sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} x^3}{\sqrt {a}}\right )}{3 b^{3/2}} \]
Antiderivative was successfully verified.
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Rule 205
Rule 275
Rule 321
Rubi steps
\begin {align*} \int \frac {x^8}{a+b x^6} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {x^2}{a+b x^2} \, dx,x,x^3\right )\\ &=\frac {x^3}{3 b}-\frac {a \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,x^3\right )}{3 b}\\ &=\frac {x^3}{3 b}-\frac {\sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} x^3}{\sqrt {a}}\right )}{3 b^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 40, normalized size = 1.00 \[ \frac {x^3}{3 b}-\frac {\sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} x^3}{\sqrt {a}}\right )}{3 b^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 89, normalized size = 2.22 \[ \left [\frac {2 \, x^{3} + \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{6} - 2 \, b x^{3} \sqrt {-\frac {a}{b}} - a}{b x^{6} + a}\right )}{6 \, b}, \frac {x^{3} - \sqrt {\frac {a}{b}} \arctan \left (\frac {b x^{3} \sqrt {\frac {a}{b}}}{a}\right )}{3 \, b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 31, normalized size = 0.78 \[ \frac {x^{3}}{3 \, b} - \frac {a \arctan \left (\frac {b x^{3}}{\sqrt {a b}}\right )}{3 \, \sqrt {a b} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 32, normalized size = 0.80 \[ \frac {x^{3}}{3 b}-\frac {a \arctan \left (\frac {b \,x^{3}}{\sqrt {a b}}\right )}{3 \sqrt {a b}\, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.27, size = 31, normalized size = 0.78 \[ \frac {x^{3}}{3 \, b} - \frac {a \arctan \left (\frac {b x^{3}}{\sqrt {a b}}\right )}{3 \, \sqrt {a b} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.02, size = 28, normalized size = 0.70 \[ \frac {x^3}{3\,b}-\frac {\sqrt {a}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x^3}{\sqrt {a}}\right )}{3\,b^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.34, size = 63, normalized size = 1.58 \[ \frac {\sqrt {- \frac {a}{b^{3}}} \log {\left (- b \sqrt {- \frac {a}{b^{3}}} + x^{3} \right )}}{6} - \frac {\sqrt {- \frac {a}{b^{3}}} \log {\left (b \sqrt {- \frac {a}{b^{3}}} + x^{3} \right )}}{6} + \frac {x^{3}}{3 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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