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 = 0.01, 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 = {281, 327, 211}
\begin {gather*} \frac {x^3}{3 b}-\frac {\sqrt {a} \text {ArcTan}\left (\frac {\sqrt {b} x^3}{\sqrt {a}}\right )}{3 b^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 281
Rule 327
Rubi steps
\begin {align*} \int \frac {x^8}{a+b x^6} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {x^2}{a+b x^2} \, dx,x,x^3\right )\\ &=\frac {x^3}{3 b}-\frac {a \text {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 = 0.01, size = 40, normalized size = 1.00 \begin {gather*} \frac {x^3}{3 b}-\frac {\sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} x^3}{\sqrt {a}}\right )}{3 b^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 32, normalized size = 0.80
method | result | size |
default | \(\frac {x^{3}}{3 b}-\frac {a \arctan \left (\frac {b \,x^{3}}{\sqrt {a b}}\right )}{3 b \sqrt {a b}}\) | \(32\) |
risch | \(\frac {x^{3}}{3 b}+\frac {\sqrt {-a b}\, \ln \left (b \,x^{3}-\sqrt {-a b}\right )}{6 b^{2}}-\frac {\sqrt {-a b}\, \ln \left (b \,x^{3}+\sqrt {-a b}\right )}{6 b^{2}}\) | \(60\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 31, normalized size = 0.78 \begin {gather*} \frac {x^{3}}{3 \, b} - \frac {a \arctan \left (\frac {b x^{3}}{\sqrt {a b}}\right )}{3 \, \sqrt {a b} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 89, normalized size = 2.22 \begin {gather*} \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 ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.11, size = 63, normalized size = 1.58 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.87, size = 31, normalized size = 0.78 \begin {gather*} \frac {x^{3}}{3 \, b} - \frac {a \arctan \left (\frac {b x^{3}}{\sqrt {a b}}\right )}{3 \, \sqrt {a b} b} \end {gather*}
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 \begin {gather*} \frac {x^3}{3\,b}-\frac {\sqrt {a}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x^3}{\sqrt {a}}\right )}{3\,b^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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