Optimal. Leaf size=49 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b} x^3}{\sqrt {a}}\right )}{6 \sqrt {a} b^{3/2}}-\frac {x^3}{6 b \left (a+b x^6\right )} \]
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Rubi [A] time = 0.02, antiderivative size = 49, 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, 288, 205} \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b} x^3}{\sqrt {a}}\right )}{6 \sqrt {a} b^{3/2}}-\frac {x^3}{6 b \left (a+b x^6\right )} \]
Antiderivative was successfully verified.
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Rule 205
Rule 275
Rule 288
Rubi steps
\begin {align*} \int \frac {x^8}{\left (a+b x^6\right )^2} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {x^2}{\left (a+b x^2\right )^2} \, dx,x,x^3\right )\\ &=-\frac {x^3}{6 b \left (a+b x^6\right )}+\frac {\operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,x^3\right )}{6 b}\\ &=-\frac {x^3}{6 b \left (a+b x^6\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {b} x^3}{\sqrt {a}}\right )}{6 \sqrt {a} b^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 49, normalized size = 1.00 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b} x^3}{\sqrt {a}}\right )}{6 \sqrt {a} b^{3/2}}-\frac {x^3}{6 b \left (a+b x^6\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 128, normalized size = 2.61 \[ \left [-\frac {2 \, a b x^{3} + {\left (b x^{6} + a\right )} \sqrt {-a b} \log \left (\frac {b x^{6} - 2 \, \sqrt {-a b} x^{3} - a}{b x^{6} + a}\right )}{12 \, {\left (a b^{3} x^{6} + a^{2} b^{2}\right )}}, -\frac {a b x^{3} - {\left (b x^{6} + a\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x^{3}}{a}\right )}{6 \, {\left (a b^{3} x^{6} + a^{2} b^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 39, normalized size = 0.80 \[ -\frac {x^{3}}{6 \, {\left (b x^{6} + a\right )} b} + \frac {\arctan \left (\frac {b x^{3}}{\sqrt {a b}}\right )}{6 \, \sqrt {a b} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 40, normalized size = 0.82 \[ -\frac {x^{3}}{6 \left (b \,x^{6}+a \right ) b}+\frac {\arctan \left (\frac {b \,x^{3}}{\sqrt {a b}}\right )}{6 \sqrt {a b}\, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.40, size = 40, normalized size = 0.82 \[ -\frac {x^{3}}{6 \, {\left (b^{2} x^{6} + a b\right )}} + \frac {\arctan \left (\frac {b x^{3}}{\sqrt {a b}}\right )}{6 \, \sqrt {a b} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 37, normalized size = 0.76 \[ \frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x^3}{\sqrt {a}}\right )}{6\,\sqrt {a}\,b^{3/2}}-\frac {x^3}{6\,b\,\left (b\,x^6+a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.61, size = 83, normalized size = 1.69 \[ - \frac {x^{3}}{6 a b + 6 b^{2} x^{6}} - \frac {\sqrt {- \frac {1}{a b^{3}}} \log {\left (- a b \sqrt {- \frac {1}{a b^{3}}} + x^{3} \right )}}{12} + \frac {\sqrt {- \frac {1}{a b^{3}}} \log {\left (a b \sqrt {- \frac {1}{a b^{3}}} + x^{3} \right )}}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
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