Optimal. Leaf size=123 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{\sqrt {3} \sqrt [3]{a}}\right )}{2 \sqrt {3} \sqrt [3]{a} b^{2/3}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{6 \sqrt [3]{a} b^{2/3}}+\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{12 \sqrt [3]{a} b^{2/3}} \]
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Rubi [A]
time = 0.06, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {281, 298, 31,
648, 631, 210, 642} \begin {gather*} \frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{12 \sqrt [3]{a} b^{2/3}}-\frac {\text {ArcTan}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{\sqrt {3} \sqrt [3]{a}}\right )}{2 \sqrt {3} \sqrt [3]{a} b^{2/3}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{6 \sqrt [3]{a} b^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 210
Rule 281
Rule 298
Rule 631
Rule 642
Rule 648
Rubi steps
\begin {align*} \int \frac {x^3}{a+b x^6} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x}{a+b x^3} \, dx,x,x^2\right )\\ &=-\frac {\text {Subst}\left (\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,x^2\right )}{6 \sqrt [3]{a} \sqrt [3]{b}}+\frac {\text {Subst}\left (\int \frac {\sqrt [3]{a}+\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,x^2\right )}{6 \sqrt [3]{a} \sqrt [3]{b}}\\ &=-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{6 \sqrt [3]{a} b^{2/3}}+\frac {\text {Subst}\left (\int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,x^2\right )}{12 \sqrt [3]{a} b^{2/3}}+\frac {\text {Subst}\left (\int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,x^2\right )}{4 \sqrt [3]{b}}\\ &=-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{6 \sqrt [3]{a} b^{2/3}}+\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{12 \sqrt [3]{a} b^{2/3}}+\frac {\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x^2}{\sqrt [3]{a}}\right )}{2 \sqrt [3]{a} b^{2/3}}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{\sqrt {3} \sqrt [3]{a}}\right )}{2 \sqrt {3} \sqrt [3]{a} b^{2/3}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{6 \sqrt [3]{a} b^{2/3}}+\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{12 \sqrt [3]{a} b^{2/3}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 154, normalized size = 1.25 \begin {gather*} \frac {-2 \sqrt {3} \tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )-2 \sqrt {3} \tan ^{-1}\left (\sqrt {3}+\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )-2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )+\log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )+\log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{12 \sqrt [3]{a} b^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 97, normalized size = 0.79
method | result | size |
risch | \(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{3} b^{2} a +1\right )}{\sum }\textit {\_R} \ln \left (-b \,x^{2} \textit {\_R} +1\right )\right )}{6}\) | \(29\) |
default | \(-\frac {\ln \left (x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{4}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{2}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{12 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{2}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\) | \(97\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 104, normalized size = 0.85 \begin {gather*} \frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{2} - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{6 \, b \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {\log \left (x^{4} - x^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{12 \, b \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {\log \left (x^{2} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{6 \, b \left (\frac {a}{b}\right )^{\frac {1}{3}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 318, normalized size = 2.59 \begin {gather*} \left [\frac {3 \, \sqrt {\frac {1}{3}} a b \sqrt {\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} \log \left (\frac {2 \, b^{2} x^{6} - 3 \, \left (-a b^{2}\right )^{\frac {2}{3}} x^{2} - a b + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, \left (-a b^{2}\right )^{\frac {2}{3}} x^{4} + a b x^{2} + \left (-a b^{2}\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}}}{b x^{6} + a}\right ) + \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b^{2} x^{4} + \left (-a b^{2}\right )^{\frac {1}{3}} b x^{2} + \left (-a b^{2}\right )^{\frac {2}{3}}\right ) - 2 \, \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b x^{2} - \left (-a b^{2}\right )^{\frac {1}{3}}\right )}{12 \, a b^{2}}, \frac {6 \, \sqrt {\frac {1}{3}} a b \sqrt {-\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, b x^{2} + \left (-a b^{2}\right )^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}}}{b}\right ) + \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b^{2} x^{4} + \left (-a b^{2}\right )^{\frac {1}{3}} b x^{2} + \left (-a b^{2}\right )^{\frac {2}{3}}\right ) - 2 \, \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b x^{2} - \left (-a b^{2}\right )^{\frac {1}{3}}\right )}{12 \, a b^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.08, size = 26, normalized size = 0.21 \begin {gather*} \operatorname {RootSum} {\left (216 t^{3} a b^{2} + 1, \left ( t \mapsto t \log {\left (36 t^{2} a b + x^{2} \right )} \right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.70, size = 118, normalized size = 0.96 \begin {gather*} -\frac {\left (-\frac {a}{b}\right )^{\frac {2}{3}} \log \left ({\left | x^{2} - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{6 \, a} - \frac {\sqrt {3} \left (-a b^{2}\right )^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{2} + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{6 \, a b^{2}} + \frac {\left (-a b^{2}\right )^{\frac {2}{3}} \log \left (x^{4} + x^{2} \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{12 \, a b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.31, size = 118, normalized size = 0.96 \begin {gather*} \frac {\ln \left (a+{\left (-a\right )}^{2/3}\,b^{1/3}\,x^2\right )}{6\,{\left (-a\right )}^{1/3}\,b^{2/3}}+\frac {\ln \left (a\,b^2+\frac {{\left (-a\right )}^{2/3}\,b^{7/3}\,x^2\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{12\,{\left (-a\right )}^{1/3}\,b^{2/3}}-\frac {\ln \left (a\,b^2-\frac {{\left (-a\right )}^{2/3}\,b^{7/3}\,x^2\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{12\,{\left (-a\right )}^{1/3}\,b^{2/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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