Optimal. Leaf size=142 \[ -\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{18 a^{4/3} b^{2/3}}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{\sqrt {3} \sqrt [3]{a}}\right )}{6 \sqrt {3} a^{4/3} b^{2/3}}+\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{36 a^{4/3} b^{2/3}}+\frac {x^4}{6 a \left (a+b x^6\right )} \]
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Rubi [A] time = 0.11, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {275, 290, 292, 31, 634, 617, 204, 628} \[ -\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{18 a^{4/3} b^{2/3}}+\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{36 a^{4/3} b^{2/3}}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{\sqrt {3} \sqrt [3]{a}}\right )}{6 \sqrt {3} a^{4/3} b^{2/3}}+\frac {x^4}{6 a \left (a+b x^6\right )} \]
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 275
Rule 290
Rule 292
Rule 617
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {x^3}{\left (a+b x^6\right )^2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x}{\left (a+b x^3\right )^2} \, dx,x,x^2\right )\\ &=\frac {x^4}{6 a \left (a+b x^6\right )}+\frac {\operatorname {Subst}\left (\int \frac {x}{a+b x^3} \, dx,x,x^2\right )}{6 a}\\ &=\frac {x^4}{6 a \left (a+b x^6\right )}-\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,x^2\right )}{18 a^{4/3} \sqrt [3]{b}}+\frac {\operatorname {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 )}{18 a^{4/3} \sqrt [3]{b}}\\ &=\frac {x^4}{6 a \left (a+b x^6\right )}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{18 a^{4/3} b^{2/3}}+\frac {\operatorname {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 )}{36 a^{4/3} b^{2/3}}+\frac {\operatorname {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 )}{12 a \sqrt [3]{b}}\\ &=\frac {x^4}{6 a \left (a+b x^6\right )}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{18 a^{4/3} b^{2/3}}+\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{36 a^{4/3} b^{2/3}}+\frac {\operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x^2}{\sqrt [3]{a}}\right )}{6 a^{4/3} b^{2/3}}\\ &=\frac {x^4}{6 a \left (a+b x^6\right )}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{\sqrt {3} \sqrt [3]{a}}\right )}{6 \sqrt {3} a^{4/3} b^{2/3}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{18 a^{4/3} b^{2/3}}+\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{36 a^{4/3} b^{2/3}}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 195, normalized size = 1.37 \[ \frac {-\frac {2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{b^{2/3}}+\frac {\log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{b^{2/3}}+\frac {\log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{b^{2/3}}-\frac {2 \sqrt {3} \tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{b^{2/3}}-\frac {2 \sqrt {3} \tan ^{-1}\left (\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}+\sqrt {3}\right )}{b^{2/3}}+\frac {6 \sqrt [3]{a} x^4}{a+b x^6}}{36 a^{4/3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 416, normalized size = 2.93 \[ \left [\frac {6 \, a b^{2} x^{4} + 3 \, \sqrt {\frac {1}{3}} {\left (a b^{2} x^{6} + a^{2} b\right )} \sqrt {\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} \log \left (\frac {2 \, b^{2} x^{6} - 3 \, \left (-a b^{2}\right )^{\frac {2}{3}} x^{2} - a b + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, \left (-a b^{2}\right )^{\frac {2}{3}} x^{4} + a b x^{2} + \left (-a b^{2}\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}}}{b x^{6} + a}\right ) + {\left (b x^{6} + a\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b^{2} x^{4} + \left (-a b^{2}\right )^{\frac {1}{3}} b x^{2} + \left (-a b^{2}\right )^{\frac {2}{3}}\right ) - 2 \, {\left (b x^{6} + a\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b x^{2} - \left (-a b^{2}\right )^{\frac {1}{3}}\right )}{36 \, {\left (a^{2} b^{3} x^{6} + a^{3} b^{2}\right )}}, \frac {6 \, a b^{2} x^{4} + 6 \, \sqrt {\frac {1}{3}} {\left (a b^{2} x^{6} + a^{2} b\right )} \sqrt {-\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, b x^{2} + \left (-a b^{2}\right )^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}}}{b}\right ) + {\left (b x^{6} + a\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b^{2} x^{4} + \left (-a b^{2}\right )^{\frac {1}{3}} b x^{2} + \left (-a b^{2}\right )^{\frac {2}{3}}\right ) - 2 \, {\left (b x^{6} + a\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b x^{2} - \left (-a b^{2}\right )^{\frac {1}{3}}\right )}{36 \, {\left (a^{2} b^{3} x^{6} + a^{3} b^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 135, normalized size = 0.95 \[ \frac {x^{4}}{6 \, {\left (b x^{6} + a\right )} a} - \frac {\left (-\frac {a}{b}\right )^{\frac {2}{3}} \log \left ({\left | x^{2} - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{18 \, a^{2}} - \frac {\sqrt {3} \left (-a b^{2}\right )^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{2} + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{18 \, a^{2} b^{2}} + \frac {\left (-a b^{2}\right )^{\frac {2}{3}} \log \left (x^{4} + x^{2} \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{36 \, a^{2} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 123, normalized size = 0.87 \[ \frac {x^{4}}{6 \left (b \,x^{6}+a \right ) a}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{2}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{18 \left (\frac {a}{b}\right )^{\frac {1}{3}} a b}-\frac {\ln \left (x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{18 \left (\frac {a}{b}\right )^{\frac {1}{3}} a b}+\frac {\ln \left (x^{4}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{2}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{36 \left (\frac {a}{b}\right )^{\frac {1}{3}} a b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.39, size = 130, normalized size = 0.92 \[ \frac {x^{4}}{6 \, {\left (a b x^{6} + a^{2}\right )}} + \frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{2} - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{18 \, a b \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {\log \left (x^{4} - x^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{36 \, a b \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {\log \left (x^{2} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{18 \, a b \left (\frac {a}{b}\right )^{\frac {1}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.34, size = 146, normalized size = 1.03 \[ \frac {x^4}{6\,a\,\left (b\,x^6+a\right )}+\frac {{\left (-1\right )}^{1/3}\,\ln \left (\frac {b^2}{81\,a^3}-\frac {{\left (-1\right )}^{1/3}\,b^{7/3}\,x^2}{81\,a^{10/3}}\right )}{18\,a^{4/3}\,b^{2/3}}-\frac {{\left (-1\right )}^{1/3}\,\ln \left ({\left (-1\right )}^{2/3}\,a^{1/3}-2\,b^{1/3}\,x^2+{\left (-1\right )}^{1/6}\,\sqrt {3}\,a^{1/3}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{18\,a^{4/3}\,b^{2/3}}+\frac {{\left (-1\right )}^{1/3}\,\ln \left (2\,b^{1/3}\,x^2-{\left (-1\right )}^{2/3}\,a^{1/3}+{\left (-1\right )}^{1/6}\,\sqrt {3}\,a^{1/3}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{18\,a^{4/3}\,b^{2/3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.67, size = 46, normalized size = 0.32 \[ \frac {x^{4}}{6 a^{2} + 6 a b x^{6}} + \operatorname {RootSum} {\left (5832 t^{3} a^{4} b^{2} + 1, \left (t \mapsto t \log {\left (324 t^{2} a^{3} b + x^{2} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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