Optimal. Leaf size=232 \[ -\frac {5 \log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{24 \sqrt {3} a^{11/6} \sqrt [6]{b}}+\frac {5 \log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{24 \sqrt {3} a^{11/6} \sqrt [6]{b}}+\frac {5 \tan ^{-1}\left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{18 a^{11/6} \sqrt [6]{b}}-\frac {5 \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{11/6} \sqrt [6]{b}}+\frac {5 \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{a}+2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{11/6} \sqrt [6]{b}}+\frac {x}{6 a \left (a+b x^6\right )} \]
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Rubi [A] time = 0.40, antiderivative size = 232, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.778, Rules used = {199, 209, 634, 618, 204, 628, 205} \[ -\frac {5 \log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{24 \sqrt {3} a^{11/6} \sqrt [6]{b}}+\frac {5 \log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{24 \sqrt {3} a^{11/6} \sqrt [6]{b}}+\frac {5 \tan ^{-1}\left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{18 a^{11/6} \sqrt [6]{b}}-\frac {5 \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{11/6} \sqrt [6]{b}}+\frac {5 \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{a}+2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{11/6} \sqrt [6]{b}}+\frac {x}{6 a \left (a+b x^6\right )} \]
Antiderivative was successfully verified.
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Rule 199
Rule 204
Rule 205
Rule 209
Rule 618
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b x^6\right )^2} \, dx &=\frac {x}{6 a \left (a+b x^6\right )}+\frac {5 \int \frac {1}{a+b x^6} \, dx}{6 a}\\ &=\frac {x}{6 a \left (a+b x^6\right )}+\frac {5 \int \frac {\sqrt [6]{a}-\frac {1}{2} \sqrt {3} \sqrt [6]{b} x}{\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx}{18 a^{11/6}}+\frac {5 \int \frac {\sqrt [6]{a}+\frac {1}{2} \sqrt {3} \sqrt [6]{b} x}{\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx}{18 a^{11/6}}+\frac {5 \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x^2} \, dx}{18 a^{5/3}}\\ &=\frac {x}{6 a \left (a+b x^6\right )}+\frac {5 \tan ^{-1}\left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{18 a^{11/6} \sqrt [6]{b}}+\frac {5 \int \frac {1}{\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx}{72 a^{5/3}}+\frac {5 \int \frac {1}{\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx}{72 a^{5/3}}-\frac {5 \int \frac {-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b}+2 \sqrt [3]{b} x}{\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx}{24 \sqrt {3} a^{11/6} \sqrt [6]{b}}+\frac {5 \int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [6]{b}+2 \sqrt [3]{b} x}{\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx}{24 \sqrt {3} a^{11/6} \sqrt [6]{b}}\\ &=\frac {x}{6 a \left (a+b x^6\right )}+\frac {5 \tan ^{-1}\left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{18 a^{11/6} \sqrt [6]{b}}-\frac {5 \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{24 \sqrt {3} a^{11/6} \sqrt [6]{b}}+\frac {5 \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{24 \sqrt {3} a^{11/6} \sqrt [6]{b}}+\frac {5 \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1-\frac {2 \sqrt [6]{b} x}{\sqrt {3} \sqrt [6]{a}}\right )}{36 \sqrt {3} a^{11/6} \sqrt [6]{b}}-\frac {5 \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1+\frac {2 \sqrt [6]{b} x}{\sqrt {3} \sqrt [6]{a}}\right )}{36 \sqrt {3} a^{11/6} \sqrt [6]{b}}\\ &=\frac {x}{6 a \left (a+b x^6\right )}+\frac {5 \tan ^{-1}\left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{18 a^{11/6} \sqrt [6]{b}}-\frac {5 \tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{11/6} \sqrt [6]{b}}+\frac {5 \tan ^{-1}\left (\sqrt {3}+\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{11/6} \sqrt [6]{b}}-\frac {5 \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{24 \sqrt {3} a^{11/6} \sqrt [6]{b}}+\frac {5 \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{24 \sqrt {3} a^{11/6} \sqrt [6]{b}}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 192, normalized size = 0.83 \[ \frac {\frac {12 a^{5/6} x}{a+b x^6}-\frac {5 \sqrt {3} \log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\sqrt [6]{b}}+\frac {5 \sqrt {3} \log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\sqrt [6]{b}}+\frac {20 \tan ^{-1}\left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{\sqrt [6]{b}}-\frac {10 \tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{\sqrt [6]{b}}+\frac {10 \tan ^{-1}\left (\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}+\sqrt {3}\right )}{\sqrt [6]{b}}}{72 a^{11/6}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.87, size = 423, normalized size = 1.82 \[ \frac {20 \, \sqrt {3} {\left (a b x^{6} + a^{2}\right )} \left (-\frac {1}{a^{11} b}\right )^{\frac {1}{6}} \arctan \left (-\frac {2}{3} \, \sqrt {3} a^{9} b x \left (-\frac {1}{a^{11} b}\right )^{\frac {5}{6}} + \frac {2}{3} \, \sqrt {3} \sqrt {a^{4} \left (-\frac {1}{a^{11} b}\right )^{\frac {1}{3}} + a^{2} x \left (-\frac {1}{a^{11} b}\right )^{\frac {1}{6}} + x^{2}} a^{9} b \left (-\frac {1}{a^{11} b}\right )^{\frac {5}{6}} + \frac {1}{3} \, \sqrt {3}\right ) + 20 \, \sqrt {3} {\left (a b x^{6} + a^{2}\right )} \left (-\frac {1}{a^{11} b}\right )^{\frac {1}{6}} \arctan \left (-\frac {2}{3} \, \sqrt {3} a^{9} b x \left (-\frac {1}{a^{11} b}\right )^{\frac {5}{6}} + \frac {2}{3} \, \sqrt {3} \sqrt {a^{4} \left (-\frac {1}{a^{11} b}\right )^{\frac {1}{3}} - a^{2} x \left (-\frac {1}{a^{11} b}\right )^{\frac {1}{6}} + x^{2}} a^{9} b \left (-\frac {1}{a^{11} b}\right )^{\frac {5}{6}} - \frac {1}{3} \, \sqrt {3}\right ) + 5 \, {\left (a b x^{6} + a^{2}\right )} \left (-\frac {1}{a^{11} b}\right )^{\frac {1}{6}} \log \left (a^{4} \left (-\frac {1}{a^{11} b}\right )^{\frac {1}{3}} + a^{2} x \left (-\frac {1}{a^{11} b}\right )^{\frac {1}{6}} + x^{2}\right ) - 5 \, {\left (a b x^{6} + a^{2}\right )} \left (-\frac {1}{a^{11} b}\right )^{\frac {1}{6}} \log \left (a^{4} \left (-\frac {1}{a^{11} b}\right )^{\frac {1}{3}} - a^{2} x \left (-\frac {1}{a^{11} b}\right )^{\frac {1}{6}} + x^{2}\right ) + 10 \, {\left (a b x^{6} + a^{2}\right )} \left (-\frac {1}{a^{11} b}\right )^{\frac {1}{6}} \log \left (a^{2} \left (-\frac {1}{a^{11} b}\right )^{\frac {1}{6}} + x\right ) - 10 \, {\left (a b x^{6} + a^{2}\right )} \left (-\frac {1}{a^{11} b}\right )^{\frac {1}{6}} \log \left (-a^{2} \left (-\frac {1}{a^{11} b}\right )^{\frac {1}{6}} + x\right ) + 12 \, x}{72 \, {\left (a b x^{6} + a^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 205, normalized size = 0.88 \[ \frac {x}{6 \, {\left (b x^{6} + a\right )} a} + \frac {5 \, \sqrt {3} \left (a b^{5}\right )^{\frac {1}{6}} \log \left (x^{2} + \sqrt {3} x \left (\frac {a}{b}\right )^{\frac {1}{6}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{72 \, a^{2} b} - \frac {5 \, \sqrt {3} \left (a b^{5}\right )^{\frac {1}{6}} \log \left (x^{2} - \sqrt {3} x \left (\frac {a}{b}\right )^{\frac {1}{6}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{72 \, a^{2} b} + \frac {5 \, \left (a b^{5}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \, x + \sqrt {3} \left (\frac {a}{b}\right )^{\frac {1}{6}}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{36 \, a^{2} b} + \frac {5 \, \left (a b^{5}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \, x - \sqrt {3} \left (\frac {a}{b}\right )^{\frac {1}{6}}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{36 \, a^{2} b} + \frac {5 \, \left (a b^{5}\right )^{\frac {1}{6}} \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{18 \, a^{2} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.45, size = 343, normalized size = 1.48 \[ \frac {\left (\frac {a}{b}\right )^{\frac {1}{3}} x}{18 \left (x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) a^{2}}-\frac {\left (\frac {a}{b}\right )^{\frac {1}{3}} x}{36 \left (x^{2}-\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) a^{2}}-\frac {\left (\frac {a}{b}\right )^{\frac {1}{3}} x}{36 \left (x^{2}+\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) a^{2}}+\frac {5 \left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{18 a^{2}}+\frac {5 \left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{36 a^{2}}+\frac {5 \left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{36 a^{2}}-\frac {5 \sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \ln \left (x^{2}-\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{72 a^{2}}+\frac {5 \sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \ln \left (x^{2}+\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{72 a^{2}}+\frac {\sqrt {\frac {a}{b}}\, \sqrt {3}}{36 \left (x^{2}-\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) a^{2}}-\frac {\sqrt {\frac {a}{b}}\, \sqrt {3}}{36 \left (x^{2}+\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.33, size = 204, normalized size = 0.88 \[ \frac {x}{6 \, {\left (a b x^{6} + a^{2}\right )}} + \frac {5 \, {\left (\frac {\sqrt {3} \log \left (b^{\frac {1}{3}} x^{2} + \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} x + a^{\frac {1}{3}}\right )}{a^{\frac {5}{6}} b^{\frac {1}{6}}} - \frac {\sqrt {3} \log \left (b^{\frac {1}{3}} x^{2} - \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} x + a^{\frac {1}{3}}\right )}{a^{\frac {5}{6}} b^{\frac {1}{6}}} + \frac {4 \, \arctan \left (\frac {b^{\frac {1}{3}} x}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{a^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} + \frac {2 \, \arctan \left (\frac {2 \, b^{\frac {1}{3}} x + \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{a^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} + \frac {2 \, \arctan \left (\frac {2 \, b^{\frac {1}{3}} x - \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{a^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}}{72 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.14, size = 242, normalized size = 1.04 \[ \frac {x}{6\,a\,\left (b\,x^6+a\right )}-\frac {\mathrm {atan}\left (\frac {b^{1/6}\,x\,1{}\mathrm {i}}{{\left (-a\right )}^{1/6}}\right )\,5{}\mathrm {i}}{18\,{\left (-a\right )}^{11/6}\,b^{1/6}}+\frac {\mathrm {atan}\left (\frac {b^{29/6}\,x\,3125{}\mathrm {i}}{7776\,{\left (-a\right )}^{35/6}\,\left (\frac {3125\,b^{14/3}}{7776\,{\left (-a\right )}^{17/3}}-\frac {\sqrt {3}\,b^{14/3}\,3125{}\mathrm {i}}{7776\,{\left (-a\right )}^{17/3}}\right )}-\frac {3125\,\sqrt {3}\,b^{29/6}\,x}{7776\,{\left (-a\right )}^{35/6}\,\left (\frac {3125\,b^{14/3}}{7776\,{\left (-a\right )}^{17/3}}-\frac {\sqrt {3}\,b^{14/3}\,3125{}\mathrm {i}}{7776\,{\left (-a\right )}^{17/3}}\right )}\right )\,\left (5+\sqrt {3}\,5{}\mathrm {i}\right )\,1{}\mathrm {i}}{36\,{\left (-a\right )}^{11/6}\,b^{1/6}}-\frac {\mathrm {atan}\left (\frac {b^{29/6}\,x\,3125{}\mathrm {i}}{7776\,{\left (-a\right )}^{35/6}\,\left (\frac {3125\,b^{14/3}}{7776\,{\left (-a\right )}^{17/3}}+\frac {\sqrt {3}\,b^{14/3}\,3125{}\mathrm {i}}{7776\,{\left (-a\right )}^{17/3}}\right )}+\frac {3125\,\sqrt {3}\,b^{29/6}\,x}{7776\,{\left (-a\right )}^{35/6}\,\left (\frac {3125\,b^{14/3}}{7776\,{\left (-a\right )}^{17/3}}+\frac {\sqrt {3}\,b^{14/3}\,3125{}\mathrm {i}}{7776\,{\left (-a\right )}^{17/3}}\right )}\right )\,\left (-5+\sqrt {3}\,5{}\mathrm {i}\right )\,1{}\mathrm {i}}{36\,{\left (-a\right )}^{11/6}\,b^{1/6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.72, size = 39, normalized size = 0.17 \[ \frac {x}{6 a^{2} + 6 a b x^{6}} + \operatorname {RootSum} {\left (2176782336 t^{6} a^{11} b + 15625, \left (t \mapsto t \log {\left (\frac {36 t a^{2}}{5} + x \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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