Optimal. Leaf size=234 \[ \frac{\log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{24 \sqrt{3} a^{7/6} b^{5/6}}-\frac{\log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{24 \sqrt{3} a^{7/6} b^{5/6}}+\frac{\tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{18 a^{7/6} b^{5/6}}-\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{7/6} b^{5/6}}+\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}+2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{7/6} b^{5/6}}+\frac{x^5}{6 a \left (a+b x^6\right )} \]
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Rubi [A] time = 0.483183, antiderivative size = 234, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538, Rules used = {290, 295, 634, 618, 204, 628, 205} \[ \frac{\log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{24 \sqrt{3} a^{7/6} b^{5/6}}-\frac{\log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{24 \sqrt{3} a^{7/6} b^{5/6}}+\frac{\tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{18 a^{7/6} b^{5/6}}-\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{7/6} b^{5/6}}+\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}+2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{7/6} b^{5/6}}+\frac{x^5}{6 a \left (a+b x^6\right )} \]
Antiderivative was successfully verified.
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Rule 290
Rule 295
Rule 634
Rule 618
Rule 204
Rule 628
Rule 205
Rubi steps
\begin{align*} \int \frac{x^4}{\left (a+b x^6\right )^2} \, dx &=\frac{x^5}{6 a \left (a+b x^6\right )}+\frac{\int \frac{x^4}{a+b x^6} \, dx}{6 a}\\ &=\frac{x^5}{6 a \left (a+b x^6\right )}+\frac{\int \frac{-\frac{\sqrt [6]{a}}{2}+\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^{7/6} b^{2/3}}+\frac{\int \frac{-\frac{\sqrt [6]{a}}{2}-\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^{7/6} b^{2/3}}+\frac{\int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x^2} \, dx}{18 a b^{2/3}}\\ &=\frac{x^5}{6 a \left (a+b x^6\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{18 a^{7/6} b^{5/6}}+\frac{\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^{7/6} b^{5/6}}-\frac{\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^{7/6} b^{5/6}}+\frac{\int \frac{1}{\sqrt [3]{a}-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx}{72 a b^{2/3}}+\frac{\int \frac{1}{\sqrt [3]{a}+\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx}{72 a b^{2/3}}\\ &=\frac{x^5}{6 a \left (a+b x^6\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{18 a^{7/6} b^{5/6}}+\frac{\log \left (\sqrt [3]{a}-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{24 \sqrt{3} a^{7/6} b^{5/6}}-\frac{\log \left (\sqrt [3]{a}+\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{24 \sqrt{3} a^{7/6} b^{5/6}}+\frac{\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^{7/6} b^{5/6}}-\frac{\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^{7/6} b^{5/6}}\\ &=\frac{x^5}{6 a \left (a+b x^6\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{18 a^{7/6} b^{5/6}}-\frac{\tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{7/6} b^{5/6}}+\frac{\tan ^{-1}\left (\sqrt{3}+\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{7/6} b^{5/6}}+\frac{\log \left (\sqrt [3]{a}-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{24 \sqrt{3} a^{7/6} b^{5/6}}-\frac{\log \left (\sqrt [3]{a}+\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{24 \sqrt{3} a^{7/6} b^{5/6}}\\ \end{align*}
Mathematica [A] time = 0.110538, size = 193, normalized size = 0.82 \[ \frac{\frac{\sqrt{3} \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{b^{5/6}}-\frac{\sqrt{3} \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{b^{5/6}}+\frac{4 \tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{b^{5/6}}-\frac{2 \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{b^{5/6}}+\frac{2 \tan ^{-1}\left (\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}+\sqrt{3}\right )}{b^{5/6}}+\frac{12 \sqrt [6]{a} x^5}{a+b x^6}}{72 a^{7/6}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.232, size = 346, normalized size = 1.5 \begin{align*}{\frac{x}{18\,ab} \left ({x}^{2}+\sqrt{3}\sqrt [6]{{\frac{a}{b}}}x+\sqrt [3]{{\frac{a}{b}}} \right ) ^{-1}}+{\frac{\sqrt{3}}{36\,ab}\sqrt [6]{{\frac{a}{b}}} \left ({x}^{2}+\sqrt{3}\sqrt [6]{{\frac{a}{b}}}x+\sqrt [3]{{\frac{a}{b}}} \right ) ^{-1}}-{\frac{\sqrt{3}}{72\,{a}^{2}} \left ({\frac{a}{b}} \right ) ^{{\frac{5}{6}}}\ln \left ({x}^{2}+\sqrt{3}\sqrt [6]{{\frac{a}{b}}}x+\sqrt [3]{{\frac{a}{b}}} \right ) }+{\frac{1}{36\,ab}\arctan \left ( 2\,{x{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}+\sqrt{3} \right ){\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}+{\frac{x}{18\,ab} \left ({x}^{2}+\sqrt [3]{{\frac{a}{b}}} \right ) ^{-1}}+{\frac{1}{18\,ab}\arctan \left ({x{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}+{\frac{x}{18\,ab} \left ({x}^{2}-\sqrt{3}\sqrt [6]{{\frac{a}{b}}}x+\sqrt [3]{{\frac{a}{b}}} \right ) ^{-1}}-{\frac{\sqrt{3}}{36\,ab}\sqrt [6]{{\frac{a}{b}}} \left ({x}^{2}-\sqrt{3}\sqrt [6]{{\frac{a}{b}}}x+\sqrt [3]{{\frac{a}{b}}} \right ) ^{-1}}+{\frac{\sqrt{3}}{72\,{a}^{2}} \left ({\frac{a}{b}} \right ) ^{{\frac{5}{6}}}\ln \left ({x}^{2}-\sqrt{3}\sqrt [6]{{\frac{a}{b}}}x+\sqrt [3]{{\frac{a}{b}}} \right ) }+{\frac{1}{36\,ab}\arctan \left ( 2\,{x{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}-\sqrt{3} \right ){\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.60936, size = 1166, normalized size = 4.98 \begin{align*} \frac{12 \, x^{5} - 4 \, \sqrt{3}{\left (a b x^{6} + a^{2}\right )} \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{1}{6}} \arctan \left (-\frac{2}{3} \, \sqrt{3} a b x \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{1}{6}} + \frac{2}{3} \, \sqrt{3} \sqrt{a^{6} b^{4} x \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{5}{6}} - a^{5} b^{3} \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{2}{3}} + x^{2}} a b \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{1}{6}} + \frac{1}{3} \, \sqrt{3}\right ) - 4 \, \sqrt{3}{\left (a b x^{6} + a^{2}\right )} \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{1}{6}} \arctan \left (-\frac{2}{3} \, \sqrt{3} a b x \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{1}{6}} + \frac{2}{3} \, \sqrt{3} \sqrt{-a^{6} b^{4} x \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{5}{6}} - a^{5} b^{3} \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{2}{3}} + x^{2}} a b \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{1}{6}} - \frac{1}{3} \, \sqrt{3}\right ) +{\left (a b x^{6} + a^{2}\right )} \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{1}{6}} \log \left (a^{6} b^{4} x \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{5}{6}} - a^{5} b^{3} \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{2}{3}} + x^{2}\right ) -{\left (a b x^{6} + a^{2}\right )} \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{1}{6}} \log \left (-a^{6} b^{4} x \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{5}{6}} - a^{5} b^{3} \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{2}{3}} + x^{2}\right ) + 2 \,{\left (a b x^{6} + a^{2}\right )} \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{1}{6}} \log \left (a^{6} b^{4} \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{5}{6}} + x\right ) - 2 \,{\left (a b x^{6} + a^{2}\right )} \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{1}{6}} \log \left (-a^{6} b^{4} \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{5}{6}} + x\right )}{72 \,{\left (a b x^{6} + a^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.90611, size = 46, normalized size = 0.2 \begin{align*} \frac{x^{5}}{6 a^{2} + 6 a b x^{6}} + \operatorname{RootSum}{\left (2176782336 t^{6} a^{7} b^{5} + 1, \left ( t \mapsto t \log{\left (60466176 t^{5} a^{6} b^{4} + x \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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