Optimal. Leaf size=220 \[ \frac{\sqrt [6]{a} \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{4 \sqrt{3} b^{7/6}}-\frac{\sqrt [6]{a} \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{4 \sqrt{3} b^{7/6}}-\frac{\sqrt [6]{a} \tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 b^{7/6}}+\frac{\sqrt [6]{a} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 b^{7/6}}-\frac{\sqrt [6]{a} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}+2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 b^{7/6}}+\frac{x}{b} \]
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Rubi [A] time = 0.423248, antiderivative size = 220, 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 = {321, 209, 634, 618, 204, 628, 205} \[ \frac{\sqrt [6]{a} \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{4 \sqrt{3} b^{7/6}}-\frac{\sqrt [6]{a} \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{4 \sqrt{3} b^{7/6}}-\frac{\sqrt [6]{a} \tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 b^{7/6}}+\frac{\sqrt [6]{a} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 b^{7/6}}-\frac{\sqrt [6]{a} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}+2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 b^{7/6}}+\frac{x}{b} \]
Antiderivative was successfully verified.
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Rule 321
Rule 209
Rule 634
Rule 618
Rule 204
Rule 628
Rule 205
Rubi steps
\begin{align*} \int \frac{x^6}{a+b x^6} \, dx &=\frac{x}{b}-\frac{a \int \frac{1}{a+b x^6} \, dx}{b}\\ &=\frac{x}{b}-\frac{\sqrt [6]{a} \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}{3 b}-\frac{\sqrt [6]{a} \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}{3 b}-\frac{\sqrt [3]{a} \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x^2} \, dx}{3 b}\\ &=\frac{x}{b}-\frac{\sqrt [6]{a} \tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 b^{7/6}}+\frac{\sqrt [6]{a} \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}{4 \sqrt{3} b^{7/6}}-\frac{\sqrt [6]{a} \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}{4 \sqrt{3} b^{7/6}}-\frac{\sqrt [3]{a} \int \frac{1}{\sqrt [3]{a}-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx}{12 b}-\frac{\sqrt [3]{a} \int \frac{1}{\sqrt [3]{a}+\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx}{12 b}\\ &=\frac{x}{b}-\frac{\sqrt [6]{a} \tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 b^{7/6}}+\frac{\sqrt [6]{a} \log \left (\sqrt [3]{a}-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{4 \sqrt{3} b^{7/6}}-\frac{\sqrt [6]{a} \log \left (\sqrt [3]{a}+\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{4 \sqrt{3} b^{7/6}}-\frac{\sqrt [6]{a} \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 )}{6 \sqrt{3} b^{7/6}}+\frac{\sqrt [6]{a} \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 )}{6 \sqrt{3} b^{7/6}}\\ &=\frac{x}{b}-\frac{\sqrt [6]{a} \tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 b^{7/6}}+\frac{\sqrt [6]{a} \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 b^{7/6}}-\frac{\sqrt [6]{a} \tan ^{-1}\left (\sqrt{3}+\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 b^{7/6}}+\frac{\sqrt [6]{a} \log \left (\sqrt [3]{a}-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{4 \sqrt{3} b^{7/6}}-\frac{\sqrt [6]{a} \log \left (\sqrt [3]{a}+\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{4 \sqrt{3} b^{7/6}}\\ \end{align*}
Mathematica [A] time = 0.0286149, size = 182, normalized size = 0.83 \[ \frac{\sqrt{3} \sqrt [6]{a} \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )-\sqrt{3} \sqrt [6]{a} \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )-4 \sqrt [6]{a} \tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )+2 \sqrt [6]{a} \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )-2 \sqrt [6]{a} \tan ^{-1}\left (\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}+\sqrt{3}\right )+12 \sqrt [6]{b} x}{12 b^{7/6}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.06, size = 164, normalized size = 0.8 \begin{align*}{\frac{x}{b}}-{\frac{\sqrt{3}}{12\,b}\sqrt [6]{{\frac{a}{b}}}\ln \left ({x}^{2}+\sqrt{3}\sqrt [6]{{\frac{a}{b}}}x+\sqrt [3]{{\frac{a}{b}}} \right ) }-{\frac{1}{6\,b}\sqrt [6]{{\frac{a}{b}}}\arctan \left ( 2\,{x{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}+\sqrt{3} \right ) }-{\frac{1}{3\,b}\sqrt [6]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \right ) }+{\frac{\sqrt{3}}{12\,b}\sqrt [6]{{\frac{a}{b}}}\ln \left ({x}^{2}-\sqrt{3}\sqrt [6]{{\frac{a}{b}}}x+\sqrt [3]{{\frac{a}{b}}} \right ) }-{\frac{1}{6\,b}\sqrt [6]{{\frac{a}{b}}}\arctan \left ( 2\,{x{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}-\sqrt{3} \right ) } \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 = 1.81859, size = 803, normalized size = 3.65 \begin{align*} -\frac{4 \, \sqrt{3} b \left (-\frac{a}{b^{7}}\right )^{\frac{1}{6}} \arctan \left (-\frac{2 \, \sqrt{3} b^{6} x \left (-\frac{a}{b^{7}}\right )^{\frac{5}{6}} - 2 \, \sqrt{3} \sqrt{b^{2} \left (-\frac{a}{b^{7}}\right )^{\frac{1}{3}} + b x \left (-\frac{a}{b^{7}}\right )^{\frac{1}{6}} + x^{2}} b^{6} \left (-\frac{a}{b^{7}}\right )^{\frac{5}{6}} - \sqrt{3} a}{3 \, a}\right ) + 4 \, \sqrt{3} b \left (-\frac{a}{b^{7}}\right )^{\frac{1}{6}} \arctan \left (-\frac{2 \, \sqrt{3} b^{6} x \left (-\frac{a}{b^{7}}\right )^{\frac{5}{6}} - 2 \, \sqrt{3} \sqrt{b^{2} \left (-\frac{a}{b^{7}}\right )^{\frac{1}{3}} - b x \left (-\frac{a}{b^{7}}\right )^{\frac{1}{6}} + x^{2}} b^{6} \left (-\frac{a}{b^{7}}\right )^{\frac{5}{6}} + \sqrt{3} a}{3 \, a}\right ) + b \left (-\frac{a}{b^{7}}\right )^{\frac{1}{6}} \log \left (b^{2} \left (-\frac{a}{b^{7}}\right )^{\frac{1}{3}} + b x \left (-\frac{a}{b^{7}}\right )^{\frac{1}{6}} + x^{2}\right ) - b \left (-\frac{a}{b^{7}}\right )^{\frac{1}{6}} \log \left (b^{2} \left (-\frac{a}{b^{7}}\right )^{\frac{1}{3}} - b x \left (-\frac{a}{b^{7}}\right )^{\frac{1}{6}} + x^{2}\right ) + 2 \, b \left (-\frac{a}{b^{7}}\right )^{\frac{1}{6}} \log \left (b \left (-\frac{a}{b^{7}}\right )^{\frac{1}{6}} + x\right ) - 2 \, b \left (-\frac{a}{b^{7}}\right )^{\frac{1}{6}} \log \left (-b \left (-\frac{a}{b^{7}}\right )^{\frac{1}{6}} + x\right ) - 12 \, x}{12 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.321754, size = 22, normalized size = 0.1 \begin{align*} \operatorname{RootSum}{\left (46656 t^{6} b^{7} + a, \left ( t \mapsto t \log{\left (- 6 t b + x \right )} \right )\right )} + \frac{x}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17692, size = 243, normalized size = 1.1 \begin{align*} \frac{x}{b} - \frac{\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 )}{12 \, b^{2}} + \frac{\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 )}{12 \, b^{2}} - \frac{\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 )}{6 \, b^{2}} - \frac{\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 )}{6 \, b^{2}} - \frac{\left (a b^{5}\right )^{\frac{1}{6}} \arctan \left (\frac{x}{\left (\frac{a}{b}\right )^{\frac{1}{6}}}\right )}{3 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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