Optimal. Leaf size=301 \[ -\frac{\left (1-\sqrt{5}\right ) \log \left (2 a^{2/5}-\left (1-\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 b^{2/5} x^2\right )}{20 a^{4/5} \sqrt [5]{b}}-\frac{\left (1+\sqrt{5}\right ) \log \left (2 a^{2/5}-\left (1+\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 b^{2/5} x^2\right )}{20 a^{4/5} \sqrt [5]{b}}+\frac{\log \left (\sqrt [5]{a}+\sqrt [5]{b} x\right )}{5 a^{4/5} \sqrt [5]{b}}-\frac{\sqrt{\frac{1}{2} \left (5+\sqrt{5}\right )} \tan ^{-1}\left (\frac{\left (1-\sqrt{5}\right ) \sqrt [5]{a}-4 \sqrt [5]{b} x}{\sqrt{2 \left (5+\sqrt{5}\right )} \sqrt [5]{a}}\right )}{5 a^{4/5} \sqrt [5]{b}}-\frac{\sqrt{\frac{1}{2} \left (5-\sqrt{5}\right )} \tan ^{-1}\left (\frac{\left (1+\sqrt{5}\right ) \sqrt [5]{a}-4 \sqrt [5]{b} x}{\sqrt{2 \left (5-\sqrt{5}\right )} \sqrt [5]{a}}\right )}{5 a^{4/5} \sqrt [5]{b}} \]
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Rubi [A] time = 0.63845, antiderivative size = 305, normalized size of antiderivative = 1.01, number of steps used = 6, number of rules used = 6, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {201, 634, 618, 204, 628, 31} \[ -\frac{\left (1-\sqrt{5}\right ) \log \left (a^{2/5}-\frac{1}{2} \left (1-\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+b^{2/5} x^2\right )}{20 a^{4/5} \sqrt [5]{b}}-\frac{\left (1+\sqrt{5}\right ) \log \left (a^{2/5}-\frac{1}{2} \left (1+\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+b^{2/5} x^2\right )}{20 a^{4/5} \sqrt [5]{b}}+\frac{\log \left (\sqrt [5]{a}+\sqrt [5]{b} x\right )}{5 a^{4/5} \sqrt [5]{b}}+\frac{\sqrt{\frac{1}{2} \left (5+\sqrt{5}\right )} \tan ^{-1}\left (\frac{2 \sqrt{\frac{2}{5+\sqrt{5}}} \sqrt [5]{b} x}{\sqrt [5]{a}}+\sqrt{\frac{1}{5} \left (5-2 \sqrt{5}\right )}\right )}{5 a^{4/5} \sqrt [5]{b}}-\frac{\sqrt{\frac{1}{2} \left (5-\sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{1}{5} \left (5+2 \sqrt{5}\right )}-\frac{\sqrt{\frac{2}{5} \left (5+\sqrt{5}\right )} \sqrt [5]{b} x}{\sqrt [5]{a}}\right )}{5 a^{4/5} \sqrt [5]{b}} \]
Warning: Unable to verify antiderivative.
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Rule 201
Rule 634
Rule 618
Rule 204
Rule 628
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{a+b x^5} \, dx &=\frac{2 \int \frac{\sqrt [5]{a}-\frac{1}{4} \left (1-\sqrt{5}\right ) \sqrt [5]{b} x}{a^{2/5}-\frac{1}{2} \left (1-\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+b^{2/5} x^2} \, dx}{5 a^{4/5}}+\frac{2 \int \frac{\sqrt [5]{a}-\frac{1}{4} \left (1+\sqrt{5}\right ) \sqrt [5]{b} x}{a^{2/5}-\frac{1}{2} \left (1+\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+b^{2/5} x^2} \, dx}{5 a^{4/5}}+\frac{\int \frac{1}{\sqrt [5]{a}+\sqrt [5]{b} x} \, dx}{5 a^{4/5}}\\ &=\frac{\log \left (\sqrt [5]{a}+\sqrt [5]{b} x\right )}{5 a^{4/5} \sqrt [5]{b}}+\frac{\left (5-\sqrt{5}\right ) \int \frac{1}{a^{2/5}-\frac{1}{2} \left (1+\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+b^{2/5} x^2} \, dx}{20 a^{3/5}}+\frac{\left (5+\sqrt{5}\right ) \int \frac{1}{a^{2/5}-\frac{1}{2} \left (1-\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+b^{2/5} x^2} \, dx}{20 a^{3/5}}-\frac{\left (1-\sqrt{5}\right ) \int \frac{-\frac{1}{2} \left (1-\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b}+2 b^{2/5} x}{a^{2/5}-\frac{1}{2} \left (1-\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+b^{2/5} x^2} \, dx}{20 a^{4/5} \sqrt [5]{b}}-\frac{\left (1+\sqrt{5}\right ) \int \frac{-\frac{1}{2} \left (1+\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b}+2 b^{2/5} x}{a^{2/5}-\frac{1}{2} \left (1+\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+b^{2/5} x^2} \, dx}{20 a^{4/5} \sqrt [5]{b}}\\ &=\frac{\log \left (\sqrt [5]{a}+\sqrt [5]{b} x\right )}{5 a^{4/5} \sqrt [5]{b}}-\frac{\left (1+\sqrt{5}\right ) \log \left (2 a^{2/5}-\sqrt [5]{a} \sqrt [5]{b} x-\sqrt{5} \sqrt [5]{a} \sqrt [5]{b} x+2 b^{2/5} x^2\right )}{20 a^{4/5} \sqrt [5]{b}}-\frac{\left (1-\sqrt{5}\right ) \log \left (2 a^{2/5}-\sqrt [5]{a} \sqrt [5]{b} x+\sqrt{5} \sqrt [5]{a} \sqrt [5]{b} x+2 b^{2/5} x^2\right )}{20 a^{4/5} \sqrt [5]{b}}-\frac{\left (5-\sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{2} \left (5-\sqrt{5}\right ) a^{2/5} b^{2/5}-x^2} \, dx,x,-\frac{1}{2} \left (1+\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b}+2 b^{2/5} x\right )}{10 a^{3/5}}-\frac{\left (5+\sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{2} \left (5+\sqrt{5}\right ) a^{2/5} b^{2/5}-x^2} \, dx,x,-\frac{1}{2} \left (1-\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b}+2 b^{2/5} x\right )}{10 a^{3/5}}\\ &=-\frac{\sqrt{\frac{1}{2} \left (5-\sqrt{5}\right )} \tan ^{-1}\left (\frac{\sqrt{5+\sqrt{5}} \left (\left (1+\sqrt{5}\right ) \sqrt [5]{a}-4 \sqrt [5]{b} x\right )}{2 \sqrt{10} \sqrt [5]{a}}\right )}{5 a^{4/5} \sqrt [5]{b}}+\frac{\sqrt{\frac{1}{2} \left (5+\sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{1}{5} \left (5-2 \sqrt{5}\right )}+\frac{2 \sqrt{\frac{2}{5+\sqrt{5}}} \sqrt [5]{b} x}{\sqrt [5]{a}}\right )}{5 a^{4/5} \sqrt [5]{b}}+\frac{\log \left (\sqrt [5]{a}+\sqrt [5]{b} x\right )}{5 a^{4/5} \sqrt [5]{b}}-\frac{\left (1+\sqrt{5}\right ) \log \left (2 a^{2/5}-\sqrt [5]{a} \sqrt [5]{b} x-\sqrt{5} \sqrt [5]{a} \sqrt [5]{b} x+2 b^{2/5} x^2\right )}{20 a^{4/5} \sqrt [5]{b}}-\frac{\left (1-\sqrt{5}\right ) \log \left (2 a^{2/5}-\sqrt [5]{a} \sqrt [5]{b} x+\sqrt{5} \sqrt [5]{a} \sqrt [5]{b} x+2 b^{2/5} x^2\right )}{20 a^{4/5} \sqrt [5]{b}}\\ \end{align*}
Mathematica [A] time = 0.148791, size = 311, normalized size = 1.03 \[ -\frac{\left (1-\sqrt{5}\right ) \log \left (a^{2/5}-\frac{1}{2} \left (1-\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+b^{2/5} x^2\right )}{20 a^{4/5} \sqrt [5]{b}}-\frac{\left (1+\sqrt{5}\right ) \log \left (a^{2/5}-\frac{1}{2} \left (1+\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+b^{2/5} x^2\right )}{20 a^{4/5} \sqrt [5]{b}}+\frac{\log \left (\sqrt [5]{a}+\sqrt [5]{b} x\right )}{5 a^{4/5} \sqrt [5]{b}}+\frac{2 \sqrt{\frac{5}{8}+\frac{\sqrt{5}}{8}} \tan ^{-1}\left (\frac{\sqrt [5]{b} x-\frac{1}{4} \left (1-\sqrt{5}\right ) \sqrt [5]{a}}{\sqrt{\frac{5}{8}+\frac{\sqrt{5}}{8}} \sqrt [5]{a}}\right )}{5 a^{4/5} \sqrt [5]{b}}+\frac{2 \sqrt{\frac{5}{8}-\frac{\sqrt{5}}{8}} \tan ^{-1}\left (\frac{\sqrt [5]{b} x-\frac{1}{4} \left (1+\sqrt{5}\right ) \sqrt [5]{a}}{\sqrt{\frac{5}{8}-\frac{\sqrt{5}}{8}} \sqrt [5]{a}}\right )}{5 a^{4/5} \sqrt [5]{b}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.079, size = 895, normalized size = 3. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.58337, size = 429, normalized size = 1.43 \begin{align*} \frac{\sqrt{5}{\left (\sqrt{5} - 1\right )} \log \left (\frac{4 \, b^{\frac{2}{5}} x - a^{\frac{1}{5}} b^{\frac{1}{5}}{\left (\sqrt{5} + 1\right )} - a^{\frac{1}{5}} b^{\frac{1}{5}} \sqrt{2 \, \sqrt{5} - 10}}{4 \, b^{\frac{2}{5}} x - a^{\frac{1}{5}} b^{\frac{1}{5}}{\left (\sqrt{5} + 1\right )} + a^{\frac{1}{5}} b^{\frac{1}{5}} \sqrt{2 \, \sqrt{5} - 10}}\right )}{10 \, a^{\frac{4}{5}} b^{\frac{1}{5}} \sqrt{2 \, \sqrt{5} - 10}} + \frac{\sqrt{5}{\left (\sqrt{5} + 1\right )} \log \left (\frac{4 \, b^{\frac{2}{5}} x + a^{\frac{1}{5}} b^{\frac{1}{5}}{\left (\sqrt{5} - 1\right )} - a^{\frac{1}{5}} b^{\frac{1}{5}} \sqrt{-2 \, \sqrt{5} - 10}}{4 \, b^{\frac{2}{5}} x + a^{\frac{1}{5}} b^{\frac{1}{5}}{\left (\sqrt{5} - 1\right )} + a^{\frac{1}{5}} b^{\frac{1}{5}} \sqrt{-2 \, \sqrt{5} - 10}}\right )}{10 \, a^{\frac{4}{5}} b^{\frac{1}{5}} \sqrt{-2 \, \sqrt{5} - 10}} - \frac{{\left (\sqrt{5} + 3\right )} \log \left (2 \, b^{\frac{2}{5}} x^{2} - a^{\frac{1}{5}} b^{\frac{1}{5}} x{\left (\sqrt{5} + 1\right )} + 2 \, a^{\frac{2}{5}}\right )}{10 \, a^{\frac{4}{5}} b^{\frac{1}{5}}{\left (\sqrt{5} + 1\right )}} - \frac{{\left (\sqrt{5} - 3\right )} \log \left (2 \, b^{\frac{2}{5}} x^{2} + a^{\frac{1}{5}} b^{\frac{1}{5}} x{\left (\sqrt{5} - 1\right )} + 2 \, a^{\frac{2}{5}}\right )}{10 \, a^{\frac{4}{5}} b^{\frac{1}{5}}{\left (\sqrt{5} - 1\right )}} + \frac{\log \left (b^{\frac{1}{5}} x + a^{\frac{1}{5}}\right )}{5 \, a^{\frac{4}{5}} b^{\frac{1}{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.190378, size = 20, normalized size = 0.07 \begin{align*} \operatorname{RootSum}{\left (3125 t^{5} a^{4} b - 1, \left ( t \mapsto t \log{\left (5 t a + x \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15769, size = 371, normalized size = 1.23 \begin{align*} -\frac{\left (-\frac{a}{b}\right )^{\frac{1}{5}} \log \left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{5}} \right |}\right )}{5 \, a} + \frac{\left (-a b^{4}\right )^{\frac{1}{5}} \sqrt{2 \, \sqrt{5} + 10} \arctan \left (-\frac{{\left (\sqrt{5} - 1\right )} \left (-\frac{a}{b}\right )^{\frac{1}{5}} - 4 \, x}{\sqrt{2 \, \sqrt{5} + 10} \left (-\frac{a}{b}\right )^{\frac{1}{5}}}\right )}{10 \, a b} + \frac{\left (-a b^{4}\right )^{\frac{1}{5}} \sqrt{-2 \, \sqrt{5} + 10} \arctan \left (\frac{{\left (\sqrt{5} + 1\right )} \left (-\frac{a}{b}\right )^{\frac{1}{5}} + 4 \, x}{\sqrt{-2 \, \sqrt{5} + 10} \left (-\frac{a}{b}\right )^{\frac{1}{5}}}\right )}{10 \, a b} + \frac{\left (-a b^{4}\right )^{\frac{1}{5}} \log \left (x^{2} + \frac{1}{2} \, x{\left (\sqrt{5} \left (-\frac{a}{b}\right )^{\frac{1}{5}} + \left (-\frac{a}{b}\right )^{\frac{1}{5}}\right )} + \left (-\frac{a}{b}\right )^{\frac{2}{5}}\right )}{5 \, a b{\left (\sqrt{5} - 1\right )}} - \frac{\left (-a b^{4}\right )^{\frac{1}{5}} \log \left (x^{2} - \frac{1}{2} \, x{\left (\sqrt{5} \left (-\frac{a}{b}\right )^{\frac{1}{5}} - \left (-\frac{a}{b}\right )^{\frac{1}{5}}\right )} + \left (-\frac{a}{b}\right )^{\frac{2}{5}}\right )}{5 \, a b{\left (\sqrt{5} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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