Optimal. Leaf size=310 \[ \frac{\left (1-\sqrt{5}\right ) \sqrt [5]{b} \log \left (a^{2/5} x^2-\frac{1}{2} \left (1-\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+b^{2/5}\right )}{20 a^{6/5}}+\frac{\left (1+\sqrt{5}\right ) \sqrt [5]{b} \log \left (a^{2/5} x^2-\frac{1}{2} \left (1+\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+b^{2/5}\right )}{20 a^{6/5}}-\frac{\sqrt [5]{b} \log \left (\sqrt [5]{a} x+\sqrt [5]{b}\right )}{5 a^{6/5}}-\frac{\sqrt{\frac{1}{2} \left (5+\sqrt{5}\right )} \sqrt [5]{b} \tan ^{-1}\left (\frac{2 \sqrt{\frac{2}{5+\sqrt{5}}} \sqrt [5]{a} x}{\sqrt [5]{b}}+\sqrt{\frac{1}{5} \left (5-2 \sqrt{5}\right )}\right )}{5 a^{6/5}}+\frac{\sqrt{\frac{1}{2} \left (5-\sqrt{5}\right )} \sqrt [5]{b} \tan ^{-1}\left (\sqrt{\frac{1}{5} \left (5+2 \sqrt{5}\right )}-\frac{\sqrt{\frac{2}{5} \left (5+\sqrt{5}\right )} \sqrt [5]{a} x}{\sqrt [5]{b}}\right )}{5 a^{6/5}}+\frac{x}{a} \]
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Rubi [A] time = 0.65755, antiderivative size = 310, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.889, Rules used = {193, 321, 201, 634, 618, 204, 628, 31} \[ \frac{\left (1-\sqrt{5}\right ) \sqrt [5]{b} \log \left (a^{2/5} x^2-\frac{1}{2} \left (1-\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+b^{2/5}\right )}{20 a^{6/5}}+\frac{\left (1+\sqrt{5}\right ) \sqrt [5]{b} \log \left (a^{2/5} x^2-\frac{1}{2} \left (1+\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+b^{2/5}\right )}{20 a^{6/5}}-\frac{\sqrt [5]{b} \log \left (\sqrt [5]{a} x+\sqrt [5]{b}\right )}{5 a^{6/5}}-\frac{\sqrt{\frac{1}{2} \left (5+\sqrt{5}\right )} \sqrt [5]{b} \tan ^{-1}\left (\frac{2 \sqrt{\frac{2}{5+\sqrt{5}}} \sqrt [5]{a} x}{\sqrt [5]{b}}+\sqrt{\frac{1}{5} \left (5-2 \sqrt{5}\right )}\right )}{5 a^{6/5}}+\frac{\sqrt{\frac{1}{2} \left (5-\sqrt{5}\right )} \sqrt [5]{b} \tan ^{-1}\left (\sqrt{\frac{1}{5} \left (5+2 \sqrt{5}\right )}-\frac{\sqrt{\frac{2}{5} \left (5+\sqrt{5}\right )} \sqrt [5]{a} x}{\sqrt [5]{b}}\right )}{5 a^{6/5}}+\frac{x}{a} \]
Antiderivative was successfully verified.
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Rule 193
Rule 321
Rule 201
Rule 634
Rule 618
Rule 204
Rule 628
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{a+\frac{b}{x^5}} \, dx &=\int \frac{x^5}{b+a x^5} \, dx\\ &=\frac{x}{a}-\frac{b \int \frac{1}{b+a x^5} \, dx}{a}\\ &=\frac{x}{a}-\frac{\sqrt [5]{b} \int \frac{1}{\sqrt [5]{b}+\sqrt [5]{a} x} \, dx}{5 a}-\frac{\left (2 \sqrt [5]{b}\right ) \int \frac{\sqrt [5]{b}-\frac{1}{4} \left (1-\sqrt{5}\right ) \sqrt [5]{a} x}{b^{2/5}-\frac{1}{2} \left (1-\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+a^{2/5} x^2} \, dx}{5 a}-\frac{\left (2 \sqrt [5]{b}\right ) \int \frac{\sqrt [5]{b}-\frac{1}{4} \left (1+\sqrt{5}\right ) \sqrt [5]{a} x}{b^{2/5}-\frac{1}{2} \left (1+\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+a^{2/5} x^2} \, dx}{5 a}\\ &=\frac{x}{a}-\frac{\sqrt [5]{b} \log \left (\sqrt [5]{b}+\sqrt [5]{a} x\right )}{5 a^{6/5}}+\frac{\left (\left (1-\sqrt{5}\right ) \sqrt [5]{b}\right ) \int \frac{-\frac{1}{2} \left (1-\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b}+2 a^{2/5} x}{b^{2/5}-\frac{1}{2} \left (1-\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+a^{2/5} x^2} \, dx}{20 a^{6/5}}+\frac{\left (\left (1+\sqrt{5}\right ) \sqrt [5]{b}\right ) \int \frac{-\frac{1}{2} \left (1+\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b}+2 a^{2/5} x}{b^{2/5}-\frac{1}{2} \left (1+\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+a^{2/5} x^2} \, dx}{20 a^{6/5}}-\frac{\left (\left (5-\sqrt{5}\right ) b^{2/5}\right ) \int \frac{1}{b^{2/5}-\frac{1}{2} \left (1+\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+a^{2/5} x^2} \, dx}{20 a}-\frac{\left (\left (5+\sqrt{5}\right ) b^{2/5}\right ) \int \frac{1}{b^{2/5}-\frac{1}{2} \left (1-\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+a^{2/5} x^2} \, dx}{20 a}\\ &=\frac{x}{a}-\frac{\sqrt [5]{b} \log \left (\sqrt [5]{b}+\sqrt [5]{a} x\right )}{5 a^{6/5}}+\frac{\left (1+\sqrt{5}\right ) \sqrt [5]{b} \log \left (2 b^{2/5}-\sqrt [5]{a} \sqrt [5]{b} x-\sqrt{5} \sqrt [5]{a} \sqrt [5]{b} x+2 a^{2/5} x^2\right )}{20 a^{6/5}}+\frac{\left (1-\sqrt{5}\right ) \sqrt [5]{b} \log \left (2 b^{2/5}-\sqrt [5]{a} \sqrt [5]{b} x+\sqrt{5} \sqrt [5]{a} \sqrt [5]{b} x+2 a^{2/5} x^2\right )}{20 a^{6/5}}+\frac{\left (\left (5-\sqrt{5}\right ) b^{2/5}\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 a^{2/5} x\right )}{10 a}+\frac{\left (\left (5+\sqrt{5}\right ) b^{2/5}\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 a^{2/5} x\right )}{10 a}\\ &=\frac{x}{a}+\frac{\sqrt{\frac{1}{2} \left (5-\sqrt{5}\right )} \sqrt [5]{b} \tan ^{-1}\left (\frac{\sqrt{5+\sqrt{5}} \left (\left (1+\sqrt{5}\right ) \sqrt [5]{b}-4 \sqrt [5]{a} x\right )}{2 \sqrt{10} \sqrt [5]{b}}\right )}{5 a^{6/5}}-\frac{\sqrt{\frac{1}{2} \left (5+\sqrt{5}\right )} \sqrt [5]{b} \tan ^{-1}\left (\sqrt{\frac{1}{5} \left (5-2 \sqrt{5}\right )}+\frac{2 \sqrt{\frac{2}{5+\sqrt{5}}} \sqrt [5]{a} x}{\sqrt [5]{b}}\right )}{5 a^{6/5}}-\frac{\sqrt [5]{b} \log \left (\sqrt [5]{b}+\sqrt [5]{a} x\right )}{5 a^{6/5}}+\frac{\left (1+\sqrt{5}\right ) \sqrt [5]{b} \log \left (2 b^{2/5}-\sqrt [5]{a} \sqrt [5]{b} x-\sqrt{5} \sqrt [5]{a} \sqrt [5]{b} x+2 a^{2/5} x^2\right )}{20 a^{6/5}}+\frac{\left (1-\sqrt{5}\right ) \sqrt [5]{b} \log \left (2 b^{2/5}-\sqrt [5]{a} \sqrt [5]{b} x+\sqrt{5} \sqrt [5]{a} \sqrt [5]{b} x+2 a^{2/5} x^2\right )}{20 a^{6/5}}\\ \end{align*}
Mathematica [A] time = 0.184218, size = 267, normalized size = 0.86 \[ \frac{-\left (\sqrt{5}-1\right ) \sqrt [5]{b} \log \left (a^{2/5} x^2+\frac{1}{2} \left (\sqrt{5}-1\right ) \sqrt [5]{a} \sqrt [5]{b} x+b^{2/5}\right )+\left (1+\sqrt{5}\right ) \sqrt [5]{b} \log \left (a^{2/5} x^2-\frac{1}{2} \left (1+\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+b^{2/5}\right )-4 \sqrt [5]{b} \log \left (\sqrt [5]{a} x+\sqrt [5]{b}\right )-2 \sqrt{2 \left (5+\sqrt{5}\right )} \sqrt [5]{b} \tan ^{-1}\left (\frac{4 \sqrt [5]{a} x+\left (\sqrt{5}-1\right ) \sqrt [5]{b}}{\sqrt{2 \left (5+\sqrt{5}\right )} \sqrt [5]{b}}\right )-2 \sqrt{10-2 \sqrt{5}} \sqrt [5]{b} \tan ^{-1}\left (\frac{4 \sqrt [5]{a} x-\left (1+\sqrt{5}\right ) \sqrt [5]{b}}{\sqrt{10-2 \sqrt{5}} \sqrt [5]{b}}\right )+20 \sqrt [5]{a} x}{20 a^{6/5}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.059, size = 911, normalized size = 2.9 \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 = 1.47183, size = 441, normalized size = 1.42 \begin{align*} -\frac{\frac{\sqrt{5} b^{\frac{1}{5}}{\left (\sqrt{5} - 1\right )} \log \left (\frac{4 \, a^{\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 \, a^{\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 )}{a^{\frac{1}{5}} \sqrt{2 \, \sqrt{5} - 10}} + \frac{\sqrt{5} b^{\frac{1}{5}}{\left (\sqrt{5} + 1\right )} \log \left (\frac{4 \, a^{\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 \, a^{\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 )}{a^{\frac{1}{5}} \sqrt{-2 \, \sqrt{5} - 10}} - \frac{b^{\frac{1}{5}}{\left (\sqrt{5} + 3\right )} \log \left (2 \, a^{\frac{2}{5}} x^{2} - a^{\frac{1}{5}} b^{\frac{1}{5}} x{\left (\sqrt{5} + 1\right )} + 2 \, b^{\frac{2}{5}}\right )}{a^{\frac{1}{5}}{\left (\sqrt{5} + 1\right )}} - \frac{b^{\frac{1}{5}}{\left (\sqrt{5} - 3\right )} \log \left (2 \, a^{\frac{2}{5}} x^{2} + a^{\frac{1}{5}} b^{\frac{1}{5}} x{\left (\sqrt{5} - 1\right )} + 2 \, b^{\frac{2}{5}}\right )}{a^{\frac{1}{5}}{\left (\sqrt{5} - 1\right )}} + \frac{2 \, b^{\frac{1}{5}} \log \left (a^{\frac{1}{5}} x + b^{\frac{1}{5}}\right )}{a^{\frac{1}{5}}}}{10 \, a} + \frac{x}{a} \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 = 0.321151, size = 22, normalized size = 0.07 \begin{align*} \operatorname{RootSum}{\left (3125 t^{5} a^{6} + b, \left ( t \mapsto t \log{\left (- 5 t a + x \right )} \right )\right )} + \frac{x}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11912, size = 362, normalized size = 1.17 \begin{align*} \frac{\left (-\frac{b}{a}\right )^{\frac{1}{5}} \log \left ({\left | x - \left (-\frac{b}{a}\right )^{\frac{1}{5}} \right |}\right )}{5 \, a} + \frac{x}{a} - \frac{\left (-a^{4} b\right )^{\frac{1}{5}} \sqrt{2 \, \sqrt{5} + 10} \arctan \left (-\frac{{\left (\sqrt{5} - 1\right )} \left (-\frac{b}{a}\right )^{\frac{1}{5}} - 4 \, x}{\sqrt{2 \, \sqrt{5} + 10} \left (-\frac{b}{a}\right )^{\frac{1}{5}}}\right )}{10 \, a^{2}} - \frac{\left (-a^{4} b\right )^{\frac{1}{5}} \sqrt{-2 \, \sqrt{5} + 10} \arctan \left (\frac{{\left (\sqrt{5} + 1\right )} \left (-\frac{b}{a}\right )^{\frac{1}{5}} + 4 \, x}{\sqrt{-2 \, \sqrt{5} + 10} \left (-\frac{b}{a}\right )^{\frac{1}{5}}}\right )}{10 \, a^{2}} - \frac{\left (-a^{4} b\right )^{\frac{1}{5}} \log \left (x^{2} + \frac{1}{2} \, x{\left (\sqrt{5} \left (-\frac{b}{a}\right )^{\frac{1}{5}} + \left (-\frac{b}{a}\right )^{\frac{1}{5}}\right )} + \left (-\frac{b}{a}\right )^{\frac{2}{5}}\right )}{5 \, a^{2}{\left (\sqrt{5} - 1\right )}} + \frac{\left (-a^{4} b\right )^{\frac{1}{5}} \log \left (x^{2} - \frac{1}{2} \, x{\left (\sqrt{5} \left (-\frac{b}{a}\right )^{\frac{1}{5}} - \left (-\frac{b}{a}\right )^{\frac{1}{5}}\right )} + \left (-\frac{b}{a}\right )^{\frac{2}{5}}\right )}{5 \, a^{2}{\left (\sqrt{5} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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