Optimal. Leaf size=70 \[ \frac{1}{2} a^{3/4} \tan ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )-\frac{1}{2} a^{3/4} \tanh ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )+\frac{1}{3} \left (a+b x^4\right )^{3/4} \]
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Rubi [A] time = 0.0463159, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {266, 50, 63, 298, 203, 206} \[ \frac{1}{2} a^{3/4} \tan ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )-\frac{1}{2} a^{3/4} \tanh ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )+\frac{1}{3} \left (a+b x^4\right )^{3/4} \]
Antiderivative was successfully verified.
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Rule 266
Rule 50
Rule 63
Rule 298
Rule 203
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (a+b x^4\right )^{3/4}}{x} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{(a+b x)^{3/4}}{x} \, dx,x,x^4\right )\\ &=\frac{1}{3} \left (a+b x^4\right )^{3/4}+\frac{1}{4} a \operatorname{Subst}\left (\int \frac{1}{x \sqrt [4]{a+b x}} \, dx,x,x^4\right )\\ &=\frac{1}{3} \left (a+b x^4\right )^{3/4}+\frac{a \operatorname{Subst}\left (\int \frac{x^2}{-\frac{a}{b}+\frac{x^4}{b}} \, dx,x,\sqrt [4]{a+b x^4}\right )}{b}\\ &=\frac{1}{3} \left (a+b x^4\right )^{3/4}-\frac{1}{2} a \operatorname{Subst}\left (\int \frac{1}{\sqrt{a}-x^2} \, dx,x,\sqrt [4]{a+b x^4}\right )+\frac{1}{2} a \operatorname{Subst}\left (\int \frac{1}{\sqrt{a}+x^2} \, dx,x,\sqrt [4]{a+b x^4}\right )\\ &=\frac{1}{3} \left (a+b x^4\right )^{3/4}+\frac{1}{2} a^{3/4} \tan ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )-\frac{1}{2} a^{3/4} \tanh ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )\\ \end{align*}
Mathematica [A] time = 0.0181698, size = 70, normalized size = 1. \[ \frac{1}{2} a^{3/4} \tan ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )-\frac{1}{2} a^{3/4} \tanh ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )+\frac{1}{3} \left (a+b x^4\right )^{3/4} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.024, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x} \left ( b{x}^{4}+a \right ) ^{{\frac{3}{4}}}}\, dx \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 = 1.9279, size = 352, normalized size = 5.03 \begin{align*} -{\left (a^{3}\right )}^{\frac{1}{4}} \arctan \left (-\frac{{\left (b x^{4} + a\right )}^{\frac{1}{4}}{\left (a^{3}\right )}^{\frac{1}{4}} a^{2} - \sqrt{\sqrt{b x^{4} + a} a^{4} + \sqrt{a^{3}} a^{3}}{\left (a^{3}\right )}^{\frac{1}{4}}}{a^{3}}\right ) - \frac{1}{4} \,{\left (a^{3}\right )}^{\frac{1}{4}} \log \left ({\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{2} +{\left (a^{3}\right )}^{\frac{3}{4}}\right ) + \frac{1}{4} \,{\left (a^{3}\right )}^{\frac{1}{4}} \log \left ({\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{2} -{\left (a^{3}\right )}^{\frac{3}{4}}\right ) + \frac{1}{3} \,{\left (b x^{4} + a\right )}^{\frac{3}{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.72361, size = 44, normalized size = 0.63 \begin{align*} - \frac{b^{\frac{3}{4}} x^{3} \Gamma \left (- \frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, - \frac{3}{4} \\ \frac{1}{4} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{4}}} \right )}}{4 \Gamma \left (\frac{1}{4}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.14142, size = 250, normalized size = 3.57 \begin{align*} -\frac{1}{4} \, \sqrt{2} \left (-a\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (-a\right )^{\frac{1}{4}} + 2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}\right )}}{2 \, \left (-a\right )^{\frac{1}{4}}}\right ) - \frac{1}{4} \, \sqrt{2} \left (-a\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (-a\right )^{\frac{1}{4}} - 2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}\right )}}{2 \, \left (-a\right )^{\frac{1}{4}}}\right ) + \frac{1}{8} \, \sqrt{2} \left (-a\right )^{\frac{3}{4}} \log \left (\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-a\right )^{\frac{1}{4}} + \sqrt{b x^{4} + a} + \sqrt{-a}\right ) - \frac{1}{8} \, \sqrt{2} \left (-a\right )^{\frac{3}{4}} \log \left (-\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-a\right )^{\frac{1}{4}} + \sqrt{b x^{4} + a} + \sqrt{-a}\right ) + \frac{1}{3} \,{\left (b x^{4} + a\right )}^{\frac{3}{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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