Optimal. Leaf size=75 \[ -\frac{b \tan ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 a^{3/4}}-\frac{b \tanh ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 a^{3/4}}-\frac{\sqrt [4]{a+b x^4}}{4 x^4} \]
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Rubi [A] time = 0.044144, antiderivative size = 75, 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, 47, 63, 212, 206, 203} \[ -\frac{b \tan ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 a^{3/4}}-\frac{b \tanh ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 a^{3/4}}-\frac{\sqrt [4]{a+b x^4}}{4 x^4} \]
Antiderivative was successfully verified.
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Rule 266
Rule 47
Rule 63
Rule 212
Rule 206
Rule 203
Rubi steps
\begin{align*} \int \frac{\sqrt [4]{a+b x^4}}{x^5} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{\sqrt [4]{a+b x}}{x^2} \, dx,x,x^4\right )\\ &=-\frac{\sqrt [4]{a+b x^4}}{4 x^4}+\frac{1}{16} b \operatorname{Subst}\left (\int \frac{1}{x (a+b x)^{3/4}} \, dx,x,x^4\right )\\ &=-\frac{\sqrt [4]{a+b x^4}}{4 x^4}+\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^4}{b}} \, dx,x,\sqrt [4]{a+b x^4}\right )\\ &=-\frac{\sqrt [4]{a+b x^4}}{4 x^4}-\frac{b \operatorname{Subst}\left (\int \frac{1}{\sqrt{a}-x^2} \, dx,x,\sqrt [4]{a+b x^4}\right )}{8 \sqrt{a}}-\frac{b \operatorname{Subst}\left (\int \frac{1}{\sqrt{a}+x^2} \, dx,x,\sqrt [4]{a+b x^4}\right )}{8 \sqrt{a}}\\ &=-\frac{\sqrt [4]{a+b x^4}}{4 x^4}-\frac{b \tan ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 a^{3/4}}-\frac{b \tanh ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 a^{3/4}}\\ \end{align*}
Mathematica [C] time = 0.0082482, size = 37, normalized size = 0.49 \[ \frac{b \left (a+b x^4\right )^{5/4} \, _2F_1\left (\frac{5}{4},2;\frac{9}{4};\frac{b x^4}{a}+1\right )}{5 a^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.032, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{5}}\sqrt [4]{b{x}^{4}+a}}\, 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 = 1.65971, size = 423, normalized size = 5.64 \begin{align*} \frac{4 \, \left (\frac{b^{4}}{a^{3}}\right )^{\frac{1}{4}} x^{4} \arctan \left (-\frac{{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{2} b \left (\frac{b^{4}}{a^{3}}\right )^{\frac{3}{4}} - \sqrt{\sqrt{b x^{4} + a} b^{2} + a^{2} \sqrt{\frac{b^{4}}{a^{3}}}} a^{2} \left (\frac{b^{4}}{a^{3}}\right )^{\frac{3}{4}}}{b^{4}}\right ) - \left (\frac{b^{4}}{a^{3}}\right )^{\frac{1}{4}} x^{4} \log \left ({\left (b x^{4} + a\right )}^{\frac{1}{4}} b + a \left (\frac{b^{4}}{a^{3}}\right )^{\frac{1}{4}}\right ) + \left (\frac{b^{4}}{a^{3}}\right )^{\frac{1}{4}} x^{4} \log \left ({\left (b x^{4} + a\right )}^{\frac{1}{4}} b - a \left (\frac{b^{4}}{a^{3}}\right )^{\frac{1}{4}}\right ) - 4 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{16 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 2.02204, size = 41, normalized size = 0.55 \begin{align*} - \frac{\sqrt [4]{b} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{4}}} \right )}}{4 x^{3} \Gamma \left (\frac{7}{4}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.19041, size = 277, normalized size = 3.69 \begin{align*} -\frac{1}{32} \, b{\left (\frac{2 \, \sqrt{2} \left (-a\right )^{\frac{1}{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 )}{a} + \frac{2 \, \sqrt{2} \left (-a\right )^{\frac{1}{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 )}{a} + \frac{\sqrt{2} \left (-a\right )^{\frac{1}{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 )}{a} - \frac{\sqrt{2} \left (-a\right )^{\frac{1}{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 )}{a} + \frac{8 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{b x^{4}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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