Optimal. Leaf size=101 \[ \frac{3 b^2 \tan ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{7/4}}+\frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{7/4}}-\frac{b \sqrt [4]{a+b x^4}}{32 a x^4}-\frac{\sqrt [4]{a+b x^4}}{8 x^8} \]
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Rubi [A] time = 0.0593026, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.467, Rules used = {266, 47, 51, 63, 212, 206, 203} \[ \frac{3 b^2 \tan ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{7/4}}+\frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{7/4}}-\frac{b \sqrt [4]{a+b x^4}}{32 a x^4}-\frac{\sqrt [4]{a+b x^4}}{8 x^8} \]
Antiderivative was successfully verified.
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Rule 266
Rule 47
Rule 51
Rule 63
Rule 212
Rule 206
Rule 203
Rubi steps
\begin{align*} \int \frac{\sqrt [4]{a+b x^4}}{x^9} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{\sqrt [4]{a+b x}}{x^3} \, dx,x,x^4\right )\\ &=-\frac{\sqrt [4]{a+b x^4}}{8 x^8}+\frac{1}{32} b \operatorname{Subst}\left (\int \frac{1}{x^2 (a+b x)^{3/4}} \, dx,x,x^4\right )\\ &=-\frac{\sqrt [4]{a+b x^4}}{8 x^8}-\frac{b \sqrt [4]{a+b x^4}}{32 a x^4}-\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{x (a+b x)^{3/4}} \, dx,x,x^4\right )}{128 a}\\ &=-\frac{\sqrt [4]{a+b x^4}}{8 x^8}-\frac{b \sqrt [4]{a+b x^4}}{32 a x^4}-\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^4}{b}} \, dx,x,\sqrt [4]{a+b x^4}\right )}{32 a}\\ &=-\frac{\sqrt [4]{a+b x^4}}{8 x^8}-\frac{b \sqrt [4]{a+b x^4}}{32 a x^4}+\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a}-x^2} \, dx,x,\sqrt [4]{a+b x^4}\right )}{64 a^{3/2}}+\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a}+x^2} \, dx,x,\sqrt [4]{a+b x^4}\right )}{64 a^{3/2}}\\ &=-\frac{\sqrt [4]{a+b x^4}}{8 x^8}-\frac{b \sqrt [4]{a+b x^4}}{32 a x^4}+\frac{3 b^2 \tan ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{7/4}}+\frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{7/4}}\\ \end{align*}
Mathematica [C] time = 0.0086602, size = 39, normalized size = 0.39 \[ -\frac{b^2 \left (a+b x^4\right )^{5/4} \, _2F_1\left (\frac{5}{4},3;\frac{9}{4};\frac{b x^4}{a}+1\right )}{5 a^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.033, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{9}}\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 = 1.66025, size = 489, normalized size = 4.84 \begin{align*} -\frac{12 \, a \left (\frac{b^{8}}{a^{7}}\right )^{\frac{1}{4}} x^{8} \arctan \left (-\frac{{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{5} b^{2} \left (\frac{b^{8}}{a^{7}}\right )^{\frac{3}{4}} - \sqrt{\sqrt{b x^{4} + a} b^{4} + a^{4} \sqrt{\frac{b^{8}}{a^{7}}}} a^{5} \left (\frac{b^{8}}{a^{7}}\right )^{\frac{3}{4}}}{b^{8}}\right ) - 3 \, a \left (\frac{b^{8}}{a^{7}}\right )^{\frac{1}{4}} x^{8} \log \left (3 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} b^{2} + 3 \, a^{2} \left (\frac{b^{8}}{a^{7}}\right )^{\frac{1}{4}}\right ) + 3 \, a \left (\frac{b^{8}}{a^{7}}\right )^{\frac{1}{4}} x^{8} \log \left (3 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} b^{2} - 3 \, a^{2} \left (\frac{b^{8}}{a^{7}}\right )^{\frac{1}{4}}\right ) + 4 \,{\left (b x^{4} + 4 \, a\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{128 \, a x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 2.75431, size = 41, normalized size = 0.41 \begin{align*} - \frac{\sqrt [4]{b} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{4}}} \right )}}{4 x^{7} \Gamma \left (\frac{11}{4}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.16095, size = 302, normalized size = 2.99 \begin{align*} \frac{1}{256} \, b^{2}{\left (\frac{6 \, \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^{2}} + \frac{6 \, \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^{2}} + \frac{3 \, \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^{2}} - \frac{3 \, \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^{2}} - \frac{8 \,{\left ({\left (b x^{4} + a\right )}^{\frac{5}{4}} + 3 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} a\right )}}{a b^{2} x^{8}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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