Optimal. Leaf size=98 \[ -\frac{5 b^2 \tan ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{3/4}}-\frac{5 b^2 \tanh ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{3/4}}-\frac{5 b \sqrt [4]{a+b x^4}}{32 x^4}-\frac{\left (a+b x^4\right )^{5/4}}{8 x^8} \]
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Rubi [A] time = 0.0566003, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ -\frac{5 b^2 \tan ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{3/4}}-\frac{5 b^2 \tanh ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{3/4}}-\frac{5 b \sqrt [4]{a+b x^4}}{32 x^4}-\frac{\left (a+b x^4\right )^{5/4}}{8 x^8} \]
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{\left (a+b x^4\right )^{5/4}}{x^9} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{(a+b x)^{5/4}}{x^3} \, dx,x,x^4\right )\\ &=-\frac{\left (a+b x^4\right )^{5/4}}{8 x^8}+\frac{1}{32} (5 b) \operatorname{Subst}\left (\int \frac{\sqrt [4]{a+b x}}{x^2} \, dx,x,x^4\right )\\ &=-\frac{5 b \sqrt [4]{a+b x^4}}{32 x^4}-\frac{\left (a+b x^4\right )^{5/4}}{8 x^8}+\frac{1}{128} \left (5 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{x (a+b x)^{3/4}} \, dx,x,x^4\right )\\ &=-\frac{5 b \sqrt [4]{a+b x^4}}{32 x^4}-\frac{\left (a+b x^4\right )^{5/4}}{8 x^8}+\frac{1}{32} (5 b) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^4}{b}} \, dx,x,\sqrt [4]{a+b x^4}\right )\\ &=-\frac{5 b \sqrt [4]{a+b x^4}}{32 x^4}-\frac{\left (a+b x^4\right )^{5/4}}{8 x^8}-\frac{\left (5 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a}-x^2} \, dx,x,\sqrt [4]{a+b x^4}\right )}{64 \sqrt{a}}-\frac{\left (5 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a}+x^2} \, dx,x,\sqrt [4]{a+b x^4}\right )}{64 \sqrt{a}}\\ &=-\frac{5 b \sqrt [4]{a+b x^4}}{32 x^4}-\frac{\left (a+b x^4\right )^{5/4}}{8 x^8}-\frac{5 b^2 \tan ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{3/4}}-\frac{5 b^2 \tanh ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{3/4}}\\ \end{align*}
Mathematica [C] time = 0.009595, size = 39, normalized size = 0.4 \[ -\frac{b^2 \left (a+b x^4\right )^{9/4} \, _2F_1\left (\frac{9}{4},3;\frac{13}{4};\frac{b x^4}{a}+1\right )}{9 a^3} \]
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}} \left ( b{x}^{4}+a \right ) ^{{\frac{5}{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 = 1.59556, size = 471, normalized size = 4.81 \begin{align*} \frac{20 \, \left (\frac{b^{8}}{a^{3}}\right )^{\frac{1}{4}} x^{8} \arctan \left (-\frac{\left (\frac{b^{8}}{a^{3}}\right )^{\frac{3}{4}}{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{2} b^{2} - \left (\frac{b^{8}}{a^{3}}\right )^{\frac{3}{4}} \sqrt{\sqrt{b x^{4} + a} b^{4} + \sqrt{\frac{b^{8}}{a^{3}}} a^{2}} a^{2}}{b^{8}}\right ) - 5 \, \left (\frac{b^{8}}{a^{3}}\right )^{\frac{1}{4}} x^{8} \log \left (5 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} b^{2} + 5 \, \left (\frac{b^{8}}{a^{3}}\right )^{\frac{1}{4}} a\right ) + 5 \, \left (\frac{b^{8}}{a^{3}}\right )^{\frac{1}{4}} x^{8} \log \left (5 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} b^{2} - 5 \, \left (\frac{b^{8}}{a^{3}}\right )^{\frac{1}{4}} a\right ) - 4 \,{\left (9 \, b x^{4} + 4 \, a\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{128 \, x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 4.17609, size = 41, normalized size = 0.42 \begin{align*} - \frac{b^{\frac{5}{4}} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{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 = 1.13146, size = 301, normalized size = 3.07 \begin{align*} -\frac{1}{256} \, b^{2}{\left (\frac{10 \, \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{10 \, \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{5 \, \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{5 \, \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 (9 \,{\left (b x^{4} + a\right )}^{\frac{5}{4}} - 5 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} a\right )}}{b^{2} x^{8}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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