Optimal. Leaf size=104 \[ \frac{5 a^2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{9/4}}+\frac{5 a^2 \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{9/4}}-\frac{5 a x \left (a+b x^4\right )^{3/4}}{32 b^2}+\frac{x^5 \left (a+b x^4\right )^{3/4}}{8 b} \]
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Rubi [A] time = 0.0314346, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {321, 240, 212, 206, 203} \[ \frac{5 a^2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{9/4}}+\frac{5 a^2 \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{9/4}}-\frac{5 a x \left (a+b x^4\right )^{3/4}}{32 b^2}+\frac{x^5 \left (a+b x^4\right )^{3/4}}{8 b} \]
Antiderivative was successfully verified.
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Rule 321
Rule 240
Rule 212
Rule 206
Rule 203
Rubi steps
\begin{align*} \int \frac{x^8}{\sqrt [4]{a+b x^4}} \, dx &=\frac{x^5 \left (a+b x^4\right )^{3/4}}{8 b}-\frac{(5 a) \int \frac{x^4}{\sqrt [4]{a+b x^4}} \, dx}{8 b}\\ &=-\frac{5 a x \left (a+b x^4\right )^{3/4}}{32 b^2}+\frac{x^5 \left (a+b x^4\right )^{3/4}}{8 b}+\frac{\left (5 a^2\right ) \int \frac{1}{\sqrt [4]{a+b x^4}} \, dx}{32 b^2}\\ &=-\frac{5 a x \left (a+b x^4\right )^{3/4}}{32 b^2}+\frac{x^5 \left (a+b x^4\right )^{3/4}}{8 b}+\frac{\left (5 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^4} \, dx,x,\frac{x}{\sqrt [4]{a+b x^4}}\right )}{32 b^2}\\ &=-\frac{5 a x \left (a+b x^4\right )^{3/4}}{32 b^2}+\frac{x^5 \left (a+b x^4\right )^{3/4}}{8 b}+\frac{\left (5 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{b} x^2} \, dx,x,\frac{x}{\sqrt [4]{a+b x^4}}\right )}{64 b^2}+\frac{\left (5 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{b} x^2} \, dx,x,\frac{x}{\sqrt [4]{a+b x^4}}\right )}{64 b^2}\\ &=-\frac{5 a x \left (a+b x^4\right )^{3/4}}{32 b^2}+\frac{x^5 \left (a+b x^4\right )^{3/4}}{8 b}+\frac{5 a^2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{9/4}}+\frac{5 a^2 \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{9/4}}\\ \end{align*}
Mathematica [A] time = 0.0377192, size = 87, normalized size = 0.84 \[ \frac{5 a^2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )+5 a^2 \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )+2 \sqrt [4]{b} x \left (a+b x^4\right )^{3/4} \left (4 b x^4-5 a\right )}{64 b^{9/4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.028, size = 0, normalized size = 0. \begin{align*} \int{{x}^{8}{\frac{1}{\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 = 1.60881, size = 525, normalized size = 5.05 \begin{align*} \frac{20 \, b^{2} \left (\frac{a^{8}}{b^{9}}\right )^{\frac{1}{4}} \arctan \left (-\frac{{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{6} b^{2} \left (\frac{a^{8}}{b^{9}}\right )^{\frac{1}{4}} - b^{2} x \left (\frac{a^{8}}{b^{9}}\right )^{\frac{1}{4}} \sqrt{\frac{a^{8} b^{5} x^{2} \sqrt{\frac{a^{8}}{b^{9}}} + \sqrt{b x^{4} + a} a^{12}}{x^{2}}}}{a^{8} x}\right ) + 5 \, b^{2} \left (\frac{a^{8}}{b^{9}}\right )^{\frac{1}{4}} \log \left (\frac{125 \,{\left (b^{7} x \left (\frac{a^{8}}{b^{9}}\right )^{\frac{3}{4}} +{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{6}\right )}}{x}\right ) - 5 \, b^{2} \left (\frac{a^{8}}{b^{9}}\right )^{\frac{1}{4}} \log \left (-\frac{125 \,{\left (b^{7} x \left (\frac{a^{8}}{b^{9}}\right )^{\frac{3}{4}} -{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{6}\right )}}{x}\right ) + 4 \,{\left (4 \, b x^{5} - 5 \, a x\right )}{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{128 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 2.16015, size = 37, normalized size = 0.36 \begin{align*} \frac{x^{9} \Gamma \left (\frac{9}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{9}{4} \\ \frac{13}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt [4]{a} \Gamma \left (\frac{13}{4}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{8}}{{\left (b x^{4} + a\right )}^{\frac{1}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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