Optimal. Leaf size=264 \[ -\frac{5 a^2 \log \left (\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{128 \sqrt{2} b^{9/4}}+\frac{5 a^2 \log \left (\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{128 \sqrt{2} b^{9/4}}-\frac{5 a^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{64 \sqrt{2} b^{9/4}}+\frac{5 a^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{64 \sqrt{2} b^{9/4}}-\frac{5 a x \left (a-b x^4\right )^{3/4}}{32 b^2}-\frac{x^5 \left (a-b x^4\right )^{3/4}}{8 b} \]
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Rubi [A] time = 0.135438, antiderivative size = 264, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {321, 240, 211, 1165, 628, 1162, 617, 204} \[ -\frac{5 a^2 \log \left (\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{128 \sqrt{2} b^{9/4}}+\frac{5 a^2 \log \left (\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{128 \sqrt{2} b^{9/4}}-\frac{5 a^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{64 \sqrt{2} b^{9/4}}+\frac{5 a^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{64 \sqrt{2} b^{9/4}}-\frac{5 a x \left (a-b x^4\right )^{3/4}}{32 b^2}-\frac{x^5 \left (a-b x^4\right )^{3/4}}{8 b} \]
Antiderivative was successfully verified.
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Rule 321
Rule 240
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
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-\sqrt{b} x^2}{1+b x^4} \, 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+\sqrt{b} x^2}{1+b x^4} \, 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{\left (5 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{\sqrt{b}}-\frac{\sqrt{2} x}{\sqrt [4]{b}}+x^2} \, dx,x,\frac{x}{\sqrt [4]{a-b x^4}}\right )}{128 b^{5/2}}+\frac{\left (5 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{\sqrt{b}}+\frac{\sqrt{2} x}{\sqrt [4]{b}}+x^2} \, dx,x,\frac{x}{\sqrt [4]{a-b x^4}}\right )}{128 b^{5/2}}-\frac{\left (5 a^2\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2}}{\sqrt [4]{b}}+2 x}{-\frac{1}{\sqrt{b}}-\frac{\sqrt{2} x}{\sqrt [4]{b}}-x^2} \, dx,x,\frac{x}{\sqrt [4]{a-b x^4}}\right )}{128 \sqrt{2} b^{9/4}}-\frac{\left (5 a^2\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2}}{\sqrt [4]{b}}-2 x}{-\frac{1}{\sqrt{b}}+\frac{\sqrt{2} x}{\sqrt [4]{b}}-x^2} \, dx,x,\frac{x}{\sqrt [4]{a-b x^4}}\right )}{128 \sqrt{2} 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}-\frac{5 a^2 \log \left (1+\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{128 \sqrt{2} b^{9/4}}+\frac{5 a^2 \log \left (1+\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{128 \sqrt{2} b^{9/4}}+\frac{\left (5 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{64 \sqrt{2} b^{9/4}}-\frac{\left (5 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{64 \sqrt{2} 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}-\frac{5 a^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{64 \sqrt{2} b^{9/4}}+\frac{5 a^2 \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{64 \sqrt{2} b^{9/4}}-\frac{5 a^2 \log \left (1+\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{128 \sqrt{2} b^{9/4}}+\frac{5 a^2 \log \left (1+\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{128 \sqrt{2} b^{9/4}}\\ \end{align*}
Mathematica [A] time = 0.129546, size = 245, normalized size = 0.93 \[ -\frac{5 \sqrt{2} a^2 \log \left (\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )-5 \sqrt{2} a^2 \log \left (\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )+10 \sqrt{2} a^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )-10 \sqrt{2} a^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )+32 b^{5/4} x^5 \left (a-b x^4\right )^{3/4}+40 a \sqrt [4]{b} x \left (a-b x^4\right )^{3/4}}{256 b^{9/4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.029, 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 [A] time = 1.95223, size = 544, normalized size = 2.06 \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 = 1.79394, size = 39, normalized size = 0.15 \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^{2 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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