Optimal. Leaf size=233 \[ -\frac{a \log \left (\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{16 \sqrt{2} b^{5/4}}+\frac{a \log \left (\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{16 \sqrt{2} b^{5/4}}-\frac{a \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{8 \sqrt{2} b^{5/4}}+\frac{a \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{8 \sqrt{2} b^{5/4}}-\frac{x \left (a-b x^4\right )^{3/4}}{4 b} \]
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Rubi [A] time = 0.109368, antiderivative size = 233, normalized size of antiderivative = 1., number of steps used = 11, 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} \[ -\frac{a \log \left (\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{16 \sqrt{2} b^{5/4}}+\frac{a \log \left (\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{16 \sqrt{2} b^{5/4}}-\frac{a \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{8 \sqrt{2} b^{5/4}}+\frac{a \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{8 \sqrt{2} b^{5/4}}-\frac{x \left (a-b x^4\right )^{3/4}}{4 b} \]
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^4}{\sqrt [4]{a-b x^4}} \, dx &=-\frac{x \left (a-b x^4\right )^{3/4}}{4 b}+\frac{a \int \frac{1}{\sqrt [4]{a-b x^4}} \, dx}{4 b}\\ &=-\frac{x \left (a-b x^4\right )^{3/4}}{4 b}+\frac{a \operatorname{Subst}\left (\int \frac{1}{1+b x^4} \, dx,x,\frac{x}{\sqrt [4]{a-b x^4}}\right )}{4 b}\\ &=-\frac{x \left (a-b x^4\right )^{3/4}}{4 b}+\frac{a \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 )}{8 b}+\frac{a \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 )}{8 b}\\ &=-\frac{x \left (a-b x^4\right )^{3/4}}{4 b}+\frac{a \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 )}{16 b^{3/2}}+\frac{a \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 )}{16 b^{3/2}}-\frac{a \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 )}{16 \sqrt{2} b^{5/4}}-\frac{a \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 )}{16 \sqrt{2} b^{5/4}}\\ &=-\frac{x \left (a-b x^4\right )^{3/4}}{4 b}-\frac{a \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 )}{16 \sqrt{2} b^{5/4}}+\frac{a \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 )}{16 \sqrt{2} b^{5/4}}+\frac{a \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 )}{8 \sqrt{2} b^{5/4}}-\frac{a \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 )}{8 \sqrt{2} b^{5/4}}\\ &=-\frac{x \left (a-b x^4\right )^{3/4}}{4 b}-\frac{a \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{8 \sqrt{2} b^{5/4}}+\frac{a \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{8 \sqrt{2} b^{5/4}}-\frac{a \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 )}{16 \sqrt{2} b^{5/4}}+\frac{a \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 )}{16 \sqrt{2} b^{5/4}}\\ \end{align*}
Mathematica [A] time = 0.0503804, size = 213, normalized size = 0.91 \[ -\frac{8 \sqrt [4]{b} x \left (a-b x^4\right )^{3/4}+\sqrt{2} a \log \left (\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )-\sqrt{2} a \log \left (\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )+2 \sqrt{2} a \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )-2 \sqrt{2} a \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{32 b^{5/4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.026, size = 0, normalized size = 0. \begin{align*} \int{{x}^{4}{\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 = 1.84554, size = 486, normalized size = 2.09 \begin{align*} \frac{4 \, b \left (-\frac{a^{4}}{b^{5}}\right )^{\frac{1}{4}} \arctan \left (-\frac{{\left (-b x^{4} + a\right )}^{\frac{1}{4}} a^{3} b \left (-\frac{a^{4}}{b^{5}}\right )^{\frac{1}{4}} - b x \left (-\frac{a^{4}}{b^{5}}\right )^{\frac{1}{4}} \sqrt{-\frac{a^{4} b^{3} x^{2} \sqrt{-\frac{a^{4}}{b^{5}}} - \sqrt{-b x^{4} + a} a^{6}}{x^{2}}}}{a^{4} x}\right ) - b \left (-\frac{a^{4}}{b^{5}}\right )^{\frac{1}{4}} \log \left (\frac{b^{4} x \left (-\frac{a^{4}}{b^{5}}\right )^{\frac{3}{4}} +{\left (-b x^{4} + a\right )}^{\frac{1}{4}} a^{3}}{x}\right ) + b \left (-\frac{a^{4}}{b^{5}}\right )^{\frac{1}{4}} \log \left (-\frac{b^{4} x \left (-\frac{a^{4}}{b^{5}}\right )^{\frac{3}{4}} -{\left (-b x^{4} + a\right )}^{\frac{1}{4}} a^{3}}{x}\right ) - 4 \,{\left (-b x^{4} + a\right )}^{\frac{3}{4}} x}{16 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.73734, size = 39, normalized size = 0.17 \begin{align*} \frac{x^{5} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt [4]{a} \Gamma \left (\frac{9}{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^{4}}{{\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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