Optimal. Leaf size=109 \[ \frac{a^{3/2} x \sqrt [4]{1-\frac{a}{b x^4}} E\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{4 b^{3/2} \sqrt [4]{a-b x^4}}-\frac{a \left (a-b x^4\right )^{3/4}}{4 b^2 x}-\frac{x^3 \left (a-b x^4\right )^{3/4}}{6 b} \]
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Rubi [A] time = 0.047434, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {321, 311, 313, 335, 275, 228} \[ \frac{a^{3/2} x \sqrt [4]{1-\frac{a}{b x^4}} E\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{4 b^{3/2} \sqrt [4]{a-b x^4}}-\frac{a \left (a-b x^4\right )^{3/4}}{4 b^2 x}-\frac{x^3 \left (a-b x^4\right )^{3/4}}{6 b} \]
Antiderivative was successfully verified.
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Rule 321
Rule 311
Rule 313
Rule 335
Rule 275
Rule 228
Rubi steps
\begin{align*} \int \frac{x^6}{\sqrt [4]{a-b x^4}} \, dx &=-\frac{x^3 \left (a-b x^4\right )^{3/4}}{6 b}+\frac{a \int \frac{x^2}{\sqrt [4]{a-b x^4}} \, dx}{2 b}\\ &=-\frac{a \left (a-b x^4\right )^{3/4}}{4 b^2 x}-\frac{x^3 \left (a-b x^4\right )^{3/4}}{6 b}-\frac{a^2 \int \frac{1}{x^2 \sqrt [4]{a-b x^4}} \, dx}{4 b^2}\\ &=-\frac{a \left (a-b x^4\right )^{3/4}}{4 b^2 x}-\frac{x^3 \left (a-b x^4\right )^{3/4}}{6 b}-\frac{\left (a^2 \sqrt [4]{1-\frac{a}{b x^4}} x\right ) \int \frac{1}{\sqrt [4]{1-\frac{a}{b x^4}} x^3} \, dx}{4 b^2 \sqrt [4]{a-b x^4}}\\ &=-\frac{a \left (a-b x^4\right )^{3/4}}{4 b^2 x}-\frac{x^3 \left (a-b x^4\right )^{3/4}}{6 b}+\frac{\left (a^2 \sqrt [4]{1-\frac{a}{b x^4}} x\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt [4]{1-\frac{a x^4}{b}}} \, dx,x,\frac{1}{x}\right )}{4 b^2 \sqrt [4]{a-b x^4}}\\ &=-\frac{a \left (a-b x^4\right )^{3/4}}{4 b^2 x}-\frac{x^3 \left (a-b x^4\right )^{3/4}}{6 b}+\frac{\left (a^2 \sqrt [4]{1-\frac{a}{b x^4}} x\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{1-\frac{a x^2}{b}}} \, dx,x,\frac{1}{x^2}\right )}{8 b^2 \sqrt [4]{a-b x^4}}\\ &=-\frac{a \left (a-b x^4\right )^{3/4}}{4 b^2 x}-\frac{x^3 \left (a-b x^4\right )^{3/4}}{6 b}+\frac{a^{3/2} \sqrt [4]{1-\frac{a}{b x^4}} x E\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{4 b^{3/2} \sqrt [4]{a-b x^4}}\\ \end{align*}
Mathematica [C] time = 0.0178506, size = 66, normalized size = 0.61 \[ \frac{x^3 \left (a \sqrt [4]{1-\frac{b x^4}{a}} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};\frac{b x^4}{a}\right )-a+b x^4\right )}{6 b \sqrt [4]{a-b x^4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.031, size = 0, normalized size = 0. \begin{align*} \int{{x}^{6}{\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{6}}{{\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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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (-b x^{4} + a\right )}^{\frac{3}{4}} x^{6}}{b x^{4} - a}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.31933, size = 39, normalized size = 0.36 \begin{align*} \frac{x^{7} \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{b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt [4]{a} \Gamma \left (\frac{11}{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^{6}}{{\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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