Optimal. Leaf size=63 \[ -\frac{\sqrt{b} x^3 \left (1-\frac{a}{b x^4}\right )^{3/4} \text{EllipticF}\left (\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right ),2\right )}{\sqrt{a} \left (a-b x^4\right )^{3/4}} \]
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Rubi [A] time = 0.026208, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {237, 335, 275, 232} \[ -\frac{\sqrt{b} x^3 \left (1-\frac{a}{b x^4}\right )^{3/4} F\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{\sqrt{a} \left (a-b x^4\right )^{3/4}} \]
Antiderivative was successfully verified.
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Rule 237
Rule 335
Rule 275
Rule 232
Rubi steps
\begin{align*} \int \frac{1}{\left (a-b x^4\right )^{3/4}} \, dx &=\frac{\left (\left (1-\frac{a}{b x^4}\right )^{3/4} x^3\right ) \int \frac{1}{\left (1-\frac{a}{b x^4}\right )^{3/4} x^3} \, dx}{\left (a-b x^4\right )^{3/4}}\\ &=-\frac{\left (\left (1-\frac{a}{b x^4}\right )^{3/4} x^3\right ) \operatorname{Subst}\left (\int \frac{x}{\left (1-\frac{a x^4}{b}\right )^{3/4}} \, dx,x,\frac{1}{x}\right )}{\left (a-b x^4\right )^{3/4}}\\ &=-\frac{\left (\left (1-\frac{a}{b x^4}\right )^{3/4} x^3\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-\frac{a x^2}{b}\right )^{3/4}} \, dx,x,\frac{1}{x^2}\right )}{2 \left (a-b x^4\right )^{3/4}}\\ &=-\frac{\sqrt{b} \left (1-\frac{a}{b x^4}\right )^{3/4} x^3 F\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{\sqrt{a} \left (a-b x^4\right )^{3/4}}\\ \end{align*}
Mathematica [C] time = 0.0066974, size = 47, normalized size = 0.75 \[ \frac{x \left (1-\frac{b x^4}{a}\right )^{3/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};\frac{b x^4}{a}\right )}{\left (a-b x^4\right )^{3/4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.036, size = 0, normalized size = 0. \begin{align*} \int \left ( -b{x}^{4}+a \right ) ^{-{\frac{3}{4}}}\, 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{1}{{\left (-b x^{4} + a\right )}^{\frac{3}{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{1}{4}}}{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 = 0.965834, size = 37, normalized size = 0.59 \begin{align*} \frac{x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{3}{4} \\ \frac{5}{4} \end{matrix}\middle |{\frac{b x^{4} e^{2 i \pi }}{a}} \right )}}{4 a^{\frac{3}{4}} \Gamma \left (\frac{5}{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{1}{{\left (-b x^{4} + a\right )}^{\frac{3}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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