Optimal. Leaf size=85 \[ -\frac{\left (a-b x^4\right )^{3/4}}{2 a x^2}-\frac{\sqrt{b} \sqrt [4]{1-\frac{b x^4}{a}} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{2 \sqrt{a} \sqrt [4]{a-b x^4}} \]
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Rubi [A] time = 0.0495269, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {275, 325, 229, 228} \[ -\frac{\left (a-b x^4\right )^{3/4}}{2 a x^2}-\frac{\sqrt{b} \sqrt [4]{1-\frac{b x^4}{a}} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{2 \sqrt{a} \sqrt [4]{a-b x^4}} \]
Antiderivative was successfully verified.
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Rule 275
Rule 325
Rule 229
Rule 228
Rubi steps
\begin{align*} \int \frac{1}{x^3 \sqrt [4]{a-b x^4}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt [4]{a-b x^2}} \, dx,x,x^2\right )\\ &=-\frac{\left (a-b x^4\right )^{3/4}}{2 a x^2}-\frac{b \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{a-b x^2}} \, dx,x,x^2\right )}{4 a}\\ &=-\frac{\left (a-b x^4\right )^{3/4}}{2 a x^2}-\frac{\left (b \sqrt [4]{1-\frac{b x^4}{a}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{1-\frac{b x^2}{a}}} \, dx,x,x^2\right )}{4 a \sqrt [4]{a-b x^4}}\\ &=-\frac{\left (a-b x^4\right )^{3/4}}{2 a x^2}-\frac{\sqrt{b} \sqrt [4]{1-\frac{b x^4}{a}} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{2 \sqrt{a} \sqrt [4]{a-b x^4}}\\ \end{align*}
Mathematica [C] time = 0.0109621, size = 52, normalized size = 0.61 \[ -\frac{\sqrt [4]{1-\frac{b x^4}{a}} \, _2F_1\left (-\frac{1}{2},\frac{1}{4};\frac{1}{2};\frac{b x^4}{a}\right )}{2 x^2 \sqrt [4]{a-b x^4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.026, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}}{\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{1}{{\left (-b x^{4} + a\right )}^{\frac{1}{4}} x^{3}}\,{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}}}{b x^{7} - a x^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.06809, size = 32, normalized size = 0.38 \begin{align*} - \frac{{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{1}{4} \\ \frac{1}{2} \end{matrix}\middle |{\frac{b x^{4} e^{2 i \pi }}{a}} \right )}}{2 \sqrt [4]{a} x^{2}} \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{1}{4}} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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