Optimal. Leaf size=88 \[ -\frac{2 b^{3/2} x^3 \left (1-\frac{a}{b x^4}\right )^{3/4} \text{EllipticF}\left (\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right ),2\right )}{3 a^{3/2} \left (a-b x^4\right )^{3/4}}-\frac{\sqrt [4]{a-b x^4}}{3 a x^3} \]
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Rubi [A] time = 0.0365147, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {325, 237, 335, 275, 232} \[ -\frac{2 b^{3/2} x^3 \left (1-\frac{a}{b x^4}\right )^{3/4} F\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{3 a^{3/2} \left (a-b x^4\right )^{3/4}}-\frac{\sqrt [4]{a-b x^4}}{3 a x^3} \]
Antiderivative was successfully verified.
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Rule 325
Rule 237
Rule 335
Rule 275
Rule 232
Rubi steps
\begin{align*} \int \frac{1}{x^4 \left (a-b x^4\right )^{3/4}} \, dx &=-\frac{\sqrt [4]{a-b x^4}}{3 a x^3}+\frac{(2 b) \int \frac{1}{\left (a-b x^4\right )^{3/4}} \, dx}{3 a}\\ &=-\frac{\sqrt [4]{a-b x^4}}{3 a x^3}+\frac{\left (2 b \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}{3 a \left (a-b x^4\right )^{3/4}}\\ &=-\frac{\sqrt [4]{a-b x^4}}{3 a x^3}-\frac{\left (2 b \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 )}{3 a \left (a-b x^4\right )^{3/4}}\\ &=-\frac{\sqrt [4]{a-b x^4}}{3 a x^3}-\frac{\left (b \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 )}{3 a \left (a-b x^4\right )^{3/4}}\\ &=-\frac{\sqrt [4]{a-b x^4}}{3 a x^3}-\frac{2 b^{3/2} \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 )}{3 a^{3/2} \left (a-b x^4\right )^{3/4}}\\ \end{align*}
Mathematica [C] time = 0.0096148, size = 52, normalized size = 0.59 \[ -\frac{\left (1-\frac{b x^4}{a}\right )^{3/4} \, _2F_1\left (-\frac{3}{4},\frac{3}{4};\frac{1}{4};\frac{b x^4}{a}\right )}{3 x^3 \left (a-b x^4\right )^{3/4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.018, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{4}} \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}} x^{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^{8} - a x^{4}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.42413, size = 34, normalized size = 0.39 \begin{align*} - \frac{i e^{\frac{3 i \pi }{4}}{{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{\frac{a}{b x^{4}}} \right )}}{6 b^{\frac{3}{4}} x^{6}} \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}} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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