Optimal. Leaf size=83 \[ \frac{x}{3 a \left (a+b x^4\right )^{3/4}}-\frac{2 \sqrt{b} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} \text{EllipticF}\left (\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right ),2\right )}{3 a^{3/2} \left (a+b x^4\right )^{3/4}} \]
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Rubi [A] time = 0.0314403, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454, Rules used = {199, 237, 335, 275, 231} \[ \frac{x}{3 a \left (a+b x^4\right )^{3/4}}-\frac{2 \sqrt{b} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{3 a^{3/2} \left (a+b x^4\right )^{3/4}} \]
Antiderivative was successfully verified.
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Rule 199
Rule 237
Rule 335
Rule 275
Rule 231
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b x^4\right )^{7/4}} \, dx &=\frac{x}{3 a \left (a+b x^4\right )^{3/4}}+\frac{2 \int \frac{1}{\left (a+b x^4\right )^{3/4}} \, dx}{3 a}\\ &=\frac{x}{3 a \left (a+b x^4\right )^{3/4}}+\frac{\left (2 \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{x}{3 a \left (a+b x^4\right )^{3/4}}-\frac{\left (2 \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{x}{3 a \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 )}{3 a \left (a+b x^4\right )^{3/4}}\\ &=\frac{x}{3 a \left (a+b x^4\right )^{3/4}}-\frac{2 \sqrt{b} \left (1+\frac{a}{b x^4}\right )^{3/4} x^3 F\left (\left .\frac{1}{2} \cot ^{-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 = 0.0168727, size = 56, normalized size = 0.67 \[ \frac{2 x \left (\frac{b x^4}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};-\frac{b x^4}{a}\right )+x}{3 a \left (a+b x^4\right )^{3/4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.038, size = 0, normalized size = 0. \begin{align*} \int \left ( b{x}^{4}+a \right ) ^{-{\frac{7}{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{7}{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^{2} x^{8} + 2 \, a b x^{4} + a^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.02983, size = 36, normalized size = 0.43 \begin{align*} \frac{x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{7}{4} \\ \frac{5}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac{7}{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{7}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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