Optimal. Leaf size=79 \[ \frac{2 \sqrt{b} x \sqrt [4]{\frac{a}{b x^4}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{a^{3/2} \sqrt [4]{a+b x^4}}-\frac{1}{a x \sqrt [4]{a+b x^4}} \]
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Rubi [A] time = 0.037294, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {283, 281, 335, 275, 196} \[ \frac{2 \sqrt{b} x \sqrt [4]{\frac{a}{b x^4}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{a^{3/2} \sqrt [4]{a+b x^4}}-\frac{1}{a x \sqrt [4]{a+b x^4}} \]
Antiderivative was successfully verified.
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Rule 283
Rule 281
Rule 335
Rule 275
Rule 196
Rubi steps
\begin{align*} \int \frac{1}{x^2 \left (a+b x^4\right )^{5/4}} \, dx &=-\frac{1}{a x \sqrt [4]{a+b x^4}}-\frac{(2 b) \int \frac{x^2}{\left (a+b x^4\right )^{5/4}} \, dx}{a}\\ &=-\frac{1}{a x \sqrt [4]{a+b x^4}}-\frac{\left (2 \sqrt [4]{1+\frac{a}{b x^4}} x\right ) \int \frac{1}{\left (1+\frac{a}{b x^4}\right )^{5/4} x^3} \, dx}{a \sqrt [4]{a+b x^4}}\\ &=-\frac{1}{a x \sqrt [4]{a+b x^4}}+\frac{\left (2 \sqrt [4]{1+\frac{a}{b x^4}} x\right ) \operatorname{Subst}\left (\int \frac{x}{\left (1+\frac{a x^4}{b}\right )^{5/4}} \, dx,x,\frac{1}{x}\right )}{a \sqrt [4]{a+b x^4}}\\ &=-\frac{1}{a x \sqrt [4]{a+b x^4}}+\frac{\left (\sqrt [4]{1+\frac{a}{b x^4}} x\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+\frac{a x^2}{b}\right )^{5/4}} \, dx,x,\frac{1}{x^2}\right )}{a \sqrt [4]{a+b x^4}}\\ &=-\frac{1}{a x \sqrt [4]{a+b x^4}}+\frac{2 \sqrt{b} \sqrt [4]{1+\frac{a}{b x^4}} x E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{a^{3/2} \sqrt [4]{a+b x^4}}\\ \end{align*}
Mathematica [C] time = 0.0098171, size = 52, normalized size = 0.66 \[ -\frac{\sqrt [4]{\frac{b x^4}{a}+1} \, _2F_1\left (-\frac{1}{4},\frac{5}{4};\frac{3}{4};-\frac{b x^4}{a}\right )}{a x \sqrt [4]{a+b x^4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.051, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}} \left ( b{x}^{4}+a \right ) ^{-{\frac{5}{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{5}{4}} x^{2}}\,{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^{2} x^{10} + 2 \, a b x^{6} + a^{2} x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.37135, size = 39, normalized size = 0.49 \begin{align*} \frac{\Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{5}{4} \\ \frac{3}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac{5}{4}} x \Gamma \left (\frac{3}{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{5}{4}} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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