Optimal. Leaf size=101 \[ -\frac{5 b^{5/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} \text{EllipticF}\left (\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right ),2\right )}{21 \sqrt{a} \left (a+b x^4\right )^{3/4}}-\frac{5 b \sqrt [4]{a+b x^4}}{21 x^3}-\frac{\left (a+b x^4\right )^{5/4}}{7 x^7} \]
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Rubi [A] time = 0.0441044, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {277, 237, 335, 275, 231} \[ -\frac{5 b^{5/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{21 \sqrt{a} \left (a+b x^4\right )^{3/4}}-\frac{5 b \sqrt [4]{a+b x^4}}{21 x^3}-\frac{\left (a+b x^4\right )^{5/4}}{7 x^7} \]
Antiderivative was successfully verified.
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Rule 277
Rule 237
Rule 335
Rule 275
Rule 231
Rubi steps
\begin{align*} \int \frac{\left (a+b x^4\right )^{5/4}}{x^8} \, dx &=-\frac{\left (a+b x^4\right )^{5/4}}{7 x^7}+\frac{1}{7} (5 b) \int \frac{\sqrt [4]{a+b x^4}}{x^4} \, dx\\ &=-\frac{5 b \sqrt [4]{a+b x^4}}{21 x^3}-\frac{\left (a+b x^4\right )^{5/4}}{7 x^7}+\frac{1}{21} \left (5 b^2\right ) \int \frac{1}{\left (a+b x^4\right )^{3/4}} \, dx\\ &=-\frac{5 b \sqrt [4]{a+b x^4}}{21 x^3}-\frac{\left (a+b x^4\right )^{5/4}}{7 x^7}+\frac{\left (5 b^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}{21 \left (a+b x^4\right )^{3/4}}\\ &=-\frac{5 b \sqrt [4]{a+b x^4}}{21 x^3}-\frac{\left (a+b x^4\right )^{5/4}}{7 x^7}-\frac{\left (5 b^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 )}{21 \left (a+b x^4\right )^{3/4}}\\ &=-\frac{5 b \sqrt [4]{a+b x^4}}{21 x^3}-\frac{\left (a+b x^4\right )^{5/4}}{7 x^7}-\frac{\left (5 b^2 \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 )}{42 \left (a+b x^4\right )^{3/4}}\\ &=-\frac{5 b \sqrt [4]{a+b x^4}}{21 x^3}-\frac{\left (a+b x^4\right )^{5/4}}{7 x^7}-\frac{5 b^{5/2} \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 )}{21 \sqrt{a} \left (a+b x^4\right )^{3/4}}\\ \end{align*}
Mathematica [C] time = 0.011482, size = 52, normalized size = 0.51 \[ -\frac{a \sqrt [4]{a+b x^4} \, _2F_1\left (-\frac{7}{4},-\frac{5}{4};-\frac{3}{4};-\frac{b x^4}{a}\right )}{7 x^7 \sqrt [4]{\frac{b x^4}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.029, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{8}} \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{{\left (b x^{4} + a\right )}^{\frac{5}{4}}}{x^{8}}\,{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{5}{4}}}{x^{8}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 2.82845, size = 31, normalized size = 0.31 \begin{align*} - \frac{b^{\frac{5}{4}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{4}, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{4}}} \right )}}{2 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{{\left (b x^{4} + a\right )}^{\frac{5}{4}}}{x^{8}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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