Optimal. Leaf size=152 \[ \frac{8 b^{9/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} \text{EllipticF}\left (\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right ),2\right )}{231 a^{7/2} \left (a+b x^4\right )^{3/4}}-\frac{4 b^3 \sqrt [4]{a+b x^4}}{231 a^3 x^3}+\frac{2 b^2 \sqrt [4]{a+b x^4}}{231 a^2 x^7}-\frac{b \sqrt [4]{a+b x^4}}{165 a x^{11}}-\frac{\sqrt [4]{a+b x^4}}{15 x^{15}} \]
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Rubi [A] time = 0.0775219, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {277, 325, 237, 335, 275, 231} \[ -\frac{4 b^3 \sqrt [4]{a+b x^4}}{231 a^3 x^3}+\frac{2 b^2 \sqrt [4]{a+b x^4}}{231 a^2 x^7}+\frac{8 b^{9/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{231 a^{7/2} \left (a+b x^4\right )^{3/4}}-\frac{b \sqrt [4]{a+b x^4}}{165 a x^{11}}-\frac{\sqrt [4]{a+b x^4}}{15 x^{15}} \]
Antiderivative was successfully verified.
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Rule 277
Rule 325
Rule 237
Rule 335
Rule 275
Rule 231
Rubi steps
\begin{align*} \int \frac{\sqrt [4]{a+b x^4}}{x^{16}} \, dx &=-\frac{\sqrt [4]{a+b x^4}}{15 x^{15}}+\frac{1}{15} b \int \frac{1}{x^{12} \left (a+b x^4\right )^{3/4}} \, dx\\ &=-\frac{\sqrt [4]{a+b x^4}}{15 x^{15}}-\frac{b \sqrt [4]{a+b x^4}}{165 a x^{11}}-\frac{\left (2 b^2\right ) \int \frac{1}{x^8 \left (a+b x^4\right )^{3/4}} \, dx}{33 a}\\ &=-\frac{\sqrt [4]{a+b x^4}}{15 x^{15}}-\frac{b \sqrt [4]{a+b x^4}}{165 a x^{11}}+\frac{2 b^2 \sqrt [4]{a+b x^4}}{231 a^2 x^7}+\frac{\left (4 b^3\right ) \int \frac{1}{x^4 \left (a+b x^4\right )^{3/4}} \, dx}{77 a^2}\\ &=-\frac{\sqrt [4]{a+b x^4}}{15 x^{15}}-\frac{b \sqrt [4]{a+b x^4}}{165 a x^{11}}+\frac{2 b^2 \sqrt [4]{a+b x^4}}{231 a^2 x^7}-\frac{4 b^3 \sqrt [4]{a+b x^4}}{231 a^3 x^3}-\frac{\left (8 b^4\right ) \int \frac{1}{\left (a+b x^4\right )^{3/4}} \, dx}{231 a^3}\\ &=-\frac{\sqrt [4]{a+b x^4}}{15 x^{15}}-\frac{b \sqrt [4]{a+b x^4}}{165 a x^{11}}+\frac{2 b^2 \sqrt [4]{a+b x^4}}{231 a^2 x^7}-\frac{4 b^3 \sqrt [4]{a+b x^4}}{231 a^3 x^3}-\frac{\left (8 b^4 \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}{231 a^3 \left (a+b x^4\right )^{3/4}}\\ &=-\frac{\sqrt [4]{a+b x^4}}{15 x^{15}}-\frac{b \sqrt [4]{a+b x^4}}{165 a x^{11}}+\frac{2 b^2 \sqrt [4]{a+b x^4}}{231 a^2 x^7}-\frac{4 b^3 \sqrt [4]{a+b x^4}}{231 a^3 x^3}+\frac{\left (8 b^4 \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 )}{231 a^3 \left (a+b x^4\right )^{3/4}}\\ &=-\frac{\sqrt [4]{a+b x^4}}{15 x^{15}}-\frac{b \sqrt [4]{a+b x^4}}{165 a x^{11}}+\frac{2 b^2 \sqrt [4]{a+b x^4}}{231 a^2 x^7}-\frac{4 b^3 \sqrt [4]{a+b x^4}}{231 a^3 x^3}+\frac{\left (4 b^4 \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 )}{231 a^3 \left (a+b x^4\right )^{3/4}}\\ &=-\frac{\sqrt [4]{a+b x^4}}{15 x^{15}}-\frac{b \sqrt [4]{a+b x^4}}{165 a x^{11}}+\frac{2 b^2 \sqrt [4]{a+b x^4}}{231 a^2 x^7}-\frac{4 b^3 \sqrt [4]{a+b x^4}}{231 a^3 x^3}+\frac{8 b^{9/2} \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 )}{231 a^{7/2} \left (a+b x^4\right )^{3/4}}\\ \end{align*}
Mathematica [C] time = 0.0091395, size = 51, normalized size = 0.34 \[ -\frac{\sqrt [4]{a+b x^4} \, _2F_1\left (-\frac{15}{4},-\frac{1}{4};-\frac{11}{4};-\frac{b x^4}{a}\right )}{15 x^{15} \sqrt [4]{\frac{b x^4}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.04, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{16}}\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{{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x^{16}}\,{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}}}{x^{16}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 6.4229, size = 46, normalized size = 0.3 \begin{align*} \frac{\sqrt [4]{a} \Gamma \left (- \frac{15}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{15}{4}, - \frac{1}{4} \\ - \frac{11}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{15} \Gamma \left (- \frac{11}{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{{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x^{16}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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