Optimal. Leaf size=134 \[ -\frac{4 b^2 \left (a-b x^4\right )^{3/4}}{39 a^3 x^5}-\frac{8 b^{7/2} x \sqrt [4]{1-\frac{a}{b x^4}} E\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{39 a^{7/2} \sqrt [4]{a-b x^4}}-\frac{10 b \left (a-b x^4\right )^{3/4}}{117 a^2 x^9}-\frac{\left (a-b x^4\right )^{3/4}}{13 a x^{13}} \]
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Rubi [A] time = 0.0639307, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {325, 313, 335, 275, 228} \[ -\frac{4 b^2 \left (a-b x^4\right )^{3/4}}{39 a^3 x^5}-\frac{8 b^{7/2} x \sqrt [4]{1-\frac{a}{b x^4}} E\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{39 a^{7/2} \sqrt [4]{a-b x^4}}-\frac{10 b \left (a-b x^4\right )^{3/4}}{117 a^2 x^9}-\frac{\left (a-b x^4\right )^{3/4}}{13 a x^{13}} \]
Antiderivative was successfully verified.
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Rule 325
Rule 313
Rule 335
Rule 275
Rule 228
Rubi steps
\begin{align*} \int \frac{1}{x^{14} \sqrt [4]{a-b x^4}} \, dx &=-\frac{\left (a-b x^4\right )^{3/4}}{13 a x^{13}}+\frac{(10 b) \int \frac{1}{x^{10} \sqrt [4]{a-b x^4}} \, dx}{13 a}\\ &=-\frac{\left (a-b x^4\right )^{3/4}}{13 a x^{13}}-\frac{10 b \left (a-b x^4\right )^{3/4}}{117 a^2 x^9}+\frac{\left (20 b^2\right ) \int \frac{1}{x^6 \sqrt [4]{a-b x^4}} \, dx}{39 a^2}\\ &=-\frac{\left (a-b x^4\right )^{3/4}}{13 a x^{13}}-\frac{10 b \left (a-b x^4\right )^{3/4}}{117 a^2 x^9}-\frac{4 b^2 \left (a-b x^4\right )^{3/4}}{39 a^3 x^5}+\frac{\left (8 b^3\right ) \int \frac{1}{x^2 \sqrt [4]{a-b x^4}} \, dx}{39 a^3}\\ &=-\frac{\left (a-b x^4\right )^{3/4}}{13 a x^{13}}-\frac{10 b \left (a-b x^4\right )^{3/4}}{117 a^2 x^9}-\frac{4 b^2 \left (a-b x^4\right )^{3/4}}{39 a^3 x^5}+\frac{\left (8 b^3 \sqrt [4]{1-\frac{a}{b x^4}} x\right ) \int \frac{1}{\sqrt [4]{1-\frac{a}{b x^4}} x^3} \, dx}{39 a^3 \sqrt [4]{a-b x^4}}\\ &=-\frac{\left (a-b x^4\right )^{3/4}}{13 a x^{13}}-\frac{10 b \left (a-b x^4\right )^{3/4}}{117 a^2 x^9}-\frac{4 b^2 \left (a-b x^4\right )^{3/4}}{39 a^3 x^5}-\frac{\left (8 b^3 \sqrt [4]{1-\frac{a}{b x^4}} x\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt [4]{1-\frac{a x^4}{b}}} \, dx,x,\frac{1}{x}\right )}{39 a^3 \sqrt [4]{a-b x^4}}\\ &=-\frac{\left (a-b x^4\right )^{3/4}}{13 a x^{13}}-\frac{10 b \left (a-b x^4\right )^{3/4}}{117 a^2 x^9}-\frac{4 b^2 \left (a-b x^4\right )^{3/4}}{39 a^3 x^5}-\frac{\left (4 b^3 \sqrt [4]{1-\frac{a}{b x^4}} x\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{1-\frac{a x^2}{b}}} \, dx,x,\frac{1}{x^2}\right )}{39 a^3 \sqrt [4]{a-b x^4}}\\ &=-\frac{\left (a-b x^4\right )^{3/4}}{13 a x^{13}}-\frac{10 b \left (a-b x^4\right )^{3/4}}{117 a^2 x^9}-\frac{4 b^2 \left (a-b x^4\right )^{3/4}}{39 a^3 x^5}-\frac{8 b^{7/2} \sqrt [4]{1-\frac{a}{b x^4}} x E\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{39 a^{7/2} \sqrt [4]{a-b x^4}}\\ \end{align*}
Mathematica [C] time = 0.0087717, size = 52, normalized size = 0.39 \[ -\frac{\sqrt [4]{1-\frac{b x^4}{a}} \, _2F_1\left (-\frac{13}{4},\frac{1}{4};-\frac{9}{4};\frac{b x^4}{a}\right )}{13 x^{13} \sqrt [4]{a-b x^4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.039, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{14}}{\frac{1}{\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{1}{{\left (-b x^{4} + a\right )}^{\frac{1}{4}} x^{14}}\,{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 x^{18} - a x^{14}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 3.96255, size = 34, normalized size = 0.25 \begin{align*} - \frac{i e^{- \frac{3 i \pi }{4}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{7}{2} \\ \frac{9}{2} \end{matrix}\middle |{\frac{a}{b x^{4}}} \right )}}{14 \sqrt [4]{b} x^{14}} \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{1}{4}} x^{14}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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