Optimal. Leaf size=129 \[ -\frac{4 b^2}{3 a^3 x \sqrt [4]{a+b x^4}}+\frac{8 b^{5/2} x \sqrt [4]{\frac{a}{b x^4}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{3 a^{7/2} \sqrt [4]{a+b x^4}}+\frac{2 b}{9 a^2 x^5 \sqrt [4]{a+b x^4}}-\frac{1}{9 a x^9 \sqrt [4]{a+b x^4}} \]
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Rubi [A] time = 0.0634339, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ -\frac{4 b^2}{3 a^3 x \sqrt [4]{a+b x^4}}+\frac{8 b^{5/2} x \sqrt [4]{\frac{a}{b x^4}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{3 a^{7/2} \sqrt [4]{a+b x^4}}+\frac{2 b}{9 a^2 x^5 \sqrt [4]{a+b x^4}}-\frac{1}{9 a x^9 \sqrt [4]{a+b x^4}} \]
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^{10} \left (a+b x^4\right )^{5/4}} \, dx &=-\frac{1}{9 a x^9 \sqrt [4]{a+b x^4}}-\frac{(10 b) \int \frac{1}{x^6 \left (a+b x^4\right )^{5/4}} \, dx}{9 a}\\ &=-\frac{1}{9 a x^9 \sqrt [4]{a+b x^4}}+\frac{2 b}{9 a^2 x^5 \sqrt [4]{a+b x^4}}+\frac{\left (4 b^2\right ) \int \frac{1}{x^2 \left (a+b x^4\right )^{5/4}} \, dx}{3 a^2}\\ &=-\frac{1}{9 a x^9 \sqrt [4]{a+b x^4}}+\frac{2 b}{9 a^2 x^5 \sqrt [4]{a+b x^4}}-\frac{4 b^2}{3 a^3 x \sqrt [4]{a+b x^4}}-\frac{\left (8 b^3\right ) \int \frac{x^2}{\left (a+b x^4\right )^{5/4}} \, dx}{3 a^3}\\ &=-\frac{1}{9 a x^9 \sqrt [4]{a+b x^4}}+\frac{2 b}{9 a^2 x^5 \sqrt [4]{a+b x^4}}-\frac{4 b^2}{3 a^3 x \sqrt [4]{a+b x^4}}-\frac{\left (8 b^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}{3 a^3 \sqrt [4]{a+b x^4}}\\ &=-\frac{1}{9 a x^9 \sqrt [4]{a+b x^4}}+\frac{2 b}{9 a^2 x^5 \sqrt [4]{a+b x^4}}-\frac{4 b^2}{3 a^3 x \sqrt [4]{a+b x^4}}+\frac{\left (8 b^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 )}{3 a^3 \sqrt [4]{a+b x^4}}\\ &=-\frac{1}{9 a x^9 \sqrt [4]{a+b x^4}}+\frac{2 b}{9 a^2 x^5 \sqrt [4]{a+b x^4}}-\frac{4 b^2}{3 a^3 x \sqrt [4]{a+b x^4}}+\frac{\left (4 b^2 \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 )}{3 a^3 \sqrt [4]{a+b x^4}}\\ &=-\frac{1}{9 a x^9 \sqrt [4]{a+b x^4}}+\frac{2 b}{9 a^2 x^5 \sqrt [4]{a+b x^4}}-\frac{4 b^2}{3 a^3 x \sqrt [4]{a+b x^4}}+\frac{8 b^{5/2} \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 )}{3 a^{7/2} \sqrt [4]{a+b x^4}}\\ \end{align*}
Mathematica [C] time = 0.0093826, size = 54, normalized size = 0.42 \[ -\frac{\sqrt [4]{\frac{b x^4}{a}+1} \, _2F_1\left (-\frac{9}{4},\frac{5}{4};-\frac{5}{4};-\frac{b x^4}{a}\right )}{9 a x^9 \sqrt [4]{a+b x^4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.061, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{10}} \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^{10}}\,{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^{18} + 2 \, a b x^{14} + a^{2} x^{10}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 4.5567, size = 44, normalized size = 0.34 \begin{align*} \frac{\Gamma \left (- \frac{9}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{9}{4}, \frac{5}{4} \\ - \frac{5}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac{5}{4}} x^{9} \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{5}{4}} x^{10}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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