Optimal. Leaf size=150 \[ \frac{3 a^{7/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} \text{EllipticF}\left (\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right ),2\right )}{112 b^{5/2} \left (a+b x^4\right )^{3/4}}-\frac{3 a^2 x^5 \sqrt [4]{a+b x^4}}{280 b^2}+\frac{3 a^3 x \sqrt [4]{a+b x^4}}{112 b^3}+\frac{1}{14} x^{13} \sqrt [4]{a+b x^4}+\frac{a x^9 \sqrt [4]{a+b x^4}}{140 b} \]
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Rubi [A] time = 0.0741586, antiderivative size = 150, 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 = {279, 321, 237, 335, 275, 231} \[ -\frac{3 a^2 x^5 \sqrt [4]{a+b x^4}}{280 b^2}+\frac{3 a^3 x \sqrt [4]{a+b x^4}}{112 b^3}+\frac{3 a^{7/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{112 b^{5/2} \left (a+b x^4\right )^{3/4}}+\frac{1}{14} x^{13} \sqrt [4]{a+b x^4}+\frac{a x^9 \sqrt [4]{a+b x^4}}{140 b} \]
Antiderivative was successfully verified.
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Rule 279
Rule 321
Rule 237
Rule 335
Rule 275
Rule 231
Rubi steps
\begin{align*} \int x^{12} \sqrt [4]{a+b x^4} \, dx &=\frac{1}{14} x^{13} \sqrt [4]{a+b x^4}+\frac{1}{14} a \int \frac{x^{12}}{\left (a+b x^4\right )^{3/4}} \, dx\\ &=\frac{a x^9 \sqrt [4]{a+b x^4}}{140 b}+\frac{1}{14} x^{13} \sqrt [4]{a+b x^4}-\frac{\left (9 a^2\right ) \int \frac{x^8}{\left (a+b x^4\right )^{3/4}} \, dx}{140 b}\\ &=-\frac{3 a^2 x^5 \sqrt [4]{a+b x^4}}{280 b^2}+\frac{a x^9 \sqrt [4]{a+b x^4}}{140 b}+\frac{1}{14} x^{13} \sqrt [4]{a+b x^4}+\frac{\left (3 a^3\right ) \int \frac{x^4}{\left (a+b x^4\right )^{3/4}} \, dx}{56 b^2}\\ &=\frac{3 a^3 x \sqrt [4]{a+b x^4}}{112 b^3}-\frac{3 a^2 x^5 \sqrt [4]{a+b x^4}}{280 b^2}+\frac{a x^9 \sqrt [4]{a+b x^4}}{140 b}+\frac{1}{14} x^{13} \sqrt [4]{a+b x^4}-\frac{\left (3 a^4\right ) \int \frac{1}{\left (a+b x^4\right )^{3/4}} \, dx}{112 b^3}\\ &=\frac{3 a^3 x \sqrt [4]{a+b x^4}}{112 b^3}-\frac{3 a^2 x^5 \sqrt [4]{a+b x^4}}{280 b^2}+\frac{a x^9 \sqrt [4]{a+b x^4}}{140 b}+\frac{1}{14} x^{13} \sqrt [4]{a+b x^4}-\frac{\left (3 a^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}{112 b^3 \left (a+b x^4\right )^{3/4}}\\ &=\frac{3 a^3 x \sqrt [4]{a+b x^4}}{112 b^3}-\frac{3 a^2 x^5 \sqrt [4]{a+b x^4}}{280 b^2}+\frac{a x^9 \sqrt [4]{a+b x^4}}{140 b}+\frac{1}{14} x^{13} \sqrt [4]{a+b x^4}+\frac{\left (3 a^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 )}{112 b^3 \left (a+b x^4\right )^{3/4}}\\ &=\frac{3 a^3 x \sqrt [4]{a+b x^4}}{112 b^3}-\frac{3 a^2 x^5 \sqrt [4]{a+b x^4}}{280 b^2}+\frac{a x^9 \sqrt [4]{a+b x^4}}{140 b}+\frac{1}{14} x^{13} \sqrt [4]{a+b x^4}+\frac{\left (3 a^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 )}{224 b^3 \left (a+b x^4\right )^{3/4}}\\ &=\frac{3 a^3 x \sqrt [4]{a+b x^4}}{112 b^3}-\frac{3 a^2 x^5 \sqrt [4]{a+b x^4}}{280 b^2}+\frac{a x^9 \sqrt [4]{a+b x^4}}{140 b}+\frac{1}{14} x^{13} \sqrt [4]{a+b x^4}+\frac{3 a^{7/2} \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 )}{112 b^{5/2} \left (a+b x^4\right )^{3/4}}\\ \end{align*}
Mathematica [C] time = 0.0599829, size = 105, normalized size = 0.7 \[ \frac{x \sqrt [4]{a+b x^4} \left (\sqrt [4]{\frac{b x^4}{a}+1} \left (-3 a^2 b x^4+15 a^3+2 a b^2 x^8+20 b^3 x^{12}\right )-15 a^3 \, _2F_1\left (-\frac{1}{4},\frac{1}{4};\frac{5}{4};-\frac{b x^4}{a}\right )\right )}{280 b^3 \sqrt [4]{\frac{b x^4}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.028, size = 0, normalized size = 0. \begin{align*} \int{x}^{12}\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{\left (b x^{4} + a\right )}^{\frac{1}{4}} x^{12}\,{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 ({\left (b x^{4} + a\right )}^{\frac{1}{4}} x^{12}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 2.78327, size = 39, normalized size = 0.26 \begin{align*} \frac{\sqrt [4]{a} x^{13} \Gamma \left (\frac{13}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{13}{4} \\ \frac{17}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{17}{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{\left (b x^{4} + a\right )}^{\frac{1}{4}} x^{12}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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