Optimal. Leaf size=129 \[ \frac{3 a^{5/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} \text{EllipticF}\left (\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right ),2\right )}{8 b^{5/2} \left (a+b x^4\right )^{3/4}}+\frac{3 a^2 x \sqrt [4]{a+b x^4}}{8 b^3}-\frac{3 a x^5 \sqrt [4]{a+b x^4}}{20 b^2}+\frac{x^9 \sqrt [4]{a+b x^4}}{10 b} \]
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Rubi [A] time = 0.0585965, 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 = {321, 237, 335, 275, 231} \[ \frac{3 a^2 x \sqrt [4]{a+b x^4}}{8 b^3}+\frac{3 a^{5/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{8 b^{5/2} \left (a+b x^4\right )^{3/4}}-\frac{3 a x^5 \sqrt [4]{a+b x^4}}{20 b^2}+\frac{x^9 \sqrt [4]{a+b x^4}}{10 b} \]
Antiderivative was successfully verified.
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Rule 321
Rule 237
Rule 335
Rule 275
Rule 231
Rubi steps
\begin{align*} \int \frac{x^{12}}{\left (a+b x^4\right )^{3/4}} \, dx &=\frac{x^9 \sqrt [4]{a+b x^4}}{10 b}-\frac{(9 a) \int \frac{x^8}{\left (a+b x^4\right )^{3/4}} \, dx}{10 b}\\ &=-\frac{3 a x^5 \sqrt [4]{a+b x^4}}{20 b^2}+\frac{x^9 \sqrt [4]{a+b x^4}}{10 b}+\frac{\left (3 a^2\right ) \int \frac{x^4}{\left (a+b x^4\right )^{3/4}} \, dx}{4 b^2}\\ &=\frac{3 a^2 x \sqrt [4]{a+b x^4}}{8 b^3}-\frac{3 a x^5 \sqrt [4]{a+b x^4}}{20 b^2}+\frac{x^9 \sqrt [4]{a+b x^4}}{10 b}-\frac{\left (3 a^3\right ) \int \frac{1}{\left (a+b x^4\right )^{3/4}} \, dx}{8 b^3}\\ &=\frac{3 a^2 x \sqrt [4]{a+b x^4}}{8 b^3}-\frac{3 a x^5 \sqrt [4]{a+b x^4}}{20 b^2}+\frac{x^9 \sqrt [4]{a+b x^4}}{10 b}-\frac{\left (3 a^3 \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}{8 b^3 \left (a+b x^4\right )^{3/4}}\\ &=\frac{3 a^2 x \sqrt [4]{a+b x^4}}{8 b^3}-\frac{3 a x^5 \sqrt [4]{a+b x^4}}{20 b^2}+\frac{x^9 \sqrt [4]{a+b x^4}}{10 b}+\frac{\left (3 a^3 \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 )}{8 b^3 \left (a+b x^4\right )^{3/4}}\\ &=\frac{3 a^2 x \sqrt [4]{a+b x^4}}{8 b^3}-\frac{3 a x^5 \sqrt [4]{a+b x^4}}{20 b^2}+\frac{x^9 \sqrt [4]{a+b x^4}}{10 b}+\frac{\left (3 a^3 \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 )}{16 b^3 \left (a+b x^4\right )^{3/4}}\\ &=\frac{3 a^2 x \sqrt [4]{a+b x^4}}{8 b^3}-\frac{3 a x^5 \sqrt [4]{a+b x^4}}{20 b^2}+\frac{x^9 \sqrt [4]{a+b x^4}}{10 b}+\frac{3 a^{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 )}{8 b^{5/2} \left (a+b x^4\right )^{3/4}}\\ \end{align*}
Mathematica [C] time = 0.0232043, size = 90, normalized size = 0.7 \[ \frac{-15 a^3 x \left (\frac{b x^4}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};-\frac{b x^4}{a}\right )+9 a^2 b x^5+15 a^3 x-2 a b^2 x^9+4 b^3 x^{13}}{40 b^3 \left (a+b x^4\right )^{3/4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.026, size = 0, normalized size = 0. \begin{align*} \int{{x}^{12} \left ( b{x}^{4}+a \right ) ^{-{\frac{3}{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{x^{12}}{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}\,{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{x^{12}}{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 2.16249, size = 37, normalized size = 0.29 \begin{align*} \frac{x^{13} \Gamma \left (\frac{13}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{13}{4} \\ \frac{17}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac{3}{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 \frac{x^{12}}{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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