Optimal. Leaf size=99 \[ \frac{a^{3/2} x \sqrt [4]{\frac{a}{b x^4}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{4 \sqrt{b} \sqrt [4]{a+b x^4}}+\frac{1}{6} x^3 \left (a+b x^4\right )^{3/4}+\frac{a x^3}{4 \sqrt [4]{a+b x^4}} \]
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Rubi [A] time = 0.047035, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {279, 310, 281, 335, 275, 196} \[ \frac{a^{3/2} x \sqrt [4]{\frac{a}{b x^4}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{4 \sqrt{b} \sqrt [4]{a+b x^4}}+\frac{1}{6} x^3 \left (a+b x^4\right )^{3/4}+\frac{a x^3}{4 \sqrt [4]{a+b x^4}} \]
Antiderivative was successfully verified.
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Rule 279
Rule 310
Rule 281
Rule 335
Rule 275
Rule 196
Rubi steps
\begin{align*} \int x^2 \left (a+b x^4\right )^{3/4} \, dx &=\frac{1}{6} x^3 \left (a+b x^4\right )^{3/4}+\frac{1}{2} a \int \frac{x^2}{\sqrt [4]{a+b x^4}} \, dx\\ &=\frac{a x^3}{4 \sqrt [4]{a+b x^4}}+\frac{1}{6} x^3 \left (a+b x^4\right )^{3/4}-\frac{1}{4} a^2 \int \frac{x^2}{\left (a+b x^4\right )^{5/4}} \, dx\\ &=\frac{a x^3}{4 \sqrt [4]{a+b x^4}}+\frac{1}{6} x^3 \left (a+b x^4\right )^{3/4}-\frac{\left (a^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}{4 b \sqrt [4]{a+b x^4}}\\ &=\frac{a x^3}{4 \sqrt [4]{a+b x^4}}+\frac{1}{6} x^3 \left (a+b x^4\right )^{3/4}+\frac{\left (a^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 )}{4 b \sqrt [4]{a+b x^4}}\\ &=\frac{a x^3}{4 \sqrt [4]{a+b x^4}}+\frac{1}{6} x^3 \left (a+b x^4\right )^{3/4}+\frac{\left (a^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 )}{8 b \sqrt [4]{a+b x^4}}\\ &=\frac{a x^3}{4 \sqrt [4]{a+b x^4}}+\frac{1}{6} x^3 \left (a+b x^4\right )^{3/4}+\frac{a^{3/2} \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 )}{4 \sqrt{b} \sqrt [4]{a+b x^4}}\\ \end{align*}
Mathematica [C] time = 0.0107419, size = 51, normalized size = 0.52 \[ \frac{x^3 \left (a+b x^4\right )^{3/4} \, _2F_1\left (-\frac{3}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right )}{3 \left (\frac{b x^4}{a}+1\right )^{3/4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.026, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \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{\left (b x^{4} + a\right )}^{\frac{3}{4}} x^{2}\,{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{3}{4}} x^{2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.34851, size = 39, normalized size = 0.39 \begin{align*} \frac{a^{\frac{3}{4}} x^{3} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{7}{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{3}{4}} x^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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