Optimal. Leaf size=100 \[ -\frac{5 a^2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{3/4}}+\frac{5 a^2 \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{3/4}}+\frac{1}{8} x^3 \left (a+b x^4\right )^{5/4}+\frac{5}{32} a x^3 \sqrt [4]{a+b x^4} \]
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Rubi [A] time = 0.0370888, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {279, 331, 298, 203, 206} \[ -\frac{5 a^2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{3/4}}+\frac{5 a^2 \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{3/4}}+\frac{1}{8} x^3 \left (a+b x^4\right )^{5/4}+\frac{5}{32} a x^3 \sqrt [4]{a+b x^4} \]
Antiderivative was successfully verified.
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Rule 279
Rule 331
Rule 298
Rule 203
Rule 206
Rubi steps
\begin{align*} \int x^2 \left (a+b x^4\right )^{5/4} \, dx &=\frac{1}{8} x^3 \left (a+b x^4\right )^{5/4}+\frac{1}{8} (5 a) \int x^2 \sqrt [4]{a+b x^4} \, dx\\ &=\frac{5}{32} a x^3 \sqrt [4]{a+b x^4}+\frac{1}{8} x^3 \left (a+b x^4\right )^{5/4}+\frac{1}{32} \left (5 a^2\right ) \int \frac{x^2}{\left (a+b x^4\right )^{3/4}} \, dx\\ &=\frac{5}{32} a x^3 \sqrt [4]{a+b x^4}+\frac{1}{8} x^3 \left (a+b x^4\right )^{5/4}+\frac{1}{32} \left (5 a^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{1-b x^4} \, dx,x,\frac{x}{\sqrt [4]{a+b x^4}}\right )\\ &=\frac{5}{32} a x^3 \sqrt [4]{a+b x^4}+\frac{1}{8} x^3 \left (a+b x^4\right )^{5/4}+\frac{\left (5 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{b} x^2} \, dx,x,\frac{x}{\sqrt [4]{a+b x^4}}\right )}{64 \sqrt{b}}-\frac{\left (5 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{b} x^2} \, dx,x,\frac{x}{\sqrt [4]{a+b x^4}}\right )}{64 \sqrt{b}}\\ &=\frac{5}{32} a x^3 \sqrt [4]{a+b x^4}+\frac{1}{8} x^3 \left (a+b x^4\right )^{5/4}-\frac{5 a^2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{3/4}}+\frac{5 a^2 \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{3/4}}\\ \end{align*}
Mathematica [C] time = 0.0083934, size = 52, normalized size = 0.52 \[ \frac{a x^3 \sqrt [4]{a+b x^4} \, _2F_1\left (-\frac{5}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right )}{3 \sqrt [4]{\frac{b x^4}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.027, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.67821, size = 494, normalized size = 4.94 \begin{align*} \frac{1}{32} \,{\left (4 \, b x^{7} + 9 \, a x^{3}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}} - \frac{5}{32} \, \left (\frac{a^{8}}{b^{3}}\right )^{\frac{1}{4}} \arctan \left (-\frac{\left (\frac{a^{8}}{b^{3}}\right )^{\frac{3}{4}}{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{2} b^{2} - \left (\frac{a^{8}}{b^{3}}\right )^{\frac{3}{4}} b^{2} x \sqrt{\frac{\sqrt{b x^{4} + a} a^{4} + \sqrt{\frac{a^{8}}{b^{3}}} b^{2} x^{2}}{x^{2}}}}{a^{8} x}\right ) + \frac{5}{128} \, \left (\frac{a^{8}}{b^{3}}\right )^{\frac{1}{4}} \log \left (\frac{5 \,{\left ({\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{2} + \left (\frac{a^{8}}{b^{3}}\right )^{\frac{1}{4}} b x\right )}}{x}\right ) - \frac{5}{128} \, \left (\frac{a^{8}}{b^{3}}\right )^{\frac{1}{4}} \log \left (\frac{5 \,{\left ({\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{2} - \left (\frac{a^{8}}{b^{3}}\right )^{\frac{1}{4}} b x\right )}}{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 3.0933, size = 39, normalized size = 0.39 \begin{align*} \frac{a^{\frac{5}{4}} x^{3} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{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 [B] time = 1.18454, size = 346, normalized size = 3.46 \begin{align*} \frac{1}{256} \,{\left (\frac{8 \, x^{8}{\left (\frac{9 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}{\left (b + \frac{a}{x^{4}}\right )}}{x} - \frac{5 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} b}{x}\right )}}{a^{2}} + \frac{10 \, \sqrt{2} \left (-b\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (-b\right )^{\frac{1}{4}} + \frac{2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x}\right )}}{2 \, \left (-b\right )^{\frac{1}{4}}}\right )}{b} + \frac{10 \, \sqrt{2} \left (-b\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (-b\right )^{\frac{1}{4}} - \frac{2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x}\right )}}{2 \, \left (-b\right )^{\frac{1}{4}}}\right )}{b} + \frac{5 \, \sqrt{2} \left (-b\right )^{\frac{1}{4}} \log \left (\sqrt{-b} + \frac{\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-b\right )^{\frac{1}{4}}}{x} + \frac{\sqrt{b x^{4} + a}}{x^{2}}\right )}{b} - \frac{5 \, \sqrt{2} \left (-b\right )^{\frac{1}{4}} \log \left (\sqrt{-b} - \frac{\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-b\right )^{\frac{1}{4}}}{x} + \frac{\sqrt{b x^{4} + a}}{x^{2}}\right )}{b}\right )} a^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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