Optimal. Leaf size=125 \[ -\frac{5 a^2 x \left (a+b x^4\right )^{3/4}}{128 b^2}+\frac{5 a^3 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{256 b^{9/4}}+\frac{5 a^3 \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{256 b^{9/4}}+\frac{1}{12} x^9 \left (a+b x^4\right )^{3/4}+\frac{a x^5 \left (a+b x^4\right )^{3/4}}{32 b} \]
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Rubi [A] time = 0.0450163, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {279, 321, 240, 212, 206, 203} \[ -\frac{5 a^2 x \left (a+b x^4\right )^{3/4}}{128 b^2}+\frac{5 a^3 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{256 b^{9/4}}+\frac{5 a^3 \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{256 b^{9/4}}+\frac{1}{12} x^9 \left (a+b x^4\right )^{3/4}+\frac{a x^5 \left (a+b x^4\right )^{3/4}}{32 b} \]
Antiderivative was successfully verified.
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Rule 279
Rule 321
Rule 240
Rule 212
Rule 206
Rule 203
Rubi steps
\begin{align*} \int x^8 \left (a+b x^4\right )^{3/4} \, dx &=\frac{1}{12} x^9 \left (a+b x^4\right )^{3/4}+\frac{1}{4} a \int \frac{x^8}{\sqrt [4]{a+b x^4}} \, dx\\ &=\frac{a x^5 \left (a+b x^4\right )^{3/4}}{32 b}+\frac{1}{12} x^9 \left (a+b x^4\right )^{3/4}-\frac{\left (5 a^2\right ) \int \frac{x^4}{\sqrt [4]{a+b x^4}} \, dx}{32 b}\\ &=-\frac{5 a^2 x \left (a+b x^4\right )^{3/4}}{128 b^2}+\frac{a x^5 \left (a+b x^4\right )^{3/4}}{32 b}+\frac{1}{12} x^9 \left (a+b x^4\right )^{3/4}+\frac{\left (5 a^3\right ) \int \frac{1}{\sqrt [4]{a+b x^4}} \, dx}{128 b^2}\\ &=-\frac{5 a^2 x \left (a+b x^4\right )^{3/4}}{128 b^2}+\frac{a x^5 \left (a+b x^4\right )^{3/4}}{32 b}+\frac{1}{12} x^9 \left (a+b x^4\right )^{3/4}+\frac{\left (5 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^4} \, dx,x,\frac{x}{\sqrt [4]{a+b x^4}}\right )}{128 b^2}\\ &=-\frac{5 a^2 x \left (a+b x^4\right )^{3/4}}{128 b^2}+\frac{a x^5 \left (a+b x^4\right )^{3/4}}{32 b}+\frac{1}{12} x^9 \left (a+b x^4\right )^{3/4}+\frac{\left (5 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{b} x^2} \, dx,x,\frac{x}{\sqrt [4]{a+b x^4}}\right )}{256 b^2}+\frac{\left (5 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{b} x^2} \, dx,x,\frac{x}{\sqrt [4]{a+b x^4}}\right )}{256 b^2}\\ &=-\frac{5 a^2 x \left (a+b x^4\right )^{3/4}}{128 b^2}+\frac{a x^5 \left (a+b x^4\right )^{3/4}}{32 b}+\frac{1}{12} x^9 \left (a+b x^4\right )^{3/4}+\frac{5 a^3 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{256 b^{9/4}}+\frac{5 a^3 \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{256 b^{9/4}}\\ \end{align*}
Mathematica [C] time = 0.0502763, size = 94, normalized size = 0.75 \[ \frac{x \left (a+b x^4\right )^{3/4} \left (\left (\frac{b x^4}{a}+1\right )^{3/4} \left (-5 a^2+3 a b x^4+8 b^2 x^8\right )+5 a^2 \, _2F_1\left (-\frac{3}{4},\frac{1}{4};\frac{5}{4};-\frac{b x^4}{a}\right )\right )}{96 b^2 \left (\frac{b x^4}{a}+1\right )^{3/4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.026, size = 0, normalized size = 0. \begin{align*} \int{x}^{8} \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(-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 = 1.95779, size = 567, normalized size = 4.54 \begin{align*} \frac{60 \, \left (\frac{a^{12}}{b^{9}}\right )^{\frac{1}{4}} b^{2} \arctan \left (-\frac{{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (\frac{a^{12}}{b^{9}}\right )^{\frac{1}{4}} a^{9} b^{2} - \left (\frac{a^{12}}{b^{9}}\right )^{\frac{1}{4}} b^{2} x \sqrt{\frac{\sqrt{\frac{a^{12}}{b^{9}}} a^{12} b^{5} x^{2} + \sqrt{b x^{4} + a} a^{18}}{x^{2}}}}{a^{12} x}\right ) + 15 \, \left (\frac{a^{12}}{b^{9}}\right )^{\frac{1}{4}} b^{2} \log \left (\frac{125 \,{\left ({\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{9} + \left (\frac{a^{12}}{b^{9}}\right )^{\frac{3}{4}} b^{7} x\right )}}{x}\right ) - 15 \, \left (\frac{a^{12}}{b^{9}}\right )^{\frac{1}{4}} b^{2} \log \left (\frac{125 \,{\left ({\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{9} - \left (\frac{a^{12}}{b^{9}}\right )^{\frac{3}{4}} b^{7} x\right )}}{x}\right ) + 4 \,{\left (32 \, b^{2} x^{9} + 12 \, a b x^{5} - 15 \, a^{2} x\right )}{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{1536 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 3.80278, size = 39, normalized size = 0.31 \begin{align*} \frac{a^{\frac{3}{4}} x^{9} \Gamma \left (\frac{9}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{9}{4} \\ \frac{13}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{13}{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^{8}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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