Optimal. Leaf size=149 \[ -\frac{9 a^2 x^5 \left (a+b x^4\right )^{3/4}}{512 b^2}+\frac{45 a^3 x \left (a+b x^4\right )^{3/4}}{2048 b^3}-\frac{45 a^4 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{4096 b^{13/4}}-\frac{45 a^4 \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{4096 b^{13/4}}+\frac{1}{16} x^{13} \left (a+b x^4\right )^{3/4}+\frac{a x^9 \left (a+b x^4\right )^{3/4}}{64 b} \]
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Rubi [A] time = 0.0609087, antiderivative size = 149, 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, 240, 212, 206, 203} \[ -\frac{9 a^2 x^5 \left (a+b x^4\right )^{3/4}}{512 b^2}+\frac{45 a^3 x \left (a+b x^4\right )^{3/4}}{2048 b^3}-\frac{45 a^4 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{4096 b^{13/4}}-\frac{45 a^4 \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{4096 b^{13/4}}+\frac{1}{16} x^{13} \left (a+b x^4\right )^{3/4}+\frac{a x^9 \left (a+b x^4\right )^{3/4}}{64 b} \]
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^{12} \left (a+b x^4\right )^{3/4} \, dx &=\frac{1}{16} x^{13} \left (a+b x^4\right )^{3/4}+\frac{1}{16} (3 a) \int \frac{x^{12}}{\sqrt [4]{a+b x^4}} \, dx\\ &=\frac{a x^9 \left (a+b x^4\right )^{3/4}}{64 b}+\frac{1}{16} x^{13} \left (a+b x^4\right )^{3/4}-\frac{\left (9 a^2\right ) \int \frac{x^8}{\sqrt [4]{a+b x^4}} \, dx}{64 b}\\ &=-\frac{9 a^2 x^5 \left (a+b x^4\right )^{3/4}}{512 b^2}+\frac{a x^9 \left (a+b x^4\right )^{3/4}}{64 b}+\frac{1}{16} x^{13} \left (a+b x^4\right )^{3/4}+\frac{\left (45 a^3\right ) \int \frac{x^4}{\sqrt [4]{a+b x^4}} \, dx}{512 b^2}\\ &=\frac{45 a^3 x \left (a+b x^4\right )^{3/4}}{2048 b^3}-\frac{9 a^2 x^5 \left (a+b x^4\right )^{3/4}}{512 b^2}+\frac{a x^9 \left (a+b x^4\right )^{3/4}}{64 b}+\frac{1}{16} x^{13} \left (a+b x^4\right )^{3/4}-\frac{\left (45 a^4\right ) \int \frac{1}{\sqrt [4]{a+b x^4}} \, dx}{2048 b^3}\\ &=\frac{45 a^3 x \left (a+b x^4\right )^{3/4}}{2048 b^3}-\frac{9 a^2 x^5 \left (a+b x^4\right )^{3/4}}{512 b^2}+\frac{a x^9 \left (a+b x^4\right )^{3/4}}{64 b}+\frac{1}{16} x^{13} \left (a+b x^4\right )^{3/4}-\frac{\left (45 a^4\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^4} \, dx,x,\frac{x}{\sqrt [4]{a+b x^4}}\right )}{2048 b^3}\\ &=\frac{45 a^3 x \left (a+b x^4\right )^{3/4}}{2048 b^3}-\frac{9 a^2 x^5 \left (a+b x^4\right )^{3/4}}{512 b^2}+\frac{a x^9 \left (a+b x^4\right )^{3/4}}{64 b}+\frac{1}{16} x^{13} \left (a+b x^4\right )^{3/4}-\frac{\left (45 a^4\right ) \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{b} x^2} \, dx,x,\frac{x}{\sqrt [4]{a+b x^4}}\right )}{4096 b^3}-\frac{\left (45 a^4\right ) \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{b} x^2} \, dx,x,\frac{x}{\sqrt [4]{a+b x^4}}\right )}{4096 b^3}\\ &=\frac{45 a^3 x \left (a+b x^4\right )^{3/4}}{2048 b^3}-\frac{9 a^2 x^5 \left (a+b x^4\right )^{3/4}}{512 b^2}+\frac{a x^9 \left (a+b x^4\right )^{3/4}}{64 b}+\frac{1}{16} x^{13} \left (a+b x^4\right )^{3/4}-\frac{45 a^4 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{4096 b^{13/4}}-\frac{45 a^4 \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{4096 b^{13/4}}\\ \end{align*}
Mathematica [C] time = 0.063912, size = 105, normalized size = 0.7 \[ \frac{x \left (a+b x^4\right )^{3/4} \left (\left (\frac{b x^4}{a}+1\right )^{3/4} \left (-9 a^2 b x^4+15 a^3+8 a b^2 x^8+32 b^3 x^{12}\right )-15 a^3 \, _2F_1\left (-\frac{3}{4},\frac{1}{4};\frac{5}{4};-\frac{b x^4}{a}\right )\right )}{512 b^3 \left (\frac{b x^4}{a}+1\right )^{3/4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.028, 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(-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 = 1.94293, size = 618, normalized size = 4.15 \begin{align*} -\frac{180 \, \left (\frac{a^{16}}{b^{13}}\right )^{\frac{1}{4}} b^{3} \arctan \left (-\frac{{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (\frac{a^{16}}{b^{13}}\right )^{\frac{1}{4}} a^{12} b^{3} - \left (\frac{a^{16}}{b^{13}}\right )^{\frac{1}{4}} b^{3} x \sqrt{\frac{\sqrt{\frac{a^{16}}{b^{13}}} a^{16} b^{7} x^{2} + \sqrt{b x^{4} + a} a^{24}}{x^{2}}}}{a^{16} x}\right ) + 45 \, \left (\frac{a^{16}}{b^{13}}\right )^{\frac{1}{4}} b^{3} \log \left (\frac{91125 \,{\left ({\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{12} + \left (\frac{a^{16}}{b^{13}}\right )^{\frac{3}{4}} b^{10} x\right )}}{x}\right ) - 45 \, \left (\frac{a^{16}}{b^{13}}\right )^{\frac{1}{4}} b^{3} \log \left (\frac{91125 \,{\left ({\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{12} - \left (\frac{a^{16}}{b^{13}}\right )^{\frac{3}{4}} b^{10} x\right )}}{x}\right ) - 4 \,{\left (128 \, b^{3} x^{13} + 32 \, a b^{2} x^{9} - 36 \, a^{2} b x^{5} + 45 \, a^{3} x\right )}{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{8192 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 7.57449, size = 39, normalized size = 0.26 \begin{align*} \frac{a^{\frac{3}{4}} 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 \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{3}{4}} x^{12}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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