Optimal. Leaf size=114 \[ -\frac{a \log \left (3 x^2-6^{3/4} x+\sqrt{6}\right )}{8 \sqrt [4]{6}}+\frac{a \log \left (3 x^2+6^{3/4} x+\sqrt{6}\right )}{8 \sqrt [4]{6}}-\frac{a \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{4 \sqrt [4]{6}}+\frac{a \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{4 \sqrt [4]{6}}+\frac{1}{12} d \log \left (3 x^4+2\right ) \]
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Rubi [A] time = 0.0991563, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.471, Rules used = {1876, 211, 1165, 628, 1162, 617, 204, 260} \[ -\frac{a \log \left (3 x^2-6^{3/4} x+\sqrt{6}\right )}{8 \sqrt [4]{6}}+\frac{a \log \left (3 x^2+6^{3/4} x+\sqrt{6}\right )}{8 \sqrt [4]{6}}-\frac{a \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{4 \sqrt [4]{6}}+\frac{a \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{4 \sqrt [4]{6}}+\frac{1}{12} d \log \left (3 x^4+2\right ) \]
Antiderivative was successfully verified.
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Rule 1876
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rule 260
Rubi steps
\begin{align*} \int \frac{a+d x^3}{2+3 x^4} \, dx &=\int \left (\frac{a}{2+3 x^4}+\frac{d x^3}{2+3 x^4}\right ) \, dx\\ &=a \int \frac{1}{2+3 x^4} \, dx+d \int \frac{x^3}{2+3 x^4} \, dx\\ &=\frac{1}{12} d \log \left (2+3 x^4\right )+\frac{a \int \frac{\sqrt{2}-\sqrt{3} x^2}{2+3 x^4} \, dx}{2 \sqrt{2}}+\frac{a \int \frac{\sqrt{2}+\sqrt{3} x^2}{2+3 x^4} \, dx}{2 \sqrt{2}}\\ &=\frac{1}{12} d \log \left (2+3 x^4\right )+\frac{a \int \frac{1}{\sqrt{\frac{2}{3}}-\frac{2^{3/4} x}{\sqrt [4]{3}}+x^2} \, dx}{4 \sqrt{6}}+\frac{a \int \frac{1}{\sqrt{\frac{2}{3}}+\frac{2^{3/4} x}{\sqrt [4]{3}}+x^2} \, dx}{4 \sqrt{6}}-\frac{a \int \frac{\frac{2^{3/4}}{\sqrt [4]{3}}+2 x}{-\sqrt{\frac{2}{3}}-\frac{2^{3/4} x}{\sqrt [4]{3}}-x^2} \, dx}{8 \sqrt [4]{6}}-\frac{a \int \frac{\frac{2^{3/4}}{\sqrt [4]{3}}-2 x}{-\sqrt{\frac{2}{3}}+\frac{2^{3/4} x}{\sqrt [4]{3}}-x^2} \, dx}{8 \sqrt [4]{6}}\\ &=-\frac{a \log \left (\sqrt{6}-6^{3/4} x+3 x^2\right )}{8 \sqrt [4]{6}}+\frac{a \log \left (\sqrt{6}+6^{3/4} x+3 x^2\right )}{8 \sqrt [4]{6}}+\frac{1}{12} d \log \left (2+3 x^4\right )+\frac{a \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt [4]{6} x\right )}{4 \sqrt [4]{6}}-\frac{a \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt [4]{6} x\right )}{4 \sqrt [4]{6}}\\ &=-\frac{a \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{4 \sqrt [4]{6}}+\frac{a \tan ^{-1}\left (1+\sqrt [4]{6} x\right )}{4 \sqrt [4]{6}}-\frac{a \log \left (\sqrt{6}-6^{3/4} x+3 x^2\right )}{8 \sqrt [4]{6}}+\frac{a \log \left (\sqrt{6}+6^{3/4} x+3 x^2\right )}{8 \sqrt [4]{6}}+\frac{1}{12} d \log \left (2+3 x^4\right )\\ \end{align*}
Mathematica [A] time = 0.031317, size = 108, normalized size = 0.95 \[ \frac{1}{48} \left (-6^{3/4} a \log \left (\sqrt{6} x^2-2 \sqrt [4]{6} x+2\right )+6^{3/4} a \log \left (\sqrt{6} x^2+2 \sqrt [4]{6} x+2\right )-2\ 6^{3/4} a \tan ^{-1}\left (1-\sqrt [4]{6} x\right )+2\ 6^{3/4} a \tan ^{-1}\left (\sqrt [4]{6} x+1\right )+4 d \log \left (3 x^4+2\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 125, normalized size = 1.1 \begin{align*}{\frac{a\sqrt{3}\sqrt [4]{6}\sqrt{2}}{24}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}-1 \right ) }+{\frac{a\sqrt{3}\sqrt [4]{6}\sqrt{2}}{48}\ln \left ({ \left ({x}^{2}+{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) \left ({x}^{2}-{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) ^{-1}} \right ) }+{\frac{a\sqrt{3}\sqrt [4]{6}\sqrt{2}}{24}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}+1 \right ) }+{\frac{d\ln \left ( 3\,{x}^{4}+2 \right ) }{12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.45189, size = 201, normalized size = 1.76 \begin{align*} \frac{1}{24} \cdot 3^{\frac{3}{4}} 2^{\frac{3}{4}} a \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (2 \, \sqrt{3} x + 3^{\frac{1}{4}} 2^{\frac{3}{4}}\right )}\right ) + \frac{1}{24} \cdot 3^{\frac{3}{4}} 2^{\frac{3}{4}} a \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (2 \, \sqrt{3} x - 3^{\frac{1}{4}} 2^{\frac{3}{4}}\right )}\right ) + \frac{1}{144} \cdot 3^{\frac{3}{4}} 2^{\frac{3}{4}}{\left (2 \cdot 3^{\frac{1}{4}} 2^{\frac{1}{4}} d + 3 \, a\right )} \log \left (\sqrt{3} x^{2} + 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) + \frac{1}{144} \cdot 3^{\frac{3}{4}} 2^{\frac{3}{4}}{\left (2 \cdot 3^{\frac{1}{4}} 2^{\frac{1}{4}} d - 3 \, a\right )} \log \left (\sqrt{3} x^{2} - 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.61018, size = 1041, normalized size = 9.13 \begin{align*} -\frac{4 \cdot 6^{\frac{1}{4}} \sqrt{3} \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}} a^{4} \arctan \left (-\frac{6^{\frac{3}{4}} \sqrt{3} \sqrt{2}{\left (a^{4}\right )}^{\frac{3}{4}} a^{4} x - 6^{\frac{3}{4}} \sqrt{3} \sqrt{2} \sqrt{\frac{1}{3}}{\left (a^{4}\right )}^{\frac{3}{4}} a^{4} \sqrt{\frac{3 \, a^{2} x^{2} + 6^{\frac{1}{4}} \sqrt{3} \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}} a x + \sqrt{6} \sqrt{a^{4}}}{a^{2}}} + 6 \, a^{7}}{6 \, a^{7}}\right ) + 4 \cdot 6^{\frac{1}{4}} \sqrt{3} \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}} a^{4} \arctan \left (-\frac{6^{\frac{3}{4}} \sqrt{3} \sqrt{2}{\left (a^{4}\right )}^{\frac{3}{4}} a^{4} x - 6^{\frac{3}{4}} \sqrt{3} \sqrt{2} \sqrt{\frac{1}{3}}{\left (a^{4}\right )}^{\frac{3}{4}} a^{4} \sqrt{\frac{3 \, a^{2} x^{2} - 6^{\frac{1}{4}} \sqrt{3} \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}} a x + \sqrt{6} \sqrt{a^{4}}}{a^{2}}} - 6 \, a^{7}}{6 \, a^{7}}\right ) -{\left (6^{\frac{1}{4}} \sqrt{3} \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}} a^{4} + 4 \, a^{4} d\right )} \log \left (3 \, a^{2} x^{2} + 6^{\frac{1}{4}} \sqrt{3} \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}} a x + \sqrt{6} \sqrt{a^{4}}\right ) +{\left (6^{\frac{1}{4}} \sqrt{3} \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}} a^{4} - 4 \, a^{4} d\right )} \log \left (3 \, a^{2} x^{2} - 6^{\frac{1}{4}} \sqrt{3} \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}} a x + \sqrt{6} \sqrt{a^{4}}\right )}{48 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.300339, size = 51, normalized size = 0.45 \begin{align*} \operatorname{RootSum}{\left (165888 t^{4} - 55296 t^{3} d + 6912 t^{2} d^{2} - 384 t d^{3} + 27 a^{4} + 8 d^{4}, \left ( t \mapsto t \log{\left (x + \frac{24 t - 2 d}{3 a} \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13125, size = 147, normalized size = 1.29 \begin{align*} \frac{1}{24} \cdot 6^{\frac{3}{4}} a \arctan \left (\frac{3}{4} \, \sqrt{2} \left (\frac{2}{3}\right )^{\frac{3}{4}}{\left (2 \, x + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}}\right )}\right ) + \frac{1}{24} \cdot 6^{\frac{3}{4}} a \arctan \left (\frac{3}{4} \, \sqrt{2} \left (\frac{2}{3}\right )^{\frac{3}{4}}{\left (2 \, x - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}}\right )}\right ) + \frac{1}{48} \,{\left (6^{\frac{3}{4}} a + 4 \, d\right )} \log \left (x^{2} + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) - \frac{1}{48} \,{\left (6^{\frac{3}{4}} a - 4 \, d\right )} \log \left (x^{2} - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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