Optimal. Leaf size=136 \[ \frac{b \tan ^{-1}\left (\sqrt{\frac{3}{2}} x^2\right )}{2 \sqrt{6}}+\frac{c \log \left (3 x^2-6^{3/4} x+\sqrt{6}\right )}{4\ 6^{3/4}}-\frac{c \log \left (3 x^2+6^{3/4} x+\sqrt{6}\right )}{4\ 6^{3/4}}-\frac{c \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{2\ 6^{3/4}}+\frac{c \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{2\ 6^{3/4}}+\frac{1}{12} d \log \left (3 x^4+2\right ) \]
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Rubi [A] time = 0.139748, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 12, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {1594, 1831, 297, 1162, 617, 204, 1165, 628, 1248, 635, 203, 260} \[ \frac{b \tan ^{-1}\left (\sqrt{\frac{3}{2}} x^2\right )}{2 \sqrt{6}}+\frac{c \log \left (3 x^2-6^{3/4} x+\sqrt{6}\right )}{4\ 6^{3/4}}-\frac{c \log \left (3 x^2+6^{3/4} x+\sqrt{6}\right )}{4\ 6^{3/4}}-\frac{c \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{2\ 6^{3/4}}+\frac{c \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{2\ 6^{3/4}}+\frac{1}{12} d \log \left (3 x^4+2\right ) \]
Antiderivative was successfully verified.
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Rule 1594
Rule 1831
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rule 1248
Rule 635
Rule 203
Rule 260
Rubi steps
\begin{align*} \int \frac{b x+c x^2+d x^3}{2+3 x^4} \, dx &=\int \frac{x \left (b+c x+d x^2\right )}{2+3 x^4} \, dx\\ &=\int \left (\frac{c x^2}{2+3 x^4}+\frac{x \left (b+d x^2\right )}{2+3 x^4}\right ) \, dx\\ &=c \int \frac{x^2}{2+3 x^4} \, dx+\int \frac{x \left (b+d x^2\right )}{2+3 x^4} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{b+d x}{2+3 x^2} \, dx,x,x^2\right )-\frac{c \int \frac{\sqrt{2}-\sqrt{3} x^2}{2+3 x^4} \, dx}{2 \sqrt{3}}+\frac{c \int \frac{\sqrt{2}+\sqrt{3} x^2}{2+3 x^4} \, dx}{2 \sqrt{3}}\\ &=\frac{1}{2} b \operatorname{Subst}\left (\int \frac{1}{2+3 x^2} \, dx,x,x^2\right )+\frac{1}{12} c \int \frac{1}{\sqrt{\frac{2}{3}}-\frac{2^{3/4} x}{\sqrt [4]{3}}+x^2} \, dx+\frac{1}{12} c \int \frac{1}{\sqrt{\frac{2}{3}}+\frac{2^{3/4} x}{\sqrt [4]{3}}+x^2} \, dx+\frac{c \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}{4\ 6^{3/4}}+\frac{c \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}{4\ 6^{3/4}}+\frac{1}{2} d \operatorname{Subst}\left (\int \frac{x}{2+3 x^2} \, dx,x,x^2\right )\\ &=\frac{b \tan ^{-1}\left (\sqrt{\frac{3}{2}} x^2\right )}{2 \sqrt{6}}+\frac{c \log \left (\sqrt{6}-6^{3/4} x+3 x^2\right )}{4\ 6^{3/4}}-\frac{c \log \left (\sqrt{6}+6^{3/4} x+3 x^2\right )}{4\ 6^{3/4}}+\frac{1}{12} d \log \left (2+3 x^4\right )+\frac{c \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt [4]{6} x\right )}{2\ 6^{3/4}}-\frac{c \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt [4]{6} x\right )}{2\ 6^{3/4}}\\ &=\frac{b \tan ^{-1}\left (\sqrt{\frac{3}{2}} x^2\right )}{2 \sqrt{6}}-\frac{c \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{2\ 6^{3/4}}+\frac{c \tan ^{-1}\left (1+\sqrt [4]{6} x\right )}{2\ 6^{3/4}}+\frac{c \log \left (\sqrt{6}-6^{3/4} x+3 x^2\right )}{4\ 6^{3/4}}-\frac{c \log \left (\sqrt{6}+6^{3/4} x+3 x^2\right )}{4\ 6^{3/4}}+\frac{1}{12} d \log \left (2+3 x^4\right )\\ \end{align*}
Mathematica [A] time = 0.0625569, size = 125, normalized size = 0.92 \[ \frac{1}{24} \left (-2 \sqrt [4]{6} \left (\sqrt [4]{6} b+c\right ) \tan ^{-1}\left (1-\sqrt [4]{6} x\right )+2 \sqrt [4]{6} \left (c-\sqrt [4]{6} b\right ) \tan ^{-1}\left (\sqrt [4]{6} x+1\right )+\sqrt [4]{6} c \log \left (\sqrt{6} x^2-2 \sqrt [4]{6} x+2\right )-\sqrt [4]{6} c \log \left (\sqrt{6} x^2+2 \sqrt [4]{6} x+2\right )+2 d \log \left (3 x^4+2\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 140, normalized size = 1. \begin{align*}{\frac{b\sqrt{6}}{12}\arctan \left ({\frac{{x}^{2}\sqrt{6}}{2}} \right ) }+{\frac{c\sqrt{3}{6}^{{\frac{3}{4}}}\sqrt{2}}{72}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}-1 \right ) }+{\frac{c\sqrt{3}{6}^{{\frac{3}{4}}}\sqrt{2}}{144}\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{c\sqrt{3}{6}^{{\frac{3}{4}}}\sqrt{2}}{72}\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 = 1.45402, size = 235, normalized size = 1.73 \begin{align*} \frac{1}{72} \, \sqrt{3} \sqrt{2}{\left (3^{\frac{3}{4}} 2^{\frac{3}{4}} c - 6 \, b\right )} \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}{72} \, \sqrt{3} \sqrt{2}{\left (3^{\frac{3}{4}} 2^{\frac{3}{4}} c + 6 \, b\right )} \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}{72} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (3^{\frac{1}{4}} 2^{\frac{3}{4}} d - \sqrt{3} c\right )} \log \left (\sqrt{3} x^{2} + 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) + \frac{1}{72} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (3^{\frac{1}{4}} 2^{\frac{3}{4}} d + \sqrt{3} c\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 [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.1706, size = 189, normalized size = 1.39 \begin{align*} \operatorname{RootSum}{\left (82944 t^{4} - 27648 t^{3} d + t^{2} \left (1728 b^{2} + 3456 d^{2}\right ) + t \left (- 288 b^{2} d + 288 b c^{2} - 192 d^{3}\right ) + 9 b^{4} + 12 b^{2} d^{2} - 24 b c^{2} d + 6 c^{4} + 4 d^{4}, \left ( t \mapsto t \log{\left (x + \frac{- 3456 t^{3} c^{2} + 864 t^{2} b^{3} + 864 t^{2} c^{2} d - 144 t b^{3} d - 108 t b^{2} c^{2} - 72 t c^{2} d^{2} + 9 b^{5} + 6 b^{3} d^{2} + 9 b^{2} c^{2} d - 9 b c^{4} + 2 c^{2} d^{3}}{18 b^{4} c - 3 c^{5}} \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10254, size = 167, normalized size = 1.23 \begin{align*} -\frac{1}{12} \,{\left (\sqrt{6} b - 6^{\frac{1}{4}} c\right )} \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}{12} \,{\left (\sqrt{6} b + 6^{\frac{1}{4}} c\right )} \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} \,{\left (6^{\frac{1}{4}} c - 2 \, d\right )} \log \left (x^{2} + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) + \frac{1}{24} \,{\left (6^{\frac{1}{4}} c + 2 \, 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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