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 = 0.14, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 12, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, 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 203
Rule 204
Rule 260
Rule 297
Rule 617
Rule 628
Rule 635
Rule 1162
Rule 1165
Rule 1248
Rule 1594
Rule 1831
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*}
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Mathematica [A] time = 0.08, 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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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 124, normalized size = 0.91 \[ -\frac {1}{12} \, {\left (\sqrt {6} b - 6^{\frac {1}{4}} c\right )} \arctan \left (\frac {3}{4} \, \sqrt {2} \left (\frac {2}{3}\right )^{\frac {3}{4}} {\left (2 \, x + \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}}\right )}\right ) + \frac {1}{12} \, {\left (\sqrt {6} b + 6^{\frac {1}{4}} c\right )} \arctan \left (\frac {3}{4} \, \sqrt {2} \left (\frac {2}{3}\right )^{\frac {3}{4}} {\left (2 \, x - \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}}\right )}\right ) - \frac {1}{24} \, {\left (6^{\frac {1}{4}} c - 2 \, d\right )} \log \left (x^{2} + \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}} x + \sqrt {\frac {2}{3}}\right ) + \frac {1}{24} \, {\left (6^{\frac {1}{4}} c + 2 \, d\right )} \log \left (x^{2} - \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}} x + \sqrt {\frac {2}{3}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 140, normalized size = 1.03 \[ \frac {\sqrt {6}\, b \arctan \left (\frac {\sqrt {6}\, x^{2}}{2}\right )}{12}+\frac {\sqrt {3}\, 6^{\frac {3}{4}} \sqrt {2}\, c \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, 6^{\frac {3}{4}} x}{6}-1\right )}{72}+\frac {\sqrt {3}\, 6^{\frac {3}{4}} \sqrt {2}\, c \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, 6^{\frac {3}{4}} x}{6}+1\right )}{72}+\frac {\sqrt {3}\, 6^{\frac {3}{4}} \sqrt {2}\, c \ln \left (\frac {x^{2}-\frac {\sqrt {3}\, 6^{\frac {1}{4}} \sqrt {2}\, x}{3}+\frac {\sqrt {6}}{3}}{x^{2}+\frac {\sqrt {3}\, 6^{\frac {1}{4}} \sqrt {2}\, x}{3}+\frac {\sqrt {6}}{3}}\right )}{144}+\frac {d \ln \left (3 x^{4}+2\right )}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.02, size = 174, normalized size = 1.28 \[ \frac {1}{72} \, \sqrt {3} \sqrt {2} {\left (3^{\frac {3}{4}} 2^{\frac {3}{4}} c - 6 \, b\right )} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (2 \, \sqrt {3} x + 3^{\frac {1}{4}} 2^{\frac {3}{4}}\right )}\right ) + \frac {1}{72} \, \sqrt {3} \sqrt {2} {\left (3^{\frac {3}{4}} 2^{\frac {3}{4}} c + 6 \, b\right )} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (2 \, \sqrt {3} x - 3^{\frac {1}{4}} 2^{\frac {3}{4}}\right )}\right ) + \frac {1}{72} \cdot 3^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (3^{\frac {1}{4}} 2^{\frac {3}{4}} d - \sqrt {3} c\right )} \log \left (\sqrt {3} x^{2} + 3^{\frac {1}{4}} 2^{\frac {3}{4}} x + \sqrt {2}\right ) + \frac {1}{72} \cdot 3^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (3^{\frac {1}{4}} 2^{\frac {3}{4}} d + \sqrt {3} c\right )} \log \left (\sqrt {3} x^{2} - 3^{\frac {1}{4}} 2^{\frac {3}{4}} x + \sqrt {2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.39, size = 300, normalized size = 2.21 \[ \sum _{k=1}^4\ln \left (-\mathrm {root}\left (z^4-\frac {d\,z^3}{3}+\frac {z^2\,\left (1728\,b^2+3456\,d^2\right )}{82944}-\frac {z\,\left (-288\,b\,c^2+288\,b^2\,d+192\,d^3\right )}{82944}-\frac {b\,c^2\,d}{3456}+\frac {b^2\,d^2}{6912}+\frac {d^4}{20736}+\frac {c^4}{13824}+\frac {b^4}{9216},z,k\right )\,\left (144\,b\,c+x\,\left (144\,b\,d-72\,c^2\right )-\mathrm {root}\left (z^4-\frac {d\,z^3}{3}+\frac {z^2\,\left (1728\,b^2+3456\,d^2\right )}{82944}-\frac {z\,\left (-288\,b\,c^2+288\,b^2\,d+192\,d^3\right )}{82944}-\frac {b\,c^2\,d}{3456}+\frac {b^2\,d^2}{6912}+\frac {d^4}{20736}+\frac {c^4}{13824}+\frac {b^4}{9216},z,k\right )\,b\,x\,864\right )+x\,\left (9\,b^3+6\,b\,d^2-6\,c^2\,d\right )-6\,c^3+12\,b\,c\,d\right )\,\mathrm {root}\left (z^4-\frac {d\,z^3}{3}+\frac {z^2\,\left (1728\,b^2+3456\,d^2\right )}{82944}-\frac {z\,\left (-288\,b\,c^2+288\,b^2\,d+192\,d^3\right )}{82944}-\frac {b\,c^2\,d}{3456}+\frac {b^2\,d^2}{6912}+\frac {d^4}{20736}+\frac {c^4}{13824}+\frac {b^4}{9216},z,k\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.99, size = 189, normalized size = 1.39 \[ \operatorname {RootSum} {\left (82944 t^{4} - 27648 t^{3} d + t^{2} \left (1728 b^{2} + 3456 d^{2}\right ) + t \left (- 288 b^{2} d + 288 b c^{2} - 192 d^{3}\right ) + 9 b^{4} + 12 b^{2} d^{2} - 24 b c^{2} d + 6 c^{4} + 4 d^{4}, \left (t \mapsto t \log {\left (x + \frac {- 3456 t^{3} c^{2} + 864 t^{2} b^{3} + 864 t^{2} c^{2} d - 144 t b^{3} d - 108 t b^{2} c^{2} - 72 t c^{2} d^{2} + 9 b^{5} + 6 b^{3} d^{2} + 9 b^{2} c^{2} d - 9 b c^{4} + 2 c^{2} d^{3}}{18 b^{4} c - 3 c^{5}} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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