Optimal. Leaf size=163 \[ -\frac {\left (\sqrt {6} a-2 c\right ) \log \left (3 x^2-6^{3/4} x+\sqrt {6}\right )}{8\ 6^{3/4}}+\frac {\left (\sqrt {6} a-2 c\right ) \log \left (3 x^2+6^{3/4} x+\sqrt {6}\right )}{8\ 6^{3/4}}-\frac {\left (\sqrt {6} a+2 c\right ) \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{4\ 6^{3/4}}+\frac {\left (\sqrt {6} a+2 c\right ) \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{4\ 6^{3/4}}+\frac {b \tan ^{-1}\left (\sqrt {\frac {3}{2}} x^2\right )}{2 \sqrt {6}} \]
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Rubi [A] time = 0.12, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {1876, 275, 203, 1168, 1162, 617, 204, 1165, 628} \[ -\frac {\left (\sqrt {6} a-2 c\right ) \log \left (3 x^2-6^{3/4} x+\sqrt {6}\right )}{8\ 6^{3/4}}+\frac {\left (\sqrt {6} a-2 c\right ) \log \left (3 x^2+6^{3/4} x+\sqrt {6}\right )}{8\ 6^{3/4}}-\frac {\left (\sqrt {6} a+2 c\right ) \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{4\ 6^{3/4}}+\frac {\left (\sqrt {6} a+2 c\right ) \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{4\ 6^{3/4}}+\frac {b \tan ^{-1}\left (\sqrt {\frac {3}{2}} x^2\right )}{2 \sqrt {6}} \]
Antiderivative was successfully verified.
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Rule 203
Rule 204
Rule 275
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 1168
Rule 1876
Rubi steps
\begin {align*} \int \frac {a+b x+c x^2}{2+3 x^4} \, dx &=\int \left (\frac {b x}{2+3 x^4}+\frac {a+c x^2}{2+3 x^4}\right ) \, dx\\ &=b \int \frac {x}{2+3 x^4} \, dx+\int \frac {a+c x^2}{2+3 x^4} \, dx\\ &=\frac {1}{2} b \operatorname {Subst}\left (\int \frac {1}{2+3 x^2} \, dx,x,x^2\right )+\frac {1}{12} \left (\sqrt {6} a-2 c\right ) \int \frac {\sqrt {6}-3 x^2}{2+3 x^4} \, dx+\frac {1}{12} \left (\sqrt {6} a+2 c\right ) \int \frac {\sqrt {6}+3 x^2}{2+3 x^4} \, dx\\ &=\frac {b \tan ^{-1}\left (\sqrt {\frac {3}{2}} x^2\right )}{2 \sqrt {6}}-\frac {\left (\sqrt {6} a-2 c\right ) \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\ 6^{3/4}}-\frac {\left (\sqrt {6} a-2 c\right ) \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\ 6^{3/4}}+\frac {1}{24} \left (\sqrt {6} a+2 c\right ) \int \frac {1}{\sqrt {\frac {2}{3}}-\frac {2^{3/4} x}{\sqrt [4]{3}}+x^2} \, dx+\frac {1}{24} \left (\sqrt {6} a+2 c\right ) \int \frac {1}{\sqrt {\frac {2}{3}}+\frac {2^{3/4} x}{\sqrt [4]{3}}+x^2} \, dx\\ &=\frac {b \tan ^{-1}\left (\sqrt {\frac {3}{2}} x^2\right )}{2 \sqrt {6}}-\frac {\left (\sqrt {6} a-2 c\right ) \log \left (\sqrt {6}-6^{3/4} x+3 x^2\right )}{8\ 6^{3/4}}+\frac {\left (\sqrt {6} a-2 c\right ) \log \left (\sqrt {6}+6^{3/4} x+3 x^2\right )}{8\ 6^{3/4}}+\frac {\left (\sqrt {6} a+2 c\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt [4]{6} x\right )}{4\ 6^{3/4}}-\frac {\left (\sqrt {6} a+2 c\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt [4]{6} x\right )}{4\ 6^{3/4}}\\ &=\frac {b \tan ^{-1}\left (\sqrt {\frac {3}{2}} x^2\right )}{2 \sqrt {6}}-\frac {\left (\sqrt {6} a+2 c\right ) \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{4\ 6^{3/4}}+\frac {\left (\sqrt {6} a+2 c\right ) \tan ^{-1}\left (1+\sqrt [4]{6} x\right )}{4\ 6^{3/4}}-\frac {\left (\sqrt {6} a-2 c\right ) \log \left (\sqrt {6}-6^{3/4} x+3 x^2\right )}{8\ 6^{3/4}}+\frac {\left (\sqrt {6} a-2 c\right ) \log \left (\sqrt {6}+6^{3/4} x+3 x^2\right )}{8\ 6^{3/4}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 129, normalized size = 0.79 \[ \frac {-2 \tan ^{-1}\left (1-\sqrt [4]{6} x\right ) \left (\sqrt {6} a+2 \left (\sqrt [4]{6} b+c\right )\right )+2 \tan ^{-1}\left (\sqrt [4]{6} x+1\right ) \left (\sqrt {6} a-2 \sqrt [4]{6} b+2 c\right )-\left (\sqrt {6} a-2 c\right ) \left (\log \left (\sqrt {6} x^2-2 \sqrt [4]{6} x+2\right )-\log \left (\sqrt {6} x^2+2 \sqrt [4]{6} x+2\right )\right )}{8\ 6^{3/4}} \]
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 = 0.22, size = 143, normalized size = 0.88 \[ \frac {1}{24} \, {\left (6^{\frac {3}{4}} a - 2 \, \sqrt {6} b + 2 \cdot 6^{\frac {1}{4}} c\right )} \arctan \left (\frac {3}{4} \, \sqrt {2} \left (\frac {2}{3}\right )^{\frac {3}{4}} {\left (2 \, x + \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}}\right )}\right ) + \frac {1}{24} \, {\left (6^{\frac {3}{4}} a + 2 \, \sqrt {6} b + 2 \cdot 6^{\frac {1}{4}} c\right )} \arctan \left (\frac {3}{4} \, \sqrt {2} \left (\frac {2}{3}\right )^{\frac {3}{4}} {\left (2 \, x - \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}}\right )}\right ) + \frac {1}{48} \, {\left (6^{\frac {3}{4}} a - 2 \cdot 6^{\frac {1}{4}} c\right )} \log \left (x^{2} + \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}} x + \sqrt {\frac {2}{3}}\right ) - \frac {1}{48} \, {\left (6^{\frac {3}{4}} a - 2 \cdot 6^{\frac {1}{4}} c\right )} \log \left (x^{2} - \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}} x + \sqrt {\frac {2}{3}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 241, normalized size = 1.48 \[ \frac {\sqrt {3}\, 6^{\frac {1}{4}} \sqrt {2}\, a \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, 6^{\frac {3}{4}} x}{6}-1\right )}{24}+\frac {\sqrt {3}\, 6^{\frac {1}{4}} \sqrt {2}\, a \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, 6^{\frac {3}{4}} x}{6}+1\right )}{24}+\frac {\sqrt {3}\, 6^{\frac {1}{4}} \sqrt {2}\, a \ln \left (\frac {x^{2}+\frac {\sqrt {3}\, 6^{\frac {1}{4}} \sqrt {2}\, x}{3}+\frac {\sqrt {6}}{3}}{x^{2}-\frac {\sqrt {3}\, 6^{\frac {1}{4}} \sqrt {2}\, x}{3}+\frac {\sqrt {6}}{3}}\right )}{48}+\frac {\sqrt {6}\, b \arctan \left (\frac {\sqrt {6}\, x^{2}}{2}\right )}{12}+\frac {\sqrt {3}\, 6^{\frac {3}{4}} \sqrt {2}\, c \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, 6^{\frac {3}{4}} x}{6}-1\right )}{72}+\frac {\sqrt {3}\, 6^{\frac {3}{4}} \sqrt {2}\, c \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, 6^{\frac {3}{4}} x}{6}+1\right )}{72}+\frac {\sqrt {3}\, 6^{\frac {3}{4}} \sqrt {2}\, c \ln \left (\frac {x^{2}-\frac {\sqrt {3}\, 6^{\frac {1}{4}} \sqrt {2}\, x}{3}+\frac {\sqrt {6}}{3}}{x^{2}+\frac {\sqrt {3}\, 6^{\frac {1}{4}} \sqrt {2}\, x}{3}+\frac {\sqrt {6}}{3}}\right )}{144} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.06, size = 187, normalized size = 1.15 \[ \frac {1}{48} \cdot 3^{\frac {1}{4}} 2^{\frac {3}{4}} {\left (\sqrt {3} a - \sqrt {2} c\right )} \log \left (\sqrt {3} x^{2} + 3^{\frac {1}{4}} 2^{\frac {3}{4}} x + \sqrt {2}\right ) - \frac {1}{48} \cdot 3^{\frac {1}{4}} 2^{\frac {3}{4}} {\left (\sqrt {3} a - \sqrt {2} c\right )} \log \left (\sqrt {3} x^{2} - 3^{\frac {1}{4}} 2^{\frac {3}{4}} x + \sqrt {2}\right ) + \frac {1}{24} \, {\left (3^{\frac {3}{4}} 2^{\frac {3}{4}} a - 2 \, \sqrt {3} \sqrt {2} b + 2 \cdot 3^{\frac {1}{4}} 2^{\frac {1}{4}} c\right )} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (2 \, \sqrt {3} x + 3^{\frac {1}{4}} 2^{\frac {3}{4}}\right )}\right ) + \frac {1}{24} \, {\left (3^{\frac {3}{4}} 2^{\frac {3}{4}} a + 2 \, \sqrt {3} \sqrt {2} b + 2 \cdot 3^{\frac {1}{4}} 2^{\frac {1}{4}} c\right )} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (2 \, \sqrt {3} x - 3^{\frac {1}{4}} 2^{\frac {3}{4}}\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.52, size = 270, normalized size = 1.66 \[ \sum _{k=1}^4\ln \left (9\,a\,b^2-9\,a^2\,c-\mathrm {root}\left (z^4+\frac {z^2\,\left (2304\,a\,c+1152\,b^2\right )}{55296}-\frac {z\,\left (288\,a^2\,b-192\,b\,c^2\right )}{55296}-\frac {a\,b^2\,c}{2304}+\frac {a^2\,c^2}{4608}+\frac {c^4}{13824}+\frac {b^4}{9216}+\frac {a^4}{6144},z,k\right )\,\left (\mathrm {root}\left (z^4+\frac {z^2\,\left (2304\,a\,c+1152\,b^2\right )}{55296}-\frac {z\,\left (288\,a^2\,b-192\,b\,c^2\right )}{55296}-\frac {a\,b^2\,c}{2304}+\frac {a^2\,c^2}{4608}+\frac {c^4}{13824}+\frac {b^4}{9216}+\frac {a^4}{6144},z,k\right )\,\left (864\,a-864\,b\,x\right )+144\,b\,c+x\,\left (108\,a^2-72\,c^2\right )\right )-6\,c^3+x\,\left (9\,b^3-18\,a\,b\,c\right )\right )\,\mathrm {root}\left (z^4+\frac {z^2\,\left (2304\,a\,c+1152\,b^2\right )}{55296}-\frac {z\,\left (288\,a^2\,b-192\,b\,c^2\right )}{55296}-\frac {a\,b^2\,c}{2304}+\frac {a^2\,c^2}{4608}+\frac {c^4}{13824}+\frac {b^4}{9216}+\frac {a^4}{6144},z,k\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 5.07, size = 292, normalized size = 1.79 \[ \operatorname {RootSum} {\left (55296 t^{4} + t^{2} \left (2304 a c + 1152 b^{2}\right ) + t \left (- 288 a^{2} b + 192 b c^{2}\right ) + 9 a^{4} + 12 a^{2} c^{2} - 24 a b^{2} c + 6 b^{4} + 4 c^{4}, \left (t \mapsto t \log {\left (x + \frac {- 13824 t^{3} a^{2} c + 27648 t^{3} a b^{2} + 9216 t^{3} c^{3} + 1728 t^{2} a^{3} b + 3456 t^{2} a b c^{2} - 2304 t^{2} b^{3} c + 216 t a^{5} - 576 t a^{3} c^{2} + 1296 t a^{2} b^{2} c + 288 t a b^{4} + 288 t a c^{4} + 288 t b^{2} c^{3} + 90 a^{4} b c - 90 a^{3} b^{3} + 60 a b^{3} c^{2} - 24 b^{5} c + 24 b c^{5}}{27 a^{6} - 18 a^{4} c^{2} + 144 a^{3} b^{2} c - 72 a^{2} b^{4} - 12 a^{2} c^{4} + 96 a b^{2} c^{3} - 48 b^{4} c^{2} + 8 c^{6}} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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