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 = 0.10, antiderivative size = 114, normalized size of antiderivative = 1.00, 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 204
Rule 211
Rule 260
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 1876
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*}
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Mathematica [A] time = 0.03, 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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fricas [B] time = 0.85, size = 359, normalized size = 3.15 \[ -\frac {4 \cdot 6^{\frac {1}{4}} \sqrt {3} \sqrt {2} {\left (a^{4}\right )}^{\frac {1}{4}} a^{4} \arctan \left (-\frac {6^{\frac {3}{4}} \sqrt {3} \sqrt {2} {\left (a^{4}\right )}^{\frac {3}{4}} a^{4} x - 6^{\frac {3}{4}} \sqrt {3} \sqrt {2} \sqrt {\frac {1}{3}} {\left (a^{4}\right )}^{\frac {3}{4}} a^{4} \sqrt {\frac {3 \, a^{2} x^{2} + 6^{\frac {1}{4}} \sqrt {3} \sqrt {2} {\left (a^{4}\right )}^{\frac {1}{4}} a x + \sqrt {6} \sqrt {a^{4}}}{a^{2}}} + 6 \, a^{7}}{6 \, a^{7}}\right ) + 4 \cdot 6^{\frac {1}{4}} \sqrt {3} \sqrt {2} {\left (a^{4}\right )}^{\frac {1}{4}} a^{4} \arctan \left (-\frac {6^{\frac {3}{4}} \sqrt {3} \sqrt {2} {\left (a^{4}\right )}^{\frac {3}{4}} a^{4} x - 6^{\frac {3}{4}} \sqrt {3} \sqrt {2} \sqrt {\frac {1}{3}} {\left (a^{4}\right )}^{\frac {3}{4}} a^{4} \sqrt {\frac {3 \, a^{2} x^{2} - 6^{\frac {1}{4}} \sqrt {3} \sqrt {2} {\left (a^{4}\right )}^{\frac {1}{4}} a x + \sqrt {6} \sqrt {a^{4}}}{a^{2}}} - 6 \, a^{7}}{6 \, a^{7}}\right ) - {\left (6^{\frac {1}{4}} \sqrt {3} \sqrt {2} {\left (a^{4}\right )}^{\frac {1}{4}} a^{4} + 4 \, a^{4} d\right )} \log \left (3 \, a^{2} x^{2} + 6^{\frac {1}{4}} \sqrt {3} \sqrt {2} {\left (a^{4}\right )}^{\frac {1}{4}} a x + \sqrt {6} \sqrt {a^{4}}\right ) + {\left (6^{\frac {1}{4}} \sqrt {3} \sqrt {2} {\left (a^{4}\right )}^{\frac {1}{4}} a^{4} - 4 \, a^{4} d\right )} \log \left (3 \, a^{2} x^{2} - 6^{\frac {1}{4}} \sqrt {3} \sqrt {2} {\left (a^{4}\right )}^{\frac {1}{4}} a x + \sqrt {6} \sqrt {a^{4}}\right )}{48 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 109, normalized size = 0.96 \[ \frac {1}{24} \cdot 6^{\frac {3}{4}} a \arctan \left (\frac {3}{4} \, \sqrt {2} \left (\frac {2}{3}\right )^{\frac {3}{4}} {\left (2 \, x + \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}}\right )}\right ) + \frac {1}{24} \cdot 6^{\frac {3}{4}} a \arctan \left (\frac {3}{4} \, \sqrt {2} \left (\frac {2}{3}\right )^{\frac {3}{4}} {\left (2 \, x - \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}}\right )}\right ) + \frac {1}{48} \, {\left (6^{\frac {3}{4}} a + 4 \, d\right )} \log \left (x^{2} + \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}} x + \sqrt {\frac {2}{3}}\right ) - \frac {1}{48} \, {\left (6^{\frac {3}{4}} a - 4 \, d\right )} \log \left (x^{2} - \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}} x + \sqrt {\frac {2}{3}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 125, normalized size = 1.10 \[ \frac {\sqrt {3}\, 6^{\frac {1}{4}} \sqrt {2}\, a \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, 6^{\frac {3}{4}} x}{6}-1\right )}{24}+\frac {\sqrt {3}\, 6^{\frac {1}{4}} \sqrt {2}\, a \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, 6^{\frac {3}{4}} x}{6}+1\right )}{24}+\frac {\sqrt {3}\, 6^{\frac {1}{4}} \sqrt {2}\, a \ln \left (\frac {x^{2}+\frac {\sqrt {3}\, 6^{\frac {1}{4}} \sqrt {2}\, x}{3}+\frac {\sqrt {6}}{3}}{x^{2}-\frac {\sqrt {3}\, 6^{\frac {1}{4}} \sqrt {2}\, x}{3}+\frac {\sqrt {6}}{3}}\right )}{48}+\frac {d \ln \left (3 x^{4}+2\right )}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.06, size = 149, normalized size = 1.31 \[ \frac {1}{24} \cdot 3^{\frac {3}{4}} 2^{\frac {3}{4}} a \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (2 \, \sqrt {3} x + 3^{\frac {1}{4}} 2^{\frac {3}{4}}\right )}\right ) + \frac {1}{24} \cdot 3^{\frac {3}{4}} 2^{\frac {3}{4}} a \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (2 \, \sqrt {3} x - 3^{\frac {1}{4}} 2^{\frac {3}{4}}\right )}\right ) + \frac {1}{144} \cdot 3^{\frac {3}{4}} 2^{\frac {3}{4}} {\left (2 \cdot 3^{\frac {1}{4}} 2^{\frac {1}{4}} d + 3 \, a\right )} \log \left (\sqrt {3} x^{2} + 3^{\frac {1}{4}} 2^{\frac {3}{4}} x + \sqrt {2}\right ) + \frac {1}{144} \cdot 3^{\frac {3}{4}} 2^{\frac {3}{4}} {\left (2 \cdot 3^{\frac {1}{4}} 2^{\frac {1}{4}} d - 3 \, a\right )} \log \left (\sqrt {3} x^{2} - 3^{\frac {1}{4}} 2^{\frac {3}{4}} x + \sqrt {2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.28, size = 117, normalized size = 1.03 \[ \ln \left (x-\frac {{\left (-1\right )}^{1/4}\,2^{1/4}\,3^{3/4}}{3}\right )\,\left (\frac {d}{12}-\frac {6^{1/4}\,\sqrt {\frac {3}{4}{}\mathrm {i}}\,a}{12}\right )+\ln \left (x+\frac {{\left (-1\right )}^{1/4}\,2^{1/4}\,3^{3/4}}{3}\right )\,\left (\frac {d}{12}+\frac {6^{1/4}\,\sqrt {\frac {3}{4}{}\mathrm {i}}\,a}{12}\right )+\ln \left (x-\frac {{\left (-1\right )}^{3/4}\,2^{1/4}\,3^{3/4}}{3}\right )\,\left (\frac {d}{12}+\frac {6^{1/4}\,\sqrt {-\frac {3}{4}{}\mathrm {i}}\,a}{12}\right )+\ln \left (x+\frac {{\left (-1\right )}^{3/4}\,2^{1/4}\,3^{3/4}}{3}\right )\,\left (\frac {d}{12}-\frac {6^{1/4}\,\sqrt {-\frac {3}{4}{}\mathrm {i}}\,a}{12}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.42, size = 51, normalized size = 0.45 \[ \operatorname {RootSum} {\left (165888 t^{4} - 55296 t^{3} d + 6912 t^{2} d^{2} - 384 t d^{3} + 27 a^{4} + 8 d^{4}, \left (t \mapsto t \log {\left (x + \frac {24 t - 2 d}{3 a} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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