Optimal. Leaf size=45 \[ \sqrt {2} \tan ^{-1}\left (\sqrt {6 x+8}+3\right )-\sqrt {2} \tan ^{-1}\left (3-\sqrt {2} \sqrt {3 x+4}\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {827, 1161, 618, 204} \[ \sqrt {2} \tan ^{-1}\left (\sqrt {6 x+8}+3\right )-\sqrt {2} \tan ^{-1}\left (3-\sqrt {2} \sqrt {3 x+4}\right ) \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 827
Rule 1161
Rubi steps
\begin {align*} \int \frac {3+x}{\sqrt {4+3 x} \left (1+x^2\right )} \, dx &=2 \operatorname {Subst}\left (\int \frac {5+x^2}{25-8 x^2+x^4} \, dx,x,\sqrt {4+3 x}\right )\\ &=\operatorname {Subst}\left (\int \frac {1}{5-3 \sqrt {2} x+x^2} \, dx,x,\sqrt {4+3 x}\right )+\operatorname {Subst}\left (\int \frac {1}{5+3 \sqrt {2} x+x^2} \, dx,x,\sqrt {4+3 x}\right )\\ &=-\left (2 \operatorname {Subst}\left (\int \frac {1}{-2-x^2} \, dx,x,-3 \sqrt {2}+2 \sqrt {4+3 x}\right )\right )-2 \operatorname {Subst}\left (\int \frac {1}{-2-x^2} \, dx,x,3 \sqrt {2}+2 \sqrt {4+3 x}\right )\\ &=-\sqrt {2} \tan ^{-1}\left (3-\sqrt {2} \sqrt {4+3 x}\right )+\sqrt {2} \tan ^{-1}\left (3+\sqrt {2} \sqrt {4+3 x}\right )\\ \end {align*}
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Mathematica [C] time = 0.04, size = 63, normalized size = 1.40 \[ \frac {1}{5} \left ((1-3 i) \sqrt {4-3 i} \tanh ^{-1}\left (\frac {\sqrt {3 x+4}}{\sqrt {4-3 i}}\right )+(1+3 i) \sqrt {4+3 i} \tanh ^{-1}\left (\frac {\sqrt {3 x+4}}{\sqrt {4+3 i}}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.06, size = 22, normalized size = 0.49 \[ \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (3 \, x - 1\right )}}{2 \, \sqrt {3 \, x + 4}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 63, normalized size = 1.40 \[ \sqrt {2} \arctan \left (\frac {1}{250} \cdot 25^{\frac {3}{4}} \sqrt {10} {\left (3 \cdot 25^{\frac {1}{4}} \sqrt {10} + 10 \, \sqrt {3 \, x + 4}\right )}\right ) + \sqrt {2} \arctan \left (-\frac {1}{250} \cdot 25^{\frac {3}{4}} \sqrt {10} {\left (3 \cdot 25^{\frac {1}{4}} \sqrt {10} - 10 \, \sqrt {3 \, x + 4}\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 52, normalized size = 1.16 \[ \sqrt {2}\, \arctan \left (\frac {\left (2 \sqrt {3 x +4}-3 \sqrt {2}\right ) \sqrt {2}}{2}\right )+\sqrt {2}\, \arctan \left (\frac {\left (2 \sqrt {3 x +4}+3 \sqrt {2}\right ) \sqrt {2}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x + 3}{{\left (x^{2} + 1\right )} \sqrt {3 \, x + 4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.76, size = 38, normalized size = 0.84 \[ \sqrt {2}\,\left (\mathrm {atan}\left (\frac {\sqrt {2}\,{\left (3\,x+4\right )}^{3/2}}{10}-\frac {3\,\sqrt {6\,x+8}}{10}\right )+\mathrm {atan}\left (\frac {\sqrt {6\,x+8}}{2}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x + 3}{\sqrt {3 x + 4} \left (x^{2} + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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