Optimal. Leaf size=43 \[ \frac {1}{16} \sqrt {4 x^2+3 i x} (8 x+3 i)+\frac {9}{64} i \sin ^{-1}\left (1-\frac {8 i x}{3}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {612, 619, 215} \[ \frac {1}{16} \sqrt {4 x^2+3 i x} (8 x+3 i)+\frac {9}{64} i \sin ^{-1}\left (1-\frac {8 i x}{3}\right ) \]
Antiderivative was successfully verified.
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Rule 215
Rule 612
Rule 619
Rubi steps
\begin {align*} \int \sqrt {3 i x+4 x^2} \, dx &=\frac {1}{16} (3 i+8 x) \sqrt {3 i x+4 x^2}+\frac {9}{32} \int \frac {1}{\sqrt {3 i x+4 x^2}} \, dx\\ &=\frac {1}{16} (3 i+8 x) \sqrt {3 i x+4 x^2}+\frac {3}{64} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{9}}} \, dx,x,3 i+8 x\right )\\ &=\frac {1}{16} (3 i+8 x) \sqrt {3 i x+4 x^2}+\frac {9}{64} i \sin ^{-1}\left (1-\frac {8 i x}{3}\right )\\ \end {align*}
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Mathematica [A] time = 0.05, size = 64, normalized size = 1.49 \[ \frac {1}{32} \sqrt {x (4 x+3 i)} \left (16 x-\frac {9 \sqrt [4]{-1} \sin ^{-1}\left ((1+i) \sqrt {\frac {2}{3}} \sqrt {x}\right )}{\sqrt {3-4 i x} \sqrt {x}}+6 i\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 39, normalized size = 0.91 \[ \frac {1}{256} \, \sqrt {4 \, x^{2} + 3 i \, x} {\left (128 \, x + 48 i\right )} - \frac {9}{64} \, \log \left (-2 \, x + \sqrt {4 \, x^{2} + 3 i \, x} - \frac {3}{4} i\right ) - \frac {9}{256} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.63, size = 1, normalized size = 0.02 \[ 0 \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 31, normalized size = 0.72 \[ \frac {9 \arcsinh \left (\frac {8 x}{3}+i\right )}{64}+\frac {\left (8 x +3 i\right ) \sqrt {4 x^{2}+3 i x}}{16} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.96, size = 49, normalized size = 1.14 \[ \frac {1}{2} \, \sqrt {4 \, x^{2} + 3 i \, x} x + \frac {3}{16} i \, \sqrt {4 \, x^{2} + 3 i \, x} + \frac {9}{64} \, \log \left (8 \, x + 4 \, \sqrt {4 \, x^{2} + 3 i \, x} + 3 i\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 39, normalized size = 0.91 \[ \frac {9\,\ln \left (x+\frac {\sqrt {x\,\left (4\,x+3{}\mathrm {i}\right )}}{2}+\frac {3}{8}{}\mathrm {i}\right )}{64}+\left (\frac {x}{2}+\frac {3}{16}{}\mathrm {i}\right )\,\sqrt {4\,x^2+x\,3{}\mathrm {i}} \]
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 \[ \int \sqrt {4 x^{2} + 3 i x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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