Optimal. Leaf size=43 \[ \frac{1}{16} \sqrt{4 x^2+3 i x} (8 x+3 i)+\frac{9}{64} i \sin ^{-1}\left (1-\frac{8 i x}{3}\right ) \]
[Out]
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Rubi [A] time = 0.024702, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{1}{16} \sqrt{4 x^2+3 i x} (8 x+3 i)+\frac{9}{64} i \sin ^{-1}\left (1-\frac{8 i x}{3}\right ) \]
Antiderivative was successfully verified.
[In] Int[Sqrt[(3*I)*x + 4*x^2],x]
[Out]
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Rubi in Sympy [A] time = 1.82941, size = 32, normalized size = 0.74 \[ \frac{\left (8 x + 3 i\right ) \sqrt{4 x^{2} + 3 i x}}{16} + \frac{9 \operatorname{asinh}{\left (\frac{8 x}{3} + i \right )}}{64} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((3*I*x+4*x**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0541475, size = 62, normalized size = 1.44 \[ \frac{1}{32} \sqrt{x (4 x+3 i)} \left (16 x+\frac{9 \log \left (2 \sqrt{x}+\sqrt{4 x+3 i}\right )}{\sqrt{4 x+3 i} \sqrt{x}}+6 i\right ) \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[(3*I)*x + 4*x^2],x]
[Out]
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Maple [A] time = 0.01, size = 31, normalized size = 0.7 \[{\frac{3\,i+8\,x}{16}\sqrt{3\,ix+4\,{x}^{2}}}+{\frac{9}{64}{\it Arcsinh} \left ({\frac{8\,x}{3}}+i \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((3*I*x+4*x^2)^(1/2),x)
[Out]
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Maxima [A] time = 0.806609, size = 66, normalized size = 1.53 \[ \frac{1}{2} \, \sqrt{4 \, x^{2} + 3 i \, x} x + \frac{3}{16} i \, \sqrt{4 \, x^{2} + 3 i \, x} + \frac{9}{64} \, \log \left (8 \, x + 4 \, \sqrt{4 \, x^{2} + 3 i \, x} + 3 i\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(4*x^2 + 3*I*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.217779, size = 170, normalized size = 3.95 \[ -\frac{32768 \, x^{4} + 49152 i \, x^{3} - 21888 \, x^{2} +{\left (4608 \, x^{2} - \sqrt{4 \, x^{2} + 3 i \, x}{\left (2304 \, x + 864 i\right )} + 3456 i \, x - 324\right )} \log \left (-2 \, x + \sqrt{4 \, x^{2} + 3 i \, x} - \frac{3}{4} i\right ) -{\left (16384 \, x^{3} + 18432 i \, x^{2} - 5184 \, x - 216 i\right )} \sqrt{4 \, x^{2} + 3 i \, x} - 2592 i \, x - 81}{32768 \, x^{2} - \sqrt{4 \, x^{2} + 3 i \, x}{\left (16384 \, x + 6144 i\right )} + 24576 i \, x - 2304} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(4*x^2 + 3*I*x),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{4 x^{2} + 3 i x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*I*x+4*x**2)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{4 \, x^{2} + 3 i \, x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(4*x^2 + 3*I*x),x, algorithm="giac")
[Out]