3.5 \(\int \sqrt{3 i x+4 x^2} \, dx\)

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]

((3*I + 8*x)*Sqrt[(3*I)*x + 4*x^2])/16 + ((9*I)/64)*ArcSin[1 - ((8*I)/3)*x]

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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]

((3*I + 8*x)*Sqrt[(3*I)*x + 4*x^2])/16 + ((9*I)/64)*ArcSin[1 - ((8*I)/3)*x]

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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]

(8*x + 3*I)*sqrt(4*x**2 + 3*I*x)/16 + 9*asinh(8*x/3 + I)/64

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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]

(Sqrt[x*(3*I + 4*x)]*(6*I + 16*x + (9*Log[2*Sqrt[x] + Sqrt[3*I + 4*x]])/(Sqrt[x]
*Sqrt[3*I + 4*x])))/32

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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]

1/16*(3*I+8*x)*(3*I*x+4*x^2)^(1/2)+9/64*arcsinh(8/3*x+I)

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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]

1/2*sqrt(4*x^2 + 3*I*x)*x + 3/16*I*sqrt(4*x^2 + 3*I*x) + 9/64*log(8*x + 4*sqrt(4
*x^2 + 3*I*x) + 3*I)

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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]

-(32768*x^4 + 49152*I*x^3 - 21888*x^2 + (4608*x^2 - sqrt(4*x^2 + 3*I*x)*(2304*x
+ 864*I) + 3456*I*x - 324)*log(-2*x + sqrt(4*x^2 + 3*I*x) - 3/4*I) - (16384*x^3
+ 18432*I*x^2 - 5184*x - 216*I)*sqrt(4*x^2 + 3*I*x) - 2592*I*x - 81)/(32768*x^2
- sqrt(4*x^2 + 3*I*x)*(16384*x + 6144*I) + 24576*I*x - 2304)

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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]

Integral(sqrt(4*x**2 + 3*I*x), x)

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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]

integrate(sqrt(4*x^2 + 3*I*x), x)