3.17 \(\int \frac{1}{\sqrt{3 i x+4 x^2}} \, dx\)

Optimal. Leaf size=16 \[ \frac{1}{2} i \sin ^{-1}\left (1-\frac{8 i x}{3}\right ) \]

[Out]

(I/2)*ArcSin[1 - ((8*I)/3)*x]

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Rubi [A]  time = 0.0159438, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{1}{2} i \sin ^{-1}\left (1-\frac{8 i x}{3}\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[(3*I)*x + 4*x^2],x]

[Out]

(I/2)*ArcSin[1 - ((8*I)/3)*x]

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Rubi in Sympy [A]  time = 1.36067, size = 8, normalized size = 0.5 \[ \frac{\operatorname{asinh}{\left (\frac{8 x}{3} + i \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(3*I*x+4*x**2)**(1/2),x)

[Out]

asinh(8*x/3 + I)/2

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Mathematica [B]  time = 0.0178141, size = 50, normalized size = 3.12 \[ \frac{\sqrt{x} \sqrt{4 x+3 i} \log \left (2 \sqrt{x}+\sqrt{4 x+3 i}\right )}{\sqrt{x (4 x+3 i)}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[(3*I)*x + 4*x^2],x]

[Out]

(Sqrt[x]*Sqrt[3*I + 4*x]*Log[2*Sqrt[x] + Sqrt[3*I + 4*x]])/Sqrt[x*(3*I + 4*x)]

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Maple [A]  time = 0.01, size = 10, normalized size = 0.6 \[{\frac{1}{2}{\it Arcsinh} \left ({\frac{8\,x}{3}}+i \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(3*I*x+4*x^2)^(1/2),x)

[Out]

1/2*arcsinh(8/3*x+I)

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Maxima [A]  time = 0.792402, size = 28, normalized size = 1.75 \[ \frac{1}{2} \, \log \left (8 \, x + 4 \, \sqrt{4 \, x^{2} + 3 i \, x} + 3 i\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(4*x^2 + 3*I*x),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.212793, size = 26, normalized size = 1.62 \[ -\frac{1}{2} \, \log \left (-2 \, x + \sqrt{4 \, x^{2} + 3 i \, x} - \frac{3}{4} i\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(4*x^2 + 3*I*x),x, algorithm="fricas")

[Out]

-1/2*log(-2*x + sqrt(4*x^2 + 3*I*x) - 3/4*I)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{4 x^{2} + 3 i x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(3*I*x+4*x**2)**(1/2),x)

[Out]

Integral(1/sqrt(4*x**2 + 3*I*x), x)

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(4*x^2 + 3*I*x),x, algorithm="giac")

[Out]

Exception raised: TypeError