3.18 \(\int \frac{1}{\left (3 i x+4 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=26 \[ \frac{2 (8 x+3 i)}{9 \sqrt{4 x^2+3 i x}} \]

[Out]

(2*(3*I + 8*x))/(9*Sqrt[(3*I)*x + 4*x^2])

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Rubi [A]  time = 0.0101403, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{2 (8 x+3 i)}{9 \sqrt{4 x^2+3 i x}} \]

Antiderivative was successfully verified.

[In]  Int[((3*I)*x + 4*x^2)^(-3/2),x]

[Out]

(2*(3*I + 8*x))/(9*Sqrt[(3*I)*x + 4*x^2])

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Rubi in Sympy [A]  time = 1.27876, size = 20, normalized size = 0.77 \[ \frac{16 x + 6 i}{9 \sqrt{4 x^{2} + 3 i x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(16*x + 6*I)/(9*sqrt(4*x**2 + 3*I*x))

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Mathematica [A]  time = 0.0153368, size = 24, normalized size = 0.92 \[ \frac{2 (8 x+3 i)}{9 \sqrt{x (4 x+3 i)}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(3*I + 8*x))/(9*Sqrt[x*(3*I + 4*x)])

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Maple [A]  time = 0.01, size = 21, normalized size = 0.8 \[{\frac{6\,i+16\,x}{9}{\frac{1}{\sqrt{3\,ix+4\,{x}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(3*I*x+4*x^2)^(3/2),x)

[Out]

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

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Maxima [A]  time = 0.711006, size = 38, normalized size = 1.46 \[ \frac{16 \, x}{9 \, \sqrt{4 \, x^{2} + 3 i \, x}} + \frac{2 i}{3 \, \sqrt{4 \, x^{2} + 3 i \, x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

16/9*x/sqrt(4*x^2 + 3*I*x) + 2/3*I/sqrt(4*x^2 + 3*I*x)

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Fricas [A]  time = 0.212089, size = 42, normalized size = 1.62 \[ \frac{2}{16 \, x^{2} - \sqrt{4 \, x^{2} + 3 i \, x}{\left (8 \, x + 3 i\right )} + 12 i \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/(16*x^2 - sqrt(4*x^2 + 3*I*x)*(8*x + 3*I) + 12*I*x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((4*x**2 + 3*I*x)**(-3/2), 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((4*x^2 + 3*I*x)^(-3/2),x, algorithm="giac")

[Out]

Exception raised: TypeError