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

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 = 0.0025989, 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, Rules used = {613} \[ \frac{2 (8 x+3 i)}{9 \sqrt{4 x^2+3 i x}} \]

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])

Rule 613

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[(-2*(b + 2*c*x))/((b^2 - 4*a*c)*Sqrt[a + b*x
+ c*x^2]), x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rubi steps

\begin{align*} \int \frac{1}{\left (3 i x+4 x^2\right )^{3/2}} \, dx &=\frac{2 (3 i+8 x)}{9 \sqrt{3 i x+4 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0053476, 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 = 0.123, size = 21, normalized size = 0.8 \begin{align*}{\frac{6\,i+16\,x}{9}{\frac{1}{\sqrt{3\,ix+4\,{x}^{2}}}}} \end{align*}

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 = 1.222, size = 38, normalized size = 1.46 \begin{align*} \frac{16 \, x}{9 \, \sqrt{4 \, x^{2} + 3 i \, x}} + \frac{2 i}{3 \, \sqrt{4 \, x^{2} + 3 i \, x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3*I*x+4*x^2)^(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 [B]  time = 2.64556, size = 100, normalized size = 3.85 \begin{align*} \frac{32 \, x^{2} + \sqrt{4 \, x^{2} + 3 i \, x}{\left (16 \, x + 6 i\right )} + 24 i \, x}{9 \,{\left (4 \, x^{2} + 3 i \, x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*(32*x^2 + sqrt(4*x^2 + 3*I*x)*(16*x + 6*I) + 24*I*x)/(4*x^2 + 3*I*x)

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

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 [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError