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]

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

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Rubi [A]  time = 0.00, antiderivative size = 26, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, 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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fricas [B]  time = 0.81, size = 39, normalized size = 1.50 \[ \frac {32 \, x^{2} + \sqrt {4 \, x^{2} + 3 i \, x} {\left (16 \, x + 6 i\right )} + 24 i \, x}{9 \, {\left (4 \, x^{2} + 3 i \, x\right )}} \]

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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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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 >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, replacing 0 by ` u`, a substitution variable should perhaps be purged.Warning, replacing 0 by ` u`, a sub
stitution variable should perhaps be purged.Warning, replacing 0 by ` u`, a substitution variable should perha
ps be purged.

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maple [A]  time = 0.10, size = 21, normalized size = 0.81 \[ \frac {\frac {16 x}{9}+\frac {2 i}{3}}{\sqrt {4 x^{2}+3 i x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(4*x^2+3*I*x)^(3/2),x)

[Out]

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

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maxima [A]  time = 1.37, size = 28, normalized size = 1.08 \[ \frac {16 \, x}{9 \, \sqrt {4 \, x^{2} + 3 i \, x}} + \frac {2 i}{3 \, \sqrt {4 \, x^{2} + 3 i \, x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(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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mupad [B]  time = 0.05, size = 20, normalized size = 0.77 \[ \frac {16\,x+6{}\mathrm {i}}{9\,\sqrt {4\,x^2+x\,3{}\mathrm {i}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*3i + 4*x^2)^(3/2),x)

[Out]

(16*x + 6i)/(9*(x*3i + 4*x^2)^(1/2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (4 x^{2} + 3 i x\right )^{\frac {3}{2}}}\, dx \]

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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