\(\int \frac {1}{(3 i x+4 x^2)^{3/2}} \, dx\) [18]

Optimal result

Integrand size = 15, antiderivative size = 26 \[ \int \frac {1}{\left (3 i x+4 x^2\right )^{3/2}} \, dx=\frac {2 (3 i+8 x)}{9 \sqrt {3 i x+4 x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {627} \[ \int \frac {1}{\left (3 i x+4 x^2\right )^{3/2}} \, dx=\frac {2 (8 x+3 i)}{9 \sqrt {4 x^2+3 i x}} \]

[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 627

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*} \text {integral}& = \frac {2 (3 i+8 x)}{9 \sqrt {3 i x+4 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {1}{\left (3 i x+4 x^2\right )^{3/2}} \, dx=\frac {2 (3 i+8 x)}{9 \sqrt {x (3 i+4 x)}} \]

[In]

Integrate[((3*I)*x + 4*x^2)^(-3/2),x]

[Out]

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

Maple [A] (verified)

Time = 1.80 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.73

method	result	size
risch	\(\frac {\frac {2 i}{3}+\frac {16 x}{9}}{\sqrt {x \left (3 i+4 x \right )}}\)	\(19\)
pseudoelliptic	\(\frac {\frac {2 i}{3}+\frac {16 x}{9}}{\sqrt {x \left (3 i+4 x \right )}}\)	\(19\)
default	\(\frac {\frac {2 i}{3}+\frac {16 x}{9}}{\sqrt {4 x^{2}+3 i x}}\)	\(21\)
gosper	\(\frac {2 x \left (3 i+4 x \right ) \left (3 i+8 x \right )}{9 \left (4 x^{2}+3 i x \right )^{\frac {3}{2}}}\)	\(28\)
trager	\(\frac {\left (-\frac {14}{225}+\frac {16 i}{75}\right ) \left (24 i x +32 x +12 i-9\right ) \sqrt {4 x^{2}+3 i x}}{x \left (12 i x -16 x -12 i-9\right )}\)	\(44\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (18) = 36\).

Time = 0.27 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.50 \[ \int \frac {1}{\left (3 i x+4 x^2\right )^{3/2}} \, dx=\frac {2 \, {\left (16 \, x^{2} + \sqrt {4 \, x^{2} + 3 i \, x} {\left (8 \, x + 3 i\right )} + 12 i \, x\right )}}{9 \, {\left (4 \, x^{2} + 3 i \, x\right )}} \]

[In]

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

[Out]

2/9*(16*x^2 + sqrt(4*x^2 + 3*I*x)*(8*x + 3*I) + 12*I*x)/(4*x^2 + 3*I*x)

Sympy [F]

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

[In]

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

[Out]

Integral((4*x**2 + 3*I*x)**(-3/2), x)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {1}{\left (3 i x+4 x^2\right )^{3/2}} \, dx=\frac {16 \, x}{9 \, \sqrt {4 \, x^{2} + 3 i \, x}} + \frac {2 i}{3 \, \sqrt {4 \, x^{2} + 3 i \, x}} \]

[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)

Giac [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (18) = 36\).

Time = 0.29 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.46 \[ \int \frac {1}{\left (3 i x+4 x^2\right )^{3/2}} \, dx=\frac {\sqrt {8 \, x^{2} + 2 \, \sqrt {16 \, x^{2} + 9} x} {\left (8 \, x + 3 i\right )} {\left (\frac {3 i \, x}{4 \, x^{2} + \sqrt {16 \, x^{4} + 9 \, x^{2}}} + 1\right )}}{9 \, {\left (4 \, x^{2} + 3 i \, x\right )}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.77 \[ \int \frac {1}{\left (3 i x+4 x^2\right )^{3/2}} \, dx=\frac {16\,x+6{}\mathrm {i}}{9\,\sqrt {4\,x^2+x\,3{}\mathrm {i}}} \]

[In]

int(1/(x*3i + 4*x^2)^(3/2),x)

[Out]

(16*x + 6i)/(9*(x*3i + 4*x^2)^(1/2))