3.19 \(\int \frac{1}{(3 i x+4 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=53 \[ \frac{64 (8 x+3 i)}{243 \sqrt{4 x^2+3 i x}}+\frac{2 (8 x+3 i)}{27 \left (4 x^2+3 i x\right )^{3/2}} \]

[Out]

(2*(3*I + 8*x))/(27*((3*I)*x + 4*x^2)^(3/2)) + (64*(3*I + 8*x))/(243*Sqrt[(3*I)*x + 4*x^2])

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Rubi [A]  time = 0.0074662, antiderivative size = 53, 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, Rules used = {614, 613} \[ \frac{64 (8 x+3 i)}{243 \sqrt{4 x^2+3 i x}}+\frac{2 (8 x+3 i)}{27 \left (4 x^2+3 i x\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[((3*I)*x + 4*x^2)^(-5/2),x]

[Out]

(2*(3*I + 8*x))/(27*((3*I)*x + 4*x^2)^(3/2)) + (64*(3*I + 8*x))/(243*Sqrt[(3*I)*x + 4*x^2])

Rule 614

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^(p + 1))/((p +
1)*(b^2 - 4*a*c)), x] - Dist[(2*c*(2*p + 3))/((p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2] && IntegerQ[4*p]

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 )^{5/2}} \, dx &=\frac{2 (3 i+8 x)}{27 \left (3 i x+4 x^2\right )^{3/2}}+\frac{32}{27} \int \frac{1}{\left (3 i x+4 x^2\right )^{3/2}} \, dx\\ &=\frac{2 (3 i+8 x)}{27 \left (3 i x+4 x^2\right )^{3/2}}+\frac{64 (3 i+8 x)}{243 \sqrt{3 i x+4 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0106478, size = 36, normalized size = 0.68 \[ \frac{2048 x^3+2304 i x^2-432 x+54 i}{243 (x (4 x+3 i))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((3*I)*x + 4*x^2)^(-5/2),x]

[Out]

(54*I - 432*x + (2304*I)*x^2 + 2048*x^3)/(243*(x*(3*I + 4*x))^(3/2))

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Maple [A]  time = 0.098, size = 42, normalized size = 0.8 \begin{align*}{\frac{6\,i+16\,x}{27} \left ( 3\,ix+4\,{x}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{192\,i+512\,x}{243}{\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)^(5/2),x)

[Out]

2/27*(3*I+8*x)/(3*I*x+4*x^2)^(3/2)+64/243*(3*I+8*x)/(3*I*x+4*x^2)^(1/2)

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Maxima [A]  time = 1.20834, size = 74, normalized size = 1.4 \begin{align*} \frac{512 \, x}{243 \, \sqrt{4 \, x^{2} + 3 i \, x}} + \frac{64 i}{81 \, \sqrt{4 \, x^{2} + 3 i \, x}} + \frac{16 \, x}{27 \,{\left (4 \, x^{2} + 3 i \, x\right )}^{\frac{3}{2}}} + \frac{2 i}{9 \,{\left (4 \, x^{2} + 3 i \, x\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3*I*x+4*x^2)^(5/2),x, algorithm="maxima")

[Out]

512/243*x/sqrt(4*x^2 + 3*I*x) + 64/81*I/sqrt(4*x^2 + 3*I*x) + 16/27*x/(4*x^2 + 3*I*x)^(3/2) + 2/9*I/(4*x^2 + 3
*I*x)^(3/2)

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Fricas [A]  time = 2.51191, size = 178, normalized size = 3.36 \begin{align*} \frac{4096 \, x^{4} + 6144 i \, x^{3} - 2304 \, x^{2} +{\left (2048 \, x^{3} + 2304 i \, x^{2} - 432 \, x + 54 i\right )} \sqrt{4 \, x^{2} + 3 i \, x}}{3888 \, x^{4} + 5832 i \, x^{3} - 2187 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3*I*x+4*x^2)^(5/2),x, algorithm="fricas")

[Out]

(4096*x^4 + 6144*I*x^3 - 2304*x^2 + (2048*x^3 + 2304*I*x^2 - 432*x + 54*I)*sqrt(4*x^2 + 3*I*x))/(3888*x^4 + 58
32*I*x^3 - 2187*x^2)

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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{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3*I*x+4*x**2)**(5/2),x)

[Out]

Integral((4*x**2 + 3*I*x)**(-5/2), 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)^(5/2),x, algorithm="giac")

[Out]

Exception raised: TypeError