Optimal. Leaf size=53 \[ \frac{64 (8 x+3 i)}{243 \sqrt{4 x^2+3 i x}}+\frac{2 (8 x+3 i)}{27 \left (4 x^2+3 i x\right )^{3/2}} \]
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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.
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Rule 614
Rule 613
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.
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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.
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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.
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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.
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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.
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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.
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