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 = 0.01, antiderivative size = 53, normalized size of antiderivative = 1.00, 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 613
Rule 614
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*}
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Mathematica [A] time = 0.01, 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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fricas [A] time = 1.03, size = 62, normalized size = 1.17 \[ \frac {4096 \, x^{4} + 6144 i \, x^{3} - 2304 \, x^{2} + {\left (2048 \, x^{3} + 2304 i \, x^{2} - 432 \, x + 54 i\right )} \sqrt {4 \, x^{2} + 3 i \, x}}{3888 \, x^{4} + 5832 i \, x^{3} - 2187 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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maple [A] time = 0.10, size = 42, normalized size = 0.79 \[ \frac {\frac {16 x}{27}+\frac {2 i}{9}}{\left (4 x^{2}+3 i x \right )^{\frac {3}{2}}}+\frac {\frac {512 x}{243}+\frac {64 i}{81}}{\sqrt {4 x^{2}+3 i x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.37, size = 55, normalized size = 1.04 \[ \frac {512 \, x}{243 \, \sqrt {4 \, x^{2} + 3 i \, x}} + \frac {64 i}{81 \, \sqrt {4 \, x^{2} + 3 i \, x}} + \frac {16 \, x}{27 \, {\left (4 \, x^{2} + 3 i \, x\right )}^{\frac {3}{2}}} + \frac {2 i}{9 \, {\left (4 \, x^{2} + 3 i \, x\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 31, normalized size = 0.58 \[ \frac {\left (16\,x+6{}\mathrm {i}\right )\,\left (128\,x^2+x\,96{}\mathrm {i}+9\right )}{243\,{\left (4\,x^2+x\,3{}\mathrm {i}\right )}^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (4 x^{2} + 3 i x\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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