Integrand size = 15, antiderivative size = 69 \[ \int \left (3 i x+4 x^2\right )^{3/2} \, dx=\frac {27 (3 i+8 x) \sqrt {3 i x+4 x^2}}{1024}+\frac {1}{32} (3 i+8 x) \left (3 i x+4 x^2\right )^{3/2}+\frac {243 i \arcsin \left (1-\frac {8 i x}{3}\right )}{4096} \]
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Time = 0.01 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {626, 633, 221} \[ \int \left (3 i x+4 x^2\right )^{3/2} \, dx=\frac {243 i \arcsin \left (1-\frac {8 i x}{3}\right )}{4096}+\frac {1}{32} (8 x+3 i) \left (4 x^2+3 i x\right )^{3/2}+\frac {27 (8 x+3 i) \sqrt {4 x^2+3 i x}}{1024} \]
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Rule 221
Rule 626
Rule 633
Rubi steps \begin{align*} \text {integral}& = \frac {1}{32} (3 i+8 x) \left (3 i x+4 x^2\right )^{3/2}+\frac {27}{64} \int \sqrt {3 i x+4 x^2} \, dx \\ & = \frac {27 (3 i+8 x) \sqrt {3 i x+4 x^2}}{1024}+\frac {1}{32} (3 i+8 x) \left (3 i x+4 x^2\right )^{3/2}+\frac {243 \int \frac {1}{\sqrt {3 i x+4 x^2}} \, dx}{2048} \\ & = \frac {27 (3 i+8 x) \sqrt {3 i x+4 x^2}}{1024}+\frac {1}{32} (3 i+8 x) \left (3 i x+4 x^2\right )^{3/2}+\frac {81 \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{9}}} \, dx,x,3 i+8 x\right )}{4096} \\ & = \frac {27 (3 i+8 x) \sqrt {3 i x+4 x^2}}{1024}+\frac {1}{32} (3 i+8 x) \left (3 i x+4 x^2\right )^{3/2}+\frac {243 i \sin ^{-1}\left (1-\frac {8 i x}{3}\right )}{4096} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.20 \[ \int \left (3 i x+4 x^2\right )^{3/2} \, dx=\frac {2 x \left (-243+108 i x-3744 x^2+7680 i x^3+4096 x^4\right )-243 \sqrt {x} \sqrt {3 i+4 x} \log \left (-2 \sqrt {x}+\sqrt {3 i+4 x}\right )}{2048 \sqrt {x (3 i+4 x)}} \]
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Time = 2.05 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.68
method | result | size |
risch | \(\frac {\left (1024 x^{3}+1152 i x^{2}-72 x +81 i\right ) \left (3 i+4 x \right ) x}{1024 \sqrt {x \left (3 i+4 x \right )}}+\frac {243 \,\operatorname {arcsinh}\left (i+\frac {8 x}{3}\right )}{4096}\) | \(47\) |
default | \(\frac {\left (3 i+8 x \right ) \left (4 x^{2}+3 i x \right )^{\frac {3}{2}}}{32}+\frac {27 \left (3 i+8 x \right ) \sqrt {4 x^{2}+3 i x}}{1024}+\frac {243 \,\operatorname {arcsinh}\left (i+\frac {8 x}{3}\right )}{4096}\) | \(51\) |
trager | \(\left (\frac {9}{8} i x^{2}+x^{3}+\frac {81}{1024} i-\frac {9}{128} x \right ) \sqrt {4 x^{2}+3 i x}-\frac {243 \ln \left (-440 x -144-165 i-192 i \sqrt {4 x^{2}+3 i x}+384 i x +220 \sqrt {4 x^{2}+3 i x}\right )}{4096}\) | \(73\) |
pseudoelliptic | \(\frac {729 \left (-\frac {27 \ln \left (\frac {-2 x +\sqrt {x \left (3 i+4 x \right )}}{x}\right )}{512}+\frac {27 \ln \left (\frac {\sqrt {x \left (3 i+4 x \right )}+2 x}{x}\right )}{512}+\left (i x^{2}+\frac {8}{9} x^{3}+\frac {9}{128} i-\frac {1}{16} x \right ) \sqrt {x \left (3 i+4 x \right )}\right ) x^{4}}{8 \left (\sqrt {x \left (3 i+4 x \right )}+2 x \right )^{4} \left (2 x -\sqrt {x \left (3 i+4 x \right )}\right )^{4}}\) | \(111\) |
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Time = 0.28 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.71 \[ \int \left (3 i x+4 x^2\right )^{3/2} \, dx=\frac {1}{1024} \, {\left (1024 \, x^{3} + 1152 i \, x^{2} - 72 \, x + 81 i\right )} \sqrt {4 \, x^{2} + 3 i \, x} - \frac {243}{4096} \, \log \left (-2 \, x + \sqrt {4 \, x^{2} + 3 i \, x} - \frac {3}{4} i\right ) - \frac {567}{32768} \]
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Time = 0.52 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.33 \[ \int \left (3 i x+4 x^2\right )^{3/2} \, dx=4 \sqrt {4 x^{2} + 3 i x} \left (\frac {x^{3}}{4} + \frac {i x^{2}}{32} + \frac {15 x}{512} - \frac {135 i}{4096}\right ) + 3 i \left (\sqrt {4 x^{2} + 3 i x} \left (\frac {x^{2}}{3} + \frac {i x}{16} + \frac {9}{128}\right ) - \frac {27 i \operatorname {asinh}{\left (\frac {8 x}{3} + i \right )}}{512}\right ) - \frac {405 \operatorname {asinh}{\left (\frac {8 x}{3} + i \right )}}{4096} \]
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Time = 0.27 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.10 \[ \int \left (3 i x+4 x^2\right )^{3/2} \, dx=\frac {1}{4} \, {\left (4 \, x^{2} + 3 i \, x\right )}^{\frac {3}{2}} x + \frac {3}{32} i \, {\left (4 \, x^{2} + 3 i \, x\right )}^{\frac {3}{2}} + \frac {27}{128} \, \sqrt {4 \, x^{2} + 3 i \, x} x + \frac {81}{1024} i \, \sqrt {4 \, x^{2} + 3 i \, x} + \frac {243}{4096} \, \log \left (8 \, x + 4 \, \sqrt {4 \, x^{2} + 3 i \, x} + 3 i\right ) \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 120 vs. \(2 (45) = 90\).
Time = 0.28 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.74 \[ \int \left (3 i x+4 x^2\right )^{3/2} \, dx=\frac {1}{2048} \, {\left (8 \, {\left (16 \, {\left (8 \, x + 9 i\right )} x - 9\right )} x + 81 i\right )} \sqrt {8 \, x^{2} + 2 \, \sqrt {16 \, x^{2} + 9} x} {\left (\frac {3 i \, x}{4 \, x^{2} + \sqrt {16 \, x^{4} + 9 \, x^{2}}} + 1\right )} - \frac {243}{4096} \, \log \left (2 \, \sqrt {8 \, x^{2} + 2 \, \sqrt {16 \, x^{2} + 9} x} {\left (\frac {3 i \, x}{4 \, x^{2} + \sqrt {16 \, x^{4} + 9 \, x^{2}}} + 1\right )} - 8 \, x - 3 i\right ) \]
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Time = 0.16 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.87 \[ \int \left (3 i x+4 x^2\right )^{3/2} \, dx=\frac {243\,\ln \left (x+\frac {\sqrt {x\,\left (4\,x+3{}\mathrm {i}\right )}}{2}+\frac {3}{8}{}\mathrm {i}\right )}{4096}+\frac {\left (4\,x+\frac {3}{2}{}\mathrm {i}\right )\,{\left (4\,x^2+x\,3{}\mathrm {i}\right )}^{3/2}}{16}+\frac {27\,\left (\frac {x}{2}+\frac {3}{16}{}\mathrm {i}\right )\,\sqrt {4\,x^2+x\,3{}\mathrm {i}}}{64} \]
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