Integrand size = 15, antiderivative size = 79 \[ \int \frac {1}{\left (3 i x+4 x^2\right )^{7/2}} \, dx=\frac {2 (3 i+8 x)}{45 \left (3 i x+4 x^2\right )^{5/2}}+\frac {128 (3 i+8 x)}{1215 \left (3 i x+4 x^2\right )^{3/2}}+\frac {4096 (3 i+8 x)}{10935 \sqrt {3 i x+4 x^2}} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {628, 627} \[ \int \frac {1}{\left (3 i x+4 x^2\right )^{7/2}} \, dx=\frac {4096 (8 x+3 i)}{10935 \sqrt {4 x^2+3 i x}}+\frac {128 (8 x+3 i)}{1215 \left (4 x^2+3 i x\right )^{3/2}}+\frac {2 (8 x+3 i)}{45 \left (4 x^2+3 i x\right )^{5/2}} \]
[In]
[Out]
Rule 627
Rule 628
Rubi steps \begin{align*} \text {integral}& = \frac {2 (3 i+8 x)}{45 \left (3 i x+4 x^2\right )^{5/2}}+\frac {64}{45} \int \frac {1}{\left (3 i x+4 x^2\right )^{5/2}} \, dx \\ & = \frac {2 (3 i+8 x)}{45 \left (3 i x+4 x^2\right )^{5/2}}+\frac {128 (3 i+8 x)}{1215 \left (3 i x+4 x^2\right )^{3/2}}+\frac {2048 \int \frac {1}{\left (3 i x+4 x^2\right )^{3/2}} \, dx}{1215} \\ & = \frac {2 (3 i+8 x)}{45 \left (3 i x+4 x^2\right )^{5/2}}+\frac {128 (3 i+8 x)}{1215 \left (3 i x+4 x^2\right )^{3/2}}+\frac {4096 (3 i+8 x)}{10935 \sqrt {3 i x+4 x^2}} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.61 \[ \int \frac {1}{\left (3 i x+4 x^2\right )^{7/2}} \, dx=\frac {1458 i-6480 x-69120 i x^2-552960 x^3+983040 i x^4+524288 x^5}{10935 (x (3 i+4 x))^{5/2}} \]
[In]
[Out]
Time = 1.90 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.63
method | result | size |
gosper | \(\frac {2 x \left (3 i+4 x \right ) \left (262144 x^{5}+491520 i x^{4}-276480 x^{3}-34560 i x^{2}-3240 x +729 i\right )}{10935 \left (4 x^{2}+3 i x \right )^{\frac {7}{2}}}\) | \(50\) |
risch | \(\frac {\frac {524288}{10935} x^{5}+\frac {65536}{729} i x^{4}-\frac {4096}{81} x^{3}-\frac {512}{81} i x^{2}-\frac {16}{27} x +\frac {2}{15} i}{x^{2} \left (3 i+4 x \right )^{2} \sqrt {x \left (3 i+4 x \right )}}\) | \(52\) |
pseudoelliptic | \(\frac {\frac {524288}{10935} x^{5}+\frac {65536}{729} i x^{4}-\frac {4096}{81} x^{3}-\frac {512}{81} i x^{2}-\frac {16}{27} x +\frac {2}{15} i}{x^{2} \left (3 i+4 x \right )^{2} \sqrt {x \left (3 i+4 x \right )}}\) | \(52\) |
default | \(\frac {\frac {2 i}{15}+\frac {16 x}{45}}{\left (4 x^{2}+3 i x \right )^{\frac {5}{2}}}+\frac {\frac {128 i}{405}+\frac {1024 x}{1215}}{\left (4 x^{2}+3 i x \right )^{\frac {3}{2}}}+\frac {\frac {4096 i}{3645}+\frac {32768 x}{10935}}{\sqrt {4 x^{2}+3 i x}}\) | \(62\) |
trager | \(\frac {\left (\frac {1054}{4271484375}+\frac {224 i}{1423828125}\right ) \left (12288000000 i x^{5}+16384000000 x^{5}+30720000000 i x^{4}-23040000000 x^{4}-12960000000 i x^{3}-17280000000 x^{3}-2160000000 i x^{2}+1620000000 x^{2}-151875000 i x -202500000 x +45562500 i-34171875\right ) \sqrt {4 x^{2}+3 i x}}{\left (12 i x -16 x -12 i-9\right )^{3} x^{3}}\) | \(88\) |
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.05 \[ \int \frac {1}{\left (3 i x+4 x^2\right )^{7/2}} \, dx=\frac {2 \, {\left (524288 \, x^{6} + 1179648 i \, x^{5} - 884736 \, x^{4} - 221184 i \, x^{3} + {\left (262144 \, x^{5} + 491520 i \, x^{4} - 276480 \, x^{3} - 34560 i \, x^{2} - 3240 \, x + 729 i\right )} \sqrt {4 \, x^{2} + 3 i \, x}\right )}}{10935 \, {\left (64 \, x^{6} + 144 i \, x^{5} - 108 \, x^{4} - 27 i \, x^{3}\right )}} \]
[In]
[Out]
\[ \int \frac {1}{\left (3 i x+4 x^2\right )^{7/2}} \, dx=\int \frac {1}{\left (4 x^{2} + 3 i x\right )^{\frac {7}{2}}}\, dx \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.04 \[ \int \frac {1}{\left (3 i x+4 x^2\right )^{7/2}} \, dx=\frac {32768 \, x}{10935 \, \sqrt {4 \, x^{2} + 3 i \, x}} + \frac {4096 i}{3645 \, \sqrt {4 \, x^{2} + 3 i \, x}} + \frac {1024 \, x}{1215 \, {\left (4 \, x^{2} + 3 i \, x\right )}^{\frac {3}{2}}} + \frac {128 i}{405 \, {\left (4 \, x^{2} + 3 i \, x\right )}^{\frac {3}{2}}} + \frac {16 \, x}{45 \, {\left (4 \, x^{2} + 3 i \, x\right )}^{\frac {5}{2}}} + \frac {2 i}{15 \, {\left (4 \, x^{2} + 3 i \, x\right )}^{\frac {5}{2}}} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.06 \[ \int \frac {1}{\left (3 i x+4 x^2\right )^{7/2}} \, dx=\frac {{\left (8 \, {\left (32 \, {\left (8 \, {\left (16 \, {\left (8 \, x + 15 i\right )} x - 135\right )} x - 135 i\right )} x - 405\right )} x + 729 i\right )} \sqrt {8 \, x^{2} + 2 \, \sqrt {16 \, x^{2} + 9} x} {\left (\frac {3 i \, x}{4 \, x^{2} + \sqrt {16 \, x^{4} + 9 \, x^{2}}} + 1\right )}}{10935 \, {\left (4 \, x^{2} + 3 i \, x\right )}^{3}} \]
[In]
[Out]
Time = 9.23 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.51 \[ \int \frac {1}{\left (3 i x+4 x^2\right )^{7/2}} \, dx=-\frac {-524288\,x^5-x^4\,983040{}\mathrm {i}+552960\,x^3+x^2\,69120{}\mathrm {i}+6480\,x-1458{}\mathrm {i}}{10935\,{\left (x\,\left (4\,x+3{}\mathrm {i}\right )\right )}^{5/2}} \]
[In]
[Out]