Integrand size = 25, antiderivative size = 33 \[ \int \frac {1}{(a-i a x)^{7/4} \sqrt [4]{a+i a x}} \, dx=-\frac {2 i (a+i a x)^{3/4}}{3 a^2 (a-i a x)^{3/4}} \]
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Time = 0.00 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {37} \[ \int \frac {1}{(a-i a x)^{7/4} \sqrt [4]{a+i a x}} \, dx=-\frac {2 i (a+i a x)^{3/4}}{3 a^2 (a-i a x)^{3/4}} \]
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Rule 37
Rubi steps \begin{align*} \text {integral}& = -\frac {2 i (a+i a x)^{3/4}}{3 a^2 (a-i a x)^{3/4}} \\ \end{align*}
Time = 4.98 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(a-i a x)^{7/4} \sqrt [4]{a+i a x}} \, dx=-\frac {2 i (a+i a x)^{3/4}}{3 a^2 (a-i a x)^{3/4}} \]
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Time = 0.18 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.94
method | result | size |
risch | \(\frac {\frac {2 x}{3}-\frac {2 i}{3}}{a \left (-a \left (i x -1\right )\right )^{\frac {3}{4}} \left (a \left (i x +1\right )\right )^{\frac {1}{4}}}\) | \(31\) |
gosper | \(\frac {2 i \left (x +i\right ) \left (-x +i\right )}{3 \left (-i a x +a \right )^{\frac {7}{4}} \left (i a x +a \right )^{\frac {1}{4}}}\) | \(32\) |
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none
Time = 0.22 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.94 \[ \int \frac {1}{(a-i a x)^{7/4} \sqrt [4]{a+i a x}} \, dx=\frac {2 \, {\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {1}{4}}}{3 \, {\left (a^{3} x + i \, a^{3}\right )}} \]
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\[ \int \frac {1}{(a-i a x)^{7/4} \sqrt [4]{a+i a x}} \, dx=\int \frac {1}{\sqrt [4]{i a \left (x - i\right )} \left (- i a \left (x + i\right )\right )^{\frac {7}{4}}}\, dx \]
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\[ \int \frac {1}{(a-i a x)^{7/4} \sqrt [4]{a+i a x}} \, dx=\int { \frac {1}{{\left (i \, a x + a\right )}^{\frac {1}{4}} {\left (-i \, a x + a\right )}^{\frac {7}{4}}} \,d x } \]
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Exception generated. \[ \int \frac {1}{(a-i a x)^{7/4} \sqrt [4]{a+i a x}} \, dx=\text {Exception raised: TypeError} \]
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Time = 0.61 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.15 \[ \int \frac {1}{(a-i a x)^{7/4} \sqrt [4]{a+i a x}} \, dx=-\frac {2\,\left (x-\mathrm {i}\right )\,{\left (-a\,\left (-1+x\,1{}\mathrm {i}\right )\right )}^{1/4}}{3\,a^2\,\left (-1+x\,1{}\mathrm {i}\right )\,{\left (a\,\left (1+x\,1{}\mathrm {i}\right )\right )}^{1/4}} \]
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