Integrand size = 25, antiderivative size = 67 \[ \int \frac {1}{(a-i a x)^{7/4} (a+i a x)^{5/4}} \, dx=-\frac {2 i}{3 a^2 (a-i a x)^{3/4} \sqrt [4]{a+i a x}}+\frac {4 i \sqrt [4]{a-i a x}}{3 a^3 \sqrt [4]{a+i a x}} \]
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Time = 0.01 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {47, 37} \[ \int \frac {1}{(a-i a x)^{7/4} (a+i a x)^{5/4}} \, dx=\frac {4 i \sqrt [4]{a-i a x}}{3 a^3 \sqrt [4]{a+i a x}}-\frac {2 i}{3 a^2 (a-i a x)^{3/4} \sqrt [4]{a+i a x}} \]
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Rule 37
Rule 47
Rubi steps \begin{align*} \text {integral}& = -\frac {2 i}{3 a^2 (a-i a x)^{3/4} \sqrt [4]{a+i a x}}+\frac {2 \int \frac {1}{(a-i a x)^{3/4} (a+i a x)^{5/4}} \, dx}{3 a} \\ & = -\frac {2 i}{3 a^2 (a-i a x)^{3/4} \sqrt [4]{a+i a x}}+\frac {4 i \sqrt [4]{a-i a x}}{3 a^3 \sqrt [4]{a+i a x}} \\ \end{align*}
Time = 2.46 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.57 \[ \int \frac {1}{(a-i a x)^{7/4} (a+i a x)^{5/4}} \, dx=\frac {2 i+4 x}{3 a^2 (a-i a x)^{3/4} \sqrt [4]{a+i a x}} \]
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Time = 0.20 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.49
method | result | size |
risch | \(\frac {\frac {4 x}{3}+\frac {2 i}{3}}{a^{2} \left (-a \left (i x -1\right )\right )^{\frac {3}{4}} \left (a \left (i x +1\right )\right )^{\frac {1}{4}}}\) | \(33\) |
gosper | \(-\frac {2 \left (x +i\right ) \left (-x +i\right ) \left (2 x +i\right )}{3 \left (-i a x +a \right )^{\frac {7}{4}} \left (i a x +a \right )^{\frac {5}{4}}}\) | \(37\) |
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none
Time = 0.23 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.54 \[ \int \frac {1}{(a-i a x)^{7/4} (a+i a x)^{5/4}} \, dx=\frac {2 \, {\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {1}{4}} {\left (2 \, x + i\right )}}{3 \, {\left (a^{4} x^{2} + a^{4}\right )}} \]
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\[ \int \frac {1}{(a-i a x)^{7/4} (a+i a x)^{5/4}} \, dx=\int \frac {1}{\left (i a \left (x - i\right )\right )^{\frac {5}{4}} \left (- i a \left (x + i\right )\right )^{\frac {7}{4}}}\, dx \]
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\[ \int \frac {1}{(a-i a x)^{7/4} (a+i a x)^{5/4}} \, dx=\int { \frac {1}{{\left (i \, a x + a\right )}^{\frac {5}{4}} {\left (-i \, a x + a\right )}^{\frac {7}{4}}} \,d x } \]
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Exception generated. \[ \int \frac {1}{(a-i a x)^{7/4} (a+i a x)^{5/4}} \, dx=\text {Exception raised: TypeError} \]
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Time = 0.66 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.60 \[ \int \frac {1}{(a-i a x)^{7/4} (a+i a x)^{5/4}} \, dx=-\frac {2\,\left (2\,x+1{}\mathrm {i}\right )\,{\left (-a\,\left (-1+x\,1{}\mathrm {i}\right )\right )}^{1/4}}{3\,a^3\,\left (-1+x\,1{}\mathrm {i}\right )\,{\left (a\,\left (1+x\,1{}\mathrm {i}\right )\right )}^{1/4}} \]
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