Integrand size = 25, antiderivative size = 100 \[ \int \frac {1}{(a-i a x)^{15/4} \sqrt [4]{a+i a x}} \, dx=-\frac {2 i (a+i a x)^{3/4}}{11 a^2 (a-i a x)^{11/4}}-\frac {8 i (a+i a x)^{3/4}}{77 a^3 (a-i a x)^{7/4}}-\frac {16 i (a+i a x)^{3/4}}{231 a^4 (a-i a x)^{3/4}} \]
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Time = 0.01 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {47, 37} \[ \int \frac {1}{(a-i a x)^{15/4} \sqrt [4]{a+i a x}} \, dx=-\frac {16 i (a+i a x)^{3/4}}{231 a^4 (a-i a x)^{3/4}}-\frac {8 i (a+i a x)^{3/4}}{77 a^3 (a-i a x)^{7/4}}-\frac {2 i (a+i a x)^{3/4}}{11 a^2 (a-i a x)^{11/4}} \]
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Rule 37
Rule 47
Rubi steps \begin{align*} \text {integral}& = -\frac {2 i (a+i a x)^{3/4}}{11 a^2 (a-i a x)^{11/4}}+\frac {4 \int \frac {1}{(a-i a x)^{11/4} \sqrt [4]{a+i a x}} \, dx}{11 a} \\ & = -\frac {2 i (a+i a x)^{3/4}}{11 a^2 (a-i a x)^{11/4}}-\frac {8 i (a+i a x)^{3/4}}{77 a^3 (a-i a x)^{7/4}}+\frac {8 \int \frac {1}{(a-i a x)^{7/4} \sqrt [4]{a+i a x}} \, dx}{77 a^2} \\ & = -\frac {2 i (a+i a x)^{3/4}}{11 a^2 (a-i a x)^{11/4}}-\frac {8 i (a+i a x)^{3/4}}{77 a^3 (a-i a x)^{7/4}}-\frac {16 i (a+i a x)^{3/4}}{231 a^4 (a-i a x)^{3/4}} \\ \end{align*}
Time = 6.85 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.52 \[ \int \frac {1}{(a-i a x)^{15/4} \sqrt [4]{a+i a x}} \, dx=\frac {2 (a+i a x)^{3/4} \left (41 i+28 x-8 i x^2\right )}{231 a^4 (i+x)^2 (a-i a x)^{3/4}} \]
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Time = 0.19 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.43
method | result | size |
gosper | \(-\frac {2 \left (x +i\right ) \left (-x +i\right ) \left (8 i x^{2}-28 x -41 i\right )}{231 \left (-i a x +a \right )^{\frac {15}{4}} \left (i a x +a \right )^{\frac {1}{4}}}\) | \(43\) |
risch | \(\frac {\frac {16}{231} x^{3}+\frac {40}{231} i x^{2}-\frac {26}{231} x +\frac {82}{231} i}{a^{3} \left (-a \left (i x -1\right )\right )^{\frac {3}{4}} \left (a \left (i x +1\right )\right )^{\frac {1}{4}} \left (x +i\right )^{2}}\) | \(50\) |
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none
Time = 0.22 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.57 \[ \int \frac {1}{(a-i a x)^{15/4} \sqrt [4]{a+i a x}} \, dx=\frac {2 \, {\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {1}{4}} {\left (8 \, x^{2} + 28 i \, x - 41\right )}}{231 \, {\left (a^{5} x^{3} + 3 i \, a^{5} x^{2} - 3 \, a^{5} x - i \, a^{5}\right )}} \]
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Timed out. \[ \int \frac {1}{(a-i a x)^{15/4} \sqrt [4]{a+i a x}} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{(a-i a x)^{15/4} \sqrt [4]{a+i a x}} \, dx=\int { \frac {1}{{\left (i \, a x + a\right )}^{\frac {1}{4}} {\left (-i \, a x + a\right )}^{\frac {15}{4}}} \,d x } \]
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Exception generated. \[ \int \frac {1}{(a-i a x)^{15/4} \sqrt [4]{a+i a x}} \, dx=\text {Exception raised: TypeError} \]
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Time = 0.81 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.51 \[ \int \frac {1}{(a-i a x)^{15/4} \sqrt [4]{a+i a x}} \, dx=\frac {{\left (x-\mathrm {i}\right )}^4\,{\left (-a\,\left (-1+x\,1{}\mathrm {i}\right )\right )}^{1/4}\,\left (8\,x^2+x\,28{}\mathrm {i}-41\right )\,2{}\mathrm {i}}{231\,a^4\,{\left (x^2+1\right )}^3\,{\left (a\,\left (1+x\,1{}\mathrm {i}\right )\right )}^{1/4}} \]
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