Integrand size = 25, antiderivative size = 67 \[ \int \frac {1}{(a-i a x)^{11/4} \sqrt [4]{a+i a x}} \, dx=-\frac {2 i (a+i a x)^{3/4}}{7 a^2 (a-i a x)^{7/4}}-\frac {4 i (a+i a x)^{3/4}}{21 a^3 (a-i a x)^{3/4}} \]
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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} \[ \int \frac {1}{(a-i a x)^{11/4} \sqrt [4]{a+i a x}} \, dx=-\frac {4 i (a+i a x)^{3/4}}{21 a^3 (a-i a x)^{3/4}}-\frac {2 i (a+i a x)^{3/4}}{7 a^2 (a-i a x)^{7/4}} \]
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Rule 37
Rule 47
Rubi steps \begin{align*} \text {integral}& = -\frac {2 i (a+i a x)^{3/4}}{7 a^2 (a-i a x)^{7/4}}+\frac {2 \int \frac {1}{(a-i a x)^{7/4} \sqrt [4]{a+i a x}} \, dx}{7 a} \\ & = -\frac {2 i (a+i a x)^{3/4}}{7 a^2 (a-i a x)^{7/4}}-\frac {4 i (a+i a x)^{3/4}}{21 a^3 (a-i a x)^{3/4}} \\ \end{align*}
Time = 6.18 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.67 \[ \int \frac {1}{(a-i a x)^{11/4} \sqrt [4]{a+i a x}} \, dx=\frac {2 (5-2 i x) (a+i a x)^{3/4}}{21 a^3 (i+x) (a-i a x)^{3/4}} \]
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Time = 0.53 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.55
method | result | size |
gosper | \(\frac {2 \left (x +i\right ) \left (-x +i\right ) \left (2 x +5 i\right )}{21 \left (-i a x +a \right )^{\frac {11}{4}} \left (i a x +a \right )^{\frac {1}{4}}}\) | \(37\) |
risch | \(\frac {\frac {4}{21} x^{2}+\frac {2}{7} i x +\frac {10}{21}}{a^{2} \left (-a \left (i x -1\right )\right )^{\frac {3}{4}} \left (a \left (i x +1\right )\right )^{\frac {1}{4}} \left (x +i\right )}\) | \(44\) |
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none
Time = 0.23 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.66 \[ \int \frac {1}{(a-i a x)^{11/4} \sqrt [4]{a+i a x}} \, dx=\frac {2 \, {\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {1}{4}} {\left (2 \, x + 5 i\right )}}{21 \, {\left (a^{4} x^{2} + 2 i \, a^{4} x - a^{4}\right )}} \]
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\[ \int \frac {1}{(a-i a x)^{11/4} \sqrt [4]{a+i a x}} \, dx=\int \frac {1}{\sqrt [4]{i a \left (x - i\right )} \left (- i a \left (x + i\right )\right )^{\frac {11}{4}}}\, dx \]
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\[ \int \frac {1}{(a-i a x)^{11/4} \sqrt [4]{a+i a x}} \, dx=\int { \frac {1}{{\left (i \, a x + a\right )}^{\frac {1}{4}} {\left (-i \, a x + a\right )}^{\frac {11}{4}}} \,d x } \]
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Exception generated. \[ \int \frac {1}{(a-i a x)^{11/4} \sqrt [4]{a+i a x}} \, dx=\text {Exception raised: TypeError} \]
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Time = 0.75 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.69 \[ \int \frac {1}{(a-i a x)^{11/4} \sqrt [4]{a+i a x}} \, dx=-\frac {{\left (-a\,\left (-1+x\,1{}\mathrm {i}\right )\right )}^{1/4}\,\left (2\,x^2+x\,3{}\mathrm {i}+5\right )\,2{}\mathrm {i}}{21\,a^3\,{\left (-1+x\,1{}\mathrm {i}\right )}^2\,{\left (a\,\left (1+x\,1{}\mathrm {i}\right )\right )}^{1/4}} \]
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