Integrand size = 25, antiderivative size = 79 \[ \int \frac {\sqrt [4]{a-i a x}}{(a+i a x)^{7/4}} \, dx=\frac {4 i \sqrt [4]{a-i a x}}{3 a (a+i a x)^{3/4}}-\frac {2 \left (1+x^2\right )^{3/4} \operatorname {EllipticF}\left (\frac {\arctan (x)}{2},2\right )}{3 (a-i a x)^{3/4} (a+i a x)^{3/4}} \]
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Time = 0.01 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {49, 42, 239, 237} \[ \int \frac {\sqrt [4]{a-i a x}}{(a+i a x)^{7/4}} \, dx=\frac {4 i \sqrt [4]{a-i a x}}{3 a (a+i a x)^{3/4}}-\frac {2 \left (x^2+1\right )^{3/4} \operatorname {EllipticF}\left (\frac {\arctan (x)}{2},2\right )}{3 (a-i a x)^{3/4} (a+i a x)^{3/4}} \]
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Rule 42
Rule 49
Rule 237
Rule 239
Rubi steps \begin{align*} \text {integral}& = \frac {4 i \sqrt [4]{a-i a x}}{3 a (a+i a x)^{3/4}}-\frac {1}{3} \int \frac {1}{(a-i a x)^{3/4} (a+i a x)^{3/4}} \, dx \\ & = \frac {4 i \sqrt [4]{a-i a x}}{3 a (a+i a x)^{3/4}}-\frac {\left (a^2+a^2 x^2\right )^{3/4} \int \frac {1}{\left (a^2+a^2 x^2\right )^{3/4}} \, dx}{3 (a-i a x)^{3/4} (a+i a x)^{3/4}} \\ & = \frac {4 i \sqrt [4]{a-i a x}}{3 a (a+i a x)^{3/4}}-\frac {\left (1+x^2\right )^{3/4} \int \frac {1}{\left (1+x^2\right )^{3/4}} \, dx}{3 (a-i a x)^{3/4} (a+i a x)^{3/4}} \\ & = \frac {4 i \sqrt [4]{a-i a x}}{3 a (a+i a x)^{3/4}}-\frac {2 \left (1+x^2\right )^{3/4} F\left (\left .\frac {1}{2} \tan ^{-1}(x)\right |2\right )}{3 (a-i a x)^{3/4} (a+i a x)^{3/4}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.02 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.89 \[ \int \frac {\sqrt [4]{a-i a x}}{(a+i a x)^{7/4}} \, dx=\frac {i \sqrt [4]{2} (1+i x)^{3/4} (a-i a x)^{5/4} \operatorname {Hypergeometric2F1}\left (\frac {5}{4},\frac {7}{4},\frac {9}{4},\frac {1}{2}-\frac {i x}{2}\right )}{5 a^2 (a+i a x)^{3/4}} \]
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\[\int \frac {\left (-i a x +a \right )^{\frac {1}{4}}}{\left (i a x +a \right )^{\frac {7}{4}}}d x\]
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\[ \int \frac {\sqrt [4]{a-i a x}}{(a+i a x)^{7/4}} \, dx=\int { \frac {{\left (-i \, a x + a\right )}^{\frac {1}{4}}}{{\left (i \, a x + a\right )}^{\frac {7}{4}}} \,d x } \]
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\[ \int \frac {\sqrt [4]{a-i a x}}{(a+i a x)^{7/4}} \, dx=\int \frac {\sqrt [4]{- i a \left (x + i\right )}}{\left (i a \left (x - i\right )\right )^{\frac {7}{4}}}\, dx \]
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\[ \int \frac {\sqrt [4]{a-i a x}}{(a+i a x)^{7/4}} \, dx=\int { \frac {{\left (-i \, a x + a\right )}^{\frac {1}{4}}}{{\left (i \, a x + a\right )}^{\frac {7}{4}}} \,d x } \]
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\[ \int \frac {\sqrt [4]{a-i a x}}{(a+i a x)^{7/4}} \, dx=\int { \frac {{\left (-i \, a x + a\right )}^{\frac {1}{4}}}{{\left (i \, a x + a\right )}^{\frac {7}{4}}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt [4]{a-i a x}}{(a+i a x)^{7/4}} \, dx=\int \frac {{\left (a-a\,x\,1{}\mathrm {i}\right )}^{1/4}}{{\left (a+a\,x\,1{}\mathrm {i}\right )}^{7/4}} \,d x \]
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