Integrand size = 25, antiderivative size = 137 \[ \int \frac {(a-i a x)^{7/4}}{(a+i a x)^{5/4}} \, dx=-\frac {14 a x}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}+\frac {4 i (a-i a x)^{7/4}}{a \sqrt [4]{a+i a x}}+\frac {14 i (a-i a x)^{3/4} (a+i a x)^{3/4}}{3 a}+\frac {14 a \sqrt [4]{1+x^2} E\left (\left .\frac {\arctan (x)}{2}\right |2\right )}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \]
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Time = 0.02 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {49, 52, 42, 235, 233, 202} \[ \int \frac {(a-i a x)^{7/4}}{(a+i a x)^{5/4}} \, dx=\frac {14 a \sqrt [4]{x^2+1} E\left (\left .\frac {\arctan (x)}{2}\right |2\right )}{\sqrt [4]{a+i a x} \sqrt [4]{a-i a x}}+\frac {4 i (a-i a x)^{7/4}}{a \sqrt [4]{a+i a x}}+\frac {14 i (a+i a x)^{3/4} (a-i a x)^{3/4}}{3 a}-\frac {14 a x}{\sqrt [4]{a+i a x} \sqrt [4]{a-i a x}} \]
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Rule 42
Rule 49
Rule 52
Rule 202
Rule 233
Rule 235
Rubi steps \begin{align*} \text {integral}& = \frac {4 i (a-i a x)^{7/4}}{a \sqrt [4]{a+i a x}}-7 \int \frac {(a-i a x)^{3/4}}{\sqrt [4]{a+i a x}} \, dx \\ & = \frac {4 i (a-i a x)^{7/4}}{a \sqrt [4]{a+i a x}}+\frac {14 i (a-i a x)^{3/4} (a+i a x)^{3/4}}{3 a}-(7 a) \int \frac {1}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \, dx \\ & = \frac {4 i (a-i a x)^{7/4}}{a \sqrt [4]{a+i a x}}+\frac {14 i (a-i a x)^{3/4} (a+i a x)^{3/4}}{3 a}-\frac {\left (7 a \sqrt [4]{a^2+a^2 x^2}\right ) \int \frac {1}{\sqrt [4]{a^2+a^2 x^2}} \, dx}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \\ & = \frac {4 i (a-i a x)^{7/4}}{a \sqrt [4]{a+i a x}}+\frac {14 i (a-i a x)^{3/4} (a+i a x)^{3/4}}{3 a}-\frac {\left (7 a \sqrt [4]{1+x^2}\right ) \int \frac {1}{\sqrt [4]{1+x^2}} \, dx}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \\ & = -\frac {14 a x}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}+\frac {4 i (a-i a x)^{7/4}}{a \sqrt [4]{a+i a x}}+\frac {14 i (a-i a x)^{3/4} (a+i a x)^{3/4}}{3 a}+\frac {\left (7 a \sqrt [4]{1+x^2}\right ) \int \frac {1}{\left (1+x^2\right )^{5/4}} \, dx}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \\ & = -\frac {14 a x}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}+\frac {4 i (a-i a x)^{7/4}}{a \sqrt [4]{a+i a x}}+\frac {14 i (a-i a x)^{3/4} (a+i a x)^{3/4}}{3 a}+\frac {14 a \sqrt [4]{1+x^2} E\left (\left .\frac {1}{2} \tan ^{-1}(x)\right |2\right )}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \\ \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 = 0.51 \[ \int \frac {(a-i a x)^{7/4}}{(a+i a x)^{5/4}} \, dx=\frac {i 2^{3/4} \sqrt [4]{1+i x} (a-i a x)^{11/4} \operatorname {Hypergeometric2F1}\left (\frac {5}{4},\frac {11}{4},\frac {15}{4},\frac {1}{2}-\frac {i x}{2}\right )}{11 a^2 \sqrt [4]{a+i a x}} \]
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Result contains higher order function than in optimal. Order 5 vs. order 4.
Time = 0.22 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.70
method | result | size |
risch | \(\frac {2 i \left (x^{2}-12 i x +13\right ) a}{3 \left (-a \left (i x -1\right )\right )^{\frac {1}{4}} \left (a \left (i x +1\right )\right )^{\frac {1}{4}}}-\frac {7 x {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{4},\frac {1}{2};\frac {3}{2};-x^{2}\right ) a \left (-a^{2} \left (i x -1\right ) \left (i x +1\right )\right )^{\frac {1}{4}}}{\left (a^{2}\right )^{\frac {1}{4}} \left (-a \left (i x -1\right )\right )^{\frac {1}{4}} \left (a \left (i x +1\right )\right )^{\frac {1}{4}}}\) | \(96\) |
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\[ \int \frac {(a-i a x)^{7/4}}{(a+i a x)^{5/4}} \, dx=\int { \frac {{\left (-i \, a x + a\right )}^{\frac {7}{4}}}{{\left (i \, a x + a\right )}^{\frac {5}{4}}} \,d x } \]
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\[ \int \frac {(a-i a x)^{7/4}}{(a+i a x)^{5/4}} \, dx=\int \frac {\left (- i a \left (x + i\right )\right )^{\frac {7}{4}}}{\left (i a \left (x - i\right )\right )^{\frac {5}{4}}}\, dx \]
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\[ \int \frac {(a-i a x)^{7/4}}{(a+i a x)^{5/4}} \, dx=\int { \frac {{\left (-i \, a x + a\right )}^{\frac {7}{4}}}{{\left (i \, a x + a\right )}^{\frac {5}{4}}} \,d x } \]
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Exception generated. \[ \int \frac {(a-i a x)^{7/4}}{(a+i a x)^{5/4}} \, dx=\text {Exception raised: RuntimeError} \]
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Timed out. \[ \int \frac {(a-i a x)^{7/4}}{(a+i a x)^{5/4}} \, dx=\int \frac {{\left (a-a\,x\,1{}\mathrm {i}\right )}^{7/4}}{{\left (a+a\,x\,1{}\mathrm {i}\right )}^{5/4}} \,d x \]
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