Integrand size = 25, antiderivative size = 100 \[ \int \frac {1}{(a-i a x)^{7/4} (a+i a x)^{9/4}} \, dx=-\frac {2 i}{3 a^2 (a-i a x)^{3/4} (a+i a x)^{5/4}}+\frac {8 i \sqrt [4]{a-i a x}}{15 a^3 (a+i a x)^{5/4}}+\frac {16 i \sqrt [4]{a-i a x}}{15 a^4 \sqrt [4]{a+i a x}} \]
[Out]
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} \[ \int \frac {1}{(a-i a x)^{7/4} (a+i a x)^{9/4}} \, dx=\frac {16 i \sqrt [4]{a-i a x}}{15 a^4 \sqrt [4]{a+i a x}}+\frac {8 i \sqrt [4]{a-i a x}}{15 a^3 (a+i a x)^{5/4}}-\frac {2 i}{3 a^2 (a+i a x)^{5/4} (a-i a x)^{3/4}} \]
[In]
[Out]
Rule 37
Rule 47
Rubi steps \begin{align*} \text {integral}& = -\frac {2 i}{3 a^2 (a-i a x)^{3/4} (a+i a x)^{5/4}}+\frac {4 \int \frac {1}{(a-i a x)^{3/4} (a+i a x)^{9/4}} \, dx}{3 a} \\ & = -\frac {2 i}{3 a^2 (a-i a x)^{3/4} (a+i a x)^{5/4}}+\frac {8 i \sqrt [4]{a-i a x}}{15 a^3 (a+i a x)^{5/4}}+\frac {8 \int \frac {1}{(a-i a x)^{3/4} (a+i a x)^{5/4}} \, dx}{15 a^2} \\ & = -\frac {2 i}{3 a^2 (a-i a x)^{3/4} (a+i a x)^{5/4}}+\frac {8 i \sqrt [4]{a-i a x}}{15 a^3 (a+i a x)^{5/4}}+\frac {16 i \sqrt [4]{a-i a x}}{15 a^4 \sqrt [4]{a+i a x}} \\ \end{align*}
Time = 5.19 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.50 \[ \int \frac {1}{(a-i a x)^{7/4} (a+i a x)^{9/4}} \, dx=\frac {2 \left (7-4 i x+8 x^2\right )}{15 a^3 (-i+x) (a-i a x)^{3/4} \sqrt [4]{a+i a x}} \]
[In]
[Out]
Time = 0.21 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.43
method | result | size |
gosper | \(-\frac {2 \left (x +i\right ) \left (-x +i\right ) \left (8 i x^{2}+4 x +7 i\right )}{15 \left (-i a x +a \right )^{\frac {7}{4}} \left (i a x +a \right )^{\frac {9}{4}}}\) | \(43\) |
risch | \(\frac {\frac {16}{15} x^{2}-\frac {8}{15} i x +\frac {14}{15}}{a^{3} \left (-a \left (i x -1\right )\right )^{\frac {3}{4}} \left (a \left (i x +1\right )\right )^{\frac {1}{4}} \left (x -i\right )}\) | \(44\) |
[In]
[Out]
none
Time = 0.22 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.56 \[ \int \frac {1}{(a-i a x)^{7/4} (a+i a x)^{9/4}} \, dx=\frac {2 \, {\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {1}{4}} {\left (8 \, x^{2} - 4 i \, x + 7\right )}}{15 \, {\left (a^{5} x^{3} - i \, a^{5} x^{2} + a^{5} x - i \, a^{5}\right )}} \]
[In]
[Out]
\[ \int \frac {1}{(a-i a x)^{7/4} (a+i a x)^{9/4}} \, dx=\int \frac {1}{\left (i a \left (x - i\right )\right )^{\frac {9}{4}} \left (- i a \left (x + i\right )\right )^{\frac {7}{4}}}\, dx \]
[In]
[Out]
Exception generated. \[ \int \frac {1}{(a-i a x)^{7/4} (a+i a x)^{9/4}} \, dx=\text {Exception raised: RuntimeError} \]
[In]
[Out]
Exception generated. \[ \int \frac {1}{(a-i a x)^{7/4} (a+i a x)^{9/4}} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
Time = 0.60 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.45 \[ \int \frac {1}{(a-i a x)^{7/4} (a+i a x)^{9/4}} \, dx=\frac {2\,{\left (-a\,\left (-1+x\,1{}\mathrm {i}\right )\right )}^{1/4}\,\left (x^2\,8{}\mathrm {i}+4\,x+7{}\mathrm {i}\right )}{15\,a^4\,\left (x^2+1\right )\,{\left (a\,\left (1+x\,1{}\mathrm {i}\right )\right )}^{1/4}} \]
[In]
[Out]