Optimal. Leaf size=67 \[ -\frac {4 i \sqrt [4]{a+i a x}}{5 a^3 \sqrt [4]{a-i a x}}-\frac {2 i \sqrt [4]{a+i a x}}{5 a^2 (a-i a x)^{5/4}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {45, 37} \[ -\frac {4 i \sqrt [4]{a+i a x}}{5 a^3 \sqrt [4]{a-i a x}}-\frac {2 i \sqrt [4]{a+i a x}}{5 a^2 (a-i a x)^{5/4}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 37
Rule 45
Rubi steps
\begin {align*} \int \frac {1}{(a-i a x)^{9/4} (a+i a x)^{3/4}} \, dx &=-\frac {2 i \sqrt [4]{a+i a x}}{5 a^2 (a-i a x)^{5/4}}+\frac {2 \int \frac {1}{(a-i a x)^{5/4} (a+i a x)^{3/4}} \, dx}{5 a}\\ &=-\frac {2 i \sqrt [4]{a+i a x}}{5 a^2 (a-i a x)^{5/4}}-\frac {4 i \sqrt [4]{a+i a x}}{5 a^3 \sqrt [4]{a-i a x}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 45, normalized size = 0.67 \[ \frac {2 (3-2 i x) \sqrt [4]{a+i a x}}{5 a^3 (x+i) \sqrt [4]{a-i a x}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.44, size = 44, normalized size = 0.66 \[ \frac {{\left (i \, a x + a\right )}^{\frac {1}{4}} {\left (-i \, a x + a\right )}^{\frac {3}{4}} {\left (4 \, x + 6 i\right )}}{5 \, a^{4} x^{2} + 10 i \, a^{4} x - 5 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {9}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 44, normalized size = 0.66 \[ \frac {\frac {4}{5} x^{2}+\frac {2}{5} i x +\frac {6}{5}}{\left (\left (i x +1\right ) a \right )^{\frac {3}{4}} \left (-\left (i x -1\right ) a \right )^{\frac {1}{4}} \left (x +i\right ) a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {9}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (a-a\,x\,1{}\mathrm {i}\right )}^{9/4}\,{\left (a+a\,x\,1{}\mathrm {i}\right )}^{3/4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (i a \left (x - i\right )\right )^{\frac {3}{4}} \left (- i a \left (x + i\right )\right )^{\frac {9}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________