Optimal. Leaf size=33 \[ \frac{2 i (a-i a x)^{5/4}}{5 a^2 (a+i a x)^{5/4}} \]
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Rubi [A] time = 0.0032964, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {37} \[ \frac{2 i (a-i a x)^{5/4}}{5 a^2 (a+i a x)^{5/4}} \]
Antiderivative was successfully verified.
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Rule 37
Rubi steps
\begin{align*} \int \frac{\sqrt [4]{a-i a x}}{(a+i a x)^{9/4}} \, dx &=\frac{2 i (a-i a x)^{5/4}}{5 a^2 (a+i a x)^{5/4}}\\ \end{align*}
Mathematica [A] time = 0.011016, size = 33, normalized size = 1. \[ \frac{2 i (a-i a x)^{5/4}}{5 a^2 (a+i a x)^{5/4}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.034, size = 50, normalized size = 1.5 \begin{align*}{\frac{4\,ix+2\,{x}^{2}-2}{5\,{a}^{2} \left ( -1+ix \right ) \left ( x-i \right ) }\sqrt [4]{-a \left ( -1+ix \right ) }{\frac{1}{\sqrt [4]{a \left ( 1+ix \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{{\left (i \, a x + a\right )}^{\frac{9}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.985, size = 113, normalized size = 3.42 \begin{align*} -\frac{{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}{\left (2 \, x + 2 i\right )}}{5 \, a^{3} x^{2} - 10 i \, a^{3} x - 5 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt [4]{- a \left (i x - 1\right )}}{\left (a \left (i x + 1\right )\right )^{\frac{9}{4}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11823, size = 46, normalized size = 1.39 \begin{align*} -\frac{{\left (-i \, a x + a\right )}^{\frac{1}{4}}{\left (-\frac{4 i \, a}{i \, a x + a} + 2 i\right )}}{5 \,{\left (i \, a x + a\right )}^{\frac{1}{4}} a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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