Optimal. Leaf size=31 \[ -\frac{2 i \sqrt [4]{a+i a x}}{a^2 \sqrt [4]{a-i a x}} \]
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Rubi [A] time = 0.0031736, antiderivative size = 31, 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} \[ -\frac{2 i \sqrt [4]{a+i a x}}{a^2 \sqrt [4]{a-i a x}} \]
Antiderivative was successfully verified.
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Rule 37
Rubi steps
\begin{align*} \int \frac{1}{(a-i a x)^{5/4} (a+i a x)^{3/4}} \, dx &=-\frac{2 i \sqrt [4]{a+i a x}}{a^2 \sqrt [4]{a-i a x}}\\ \end{align*}
Mathematica [A] time = 0.0126921, size = 31, normalized size = 1. \[ -\frac{2 i \sqrt [4]{a+i a x}}{a^2 \sqrt [4]{a-i a x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.029, size = 31, normalized size = 1. \begin{align*} 2\,{\frac{x-i}{a \left ( a \left ( 1+ix \right ) \right ) ^{3/4}\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{1}{{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{5}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54243, size = 76, normalized size = 2.45 \begin{align*} \frac{2 \,{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}}{a^{3} x + i \, 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{1}{\left (a \left (i x + 1\right )\right )^{\frac{3}{4}} \left (- a \left (i x - 1\right )\right )^{\frac{5}{4}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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