Optimal. Leaf size=67 \[ -\frac{4 i \sqrt [4]{a+i a x}}{5 a^3 \sqrt [4]{a-i a x}}-\frac{2 i \sqrt [4]{a+i a x}}{5 a^2 (a-i a x)^{5/4}} \]
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Rubi [A] time = 0.0098718, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, 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.
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Rule 45
Rule 37
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 = 0.0179804, 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.
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Maple [A] time = 0.034, size = 44, normalized size = 0.7 \begin{align*}{\frac{4\,{x}^{2}+6+2\,ix}{5\,{a}^{2} \left ( x+i \right ) } \left ( a \left ( 1+ix \right ) \right ) ^{-{\frac{3}{4}}}{\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{1}{{\left (i \, a x + a\right )}^{\frac{3}{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 [A] time = 1.58109, size = 112, normalized size = 1.67 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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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