Optimal. Leaf size=33 \[ \frac{2 i (a-i a x)^{3/4}}{3 a^2 (a+i a x)^{3/4}} \]
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Rubi [A] time = 0.0034473, 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} \[ \frac{2 i (a-i a x)^{3/4}}{3 a^2 (a+i a x)^{3/4}} \]
Antiderivative was successfully verified.
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Rule 37
Rubi steps
\begin{align*} \int \frac{1}{\sqrt [4]{a-i a x} (a+i a x)^{7/4}} \, dx &=\frac{2 i (a-i a x)^{3/4}}{3 a^2 (a+i a x)^{3/4}}\\ \end{align*}
Mathematica [A] time = 0.0106085, size = 33, normalized size = 1. \[ \frac{2 i (a-i a x)^{3/4}}{3 a^2 (a+i a x)^{3/4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 31, normalized size = 0.9 \begin{align*}{\frac{2\,x+2\,i}{3\,a} \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{7}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55673, size = 78, normalized size = 2.36 \begin{align*} \frac{2 \,{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}}{3 \,{\left (a^{3} x - i \, a^{3}\right )}} \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{7}{4}} \sqrt [4]{- a \left (i x - 1\right )}}\, 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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