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 = 0.00, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {37} \begin {gather*} \frac {2 i (a-i a x)^{5/4}}{5 a^2 (a+i a x)^{5/4}} \end {gather*}
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*}
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Mathematica [A] time = 0.01, size = 33, normalized size = 1.00 \begin {gather*} \frac {2 i (a-i a x)^{5/4}}{5 a^2 (a+i a x)^{5/4}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.11, size = 33, normalized size = 1.00 \begin {gather*} \frac {2 i (a-i a x)^{5/4}}{5 a^2 (a+i a x)^{5/4}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.43, size = 45, normalized size = 1.36 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 50, normalized size = 1.52 \begin {gather*} \frac {2 \left (-\left (i x -1\right ) a \right )^{\frac {1}{4}} \left (x^{2}+2 i x -1\right )}{5 \left (i x -1\right ) \left (\left (i x +1\right ) a \right )^{\frac {1}{4}} \left (x -i\right ) a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (-i \, a x + a\right )}^{\frac {1}{4}}}{{\left (i \, a x + a\right )}^{\frac {9}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.55, size = 38, normalized size = 1.15 \begin {gather*} -\frac {2\,\left (-1+x\,1{}\mathrm {i}\right )\,{\left (-a\,\left (-1+x\,1{}\mathrm {i}\right )\right )}^{1/4}}{5\,a^2\,\left (x-\mathrm {i}\right )\,{\left (a\,\left (1+x\,1{}\mathrm {i}\right )\right )}^{1/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [4]{- i a \left (x + i\right )}}{\left (i a \left (x - i\right )\right )^{\frac {9}{4}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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