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 = 0.08, 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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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 49 vs. \(2 (24 ) = 48\).
time = 0.15, size = 50, normalized size = 1.52
method | result | size |
risch | \(\frac {2 \left (-a \left (i x -1\right )\right )^{\frac {1}{4}} \left (x^{2}+2 i x -1\right )}{5 a^{2} \left (i x -1\right ) \left (a \left (i x +1\right )\right )^{\frac {1}{4}} \left (x -i\right )}\) | \(50\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.76, size = 42, normalized size = 1.27 \begin {gather*} -\frac {2 \, {\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {1}{4}} {\left (x + i\right )}}{5 \, {\left (a^{3} x^{2} - 2 i \, a^{3} x - a^{3}\right )}} \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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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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