Optimal. Leaf size=65 \[ -\frac {2 i}{a^2 \sqrt [4]{a-i a x} (a+i a x)^{3/4}}+\frac {4 i (a-i a x)^{3/4}}{3 a^3 (a+i a x)^{3/4}} \]
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Rubi [A]
time = 0.01, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {47, 37}
\begin {gather*} \frac {4 i (a-i a x)^{3/4}}{3 a^3 (a+i a x)^{3/4}}-\frac {2 i}{a^2 \sqrt [4]{a-i a x} (a+i a x)^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rubi steps
\begin {align*} \int \frac {1}{(a-i a x)^{5/4} (a+i a x)^{7/4}} \, dx &=-\frac {2 i}{a^2 \sqrt [4]{a-i a x} (a+i a x)^{3/4}}+\frac {2 \int \frac {1}{\sqrt [4]{a-i a x} (a+i a x)^{7/4}} \, dx}{a}\\ &=-\frac {2 i}{a^2 \sqrt [4]{a-i a x} (a+i a x)^{3/4}}+\frac {4 i (a-i a x)^{3/4}}{3 a^3 (a+i a x)^{3/4}}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 45, normalized size = 0.69 \begin {gather*} \frac {2 (1+2 i x) (a-i a x)^{3/4}}{3 a^3 (i+x) (a+i a x)^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [F(-1)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.15, size = 33, normalized size = 0.51
method | result | size |
risch | \(\frac {-\frac {2 i}{3}+\frac {4 x}{3}}{a^{2} \left (a \left (i x +1\right )\right )^{\frac {3}{4}} \left (-a \left (i x -1\right )\right )^{\frac {1}{4}}}\) | \(33\) |
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 = 0.29, size = 36, normalized size = 0.55 \begin {gather*} \frac {2 \, {\left (i \, a x + a\right )}^{\frac {1}{4}} {\left (-i \, a x + a\right )}^{\frac {3}{4}} {\left (2 \, x - i\right )}}{3 \, {\left (a^{4} x^{2} + a^{4}\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 {1}{\left (i a \left (x - i\right )\right )^{\frac {7}{4}} \left (- i a \left (x + i\right )\right )^{\frac {5}{4}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] N/A
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 [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{{\left (a-a\,x\,1{}\mathrm {i}\right )}^{5/4}\,{\left (a+a\,x\,1{}\mathrm {i}\right )}^{7/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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