Optimal. Leaf size=65 \[ \frac {4 i (a-i a x)^{3/4}}{3 a^3 (a+i a x)^{3/4}}-\frac {2 i}{a^2 \sqrt [4]{a-i a x} (a+i a x)^{3/4}} \]
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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 = {45, 37} \[ \frac {4 i (a-i a x)^{3/4}}{3 a^3 (a+i a x)^{3/4}}-\frac {2 i}{a^2 \sqrt [4]{a-i a x} (a+i a x)^{3/4}} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
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 = 0.02, size = 38, normalized size = 0.58 \[ \frac {4 x-2 i}{3 a^2 \sqrt [4]{a-i a x} (a+i a x)^{3/4}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 36, normalized size = 0.55 \[ \frac {{\left (i \, a x + a\right )}^{\frac {1}{4}} {\left (-i \, a x + a\right )}^{\frac {3}{4}} {\left (4 \, x - 2 i\right )}}{3 \, {\left (a^{4} x^{2} + a^{4}\right )}} \]
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 \[ \int \frac {1}{{\left (i \, a x + a\right )}^{\frac {7}{4}} {\left (-i \, a x + a\right )}^{\frac {5}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 33, normalized size = 0.51 \[ \frac {\frac {4 x}{3}-\frac {2 i}{3}}{\left (\left (i x +1\right ) a \right )^{\frac {3}{4}} \left (-\left (i x -1\right ) a \right )^{\frac {1}{4}} a^{2}} \]
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 \[ \int \frac {1}{{\left (i \, a x + a\right )}^{\frac {7}{4}} {\left (-i \, a x + a\right )}^{\frac {5}{4}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (a-a\,x\,1{}\mathrm {i}\right )}^{5/4}\,{\left (a+a\,x\,1{}\mathrm {i}\right )}^{7/4}} \,d x \]
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 \[ \int \frac {1}{\left (i a \left (x - i\right )\right )^{\frac {7}{4}} \left (- i a \left (x + i\right )\right )^{\frac {5}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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