Optimal. Leaf size=100 \[ \frac {16 i \sqrt [4]{a-i a x}}{15 a^4 \sqrt [4]{a+i a x}}+\frac {8 i \sqrt [4]{a-i a x}}{15 a^3 (a+i a x)^{5/4}}-\frac {2 i}{3 a^2 (a+i a x)^{5/4} (a-i a x)^{3/4}} \]
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Rubi [A] time = 0.02, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {45, 37} \[ \frac {16 i \sqrt [4]{a-i a x}}{15 a^4 \sqrt [4]{a+i a x}}+\frac {8 i \sqrt [4]{a-i a x}}{15 a^3 (a+i a x)^{5/4}}-\frac {2 i}{3 a^2 (a+i a x)^{5/4} (a-i a x)^{3/4}} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rubi steps
\begin {align*} \int \frac {1}{(a-i a x)^{7/4} (a+i a x)^{9/4}} \, dx &=-\frac {2 i}{3 a^2 (a-i a x)^{3/4} (a+i a x)^{5/4}}+\frac {4 \int \frac {1}{(a-i a x)^{3/4} (a+i a x)^{9/4}} \, dx}{3 a}\\ &=-\frac {2 i}{3 a^2 (a-i a x)^{3/4} (a+i a x)^{5/4}}+\frac {8 i \sqrt [4]{a-i a x}}{15 a^3 (a+i a x)^{5/4}}+\frac {8 \int \frac {1}{(a-i a x)^{3/4} (a+i a x)^{5/4}} \, dx}{15 a^2}\\ &=-\frac {2 i}{3 a^2 (a-i a x)^{3/4} (a+i a x)^{5/4}}+\frac {8 i \sqrt [4]{a-i a x}}{15 a^3 (a+i a x)^{5/4}}+\frac {16 i \sqrt [4]{a-i a x}}{15 a^4 \sqrt [4]{a+i a x}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 50, normalized size = 0.50 \[ \frac {2 \left (8 x^2-4 i x+7\right )}{15 a^3 (x-i) (a-i a x)^{3/4} \sqrt [4]{a+i a x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 58, normalized size = 0.58 \[ \frac {2 \, {\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {1}{4}} {\left (8 \, x^{2} - 4 i \, x + 7\right )}}{15 \, a^{5} x^{3} - 15 i \, a^{5} x^{2} + 15 \, a^{5} x - 15 i \, a^{5}} \]
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 \[ \int \frac {1}{{\left (i \, a x + a\right )}^{\frac {9}{4}} {\left (-i \, a x + a\right )}^{\frac {7}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 44, normalized size = 0.44 \[ \frac {\frac {16}{15} x^{2}-\frac {8}{15} i x +\frac {14}{15}}{\left (-\left (i x -1\right ) a \right )^{\frac {3}{4}} \left (\left (i x +1\right ) a \right )^{\frac {1}{4}} \left (x -i\right ) a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.53, size = 45, normalized size = 0.45 \[ \frac {2\,{\left (-a\,\left (-1+x\,1{}\mathrm {i}\right )\right )}^{1/4}\,\left (x^2\,8{}\mathrm {i}+4\,x+7{}\mathrm {i}\right )}{15\,a^4\,\left (x^2+1\right )\,{\left (a\,\left (1+x\,1{}\mathrm {i}\right )\right )}^{1/4}} \]
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 \[ \int \frac {1}{\left (i a \left (x - i\right )\right )^{\frac {9}{4}} \left (- i a \left (x + i\right )\right )^{\frac {7}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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