Optimal. Leaf size=67 \[ -\frac {4 i (a+i a x)^{3/4}}{21 a^3 (a-i a x)^{3/4}}-\frac {2 i (a+i a x)^{3/4}}{7 a^2 (a-i a x)^{7/4}} \]
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Rubi [A] time = 0.01, antiderivative size = 67, 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} \[ -\frac {4 i (a+i a x)^{3/4}}{21 a^3 (a-i a x)^{3/4}}-\frac {2 i (a+i a x)^{3/4}}{7 a^2 (a-i a x)^{7/4}} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rubi steps
\begin {align*} \int \frac {1}{(a-i a x)^{11/4} \sqrt [4]{a+i a x}} \, dx &=-\frac {2 i (a+i a x)^{3/4}}{7 a^2 (a-i a x)^{7/4}}+\frac {2 \int \frac {1}{(a-i a x)^{7/4} \sqrt [4]{a+i a x}} \, dx}{7 a}\\ &=-\frac {2 i (a+i a x)^{3/4}}{7 a^2 (a-i a x)^{7/4}}-\frac {4 i (a+i a x)^{3/4}}{21 a^3 (a-i a x)^{3/4}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 45, normalized size = 0.67 \[ \frac {2 (5-2 i x) (a+i a x)^{3/4}}{21 a^3 (x+i) (a-i a x)^{3/4}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 44, normalized size = 0.66 \[ \frac {{\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {1}{4}} {\left (4 \, x + 10 i\right )}}{21 \, a^{4} x^{2} + 42 i \, a^{4} x - 21 \, a^{4}} \]
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 \[ \int \frac {1}{{\left (i \, a x + a\right )}^{\frac {1}{4}} {\left (-i \, a x + a\right )}^{\frac {11}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 44, normalized size = 0.66 \[ \frac {\frac {4}{21} x^{2}+\frac {2}{7} i x +\frac {10}{21}}{\left (-\left (i x -1\right ) a \right )^{\frac {3}{4}} \left (\left (i x +1\right ) a \right )^{\frac {1}{4}} \left (x +i\right ) a^{2}} \]
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 \[ \int \frac {1}{{\left (i \, a x + a\right )}^{\frac {1}{4}} {\left (-i \, a x + a\right )}^{\frac {11}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.67, size = 46, normalized size = 0.69 \[ -\frac {{\left (-a\,\left (-1+x\,1{}\mathrm {i}\right )\right )}^{1/4}\,\left (2\,x^2+x\,3{}\mathrm {i}+5\right )\,2{}\mathrm {i}}{21\,a^3\,{\left (-1+x\,1{}\mathrm {i}\right )}^2\,{\left (a\,\left (1+x\,1{}\mathrm {i}\right )\right )}^{1/4}} \]
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 \[ \int \frac {1}{\sqrt [4]{i a \left (x - i\right )} \left (- i a \left (x + i\right )\right )^{\frac {11}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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