Optimal. Leaf size=100 \[ -\frac {2 i \sqrt [4]{a+i a x}}{9 a^2 (a-i a x)^{9/4}}-\frac {8 i \sqrt [4]{a+i a x}}{45 a^3 (a-i a x)^{5/4}}-\frac {16 i \sqrt [4]{a+i a x}}{45 a^4 \sqrt [4]{a-i a x}} \]
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Rubi [A]
time = 0.01, 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 = {47, 37}
\begin {gather*} -\frac {16 i \sqrt [4]{a+i a x}}{45 a^4 \sqrt [4]{a-i a x}}-\frac {8 i \sqrt [4]{a+i a x}}{45 a^3 (a-i a x)^{5/4}}-\frac {2 i \sqrt [4]{a+i a x}}{9 a^2 (a-i a x)^{9/4}} \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)^{13/4} (a+i a x)^{3/4}} \, dx &=-\frac {2 i \sqrt [4]{a+i a x}}{9 a^2 (a-i a x)^{9/4}}+\frac {4 \int \frac {1}{(a-i a x)^{9/4} (a+i a x)^{3/4}} \, dx}{9 a}\\ &=-\frac {2 i \sqrt [4]{a+i a x}}{9 a^2 (a-i a x)^{9/4}}-\frac {8 i \sqrt [4]{a+i a x}}{45 a^3 (a-i a x)^{5/4}}+\frac {8 \int \frac {1}{(a-i a x)^{5/4} (a+i a x)^{3/4}} \, dx}{45 a^2}\\ &=-\frac {2 i \sqrt [4]{a+i a x}}{9 a^2 (a-i a x)^{9/4}}-\frac {8 i \sqrt [4]{a+i a x}}{45 a^3 (a-i a x)^{5/4}}-\frac {16 i \sqrt [4]{a+i a x}}{45 a^4 \sqrt [4]{a-i a x}}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 52, normalized size = 0.52 \begin {gather*} \frac {2 \sqrt [4]{a+i a x} \left (17 i+20 x-8 i x^2\right )}{45 a^4 (i+x)^2 \sqrt [4]{a-i a x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 50, normalized size = 0.50
method | result | size |
risch | \(\frac {\frac {16}{45} x^{3}+\frac {8}{15} i x^{2}+\frac {2}{15} x +\frac {34}{45} i}{a^{3} \left (a \left (i x +1\right )\right )^{\frac {3}{4}} \left (-a \left (i x -1\right )\right )^{\frac {1}{4}} \left (x +i\right )^{2}}\) | \(50\) |
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 = 0.75, size = 57, normalized size = 0.57 \begin {gather*} \frac {2 \, {\left (i \, a x + a\right )}^{\frac {1}{4}} {\left (-i \, a x + a\right )}^{\frac {3}{4}} {\left (8 \, x^{2} + 20 i \, x - 17\right )}}{45 \, {\left (a^{5} x^{3} + 3 i \, a^{5} x^{2} - 3 \, a^{5} x - i \, a^{5}\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 {3}{4}} \left (- i a \left (x + i\right )\right )^{\frac {13}{4}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 = -0.01 \begin {gather*} \int \frac {1}{{\left (a-a\,x\,1{}\mathrm {i}\right )}^{13/4}\,{\left (a+a\,x\,1{}\mathrm {i}\right )}^{3/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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