Optimal. Leaf size=100 \[ -\frac {2 i}{5 a^2 (a-i a x)^{5/4} (a+i a x)^{3/4}}-\frac {8 i}{5 a^3 \sqrt [4]{a-i a x} (a+i a x)^{3/4}}+\frac {16 i (a-i a x)^{3/4}}{15 a^4 (a+i a x)^{3/4}} \]
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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 (a-i a x)^{3/4}}{15 a^4 (a+i a x)^{3/4}}-\frac {8 i}{5 a^3 (a+i a x)^{3/4} \sqrt [4]{a-i a x}}-\frac {2 i}{5 a^2 (a+i a x)^{3/4} (a-i a x)^{5/4}} \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)^{9/4} (a+i a x)^{7/4}} \, dx &=-\frac {2 i}{5 a^2 (a-i a x)^{5/4} (a+i a x)^{3/4}}+\frac {4 \int \frac {1}{(a-i a x)^{5/4} (a+i a x)^{7/4}} \, dx}{5 a}\\ &=-\frac {2 i}{5 a^2 (a-i a x)^{5/4} (a+i a x)^{3/4}}-\frac {8 i}{5 a^3 \sqrt [4]{a-i a x} (a+i a x)^{3/4}}+\frac {8 \int \frac {1}{\sqrt [4]{a-i a x} (a+i a x)^{7/4}} \, dx}{5 a^2}\\ &=-\frac {2 i}{5 a^2 (a-i a x)^{5/4} (a+i a x)^{3/4}}-\frac {8 i}{5 a^3 \sqrt [4]{a-i a x} (a+i a x)^{3/4}}+\frac {16 i (a-i a x)^{3/4}}{15 a^4 (a+i a x)^{3/4}}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 52, normalized size = 0.52 \begin {gather*} \frac {2 i (a-i a x)^{3/4} \left (7+4 i x+8 x^2\right )}{15 a^4 (i+x)^2 (a+i a x)^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 44, normalized size = 0.44
method | result | size |
risch | \(\frac {\frac {16}{15} x^{2}+\frac {8}{15} i x +\frac {14}{15}}{a^{3} \left (a \left (i x +1\right )\right )^{\frac {3}{4}} \left (-a \left (i x -1\right )\right )^{\frac {1}{4}} \left (x +i\right )}\) | \(44\) |
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 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.61, size = 56, normalized size = 0.56 \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} + 4 i \, x + 7\right )}}{15 \, {\left (a^{5} x^{3} + i \, a^{5} x^{2} + 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 {7}{4}} \left (- i a \left (x + i\right )\right )^{\frac {9}{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 )}^{9/4}\,{\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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