Optimal. Leaf size=139 \[ \frac {4 i (a-i a x)^{9/4}}{3 a (a+i a x)^{3/4}}+10 i \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}+\frac {2 i (a-i a x)^{5/4} \sqrt [4]{a+i a x}}{a}-\frac {10 a^2 \left (1+x^2\right )^{3/4} F\left (\left .\frac {1}{2} \tan ^{-1}(x)\right |2\right )}{(a-i a x)^{3/4} (a+i a x)^{3/4}} \]
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Rubi [A]
time = 0.02, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {49, 52, 42, 239,
237} \begin {gather*} -\frac {10 a^2 \left (x^2+1\right )^{3/4} F\left (\left .\frac {1}{2} \tan ^{-1}(x)\right |2\right )}{(a+i a x)^{3/4} (a-i a x)^{3/4}}+\frac {4 i (a-i a x)^{9/4}}{3 a (a+i a x)^{3/4}}+\frac {2 i \sqrt [4]{a+i a x} (a-i a x)^{5/4}}{a}+10 i \sqrt [4]{a+i a x} \sqrt [4]{a-i a x} \end {gather*}
Antiderivative was successfully verified.
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Rule 42
Rule 49
Rule 52
Rule 237
Rule 239
Rubi steps
\begin {align*} \int \frac {(a-i a x)^{9/4}}{(a+i a x)^{7/4}} \, dx &=\frac {4 i (a-i a x)^{9/4}}{3 a (a+i a x)^{3/4}}-3 \int \frac {(a-i a x)^{5/4}}{(a+i a x)^{3/4}} \, dx\\ &=\frac {4 i (a-i a x)^{9/4}}{3 a (a+i a x)^{3/4}}+\frac {2 i (a-i a x)^{5/4} \sqrt [4]{a+i a x}}{a}-(5 a) \int \frac {\sqrt [4]{a-i a x}}{(a+i a x)^{3/4}} \, dx\\ &=\frac {4 i (a-i a x)^{9/4}}{3 a (a+i a x)^{3/4}}+10 i \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}+\frac {2 i (a-i a x)^{5/4} \sqrt [4]{a+i a x}}{a}-\left (5 a^2\right ) \int \frac {1}{(a-i a x)^{3/4} (a+i a x)^{3/4}} \, dx\\ &=\frac {4 i (a-i a x)^{9/4}}{3 a (a+i a x)^{3/4}}+10 i \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}+\frac {2 i (a-i a x)^{5/4} \sqrt [4]{a+i a x}}{a}-\frac {\left (5 a^2 \left (a^2+a^2 x^2\right )^{3/4}\right ) \int \frac {1}{\left (a^2+a^2 x^2\right )^{3/4}} \, dx}{(a-i a x)^{3/4} (a+i a x)^{3/4}}\\ &=\frac {4 i (a-i a x)^{9/4}}{3 a (a+i a x)^{3/4}}+10 i \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}+\frac {2 i (a-i a x)^{5/4} \sqrt [4]{a+i a x}}{a}-\frac {\left (5 a^2 \left (1+x^2\right )^{3/4}\right ) \int \frac {1}{\left (1+x^2\right )^{3/4}} \, dx}{(a-i a x)^{3/4} (a+i a x)^{3/4}}\\ &=\frac {4 i (a-i a x)^{9/4}}{3 a (a+i a x)^{3/4}}+10 i \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}+\frac {2 i (a-i a x)^{5/4} \sqrt [4]{a+i a x}}{a}-\frac {10 a^2 \left (1+x^2\right )^{3/4} F\left (\left .\frac {1}{2} \tan ^{-1}(x)\right |2\right )}{(a-i a x)^{3/4} (a+i a x)^{3/4}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.04, size = 70, normalized size = 0.50 \begin {gather*} \frac {i \sqrt [4]{2} (1+i x)^{3/4} (a-i a x)^{13/4} \, _2F_1\left (\frac {7}{4},\frac {13}{4};\frac {17}{4};\frac {1}{2}-\frac {i x}{2}\right )}{13 a^2 (a+i a x)^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [F(-1)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (-i a x +a \right )^{\frac {9}{4}}}{\left (i a x +a \right )^{\frac {7}{4}}}\, dx\]
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 [F]
time = 0.31, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 {\left (- i a \left (x + i\right )\right )^{\frac {9}{4}}}{\left (i a \left (x - i\right )\right )^{\frac {7}{4}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] N/A
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Could not integrate} \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 {{\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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