Optimal. Leaf size=76 \[ \frac {2 a \left (x^2+1\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \tan ^{-1}(x),2\right )}{(a-i a x)^{3/4} (a+i a x)^{3/4}}-\frac {2 i \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}{a} \]
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Rubi [A] time = 0.01, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {50, 42, 233, 231} \[ \frac {2 a \left (x^2+1\right )^{3/4} F\left (\left .\frac {1}{2} \tan ^{-1}(x)\right |2\right )}{(a-i a x)^{3/4} (a+i a x)^{3/4}}-\frac {2 i \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}{a} \]
Antiderivative was successfully verified.
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Rule 42
Rule 50
Rule 231
Rule 233
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{a-i a x}}{(a+i a x)^{3/4}} \, dx &=-\frac {2 i \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}{a}+a \int \frac {1}{(a-i a x)^{3/4} (a+i a x)^{3/4}} \, dx\\ &=-\frac {2 i \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}{a}+\frac {\left (a \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 {2 i \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}{a}+\frac {\left (a \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 {2 i \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}{a}+\frac {2 a \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] time = 0.02, size = 70, normalized size = 0.92 \[ \frac {2 i \sqrt [4]{2} (1+i x)^{3/4} (a-i a x)^{5/4} \, _2F_1\left (\frac {3}{4},\frac {5}{4};\frac {9}{4};\frac {1}{2}-\frac {i x}{2}\right )}{5 a (a+i a x)^{3/4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ \frac {a {\rm integral}\left (\frac {{\left (i \, a x + a\right )}^{\frac {1}{4}} {\left (-i \, a x + a\right )}^{\frac {1}{4}}}{a x^{2} + a}, x\right ) - 2 i \, {\left (i \, a x + a\right )}^{\frac {1}{4}} {\left (-i \, a x + a\right )}^{\frac {1}{4}}}{a} \]
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 \[ \int \frac {{\left (-i \, a x + a\right )}^{\frac {1}{4}}}{{\left (i \, a x + a\right )}^{\frac {3}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int \frac {\left (-i a x +a \right )^{\frac {1}{4}}}{\left (i a x +a \right )^{\frac {3}{4}}}\, dx \]
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 \[ \int \frac {{\left (-i \, a x + a\right )}^{\frac {1}{4}}}{{\left (i \, a x + a\right )}^{\frac {3}{4}}}\,{d x} \]
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 \[ \int \frac {{\left (a-a\,x\,1{}\mathrm {i}\right )}^{1/4}}{{\left (a+a\,x\,1{}\mathrm {i}\right )}^{3/4}} \,d x \]
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 \[ \int \frac {\sqrt [4]{- i a \left (x + i\right )}}{\left (i a \left (x - i\right )\right )^{\frac {3}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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