Optimal. Leaf size=79 \[ \frac {4 i \sqrt [4]{a-i a x}}{3 a (a+i a x)^{3/4}}-\frac {2 \left (x^2+1\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \tan ^{-1}(x),2\right )}{3 (a-i a x)^{3/4} (a+i a x)^{3/4}} \]
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Rubi [A] time = 0.01, antiderivative size = 79, 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 = {47, 42, 233, 231} \[ \frac {4 i \sqrt [4]{a-i a x}}{3 a (a+i a x)^{3/4}}-\frac {2 \left (x^2+1\right )^{3/4} F\left (\left .\frac {1}{2} \tan ^{-1}(x)\right |2\right )}{3 (a-i a x)^{3/4} (a+i a x)^{3/4}} \]
Antiderivative was successfully verified.
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Rule 42
Rule 47
Rule 231
Rule 233
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{a-i a x}}{(a+i a x)^{7/4}} \, dx &=\frac {4 i \sqrt [4]{a-i a x}}{3 a (a+i a x)^{3/4}}-\frac {1}{3} \int \frac {1}{(a-i a x)^{3/4} (a+i a x)^{3/4}} \, dx\\ &=\frac {4 i \sqrt [4]{a-i a x}}{3 a (a+i a x)^{3/4}}-\frac {\left (a^2+a^2 x^2\right )^{3/4} \int \frac {1}{\left (a^2+a^2 x^2\right )^{3/4}} \, dx}{3 (a-i a x)^{3/4} (a+i a x)^{3/4}}\\ &=\frac {4 i \sqrt [4]{a-i a x}}{3 a (a+i a x)^{3/4}}-\frac {\left (1+x^2\right )^{3/4} \int \frac {1}{\left (1+x^2\right )^{3/4}} \, dx}{3 (a-i a x)^{3/4} (a+i a x)^{3/4}}\\ &=\frac {4 i \sqrt [4]{a-i a x}}{3 a (a+i a x)^{3/4}}-\frac {2 \left (1+x^2\right )^{3/4} F\left (\left .\frac {1}{2} \tan ^{-1}(x)\right |2\right )}{3 (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 = 0.89 \[ \frac {i \sqrt [4]{2} (1+i x)^{3/4} (a-i a x)^{5/4} \, _2F_1\left (\frac {5}{4},\frac {7}{4};\frac {9}{4};\frac {1}{2}-\frac {i x}{2}\right )}{5 a^2 (a+i a x)^{3/4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ \frac {3 \, {\left (a^{2} x - i \, a^{2}\right )} {\rm integral}\left (-\frac {{\left (i \, a x + a\right )}^{\frac {1}{4}} {\left (-i \, a x + a\right )}^{\frac {1}{4}}}{3 \, {\left (a^{2} x^{2} + a^{2}\right )}}, x\right ) + 4 \, {\left (i \, a x + a\right )}^{\frac {1}{4}} {\left (-i \, a x + a\right )}^{\frac {1}{4}}}{3 \, {\left (a^{2} x - i \, a^{2}\right )}} \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.07, size = 0, normalized size = 0.00 \[ \int \frac {\left (-i a x +a \right )^{\frac {1}{4}}}{\left (i a x +a \right )^{\frac {7}{4}}}\, dx \]
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 \[ \int \frac {{\left (-i \, a x + a\right )}^{\frac {1}{4}}}{{\left (i \, a x + a\right )}^{\frac {7}{4}}}\,{d x} \]
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 \[ \int \frac {{\left (a-a\,x\,1{}\mathrm {i}\right )}^{1/4}}{{\left (a+a\,x\,1{}\mathrm {i}\right )}^{7/4}} \,d x \]
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 \[ \int \frac {\sqrt [4]{- i a \left (x + i\right )}}{\left (i a \left (x - i\right )\right )^{\frac {7}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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