Optimal. Leaf size=154 \[ \frac {14 x}{65 a^6 \left (x^2+1\right ) \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}+\frac {42 \sqrt [4]{x^2+1} E\left (\left .\frac {1}{2} \tan ^{-1}(x)\right |2\right )}{65 a^6 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac {2 i}{13 a^3 (a-i a x)^{9/4} (a+i a x)^{5/4}}-\frac {2 i}{13 a^2 (a-i a x)^{13/4} (a+i a x)^{5/4}} \]
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Rubi [A] time = 0.04, antiderivative size = 154, 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 = {51, 42, 199, 197, 196} \[ \frac {14 x}{65 a^6 \left (x^2+1\right ) \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}+\frac {42 \sqrt [4]{x^2+1} E\left (\left .\frac {1}{2} \tan ^{-1}(x)\right |2\right )}{65 a^6 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac {2 i}{13 a^3 (a-i a x)^{9/4} (a+i a x)^{5/4}}-\frac {2 i}{13 a^2 (a-i a x)^{13/4} (a+i a x)^{5/4}} \]
Antiderivative was successfully verified.
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Rule 42
Rule 51
Rule 196
Rule 197
Rule 199
Rubi steps
\begin {align*} \int \frac {1}{(a-i a x)^{17/4} (a+i a x)^{9/4}} \, dx &=-\frac {2 i}{13 a^2 (a-i a x)^{13/4} (a+i a x)^{5/4}}+\frac {9 \int \frac {1}{(a-i a x)^{13/4} (a+i a x)^{9/4}} \, dx}{13 a}\\ &=-\frac {2 i}{13 a^2 (a-i a x)^{13/4} (a+i a x)^{5/4}}-\frac {2 i}{13 a^3 (a-i a x)^{9/4} (a+i a x)^{5/4}}+\frac {7 \int \frac {1}{(a-i a x)^{9/4} (a+i a x)^{9/4}} \, dx}{13 a^2}\\ &=-\frac {2 i}{13 a^2 (a-i a x)^{13/4} (a+i a x)^{5/4}}-\frac {2 i}{13 a^3 (a-i a x)^{9/4} (a+i a x)^{5/4}}+\frac {\left (7 \sqrt [4]{a^2+a^2 x^2}\right ) \int \frac {1}{\left (a^2+a^2 x^2\right )^{9/4}} \, dx}{13 a^2 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ &=-\frac {2 i}{13 a^2 (a-i a x)^{13/4} (a+i a x)^{5/4}}-\frac {2 i}{13 a^3 (a-i a x)^{9/4} (a+i a x)^{5/4}}+\frac {14 x}{65 a^6 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x} \left (1+x^2\right )}+\frac {\left (21 \sqrt [4]{a^2+a^2 x^2}\right ) \int \frac {1}{\left (a^2+a^2 x^2\right )^{5/4}} \, dx}{65 a^4 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ &=-\frac {2 i}{13 a^2 (a-i a x)^{13/4} (a+i a x)^{5/4}}-\frac {2 i}{13 a^3 (a-i a x)^{9/4} (a+i a x)^{5/4}}+\frac {14 x}{65 a^6 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x} \left (1+x^2\right )}+\frac {\left (21 \sqrt [4]{1+x^2}\right ) \int \frac {1}{\left (1+x^2\right )^{5/4}} \, dx}{65 a^6 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ &=-\frac {2 i}{13 a^2 (a-i a x)^{13/4} (a+i a x)^{5/4}}-\frac {2 i}{13 a^3 (a-i a x)^{9/4} (a+i a x)^{5/4}}+\frac {14 x}{65 a^6 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x} \left (1+x^2\right )}+\frac {42 \sqrt [4]{1+x^2} E\left (\left .\frac {1}{2} \tan ^{-1}(x)\right |2\right )}{65 a^6 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 70, normalized size = 0.45 \[ -\frac {i \sqrt [4]{1+i x} \, _2F_1\left (-\frac {13}{4},\frac {9}{4};-\frac {9}{4};\frac {1}{2}-\frac {i x}{2}\right )}{13 \sqrt [4]{2} a^3 (a-i a x)^{13/4} \sqrt [4]{a+i a x}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ \frac {{\left (42 \, x^{5} + 84 i \, x^{4} + 14 \, x^{3} + 112 i \, x^{2} - 46 \, x + 20 i\right )} {\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {3}{4}} + {\left (65 \, a^{8} x^{6} + 130 i \, a^{8} x^{5} + 65 \, a^{8} x^{4} + 260 i \, a^{8} x^{3} - 65 \, a^{8} x^{2} + 130 i \, a^{8} x - 65 \, a^{8}\right )} {\rm integral}\left (-\frac {21 \, {\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {3}{4}}}{65 \, {\left (a^{8} x^{2} + a^{8}\right )}}, x\right )}{65 \, a^{8} x^{6} + 130 i \, a^{8} x^{5} + 65 \, a^{8} x^{4} + 260 i \, a^{8} x^{3} - 65 \, a^{8} x^{2} + 130 i \, a^{8} x - 65 \, a^{8}} \]
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 \[ \int \frac {1}{{\left (i \, a x + a\right )}^{\frac {9}{4}} {\left (-i \, a x + a\right )}^{\frac {17}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.10, size = 130, normalized size = 0.84 \[ -\frac {21 \left (-\left (i x -1\right ) \left (i x +1\right ) a^{2}\right )^{\frac {1}{4}} x \hypergeom \left (\left [\frac {1}{4}, \frac {1}{2}\right ], \left [\frac {3}{2}\right ], -x^{2}\right )}{65 \left (a^{2}\right )^{\frac {1}{4}} \left (-\left (i x -1\right ) a \right )^{\frac {1}{4}} \left (\left (i x +1\right ) a \right )^{\frac {1}{4}} a^{6}}+\frac {\frac {42}{65} x^{5}+\frac {84}{65} i x^{4}+\frac {14}{65} x^{3}+\frac {112}{65} i x^{2}-\frac {46}{65} x +\frac {4}{13} i}{\left (x -i\right ) \left (x +i\right )^{3} \left (-\left (i x -1\right ) a \right )^{\frac {1}{4}} \left (\left (i x +1\right ) a \right )^{\frac {1}{4}} a^{6}} \]
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 \[ \text {Exception raised: RuntimeError} \]
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 \[ \int \frac {1}{{\left (a-a\,x\,1{}\mathrm {i}\right )}^{17/4}\,{\left (a+a\,x\,1{}\mathrm {i}\right )}^{9/4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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