Optimal. Leaf size=115 \[ \frac {2 \sqrt [4]{x^2+1} E\left (\left .\frac {1}{2} \tan ^{-1}(x)\right |2\right )}{3 a^4 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac {2 i}{9 a^3 (a-i a x)^{5/4} \sqrt [4]{a+i a x}}-\frac {2 i}{9 a^2 (a-i a x)^{9/4} \sqrt [4]{a+i a x}} \]
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Rubi [A] time = 0.03, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {51, 42, 197, 196} \[ \frac {2 \sqrt [4]{x^2+1} E\left (\left .\frac {1}{2} \tan ^{-1}(x)\right |2\right )}{3 a^4 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac {2 i}{9 a^3 (a-i a x)^{5/4} \sqrt [4]{a+i a x}}-\frac {2 i}{9 a^2 (a-i a x)^{9/4} \sqrt [4]{a+i a x}} \]
Antiderivative was successfully verified.
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Rule 42
Rule 51
Rule 196
Rule 197
Rubi steps
\begin {align*} \int \frac {1}{(a-i a x)^{13/4} (a+i a x)^{5/4}} \, dx &=-\frac {2 i}{9 a^2 (a-i a x)^{9/4} \sqrt [4]{a+i a x}}+\frac {5 \int \frac {1}{(a-i a x)^{9/4} (a+i a x)^{5/4}} \, dx}{9 a}\\ &=-\frac {2 i}{9 a^2 (a-i a x)^{9/4} \sqrt [4]{a+i a x}}-\frac {2 i}{9 a^3 (a-i a x)^{5/4} \sqrt [4]{a+i a x}}+\frac {\int \frac {1}{(a-i a x)^{5/4} (a+i a x)^{5/4}} \, dx}{3 a^2}\\ &=-\frac {2 i}{9 a^2 (a-i a x)^{9/4} \sqrt [4]{a+i a x}}-\frac {2 i}{9 a^3 (a-i a x)^{5/4} \sqrt [4]{a+i a x}}+\frac {\sqrt [4]{a^2+a^2 x^2} \int \frac {1}{\left (a^2+a^2 x^2\right )^{5/4}} \, dx}{3 a^2 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ &=-\frac {2 i}{9 a^2 (a-i a x)^{9/4} \sqrt [4]{a+i a x}}-\frac {2 i}{9 a^3 (a-i a x)^{5/4} \sqrt [4]{a+i a x}}+\frac {\sqrt [4]{1+x^2} \int \frac {1}{\left (1+x^2\right )^{5/4}} \, dx}{3 a^4 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ &=-\frac {2 i}{9 a^2 (a-i a x)^{9/4} \sqrt [4]{a+i a x}}-\frac {2 i}{9 a^3 (a-i a x)^{5/4} \sqrt [4]{a+i a x}}+\frac {2 \sqrt [4]{1+x^2} E\left (\left .\frac {1}{2} \tan ^{-1}(x)\right |2\right )}{3 a^4 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 70, normalized size = 0.61 \[ -\frac {i 2^{3/4} \sqrt [4]{1+i x} \, _2F_1\left (-\frac {9}{4},\frac {5}{4};-\frac {5}{4};\frac {1}{2}-\frac {i x}{2}\right )}{9 a^2 (a-i a x)^{9/4} \sqrt [4]{a+i a x}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ \frac {{\left (6 \, x^{3} + 12 i \, x^{2} - 4 \, x + 4 i\right )} {\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {3}{4}} + {\left (9 \, a^{6} x^{4} + 18 i \, a^{6} x^{3} + 18 i \, a^{6} x - 9 \, a^{6}\right )} {\rm integral}\left (-\frac {{\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {3}{4}}}{3 \, {\left (a^{6} x^{2} + a^{6}\right )}}, x\right )}{9 \, a^{6} x^{4} + 18 i \, a^{6} x^{3} + 18 i \, a^{6} x - 9 \, a^{6}} \]
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 \[ \int \frac {1}{{\left (i \, a x + a\right )}^{\frac {5}{4}} {\left (-i \, a x + a\right )}^{\frac {13}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.08, size = 113, normalized size = 0.98 \[ -\frac {\left (-\left (i x -1\right ) \left (i x +1\right ) a^{2}\right )^{\frac {1}{4}} x \hypergeom \left (\left [\frac {1}{4}, \frac {1}{2}\right ], \left [\frac {3}{2}\right ], -x^{2}\right )}{3 \left (a^{2}\right )^{\frac {1}{4}} \left (-\left (i x -1\right ) a \right )^{\frac {1}{4}} \left (\left (i x +1\right ) a \right )^{\frac {1}{4}} a^{4}}+\frac {\frac {2}{3} x^{3}+\frac {4}{3} i x^{2}-\frac {4}{9} x +\frac {4}{9} i}{\left (x +i\right )^{2} \left (-\left (i x -1\right ) a \right )^{\frac {1}{4}} \left (\left (i x +1\right ) a \right )^{\frac {1}{4}} a^{4}} \]
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 \[ \int \frac {1}{{\left (a-a\,x\,1{}\mathrm {i}\right )}^{13/4}\,{\left (a+a\,x\,1{}\mathrm {i}\right )}^{5/4}} \,d x \]
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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