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 = 0.0264505, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, 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 51
Rule 42
Rule 197
Rule 196
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*}
Mathematica [C] time = 0.0259067, 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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Maple [C] time = 0.066, size = 113, normalized size = 1. \begin{align*}{\frac{12\,i{x}^{2}+6\,{x}^{3}-4\,x+4\,i}{9\, \left ( x+i \right ) ^{2}{a}^{4}}{\frac{1}{\sqrt [4]{-a \left ( -1+ix \right ) }}}{\frac{1}{\sqrt [4]{a \left ( 1+ix \right ) }}}}-{\frac{x}{3\,{a}^{4}}{\mbox{$_2$F$_1$}({\frac{1}{4}},{\frac{1}{2}};\,{\frac{3}{2}};\,-{x}^{2})}\sqrt [4]{-{a}^{2} \left ( -1+ix \right ) \left ( 1+ix \right ) }{\frac{1}{\sqrt [4]{{a}^{2}}}}{\frac{1}{\sqrt [4]{-a \left ( -1+ix \right ) }}}{\frac{1}{\sqrt [4]{a \left ( 1+ix \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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