Optimal. Leaf size=82 \[ \frac{6 \sqrt [4]{x^2+1} E\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{5 a^3 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}+\frac{2 i}{5 a^2 \sqrt [4]{a-i a x} (a+i a x)^{5/4}} \]
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Rubi [A] time = 0.0246731, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {51, 46, 42, 197, 196} \[ \frac{6 \sqrt [4]{x^2+1} E\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{5 a^3 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}+\frac{2 i}{5 a^2 \sqrt [4]{a-i a x} (a+i a x)^{5/4}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 46
Rule 42
Rule 197
Rule 196
Rubi steps
\begin{align*} \int \frac{1}{(a-i a x)^{5/4} (a+i a x)^{9/4}} \, dx &=-\frac{2 i}{a^2 \sqrt [4]{a-i a x} (a+i a x)^{5/4}}+\frac{3 \int \frac{1}{\sqrt [4]{a-i a x} (a+i a x)^{9/4}} \, dx}{a}\\ &=\frac{2 i}{5 a^2 \sqrt [4]{a-i a x} (a+i a x)^{5/4}}+\frac{3 \int \frac{1}{(a-i a x)^{5/4} (a+i a x)^{5/4}} \, dx}{5 a}\\ &=\frac{2 i}{5 a^2 \sqrt [4]{a-i a x} (a+i a x)^{5/4}}+\frac{\left (3 \sqrt [4]{a^2+a^2 x^2}\right ) \int \frac{1}{\left (a^2+a^2 x^2\right )^{5/4}} \, dx}{5 a \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ &=\frac{2 i}{5 a^2 \sqrt [4]{a-i a x} (a+i a x)^{5/4}}+\frac{\left (3 \sqrt [4]{1+x^2}\right ) \int \frac{1}{\left (1+x^2\right )^{5/4}} \, dx}{5 a^3 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ &=\frac{2 i}{5 a^2 \sqrt [4]{a-i a x} (a+i a x)^{5/4}}+\frac{6 \sqrt [4]{1+x^2} E\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{5 a^3 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ \end{align*}
Mathematica [C] time = 0.0256838, size = 68, normalized size = 0.83 \[ -\frac{i \sqrt [4]{1+i x} \, _2F_1\left (-\frac{1}{4},\frac{9}{4};\frac{3}{4};\frac{1}{2}-\frac{i x}{2}\right )}{\sqrt [4]{2} a^3 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.056, size = 107, normalized size = 1.3 \begin{align*}{\frac{-6\,ix+6\,{x}^{2}+2}{ \left ( 5\,x-5\,i \right ){a}^{3}}{\frac{1}{\sqrt [4]{-a \left ( -1+ix \right ) }}}{\frac{1}{\sqrt [4]{a \left ( 1+ix \right ) }}}}-{\frac{3\,x}{5\,{a}^{3}}{\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{2 \,{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}{\left (3 \, x^{2} - 3 i \, x + 1\right )} +{\left (5 \, a^{5} x^{3} - 5 i \, a^{5} x^{2} + 5 \, a^{5} x - 5 i \, a^{5}\right )}{\rm integral}\left (-\frac{3 \,{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}}{5 \,{\left (a^{5} x^{2} + a^{5}\right )}}, x\right )}{5 \, a^{5} x^{3} - 5 i \, a^{5} x^{2} + 5 \, a^{5} x - 5 i \, a^{5}} \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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