Optimal. Leaf size=82 \[ \frac{2 \sqrt [4]{x^2+1} E\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{5 a^2 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac{4 i}{5 a (a-i a x)^{5/4} \sqrt [4]{a+i a x}} \]
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Rubi [A] time = 0.0165606, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {46, 42, 197, 196} \[ \frac{2 \sqrt [4]{x^2+1} E\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{5 a^2 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac{4 i}{5 a (a-i a x)^{5/4} \sqrt [4]{a+i a x}} \]
Antiderivative was successfully verified.
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Rule 46
Rule 42
Rule 197
Rule 196
Rubi steps
\begin{align*} \int \frac{1}{(a-i a x)^{9/4} \sqrt [4]{a+i a x}} \, dx &=-\frac{4 i}{5 a (a-i a x)^{5/4} \sqrt [4]{a+i a x}}+\frac{1}{5} \int \frac{1}{(a-i a x)^{5/4} (a+i a x)^{5/4}} \, dx\\ &=-\frac{4 i}{5 a (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}{5 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ &=-\frac{4 i}{5 a (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}{5 a^2 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ &=-\frac{4 i}{5 a (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 )}{5 a^2 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ \end{align*}
Mathematica [C] time = 0.0258731, size = 70, normalized size = 0.85 \[ -\frac{2 i 2^{3/4} \sqrt [4]{1+i x} \, _2F_1\left (-\frac{5}{4},\frac{1}{4};-\frac{1}{4};\frac{1}{2}-\frac{i x}{2}\right )}{5 a (a-i a x)^{5/4} \sqrt [4]{a+i a x}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.053, size = 105, normalized size = 1.3 \begin{align*}{\frac{2\,{x}^{2}+4+2\,ix}{ \left ( 5\,x+5\,i \right ){a}^{2}}{\frac{1}{\sqrt [4]{-a \left ( -1+ix \right ) }}}{\frac{1}{\sqrt [4]{a \left ( 1+ix \right ) }}}}-{\frac{x}{5\,{a}^{2}}{\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{9}{4}}}\,{d x} \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 (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}{\left (2 \, x + 4 i\right )} +{\left (5 \, a^{4} x^{2} + 10 i \, a^{4} x - 5 \, a^{4}\right )}{\rm integral}\left (-\frac{{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}}{5 \,{\left (a^{4} x^{2} + a^{4}\right )}}, x\right )}{5 \, a^{4} x^{2} + 10 i \, a^{4} x - 5 \, a^{4}} \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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