Optimal. Leaf size=141 \[ -\frac{42 \sqrt [4]{x^2+1} E\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{5 \sqrt [4]{a+i a x} \sqrt [4]{a-i a x}}+\frac{4 i (a-i a x)^{7/4}}{5 a (a+i a x)^{5/4}}-\frac{28 i (a-i a x)^{3/4}}{5 a \sqrt [4]{a+i a x}}+\frac{42 x}{5 \sqrt [4]{a+i a x} \sqrt [4]{a-i a x}} \]
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Rubi [A] time = 0.0303233, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {47, 42, 229, 227, 196} \[ -\frac{42 \sqrt [4]{x^2+1} E\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{5 \sqrt [4]{a+i a x} \sqrt [4]{a-i a x}}+\frac{4 i (a-i a x)^{7/4}}{5 a (a+i a x)^{5/4}}-\frac{28 i (a-i a x)^{3/4}}{5 a \sqrt [4]{a+i a x}}+\frac{42 x}{5 \sqrt [4]{a+i a x} \sqrt [4]{a-i a x}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 42
Rule 229
Rule 227
Rule 196
Rubi steps
\begin{align*} \int \frac{(a-i a x)^{7/4}}{(a+i a x)^{9/4}} \, dx &=\frac{4 i (a-i a x)^{7/4}}{5 a (a+i a x)^{5/4}}-\frac{7}{5} \int \frac{(a-i a x)^{3/4}}{(a+i a x)^{5/4}} \, dx\\ &=\frac{4 i (a-i a x)^{7/4}}{5 a (a+i a x)^{5/4}}-\frac{28 i (a-i a x)^{3/4}}{5 a \sqrt [4]{a+i a x}}+\frac{21}{5} \int \frac{1}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \, dx\\ &=\frac{4 i (a-i a x)^{7/4}}{5 a (a+i a x)^{5/4}}-\frac{28 i (a-i a x)^{3/4}}{5 a \sqrt [4]{a+i a x}}+\frac{\left (21 \sqrt [4]{a^2+a^2 x^2}\right ) \int \frac{1}{\sqrt [4]{a^2+a^2 x^2}} \, dx}{5 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ &=\frac{4 i (a-i a x)^{7/4}}{5 a (a+i a x)^{5/4}}-\frac{28 i (a-i a x)^{3/4}}{5 a \sqrt [4]{a+i a x}}+\frac{\left (21 \sqrt [4]{1+x^2}\right ) \int \frac{1}{\sqrt [4]{1+x^2}} \, dx}{5 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ &=\frac{4 i (a-i a x)^{7/4}}{5 a (a+i a x)^{5/4}}+\frac{42 x}{5 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac{28 i (a-i a x)^{3/4}}{5 a \sqrt [4]{a+i a x}}-\frac{\left (21 \sqrt [4]{1+x^2}\right ) \int \frac{1}{\left (1+x^2\right )^{5/4}} \, dx}{5 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ &=\frac{4 i (a-i a x)^{7/4}}{5 a (a+i a x)^{5/4}}+\frac{42 x}{5 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac{28 i (a-i a x)^{3/4}}{5 a \sqrt [4]{a+i a x}}-\frac{42 \sqrt [4]{1+x^2} E\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{5 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ \end{align*}
Mathematica [C] time = 0.0378105, size = 70, normalized size = 0.5 \[ \frac{i \sqrt [4]{1+i x} (a-i a x)^{11/4} \, _2F_1\left (\frac{9}{4},\frac{11}{4};\frac{15}{4};\frac{1}{2}-\frac{i x}{2}\right )}{11 \sqrt [4]{2} a^3 \sqrt [4]{a+i a x}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.05, size = 101, normalized size = 0.7 \begin{align*} -{\frac{32\,{x}^{2}+24+8\,ix}{5\,x-5\,i}{\frac{1}{\sqrt [4]{-a \left ( -1+ix \right ) }}}{\frac{1}{\sqrt [4]{a \left ( 1+ix \right ) }}}}+{\frac{21\,x}{5}{\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{{\left (-i \, a x + a\right )}^{\frac{7}{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{2 \,{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}{\left (5 \, x^{2} - 30 i \, x - 21\right )} + 5 \,{\left (a^{2} x^{3} - 2 i \, a^{2} x^{2} - a^{2} x\right )}{\rm integral}\left (\frac{42 \,{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}}{5 \,{\left (a^{2} x^{4} + a^{2} x^{2}\right )}}, x\right )}{5 \,{\left (a^{2} x^{3} - 2 i \, a^{2} x^{2} - a^{2} x\right )}} \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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