Optimal. Leaf size=121 \[ \frac{14 x}{45 a^5 \left (x^2+1\right ) \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}+\frac{14 \sqrt [4]{x^2+1} E\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{15 a^5 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac{2 i}{9 a^2 (a-i a x)^{9/4} (a+i a x)^{5/4}} \]
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Rubi [A] time = 0.0257717, antiderivative size = 121, 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, 42, 199, 197, 196} \[ \frac{14 x}{45 a^5 \left (x^2+1\right ) \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}+\frac{14 \sqrt [4]{x^2+1} E\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{15 a^5 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac{2 i}{9 a^2 (a-i a x)^{9/4} (a+i a x)^{5/4}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 42
Rule 199
Rule 197
Rule 196
Rubi steps
\begin{align*} \int \frac{1}{(a-i a x)^{13/4} (a+i a x)^{9/4}} \, dx &=-\frac{2 i}{9 a^2 (a-i a x)^{9/4} (a+i a x)^{5/4}}+\frac{7 \int \frac{1}{(a-i a x)^{9/4} (a+i a x)^{9/4}} \, dx}{9 a}\\ &=-\frac{2 i}{9 a^2 (a-i a x)^{9/4} (a+i a x)^{5/4}}+\frac{\left (7 \sqrt [4]{a^2+a^2 x^2}\right ) \int \frac{1}{\left (a^2+a^2 x^2\right )^{9/4}} \, dx}{9 a \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ &=-\frac{2 i}{9 a^2 (a-i a x)^{9/4} (a+i a x)^{5/4}}+\frac{14 x}{45 a^5 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x} \left (1+x^2\right )}+\frac{\left (7 \sqrt [4]{a^2+a^2 x^2}\right ) \int \frac{1}{\left (a^2+a^2 x^2\right )^{5/4}} \, dx}{15 a^3 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ &=-\frac{2 i}{9 a^2 (a-i a x)^{9/4} (a+i a x)^{5/4}}+\frac{14 x}{45 a^5 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x} \left (1+x^2\right )}+\frac{\left (7 \sqrt [4]{1+x^2}\right ) \int \frac{1}{\left (1+x^2\right )^{5/4}} \, dx}{15 a^5 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ &=-\frac{2 i}{9 a^2 (a-i a x)^{9/4} (a+i a x)^{5/4}}+\frac{14 x}{45 a^5 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x} \left (1+x^2\right )}+\frac{14 \sqrt [4]{1+x^2} E\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{15 a^5 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ \end{align*}
Mathematica [C] time = 0.0279646, size = 70, normalized size = 0.58 \[ -\frac{i \sqrt [4]{1+i x} \, _2F_1\left (-\frac{9}{4},\frac{9}{4};-\frac{5}{4};\frac{1}{2}-\frac{i x}{2}\right )}{9 \sqrt [4]{2} a^3 (a-i a x)^{9/4} \sqrt [4]{a+i a x}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.077, size = 124, normalized size = 1. \begin{align*}{\frac{42\,i{x}^{3}+42\,{x}^{4}+56\,ix+56\,{x}^{2}+10}{ \left ( 45\,x-45\,i \right ) \left ( x+i \right ) ^{2}{a}^{5}}{\frac{1}{\sqrt [4]{-a \left ( -1+ix \right ) }}}{\frac{1}{\sqrt [4]{a \left ( 1+ix \right ) }}}}-{\frac{7\,x}{15\,{a}^{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(-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 (42 \, x^{4} + 42 i \, x^{3} + 56 \, x^{2} + 56 i \, x + 10\right )}{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}} +{\left (45 \, a^{7} x^{5} + 45 i \, a^{7} x^{4} + 90 \, a^{7} x^{3} + 90 i \, a^{7} x^{2} + 45 \, a^{7} x + 45 i \, a^{7}\right )}{\rm integral}\left (-\frac{7 \,{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}}{15 \,{\left (a^{7} x^{2} + a^{7}\right )}}, x\right )}{45 \, a^{7} x^{5} + 45 i \, a^{7} x^{4} + 90 \, a^{7} x^{3} + 90 i \, a^{7} x^{2} + 45 \, a^{7} x + 45 i \, a^{7}} \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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