Optimal. Leaf size=137 \[ \frac{14 a \sqrt [4]{x^2+1} E\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{\sqrt [4]{a+i a x} \sqrt [4]{a-i a x}}+\frac{4 i (a-i a x)^{7/4}}{a \sqrt [4]{a+i a x}}+\frac{14 i (a+i a x)^{3/4} (a-i a x)^{3/4}}{3 a}-\frac{14 a x}{\sqrt [4]{a+i a x} \sqrt [4]{a-i a x}} \]
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Rubi [A] time = 0.0320576, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {47, 50, 42, 229, 227, 196} \[ \frac{14 a \sqrt [4]{x^2+1} E\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{\sqrt [4]{a+i a x} \sqrt [4]{a-i a x}}+\frac{4 i (a-i a x)^{7/4}}{a \sqrt [4]{a+i a x}}+\frac{14 i (a+i a x)^{3/4} (a-i a x)^{3/4}}{3 a}-\frac{14 a x}{\sqrt [4]{a+i a x} \sqrt [4]{a-i a x}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 42
Rule 229
Rule 227
Rule 196
Rubi steps
\begin{align*} \int \frac{(a-i a x)^{7/4}}{(a+i a x)^{5/4}} \, dx &=\frac{4 i (a-i a x)^{7/4}}{a \sqrt [4]{a+i a x}}-7 \int \frac{(a-i a x)^{3/4}}{\sqrt [4]{a+i a x}} \, dx\\ &=\frac{4 i (a-i a x)^{7/4}}{a \sqrt [4]{a+i a x}}+\frac{14 i (a-i a x)^{3/4} (a+i a x)^{3/4}}{3 a}-(7 a) \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}}{a \sqrt [4]{a+i a x}}+\frac{14 i (a-i a x)^{3/4} (a+i a x)^{3/4}}{3 a}-\frac{\left (7 a \sqrt [4]{a^2+a^2 x^2}\right ) \int \frac{1}{\sqrt [4]{a^2+a^2 x^2}} \, dx}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ &=\frac{4 i (a-i a x)^{7/4}}{a \sqrt [4]{a+i a x}}+\frac{14 i (a-i a x)^{3/4} (a+i a x)^{3/4}}{3 a}-\frac{\left (7 a \sqrt [4]{1+x^2}\right ) \int \frac{1}{\sqrt [4]{1+x^2}} \, dx}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ &=-\frac{14 a x}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}+\frac{4 i (a-i a x)^{7/4}}{a \sqrt [4]{a+i a x}}+\frac{14 i (a-i a x)^{3/4} (a+i a x)^{3/4}}{3 a}+\frac{\left (7 a \sqrt [4]{1+x^2}\right ) \int \frac{1}{\left (1+x^2\right )^{5/4}} \, dx}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ &=-\frac{14 a x}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}+\frac{4 i (a-i a x)^{7/4}}{a \sqrt [4]{a+i a x}}+\frac{14 i (a-i a x)^{3/4} (a+i a x)^{3/4}}{3 a}+\frac{14 a \sqrt [4]{1+x^2} E\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ \end{align*}
Mathematica [C] time = 0.0283203, size = 70, normalized size = 0.51 \[ \frac{i 2^{3/4} \sqrt [4]{1+i x} (a-i a x)^{11/4} \, _2F_1\left (\frac{5}{4},\frac{11}{4};\frac{15}{4};\frac{1}{2}-\frac{i x}{2}\right )}{11 a^2 \sqrt [4]{a+i a x}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.051, size = 96, normalized size = 0.7 \begin{align*}{{\frac{2\,i}{3}} \left ({x}^{2}+13-12\,ix \right ) a{\frac{1}{\sqrt [4]{-a \left ( -1+ix \right ) }}}{\frac{1}{\sqrt [4]{a \left ( 1+ix \right ) }}}}-7\,{\frac{x{\mbox{$_2$F$_1$}(1/4,1/2;\,3/2;\,-{x}^{2})}a\sqrt [4]{-{a}^{2} \left ( -1+ix \right ) \left ( 1+ix \right ) }}{\sqrt [4]{{a}^{2}}\sqrt [4]{-a \left ( -1+ix \right ) }\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{5}{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 i \, x^{2} - 16 \, x + 42 i\right )} + 3 \,{\left (a x^{2} - i \, a x\right )}{\rm integral}\left (-\frac{14 \,{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}}{a x^{4} + a x^{2}}, x\right )}{3 \,{\left (a x^{2} - i \, a 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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