Optimal. Leaf size=144 \[ -\frac{14 a^2 \sqrt [4]{x^2+1} E\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{5 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}+\frac{14 a^2 x}{5 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac{14}{15} i (a-i a x)^{3/4} (a+i a x)^{3/4}-\frac{2 i (a-i a x)^{7/4} (a+i a x)^{3/4}}{5 a} \]
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Rubi [A] time = 0.11211, antiderivative size = 144, 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 \[ -\frac{14 a^2 \sqrt [4]{x^2+1} E\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{5 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}+\frac{14 a^2 x}{5 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac{14}{15} i (a-i a x)^{3/4} (a+i a x)^{3/4}-\frac{2 i (a-i a x)^{7/4} (a+i a x)^{3/4}}{5 a} \]
Antiderivative was successfully verified.
[In] Int[(a - I*a*x)^(7/4)/(a + I*a*x)^(1/4),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{7 a^{2} \left (- i a x + a\right )^{\frac{3}{4}} \left (i a x + a\right )^{\frac{3}{4}} \int \frac{1}{\sqrt [4]{a^{2} x^{2} + a^{2}}}\, dx}{5 \left (a^{2} x^{2} + a^{2}\right )^{\frac{3}{4}}} - \frac{14 i \left (- i a x + a\right )^{\frac{3}{4}} \left (i a x + a\right )^{\frac{3}{4}}}{15} - \frac{2 i \left (- i a x + a\right )^{\frac{7}{4}} \left (i a x + a\right )^{\frac{3}{4}}}{5 a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a-I*a*x)**(7/4)/(a+I*a*x)**(1/4),x)
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Mathematica [C] time = 0.0816059, size = 84, normalized size = 0.58 \[ \frac{2 a (a-i a x)^{3/4} \left (7 i 2^{3/4} \sqrt [4]{1+i x} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};\frac{1}{2}-\frac{i x}{2}\right )-3 i x^2+7 x-10 i\right )}{15 \sqrt [4]{a+i a x}} \]
Antiderivative was successfully verified.
[In] Integrate[(a - I*a*x)^(7/4)/(a + I*a*x)^(1/4),x]
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Maple [C] time = 0.266, size = 104, normalized size = 0.7 \[ -{\frac{ \left ( 20\,i+6\,x \right ) \left ( x+i \right ) \left ( x-i \right ){a}^{2}}{15}{\frac{1}{\sqrt [4]{-a \left ( -1+ix \right ) }}}{\frac{1}{\sqrt [4]{a \left ( 1+ix \right ) }}}}+{\frac{7\,x{a}^{2}}{5}{\mbox{$_2$F$_1$}({\frac{1}{4}},{\frac{1}{2}};\,{\frac{3}{2}};\,-{x}^{2})}\sqrt [4]{-{a}^{2} \left ( -1+ix \right ) \left ( 1+ix \right ) }{\frac{1}{\sqrt [4]{{a}^{2}}}}{\frac{1}{\sqrt [4]{-a \left ( -1+ix \right ) }}}{\frac{1}{\sqrt [4]{a \left ( 1+ix \right ) }}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a-I*a*x)^(7/4)/(a+I*a*x)^(1/4),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-i \, a x + a\right )}^{\frac{7}{4}}}{{\left (i \, a x + a\right )}^{\frac{1}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-I*a*x + a)^(7/4)/(I*a*x + a)^(1/4),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[ -\frac{2 \,{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}{\left (3 \, x^{2} + 10 i \, x - 21\right )} - 15 \, x{\rm integral}\left (\frac{14 \,{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}}{5 \,{\left (x^{4} + x^{2}\right )}}, x\right )}{15 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-I*a*x + a)^(7/4)/(I*a*x + a)^(1/4),x, algorithm="fricas")
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a-I*a*x)**(7/4)/(a+I*a*x)**(1/4),x)
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-I*a*x + a)^(7/4)/(I*a*x + a)^(1/4),x, algorithm="giac")
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