Optimal. Leaf size=148 \[ \frac{2 \sqrt [4]{x^2+1} E\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{39 a^4 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac{10 i (a+i a x)^{3/4}}{117 a^3 (a-i a x)^{9/4}}-\frac{4 i}{39 a^3 (a-i a x)^{5/4} \sqrt [4]{a+i a x}}-\frac{2 i (a+i a x)^{3/4}}{13 a^2 (a-i a x)^{13/4}} \]
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Rubi [A] time = 0.129395, antiderivative size = 148, 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 \[ \frac{2 \sqrt [4]{x^2+1} E\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{39 a^4 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac{10 i (a+i a x)^{3/4}}{117 a^3 (a-i a x)^{9/4}}-\frac{4 i}{39 a^3 (a-i a x)^{5/4} \sqrt [4]{a+i a x}}-\frac{2 i (a+i a x)^{3/4}}{13 a^2 (a-i a x)^{13/4}} \]
Antiderivative was successfully verified.
[In] Int[1/((a - I*a*x)^(17/4)*(a + I*a*x)^(1/4)),x]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{2 i \left (i a x + a\right )^{\frac{3}{4}}}{13 a^{2} \left (- i a x + a\right )^{\frac{13}{4}}} - \frac{4 i}{39 a^{3} \left (- i a x + a\right )^{\frac{5}{4}} \sqrt [4]{i a x + a}} - \frac{10 i \left (i a x + a\right )^{\frac{3}{4}}}{117 a^{3} \left (- i a x + a\right )^{\frac{9}{4}}} - \frac{\left (- i a x + a\right )^{\frac{3}{4}} \left (i a x + a\right )^{\frac{3}{4}} \int \frac{1}{\sqrt [4]{a^{2} x^{2} + a^{2}}}\, dx}{39 a^{4} \left (a^{2} x^{2} + a^{2}\right )^{\frac{3}{4}}} + \frac{2 x \left (- i a x + a\right )^{\frac{3}{4}} \left (i a x + a\right )^{\frac{3}{4}}}{39 a^{6} \left (x^{2} + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a-I*a*x)**(17/4)/(a+I*a*x)**(1/4),x)
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Mathematica [C] time = 0.138672, size = 102, normalized size = 0.69 \[ -\frac{2 \left (2^{3/4} \sqrt [4]{1+i x} (x+i)^4 \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};\frac{1}{2}-\frac{i x}{2}\right )-3 x^4-9 i x^3+8 x^2+20\right )}{117 a^4 (x+i)^3 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a - I*a*x)^(17/4)*(a + I*a*x)^(1/4)),x]
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Maple [C] time = 0.1, size = 114, normalized size = 0.8 \[{\frac{18\,i{x}^{3}+6\,{x}^{4}-40-16\,{x}^{2}}{117\, \left ( x+i \right ) ^{3}{a}^{4}}{\frac{1}{\sqrt [4]{-a \left ( -1+ix \right ) }}}{\frac{1}{\sqrt [4]{a \left ( 1+ix \right ) }}}}-{\frac{x}{39\,{a}^{4}}{\mbox{$_2$F$_1$}({\frac{1}{4}},{\frac{1}{2}};\,{\frac{3}{2}};\,-{x}^{2})}\sqrt [4]{-{a}^{2} \left ( -1+ix \right ) \left ( 1+ix \right ) }{\frac{1}{\sqrt [4]{{a}^{2}}}}{\frac{1}{\sqrt [4]{-a \left ( -1+ix \right ) }}}{\frac{1}{\sqrt [4]{a \left ( 1+ix \right ) }}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a-I*a*x)^(17/4)/(a+I*a*x)^(1/4),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{17}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((I*a*x + a)^(1/4)*(-I*a*x + a)^(17/4)),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[ \frac{{\left (6 \, x^{3} + 24 i \, x^{2} - 40 \, x - 40 i\right )}{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}} +{\left (117 \, a^{6} x^{4} + 468 i \, a^{6} x^{3} - 702 \, a^{6} x^{2} - 468 i \, a^{6} x + 117 \, a^{6}\right )}{\rm integral}\left (-\frac{{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}}{39 \,{\left (a^{6} x^{2} + a^{6}\right )}}, x\right )}{117 \, a^{6} x^{4} + 468 i \, a^{6} x^{3} - 702 \, a^{6} x^{2} - 468 i \, a^{6} x + 117 \, a^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((I*a*x + a)^(1/4)*(-I*a*x + a)^(17/4)),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(1/(a-I*a*x)**(17/4)/(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(1/((I*a*x + a)^(1/4)*(-I*a*x + a)^(17/4)),x, algorithm="giac")
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