Optimal. Leaf size=82 \[ \frac{2 \sqrt [4]{x^2+1} E\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{5 a^2 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}+\frac{4 i}{5 a \sqrt [4]{a-i a x} (a+i a x)^{5/4}} \]
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Rubi [A] time = 0.0621999, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16 \[ \frac{2 \sqrt [4]{x^2+1} E\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{5 a^2 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}+\frac{4 i}{5 a \sqrt [4]{a-i a x} (a+i a x)^{5/4}} \]
Antiderivative was successfully verified.
[In] Int[1/((a - I*a*x)^(1/4)*(a + I*a*x)^(9/4)),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{4 i}{5 a \sqrt [4]{- i a x + a} \left (i a x + a\right )^{\frac{5}{4}}} - \frac{\left (- i a x + a\right )^{\frac{3}{4}} \left (i a x + a\right )^{\frac{3}{4}} \int \frac{1}{\sqrt [4]{a^{2} x^{2} + a^{2}}}\, dx}{5 a^{2} \left (a^{2} x^{2} + a^{2}\right )^{\frac{3}{4}}} + \frac{2 x \left (- i a x + a\right )^{\frac{3}{4}} \left (i a x + a\right )^{\frac{3}{4}}}{5 a^{4} \left (x^{2} + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a-I*a*x)**(1/4)/(a+I*a*x)**(9/4),x)
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Mathematica [C] time = 0.0821035, size = 84, normalized size = 1.02 \[ \frac{2 (a-i a x)^{3/4} \left (-2^{3/4} (1+i x)^{5/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};\frac{1}{2}-\frac{i x}{2}\right )+3 i x+6\right )}{15 a^3 (x-i) \sqrt [4]{a+i a x}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a - I*a*x)^(1/4)*(a + I*a*x)^(9/4)),x]
[Out]
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Maple [C] time = 0.059, size = 105, normalized size = 1.3 \[{\frac{2\,{x}^{2}+4-2\,ix}{ \left ( 5\,x-5\,i \right ){a}^{2}}{\frac{1}{\sqrt [4]{-a \left ( -1+ix \right ) }}}{\frac{1}{\sqrt [4]{a \left ( 1+ix \right ) }}}}-{\frac{x}{5\,{a}^{2}}{\mbox{$_2$F$_1$}({\frac{1}{4}},{\frac{1}{2}};\,{\frac{3}{2}};\,-{x}^{2})}\sqrt [4]{-{a}^{2} \left ( -1+ix \right ) \left ( 1+ix \right ) }{\frac{1}{\sqrt [4]{{a}^{2}}}}{\frac{1}{\sqrt [4]{-a \left ( -1+ix \right ) }}}{\frac{1}{\sqrt [4]{a \left ( 1+ix \right ) }}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a-I*a*x)^(1/4)/(a+I*a*x)^(9/4),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (i \, a x + a\right )}^{\frac{9}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((I*a*x + a)^(9/4)*(-I*a*x + a)^(1/4)),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[ \frac{{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}{\left (2 \, x - 4 i\right )} +{\left (5 \, a^{4} x^{2} - 10 i \, a^{4} x - 5 \, a^{4}\right )}{\rm integral}\left (-\frac{{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}}{5 \,{\left (a^{4} x^{2} + a^{4}\right )}}, x\right )}{5 \, a^{4} x^{2} - 10 i \, a^{4} x - 5 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((I*a*x + a)^(9/4)*(-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(1/(a-I*a*x)**(1/4)/(a+I*a*x)**(9/4),x)
[Out]
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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)^(9/4)*(-I*a*x + a)^(1/4)),x, algorithm="giac")
[Out]