Optimal. Leaf size=33 \[ -\frac{2 i (a+i a x)^{3/4}}{3 a^2 (a-i a x)^{3/4}} \]
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Rubi [A] time = 0.0230548, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04 \[ -\frac{2 i (a+i a x)^{3/4}}{3 a^2 (a-i a x)^{3/4}} \]
Antiderivative was successfully verified.
[In] Int[1/((a - I*a*x)^(7/4)*(a + I*a*x)^(1/4)),x]
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Rubi in Sympy [A] time = 5.7752, size = 29, normalized size = 0.88 \[ - \frac{2 i \left (i a x + a\right )^{\frac{3}{4}}}{3 a^{2} \left (- i a x + a\right )^{\frac{3}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a-I*a*x)**(7/4)/(a+I*a*x)**(1/4),x)
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Mathematica [A] time = 0.0331662, size = 33, normalized size = 1. \[ -\frac{2 i (a+i a x)^{3/4}}{3 a^2 (a-i a x)^{3/4}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a - I*a*x)^(7/4)*(a + I*a*x)^(1/4)),x]
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Maple [A] time = 0.074, size = 31, normalized size = 0.9 \[{\frac{2\,x-2\,i}{3\,a} \left ( -a \left ( -1+ix \right ) \right ) ^{-{\frac{3}{4}}}{\frac{1}{\sqrt [4]{a \left ( 1+ix \right ) }}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a-I*a*x)^(7/4)/(a+I*a*x)^(1/4),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{7}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((I*a*x + a)^(1/4)*(-I*a*x + a)^(7/4)),x, algorithm="maxima")
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Fricas [A] time = 0.205049, size = 35, normalized size = 1.06 \[ \frac{2 \, x - 2 i}{3 \,{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}} a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((I*a*x + a)^(1/4)*(-I*a*x + a)^(7/4)),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)**(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(1/((I*a*x + a)^(1/4)*(-I*a*x + a)^(7/4)),x, algorithm="giac")
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