Optimal. Leaf size=31 \[ -\frac{2 i \sqrt [4]{a+i a x}}{a^2 \sqrt [4]{a-i a x}} \]
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Rubi [A] time = 0.0233917, antiderivative size = 31, 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 \[ -\frac{2 i \sqrt [4]{a+i a x}}{a^2 \sqrt [4]{a-i a x}} \]
Antiderivative was successfully verified.
[In] Int[1/((a - I*a*x)^(5/4)*(a + I*a*x)^(3/4)),x]
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Rubi in Sympy [A] time = 5.39696, size = 27, normalized size = 0.87 \[ - \frac{2 i \sqrt [4]{i a x + a}}{a^{2} \sqrt [4]{- i a x + a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a-I*a*x)**(5/4)/(a+I*a*x)**(3/4),x)
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Mathematica [A] time = 0.0304198, size = 31, normalized size = 1. \[ -\frac{2 i \sqrt [4]{a+i a x}}{a^2 \sqrt [4]{a-i a x}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a - I*a*x)^(5/4)*(a + I*a*x)^(3/4)),x]
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Maple [A] time = 0.053, size = 31, normalized size = 1. \[ 2\,{\frac{x-i}{a \left ( a \left ( 1+ix \right ) \right ) ^{3/4}\sqrt [4]{-a \left ( -1+ix \right ) }}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a-I*a*x)^(5/4)/(a+I*a*x)^(3/4),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{5}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((I*a*x + a)^(3/4)*(-I*a*x + a)^(5/4)),x, algorithm="maxima")
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Fricas [A] time = 0.205152, size = 42, normalized size = 1.35 \[ \frac{2 \,{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}}{a^{3} x + i \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((I*a*x + a)^(3/4)*(-I*a*x + a)^(5/4)),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a \left (i x + 1\right )\right )^{\frac{3}{4}} \left (- a \left (i x - 1\right )\right )^{\frac{5}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(a-I*a*x)**(5/4)/(a+I*a*x)**(3/4),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)^(3/4)*(-I*a*x + a)^(5/4)),x, algorithm="giac")
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