Optimal. Leaf size=65 \[ \frac{4 i (a-i a x)^{3/4}}{3 a^3 (a+i a x)^{3/4}}-\frac{2 i}{a^2 \sqrt [4]{a-i a x} (a+i a x)^{3/4}} \]
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Rubi [A] time = 0.0515329, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08 \[ \frac{4 i (a-i a x)^{3/4}}{3 a^3 (a+i a x)^{3/4}}-\frac{2 i}{a^2 \sqrt [4]{a-i a x} (a+i a x)^{3/4}} \]
Antiderivative was successfully verified.
[In] Int[1/((a - I*a*x)^(5/4)*(a + I*a*x)^(7/4)),x]
[Out]
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Rubi in Sympy [A] time = 10.9833, size = 56, normalized size = 0.86 \[ \frac{2 i}{3 a^{2} \sqrt [4]{- i a x + a} \left (i a x + a\right )^{\frac{3}{4}}} - \frac{4 i \sqrt [4]{i a x + a}}{3 a^{3} \sqrt [4]{- i a x + a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a-I*a*x)**(5/4)/(a+I*a*x)**(7/4),x)
[Out]
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Mathematica [A] time = 0.0436556, size = 47, normalized size = 0.72 \[ -\frac{2 i (2 x-i) \sqrt [4]{a+i a x}}{3 a^3 (x-i) \sqrt [4]{a-i a x}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a - I*a*x)^(5/4)*(a + I*a*x)^(7/4)),x]
[Out]
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Maple [A] time = 0.065, size = 33, normalized size = 0.5 \[{\frac{4\,x-2\,i}{3\,{a}^{2}} \left ( a \left ( 1+ix \right ) \right ) ^{-{\frac{3}{4}}}{\frac{1}{\sqrt [4]{-a \left ( -1+ix \right ) }}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a-I*a*x)^(5/4)/(a+I*a*x)^(7/4),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (i \, a x + a\right )}^{\frac{7}{4}}{\left (-i \, a x + a\right )}^{\frac{5}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((I*a*x + a)^(7/4)*(-I*a*x + a)^(5/4)),x, algorithm="maxima")
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Fricas [A] time = 0.21455, size = 35, normalized size = 0.54 \[ \frac{4 \, x - 2 i}{3 \,{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((I*a*x + a)^(7/4)*(-I*a*x + a)^(5/4)),x, algorithm="fricas")
[Out]
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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)**(5/4)/(a+I*a*x)**(7/4),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)^(7/4)*(-I*a*x + a)^(5/4)),x, algorithm="giac")
[Out]