Optimal. Leaf size=67 \[ \frac{4 i \sqrt [4]{a-i a x}}{5 a^3 \sqrt [4]{a+i a x}}+\frac{2 i \sqrt [4]{a-i a x}}{5 a^2 (a+i a x)^{5/4}} \]
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Rubi [A] time = 0.0511653, antiderivative size = 67, 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 \[ \frac{4 i \sqrt [4]{a-i a x}}{5 a^3 \sqrt [4]{a+i a x}}+\frac{2 i \sqrt [4]{a-i a x}}{5 a^2 (a+i a x)^{5/4}} \]
Antiderivative was successfully verified.
[In] Int[1/((a - I*a*x)^(3/4)*(a + I*a*x)^(9/4)),x]
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Rubi in Sympy [A] time = 11.6826, size = 56, normalized size = 0.84 \[ \frac{2 i \sqrt [4]{- i a x + a}}{5 a^{2} \left (i a x + a\right )^{\frac{5}{4}}} + \frac{4 i \sqrt [4]{- i a x + a}}{5 a^{3} \sqrt [4]{i a x + a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a-I*a*x)**(3/4)/(a+I*a*x)**(9/4),x)
[Out]
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Mathematica [A] time = 0.0377538, size = 45, normalized size = 0.67 \[ \frac{2 (2 x-3 i) \sqrt [4]{a-i a x} (a+i a x)^{3/4}}{5 a^4 (x-i)^2} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a - I*a*x)^(3/4)*(a + I*a*x)^(9/4)),x]
[Out]
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Maple [A] time = 0.061, size = 44, normalized size = 0.7 \[{\frac{4\,{x}^{2}+6-2\,ix}{5\,{a}^{2} \left ( x-i \right ) } \left ( -a \left ( -1+ix \right ) \right ) ^{-{\frac{3}{4}}}{\frac{1}{\sqrt [4]{a \left ( 1+ix \right ) }}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a-I*a*x)^(3/4)/(a+I*a*x)^(9/4),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (i \, a x + a\right )}^{\frac{9}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((I*a*x + a)^(9/4)*(-I*a*x + a)^(3/4)),x, algorithm="maxima")
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Fricas [A] time = 0.237875, size = 59, normalized size = 0.88 \[ \frac{{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}{\left (4 \, x - 6 i\right )}}{5 \, a^{4} x^{2} - 10 i \, a^{4} x - 5 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((I*a*x + a)^(9/4)*(-I*a*x + a)^(3/4)),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)**(3/4)/(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)^(3/4)),x, algorithm="giac")
[Out]