Optimal. Leaf size=133 \[ \frac{32 i \sqrt [4]{a-i a x}}{35 a^5 \sqrt [4]{a+i a x}}+\frac{16 i \sqrt [4]{a-i a x}}{35 a^4 (a+i a x)^{5/4}}-\frac{4 i}{7 a^3 (a+i a x)^{5/4} (a-i a x)^{3/4}}-\frac{2 i}{7 a^2 (a+i a x)^{5/4} (a-i a x)^{7/4}} \]
[Out]
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Rubi [A] time = 0.114492, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08 \[ \frac{32 i \sqrt [4]{a-i a x}}{35 a^5 \sqrt [4]{a+i a x}}+\frac{16 i \sqrt [4]{a-i a x}}{35 a^4 (a+i a x)^{5/4}}-\frac{4 i}{7 a^3 (a+i a x)^{5/4} (a-i a x)^{3/4}}-\frac{2 i}{7 a^2 (a+i a x)^{5/4} (a-i a x)^{7/4}} \]
Antiderivative was successfully verified.
[In] Int[1/((a - I*a*x)^(11/4)*(a + I*a*x)^(9/4)),x]
[Out]
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Rubi in Sympy [A] time = 23.8811, size = 114, normalized size = 0.86 \[ \frac{2 i}{5 a^{2} \left (- i a x + a\right )^{\frac{7}{4}} \left (i a x + a\right )^{\frac{5}{4}}} + \frac{12 i}{5 a^{3} \left (- i a x + a\right )^{\frac{7}{4}} \sqrt [4]{i a x + a}} - \frac{48 i \left (i a x + a\right )^{\frac{3}{4}}}{35 a^{4} \left (- i a x + a\right )^{\frac{7}{4}}} - \frac{32 i \left (i a x + a\right )^{\frac{3}{4}}}{35 a^{5} \left (- i a x + a\right )^{\frac{3}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a-I*a*x)**(11/4)/(a+I*a*x)**(9/4),x)
[Out]
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Mathematica [A] time = 0.062245, size = 64, normalized size = 0.48 \[ \frac{2 \left (-16 i x^3+8 x^2-22 i x+9\right ) (a+i a x)^{3/4}}{35 a^5 (x-i)^2 (x+i) (a-i a x)^{3/4}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a - I*a*x)^(11/4)*(a + I*a*x)^(9/4)),x]
[Out]
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Maple [A] time = 0.08, size = 56, normalized size = 0.4 \[{\frac{32\,{x}^{3}+16\,i{x}^{2}+44\,x+18\,i}{35\,{a}^{4} \left ( x-i \right ) \left ( x+i \right ) } \left ( -a \left ( -1+ix \right ) \right ) ^{-{\frac{3}{4}}}{\frac{1}{\sqrt [4]{a \left ( 1+ix \right ) }}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a-I*a*x)^(11/4)/(a+I*a*x)^(9/4),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (i \, a x + a\right )}^{\frac{9}{4}}{\left (-i \, a x + a\right )}^{\frac{11}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((I*a*x + a)^(9/4)*(-I*a*x + a)^(11/4)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.243909, size = 62, normalized size = 0.47 \[ \frac{32 \, x^{3} + 16 i \, x^{2} + 44 \, x + 18 i}{35 \,{\left (a^{4} x^{2} + a^{4}\right )}{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((I*a*x + a)^(9/4)*(-I*a*x + a)^(11/4)),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)**(11/4)/(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)^(11/4)),x, algorithm="giac")
[Out]