Optimal. Leaf size=100 \[ -\frac{16 i (a+i a x)^{3/4}}{231 a^4 (a-i a x)^{3/4}}-\frac{8 i (a+i a x)^{3/4}}{77 a^3 (a-i a x)^{7/4}}-\frac{2 i (a+i a x)^{3/4}}{11 a^2 (a-i a x)^{11/4}} \]
[Out]
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Rubi [A] time = 0.0814795, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08 \[ -\frac{16 i (a+i a x)^{3/4}}{231 a^4 (a-i a x)^{3/4}}-\frac{8 i (a+i a x)^{3/4}}{77 a^3 (a-i a x)^{7/4}}-\frac{2 i (a+i a x)^{3/4}}{11 a^2 (a-i a x)^{11/4}} \]
Antiderivative was successfully verified.
[In] Int[1/((a - I*a*x)^(15/4)*(a + I*a*x)^(1/4)),x]
[Out]
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Rubi in Sympy [A] time = 18.3467, size = 87, normalized size = 0.87 \[ - \frac{2 i \left (i a x + a\right )^{\frac{3}{4}}}{11 a^{2} \left (- i a x + a\right )^{\frac{11}{4}}} - \frac{8 i \left (i a x + a\right )^{\frac{3}{4}}}{77 a^{3} \left (- i a x + a\right )^{\frac{7}{4}}} - \frac{16 i \left (i a x + a\right )^{\frac{3}{4}}}{231 a^{4} \left (- i a x + a\right )^{\frac{3}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a-I*a*x)**(15/4)/(a+I*a*x)**(1/4),x)
[Out]
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Mathematica [A] time = 0.0524535, size = 52, normalized size = 0.52 \[ \frac{2 \left (-8 i x^2+28 x+41 i\right ) (a+i a x)^{3/4}}{231 a^4 (x+i)^2 (a-i a x)^{3/4}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a - I*a*x)^(15/4)*(a + I*a*x)^(1/4)),x]
[Out]
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Maple [A] time = 0.068, size = 50, normalized size = 0.5 \[{\frac{40\,i{x}^{2}+16\,{x}^{3}-26\,x+82\,i}{231\,{a}^{3} \left ( x+i \right ) ^{2}} \left ( -a \left ( -1+ix \right ) \right ) ^{-{\frac{3}{4}}}{\frac{1}{\sqrt [4]{a \left ( 1+ix \right ) }}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a-I*a*x)^(15/4)/(a+I*a*x)^(1/4),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{15}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((I*a*x + a)^(1/4)*(-I*a*x + a)^(15/4)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.205212, size = 73, normalized size = 0.73 \[ \frac{16 \, x^{3} + 40 i \, x^{2} - 26 \, x + 82 i}{{\left (231 \, a^{3} x^{2} + 462 i \, a^{3} x - 231 \, a^{3}\right )}{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((I*a*x + a)^(1/4)*(-I*a*x + a)^(15/4)),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)**(15/4)/(a+I*a*x)**(1/4),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)^(1/4)*(-I*a*x + a)^(15/4)),x, algorithm="giac")
[Out]