Optimal. Leaf size=133 \[ -\frac{32 i (a+i a x)^{3/4}}{1155 a^5 (a-i a x)^{3/4}}-\frac{16 i (a+i a x)^{3/4}}{385 a^4 (a-i a x)^{7/4}}-\frac{4 i (a+i a x)^{3/4}}{55 a^3 (a-i a x)^{11/4}}-\frac{2 i (a+i a x)^{3/4}}{15 a^2 (a-i a x)^{15/4}} \]
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Rubi [A] time = 0.116078, 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 \[ -\frac{32 i (a+i a x)^{3/4}}{1155 a^5 (a-i a x)^{3/4}}-\frac{16 i (a+i a x)^{3/4}}{385 a^4 (a-i a x)^{7/4}}-\frac{4 i (a+i a x)^{3/4}}{55 a^3 (a-i a x)^{11/4}}-\frac{2 i (a+i a x)^{3/4}}{15 a^2 (a-i a x)^{15/4}} \]
Antiderivative was successfully verified.
[In] Int[1/((a - I*a*x)^(19/4)*(a + I*a*x)^(1/4)),x]
[Out]
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Rubi in Sympy [A] time = 25.8857, size = 116, normalized size = 0.87 \[ - \frac{2 i \left (i a x + a\right )^{\frac{3}{4}}}{15 a^{2} \left (- i a x + a\right )^{\frac{15}{4}}} - \frac{4 i \left (i a x + a\right )^{\frac{3}{4}}}{55 a^{3} \left (- i a x + a\right )^{\frac{11}{4}}} - \frac{16 i \left (i a x + a\right )^{\frac{3}{4}}}{385 a^{4} \left (- i a x + a\right )^{\frac{7}{4}}} - \frac{32 i \left (i a x + a\right )^{\frac{3}{4}}}{1155 a^{5} \left (- i a x + a\right )^{\frac{3}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a-I*a*x)**(19/4)/(a+I*a*x)**(1/4),x)
[Out]
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Mathematica [A] time = 0.0611999, size = 57, normalized size = 0.43 \[ \frac{2 \left (-16 i x^3+72 x^2+138 i x-159\right ) (a+i a x)^{3/4}}{1155 a^5 (x+i)^3 (a-i a x)^{3/4}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a - I*a*x)^(19/4)*(a + I*a*x)^(1/4)),x]
[Out]
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Maple [A] time = 0.072, size = 55, normalized size = 0.4 \[{\frac{112\,i{x}^{3}+32\,{x}^{4}-42\,ix-318-132\,{x}^{2}}{1155\,{a}^{4} \left ( x+i \right ) ^{3}} \left ( -a \left ( -1+ix \right ) \right ) ^{-{\frac{3}{4}}}{\frac{1}{\sqrt [4]{a \left ( 1+ix \right ) }}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a-I*a*x)^(19/4)/(a+I*a*x)^(1/4),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{19}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((I*a*x + a)^(1/4)*(-I*a*x + a)^(19/4)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.21272, size = 90, normalized size = 0.68 \[ \frac{32 \, x^{4} + 112 i \, x^{3} - 132 \, x^{2} - 42 i \, x - 318}{{\left (1155 \, a^{4} x^{3} + 3465 i \, a^{4} x^{2} - 3465 \, a^{4} x - 1155 i \, a^{4}\right )}{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((I*a*x + a)^(1/4)*(-I*a*x + a)^(19/4)),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)**(19/4)/(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)^(19/4)),x, algorithm="giac")
[Out]