3.1225 \(\int \frac{\sqrt [4]{a-i a x}}{(a+i a x)^{9/4}} \, dx\)

Optimal. Leaf size=33 \[ \frac{2 i (a-i a x)^{5/4}}{5 a^2 (a+i a x)^{5/4}} \]

[Out]

(((2*I)/5)*(a - I*a*x)^(5/4))/(a^2*(a + I*a*x)^(5/4))

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Rubi [A]  time = 0.0234605, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04 \[ \frac{2 i (a-i a x)^{5/4}}{5 a^2 (a+i a x)^{5/4}} \]

Antiderivative was successfully verified.

[In]  Int[(a - I*a*x)^(1/4)/(a + I*a*x)^(9/4),x]

[Out]

(((2*I)/5)*(a - I*a*x)^(5/4))/(a^2*(a + I*a*x)^(5/4))

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Rubi in Sympy [A]  time = 5.88938, size = 27, normalized size = 0.82 \[ \frac{2 i \left (- i a x + a\right )^{\frac{5}{4}}}{5 a^{2} \left (i a x + a\right )^{\frac{5}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a-I*a*x)**(1/4)/(a+I*a*x)**(9/4),x)

[Out]

2*I*(-I*a*x + a)**(5/4)/(5*a**2*(I*a*x + a)**(5/4))

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Mathematica [A]  time = 0.0303027, size = 43, normalized size = 1.3 \[ -\frac{2 (x+i) \sqrt [4]{a-i a x} (a+i a x)^{3/4}}{5 a^3 (x-i)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a - I*a*x)^(1/4)/(a + I*a*x)^(9/4),x]

[Out]

(-2*(I + x)*(a - I*a*x)^(1/4)*(a + I*a*x)^(3/4))/(5*a^3*(-I + x)^2)

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Maple [B]  time = 0.062, size = 50, normalized size = 1.5 \[{\frac{2\,{x}^{2}-2+4\,ix}{5\,{a}^{2} \left ( -1+ix \right ) \left ( x-i \right ) }\sqrt [4]{-a \left ( -1+ix \right ) }{\frac{1}{\sqrt [4]{a \left ( 1+ix \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a-I*a*x)^(1/4)/(a+I*a*x)^(9/4),x)

[Out]

2/5/a^2*(-a*(-1+I*x))^(1/4)/(-1+I*x)/(a*(1+I*x))^(1/4)*(x^2-1+2*I*x)/(x-I)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{{\left (i \, a x + a\right )}^{\frac{9}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-I*a*x + a)^(1/4)/(I*a*x + a)^(9/4),x, algorithm="maxima")

[Out]

integrate((-I*a*x + a)^(1/4)/(I*a*x + a)^(9/4), x)

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Fricas [A]  time = 0.225519, size = 61, normalized size = 1.85 \[ -\frac{{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}{\left (2 \, x + 2 i\right )}}{5 \, a^{3} x^{2} - 10 i \, a^{3} x - 5 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-I*a*x + a)^(1/4)/(I*a*x + a)^(9/4),x, algorithm="fricas")

[Out]

-(I*a*x + a)^(3/4)*(-I*a*x + a)^(1/4)*(2*x + 2*I)/(5*a^3*x^2 - 10*I*a^3*x - 5*a^
3)

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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((a-I*a*x)**(1/4)/(a+I*a*x)**(9/4),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.224942, size = 46, normalized size = 1.39 \[ -\frac{{\left (-i \, a x + a\right )}^{\frac{1}{4}}{\left (-\frac{4 i \, a}{i \, a x + a} + 2 i\right )}}{5 \,{\left (i \, a x + a\right )}^{\frac{1}{4}} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-I*a*x + a)^(1/4)/(I*a*x + a)^(9/4),x, algorithm="giac")

[Out]

-1/5*(-I*a*x + a)^(1/4)*(-4*I*a/(I*a*x + a) + 2*I)/((I*a*x + a)^(1/4)*a^2)