3.1197 \(\int \frac{1}{(a-i a x)^{5/4} (a+i a x)^{7/4}} \, dx\)

Optimal. Leaf size=65 \[ \frac{4 i (a-i a x)^{3/4}}{3 a^3 (a+i a x)^{3/4}}-\frac{2 i}{a^2 \sqrt [4]{a-i a x} (a+i a x)^{3/4}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0515329, antiderivative size = 65, 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 \[ \frac{4 i (a-i a x)^{3/4}}{3 a^3 (a+i a x)^{3/4}}-\frac{2 i}{a^2 \sqrt [4]{a-i a x} (a+i a x)^{3/4}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a - I*a*x)^(5/4)*(a + I*a*x)^(7/4)),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 10.9833, size = 56, normalized size = 0.86 \[ \frac{2 i}{3 a^{2} \sqrt [4]{- i a x + a} \left (i a x + a\right )^{\frac{3}{4}}} - \frac{4 i \sqrt [4]{i a x + a}}{3 a^{3} \sqrt [4]{- i a x + a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a-I*a*x)**(5/4)/(a+I*a*x)**(7/4),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0436556, size = 47, normalized size = 0.72 \[ -\frac{2 i (2 x-i) \sqrt [4]{a+i a x}}{3 a^3 (x-i) \sqrt [4]{a-i a x}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a - I*a*x)^(5/4)*(a + I*a*x)^(7/4)),x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.065, size = 33, normalized size = 0.5 \[{\frac{4\,x-2\,i}{3\,{a}^{2}} \left ( a \left ( 1+ix \right ) \right ) ^{-{\frac{3}{4}}}{\frac{1}{\sqrt [4]{-a \left ( -1+ix \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a-I*a*x)^(5/4)/(a+I*a*x)^(7/4),x)

[Out]

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

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (i \, a x + a\right )}^{\frac{7}{4}}{\left (-i \, a x + a\right )}^{\frac{5}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((I*a*x + a)^(7/4)*(-I*a*x + a)^(5/4)),x, algorithm="maxima")

[Out]

integrate(1/((I*a*x + a)^(7/4)*(-I*a*x + a)^(5/4)), x)

_______________________________________________________________________________________

Fricas [A]  time = 0.21455, size = 35, normalized size = 0.54 \[ \frac{4 \, x - 2 i}{3 \,{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((I*a*x + a)^(7/4)*(-I*a*x + a)^(5/4)),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

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)**(5/4)/(a+I*a*x)**(7/4),x)

[Out]

Timed out

_______________________________________________________________________________________

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)^(7/4)*(-I*a*x + a)^(5/4)),x, algorithm="giac")

[Out]

Exception raised: TypeError