3.1188 \(\int \frac{1}{(a-i a x)^{13/4} (a+i a x)^{3/4}} \, dx\)

Optimal. Leaf size=100 \[ -\frac{16 i \sqrt [4]{a+i a x}}{45 a^4 \sqrt [4]{a-i a x}}-\frac{8 i \sqrt [4]{a+i a x}}{45 a^3 (a-i a x)^{5/4}}-\frac{2 i \sqrt [4]{a+i a x}}{9 a^2 (a-i a x)^{9/4}} \]

[Out]

(((-2*I)/9)*(a + I*a*x)^(1/4))/(a^2*(a - I*a*x)^(9/4)) - (((8*I)/45)*(a + I*a*x)
^(1/4))/(a^3*(a - I*a*x)^(5/4)) - (((16*I)/45)*(a + I*a*x)^(1/4))/(a^4*(a - I*a*
x)^(1/4))

_______________________________________________________________________________________

Rubi [A]  time = 0.0809087, 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 \[ -\frac{16 i \sqrt [4]{a+i a x}}{45 a^4 \sqrt [4]{a-i a x}}-\frac{8 i \sqrt [4]{a+i a x}}{45 a^3 (a-i a x)^{5/4}}-\frac{2 i \sqrt [4]{a+i a x}}{9 a^2 (a-i a x)^{9/4}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a - I*a*x)^(13/4)*(a + I*a*x)^(3/4)),x]

[Out]

(((-2*I)/9)*(a + I*a*x)^(1/4))/(a^2*(a - I*a*x)^(9/4)) - (((8*I)/45)*(a + I*a*x)
^(1/4))/(a^3*(a - I*a*x)^(5/4)) - (((16*I)/45)*(a + I*a*x)^(1/4))/(a^4*(a - I*a*
x)^(1/4))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 18.3032, size = 87, normalized size = 0.87 \[ - \frac{2 i \sqrt [4]{i a x + a}}{9 a^{2} \left (- i a x + a\right )^{\frac{9}{4}}} - \frac{8 i \sqrt [4]{i a x + a}}{45 a^{3} \left (- i a x + a\right )^{\frac{5}{4}}} - \frac{16 i \sqrt [4]{i a x + a}}{45 a^{4} \sqrt [4]{- i a x + a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a-I*a*x)**(13/4)/(a+I*a*x)**(3/4),x)

[Out]

-2*I*(I*a*x + a)**(1/4)/(9*a**2*(-I*a*x + a)**(9/4)) - 8*I*(I*a*x + a)**(1/4)/(4
5*a**3*(-I*a*x + a)**(5/4)) - 16*I*(I*a*x + a)**(1/4)/(45*a**4*(-I*a*x + a)**(1/
4))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0497746, size = 52, normalized size = 0.52 \[ \frac{2 \left (-8 i x^2+20 x+17 i\right ) \sqrt [4]{a+i a x}}{45 a^4 (x+i)^2 \sqrt [4]{a-i a x}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a - I*a*x)^(13/4)*(a + I*a*x)^(3/4)),x]

[Out]

(2*(a + I*a*x)^(1/4)*(17*I + 20*x - (8*I)*x^2))/(45*a^4*(I + x)^2*(a - I*a*x)^(1
/4))

_______________________________________________________________________________________

Maple [A]  time = 0.065, size = 50, normalized size = 0.5 \[{\frac{24\,i{x}^{2}+16\,{x}^{3}+6\,x+34\,i}{45\,{a}^{3} \left ( x+i \right ) ^{2}} \left ( a \left ( 1+ix \right ) \right ) ^{-{\frac{3}{4}}}{\frac{1}{\sqrt [4]{-a \left ( -1+ix \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a-I*a*x)^(13/4)/(a+I*a*x)^(3/4),x)

[Out]

2/45/a^3/(a*(1+I*x))^(3/4)/(-a*(-1+I*x))^(1/4)*(12*I*x^2+8*x^3+3*x+17*I)/(x+I)^2

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{13}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((I*a*x + a)^(3/4)*(-I*a*x + a)^(13/4)),x, algorithm="maxima")

[Out]

integrate(1/((I*a*x + a)^(3/4)*(-I*a*x + a)^(13/4)), x)

_______________________________________________________________________________________

Fricas [A]  time = 0.211563, size = 78, normalized size = 0.78 \[ \frac{2 \,{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}{\left (8 \, x^{2} + 20 i \, x - 17\right )}}{45 \, a^{5} x^{3} + 135 i \, a^{5} x^{2} - 135 \, a^{5} x - 45 i \, a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((I*a*x + a)^(3/4)*(-I*a*x + a)^(13/4)),x, algorithm="fricas")

[Out]

2*(I*a*x + a)^(1/4)*(-I*a*x + a)^(3/4)*(8*x^2 + 20*I*x - 17)/(45*a^5*x^3 + 135*I
*a^5*x^2 - 135*a^5*x - 45*I*a^5)

_______________________________________________________________________________________

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)**(13/4)/(a+I*a*x)**(3/4),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)^(3/4)*(-I*a*x + a)^(13/4)),x, algorithm="giac")

[Out]

Exception raised: TypeError