3.1207 \(\int \frac{(a-i a x)^{3/4}}{(a+i a x)^{5/4}} \, dx\)

Optimal. Leaf size=102 \[ \frac{6 \sqrt [4]{x^2+1} E\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac{6 x}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}+\frac{4 i (a-i a x)^{3/4}}{a \sqrt [4]{a+i a x}} \]

[Out]

(-6*x)/((a - I*a*x)^(1/4)*(a + I*a*x)^(1/4)) + ((4*I)*(a - I*a*x)^(3/4))/(a*(a +
I*a*x)^(1/4)) + (6*(1 + x^2)^(1/4)*EllipticE[ArcTan[x]/2, 2])/((a - I*a*x)^(1/4
)*(a + I*a*x)^(1/4))

_______________________________________________________________________________________

Rubi [A]  time = 0.0712439, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{6 \sqrt [4]{x^2+1} E\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac{6 x}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}+\frac{4 i (a-i a x)^{3/4}}{a \sqrt [4]{a+i a x}} \]

Antiderivative was successfully verified.

[In]  Int[(a - I*a*x)^(3/4)/(a + I*a*x)^(5/4),x]

[Out]

(-6*x)/((a - I*a*x)^(1/4)*(a + I*a*x)^(1/4)) + ((4*I)*(a - I*a*x)^(3/4))/(a*(a +
I*a*x)^(1/4)) + (6*(1 + x^2)^(1/4)*EllipticE[ArcTan[x]/2, 2])/((a - I*a*x)^(1/4
)*(a + I*a*x)^(1/4))

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{3 a^{2} \left (- i a x + a\right )^{\frac{3}{4}} \left (i a x + a\right )^{\frac{3}{4}} \int \frac{1}{\left (a^{2} x^{2} + a^{2}\right )^{\frac{5}{4}}}\, dx}{\left (a^{2} x^{2} + a^{2}\right )^{\frac{3}{4}}} + \frac{4 i \left (- i a x + a\right )^{\frac{3}{4}}}{a \sqrt [4]{i a x + a}} - \frac{6 x \left (- i a x + a\right )^{\frac{3}{4}} \left (i a x + a\right )^{\frac{3}{4}}}{a^{2} \left (x^{2} + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a-I*a*x)**(3/4)/(a+I*a*x)**(5/4),x)

[Out]

3*a**2*(-I*a*x + a)**(3/4)*(I*a*x + a)**(3/4)*Integral((a**2*x**2 + a**2)**(-5/4
), x)/(a**2*x**2 + a**2)**(3/4) + 4*I*(-I*a*x + a)**(3/4)/(a*(I*a*x + a)**(1/4))
- 6*x*(-I*a*x + a)**(3/4)*(I*a*x + a)**(3/4)/(a**2*(x**2 + 1))

_______________________________________________________________________________________

Mathematica [C]  time = 0.0523255, size = 71, normalized size = 0.7 \[ -\frac{2 i (a-i a x)^{3/4} \left (-2+2^{3/4} \sqrt [4]{1+i x} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};\frac{1}{2}-\frac{i x}{2}\right )\right )}{a \sqrt [4]{a+i a x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a - I*a*x)^(3/4)/(a + I*a*x)^(5/4),x]

[Out]

((-2*I)*(a - I*a*x)^(3/4)*(-2 + 2^(3/4)*(1 + I*x)^(1/4)*Hypergeometric2F1[1/4, 3
/4, 7/4, 1/2 - (I/2)*x]))/(a*(a + I*a*x)^(1/4))

_______________________________________________________________________________________

Maple [C]  time = 0.065, size = 88, normalized size = 0.9 \[ 4\,{\frac{x+i}{\sqrt [4]{-a \left ( -1+ix \right ) }\sqrt [4]{a \left ( 1+ix \right ) }}}-3\,{\frac{x{\mbox{$_2$F$_1$}(1/4,1/2;\,3/2;\,-{x}^{2})}\sqrt [4]{-{a}^{2} \left ( -1+ix \right ) \left ( 1+ix \right ) }}{\sqrt [4]{{a}^{2}}\sqrt [4]{-a \left ( -1+ix \right ) }\sqrt [4]{a \left ( 1+ix \right ) }}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a-I*a*x)^(3/4)/(a+I*a*x)^(5/4),x)

[Out]

4*(x+I)/(-a*(-1+I*x))^(1/4)/(a*(1+I*x))^(1/4)-3/(a^2)^(1/4)*x*hypergeom([1/4,1/2
],[3/2],-x^2)*(-a^2*(-1+I*x)*(1+I*x))^(1/4)/(-a*(-1+I*x))^(1/4)/(a*(1+I*x))^(1/4
)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-i \, a x + a\right )}^{\frac{3}{4}}}{{\left (i \, a x + a\right )}^{\frac{5}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-I*a*x + a)^(3/4)/(I*a*x + a)^(5/4),x, algorithm="maxima")

[Out]

integrate((-I*a*x + a)^(3/4)/(I*a*x + a)^(5/4), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[ -\frac{{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}{\left (2 \, x - 6 i\right )} -{\left (a^{2} x^{2} - i \, a^{2} x\right )}{\rm integral}\left (-\frac{6 \,{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}}{a^{2} x^{4} + a^{2} x^{2}}, x\right )}{a^{2} x^{2} - i \, a^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-I*a*x + a)^(3/4)/(I*a*x + a)^(5/4),x, algorithm="fricas")

[Out]

-((I*a*x + a)^(3/4)*(-I*a*x + a)^(3/4)*(2*x - 6*I) - (a^2*x^2 - I*a^2*x)*integra
l(-6*(I*a*x + a)^(3/4)*(-I*a*x + a)^(3/4)/(a^2*x^4 + a^2*x^2), x))/(a^2*x^2 - I*
a^2*x)

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (- a \left (i x - 1\right )\right )^{\frac{3}{4}}}{\left (a \left (i x + 1\right )\right )^{\frac{5}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a-I*a*x)**(3/4)/(a+I*a*x)**(5/4),x)

[Out]

Integral((-a*(I*x - 1))**(3/4)/(a*(I*x + 1))**(5/4), x)

_______________________________________________________________________________________

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((-I*a*x + a)^(3/4)/(I*a*x + a)^(5/4),x, algorithm="giac")

[Out]

Exception raised: TypeError