3.1209 \(\int \frac{1}{(a-i a x)^{5/4} (a+i a x)^{5/4}} \, dx\)

Optimal. Leaf size=46 \[ \frac{2 \sqrt [4]{x^2+1} E\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{a^2 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \]

[Out]

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

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Rubi [A]  time = 0.0328834, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12 \[ \frac{2 \sqrt [4]{x^2+1} E\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{a^2 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-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)

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Mathematica [C]  time = 0.0805656, size = 79, normalized size = 1.72 \[ \frac{6 x-2\ 2^{3/4} \sqrt [4]{1+i x} (x+i) \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};\frac{1}{2}-\frac{i x}{2}\right )}{3 a^2 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.057, size = 91, normalized size = 2. \[ 2\,{\frac{x}{\sqrt [4]{-a \left ( -1+ix \right ) }\sqrt [4]{a \left ( 1+ix \right ) }{a}^{2}}}-{\frac{x}{{a}^{2}}{\mbox{$_2$F$_1$}({\frac{1}{4}},{\frac{1}{2}};\,{\frac{3}{2}};\,-{x}^{2})}\sqrt [4]{-{a}^{2} \left ( -1+ix \right ) \left ( 1+ix \right ) }{\frac{1}{\sqrt [4]{{a}^{2}}}}{\frac{1}{\sqrt [4]{-a \left ( -1+ix \right ) }}}{\frac{1}{\sqrt [4]{a \left ( 1+ix \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

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

[Out]

Exception raised: TypeError