3.1172 \(\int \frac{(a-i a x)^{3/4}}{\sqrt [4]{a+i a x}} \, dx\)

Optimal. Leaf size=106 \[ -\frac{2 a \sqrt [4]{x^2+1} E\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}+\frac{2 a x}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac{2 i (a-i a x)^{3/4} (a+i a x)^{3/4}}{3 a} \]

[Out]

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

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Rubi [A]  time = 0.07669, antiderivative size = 106, 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 \[ -\frac{2 a \sqrt [4]{x^2+1} E\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}+\frac{2 a x}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac{2 i (a-i a x)^{3/4} (a+i a x)^{3/4}}{3 a} \]

Antiderivative was successfully verified.

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

[Out]

(2*a*x)/((a - I*a*x)^(1/4)*(a + I*a*x)^(1/4)) - (((2*I)/3)*(a - I*a*x)^(3/4)*(a
+ I*a*x)^(3/4))/a - (2*a*(1 + x^2)^(1/4)*EllipticE[ArcTan[x]/2, 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. \[ - \frac{a^{3} \left (- i a x + a\right )^{\frac{3}{4}} \left (i a x + a\right )^{\frac{3}{4}} \int \frac{1}{\left (a^{2} x^{2} + a^{2}\right )^{\frac{5}{4}}}\, dx}{\left (a^{2} x^{2} + a^{2}\right )^{\frac{3}{4}}} + \frac{2 x \left (- i a x + a\right )^{\frac{3}{4}} \left (i a x + a\right )^{\frac{3}{4}}}{a \left (x^{2} + 1\right )} - \frac{2 i \left (- i a x + a\right )^{\frac{3}{4}} \left (i a x + a\right )^{\frac{3}{4}}}{3 a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0532324, size = 74, normalized size = 0.7 \[ \frac{2 (a-i a x)^{3/4} \left (i 2^{3/4} \sqrt [4]{1+i x} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};\frac{1}{2}-\frac{i x}{2}\right )+x-i\right )}{3 \sqrt [4]{a+i a x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.058, size = 94, normalized size = 0.9 \[{-{\frac{2\,i}{3}} \left ( x+i \right ) \left ( x-i \right ) a{\frac{1}{\sqrt [4]{-a \left ( -1+ix \right ) }}}{\frac{1}{\sqrt [4]{a \left ( 1+ix \right ) }}}}+{ax{\mbox{$_2$F$_1$}({\frac{1}{4}},{\frac{1}{2}};\,{\frac{3}{2}};\,-{x}^{2})}\sqrt [4]{-{a}^{2} \left ( -1+ix \right ) \left ( 1+ix \right ) }{\frac{1}{\sqrt [4]{{a}^{2}}}}{\frac{1}{\sqrt [4]{-a \left ( -1+ix \right ) }}}{\frac{1}{\sqrt [4]{a \left ( 1+ix \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*I*(x+I)*(x-I)*a/(-a*(-1+I*x))^(1/4)/(a*(1+I*x))^(1/4)+1/(a^2)^(1/4)*x*hyper
geom([1/4,1/2],[3/2],-x^2)*a*(-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. \[ \int \frac{{\left (-i \, a x + a\right )}^{\frac{3}{4}}}{{\left (i \, a x + a\right )}^{\frac{1}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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((-I*a*x + a)^(3/4)/(I*a*x + a)^(1/4),x, algorithm="giac")

[Out]

Exception raised: TypeError