3.1190 \(\int \frac{\sqrt [4]{a-i a x}}{(a+i a x)^{3/4}} \, dx\)

Optimal. Leaf size=76 \[ \frac{2 a \left (x^2+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{(a-i a x)^{3/4} (a+i a x)^{3/4}}-\frac{2 i \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}{a} \]

[Out]

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

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Rubi [A]  time = 0.0598307, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16 \[ \frac{2 a \left (x^2+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{(a-i a x)^{3/4} (a+i a x)^{3/4}}-\frac{2 i \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}{a} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 14.3364, size = 63, normalized size = 0.83 \[ - \frac{2 i \sqrt [4]{- i a x + a} \sqrt [4]{i a x + a}}{a} + \frac{2 \sqrt [4]{- i a x + a} \sqrt [4]{i a x + a} F\left (\frac{\operatorname{atan}{\left (x \right )}}{2}\middle | 2\right )}{a \sqrt [4]{x^{2} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*I*(-I*a*x + a)**(1/4)*(I*a*x + a)**(1/4)/a + 2*(-I*a*x + a)**(1/4)*(I*a*x + a
)**(1/4)*elliptic_f(atan(x)/2, 2)/(a*(x**2 + 1)**(1/4))

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Mathematica [C]  time = 0.0509695, size = 72, normalized size = 0.95 \[ \frac{2 \sqrt [4]{a-i a x} \left (i \sqrt [4]{2} (1+i x)^{3/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};\frac{1}{2}-\frac{i x}{2}\right )+x-i\right )}{(a+i a x)^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(a - I*a*x)^(1/4)*(-I + x + I*2^(1/4)*(1 + I*x)^(3/4)*Hypergeometric2F1[1/4,
3/4, 5/4, 1/2 - (I/2)*x]))/(a + I*a*x)^(3/4)

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Maple [F]  time = 0.059, size = 0, normalized size = 0. \[ \int{1\sqrt [4]{a-iax} \left ( a+iax \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[ \frac{a{\rm integral}\left (\frac{{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{a x^{2} + a}, x\right ) - 2 i \,{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: TypeError