3.1201 \(\int \frac{\sqrt [4]{a-i a x}}{(a+i a x)^{7/4}} \, dx\)

Optimal. Leaf size=79 \[ \frac{4 i \sqrt [4]{a-i a x}}{3 a (a+i a x)^{3/4}}-\frac{2 \left (x^2+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{3 (a-i a x)^{3/4} (a+i a x)^{3/4}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0580244, antiderivative size = 79, 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 \[ \frac{4 i \sqrt [4]{a-i a x}}{3 a (a+i a x)^{3/4}}-\frac{2 \left (x^2+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{3 (a-i a x)^{3/4} (a+i a x)^{3/4}} \]

Antiderivative was successfully verified.

[In]  Int[(a - I*a*x)^(1/4)/(a + I*a*x)^(7/4),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 13.3636, size = 68, normalized size = 0.86 \[ \frac{4 i \sqrt [4]{- i a x + a}}{3 a \left (i a x + a\right )^{\frac{3}{4}}} - \frac{2 \sqrt [4]{- i a x + a} \sqrt [4]{i a x + a} F\left (\frac{\operatorname{atan}{\left (x \right )}}{2}\middle | 2\right )}{3 a^{2} \sqrt [4]{x^{2} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a-I*a*x)**(1/4)/(a+I*a*x)**(7/4),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [C]  time = 0.0508079, size = 73, normalized size = 0.92 \[ -\frac{2 i \sqrt [4]{a-i a x} \left (-2+\sqrt [4]{2} (1+i x)^{3/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};\frac{1}{2}-\frac{i x}{2}\right )\right )}{3 a (a+i a x)^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a - I*a*x)^(1/4)/(a + I*a*x)^(7/4),x]

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.06, size = 0, normalized size = 0. \[ \int{1\sqrt [4]{a-iax} \left ( a+iax \right ) ^{-{\frac{7}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a-I*a*x)^(1/4)/(a+I*a*x)^(7/4),x)

[Out]

int((a-I*a*x)^(1/4)/(a+I*a*x)^(7/4),x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{{\left (i \, a x + a\right )}^{\frac{7}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-I*a*x + a)^(1/4)/(I*a*x + a)^(7/4),x, algorithm="maxima")

[Out]

integrate((-I*a*x + a)^(1/4)/(I*a*x + a)^(7/4), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[ \frac{3 \,{\left (a^{2} x - i \, a^{2}\right )}{\rm integral}\left (-\frac{{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{3 \,{\left (a^{2} x^{2} + a^{2}\right )}}, x\right ) + 4 \,{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{3 \,{\left (a^{2} x - i \, a^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-I*a*x + a)^(1/4)/(I*a*x + a)^(7/4),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt [4]{- a \left (i x - 1\right )}}{\left (a \left (i x + 1\right )\right )^{\frac{7}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a-I*a*x)**(1/4)/(a+I*a*x)**(7/4),x)

[Out]

Integral((-a*(I*x - 1))**(1/4)/(a*(I*x + 1))**(7/4), 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)^(1/4)/(I*a*x + a)^(7/4),x, algorithm="giac")

[Out]

Exception raised: TypeError