3.1192 \(\int \frac{1}{(a-i a x)^{7/4} (a+i a x)^{3/4}} \, dx\)

Optimal. Leaf size=82 \[ \frac{2 \left (x^2+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{3 a (a-i a x)^{3/4} (a+i a x)^{3/4}}-\frac{2 i \sqrt [4]{a+i a x}}{3 a^2 (a-i a x)^{3/4}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.060776, antiderivative size = 82, 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 \[ \frac{2 \left (x^2+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{3 a (a-i a x)^{3/4} (a+i a x)^{3/4}}-\frac{2 i \sqrt [4]{a+i a x}}{3 a^2 (a-i a x)^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 14.2798, size = 70, normalized size = 0.85 \[ - \frac{2 i \sqrt [4]{i a x + a}}{3 a^{2} \left (- i a x + a\right )^{\frac{3}{4}}} + \frac{2 \sqrt [4]{- i a x + a} \sqrt [4]{i a x + a} F\left (\frac{\operatorname{atan}{\left (x \right )}}{2}\middle | 2\right )}{3 a^{3} \sqrt [4]{x^{2} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a-I*a*x)**(7/4)/(a+I*a*x)**(3/4),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [C]  time = 0.0737164, size = 79, normalized size = 0.96 \[ \frac{2 \left (\sqrt [4]{2} (1+i x)^{3/4} (x+i) \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};\frac{1}{2}-\frac{i x}{2}\right )+x-i\right )}{3 a (a-i a x)^{3/4} (a+i a x)^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.075, size = 0, normalized size = 0. \[ \int{1 \left ( a-iax \right ) ^{-{\frac{7}{4}}} \left ( a+iax \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int(1/(a-I*a*x)^(7/4)/(a+I*a*x)^(3/4),x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{7}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((I*a*x + a)^(3/4)*(-I*a*x + a)^(7/4)),x, algorithm="maxima")

[Out]

integrate(1/((I*a*x + a)^(3/4)*(-I*a*x + a)^(7/4)), x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((I*a*x + a)^(3/4)*(-I*a*x + a)^(7/4)),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

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)**(7/4)/(a+I*a*x)**(3/4),x)

[Out]

Timed out

_______________________________________________________________________________________

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)^(3/4)*(-I*a*x + a)^(7/4)),x, algorithm="giac")

[Out]

Exception raised: TypeError