3.1203 \(\int \frac{1}{(a-i a x)^{7/4} (a+i a x)^{7/4}} \, dx\)

Optimal. Leaf size=81 \[ \frac{2 \left (x^2+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{3 a^2 (a-i a x)^{3/4} (a+i a x)^{3/4}}+\frac{2 x}{3 a^2 (a-i a x)^{3/4} (a+i a x)^{3/4}} \]

[Out]

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

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Rubi [A]  time = 0.0491225, antiderivative size = 81, 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 \[ \frac{2 \left (x^2+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{3 a^2 (a-i a x)^{3/4} (a+i a x)^{3/4}}+\frac{2 x}{3 a^2 (a-i a x)^{3/4} (a+i a x)^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 10.2986, size = 75, normalized size = 0.93 \[ \frac{2 x \sqrt [4]{- i a x + a} \sqrt [4]{i a x + a}}{3 a^{4} \left (x^{2} + 1\right )} + \frac{2 \sqrt [4]{- i a x + a} \sqrt [4]{i a x + a} F\left (\frac{\operatorname{atan}{\left (x \right )}}{2}\middle | 2\right )}{3 a^{4} \sqrt [4]{x^{2} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0835872, size = 76, normalized size = 0.94 \[ \frac{2 \left (x+\sqrt [4]{2} (1+i x)^{3/4} (x+i) \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};\frac{1}{2}-\frac{i x}{2}\right )\right )}{3 a^2 (a-i a x)^{3/4} (a+i a x)^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.069, size = 0, normalized size = 0. \[ \int{1 \left ( a-iax \right ) ^{-{\frac{7}{4}}} \left ( a+iax \right ) ^{-{\frac{7}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(3*(a^4*x^2 + a^4)*integral(1/3*(I*a*x + a)^(1/4)*(-I*a*x + a)^(1/4)/(a^4*x^
2 + a^4), x) + 2*(I*a*x + a)^(1/4)*(-I*a*x + a)^(1/4)*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)**(7/4)/(a+I*a*x)**(7/4),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)^(7/4)*(-I*a*x + a)^(7/4)),x, algorithm="giac")

[Out]

Exception raised: TypeError