3.1193 \(\int \frac{1}{(a-i a x)^{11/4} (a+i a x)^{3/4}} \, dx\)

Optimal. Leaf size=115 \[ -\frac{2 i \sqrt [4]{a+i a x}}{7 a^3 (a-i a x)^{3/4}}+\frac{2 \left (x^2+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{7 a^2 (a-i a x)^{3/4} (a+i a x)^{3/4}}-\frac{2 i \sqrt [4]{a+i a x}}{7 a^2 (a-i a x)^{7/4}} \]

[Out]

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

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Rubi [A]  time = 0.0923829, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16 \[ -\frac{2 i \sqrt [4]{a+i a x}}{7 a^3 (a-i a x)^{3/4}}+\frac{2 \left (x^2+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{7 a^2 (a-i a x)^{3/4} (a+i a x)^{3/4}}-\frac{2 i \sqrt [4]{a+i a x}}{7 a^2 (a-i a x)^{7/4}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a - I*a*x)^(11/4)*(a + I*a*x)^(3/4)),x]

[Out]

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

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Rubi in Sympy [A]  time = 21.1772, size = 99, normalized size = 0.86 \[ - \frac{2 i \sqrt [4]{i a x + a}}{7 a^{2} \left (- i a x + a\right )^{\frac{7}{4}}} - \frac{2 i \sqrt [4]{i a x + a}}{7 a^{3} \left (- i a x + a\right )^{\frac{3}{4}}} + \frac{2 \sqrt [4]{- i a x + a} \sqrt [4]{i a x + a} F\left (\frac{\operatorname{atan}{\left (x \right )}}{2}\middle | 2\right )}{7 a^{4} \sqrt [4]{x^{2} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a-I*a*x)**(11/4)/(a+I*a*x)**(3/4),x)

[Out]

-2*I*(I*a*x + a)**(1/4)/(7*a**2*(-I*a*x + a)**(7/4)) - 2*I*(I*a*x + a)**(1/4)/(7
*a**3*(-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)/(7*a**4*(x**2 + 1)**(1/4))

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Mathematica [C]  time = 0.104467, size = 93, normalized size = 0.81 \[ \frac{2 \left (\sqrt [4]{2} (1+i x)^{3/4} (x+i)^2 \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};\frac{1}{2}-\frac{i x}{2}\right )+x^2+i x+2\right )}{7 a^2 (x+i) (a-i a x)^{3/4} (a+i a x)^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a - I*a*x)^(11/4)*(a + I*a*x)^(3/4)),x]

[Out]

(2*(2 + I*x + x^2 + 2^(1/4)*(1 + I*x)^(3/4)*(I + x)^2*Hypergeometric2F1[1/4, 3/4
, 5/4, 1/2 - (I/2)*x]))/(7*a^2*(I + x)*(a - I*a*x)^(3/4)*(a + I*a*x)^(3/4))

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Maple [F]  time = 0.079, size = 0, normalized size = 0. \[ \int{1 \left ( a-iax \right ) ^{-{\frac{11}{4}}} \left ( a+iax \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a-I*a*x)^(11/4)/(a+I*a*x)^(3/4),x)

[Out]

int(1/(a-I*a*x)^(11/4)/(a+I*a*x)^(3/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{11}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((I*a*x + a)^(3/4)*(-I*a*x + a)^(11/4)),x, algorithm="maxima")

[Out]

integrate(1/((I*a*x + a)^(3/4)*(-I*a*x + a)^(11/4)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[ \frac{{\left (7 \, a^{4} x^{2} + 14 i \, a^{4} x - 7 \, a^{4}\right )}{\rm integral}\left (\frac{{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{7 \,{\left (a^{4} x^{2} + a^{4}\right )}}, x\right ) +{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}{\left (2 \, x + 4 i\right )}}{7 \, a^{4} x^{2} + 14 i \, a^{4} x - 7 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((I*a*x + a)^(3/4)*(-I*a*x + a)^(11/4)),x, algorithm="fricas")

[Out]

((7*a^4*x^2 + 14*I*a^4*x - 7*a^4)*integral(1/7*(I*a*x + a)^(1/4)*(-I*a*x + a)^(1
/4)/(a^4*x^2 + a^4), x) + (I*a*x + a)^(1/4)*(-I*a*x + a)^(1/4)*(2*x + 4*I))/(7*a
^4*x^2 + 14*I*a^4*x - 7*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)**(11/4)/(a+I*a*x)**(3/4),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)^(3/4)*(-I*a*x + a)^(11/4)),x, algorithm="giac")

[Out]

Exception raised: TypeError