3.1211 \(\int \frac{1}{(a-i a x)^{13/4} (a+i a x)^{5/4}} \, dx\)

Optimal. Leaf size=115 \[ \frac{2 \sqrt [4]{x^2+1} E\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{3 a^4 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac{2 i}{9 a^3 (a-i a x)^{5/4} \sqrt [4]{a+i a x}}-\frac{2 i}{9 a^2 (a-i a x)^{9/4} \sqrt [4]{a+i a x}} \]

[Out]

((-2*I)/9)/(a^2*(a - I*a*x)^(9/4)*(a + I*a*x)^(1/4)) - ((2*I)/9)/(a^3*(a - I*a*x
)^(5/4)*(a + I*a*x)^(1/4)) + (2*(1 + x^2)^(1/4)*EllipticE[ArcTan[x]/2, 2])/(3*a^
4*(a - I*a*x)^(1/4)*(a + I*a*x)^(1/4))

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Rubi [A]  time = 0.0954007, 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 \[ \frac{2 \sqrt [4]{x^2+1} E\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{3 a^4 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac{2 i}{9 a^3 (a-i a x)^{5/4} \sqrt [4]{a+i a x}}-\frac{2 i}{9 a^2 (a-i a x)^{9/4} \sqrt [4]{a+i a x}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a - I*a*x)^(13/4)*(a + I*a*x)^(5/4)),x]

[Out]

((-2*I)/9)/(a^2*(a - I*a*x)^(9/4)*(a + I*a*x)^(1/4)) - ((2*I)/9)/(a^3*(a - I*a*x
)^(5/4)*(a + I*a*x)^(1/4)) + (2*(1 + x^2)^(1/4)*EllipticE[ArcTan[x]/2, 2])/(3*a^
4*(a - I*a*x)^(1/4)*(a + I*a*x)^(1/4))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{2 i}{a^{2} \left (- i a x + a\right )^{\frac{9}{4}} \sqrt [4]{i a x + a}} - \frac{4 i}{3 a^{3} \left (- i a x + a\right )^{\frac{5}{4}} \sqrt [4]{i a x + a}} - \frac{10 i \left (i a x + a\right )^{\frac{3}{4}}}{9 a^{3} \left (- i a x + a\right )^{\frac{9}{4}}} - \frac{\left (- i a x + a\right )^{\frac{3}{4}} \left (i a x + a\right )^{\frac{3}{4}} \int \frac{1}{\sqrt [4]{a^{2} x^{2} + a^{2}}}\, dx}{3 a^{4} \left (a^{2} x^{2} + a^{2}\right )^{\frac{3}{4}}} + \frac{2 x \left (- i a x + a\right )^{\frac{3}{4}} \left (i a x + a\right )^{\frac{3}{4}}}{3 a^{6} \left (x^{2} + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a-I*a*x)**(13/4)/(a+I*a*x)**(5/4),x)

[Out]

2*I/(a**2*(-I*a*x + a)**(9/4)*(I*a*x + a)**(1/4)) - 4*I/(3*a**3*(-I*a*x + a)**(5
/4)*(I*a*x + a)**(1/4)) - 10*I*(I*a*x + a)**(3/4)/(9*a**3*(-I*a*x + a)**(9/4)) -
(-I*a*x + a)**(3/4)*(I*a*x + a)**(3/4)*Integral((a**2*x**2 + a**2)**(-1/4), x)/
(3*a**4*(a**2*x**2 + a**2)**(3/4)) + 2*x*(-I*a*x + a)**(3/4)*(I*a*x + a)**(3/4)/
(3*a**6*(x**2 + 1))

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Mathematica [C]  time = 0.129478, size = 103, normalized size = 0.9 \[ \frac{-2\ 2^{3/4} \sqrt [4]{1+i x} (x+i)^3 \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};\frac{1}{2}-\frac{i x}{2}\right )+6 x^3+12 i x^2-4 x+4 i}{9 a^4 (x+i)^2 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a - I*a*x)^(13/4)*(a + I*a*x)^(5/4)),x]

[Out]

(4*I - 4*x + (12*I)*x^2 + 6*x^3 - 2*2^(3/4)*(1 + I*x)^(1/4)*(I + x)^3*Hypergeome
tric2F1[1/4, 3/4, 7/4, 1/2 - (I/2)*x])/(9*a^4*(I + x)^2*(a - I*a*x)^(1/4)*(a + I
*a*x)^(1/4))

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Maple [C]  time = 0.102, size = 113, normalized size = 1. \[{\frac{12\,i{x}^{2}+6\,{x}^{3}-4\,x+4\,i}{9\, \left ( x+i \right ) ^{2}{a}^{4}}{\frac{1}{\sqrt [4]{-a \left ( -1+ix \right ) }}}{\frac{1}{\sqrt [4]{a \left ( 1+ix \right ) }}}}-{\frac{x}{3\,{a}^{4}}{\mbox{$_2$F$_1$}({\frac{1}{4}},{\frac{1}{2}};\,{\frac{3}{2}};\,-{x}^{2})}\sqrt [4]{-{a}^{2} \left ( -1+ix \right ) \left ( 1+ix \right ) }{\frac{1}{\sqrt [4]{{a}^{2}}}}{\frac{1}{\sqrt [4]{-a \left ( -1+ix \right ) }}}{\frac{1}{\sqrt [4]{a \left ( 1+ix \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a-I*a*x)^(13/4)/(a+I*a*x)^(5/4),x)

[Out]

2/9*(6*I*x^2+3*x^3-2*x+2*I)/(x+I)^2/a^4/(-a*(-1+I*x))^(1/4)/(a*(1+I*x))^(1/4)-1/
3/(a^2)^(1/4)*x*hypergeom([1/4,1/2],[3/2],-x^2)/a^4*(-a^2*(-1+I*x)*(1+I*x))^(1/4
)/(-a*(-1+I*x))^(1/4)/(a*(1+I*x))^(1/4)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((I*a*x + a)^(5/4)*(-I*a*x + a)^(13/4)),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[ \frac{{\left (6 \, x^{3} + 12 i \, x^{2} - 4 \, x + 4 i\right )}{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}} +{\left (9 \, a^{6} x^{4} + 18 i \, a^{6} x^{3} + 18 i \, a^{6} x - 9 \, a^{6}\right )}{\rm integral}\left (-\frac{{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}}{3 \,{\left (a^{6} x^{2} + a^{6}\right )}}, x\right )}{9 \, a^{6} x^{4} + 18 i \, a^{6} x^{3} + 18 i \, a^{6} x - 9 \, a^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((I*a*x + a)^(5/4)*(-I*a*x + a)^(13/4)),x, algorithm="fricas")

[Out]

((6*x^3 + 12*I*x^2 - 4*x + 4*I)*(I*a*x + a)^(3/4)*(-I*a*x + a)^(3/4) + (9*a^6*x^
4 + 18*I*a^6*x^3 + 18*I*a^6*x - 9*a^6)*integral(-1/3*(I*a*x + a)^(3/4)*(-I*a*x +
a)^(3/4)/(a^6*x^2 + a^6), x))/(9*a^6*x^4 + 18*I*a^6*x^3 + 18*I*a^6*x - 9*a^6)

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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)**(13/4)/(a+I*a*x)**(5/4),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)^(5/4)*(-I*a*x + a)^(13/4)),x, algorithm="giac")

[Out]

Exception raised: TypeError