3.1175 \(\int \frac{1}{(a-i a x)^{9/4} \sqrt [4]{a+i a x}} \, dx\)

Optimal. Leaf size=82 \[ \frac{2 \sqrt [4]{x^2+1} E\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{5 a^2 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac{4 i}{5 a (a-i a x)^{5/4} \sqrt [4]{a+i a x}} \]

[Out]

((-4*I)/5)/(a*(a - I*a*x)^(5/4)*(a + I*a*x)^(1/4)) + (2*(1 + x^2)^(1/4)*Elliptic
E[ArcTan[x]/2, 2])/(5*a^2*(a - I*a*x)^(1/4)*(a + I*a*x)^(1/4))

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Rubi [A]  time = 0.0643351, 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 \[ \frac{2 \sqrt [4]{x^2+1} E\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{5 a^2 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac{4 i}{5 a (a-i a x)^{5/4} \sqrt [4]{a+i a x}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a - I*a*x)^(9/4)*(a + I*a*x)^(1/4)),x]

[Out]

((-4*I)/5)/(a*(a - I*a*x)^(5/4)*(a + I*a*x)^(1/4)) + (2*(1 + x^2)^(1/4)*Elliptic
E[ArcTan[x]/2, 2])/(5*a^2*(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. \[ - \frac{4 i}{5 a \left (- i a x + a\right )^{\frac{5}{4}} \sqrt [4]{i a x + a}} - \frac{\left (- i a x + a\right )^{\frac{3}{4}} \left (i a x + a\right )^{\frac{3}{4}} \int \frac{1}{\sqrt [4]{a^{2} x^{2} + a^{2}}}\, dx}{5 a^{2} \left (a^{2} x^{2} + a^{2}\right )^{\frac{3}{4}}} + \frac{2 x \left (- i a x + a\right )^{\frac{3}{4}} \left (i a x + a\right )^{\frac{3}{4}}}{5 a^{4} \left (x^{2} + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a-I*a*x)**(9/4)/(a+I*a*x)**(1/4),x)

[Out]

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

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Mathematica [C]  time = 0.103856, size = 97, normalized size = 1.18 \[ \frac{6 \left (x^2+i x+2\right )-2\ 2^{3/4} \sqrt [4]{1+i x} (x+i)^2 \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};\frac{1}{2}-\frac{i x}{2}\right )}{15 a^2 (x+i) \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a - I*a*x)^(9/4)*(a + I*a*x)^(1/4)),x]

[Out]

(6*(2 + I*x + x^2) - 2*2^(3/4)*(1 + I*x)^(1/4)*(I + x)^2*Hypergeometric2F1[1/4,
3/4, 7/4, 1/2 - (I/2)*x])/(15*a^2*(I + x)*(a - I*a*x)^(1/4)*(a + I*a*x)^(1/4))

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Maple [C]  time = 0.193, size = 105, normalized size = 1.3 \[{\frac{2\,{x}^{2}+4+2\,ix}{ \left ( 5\,x+5\,i \right ){a}^{2}}{\frac{1}{\sqrt [4]{-a \left ( -1+ix \right ) }}}{\frac{1}{\sqrt [4]{a \left ( 1+ix \right ) }}}}-{\frac{x}{5\,{a}^{2}}{\mbox{$_2$F$_1$}({\frac{1}{4}},{\frac{1}{2}};\,{\frac{3}{2}};\,-{x}^{2})}\sqrt [4]{-{a}^{2} \left ( -1+ix \right ) \left ( 1+ix \right ) }{\frac{1}{\sqrt [4]{{a}^{2}}}}{\frac{1}{\sqrt [4]{-a \left ( -1+ix \right ) }}}{\frac{1}{\sqrt [4]{a \left ( 1+ix \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a-I*a*x)^(9/4)/(a+I*a*x)^(1/4),x)

[Out]

2/5*(x^2+2+I*x)/(x+I)/a^2/(-a*(-1+I*x))^(1/4)/(a*(1+I*x))^(1/4)-1/5/(a^2)^(1/4)*
x*hypergeom([1/4,1/2],[3/2],-x^2)/a^2*(-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. \[ \int \frac{1}{{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{9}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((I*a*x + a)^(1/4)*(-I*a*x + a)^(9/4)),x, algorithm="maxima")

[Out]

integrate(1/((I*a*x + a)^(1/4)*(-I*a*x + a)^(9/4)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[ \frac{{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}{\left (2 \, x + 4 i\right )} +{\left (5 \, a^{4} x^{2} + 10 i \, a^{4} x - 5 \, a^{4}\right )}{\rm integral}\left (-\frac{{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}}{5 \,{\left (a^{4} x^{2} + a^{4}\right )}}, x\right )}{5 \, a^{4} x^{2} + 10 i \, a^{4} x - 5 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((I*a*x + a)^(1/4)*(-I*a*x + a)^(9/4)),x, algorithm="fricas")

[Out]

((I*a*x + a)^(3/4)*(-I*a*x + a)^(3/4)*(2*x + 4*I) + (5*a^4*x^2 + 10*I*a^4*x - 5*
a^4)*integral(-1/5*(I*a*x + a)^(3/4)*(-I*a*x + a)^(3/4)/(a^4*x^2 + a^4), x))/(5*
a^4*x^2 + 10*I*a^4*x - 5*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)**(9/4)/(a+I*a*x)**(1/4),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)^(1/4)*(-I*a*x + a)^(9/4)),x, algorithm="giac")

[Out]

Exception raised: TypeError