3.1218 \(\int \frac{(a-i a x)^{3/4}}{(a+i a x)^{9/4}} \, dx\)

Optimal. Leaf size=115 \[ -\frac{6 \sqrt [4]{x^2+1} E\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{5 a \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac{6 i}{5 a \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}+\frac{4 i (a-i a x)^{3/4}}{5 a (a+i a x)^{5/4}} \]

[Out]

(((4*I)/5)*(a - I*a*x)^(3/4))/(a*(a + I*a*x)^(5/4)) - ((6*I)/5)/(a*(a - I*a*x)^(
1/4)*(a + I*a*x)^(1/4)) - (6*(1 + x^2)^(1/4)*EllipticE[ArcTan[x]/2, 2])/(5*a*(a
- I*a*x)^(1/4)*(a + I*a*x)^(1/4))

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Rubi [A]  time = 0.091102, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{6 \sqrt [4]{x^2+1} E\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{5 a \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac{6 i}{5 a \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}+\frac{4 i (a-i a x)^{3/4}}{5 a (a+i a x)^{5/4}} \]

Antiderivative was successfully verified.

[In]  Int[(a - I*a*x)^(3/4)/(a + I*a*x)^(9/4),x]

[Out]

(((4*I)/5)*(a - I*a*x)^(3/4))/(a*(a + I*a*x)^(5/4)) - ((6*I)/5)/(a*(a - I*a*x)^(
1/4)*(a + I*a*x)^(1/4)) - (6*(1 + x^2)^(1/4)*EllipticE[ArcTan[x]/2, 2])/(5*a*(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. \[ - \frac{3 a \left (- i a x + a\right )^{\frac{3}{4}} \left (i a x + a\right )^{\frac{3}{4}} \int \frac{1}{\left (a^{2} x^{2} + a^{2}\right )^{\frac{5}{4}}}\, dx}{5 \left (a^{2} x^{2} + a^{2}\right )^{\frac{3}{4}}} + \frac{4 i \left (- i a x + a\right )^{\frac{3}{4}}}{5 a \left (i a x + a\right )^{\frac{5}{4}}} - \frac{6 i}{5 a \sqrt [4]{- i a x + a} \sqrt [4]{i a x + a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a-I*a*x)**(3/4)/(a+I*a*x)**(9/4),x)

[Out]

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

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Mathematica [C]  time = 0.0798959, size = 83, normalized size = 0.72 \[ \frac{2 (a-i a x)^{3/4} \left (2^{3/4} (1+i x)^{5/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};\frac{1}{2}-\frac{i x}{2}\right )-3 i x-1\right )}{5 a^2 (x-i) \sqrt [4]{a+i a x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a - I*a*x)^(3/4)/(a + I*a*x)^(9/4),x]

[Out]

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

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Maple [C]  time = 0.058, size = 107, normalized size = 0.9 \[ -{\frac{6\,{x}^{2}+2+4\,ix}{ \left ( 5\,x-5\,i \right ) a}{\frac{1}{\sqrt [4]{-a \left ( -1+ix \right ) }}}{\frac{1}{\sqrt [4]{a \left ( 1+ix \right ) }}}}+{\frac{3\,x}{5\,a}{\mbox{$_2$F$_1$}({\frac{1}{4}},{\frac{1}{2}};\,{\frac{3}{2}};\,-{x}^{2})}\sqrt [4]{-{a}^{2} \left ( -1+ix \right ) \left ( 1+ix \right ) }{\frac{1}{\sqrt [4]{{a}^{2}}}}{\frac{1}{\sqrt [4]{-a \left ( -1+ix \right ) }}}{\frac{1}{\sqrt [4]{a \left ( 1+ix \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a-I*a*x)^(3/4)/(a+I*a*x)^(9/4),x)

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-I*a*x + a)^(3/4)/(I*a*x + a)^(9/4),x, algorithm="maxima")

[Out]

integrate((-I*a*x + a)^(3/4)/(I*a*x + a)^(9/4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[ -\frac{2 \,{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}{\left (5 i \, x + 3\right )} - 5 \,{\left (a^{3} x^{3} - 2 i \, a^{3} x^{2} - a^{3} x\right )}{\rm integral}\left (\frac{6 \,{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}}{5 \,{\left (a^{3} x^{4} + a^{3} x^{2}\right )}}, x\right )}{5 \,{\left (a^{3} x^{3} - 2 i \, a^{3} x^{2} - a^{3} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-I*a*x + a)^(3/4)/(I*a*x + a)^(9/4),x, algorithm="fricas")

[Out]

-1/5*(2*(I*a*x + a)^(3/4)*(-I*a*x + a)^(3/4)*(5*I*x + 3) - 5*(a^3*x^3 - 2*I*a^3*
x^2 - a^3*x)*integral(6/5*(I*a*x + a)^(3/4)*(-I*a*x + a)^(3/4)/(a^3*x^4 + a^3*x^
2), x))/(a^3*x^3 - 2*I*a^3*x^2 - a^3*x)

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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((a-I*a*x)**(3/4)/(a+I*a*x)**(9/4),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((-I*a*x + a)^(3/4)/(I*a*x + a)^(9/4),x, algorithm="giac")

[Out]

Exception raised: TypeError