3.1206 \(\int \frac{(a-i a x)^{7/4}}{(a+i a x)^{5/4}} \, dx\)

Optimal. Leaf size=137 \[ \frac{14 a \sqrt [4]{x^2+1} E\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{\sqrt [4]{a+i a x} \sqrt [4]{a-i a x}}+\frac{4 i (a-i a x)^{7/4}}{a \sqrt [4]{a+i a x}}+\frac{14 i (a+i a x)^{3/4} (a-i a x)^{3/4}}{3 a}-\frac{14 a x}{\sqrt [4]{a+i a x} \sqrt [4]{a-i a x}} \]

[Out]

(-14*a*x)/((a - I*a*x)^(1/4)*(a + I*a*x)^(1/4)) + ((4*I)*(a - I*a*x)^(7/4))/(a*(
a + I*a*x)^(1/4)) + (((14*I)/3)*(a - I*a*x)^(3/4)*(a + I*a*x)^(3/4))/a + (14*a*(
1 + x^2)^(1/4)*EllipticE[ArcTan[x]/2, 2])/((a - I*a*x)^(1/4)*(a + I*a*x)^(1/4))

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Rubi [A]  time = 0.107614, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ \frac{14 a \sqrt [4]{x^2+1} E\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{\sqrt [4]{a+i a x} \sqrt [4]{a-i a x}}+\frac{4 i (a-i a x)^{7/4}}{a \sqrt [4]{a+i a x}}+\frac{14 i (a+i a x)^{3/4} (a-i a x)^{3/4}}{3 a}-\frac{14 a x}{\sqrt [4]{a+i a x} \sqrt [4]{a-i a x}} \]

Antiderivative was successfully verified.

[In]  Int[(a - I*a*x)^(7/4)/(a + I*a*x)^(5/4),x]

[Out]

(-14*a*x)/((a - I*a*x)^(1/4)*(a + I*a*x)^(1/4)) + ((4*I)*(a - I*a*x)^(7/4))/(a*(
a + I*a*x)^(1/4)) + (((14*I)/3)*(a - I*a*x)^(3/4)*(a + I*a*x)^(3/4))/a + (14*a*(
1 + x^2)^(1/4)*EllipticE[ArcTan[x]/2, 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. \[ - \frac{7 a \left (- i a x + a\right )^{\frac{3}{4}} \left (i a x + a\right )^{\frac{3}{4}} \int \frac{1}{\sqrt [4]{a^{2} x^{2} + a^{2}}}\, dx}{\left (a^{2} x^{2} + a^{2}\right )^{\frac{3}{4}}} + \frac{4 i \left (- i a x + a\right )^{\frac{7}{4}}}{a \sqrt [4]{i a x + a}} + \frac{14 i \left (- i a x + a\right )^{\frac{3}{4}} \left (i a x + a\right )^{\frac{3}{4}}}{3 a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a-I*a*x)**(7/4)/(a+I*a*x)**(5/4),x)

[Out]

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

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Mathematica [C]  time = 0.0699576, size = 74, normalized size = 0.54 \[ -\frac{2 (a-i a x)^{3/4} \left (7 i 2^{3/4} \sqrt [4]{1+i x} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};\frac{1}{2}-\frac{i x}{2}\right )+x-13 i\right )}{3 \sqrt [4]{a+i a x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a - I*a*x)^(7/4)/(a + I*a*x)^(5/4),x]

[Out]

(-2*(a - I*a*x)^(3/4)*(-13*I + x + (7*I)*2^(3/4)*(1 + I*x)^(1/4)*Hypergeometric2
F1[1/4, 3/4, 7/4, 1/2 - (I/2)*x]))/(3*(a + I*a*x)^(1/4))

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Maple [C]  time = 0.077, size = 96, normalized size = 0.7 \[{{\frac{2\,i}{3}} \left ({x}^{2}+13-12\,ix \right ) a{\frac{1}{\sqrt [4]{-a \left ( -1+ix \right ) }}}{\frac{1}{\sqrt [4]{a \left ( 1+ix \right ) }}}}-7\,{\frac{x{\mbox{$_2$F$_1$}(1/4,1/2;\,3/2;\,-{x}^{2})}a\sqrt [4]{-{a}^{2} \left ( -1+ix \right ) \left ( 1+ix \right ) }}{\sqrt [4]{{a}^{2}}\sqrt [4]{-a \left ( -1+ix \right ) }\sqrt [4]{a \left ( 1+ix \right ) }}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a-I*a*x)^(7/4)/(a+I*a*x)^(5/4),x)

[Out]

2/3*I*(x^2+13-12*I*x)*a/(-a*(-1+I*x))^(1/4)/(a*(1+I*x))^(1/4)-7/(a^2)^(1/4)*x*hy
pergeom([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. \[ \int \frac{{\left (-i \, a x + a\right )}^{\frac{7}{4}}}{{\left (i \, a x + a\right )}^{\frac{5}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-I*a*x + a)^(7/4)/(I*a*x + a)^(5/4),x, algorithm="maxima")

[Out]

integrate((-I*a*x + a)^(7/4)/(I*a*x + a)^(5/4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[ \frac{{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}{\left (2 i \, x^{2} - 16 \, x + 42 i\right )} +{\left (3 \, a x^{2} - 3 i \, a x\right )}{\rm integral}\left (-\frac{14 \,{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}}{a x^{4} + a x^{2}}, x\right )}{3 \, a x^{2} - 3 i \, a x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-I*a*x + a)^(7/4)/(I*a*x + a)^(5/4),x, algorithm="fricas")

[Out]

((I*a*x + a)^(3/4)*(-I*a*x + a)^(3/4)*(2*I*x^2 - 16*x + 42*I) + (3*a*x^2 - 3*I*a
*x)*integral(-14*(I*a*x + a)^(3/4)*(-I*a*x + a)^(3/4)/(a*x^4 + a*x^2), x))/(3*a*
x^2 - 3*I*a*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)**(7/4)/(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((-I*a*x + a)^(7/4)/(I*a*x + a)^(5/4),x, algorithm="giac")

[Out]

Exception raised: TypeError