3.1217 \(\int \frac{(a-i a x)^{7/4}}{(a+i a x)^{9/4}} \, dx\)

Optimal. Leaf size=141 \[ -\frac{42 \sqrt [4]{x^2+1} E\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{5 \sqrt [4]{a+i a x} \sqrt [4]{a-i a x}}+\frac{4 i (a-i a x)^{7/4}}{5 a (a+i a x)^{5/4}}-\frac{28 i (a-i a x)^{3/4}}{5 a \sqrt [4]{a+i a x}}+\frac{42 x}{5 \sqrt [4]{a+i a x} \sqrt [4]{a-i a x}} \]

[Out]

(((4*I)/5)*(a - I*a*x)^(7/4))/(a*(a + I*a*x)^(5/4)) + (42*x)/(5*(a - I*a*x)^(1/4
)*(a + I*a*x)^(1/4)) - (((28*I)/5)*(a - I*a*x)^(3/4))/(a*(a + I*a*x)^(1/4)) - (4
2*(1 + x^2)^(1/4)*EllipticE[ArcTan[x]/2, 2])/(5*(a - I*a*x)^(1/4)*(a + I*a*x)^(1
/4))

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Rubi [A]  time = 0.102779, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{42 \sqrt [4]{x^2+1} E\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{5 \sqrt [4]{a+i a x} \sqrt [4]{a-i a x}}+\frac{4 i (a-i a x)^{7/4}}{5 a (a+i a x)^{5/4}}-\frac{28 i (a-i a x)^{3/4}}{5 a \sqrt [4]{a+i a x}}+\frac{42 x}{5 \sqrt [4]{a+i a x} \sqrt [4]{a-i a x}} \]

Antiderivative was successfully verified.

[In]  Int[(a - I*a*x)^(7/4)/(a + I*a*x)^(9/4),x]

[Out]

(((4*I)/5)*(a - I*a*x)^(7/4))/(a*(a + I*a*x)^(5/4)) + (42*x)/(5*(a - I*a*x)^(1/4
)*(a + I*a*x)^(1/4)) - (((28*I)/5)*(a - I*a*x)^(3/4))/(a*(a + I*a*x)^(1/4)) - (4
2*(1 + x^2)^(1/4)*EllipticE[ArcTan[x]/2, 2])/(5*(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. \[ \frac{21 \left (- i a x + a\right )^{\frac{3}{4}} \left (i a x + a\right )^{\frac{3}{4}} \int \frac{1}{\sqrt [4]{a^{2} x^{2} + a^{2}}}\, dx}{5 \left (a^{2} x^{2} + a^{2}\right )^{\frac{3}{4}}} + \frac{4 i \left (- i a x + a\right )^{\frac{7}{4}}}{5 a \left (i a x + a\right )^{\frac{5}{4}}} - \frac{28 i \left (- i a x + a\right )^{\frac{3}{4}}}{5 a \sqrt [4]{i a x + a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a-I*a*x)**(7/4)/(a+I*a*x)**(9/4),x)

[Out]

21*(-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*x**2 + a**2)**(3/4)) + 4*I*(-I*a*x + a)**(7/4)/(5*a*(I*a*x + a)**(5/4
)) - 28*I*(-I*a*x + a)**(3/4)/(5*a*(I*a*x + a)**(1/4))

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Mathematica [C]  time = 0.096575, size = 84, normalized size = 0.6 \[ \frac{2 (a-i a x)^{3/4} \left (7\ 2^{3/4} (1+i x)^{5/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};\frac{1}{2}-\frac{i x}{2}\right )-16 i x-12\right )}{5 a (x-i) \sqrt [4]{a+i a x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a - I*a*x)^(7/4)/(a + I*a*x)^(9/4),x]

[Out]

(2*(a - I*a*x)^(3/4)*(-12 - (16*I)*x + 7*2^(3/4)*(1 + I*x)^(5/4)*Hypergeometric2
F1[1/4, 3/4, 7/4, 1/2 - (I/2)*x]))/(5*a*(-I + x)*(a + I*a*x)^(1/4))

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Maple [C]  time = 0.081, size = 101, normalized size = 0.7 \[ -{\frac{32\,{x}^{2}+24+8\,ix}{5\,x-5\,i}{\frac{1}{\sqrt [4]{-a \left ( -1+ix \right ) }}}{\frac{1}{\sqrt [4]{a \left ( 1+ix \right ) }}}}+{\frac{21\,x}{5}{\mbox{$_2$F$_1$}({\frac{1}{4}},{\frac{1}{2}};\,{\frac{3}{2}};\,-{x}^{2})}\sqrt [4]{-{a}^{2} \left ( -1+ix \right ) \left ( 1+ix \right ) }{\frac{1}{\sqrt [4]{{a}^{2}}}}{\frac{1}{\sqrt [4]{-a \left ( -1+ix \right ) }}}{\frac{1}{\sqrt [4]{a \left ( 1+ix \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a-I*a*x)^(7/4)/(a+I*a*x)^(9/4),x)

[Out]

-8/5*(4*x^2+3+I*x)/(x-I)/(-a*(-1+I*x))^(1/4)/(a*(1+I*x))^(1/4)+21/5/(a^2)^(1/4)*
x*hypergeom([1/4,1/2],[3/2],-x^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. \[ \int \frac{{\left (-i \, a x + a\right )}^{\frac{7}{4}}}{{\left (i \, a x + a\right )}^{\frac{9}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-I*a*x + a)^(7/4)/(I*a*x + a)^(9/4),x, algorithm="maxima")

[Out]

integrate((-I*a*x + a)^(7/4)/(I*a*x + a)^(9/4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[ \frac{2 \,{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}{\left (5 \, x^{2} - 30 i \, x - 21\right )} + 5 \,{\left (a^{2} x^{3} - 2 i \, a^{2} x^{2} - a^{2} x\right )}{\rm integral}\left (\frac{42 \,{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}}{5 \,{\left (a^{2} x^{4} + a^{2} x^{2}\right )}}, x\right )}{5 \,{\left (a^{2} x^{3} - 2 i \, a^{2} x^{2} - a^{2} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-I*a*x + a)^(7/4)/(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*x^2 - 30*I*x - 21) + 5*(a^2*x^3 -
2*I*a^2*x^2 - a^2*x)*integral(42/5*(I*a*x + a)^(3/4)*(-I*a*x + a)^(3/4)/(a^2*x^
4 + a^2*x^2), x))/(a^2*x^3 - 2*I*a^2*x^2 - a^2*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)**(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)^(7/4)/(I*a*x + a)^(9/4),x, algorithm="giac")

[Out]

Exception raised: TypeError