3.1187 \(\int \frac{1}{(a-i a x)^{9/4} (a+i a x)^{3/4}} \, dx\)

Optimal. Leaf size=67 \[ -\frac{4 i \sqrt [4]{a+i a x}}{5 a^3 \sqrt [4]{a-i a x}}-\frac{2 i \sqrt [4]{a+i a x}}{5 a^2 (a-i a x)^{5/4}} \]

[Out]

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

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Rubi [A]  time = 0.0505762, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08 \[ -\frac{4 i \sqrt [4]{a+i a x}}{5 a^3 \sqrt [4]{a-i a x}}-\frac{2 i \sqrt [4]{a+i a x}}{5 a^2 (a-i a x)^{5/4}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a - I*a*x)^(9/4)*(a + I*a*x)^(3/4)),x]

[Out]

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

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Rubi in Sympy [A]  time = 11.4968, size = 58, normalized size = 0.87 \[ - \frac{2 i \sqrt [4]{i a x + a}}{5 a^{2} \left (- i a x + a\right )^{\frac{5}{4}}} - \frac{4 i \sqrt [4]{i a x + a}}{5 a^{3} \sqrt [4]{- i a x + a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a-I*a*x)**(9/4)/(a+I*a*x)**(3/4),x)

[Out]

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

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Mathematica [A]  time = 0.0400398, size = 45, normalized size = 0.67 \[ \frac{2 (3-2 i x) \sqrt [4]{a+i a x}}{5 a^3 (x+i) \sqrt [4]{a-i a x}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a - I*a*x)^(9/4)*(a + I*a*x)^(3/4)),x]

[Out]

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

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Maple [A]  time = 0.063, size = 44, normalized size = 0.7 \[{\frac{4\,{x}^{2}+6+2\,ix}{5\,{a}^{2} \left ( x+i \right ) } \left ( a \left ( 1+ix \right ) \right ) ^{-{\frac{3}{4}}}{\frac{1}{\sqrt [4]{-a \left ( -1+ix \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a-I*a*x)^(9/4)/(a+I*a*x)^(3/4),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{9}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((I*a*x + a)^(3/4)*(-I*a*x + a)^(9/4)),x, algorithm="maxima")

[Out]

integrate(1/((I*a*x + a)^(3/4)*(-I*a*x + a)^(9/4)), x)

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Fricas [A]  time = 0.206297, size = 59, normalized size = 0.88 \[ \frac{{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}{\left (4 \, x + 6 i\right )}}{5 \, a^{4} x^{2} + 10 i \, a^{4} x - 5 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((I*a*x + a)^(3/4)*(-I*a*x + a)^(9/4)),x, algorithm="fricas")

[Out]

(I*a*x + a)^(1/4)*(-I*a*x + a)^(3/4)*(4*x + 6*I)/(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)**(3/4),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)^(3/4)*(-I*a*x + a)^(9/4)),x, algorithm="giac")

[Out]

Exception raised: TypeError