3.1186 \(\int \frac{1}{(a-i a x)^{5/4} (a+i a x)^{3/4}} \, dx\)

Optimal. Leaf size=31 \[ -\frac{2 i \sqrt [4]{a+i a x}}{a^2 \sqrt [4]{a-i a x}} \]

[Out]

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

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Rubi [A]  time = 0.0233917, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04 \[ -\frac{2 i \sqrt [4]{a+i a x}}{a^2 \sqrt [4]{a-i a x}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a - I*a*x)^(5/4)*(a + I*a*x)^(3/4)),x]

[Out]

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

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Rubi in Sympy [A]  time = 5.39696, size = 27, normalized size = 0.87 \[ - \frac{2 i \sqrt [4]{i a x + a}}{a^{2} \sqrt [4]{- i a x + a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a-I*a*x)**(5/4)/(a+I*a*x)**(3/4),x)

[Out]

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

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Mathematica [A]  time = 0.0304198, size = 31, normalized size = 1. \[ -\frac{2 i \sqrt [4]{a+i a x}}{a^2 \sqrt [4]{a-i a x}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a - I*a*x)^(5/4)*(a + I*a*x)^(3/4)),x]

[Out]

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

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Maple [A]  time = 0.053, size = 31, normalized size = 1. \[ 2\,{\frac{x-i}{a \left ( a \left ( 1+ix \right ) \right ) ^{3/4}\sqrt [4]{-a \left ( -1+ix \right ) }}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a-I*a*x)^(5/4)/(a+I*a*x)^(3/4),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{5}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((I*a*x + a)^(3/4)*(-I*a*x + a)^(5/4)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.205152, size = 42, normalized size = 1.35 \[ \frac{2 \,{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}}{a^{3} x + i \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((I*a*x + a)^(3/4)*(-I*a*x + a)^(5/4)),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a \left (i x + 1\right )\right )^{\frac{3}{4}} \left (- a \left (i x - 1\right )\right )^{\frac{5}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a-I*a*x)**(5/4)/(a+I*a*x)**(3/4),x)

[Out]

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

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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)^(5/4)),x, algorithm="giac")

[Out]

Exception raised: TypeError