3.1214 \(\int \frac{1}{(a-i a x)^{3/4} (a+i a x)^{5/4}} \, dx\)

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 = 0.0234144, 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)^(3/4)*(a + I*a*x)^(5/4)),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 = 6.0226, size = 26, normalized size = 0.84 \[ \frac{2 i \sqrt [4]{- i a x + a}}{a^{2} \sqrt [4]{i a x + a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a-I*a*x)**(3/4)/(a+I*a*x)**(5/4),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 = 0.0312271, size = 36, normalized size = 1.16 \[ \frac{2 \sqrt [4]{a-i a x} (a+i a x)^{3/4}}{a^3 (x-i)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.054, 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)^(3/4)/(a+I*a*x)^(5/4),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. \[ \int \frac{1}{{\left (i \, a x + a\right )}^{\frac{5}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.212869, size = 42, normalized size = 1.35 \[ \frac{2 \,{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{a^{3} x - i \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/((a*(I*x + 1))**(5/4)*(-a*(I*x - 1))**(3/4)), 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)^(5/4)*(-I*a*x + a)^(3/4)),x, algorithm="giac")

[Out]

Exception raised: TypeError