3.1180 \(\int \frac{1}{(a-i a x)^{7/4} \sqrt [4]{a+i a x}} \, dx\)

Optimal. Leaf size=33 \[ -\frac{2 i (a+i a x)^{3/4}}{3 a^2 (a-i a x)^{3/4}} \]

[Out]

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

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Rubi [A]  time = 0.0230548, antiderivative size = 33, 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 \[ -\frac{2 i (a+i a x)^{3/4}}{3 a^2 (a-i a x)^{3/4}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a - I*a*x)^(7/4)*(a + I*a*x)^(1/4)),x]

[Out]

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

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Rubi in Sympy [A]  time = 5.7752, size = 29, normalized size = 0.88 \[ - \frac{2 i \left (i a x + a\right )^{\frac{3}{4}}}{3 a^{2} \left (- i a x + a\right )^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a-I*a*x)**(7/4)/(a+I*a*x)**(1/4),x)

[Out]

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

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Mathematica [A]  time = 0.0331662, size = 33, normalized size = 1. \[ -\frac{2 i (a+i a x)^{3/4}}{3 a^2 (a-i a x)^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a - I*a*x)^(7/4)*(a + I*a*x)^(1/4)),x]

[Out]

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

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Maple [A]  time = 0.074, size = 31, normalized size = 0.9 \[{\frac{2\,x-2\,i}{3\,a} \left ( -a \left ( -1+ix \right ) \right ) ^{-{\frac{3}{4}}}{\frac{1}{\sqrt [4]{a \left ( 1+ix \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a-I*a*x)^(7/4)/(a+I*a*x)^(1/4),x)

[Out]

2/3/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. \[ \int \frac{1}{{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{7}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((I*a*x + a)^(1/4)*(-I*a*x + a)^(7/4)),x, algorithm="maxima")

[Out]

integrate(1/((I*a*x + a)^(1/4)*(-I*a*x + a)^(7/4)), x)

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Fricas [A]  time = 0.205049, size = 35, normalized size = 1.06 \[ \frac{2 \, x - 2 i}{3 \,{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}} a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((I*a*x + a)^(1/4)*(-I*a*x + a)^(7/4)),x, algorithm="fricas")

[Out]

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

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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)**(7/4)/(a+I*a*x)**(1/4),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)^(1/4)*(-I*a*x + a)^(7/4)),x, algorithm="giac")

[Out]

Exception raised: TypeError