3.1196 \(\int \frac{1}{\sqrt [4]{a-i a x} (a+i a x)^{7/4}} \, dx\)

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 = 0.0234852, 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)^(1/4)*(a + I*a*x)^(7/4)),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 = 6.06922, size = 27, normalized size = 0.82 \[ \frac{2 i \left (- i a x + a\right )^{\frac{3}{4}}}{3 a^{2} \left (i a x + a\right )^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a-I*a*x)**(1/4)/(a+I*a*x)**(7/4),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 = 0.0327272, size = 38, normalized size = 1.15 \[ \frac{2 (a-i a x)^{3/4} \sqrt [4]{a+i a x}}{3 a^3 (x-i)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.053, 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)^(1/4)/(a+I*a*x)^(7/4),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. \[ \int \frac{1}{{\left (i \, a x + a\right )}^{\frac{7}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.210255, size = 35, normalized size = 1.06 \[ \frac{2 \, x + 2 i}{3 \,{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}} a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a \left (i x + 1\right )\right )^{\frac{7}{4}} \sqrt [4]{- a \left (i x - 1\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: TypeError